Electrical, Electromagnetic, and Optical Characterization of the InP/InGaAs Alloy System
A Thesis
Presented to the Faculty of the
Department of Electronics and Communications Engineering
Gokongwei College of Engineering
De La Salle University
In Partial Fulfillment of the
Requirements for the Degree of
Bachelor of Science in Electronics and Communications Engineering
by
DELA CRUZ, Juan Z.
FRANCO, Nat Y.
GARCIA, Sebastian X.
MARTINEZ, Isabella W.
RIANZARES, Max V.
April, 2017
ii
Suggestions for choosing the title of your document: The title must be reflective of its
problem. Use the following checklist to help you in making the title.
WHAT question: Will answer the following
Did you put what you are trying to investigate?
Did you indicate what you are trying to find out, determine or discover?
WHO question: Will answer who are the respondents or subjects of the study
WHERE question: Will indicate the research locale, setting or the place where the
research study is conducted or where it can be applied.
2017
All Rights Reserved. No part of this publication may be reproduced, stored in an information retrieval system,
or transmitted, in any form or by any means, electronic, mechanical, by photocopying, scanning, recording,
or otherwise, except under the terms of the applicable law.
ACKNOWLEDGMENT
Write this prior to binding your thesis if you have submitted necessary requirements and
are told by the university that you have passed.
iv
ABSTRACT
Keep your abstract short by giving the gist/nutshell of your thesis. Use the following
checklist questions to help you in crafting your abstract.
Did you briefly state what you intend to do?
Did you concisely discuss the problem statement?
Did you tersely mention the objectives in general terms?
Did you succinctly describe the methodology for the target audience?
Did you strongly describe your significant results and your conclusions?
Index Terms—alloy system, characterization, InP, InGaAs.
v
TABLE OF CONTENTS
Thesis Approval Sheet
ii
Acknowledgment
iv
Abstract
v
Table of Contents
vi
List of Figures
ix
List of Tables
x
Abbreviations
xi
Notation
xii
Glossary
xiii
Listings
xiv
Chapter 1 INTRODUCTION
1.1 Background of the Study . . . . . . . . . .
1.2 Prior Studies . . . . . . . . . . . . . . . . .
1.3 Problem Statement . . . . . . . . . . . . .
1.4 Objectives . . . . . . . . . . . . . . . . . .
1.4.1 General Objective(s) . . . . . . . .
1.4.2 Specific Objectives . . . . . . . . .
1.5 Significance of the Study . . . . . . . . . .
1.6 Assumptions, Scope and Delimitations . . .
1.6.1 Assumptions . . . . . . . . . . . .
1.6.2 Scope . . . . . . . . . . . . . . . .
1.6.3 Delimitations . . . . . . . . . . . .
1.7 Description and Methodology of the Thesis
1.8 Overview of the Thesis . . . . . . . . . . .
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Chapter 2 LITERATURE REVIEW
2.1 Existing Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Table of Contents
2.2
2.3
Lacking in the Approaches . . . . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
15
Chapter 3 THEORETICAL CONSIDERATIONS
3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 4 DESIGN CONSIDERATIONS
4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 5 METHODOLOGY
5.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 6 RESULTS AND DISCUSSIONS
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 7 CONCLUSIONS, RECOMMENDATIONS, AND FUTURE DIRECTIVES
7.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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References
40
Appendix A STUDENT RESEARCH ETHICS CLEARANCE
49
Appendix B ANSWERS TO QUESTIONS TO THIS THESIS
51
Appendix C REVISIONS TO THE PROPOSAL
60
Appendix D REVISIONS TO THE FINAL
66
Appendix E USAGE EXAMPLES
E1 Equations . . . . . . . . . .
E2 Notations . . . . . . . . . .
E2.1 Math alphabets . . .
E2.2 Vector symbols . . .
E2.3 Matrix symbols . . .
E2.4 Tensor symbols . . .
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vii
Table of Contents
E2.5
Bold math version . . . . . .
E2.5.1 Vector symbols . .
E2.5.2 Matrix symbols . .
E2.5.3 Tensor symbols . .
E3 Abbreviation . . . . . . . . . . . . .
E4 Glossary . . . . . . . . . . . . . . .
E5 Figure . . . . . . . . . . . . . . . .
E6 Table . . . . . . . . . . . . . . . . .
E7 Algorithm or Pseudocode Listing . .
E8 Program/Code Listing . . . . . . . .
E9 Referencing . . . . . . . . . . . . .
E9.1 A subsection . . . . . . . .
E9.1.1 A sub-subsection .
E10 Citing . . . . . . . . . . . . . . . .
E10.1 Books . . . . . . . . . . . .
E10.2 Booklets . . . . . . . . . . .
E10.3 Proceedings . . . . . . . . .
E10.4 In books . . . . . . . . . . .
E10.5 In proceedings . . . . . . .
E10.6 Journals . . . . . . . . . . .
E10.7 Theses/dissertations . . . . .
E10.8 Technical Reports and Others
E10.9 Miscellaneous . . . . . . . .
E11 Index . . . . . . . . . . . . . . . . .
E12 Adding Relevant PDF Pages . . . .
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Appendix F SOME LIST OF MATH SYMBOLS
114
Appendix G DISLAYING MATH EXPRESSIONS
132
Appendix H IEEE EDITORIAL STYLE MANUAL
164
Appendix I
196
IEEE CITATION REFERENCE
Appendix J IEEE PUBLICATION ABBREVIATIONS
204
Appendix K IEEE INDEX TERMS
212
Appendix L VITA
280
Index
282
viii
LIST OF FIGURES
3.1
A quadrilateral image example. . . . . . . . . . . . . . . . . . . . . . . . . .
19
E.1 A quadrilateral image example. . . . . . . . . . . . . . . . . . . . . . . . . .
E.2 Figures on top of each other. See List. E.6 for the corresponding LATEX code.
E.3 Four figures in each corner. See List. E.7 for the corresponding LATEX code. .
85
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ix
LIST OF TABLES
C.1 Summary of Revisions to the Proposal . . . . . . . . . . . . . . . . . . . . .
61
D.1 Summary of Revisions to the Thesis . . . . . . . . . . . . . . . . . . . . . .
67
E.1 Feasible triples for highly variable grid . . . . . . . . . . . . . . . . . . . . .
E.2 Calculation of y = xn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
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ABBREVIATIONS
AC
HTML
CSS
XML
Alternating Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Hyper-text Markup Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Cascading Style Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
eXtensible Markup Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
xi
NOTATION
S
U
∅
|S|
h (t)
x (t)
y (t)
a collection of distinct objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
the set containing everything . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
the set with no elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
the number of elements in the set S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
impulse response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
input signal represented in the time domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
output signal represented in the time domain . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Throughout this thesis, mathematical notations conform to ISO 80000-2 standard, e.g.
variable names are printed in italics, the only exception being acronyms like e.g. SNR,
which are printed in regular font. Constants are also set in regular font like j. Functions
are
√
also set in regular font, e.g. in sin (·). Commonly used notations are t, f , j = −1, n and
exp (·), which refer to the time variable, frequency variable, imaginary unit, nth variable,
and exponential function, respectively.
xii
GLOSSARY
matrix
a concise and useful way of uniquely representing and working
with linear transformations; a rectangular table of elements
Functional Analysis
the branch of Mathematics concerned with the study of spaces
of functions
xiii
LISTINGS
E.1
E.2
E.3
E.4
E.5
E.6
E.7
E.8
E.9
E.10
E.11
E.12
E.13
E.14
E.15
E.16
Sample LATEX code for equations and notations usage . . . .
Sample LATEX code for notations usage . . . . . . . . . . . .
Sample LATEX code for abbreviations usage . . . . . . . . . .
Sample LATEX code for glossary and notations usage . . . . .
Sample LATEX code for a single figure . . . . . . . . . . . .
Sample LATEX code for three figures on top of each other . .
Sample LATEX code for the four figures . . . . . . . . . . . .
Sample LATEX code for making typical table environment . .
Sample LATEX code for algorithm or pseudocode listing usage
Computing Fibonacci numbers . . . . . . . . . . . . . . . .
Sample LATEX code for program listing . . . . . . . . . . . .
Sample LATEX code for referencing sections . . . . . . . . .
Sample LATEX code for referencing subsections . . . . . . .
Sample LATEX code for referencing sub-subsections . . . . .
Sample LATEX code for Index usage . . . . . . . . . . . . . .
Sample LATEX code for including PDF pages . . . . . . . . .
xiv
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Chapter 1
INTRODUCTION
Contents
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Background of the Study . . . . . . . . . .
Prior Studies . . . . . . . . . . . . . . . . .
Problem Statement . . . . . . . . . . . . .
Objectives . . . . . . . . . . . . . . . . . .
1.4.1 General Objective(s) . . . . . . . .
1.4.2 Specific Objectives . . . . . . . . .
Significance of the Study . . . . . . . . . .
Assumptions, Scope and Delimitations . . .
1.6.1 Assumptions . . . . . . . . . . . .
1.6.2 Scope . . . . . . . . . . . . . . . .
1.6.3 Delimitations . . . . . . . . . . . .
Description and Methodology of the Thesis
Overview of the Thesis . . . . . . . . . . .
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1. Introduction
1.1
Background of the Study
Aside from the usual text descriptions of the background, put here figures that will cast
images to your audience about the context of your work.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
2
1. Introduction
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
3
1. Introduction
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
1.2
Prior Studies
Put here a narrative and a summary (not a duplicate) of your literature review chapter. In
this section, summarize and highlight the gap(s) found in the literature review in Chapter 2.
Preferably, a table showing the summary would be helpful.
Prior Studies or Literature Review1 (expansion of the Prior Studies) is basically about
competition. Competition.
So the suggested goals in writing the narrative of the Prior Studies in summative and
highlighted forms are, in no particular order:
1. to mention briefly the problem;
2. to show the features of the existing literature in solving the problem
3. to show the weaknesses of the solutions of existing literature
4. to show how your solution is better (can be better (for proposals))
If the suggested table will be placed, please discuss it in light of the above-mentioned items.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
1
The main difference between the Prior Studies and Literature Review is that the Prior Studies is done in
a concise manner. By the way, this is also an example of a footnote usage.
4
1. Introduction
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
1.3
Problem Statement
The problem statement needs to be very clear and to the point.
A persuasive problem statement from a contextualized and intended-audience-awareness
perspective consists of:
1. PS1: description of the ideal scenario for your intended audience
• Describe the goals, desired state, or the values that your audience considers
important and that are relevant to the problem.
2. PS2: reality of the situation
• Describe a condition that prevents the goal, state, or value discussed in PS1
from being achieved or realized at the present time.
• It is imperative to make the audience feel the pain point.
3. PS3: consequences for the audience
• Using specific details, show how the situation contains little promise of improvement unless something is done.
5
1. Introduction
After the above-mentioned items, succinctly describe your solution. Please avoid describing
your entire solution here since you will articulate and elucidate it by showing what you want
to achieve through your objectives, and how you will make it through your methodology.
A well constructed problem statement will convince your audience that the problem is real
and worth having you solve it.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
1.4
Objectives
Your objectives are the states that you desire to achieve in solving the problem. The general
objective is the main state to be achieved whereas the specific ones are sub-states to be
achieved.
1.4.1
General Objective(s)
To . . . ;
6
1. Introduction
1.4.2
Specific Objectives
1. To . . . ;
2. To . . . ;
3. To . . . ;
4. To . . . ;
5. To . . . ;
1.5
Significance of the Study
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
1.6
Assumptions, Scope and Delimitations
Bulletize your assumptions in one group, and then bulletize the scope in another, and do
the same for your delimitations. The assumptions to put here are those major facts or
7
1. Introduction
statements that are key for your proposed solution to work. Scope refers to the space(s)
for the operation of your proposed solution, whereas delimitations are the limits of the
operation of your proposed solution.
1.6.1
Assumptions
1. . . . ;
2. . . . ;
3. . . . ;
1.6.2
Scope
1. . . . ;
2. . . . ;
3. . . . ;
1.6.3
Delimitations
1. . . . ;
2. . . . ;
3. . . . ;
8
1. Introduction
1.7
Description and Methodology of the Thesis
A purpose of the description here is to re-steer/remind the panelist/reader again by tersely
describing what your thesis is about (i.e. the problem and the main goal you want to
achieve) in another way without sounding repetitive.
Your methodology is your means of achieving your stated objectives.
Note that each stated objective should have a corresponding methodology of achieving
it.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
1.8
Overview of the Thesis
Provide here a brief summary and what the reader should expect from each succeeding
chapter. Show how each chapter is connected with each other.
9
Chapter 2
LITERATURE REVIEW
Contents
2.1
2.2
2.3
Existing Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lacking in the Approaches . . . . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
11
13
15
2. Literature Review
It is to be noted that each subsection in this chapter should discuss in narrative form
each table that is presented.
2.1
Existing Work
Cite and summarize here relevant and significant literature (dissertations, theses, journals,
patents, notable conference papers) through a table and descriptions to prove that no one
has done your work yet and/or that your work is not a duplication of existing ones. Your
focus here is what has been done.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
11
2. Literature Review
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
12
2. Literature Review
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
2.2
Lacking in the Approaches
You can summarize the weaknesses of existing approaches by a tabular comparison of the
literature. Your focus here is what has not been done, i.e. what features were missed, what
solutions were not considered, what the demerits are, etc. Through these items, you then
can introduce the necessity for doing your proposed solution.
It is to be noted that degree of novelty for undergraduate thesis is lower than those for
graduate school. If a PhD dissertation/thesis has a high degree of novelty and that for an
undergraduate is low, then a master’s thesis is somewhere between the two.
Briefly include here the following in order to remind the reader why you are highlighting
the weaknesses of the solutions of existing literature.
• mentioning of the problem
• showing how your solution is better (can be better (for proposals))
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
13
2. Literature Review
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
14
2. Literature Review
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
2.3
Summary
Provide the gist of this chapter such that it reflects the contents and the message.
15
Chapter 3
THEORETICAL CONSIDERATIONS
Contents
3.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
19
3. Theoretical Considerations
Before starting the first section, provide an overview of the purpose of this chapter and
its contents, and how they are relevant to your methodology. Discuss in this chapter the
relevant theories and concepts that should support your proposed solutions.
This chapter is for providing the context to your panelist/reader. It is actually an
expanded form of the Background of the Study that you have put in Chapter 1.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
17
3. Theoretical Considerations
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
18
3. Theoretical Considerations
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Fig. 3.1
3.1
A quadrilateral image example.
Summary
Provide the gist of this chapter such that it reflects the contents and the message.
19
Chapter 4
DESIGN CONSIDERATIONS
Contents
4.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
23
4. Design Considerations
Before starting the first section, provide an overview of the purpose of this chapter and
its contents, and how they are relevant to your methodology.
Your primary goal in the Design Considerations chapter is to describe to your panelist/readers the key topics that fall further under Theoretical Considerations, but should
be placed here instead since they are geared towards your Methodology. These key topics
are those that you have directly adopted in making your solution/methodology. You can
think of the connection of the Design Considerations chapter to the Theoretical Considerations chapter in this way: if your Theoretical Considerations chapter serves as the main
foundation of a building, then the Design Considerations chapter functions as the columns.
The Design Considerations chapter is an avenue for explaining why you considered
the topics here for your proposed methodology. This chapter is different from your
methodology, because topics you discuss here are already accepted as part of the body of
knowledge, and may have not been developed by you.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
21
4. Design Considerations
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
22
4. Design Considerations
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
4.1
Summary
Provide the gist of this chapter such that it reflects the contents and message.
23
Chapter 5
METHODOLOGY
Contents
5.1
5.2
5.3
Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
25
27
30
5. Methodology
Put an overview of the contents of chapter. Mention here your methodology flow
through a figure and provide an overview of it and how your methodology achieves your
objectives. How your methodology achieves each of your specific objective is what
your panelists/examiners will be looking for. Specify how your methodology achieves
your general objective, and specific objectives. A point-by-point comparison how your
methodology achieves each of your specific objective is expected in the final Thesis.
Also make sure that you refer clearly to the chapters on the Literature Review, Theoretical Considerations, and Design Considerations showing how your methodology ties with
those that you have discussed in those chapters.
Make an overview of the contents of chapter. Put here your methodology flow through
a figure and provide an overview of it.
5.1
Implementation
Summarize the process used to create/set-up the work with an explanation of such process,
instruments, and materials that you used if any. If the description is lengthy, use condensed
bullet points.
Rule of thumb: Implementation is how you made your work; (keywords: implemented,
created, made, soldered, programmed, etc.).
If you wrote a program or made a simulation, you must state how the program or
simulation functions in this section. An algorithm or a pseudocode as shown in Table E.2
is a good example.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
25
5. Methodology
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
26
5. Methodology
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
5.2
Evaluation
Describe the procedures for evaluating the correct behavior and outcome of your work,
including what information you need to gather and how you will obtain or measure it.
27
5. Methodology
Rule of thumb: Evaluation is how you tested your work; (keywords: measured, tested,
compared, simulated, etc.).
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
28
5. Methodology
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
29
5. Methodology
5.3
Summary
Provide the gist of this chapter such that it reflects the contents and the message.
30
Chapter 6
RESULTS AND DISCUSSIONS
Contents
6.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
34
6. Results and Discussions
Show in this chapter proofs why your proposed solution works. However, presenting
results (”It worked”) without an appropriate explanation does not show thorough understanding. Aside from the data and results that you have obtained, and their explanation,
the discussion includes why components of your proposed solution work did or did not
work in accordance to what you described in the evaluation process, and how the proposed
solution performed and faired. Interpret the results and the reasons why they were obtained.
If your results are incorrect, apparent discrepancies from theory should be pointed out and
explained. In essence, what do the results mean. Citing existing publication can help you
compare your results and your explanations.
The next items below is not related to the description of this results and discussions
chapter, but serves as an opener for the LATEXportion of this template.
Here is an example of a citation for ISO 80000-2 standard [ISO, 2009]. Another one
is [Einstein, 1905] and [Croft, 1978].
In using this template, the user is expected to have a working knowledge of LATEX. A
good introduction is in [Oetiker et al., 2014]. Its latest version can be accessed at http://
www.ctan.org/tex-archive/info/lshort. See the Appendix of document_guide.pdf for
examples.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
32
6. Results and Discussions
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
33
6. Results and Discussions
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
6.1
Summary
Provide the gist of this chapter such that it reflects the contents and the message.
34
Chapter 7
CONCLUSIONS, RECOMMENDATIONS,
AND FUTURE DIRECTIVES
Contents
7.1
7.2
7.3
7.4
Concluding Remarks
Contributions . . . .
Recommendations . .
Future Prospects . . .
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38
7. Conclusions, Recommendations, and Future Directives
7.1
Concluding Remarks
In this Thesis, . . .
Put here the main points that should be known and learned about the work topic.
Summarize or give the gist of the essential principles and inferences drawn from your
results.
7.2
Contributions
The interrelated contributions and supplements that have been developed by the author(s)
in this Thesis are listed as follows. Only those that are unique to the authors’ work are
included.
• the ;
• the ;
• the ;
7.3
Recommendations
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
36
7. Conclusions, Recommendations, and Future Directives
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
37
7. Conclusions, Recommendations, and Future Directives
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
7.4
Future Prospects
There are several prospect related in this research that may be extended for further studies.
. . . So the suggested topics are listed in the following.
1. the . . . .
2. the . . . .
3. the . . . .
38
7. Conclusions, Recommendations, and Future Directives
Note that for ECE undergraduate theses, as per the directions of the thesis adviser,
Recommendations and Future Directives will be removed for the hardbound copy but will
be retained for database storage.
39
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research results. Talk at Fanstord University (this is a full UNPUBLISHED entry).
[Ünderwood et al., 2004] Ünderwood, U., Ñet, N., and P̄ot, P. (2004). Lower bounds for wishful
research results. Talk at Fanstord University (this is a minimal UNPUBLISHED entry).
[van Gennep, 1909a] van Gennep, A. (1909a). Les rites de passage. Nourry.
[van Gennep, 1909b] van Gennep, A. (1909b). Les rites de passage. Nourry.
[van Gennep, 1960] van Gennep, A. (1960). The Rites of Passage. University of Chicago Press.
[Vázques de Parga et al., 1993] Vázques de Parga, L., Lacarra, J. M., and Urı́a Rı́u, J. (1993). Las
Peregrinaciones a Santiago de Compostela. Iberdrola. Ed. facs. de la realizada en 1948–49.
[von Brandt and Hoffmann, 1987] von Brandt, A. and Hoffmann, E. (1987). Die nordischen
länder von der mitte des 11. jahrhunderts bis 1448. In Seibt, F., editor, Europa im Hoch- und
Spätmittelalter, number 2 in Handbuch der europäischen Geschichte, pages 884–917. Klett-Cotta.
[Wassenberg and Sanders, 2010] Wassenberg, J. and Sanders, P. (2010). Faster radix sort via virtual
memory and write-combining.
[Weinberg, 1967] Weinberg, S. (1967). A model of leptons. Phys. Rev. Lett., 19:1264–1266.
[Westfahl, 2004] Westfahl, G. (2004). The true frontier. pages 55–65.
[Wilde, 1899] Wilde, O. (1899). The Importance of Being Earnest: A Trivial Comedy for Serious
People. English and American drama of the Nineteenth Century. Leonard Smithers and Company.
[Winget Ltd., 1967] Winget Ltd. (1967). Detachable bulldozer attachment for dumper vehicles.
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References
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[Wood, 1961] Wood, R. H. (1961). Plastic and Elastic Design of Slabs and Plates. Thames &
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[Worman, 2002] Worman, N. (2002). The Cast of Character. University of Texas Press.
[Wright, 1963] Wright, R. C. (1963). Report Literature, pages 46–59.
[Wright, 1978a] Wright, S. (1978a). Evolution and the genetics of populations, volume 4. Univ. of
Chicago Press, Chicago.
[Wright, 1978b] Wright, S. (1978b). Variability within and among natural populations. In Evolution
and the genetics of populations, vol. 4. Univ. of Chicago Press, Chicago.
[Yoon et al., 2006] Yoon, M. S., Ryu, D., Kim, J., and Ahn, K. H. (2006). Palladium pincer complexes with reduced bond angle strain: efficient catalysts for the heck reaction. Organometallics,
25(10):2409–2411.
LATEX-comment this and the following texts after you have implemented them. See the
following references for helpful guides for the bibliography and script editing in general.
Note that the links might be unavailable, but the names can be searched in the Web.
1. IEEE Citation Reference: www.ieee.org/documents/ieeecitationref.pdf
2. IEEE Editorial Style manual: www.ieee.org/documents/style manual.pdf
3. IEEE Abbreviations for Transactions, Journals, Letters, and Magazines: www.ieee.
org/documents/trans journal names.pdf
Also in your BibTeX file, enclose letters or words that should all be in uppercase in curly
brackets. Example: IBM, Philippines, eXtensible Markup Language.
Produced: April 9, 2017, 01:19
48
Appendix A
STUDENT RESEARCH ETHICS
CLEARANCE
49
A. Student Research Ethics Clearance
RESEARCH ETHICS CLEARANCE FORM1
For Thesis Proposals
Names of Student Researcher(s):
Dela Cruz, Juan Z.
College: Gokongwei College of Engineering
Department: Electronics and Communications Engineering
Course: PhD-ECE
Expected Duration of the Project: from: April 2015
Ethical considerations
to: April 2017
None
(The Ethics Checklists may be used as guides in determining areas for ethical concern/consideration)
To the best of my knowledge, the ethical issues listed above have been addressed in the research.
Dr. Francisco D. Baltasar
_________________________________
Name and Signature of Adviser/Mentor:
Date: April 8, 2017
Noted by:
Dr. Rafael W. Sison
_________________________________
Name and Signature of the Department Chairperson:
Date: April 8, 2017
1
The same form can be used for the reports of completed projects. The appropriate heading need only be used.
50
Appendix B
ANSWERS TO QUESTIONS TO THIS
THESIS
51
B. Answers to Questions to this Thesis
B1
How important is the problem to practice?
A possible answer to this question is the summary of your Significance of the Study, and
that portion of the Problem Statement where you describe the ideal scenario for your
intended audience.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
B2 How will you know if the solution/s that you will
achieve would be better than existing ones?
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
B2.1
How will you measure the improvement/s?
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
52
B. Answers to Questions to this Thesis
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
B2.1.1
What is/are your basis/bases for the improvement/s?
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
B2.1.2
Why did you choose that/those basis/bases?
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
B2.1.3
How significant are your measure/s of the improvement/s?
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
53
B. Answers to Questions to this Thesis
B3 What is the difference of the solution/s from existing ones?
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
B3.1
How is it different from previous and existing ones?
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
B4 What are the assumptions made (that are behind for your proposed solution to work)?
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
54
B. Answers to Questions to this Thesis
B4.1 Will your proposed solution/s be sensitive to these assumptions?
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
B4.2 Can your proposed solution/s be applied to more general
cases when some of the assumptions are eliminated? If so,
how?
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
B5 What is the necessity of your approach / proposed solution/s?
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
55
B. Answers to Questions to this Thesis
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
B5.1 What will be the limits of applicability of your proposed solution/s?
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
B5.2 What will be the message of the proposed solution to
technical people? How about to non-technical managers and
business men?
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
B6 How will you know if your proposed solution/s
is/are correct?
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
56
B. Answers to Questions to this Thesis
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
B6.1 Will your results warrant the level of mathematics used
(i.e., will the end justify the means)?
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
B7 Is/are there an/ alternative way/s to get to the
same solution/s?
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
B7.1 Can you come up with illustrating examples, or even better, counter examples to your proposed solution/s?
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
57
B. Answers to Questions to this Thesis
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
B7.2 Is there an approximation that can arrive at the essentially the same proposed solution/s more easily?
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
B8 If you were the examiner of your Thesis, how
would you present the Thesis in another way? Give
your remarks, especially for your methodology and
the results and discussions.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
58
B. Answers to Questions to this Thesis
B8.1 What are the weaknesses of your Thesis, specifically your
methodology and the results and discussions?
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
59
Appendix C
REVISIONS TO THE PROPOSAL
60
C. Revisions to the Proposal
Make a table with the following columns for showing the summary of revisions to the
proposal based on the comments of the panel of examiners.
1. Panelist name
2. Comment
3. Summary of how the comment was addressed
4. Locations in the document where the changes have been reflected
TABLE C.1
Panelist
name
Dr.
Francisco
D.
Baltasar
S UMMARY OF R EVISIONS TO THE P ROPOSAL
Comment
Summary of how the comment was addressed
Locations
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Etiam
lobortis facilisis sem.
Nullam nec mi et neque
pharetra
sollicitudin.
Praesent
imperdiet
mi nec ante. Donec
ullamcorper, felis non
sodales commodo, lectus velit ultrices augue,
a dignissim nibh lectus
placerat pede. Vivamus
nunc nunc, molestie ut,
ultricies vel, semper
in, velit. Ut porttitor.
Praesent in sapien.
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Duis
fringilla tristique neque.
Sed interdum libero ut
metus.
Pellentesque
placerat. Nam rutrum
augue a leo. Morbi
sed elit sit amet ante
lobortis
sollicitudin.
Praesent blandit blandit
mauris. Praesent lectus
tellus, aliquet aliquam,
luctus a, egestas a,
turpis. Mauris lacinia
lorem sit amet ipsum.
Nunc quis urna dictum
turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit.
Etiam lobortis facilisis sem. Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus placerat pede. Vivamus
nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut
porttitor. Praesent in sapien. Lorem ipsum dolor sit amet,
consectetuer adipiscing elit. Duis fringilla tristique neque.
Sed interdum libero ut metus. Pellentesque placerat. Nam
rutrum augue a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris. Praesent lectus
tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris
lacinia lorem sit amet ipsum. Nunc quis urna dictum turpis
accumsan semper.
Sec.
5.1
on p. 25,
Sec.
5.2
on p. 27,
Fig. 3.1 on
p. 19
First itemtext
Second itemtext
Last itemtext
First itemtext
Second itemtext
Continued on next page
61
C. Revisions to the Proposal
Panelist
name
Comment
Dr. Amado
Z. Hernandez
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Etiam
lobortis facilisis sem.
Nullam nec mi et neque
pharetra
sollicitudin.
Praesent
imperdiet
mi nec ante. Donec
ullamcorper, felis non
sodales commodo, lectus velit ultrices augue,
a dignissim nibh lectus
placerat pede. Vivamus
nunc nunc, molestie ut,
ultricies vel, semper
in, velit. Ut porttitor.
Praesent in sapien.
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Duis
fringilla tristique neque.
Sed interdum libero ut
metus.
Pellentesque
placerat. Nam rutrum
augue a leo. Morbi
sed elit sit amet ante
lobortis
sollicitudin.
Praesent blandit blandit
mauris. Praesent lectus
tellus, aliquet aliquam,
luctus a, egestas a,
turpis. Mauris lacinia
lorem sit amet ipsum.
Nunc quis urna dictum
turpis accumsan semper.
Continued from previous page
Summary of how the comment was addressed
Locations
Lorem ipsum dolor sit amet, consectetuer adipiscing elit.
Etiam lobortis facilisis sem. Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus placerat pede. Vivamus
nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut
porttitor. Praesent in sapien. Lorem ipsum dolor sit amet,
consectetuer adipiscing elit. Duis fringilla tristique neque.
Sed interdum libero ut metus. Pellentesque placerat. Nam
rutrum augue a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris. Praesent lectus
tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris
lacinia lorem sit amet ipsum. Nunc quis urna dictum turpis
accumsan semper.
Sec.
5.1
on p. 25,
Sec.
5.2
on p. 27,
Fig. 3.1 on
p. 19
First itemtext
Second itemtext
Last itemtext
First itemtext
Second itemtext
Continued on next page
62
C. Revisions to the Proposal
Panelist
name
Dr. Jose Y.
Alonzo
Comment
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Etiam
lobortis facilisis sem.
Nullam nec mi et neque
pharetra
sollicitudin.
Praesent
imperdiet
mi nec ante. Donec
ullamcorper, felis non
sodales commodo, lectus velit ultrices augue,
a dignissim nibh lectus
placerat pede. Vivamus
nunc nunc, molestie ut,
ultricies vel, semper
in, velit. Ut porttitor.
Praesent in sapien.
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Duis
fringilla tristique neque.
Sed interdum libero ut
metus.
Pellentesque
placerat. Nam rutrum
augue a leo. Morbi
sed elit sit amet ante
lobortis
sollicitudin.
Praesent blandit blandit
mauris. Praesent lectus
tellus, aliquet aliquam,
luctus a, egestas a,
turpis. Mauris lacinia
lorem sit amet ipsum.
Nunc quis urna dictum
turpis accumsan semper.
Continued from previous page
Summary of how the comment was addressed
Locations
Lorem ipsum dolor sit amet, consectetuer adipiscing elit.
Etiam lobortis facilisis sem. Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus placerat pede. Vivamus
nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut
porttitor. Praesent in sapien. Lorem ipsum dolor sit amet,
consectetuer adipiscing elit. Duis fringilla tristique neque.
Sed interdum libero ut metus. Pellentesque placerat. Nam
rutrum augue a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris. Praesent lectus
tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris
lacinia lorem sit amet ipsum. Nunc quis urna dictum turpis
accumsan semper.
Sec.
5.1
on p. 25,
Sec.
5.2
on p. 27,
Fig. 3.1 on
p. 19
• First itemtext
• Second itemtext
• Last itemtext
• First itemtext
• Second itemtext
Continued on next page
63
C. Revisions to the Proposal
Panelist
name
Dr.
Mariana
X.
Mercado
Comment
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Etiam
lobortis facilisis sem.
Nullam nec mi et neque
pharetra
sollicitudin.
Praesent
imperdiet
mi nec ante. Donec
ullamcorper, felis non
sodales commodo, lectus velit ultrices augue,
a dignissim nibh lectus
placerat pede. Vivamus
nunc nunc, molestie ut,
ultricies vel, semper
in, velit. Ut porttitor.
Praesent in sapien.
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Duis
fringilla tristique neque.
Sed interdum libero ut
metus.
Pellentesque
placerat. Nam rutrum
augue a leo. Morbi
sed elit sit amet ante
lobortis
sollicitudin.
Praesent blandit blandit
mauris. Praesent lectus
tellus, aliquet aliquam,
luctus a, egestas a,
turpis. Mauris lacinia
lorem sit amet ipsum.
Nunc quis urna dictum
turpis accumsan semper.
Continued from previous page
Summary of how the comment was addressed
Locations
Lorem ipsum dolor sit amet, consectetuer adipiscing elit.
Etiam lobortis facilisis sem. Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus placerat pede. Vivamus
nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut
porttitor. Praesent in sapien. Lorem ipsum dolor sit amet,
consectetuer adipiscing elit. Duis fringilla tristique neque.
Sed interdum libero ut metus. Pellentesque placerat. Nam
rutrum augue a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris. Praesent lectus
tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris
lacinia lorem sit amet ipsum. Nunc quis urna dictum turpis
accumsan semper.
Sec.
5.1
on p. 25,
Sec.
5.2
on p. 27,
Fig. 3.1 on
p. 19
1. First itemtext
2. Second itemtext
3. Last itemtext
4. First itemtext
5. Second itemtext
Continued on next page
64
C. Revisions to the Proposal
Panelist
name
Dr. Rafael
W. Sison
Comment
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Etiam
lobortis facilisis sem.
Nullam nec mi et neque
pharetra
sollicitudin.
Praesent
imperdiet
mi nec ante. Donec
ullamcorper, felis non
sodales commodo, lectus velit ultrices augue,
a dignissim nibh lectus
placerat pede. Vivamus
nunc nunc, molestie ut,
ultricies vel, semper
in, velit. Ut porttitor.
Praesent in sapien.
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Duis
fringilla tristique neque.
Sed interdum libero ut
metus.
Pellentesque
placerat. Nam rutrum
augue a leo. Morbi
sed elit sit amet ante
lobortis
sollicitudin.
Praesent blandit blandit
mauris. Praesent lectus
tellus, aliquet aliquam,
luctus a, egestas a,
turpis. Mauris lacinia
lorem sit amet ipsum.
Nunc quis urna dictum
turpis accumsan semper.
Continued from previous page
Summary of how the comment was addressed
Lorem ipsum dolor sit amet, consectetuer adipiscing elit.
Etiam lobortis facilisis sem. Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus placerat pede. Vivamus
nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut
porttitor. Praesent in sapien. Lorem ipsum dolor sit amet,
consectetuer adipiscing elit. Duis fringilla tristique neque.
Sed interdum libero ut metus. Pellentesque placerat. Nam
rutrum augue a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris. Praesent lectus
tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris
lacinia lorem sit amet ipsum. Nunc quis urna dictum turpis
accumsan semper.
Locations
Sec.
5.1
on p. 25,
Sec.
5.2
on p. 27,
Fig. 3.1 on
p. 19
65
Appendix D
REVISIONS TO THE FINAL
66
D. Revisions to the Final
Make a table with the following columns for showing the summary of revisions to the
proposal based on the comments of the panel of examiners.
1. Panelist name
2. Comment
3. Summary of how the comment has been addressed
4. Locations in the document where the changes have been reflected
TABLE D.1
Panelist
name
Dr.
Francisco
D.
Baltasar
S UMMARY OF R EVISIONS TO THE T HESIS
Comment
Summary of how the comment has been addressed
Locations
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Etiam
lobortis facilisis sem.
Nullam nec mi et neque
pharetra
sollicitudin.
Praesent
imperdiet
mi nec ante. Donec
ullamcorper, felis non
sodales commodo, lectus velit ultrices augue,
a dignissim nibh lectus
placerat pede. Vivamus
nunc nunc, molestie ut,
ultricies vel, semper
in, velit. Ut porttitor.
Praesent in sapien.
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Duis
fringilla tristique neque.
Sed interdum libero ut
metus.
Pellentesque
placerat. Nam rutrum
augue a leo. Morbi
sed elit sit amet ante
lobortis
sollicitudin.
Praesent blandit blandit
mauris. Praesent lectus
tellus, aliquet aliquam,
luctus a, egestas a,
turpis. Mauris lacinia
lorem sit amet ipsum.
Nunc quis urna dictum
turpis accumsan semper.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit.
Etiam lobortis facilisis sem. Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus placerat pede. Vivamus
nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut
porttitor. Praesent in sapien. Lorem ipsum dolor sit amet,
consectetuer adipiscing elit. Duis fringilla tristique neque.
Sed interdum libero ut metus. Pellentesque placerat. Nam
rutrum augue a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris. Praesent lectus
tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris
lacinia lorem sit amet ipsum. Nunc quis urna dictum turpis
accumsan semper.
Sec.
5.1
on p. 25,
Sec.
5.2
on p. 27,
Fig. 3.1 on
p. 19
First itemtext
Second itemtext
Last itemtext
First itemtext
Second itemtext
Continued on next page
67
D. Revisions to the Final
Panelist
name
Comment
Dr. Amado
Z. Hernandez
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Etiam
lobortis facilisis sem.
Nullam nec mi et neque
pharetra
sollicitudin.
Praesent
imperdiet
mi nec ante. Donec
ullamcorper, felis non
sodales commodo, lectus velit ultrices augue,
a dignissim nibh lectus
placerat pede. Vivamus
nunc nunc, molestie ut,
ultricies vel, semper
in, velit. Ut porttitor.
Praesent in sapien.
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Duis
fringilla tristique neque.
Sed interdum libero ut
metus.
Pellentesque
placerat. Nam rutrum
augue a leo. Morbi
sed elit sit amet ante
lobortis
sollicitudin.
Praesent blandit blandit
mauris. Praesent lectus
tellus, aliquet aliquam,
luctus a, egestas a,
turpis. Mauris lacinia
lorem sit amet ipsum.
Nunc quis urna dictum
turpis accumsan semper.
Continued from previous page
Summary of how the comment has been addressed
Lorem ipsum dolor sit amet, consectetuer adipiscing elit.
Etiam lobortis facilisis sem. Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus placerat pede. Vivamus
nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut
porttitor. Praesent in sapien. Lorem ipsum dolor sit amet,
consectetuer adipiscing elit. Duis fringilla tristique neque.
Sed interdum libero ut metus. Pellentesque placerat. Nam
rutrum augue a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris. Praesent lectus
tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris
lacinia lorem sit amet ipsum. Nunc quis urna dictum turpis
accumsan semper.
Locations
Sec.
5.1
on p. 25,
Sec.
5.2
on p. 27,
Fig. 3.1 on
p. 19
First itemtext
Second itemtext
Last itemtext
First itemtext
Second itemtext
Continued on next page
68
D. Revisions to the Final
Panelist
name
Dr. Jose Y.
Alonzo
Comment
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Etiam
lobortis facilisis sem.
Nullam nec mi et neque
pharetra
sollicitudin.
Praesent
imperdiet
mi nec ante. Donec
ullamcorper, felis non
sodales commodo, lectus velit ultrices augue,
a dignissim nibh lectus
placerat pede. Vivamus
nunc nunc, molestie ut,
ultricies vel, semper
in, velit. Ut porttitor.
Praesent in sapien.
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Duis
fringilla tristique neque.
Sed interdum libero ut
metus.
Pellentesque
placerat. Nam rutrum
augue a leo. Morbi
sed elit sit amet ante
lobortis
sollicitudin.
Praesent blandit blandit
mauris. Praesent lectus
tellus, aliquet aliquam,
luctus a, egestas a,
turpis. Mauris lacinia
lorem sit amet ipsum.
Nunc quis urna dictum
turpis accumsan semper.
Continued from previous page
Summary of how the comment has been addressed
Lorem ipsum dolor sit amet, consectetuer adipiscing elit.
Etiam lobortis facilisis sem. Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus placerat pede. Vivamus
nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut
porttitor. Praesent in sapien. Lorem ipsum dolor sit amet,
consectetuer adipiscing elit. Duis fringilla tristique neque.
Sed interdum libero ut metus. Pellentesque placerat. Nam
rutrum augue a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris. Praesent lectus
tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris
lacinia lorem sit amet ipsum. Nunc quis urna dictum turpis
accumsan semper.
Locations
Sec.
5.1
on p. 25,
Sec.
5.2
on p. 27,
Fig. 3.1 on
p. 19
• First itemtext
• Second itemtext
• Last itemtext
• First itemtext
• Second itemtext
Continued on next page
69
D. Revisions to the Final
Panelist
name
Dr.
Mariana
X.
Mercado
Comment
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Etiam
lobortis facilisis sem.
Nullam nec mi et neque
pharetra
sollicitudin.
Praesent
imperdiet
mi nec ante. Donec
ullamcorper, felis non
sodales commodo, lectus velit ultrices augue,
a dignissim nibh lectus
placerat pede. Vivamus
nunc nunc, molestie ut,
ultricies vel, semper
in, velit. Ut porttitor.
Praesent in sapien.
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Duis
fringilla tristique neque.
Sed interdum libero ut
metus.
Pellentesque
placerat. Nam rutrum
augue a leo. Morbi
sed elit sit amet ante
lobortis
sollicitudin.
Praesent blandit blandit
mauris. Praesent lectus
tellus, aliquet aliquam,
luctus a, egestas a,
turpis. Mauris lacinia
lorem sit amet ipsum.
Nunc quis urna dictum
turpis accumsan semper.
Continued from previous page
Summary of how the comment has been addressed
Lorem ipsum dolor sit amet, consectetuer adipiscing elit.
Etiam lobortis facilisis sem. Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus placerat pede. Vivamus
nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut
porttitor. Praesent in sapien. Lorem ipsum dolor sit amet,
consectetuer adipiscing elit. Duis fringilla tristique neque.
Sed interdum libero ut metus. Pellentesque placerat. Nam
rutrum augue a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris. Praesent lectus
tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris
lacinia lorem sit amet ipsum. Nunc quis urna dictum turpis
accumsan semper.
Locations
Sec.
5.1
on p. 25,
Sec.
5.2
on p. 27,
Fig. 3.1 on
p. 19
1. First itemtext
2. Second itemtext
3. Last itemtext
4. First itemtext
5. Second itemtext
Continued on next page
70
D. Revisions to the Final
Panelist
name
Dr. Rafael
W. Sison
Comment
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Etiam
lobortis facilisis sem.
Nullam nec mi et neque
pharetra
sollicitudin.
Praesent
imperdiet
mi nec ante. Donec
ullamcorper, felis non
sodales commodo, lectus velit ultrices augue,
a dignissim nibh lectus
placerat pede. Vivamus
nunc nunc, molestie ut,
ultricies vel, semper
in, velit. Ut porttitor.
Praesent in sapien.
Lorem ipsum dolor
sit amet, consectetuer
adipiscing elit. Duis
fringilla tristique neque.
Sed interdum libero ut
metus.
Pellentesque
placerat. Nam rutrum
augue a leo. Morbi
sed elit sit amet ante
lobortis
sollicitudin.
Praesent blandit blandit
mauris. Praesent lectus
tellus, aliquet aliquam,
luctus a, egestas a,
turpis. Mauris lacinia
lorem sit amet ipsum.
Nunc quis urna dictum
turpis accumsan semper.
Continued from previous page
Summary of how the comment has been addressed
Lorem ipsum dolor sit amet, consectetuer adipiscing elit.
Etiam lobortis facilisis sem. Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus placerat pede. Vivamus
nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut
porttitor. Praesent in sapien. Lorem ipsum dolor sit amet,
consectetuer adipiscing elit. Duis fringilla tristique neque.
Sed interdum libero ut metus. Pellentesque placerat. Nam
rutrum augue a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris. Praesent lectus
tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris
lacinia lorem sit amet ipsum. Nunc quis urna dictum turpis
accumsan semper.
Locations
Sec.
5.1
on p. 25,
Sec.
5.2
on p. 27,
Fig. 3.1 on
p. 19
71
Appendix E
USAGE EXAMPLES
72
E. Usage Examples
The user is expected to have a working knowledge of LATEX. A good introduction
is in [Oetiker et al., 2014]. Its latest version can be accessed at http://www.ctan.org/
tex-archive/info/lshort.
E1
Equations
The following examples show how to typeset equations in LATEX. This section also shows
examples of the use of \gls{ } commands in conjunction with the items that are in
the notation.tex file. Please make sure that the entries in notation.tex are
those that are referenced in the LATEX document files used by this Thesis. Please
comment out unused notations and be careful with the commas and brackets in
notation.tex .
In (E.1), the output signal y (t) is the result of the convolution of the input signal x (t)
and the impulse response h (t) .
Z
+∞
y (t) = h (t) ∗ x (t) =
h (t − τ ) x (τ ) dτ
(E.1)
−∞
Other example equations are as follows.
#
"
A B V2
V1
=
I1
C D I2
(E.2)
j y k
k
1 j
−17bxc−mod(byc,17)
< mod
2
,2 ,
2
17
(E.3)
|ζ(x)3 ζ(x + iy)4 ζ(x + 2iy)| = exp
X 3 + 4 cos(ny log p) + cos(2ny log p)
n,p
npnx
≥ 1 (E.4)
73
E. Usage Examples
The verbatim LATEX code of Sec. E1 is in List. E.1.
Listing E.1: Sample LATEX code for equations and notations usage
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
The following examples show how to typeset equations in \ LaTeX . This
section also shows examples of the use of \ verb | \ gls { } | commands
in conjunction with the items that are in the \ verb | notation . tex |
file . \ textbf { Please make sure that the entries in } \ verb | notation .
tex |\ textbf { are those that are referenced in the \ LaTeX \
document files used by this \ documentType . Please comment out
unused notations and be careful with the commas and brackets in } \
verb | notation . tex |.
In ~\ eqref { eq : conv } , the output signal \ gls { not : output _ sigt } is the
result of the convolution of the input signal \ gls { not : input _ sigt }
and the impulse response \ gls { not : ir }.
\ begin { eqnarray }
y \ left ( t \ right ) = h \ left ( t \ right ) * x \ left ( t \ right ) =\ int _{ -\
infty }^{+\ infty } h \ left ( t -\ tau \ right ) x \ left ( \ tau \ right ) \
mathrm { d }\ tau
\ label { eq : conv }
\ end { eqnarray }
Other example equations are as follows .
\ begin { eqnarray }
\ left [ \ dfrac { V _{1} }{ I _{1} } \ right ] =
\ begin { bmatrix }
A & B \\
C & D
\ end { bmatrix }
\ left [ \ dfrac { V _{2} }{ I _{2} } \ right ]
\ label { eq : ABCD }
\ end { eqnarray }
\ begin { eqnarray }
\ dfrac {1}{2} < \ left \ lfloor \ mathrm { mod }\ left (\ left \ lfloor \ dfrac { y }{17}
\ right \ rfloor 2^{ -17 \ lfloor x \ rfloor - \ mathrm { mod }(\ lfloor y \
rfloor , 17) } ,2\ right ) \ right \ rfloor ,
\ end { eqnarray }
\ begin { eqnarray }
| \ zeta ( x ) ^3 \ zeta ( x + iy ) ^4 \ zeta ( x + 2 iy ) | =
\ exp \ sum _{ n , p } \ frac {3 + 4 \ cos ( ny \ log p ) + \ cos (2 ny \ log p ) }{ np ^{ nx
}} \ ge 1
\ end { eqnarray }
74
E. Usage Examples
E2
Notations
In order to use the standardized notation, the user is highly suggested to see the ISO 80000-2
standard [ISO, 2009].
See https://en.wikipedia.org/wiki/Help:Displaying a formula and https://en.wikipedia.
org/wiki/List of mathematical symbols for LATEX maths and other notations, respectively.
The following were taken from isomath-test.tex .
E2.1
Math alphabets
If there are other symbols in place of Greek letters in a math alphabet, it uses T1 or OT1
font encoding instead of OML.
mathnormal
mathit
mathrm
mathbf
mathsf
mathtt
A, B, Γ, ∆, Θ, Λ, Ξ, Π, Σ, Φ, Ψ, Ω, α, β, π, ν, ω, v, w, 0, 1, 9
A, B , Γ , ∆, Θ, Λ, Ξ , Π , Σ , Φ, Ψ , Ω , ff , fi , ß , ˚, ! , v , w , 0 , 1 , 9
A, B, Γ, ∆, Θ, Λ, Ξ, Π, Σ, Φ, Ψ, Ω, ff, fi, ß, ˚, !, v, w, 0, 1, 9
A, B, Γ, ∆, Θ, Λ, Ξ, Π, Σ, Φ, Ψ, Ω, ff , fi, ß, ˚, !, v, w, 0, 1, 9
A, B, Γ, ∆, Θ, Λ, Ξ, Π, Σ, Φ, Ψ, Ω, ff, fi, ß, ˚, !, v, w, 0, 1, 9
A, B, Γ, ∆, Θ, Λ, Ξ, Π, Σ, Φ, Ψ, Ω, ↑, ↓, ß, ˚, !, v, w, 0, 1, 9
New alphabets bold-italic, sans-serif-italic, and sans-serif-bold-italic.
mathbfit
mathsfit
mathsfbfit
A, B, Γ , ∆, Θ, Λ, Ξ, Π, Σ, Φ, Ψ , Ω, α, β, π, ν, ω, v, w, , ,
A, B , , , , , , , , , , , , , , , ! , v , w , 0, 1, 9
A, B , , , , , , , , , , , , , , , !, v , w , 0, 1, 9
Do the math alphabets match?
axαωaxαω ax ! T C T CΘΓ T CΘΓ
E2.2
Vector symbols
Alphabetic symbols for vectors are boldface italic, λ = e1 · a, while numeric ones (e.g.
the zero vector) are bold upright, a + 0 = a.
E2.3
Matrix symbols
Symbols for matrices are boldface italic, too:1 Λ = E · A.
1
However, matrix symbols are usually capital letters whereas vectors are small ones. Exceptions are
physical quantities like the force vector F or the electrical field E.
75
E. Usage Examples
E2.4
Tensor symbols
Symbols for tensors are sans-serif bold italic,
=e ·a
⇐⇒
αijl = eijk · akl .
The permittivity tensor describes the coupling of electric field and displacement:
D = 0 r E
76
E. Usage Examples
E2.5
Bold math version
The “bold” math version is selected with the commands \boldmath or \mathversion{bold}
mathnormal
mathit
mathrm
mathbf
mathsf
mathtt
A, B, Γ, ∆, Θ, Λ, Ξ, Π, Σ, Φ, Ψ, Ω, α, β, π, ν, ω, v, w, 0, 1, 9
A, B, Γ , ∆, Θ, Λ, Ξ , Π , Σ , Φ, Ψ , Ω, ff , fi , ß, ˚, ! , v , w , 0 , 1 , 9
A, B, Γ, ∆, Θ, Λ, Ξ, Π, Σ, Φ, Ψ, Ω, ff , fi, ß, ˚, !, v, w, 0, 1, 9
A, B, Γ, ∆, Θ, Λ, Ξ, Π, Σ, Φ, Ψ, Ω, ff , fi, ß, ˚, !, v, w, 0, 1, 9
A, B, Γ, ∆, Θ, Λ, Ξ, Π, Σ, Φ, Ψ, Ω, ff, fi, ß, ˚, !, v, w, 0, 1, 9
A, B, Γ, ∆, Θ, Λ, Ξ, Π, Σ, Φ, Ψ, Ω, ↑, ↓, ß, ˚, !, v, w, 0, 1, 9
New alphabets bold-italic, sans-serif-italic, and sans-serif-bold-italic.
mathbfit
mathsfit
mathsfbfit
A, B, Γ , ∆, Θ, Λ, Ξ, Π, Σ, Φ, Ψ , Ω, α, β, π, ν, ω, v, w, , ,
A, B , , , , , , , , , , , , , , , !, v , w , 0, 1, 9
A, B , , , , , , , , , , , , , , , !, v , w , 0, 1, 9
Do the math alphabets match?
axαωaxαω ax ! T C T CΘΓ T CΘΓ
E2.5.1
Vector symbols
Alphabetic symbols for vectors are boldface italic, λ = e1 · a, while numeric ones (e.g.
the zero vector) are bold upright, a + 0 = a.
E2.5.2
Matrix symbols
Symbols for matrices are boldface italic, too:2 Λ = E · A.
E2.5.3
Tensor symbols
Symbols for tensors are sans-serif bold italic,
=e ·a
⇐⇒
αijl = eijk · akl .
The permittivity tensor describes the coupling of electric field and displacement:
D = 0 r E
2
However, matrix symbols are usually capital letters whereas vectors are small ones. Exceptions are
physical quantities like the force vector F or the electrical field E.
77
E. Usage Examples
The verbatim LATEX code of Sec. E2 is in List. E.2.
Listing E.2: Sample LATEX code for notations usage
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
% A teststring with Latin and Greek letters ::
\ newcommand {\ teststring }{ %
% capital Latin letters
% A ,B ,C ,
A ,B ,
% capital Greek letters
% \ Gamma ,\ Delta ,\ Theta ,\ Lambda ,\ Xi ,\ Pi ,\ Sigma ,\ Upsilon ,\ Phi ,\ Psi ,
\ Gamma ,\ Delta ,\ Theta ,\ Lambda ,\ Xi ,\ Pi ,\ Sigma ,\ Phi ,\ Psi ,\ Omega ,
% small Greek letters
\ alpha ,\ beta ,\ pi ,\ nu ,\ omega ,
% small Latin letters :
% compare \ nu , \ omega , v , and w
v ,w ,
% digits
0 ,1 ,9
}
\ subsection { Math alphabets }
If there are other symbols in place of Greek letters in a math
alphabet , it uses T 1 or OT 1 font encoding instead of OML .
\ begin { eqnarray *}
\ mbox { mathnormal } & & \ teststring \\
\ mbox { mathit } & & \ mathit {\ teststring }\\
\ mbox { mathrm } & & \ mathrm {\ teststring }\\
\ mbox { mathbf } & & \ mathbf {\ teststring }\\
\ mbox { mathsf } & & \ mathsf {\ teststring }\\
\ mbox { mathtt } & & \ mathtt {\ teststring }
\ end { eqnarray *}
New alphabets bold - italic , sans - serif - italic , and sans - serif - bold italic .
\ begin { eqnarray *}
\ mbox { mathbfit }
& & \ mathbfit {\ teststring }\\
\ mbox { mathsfit }
& & \ mathsfit {\ teststring }\\
\ mbox { mathsfbfit } & & \ mathsfbfit {\ teststring }
\ end { eqnarray *}
%
Do the math alphabets match ?
$
\ mathnormal { a
\ mathbfit
{a
\ mathsfbfit { a x
\ quad
\ mathsfbfit { T C
\ mathbfit
{T
\ mathnormal { T
$
x \ alpha \ omega }
x \ alpha \ omega }
\ alpha \ omega }
\ Theta \ Gamma }
C \ Theta \ Gamma }
C \ Theta \ Gamma }
\ subsection { Vector symbols }
78
E. Usage Examples
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
Alphabetic symbols for vectors are boldface italic ,
$\ vec {\ lambda }=\ vec { e }_{1}\ cdot \ vec { a }$ ,
while numeric ones ( e . g . the zero vector ) are bold upright ,
$\ vec { a } + \ vec {0} = \ vec { a }$.
\ subsection { Matrix symbols }
Symbols for matrices are boldface italic , too : %
\ footnote { However , matrix symbols are usually capital letters whereas
vectors
are small ones . Exceptions are physical quantities like the force
vector $\ vec { F }$ or the electrical field $\ vec { E }$. %
}
$\ matrixsym {\ Lambda }=\ matrixsym { E }\ cdot \ matrixsym { A }.$
\ subsection { Tensor symbols }
Symbols for tensors are sans - serif bold italic ,
\[
\ tensorsym {\ alpha } = \ tensorsym { e }\ cdot \ tensorsym { a }
\ quad \ Longleftrightarrow \ quad
\ alpha _{ ijl } = e _{ ijk }\ cdot a _{ kl }.
\]
The permittivity tensor describes the coupling of electric field and
displacement : \[
\ vec { D }=\ epsilon _{0}\ tensorsym {\ epsilon }_{\ mathrm { r }}\ vec { E }\]
\ newpage
\ subsection { Bold math version }
The ‘‘ bold ’ ’ math version is selected with the commands
\ verb +\ boldmath + or \ verb +\ mathversion { bold }+
{\ boldmath
\ begin { eqnarray *}
\ mbox { mathnormal } & & \ teststring \\
\ mbox { mathit } & & \ mathit {\ teststring }\\
\ mbox { mathrm } & & \ mathrm {\ teststring }\\
\ mbox { mathbf } & & \ mathbf {\ teststring }\\
\ mbox { mathsf } & & \ mathsf {\ teststring }\\
\ mbox { mathtt } & & \ mathtt {\ teststring }
\ end { eqnarray *}
New alphabets bold - italic , sans - serif - italic , and sans - serif - bold italic .
\ begin { eqnarray *}
\ mbox { mathbfit }
& & \ mathbfit {\ teststring }\\
\ mbox { mathsfit }
& & \ mathsfit {\ teststring }\\
\ mbox { mathsfbfit } & & \ mathsfbfit {\ teststring }
\ end { eqnarray *}
%
Do the math alphabets match ?
79
E. Usage Examples
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
$
\ mathnormal { a
\ mathbfit
{a
\ mathsfbfit { a x
\ quad
\ mathsfbfit { T C
\ mathbfit
{T
\ mathnormal { T
$
x \ alpha \ omega }
x \ alpha \ omega }
\ alpha \ omega }
\ Theta \ Gamma }
C \ Theta \ Gamma }
C \ Theta \ Gamma }
\ subsection { Vector symbols }
Alphabetic symbols for vectors are boldface italic ,
$\ vec {\ lambda }=\ vec { e }_{1}\ cdot \ vec { a }$ ,
while numeric ones ( e . g . the zero vector ) are bold upright ,
$\ vec { a } + \ vec {0} = \ vec { a }$.
\ subsection { Matrix symbols }
Symbols for matrices are boldface italic , too : %
\ footnote { However , matrix symbols are usually capital letters whereas
vectors
are small ones . Exceptions are physical quantities like the force
vector $\ vec { F }$ or the electrical field $\ vec { E }$. %
}
$\ matrixsym {\ Lambda }=\ matrixsym { E }\ cdot \ matrixsym { A }.$
\ subsection { Tensor symbols }
Symbols for tensors are sans - serif bold italic ,
\[
\ tensorsym {\ alpha } = \ tensorsym { e }\ cdot \ tensorsym { a }
\ quad \ Longleftrightarrow \ quad
\ alpha _{ ijl } = e _{ ijk }\ cdot a _{ kl }.
\]
The permittivity tensor describes the coupling of electric field and
displacement : \[
\ vec { D }=\ epsilon _{0}\ tensorsym {\ epsilon }_{\ mathrm { r }}\ vec { E }\]
}
80
E. Usage Examples
E3
Abbreviation
This section shows examples of the use of LATEXcommands in conjunction with the
items that are in the abbreviation.tex and in the glossary.tex files. Please see
List. E.3. To lessen the LATEX parsing time, it is suggested that you use \acr{ } only
for the first occurrence of the word to be abbreviated.
Again please see List. E.3. Here is an example of first use: alternating current (ac).
Next use: ac. Full: alternating current (ac). Here’s an acronym referenced using \acr :
hyper-text markup language (html). And here it is again: html. If you are used to the
glossaries package, note the difference in using \gls : hyper-text markup language
(html). And again (no difference): hyper-text markup language (html). Here are some more
entries:
• extensible markup language (xml) and cascading style sheet (css).
• Next use: xml and css.
• Full form: extensible markup language (xml) and cascading style sheet (css).
• Reset again.
• Start with a capital. Hyper-text markup language (html).
• Next: Html. Full: Hyper-text markup language (html).
• Prefer capitals? Extensible markup language (XML). Next: XML. Full: extensible
markup language (XML).
• Prefer small-caps? Cascading style sheet (CSS). Next:
sheet (CSS).
CSS .
Full: cascading style
• Resetting all acronyms.
• Here are the acronyms again:
• Hyper-text markup language (HTML), extensible markup language (XML) and cascading style sheet (CSS).
• Next use: H TML, XML and CSS.
• Full form: Hyper-text markup language (HTML), extensible markup language (XML)
and cascading style sheet (CSS).
81
E. Usage Examples
• Provide your own link text: style sheet.
The verbatim LATEX code of Sec. E3 is in List. E.3.
Listing E.3: Sample LATEX code for abbreviations usage
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
Again please see List .~\ ref { lst : abbrv }. Here is an example of first use :
\ acr { ac }. Next use : \ acr { ac }. Full : \ gls { ac }. Here ’ s an acronym
referenced using \ verb | \ acr |: \ acr { html }. And here it is again : \
acr { html }. If you are used to the \ texttt { glossaries } package , note
the difference in using \ verb | \ gls |: \ gls { html }. And again ( no
difference ) : \ gls { html }. Here are some more entries :
\ begin { itemize }
\ item \ acr { xml } and \ acr { css }.
\ item Next use : \ acr { xml } and \ acr { css }.
\ item Full form : \ gls { xml } and \ gls { css }.
\ item Reset again . \ glsresetall { abbreviation }
\ item Start with a capital . \ Acr { html }.
\ item Next : \ Acr { html }. Full : \ Gls { html }.
\ item Prefer capitals ? \ renewcommand {\ acronymfont }[1]{\
MakeTextUppercase {#1}} \ Acr { xml }. Next : \ acr { xml }. Full : \ gls { xml
}.
\ item Prefer small - caps ? \ renewcommand {\ acronymfont }[1]{\ textsc {#1}}
\ Acr { css }. Next : \ acr { css }. Full : \ gls { css }.
\ item Resetting all acronyms .\ glsresetall { abbreviation }
\ item Here are the acronyms again :
\ item \ Acr { html } , \ acr { xml } and \ acr { css }.
\ item Next use : \ Acr { html } , \ acr { xml } and \ acr { css }.
\ item Full form : \ Gls { html } , \ gls { xml } and \ gls { css }.
\ item Provide your own link text : \ glslink {[ textbf ] css }{ style }
\ end { itemize }
82
E. Usage Examples
E4
Glossary
This section shows examples of the use of \gls{ } commands in conjunction with the
items that are in the glossary.tex and notation.tex files. Note that entries in
notation.tex are prefixed with “ not: ” label (see List. E.4).
Please make sure that the entries in notation.tex are those that are referenced
in the LATEX document files used by this Thesis. Please comment out unused notations
and be careful with the commas and brackets in notation.tex .
• Matrices are usually denoted by a bold capital letter, such as A. The matrix’s (i, j)th
element is usually denoted aij . Matrix I is the identity matrix.
• A set, denoted as S , is a collection of objects.
• The universal set, denoted as U , is the set of everything.
• The empty set, denoted as ∅ , contains no elements.
• Functional Analysis is seen as the study of complete normed vector spaces, i.e.,
Banach spaces.
• The cardinality of a set, denoted as |S| , is the number of elements in the set.
The verbatim LATEX code for the part of Sec. E4 is in List. E.4.
83
E. Usage Examples
Listing E.4: Sample LATEX code for glossary and notations usage
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
\ begin { itemize }
\ item \ Glspl { matrix } are usually denoted by a bold capital letter ,
such as $\ mathbfit { A }$. The \ gls { matrix } ’ s $( i , j ) $ th element is
usually denoted $ a _{ ij }$. \ Gls { matrix } $\ mathbf { I }$ is the
identity \ gls { matrix }.
\ item A set , denoted as \ gls { not : set } , is a collection of objects .
\ item The universal set ,
set of everything .
denoted as \ gls { not : universalSet } , is the
\ item The empty set , denoted as \ gls { not : emptySet } , contains no
elements .
\ item \ Gls { Functional Analysis } is seen as the study of complete
normed vector spaces , i . e . , Banach spaces .
\ item The cardinality of a set , denoted as \ gls { not : cardinality } , is
the number of elements in the set .
\ end { enumerate }
84
E. Usage Examples
E5
Figure
This section shows several ways of placing figures. PDFLATEX compatible files are PDF,
PNG, and JPG. Please see the figure subdirectory.
Fig. E.1
A quadrilateral image example.
85
E. Usage Examples
Fig. E.1 is a gray box enclosed by a dark border. List. E.5 shows the corresponding
EX code.
LAT
Listing E.5: Sample LATEX code for a single figure
1
2
3
4
5
6
7
8
9
10
\ begin { figure }[! htbp ]
\ centering
\ includegraphics [ width =0.5\ textwidth ]{ example }
\ caption { A quadrilateral image example .}
\ label { fig : example }
\ end { figure }
\ cleardoublepage
Fig .~\ ref { fig : example } is a gray box enclosed by a dark border . List .~\
ref { lst : onefig } shows the corresponding \ LaTeX \ code .
\ end { figure }
86
E. Usage Examples
(a) A sub-figure in the top row.
(b) A sub-figure in the middle row.
(c) A sub-figure in the bottom row.
Fig. E.2
Figures on top of each other. See List. E.6 for the corresponding LATEX code.
87
E. Usage Examples
Listing E.6: Sample LATEX code for three figures on top of each other
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
\ begin { figure }[! htbp ]
\ centering
\ subbottom [ A sub - figure in the top row .]{
\ includegraphics [ width =0.35\ textwidth ]{ example _ gray _ box }
\ label { fig : top }
}
\ vfill
\ subbottom [ A sub - figure in the middle row .]{
\ includegraphics [ width =0.35\ textwidth ]{ example _ gray _ box }
\ label { fig : mid }
}
\ vfill
\ subbottom [ A sub - figure in the bottom row .]{
\ includegraphics [ width =0.35\ textwidth ]{ example _ gray _ box }
\ label { fig : botm }
}
\ caption { Figures on top of each other }
\ label { fig : tmb }
\ end { figure }
88
E. Usage Examples
(a) A sub-figure in the upper-left corner.
(b) A sub-figure in the upper-right corner.
(c) A sub-figure in the lower-left corner.
(d) A sub-figure in the lower-right corner
Fig. E.3
Four figures in each corner. See List. E.7 for the corresponding LATEX code.
89
E. Usage Examples
Listing E.7: Sample LATEX code for the four figures
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
\ begin { figure }[! htbp ]
\ centering
\ subbottom [ A sub - figure in the upper - left corner .]{
\ includegraphics [ width =0.45\ textwidth ]{ example _ gray _ box }
\ label { fig : upprleft }
}
\ hfill
\ subbottom [ A sub - figure in the upper - right corner .]{
\ includegraphics [ width =0.45\ textwidth ]{ example _ gray _ box }
\ label { fig : uppright }
}
\ vfill
\ subbottom [ A sub - figure in the lower - left corner .]{
\ includegraphics [ width =0.45\ textwidth ]{ example _ gray _ box }
\ label { fig : lowerleft }
}
\ hfill
\ subbottom [ A sub - figure in the lower - right corner ]{
\ includegraphics [ width =0.45\ textwidth ]{ example _ gray _ box }
\ label { fig : lowright }
}
\ caption { Four figures in each corner . See List .~\ ref { lst : fourfigs } for
the corresponding \ LaTeX \ code .}
\ label { fig : fourfig }
\ end { figure }
90
E. Usage Examples
E6
Table
This section shows an example of placing a table (a long one). Table E.1 are the triples.
TABLE E.1
Time (s)
0
2745
5490
8235
10980
13725
16470
19215
21960
24705
27450
30195
32940
35685
38430
41175
43920
46665
49410
52155
54900
57645
60390
63135
65880
68625
71370
74115
76860
79605
82350
85095
87840
90585
93330
96075
98820
101565
104310
107055
109800
112545
115290
118035
120780
123525
F EASIBLE TRIPLES FOR HIGHLY VARIABLE GRID
Triple chosen
(1, 11, 13725)
(1, 12, 10980)
(1, 12, 13725)
(1, 12, 16470)
(1, 12, 16470)
(1, 12, 16470)
(1, 13, 16470)
(1, 12, 16470)
(1, 12, 16470)
(1, 12, 16470)
(1, 12, 16470)
(2, 2, 2745)
(1, 13, 16470)
(1, 13, 13725)
(1, 13, 10980)
(1, 12, 13725)
(1, 13, 10980)
(2, 2, 2745)
(2, 2, 2745)
(1, 12, 16470)
(1, 13, 13725)
(1, 13, 13725)
(1, 12, 13725)
(1, 13, 16470)
(1, 13, 16470)
(2, 2, 2745)
(1, 13, 13725)
(1, 12, 13725)
(1, 13, 13725)
(1, 13, 13725)
(1, 12, 13725)
(1, 12, 13725)
(1, 13, 16470)
(1, 13, 16470)
(1, 13, 13725)
(1, 13, 16470)
(1, 13, 16470)
(1, 13, 13725)
(1, 13, 16470)
(1, 13, 13725)
(1, 13, 13725)
(1, 12, 16470)
(1, 13, 16470)
(1, 13, 13725)
(1, 13, 16470)
(1, 13, 13725)
Other feasible triples
(1, 12, 10980), (1, 13, 8235), (2, 2, 0), (3, 1, 0)
(1, 13, 8235), (2, 2, 0), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(1, 13, 13725), (2, 2, 2745), (2, 3, 0), (3, 1, 0)
(1, 13, 13725), (2, 2, 2745), (2, 3, 0), (3, 1, 0)
(1, 13, 13725), (2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(1, 13, 13725), (2, 2, 2745), (2, 3, 0), (3, 1, 0)
(1, 13, 13725), (2, 2, 2745), (2, 3, 0), (3, 1, 0)
(1, 13, 13725), (2, 2, 2745), (2, 3, 0), (3, 1, 0)
(1, 13, 13725), (2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(1, 13, 10980), (2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 3, 0), (3, 1, 0)
(2, 3, 0), (3, 1, 0)
(1, 13, 13725), (2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(1, 13, 10980), (2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(1, 13, 13725), (2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
Continued on next page
91
E. Usage Examples
Time (s)
126270
129015
131760
134505
137250
139995
142740
145485
148230
150975
153720
156465
159210
161955
164700
Triple chosen
(1, 12, 16470)
(2, 2, 2745)
(2, 2, 2745)
(1, 13, 16470)
(1, 13, 13725)
(2, 2, 2745)
(2, 2, 2745)
(1, 12, 16470)
(2, 2, 2745)
(1, 13, 16470)
(1, 12, 13725)
(1, 13, 13725)
(1, 13, 13725)
(1, 13, 16470)
(1, 13, 13725)
Continued from previous page
Other feasible triples
(1, 13, 13725), (2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 3, 0), (3, 1, 0)
(2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 3, 0), (3, 1, 0)
(2, 3, 0), (3, 1, 0)
(1, 13, 13725), (2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
(2, 2, 2745), (2, 3, 0), (3, 1, 0)
92
E. Usage Examples
List. E.8 shows the corresponding LATEX code.
Listing E.8: Sample LATEX code for making typical table environment
1
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\ begin { center }
{\ scriptsize
\ begin { tabularx }{\ textwidth }{ p {0.1\ textwidth }| p {0.2\ textwidth }| p {0.5\
textwidth }}
\ caption { Feasible triples for highly variable grid } \ label { tab : triple _
grid } \\
\ hline
\ hline
\ textbf { Time ( s ) } &
\ textbf { Triple chosen } &
\ textbf { Other feasible triples } \\
\ hline
\ endfirsthead
\ multicolumn {3}{ c } %
{\ textit { Continued from previous page }} \\
\ hline
\ hline
\ textbf { Time ( s ) } &
\ textbf { Triple chosen } &
\ textbf { Other feasible triples } \\
\ hline
\ endhead
\ hline
\ multicolumn {3}{ r }{\ textit { Continued on next page }} \\
\ endfoot
\ hline
\ endlastfoot
\ hline
0 & (1 , 11 , 13725) & (1 , 12 , 10980) , (1 , 13 , 8235) , (2 , 2 , 0) , (3 , 1 , 0)
\\
2745 & (1 , 12 , 10980) & (1 , 13 , 8235) , (2 , 2 , 0) , (2 , 3 , 0) , (3 , 1 , 0)
\\
5490 & (1 , 12 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
8235 & (1 , 12 , 16470) & (1 , 13 , 13725) , (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 ,
0) \\
10980 & (1 , 12 , 16470) & (1 , 13 , 13725) , (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 ,
0) \\
13725 & (1 , 12 , 16470) & (1 , 13 , 13725) , (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 ,
0) \\
16470 & (1 , 13 , 16470) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
19215 & (1 , 12 , 16470) & (1 , 13 , 13725) , (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 ,
0) \\
21960 & (1 , 12 , 16470) & (1 , 13 , 13725) , (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 ,
0) \\
24705 & (1 , 12 , 16470) & (1 , 13 , 13725) , (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 ,
0) \\
27450 & (1 , 12 , 16470) & (1 , 13 , 13725) , (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 ,
0) \\
30195 & (2 , 2 , 2745) & (2 , 3 , 0) , (3 , 1 , 0) \\
32940 & (1 , 13 , 16470) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
35685 & (1 , 13 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
38430 & (1 , 13 , 10980) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
93
E. Usage Examples
43
44
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41175 & (1 , 12 , 13725) & (1 , 13 , 10980) , (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 ,
0) \\
43920 & (1 , 13 , 10980) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
46665 & (2 , 2 , 2745) & (2 , 3 , 0) , (3 , 1 , 0) \\
49410 & (2 , 2 , 2745) & (2 , 3 , 0) , (3 , 1 , 0) \\
52155 & (1 , 12 , 16470) & (1 , 13 , 13725) , (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 ,
0) \\
54900 & (1 , 13 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
57645 & (1 , 13 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
60390 & (1 , 12 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
63135 & (1 , 13 , 16470) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
65880 & (1 , 13 , 16470) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
68625 & (2 , 2 , 2745) & (2 , 3 , 0) , (3 , 1 , 0) \\
71370 & (1 , 13 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
74115 & (1 , 12 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
76860 & (1 , 13 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
79605 & (1 , 13 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
82350 & (1 , 12 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
85095 & (1 , 12 , 13725) & (1 , 13 , 10980) , (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 ,
0) \\
87840 & (1 , 13 , 16470) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
90585 & (1 , 13 , 16470) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
93330 & (1 , 13 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
96075 & (1 , 13 , 16470) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
98820 & (1 , 13 , 16470) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
101565 & (1 , 13 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
104310 & (1 , 13 , 16470) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
107055 & (1 , 13 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
109800 & (1 , 13 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
112545 & (1 , 12 , 16470) & (1 , 13 , 13725) , (2 , 2 , 2745) , (2 , 3 , 0) , (3 ,
1 , 0) \\
115290 & (1 , 13 , 16470) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
118035 & (1 , 13 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
120780 & (1 , 13 , 16470) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
123525 & (1 , 13 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
126270 & (1 , 12 , 16470) & (1 , 13 , 13725) , (2 , 2 , 2745) , (2 , 3 , 0) , (3 ,
1 , 0) \\
129015 & (2 , 2 , 2745) & (2 , 3 , 0) , (3 , 1 , 0) \\
131760 & (2 , 2 , 2745) & (2 , 3 , 0) , (3 , 1 , 0) \\
134505 & (1 , 13 , 16470) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
137250 & (1 , 13 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
139995 & (2 , 2 , 2745) & (2 , 3 , 0) , (3 , 1 , 0) \\
142740 & (2 , 2 , 2745) & (2 , 3 , 0) , (3 , 1 , 0) \\
145485 & (1 , 12 , 16470) & (1 , 13 , 13725) , (2 , 2 , 2745) , (2 , 3 , 0) , (3 ,
1 , 0) \\
148230 & (2 , 2 , 2745) & (2 , 3 , 0) , (3 , 1 , 0) \\
150975 & (1 , 13 , 16470) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
153720 & (1 , 12 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
156465 & (1 , 13 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
159210 & (1 , 13 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
161955 & (1 , 13 , 16470) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
164700 & (1 , 13 , 13725) & (2 , 2 , 2745) , (2 , 3 , 0) , (3 , 1 , 0) \\
\ end { tabularx }
}
\ end { center }
94
E. Usage Examples
E7
Algorithm or Pseudocode Listing
Table E.2 shows an example pseudocode. Note that if the pseudocode exceeds one page, it
can mean that its implementation is not modular. List. E.9 shows the corresponding LATEX
code.
TABLE E.2
Input(s):
n
x
Output(s):
y
:
:
nth power; n ∈ Z+
base value; x ∈ R+
:
result; y ∈ R+
C ALCULATION OF y = xn
Require: n ≥ 0 ∨ x 6= 0
Ensure: y = xn
1: y ⇐ 1
2: if n < 0 then
3:
X ⇐ 1/x
4:
N ⇐ −n
5: else
6:
X⇐x
7:
N ⇐n
8: end if
9: while N 6= 0 do
10:
if N is even then
11:
X ⇐X ×X
12:
N ⇐ N/2
13:
else {N is odd}
14:
y ⇐y×X
15:
N ⇐N −1
16:
end if
17: end while
95
E. Usage Examples
Listing E.9: Sample LATEX code for algorithm or pseudocode listing usage
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\ begin { table }[! htbp ]
\ caption { Calculation of $ y = x ^ n $}
\ label { tab : calcxn }
{\ footnotesize
\ begin { tabular }{ lll }
\ hline
\ hline
{\ bfseries Input ( s ) :} & & \\
$ n $ & : & $ n $ th power ; $ n \ in \ mathbb { Z }^{+}$ \\
$ x $ & : & base value ; $ x \ in \ mathbb { R }^{+}$ \\
\ hline
{\ bfseries Output ( s ) :} & & \\
$ y $ & : & result ; $ y \ in \ mathbb { R }^{+}$ \\
\ hline
\ hline
\\
\ end { tabular }
}
\ begin { algorithmic }[1]
{\ footnotesize
\ REQUIRE $ n \ geq 0 \ vee x \ neq 0$
\ ENSURE $ y = x ^ n $
\ STATE $ y \ Leftarrow 1$
\ IF {$ n < 0$}
\ STATE $ X \ Leftarrow 1 / x $
\ STATE $ N \ Leftarrow -n $
\ ELSE
\ STATE $ X \ Leftarrow x $
\ STATE $ N \ Leftarrow n $
\ ENDIF
\ WHILE {$ N \ neq 0$}
\ IF {$ N $ is even }
\ STATE $ X \ Leftarrow X \ times X $
\ STATE $ N \ Leftarrow N / 2$
\ ELSE [$ N $ is odd ]
\ STATE $ y \ Leftarrow y \ times X $
\ STATE $ N \ Leftarrow N - 1$
\ ENDIF
\ ENDWHILE
}
\ end { algorithmic }
\ end { table }
96
E. Usage Examples
E8
Program/Code Listing
List. E.10 is a program listing of a C code for computing Fibonacci numbers by calling the
actual code. Please see the code subdirectory.
Listing E.10: Computing Fibonacci numbers in C (./code/fibo.c)
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/* fibo . c -- It prints out the first N Fibonacci
*
numbers .
*/
# include < stdio .h >
int main ( void ) {
int n ;
int i ;
int current ;
int next ;
int twoaway ;
/*
/*
/*
/*
/*
Number of fibonacci numbers we will print */
Index of fibonacci number to be printed next */
Value of the ( i ) th fibonacci number */
Value of the ( i +1) th fibonacci number */
Value of the ( i +2) th fibonacci number */
printf ( " How many Fibonacci numbers do you want to compute ? " ) ;
scanf ( " % d " , & n ) ;
if (n <=0)
printf ( " The number should be positive .\ n " ) ;
else {
printf ( " \ n \ n \ tI \ t Fibonacci ( I ) \ n \ t = == == === == == === == == ==\ n " ) ;
next = current = 1;
for ( i =1; i <= n ; i ++) {
printf ( " \ t % d \ t
% d \ n " , i , current ) ;
twoaway = current + next ;
current = next ;
next
= twoaway ;
}
}
}
/* The output from a run of this program was :
How many Fibonacci numbers do you want to compute ? 9
I
Fibonacci ( I )
=== ==== ===== ==== =====
1
1
2
1
3
2
4
3
5
5
6
8
7
13
8
21
9
34
*/
97
E. Usage Examples
List. E.11 shows the corresponding LATEX code.
Listing E.11: Sample LATEX code for program listing
1
List .~\ ref { lst : fib _ c } is a program listing of a C code for computing
Fibonacci numbers by calling the actual code . Please see the \ verb |
code | subdirectory .
98
E. Usage Examples
E9
Referencing
Referencing chapters: This appendix is in Appendix E, which is about examples in using
various LATEX commands.
Referencing sections: This section is Sec. E9, which shows how to refer to the locations
of various labels that have been placed in the LATEX files. List. E.12 shows the corresponding
LATEX code.
Listing E.12: Sample LATEX code for referencing sections
1
Referencing sections : This section is Sec .~\ ref { sec : ref } , which shows
how to refer to the locations of various labels that have been
placed in the \ LaTeX \ files . List .~\ ref { lst : refsec } shows the
corresponding \ LaTeX \ code .
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
99
E. Usage Examples
E9.1
A subsection
Referencing subsections: This section is Sec. E9.1, which shows how to refer to a subsection.
List. E.13 shows the corresponding LATEX code.
Listing E.13: Sample LATEX code for referencing subsections
1
Referencing subsections : This section is Sec .~\ ref { sec : subsec } , which
shows how to refer to a subsection . List .~\ ref { lst : refsub } shows the
corresponding \ LaTeX \ code .
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
100
E. Usage Examples
E9.1.1
A sub-subsection
Referencing sub-subsections: This section is Sec. E9.1.1, which shows how to refer to a
sub-subsection. List. E.14 shows the corresponding LATEX code.
Listing E.14: Sample LATEX code for referencing sub-subsections
1
Referencing sub - subsections : This section is Sec .~\ ref { sec : subsubsec } ,
which shows how to refer to a sub - subsection . List .~\ ref { lst :
refsubsub } shows the corresponding \ LaTeX \ code .
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem.
Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec
ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus
placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor.
Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla
tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue
a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris.
Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit
amet ipsum. Nunc quis urna dictum turpis accumsan semper.
101
E. Usage Examples
E10
Citing
Citing bibliography content is done using BibTeX. It requires the creation of a BibTeX
file (.bib extension name), and then added in the argument of \bibliography{ } . For
each .bib file, separate them by a comma in the argument of \bibliography{ } without
the extension name. Building your BibTeX file (references.bib) can be done easily with a
tool called JabRef (www.jabref.org).
The following subsections are examples of citations.
E10.1
Books
• [‘Chicago’, 1982]
• [Aristotle, 1877]
• [Aristotle, 1907]
• [Aristotle, 1968]
• [Aristotle, 1929]
• [ABCM, 1959]
• [Augustine, 1995]
• [Averroes, 1982]
• [Butcher, 1981]
• [Chapman, 1975]
• [Cicero, 1995]
• [Coleridge, 1983]
• [Cotton et al., 1999]
• [van Gennep, 1909a]
• [van Gennep, 1909b]
• [van Gennep, 1960]
• [Gerhardt, 2000]
• [Gonzalez, 2001]
102
E. Usage Examples
• [Goossens et al., 1994]
• [Hammond, 1997]
• [Hershkovitz, 1962]
• [Hoel, 1971a]
• [Homer, 2004]
• [Knuth, 1981a]
• [Knuth, 1981b]
• [Knuth, 1973a]
• [Kullback, 1997a]
• [Kullback, 1997b]
• [Kullback, 1959]
• [Malinowski, 1972]
• [Maron, 2000]
• [Massa, 2004]
• [McColvin, 2004]
• [Nietzsche, 1988b]
• [Nietzsche, 1988a]
• [Oetiker et al., 2014]
• [Piccato, 2001]
• [Smart, 1976]
• [Vázques de Parga et al., 1993]
• [Wilde, 1899]
• [Wood, 1961]
• [Worman, 2002]
• [Wright, 1978a]
• [Lipcoll et al., 1977]
103
E. Usage Examples
E10.2
Booklets
• [Knvth, 1988]
E10.3
Proceedings
• [Oz and Yannakakis, 1983]
E10.4
In books
• [von Brandt and Hoffmann, 1987]
• [BSI, 1973a]
• [Eckstein and Zuckermann, 1960]
• [Feigl, 1958]
• [Gordon, 1975]
• [Hanson, 1967]
• [Hoel, 1971b]
• [Hyman, 1981]
• [Kant, 1968a]
• [Kant, 1968b]
• [Knuth, 1973b]
• [Knuth, 1973c]
• [Lincoll, 1977a]
• [Lincoll, 2004]
• [Lincoll, 1977b]
• [McNeill, 1963]
• [Milton, 1924]
• [Nietzsche, 1988c]
104
E. Usage Examples
• [Ogilvy, 1965]
• [Pines, 1979]
• [Ramsbottom, 1931]
• [Ranganthan, 1951]
• [Thomson, 1971]
• [Westfahl, 2004]
• [Wright, 1963]
• [Wright, 1978b]
E10.5
In proceedings
• [Chave, 1964]
• [Chomsky, 1973]
• [Moraux, 1979]
• [Oaho et al., 1983a]
• [Oaho et al., 2004]
• [Oaho et al., 1983b]
• [Salam, 1968]
E10.6
Journals
• [Aamport, 2004]
• [Aamport, 1986a]
• [Aamport, 1986b]
• [Aksın et al., 2006]
• [Angenendt, 2002]
• [Aslin, 1949]
105
E. Usage Examples
• [Baez and Lauda, 2004a]
• [Bertram and Wentworth, 1996]
• [Bry and Afflerbach, 1968]
• [Doody, 1974]
• [Einstein, 1905]
• [Fletcher and Hopkins, 1907]
• [Gillies, 1933]
• [Glashow, 1961]
• [Godfrey, 1959]
• [Hanlon, 1972]
• [Heller and Lederis, 1958]
• [Herrmann et al., 2006]
• [Hostetler et al., 1998]
• [Howells, 1966a]
• [Howells, 1966b]
• [Howells, 1951]
• [ISO, 2009]
• [Jackson, 1979]
• [Johnson, 1974]
• [Moore, 1998]
• [Moore, 1965]
• [Prufer, 1964]
• [Reese, 1958]
• [Sarfraz and Razzak, 2002]
106
E. Usage Examples
• [Shore, 1991]
• [Sigfridsson and Ryde, 1998]
• [Weinberg, 1967]
• [Yoon et al., 2006]
• [GAJ, 1986]
E10.7
Theses/dissertations
• [Croft, 1978]
• [Maguire, 1976]
• [Mann, 1968]
• [Masterly, 1988a]
• [Masterly, 1988b]
• [Phony-Baloney, 1988a]
• [Phony-Baloney, 1988b]
E10.8
Technical Reports and Others
• [‘Brunswick’, 1985]
• [BSI, 1983]
• [BSI, 1978]
• [BSI, 1976]
• [BSI, 1973b]
• [Ellis and Walton, 1971]
• [Térrific, 1988]
• [Terrific, 1988]
• [Winget Ltd., 1967]
107
E. Usage Examples
• [Ünderwood et al., 2004]
• [Ünderwood et al., 1988]
• [Downes, 1974]
• [Exchequer, 1639]
• [Pym, 1624]
• [Traquair, 1638]
E10.9
Miscellaneous
• [Almendro et al., 1998]
• [Baez and Lauda, 2004b]
• [Chiu and Chow, 1978]
• [Itzhaki, 1996]
• [Kowalik and Isard, 1995]
• [Laufenberg et al., 2006]
• [Loh, 1992]
• [Markey, 2005]
• [Missilany, 1984]
• [Padhye et al., 1999]
• [Sorace et al., 1997]
• [Wassenberg and Sanders, 2010]
• [Missilany, 2004]
108
E. Usage Examples
E11
Index
For key words or topics that are expected (or the user would like) to appear in the Index, use
index{key} , where key is an example keyword to appear in the Index. For example,
Fredholm integral and Fourier operator of the following paragraph are in the Index.
If we make a very large matrix with complex exponentials in the rows (i.e., cosine real
parts and sine imaginary parts), and increase the resolution without bound, we approach
the kernel of the Fredholm integral equation of the 2nd kind, namely the Fourier operator
that defines the continuous Fourier transform.
List. E.15 is a program listing of the above-mentioned paragraph.
Listing E.15: Sample LATEX code for Index usage
1
If we make a very large matrix with complex exponentials in the rows ( i .
e . , cosine real parts and sine imaginary parts ) , and increase the
resolution without bound , we approach the kernel of the \ index {
Fredholm integral } Fredholm integral equation of the 2 nd kind ,
namely the \ index { Fourier } Fourier operator that defines the
continuous Fourier transform .
109
E. Usage Examples
E12
Adding Relevant PDF Pages
Examples of such PDF pages are Standards, Datasheets, Specification Sheets, Application
Notes, etc. Selected PDF pages can be added (see List. E.16), but note that the options
must be tweaked. See the manual of pdfpages for other options.
Listing E.16: Sample LATEX code for including PDF pages
1
2
3
4
5
6
\ includepdf [ pages ={8 -10} , %
offset =3.5 mm -10 mm , %
scale =0.73 , %
frame , %
pagecommand ={} ,]
{./ reference / Xilinx 2015 - UltraScale - Architecture - Overview . pdf }
110
E. Usage Examples
UltraScale Architecture and Product Overview
Virtex UltraScale FPGA Feature Summary
Table 6: Virtex UltraScale FPGA Feature Summary
VU065
VU080
VU095
VU125
VU160
VU190
VU440
Logic Cells
626,640
780,000
940,800
1,253,280
1,621,200
1,879,920
4,432,680
CLB Flip-Flops
716,160
891,424
1,075,200
1,432,320
1,852,800
2,148,480
5,065,920
CLB LUTs
358,080
445,712
537,600
716,160
926,400
1,074,240
2,532,960
Maximum Distributed RAM (Mb)
Block RAM/FIFO w/ECC (36Kb each)
Total Block RAM (Mb)
4.8
3.9
4.8
9.7
12.7
14.5
28.7
1,260
1,421
1,728
2,520
3,276
3,780
2,520
88.6
44.3
50.0
60.8
88.6
115.2
132.9
CMT (1 MMCM, 2 PLLs)
10
16
16
20
30
30
30
I/O DLLs
40
64
64
80
120
120
120
Fractional PLLs
Maximum HP I/Os(1)
Maximum HR I/Os(2)
DSP Slices
5
8
8
10
15
15
0
468
780
780
780
650
650
1,404
52
52
52
104
52
52
52
600
672
768
1,200
1,560
1,800
2,880
System Monitor
1
1
1
2
3
3
3
PCIe Gen3 x8
2
4
4
4
5
6
6
150G Interlaken
3
6
6
6
8
9
0
100G Ethernet
3
4
4
6
9
9
3
GTH 16.3Gb/s Transceivers
20
32
32
40
52
60
48
GTY 30.5Gb/s Transceivers
20
32
32
40
52
60
0
Notes:
1. HP = High-performance I/O with support for I/O voltage from 1.0V to 1.8V.
2. HR = High-range I/O with support for I/O voltage from 1.2V to 3.3V.
DS890 (v2.1) April 27, 2015
Preliminary Product Specification
www.xilinx.com
8
111
E. Usage Examples
UltraScale Architecture and Product Overview
Virtex UltraScale Device-Package Combinations and Maximum I/Os
Table 7: Virtex UltraScale Device-Package Combinations and Maximum I/Os
Package
Dimensions
(mm)
VU065
VU080
VU095
VU125
VU160
VU190
VU440
Package(1)(2)(3)
HR, HP
GTH, GTY
HR, HP
GTH, GTY
HR, HP
GTH, GTY
HR, HP
GTH, GTY
HR, HP
GTH, GTY
HR, HP
GTH, GTY
HR, HP
GTH, GTY
FFVC1517
40x40
52, 468
20, 20
52, 468
20, 20
52, 468
20, 20
FFVD1517
40x40
52, 286
32, 32
52, 286
32, 32
FLVD1517
40x40
FFVB1760
42.5x42.5
FLVB1760
42.5x42.5
FFVA2104
47.5x47.5
FLVA2104
47.5x47.5
FFVB2104
47.5x47.5
FLVB2104
47.5x47.5
FLGB2104
47.5x47.5
52, 650
40, 36
52, 650
40, 36
FFVC2104
47.5x47.5
FLVC2104
47.5x47.5
FLGC2104
47.5x47.5
52, 364
52, 52
52, 364
52, 52
FLGB2377
50x50
FLGA2577
52.5x52.5
FLGA2892
55x55
52, 286
40, 32
52, 650
32, 16
52, 650
32, 16
52, 650
36, 16
52, 780
28, 24
52, 780
28, 24
52, 780
28, 24
52, 650
32, 32
52, 650
32, 32
52, 650
40, 36
52, 364
32, 32
52, 364
40, 40
52, 1248
36, 0
0, 448
60, 60
52, 1404
48, 0
Notes:
1.
2.
3.
Go to Ordering Information for package designation details.
All packages have 1.0mm ball pitch.
Packages with the same last letter and number sequence, e.g., A2104, are footprint compatible with all other UltraScale
architecture-based devices with the same sequence. The footprint compatible devices within this family are outlined. See
the UltraScale Architecture Product Selection Guide for details on inter-family migration.
DS890 (v2.1) April 27, 2015
Preliminary Product Specification
www.xilinx.com
9
112
E. Usage Examples
UltraScale Architecture and Product Overview
Virtex UltraScale+ FPGA Feature Summary
Table 8: Virtex UltraScale+ FPGA Feature Summary
VU3P
VU5P
VU7P
VU9P
VU11P
VU13P
Logic Cells
689,640
1,051,010
1,379,280
2,068,920
2,147,040
2,862,720
CLB Flip-Flops
788,160
1,201,154
1,576,320
2,364,480
2,453,760
3,271,680
CLB LUTs
394,080
600,577
788,160
1,182,240
1,226,880
1,635,840
Max. Distributed RAM (Mb)
12.0
18.3
24.1
36.1
34.8
46.4
Block RAM/FIFO w/ECC
(36Kb each)
720
1,024
1,440
2,160
2,016
2,688
Block RAM (Mb)
25.3
36.0
50.6
75.9
70.9
94.5
UltraRAM Blocks
320
470
640
960
1,152
1,536
UltraRAM (Mb)
90.0
132.2
180.0
270.0
324.0
432.0
10
20
20
30
12
16
520
832
832
832
624
832
2,280
3,474
4,560
6,840
8,928
11,904
1
2
2
3
3
4
40
80
80
120
96
128
CMTs (1 MMCM and 2 PLLs)
Max. HP I/O(1)
DSP Slices
System Monitor
GTY Transceivers 32.75Gb/s
PCIe Gen3 x16 and Gen4 x8
2
4
4
6
3
4
150G Interlaken
3
4
6
9
9
12
100G Ethernet w/RS-FEC
3
4
6
9
6
8
Notes:
1. HP = High-performance I/O with support for I/O voltage from 1.0V to 1.8V.
Virtex UltraScale+ Device-Package Combinations and Maximum I/Os
Table 9: Virtex UltraScale+ Device-Package Combinations and Maximum I/Os
Package
Dimensions
(mm)
VU3P
VU5P
VU7P
VU9P
VU11P
VU13P
HP, GTY
HP, GTY
HP, GTY
HP, GTY
HP, GTY
HP, GTY
FFVC1517
40x40
520, 40
FLVF1924
45x45
FLVA2104
47.5x47.5
832, 52
832, 52
832, 52
702, 76
702, 76
702, 76
624, 76
416, 80
416, 80
416, 104
416, 96
448, 120
448, 96
Package
(1)(2)(3)
FHVA2104
52.5x52.5(4)
FLVB2104
47.5x47.5
FHVB2104
52.5x52.5(4)
FLVC2104
47.5x47.5
FHVC2104
52.5x52.5(4)
FLVA2577
52.5x52.5
624, 64
832, 52
702, 76
416, 104
448, 128
Notes:
1. Go to Ordering Information for package designation details.
2.
3.
All packages have 1.0mm ball pitch.
Packages with the same last letter and number sequence, e.g., A2104, are footprint compatible with all other UltraScale
devices with the same sequence. The footprint compatible devices within this family are outlined.
4.
These 52.5x52.5mm overhang packages have the same PCB ball footprint as the corresponding 47.5x47.5mm packages (i.e., the
same last letter and number sequence) and are footprint compatible.
DS890 (v2.1) April 27, 2015
Preliminary Product Specification
www.xilinx.com
10
113
Appendix F
SOME LIST OF MATH SYMBOLS
114
F. Some List of Math Symbols
List of mathematical symbols
From Wikipedia, the free encyclopedia
This is a list of symbols found within all branches of mathematics to express a formula or to represent a constant.
When reading the list, it is important to recognize that a mathematical concept is independent of the symbol chosen to represent it. For many of the symbols below, the symbol is usually
synonymous with the corresponding concept (ultimately an arbitrary choice made as a result of the cumulative history of mathematics), but in some situations a different convention may
be used. For example, depending on context, the triple bar "≡" may represent congruence or a definition. Further, in mathematical logic, numerical equality is sometimes represented by
"≡" instead of "=", with the latter representing equality of well-formed formulas. In short, convention dictates the meaning.
Each symbol is shown both in HTML, whose display depends on the browser's access to an appropriate font installed on the particular device, and typeset as an image using TeX.
Contents
1
2
3
4
5
6
7
Guide
Basic symbols
Symbols based on equality
Symbols that point left or right
Brackets
Other non-letter symbols
Letter-based symbols
7.1 Letter modifiers
7.2 Symbols based on Latin letters
7.3 Symbols based on Hebrew or Greek letters
8 Variations
9 See also
10 References
11 External links
Guide
This list is organized by symbol type and is intended to facilitate finding an unfamiliar symbol by its visual appearance. For a related list organized by mathematical topic, see List of
mathematical symbols by subject. That list also includes LaTeX and HTML markup and Unicode code points for each symbol.
Basic symbols: Symbols widely used in mathematics, roughly through first-year calculus. More advanced meanings are included with some symbols listed here.
Symbols based on equality "=": Symbols derived from or similar to the equal sign, including double-headed arrows. Not surprisingly these symbols are often associated with an
equivalence relation.
Symbols that point left or right: Symbols, such as < and >, that appear to point to one side or another.
Brackets: Symbols that are placed on either side of a variable or expression, such as | x |.
Other non-letter symbols: Symbols that do not fall in any of the other categories.
Letter-based symbols: Many mathematical symbols are based on, or closely resemble, a letter in some alphabet. This section includes such symbols, including symbols that
resemble upside-down letters. Many letters have conventional meanings in various branches of mathematics and physics. These are not listed here. The See also section, below, has
several lists of such usages.
Letter modifiers: Symbols that can be placed on or next to any letter to modify the letter's meaning.
Symbols based on Latin letters, including those symbols that resemble or contain an X
Symbols based on Hebrew or Greek letters e.g. א, ב, δ, Δ, π, Π, σ, Σ, Φ. Note: symbols resembling Λ are grouped with "V" under Latin letters.
Variations: Usage in languages written right-to-left
Basic symbols
115
F. Some List of Math Symbols
Symbol
in
HTML
Symbol
in TeX
Name
Explanation
Read as
Examples
Category
addition
plus;
add
4 + 6 means the sum of 4 and 6.
2+7=9
arithmetic
+
disjoint union
the disjoint
union of ... and A1 + A2 means the disjoint union of sets A1 and A2.
...
A1 = {3, 4, 5, 6} ∧ A2 = {7, 8, 9, 10} ⇒
A1 + A2 = {(3, 1), (4, 1), (5, 1), (6, 1), (7, 2), (8, 2), (9, 2), (10, 2)}
set theory
subtraction
minus;
take;
subtract
9 − 4 means the subtraction of 4 from 9.
8−3=5
arithmetic
negative sign
−
negative;
minus;
−3 means the additive inverse of the number 3.
the opposite of
−(−5) = 5
arithmetic
set-theoretic
complement
A − B means the set that contains all the elements of A
that are not in B.
{1, 2, 4} − {1, 3, 4} = {2}
minus;
without
(∖ can also be used for set-theoretic complement as
set theory described below.)
plus-minus
plus or minus
±
The equation x = 5 ± √ 4 , has two solutions, x = 7 and x = 3.
6 ± 3 means both 6 + 3 and 6 − 3.
Note: {{sqrt|4}} was used to get √4 .
arithmetic
plus-minus
10 ± 2 or equivalently 10 ± 20% means the range from
plus or minus
10 − 2 to 10 + 2.
If a = 100 ± 1 mm, then a ≥ 99 mm and a ≤ 101 mm.
measurement
minus-plus
∓
minus or plus
6 ± (3 ∓ 5) means 6 + (3 − 5) and 6 − (3 + 5).
cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y).
3 × 4 or 3 ⋅ 4 means the multiplication of 3 by 4.
7 ⋅ 8 = 56
u ⋅ v means the dot product of vectors u and v
(1, 2, 5) ⋅ (3, 4, −1) = 6
u × v means the cross product of vectors u and v
i j k
(1, 2, 5) × (3, 4, −1) = 1 2 5 = (−22, 16, −2)
arithmetic
multiplication
times;
multiplied by
arithmetic
dot product
scalar product
×
⋅
·
dot
linear algebra
vector algebra
cross product
vector product
cross
3
linear algebra
vector algebra
4 −1
placeholder
A · means a placeholder for an argument of a function.
Indicates the functional nature of an expression without
functional assigning a specific symbol for an argument.
analysis
(silent)
division
(Obelus)
divided by;
over
÷
2 ÷ 4 = 0.5
6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3.
12 ⁄ 4 = 3
arithmetic
quotient group
⁄
|·|
mod
group theory
G / H means the quotient of group G modulo its subgroup
{0, a, 2a, b, b + a, b + 2a} / {0, b} = {{0, b}, {a, b + a}, {2a, b + 2a}}
H.
quotient set
mod
A/~ means the set of all ~ equivalence classes in A.
If we define ~ by x ~ y ⇔ x − y ∈ ℤ, then ℝ/~ = {x + n : n ∈ ℤ, x ∈ [0,1)}.
set theory
square root
the (principal)
√ x means the nonnegative number whose square is x.
square root of
√4 = 2
real numbers
√
complex square
root
the (complex) If z = r exp(iφ) is represented in polar coordinates with
square root of −π < φ ≤ π, then √ z = √ r exp(iφ/2).
√ −1 = i
complex
numbers
summation
∑
sum over ...
from ... to ... of
means
.
arithmetic
∫
indefinite
integral or
antiderivative
indefinite
integral of
∫ f(x) dx means a function whose derivative is f.
x3
2
∫ x dx = 3 + C
116
F. Some List of Math Symbols
- OR the
antiderivative of
calculus
definite integral
integral from ... b
to ... of ... with ∫ a f(x) dx means the signed area between the x-axis and
the graph of the function f between x = a and x = b.
respect to
b3 − a3
b 2
∫ a x dx = 3
calculus
line integral
∫ f ds means the integral of f along the curve C,
line/ path/ curve/ C
b
integral of ... ∫ a f(r(t)) | r'(t) | dt, where r is a parametrization of C. (If
along ...
the curve is closed, the symbol ∮ may be used instead, as
calculus described below.)
∮
Contour
integral;
closed line
integral
Similar to the integral, but used to denote a single
integration over a closed curve or loop. It is sometimes
used in physics texts involving equations regarding Gauss's
Law, and while these formulas involve a closed surface
integral, the representations describe only the first
integration of the volume over the enclosing surface.
Instances where the latter requires simultaneous double
integration, the symbol ∯ would be more appropriate. A
third related symbol is the closed volume integral, denoted
contour integral by the symbol ∰ .
of
The contour integral can also frequently be found with a
calculus
subscript capital letter C, ∮ C, denoting that a closed loop
integral is, in fact, around a contour C, or sometimes
dually appropriately, a circle C. In representations of
1
If C is a Jordan curve about 0, then ∮ C z dz = 2πi.
Gauss's Law, a subscript capital S, ∮ S, is used to denote
that the integration is over a closed surface.
therefore
∴
therefore;
so;
hence
Sometimes used in proofs before logical consequences.
All humans are mortal. Socrates is a human. ∴ Socrates is mortal.
Sometimes used in proofs before reasoning.
11 is prime ∵ it has no positive integer factors other than itself and one.
everywhere
because
∵
because;
since
everywhere
factorial
factorial
n! means the product 1 × 2 × ... × n.
combinatorics
The statement !A is true if and only if A is false.
!
logical negation A slash placed through another operator is the same as "!"
not
placed in front.
propositional
logic (The symbol ! is primarily from computer science. It is
avoided in mathematical texts, where the notation ¬A is
preferred.)
!(!A) ⇔ A
x ≠ y ⇔ !(x = y)
The statement ¬A is true if and only if A is false.
¬
˜
logical negation A slash placed through another operator is the same as "¬"
not
placed in front.
¬(¬A) ⇔ A
x ≠ y ⇔ ¬(x = y)
propositional
logic (The symbol ~ has many other uses, so ¬ or the slash
notation is preferred. Computer scientists will often use !
but this is avoided in mathematical texts.)
proportionality
∝
is proportional
to;
y ∝ x means that y = kx for some constant k.
varies as
if y = 2x, then y ∝ x.
everywhere
infinity
∞
infinity
∞ is an element of the extended number line that is greater
than all real numbers; it often occurs in limits.
numbers
(1) a + 0 := a (def.)
■
(2) a + succ(b) := succ(a + b) (def.)
□
end of proof
Proposition. 3 + 2 = 5.
Proof.
∎
QED;
Used to mark the end of a proof.
tombstone;
Halmos finality
(May also be written Q.E.D.)
symbol
▮
everywhere
‣
3 + 2 = 3 + succ(1) (definition of succ)
3 + succ(1) = succ(3 + 1) (2)
succ(3 + 1) = succ(3 + succ(0)) (definition of succ)
succ(3 + succ(0)) = succ(succ(3 + 0)) (2)
succ(succ(3 + 0)) = succ(succ(3)) (1)
succ(succ(3)) = succ(4) = 5 (definition of succ) ▮
Symbols based on equality
117
F. Some List of Math Symbols
Symbol
in
HTML
Symbol
Name
Explanation
Read as
in TeX
Examples
Category
equality
=
is equal to;
equals
means
and
represent the same math object (Both symbols have the same value).
everywhere
inequality
≠
is not equal to;
does not equal
means that
the same value).
and
do not represent the same math object (Both symbols do not have
(The forms !=, /= or <> are generally used in programming languages where ease of typing
everywhere and use of ASCII text is preferred.)
approximately equal
x ≈ y means x is approximately equal to y.
is approximately equal to
≈
π ≈ 3.14159
everywhere This may also be written ≃, ≅, ~, ♎ (Libra Symbol), or ≒.
isomorphism
G ≈ H means that group G is isomorphic (structurally identical) to group H.
Q8 / C2 ≈ V
is isomorphic to
group theory (≅ can also be used for isomorphic, as described below.)
probability distribution
has distribution
X ~ D, means the random variable X has the probability distribution D.
X ~ N(0,1), the standard normal
distribution
statistics
row equivalence
is row equivalent to
A ~ B means that B can be generated by using a series of elementary row operations on A
matrix theory
same order of magnitude
roughly similar;
poorly approximates;
is on the order of
~
2~5
m ~ n means the quantities m and n have the same order of magnitude, or general size.
8 × 9 ~ 100
(Note that ~ is used for an approximation that is poor, otherwise use ≈ .)
but π2 ≈ 10
approximation theory
similarity
is similar to[1]
△ABC ~ △DEF means triangle ABC is similar to (has the same shape) triangle DEF.
geometry
asymptotically equivalent
is asymptotically
equivalent to
f ~ g means
.
x ~ x+1
asymptotic analysis
equivalence relation
are in the same equivalence
a ~ b means
class
(and equivalently
).
1 ~ 5 mod 4
everywhere
=:
:=
≡
x := y, y =: x or x ≡ y means x is defined to be another name for y, under certain assumptions
taken in context.
definition
:⇔
is defined as;
is equal by definition to
(Some writers use ≡ to mean congruence).
everywhere
P ⇔ Q means P is defined to be logically equivalent to Q. P ⇔ Q means if and only if (iff)
≜
≝
≐
congruence
is congruent to
≅
geometry
isomorphic
△ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle
DEF.
G ≅ H means that group G is isomorphic (structurally identical) to group H.
V ≅ C2 × C2
is isomorphic to
abstract algebra (≈ can also be used for isomorphic, as described above.)
congruence relation
≡
... is congruent to ...
modulo ...
a ≡ b (mod n) means a − b is divisible by n
5 ≡ 2 (mod 3)
A ⇔ B means A is true if B is true and A is false if B is false.
x+5=y+2⇔x+3=y
modular arithmetic
⇔
↔
material equivalence
if and only if;
iff
propositional logic
Symbols that point left or right
118
F. Some List of Math Symbols
Symbol
in
HTML
Symbol
Name
Explanation
Read as
in TeX
strict inequality
means x is less than y.
is less than,
is greater than
<
means x is greater than y.
order theory
>
Examples
Category
proper subgroup
means H is a proper subgroup of G.
is a proper subgroup of
group theory
significant (strict)
inequality
x ≪ y means x is much less than y.
is much less than,
is much greater than
≪
≫
0.003 ≪ 1000000
x ≫ y means x is much greater than y.
order theory
asymptotic comparison
f ≪ g means the growth of f is asymptotically bounded by g.
is of smaller order than,
is of greater order than
(This is I. M. Vinogradov's notation. Another notation is the Big
analytic number theory O notation, which looks like f = O(g).)
x ≪ ex
absolute continuity
is absolutely continuous
with respect to
means that
i.e., whenever
is absolutely continuous with respect to ,
, we have
.
If is the counting measure on
measure, then
.
and
is the Lebesgue
measure theory
x ≤ y means x is less than or equal to y.
inequality
is less than or equal to,
is greater than or equal to
x ≥ y means x is greater than or equal to y.
(The forms <= and >= are generally used in programming
order theory languages, where ease of typing and use of ASCII text is
preferred.)
≤
subgroup
is a subgroup of
H ≤ G means H is a subgroup of G.
group theory
≥
3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5
Z≤Z
A3 ≤ S3
If
reduction
A ≤ B means the problem A can be reduced to the problem B.
is reducible to
Subscripts can be added to the ≤ to indicate what kind of
computational complexity reduction.
theory
then
congruence relation
7k ≡ 28 (mod 2) is only true if k is an even integer. Assume
... is less than ... is greater
that the problem requires k to be non-negative; the domain is
than ...
defined as 0 ≦ k ≦ ∞.
modular arithmetic
x ≦ y means that each component of vector x is less than or
≦
≧
10a ≡ 5 (mod 5) for 1 ≦ a ≦ 10
equal to each corresponding component of vector y.
vector inequality
x ≧ y means that each component of vector x is greater than or
... is less than or equal... is
equal to each corresponding component of vector y.
greater than or equal...
order theory It is important to note that x ≦ y remains true if every element is
equal. However, if the operator is changed, x ≤ y is true if and
only if x ≠ y is also true.
Karp reduction
≺
≻
is Karp reducible to;
is polynomial-time
many-one reducible to
L1 ≺ L2 means that the problem L1 is Karp reducible to L2.[2]
If L1 ≺ L2 and L2 ∈ P, then L1 ∈ P.
computational complexity
theory
Nondominated order
is nondominated by
P ≺ Q means that the element P is nondominated by element
[3]
Multi-objective Q.
optimization
If P1 ≺ Q2 then
normal subgroup
is a normal subgroup of
N ◅ G means that N is a normal subgroup of group G.
Z(G) ◅ G
I ◅ R means that I is an ideal of ring R.
(2) ◅ Z
group theory
◅
ideal
is an ideal of
▻
ring theory
antijoin
R ▻ S means the antijoin of the relations R and S, the tuples in
R for which there is not a tuple in S that is equal on their
relational algebra common attribute names.
the antijoin of
⇒
material implication
implies;
if ... then
→
propositional logic,
Heyting algebra
⊃
⊆
(→ may mean the same as ⇒, or it may have the meaning for
functions given below.)
set theory
x = 2 ⇒ x2 = 4 is true, but x2 = 4 ⇒ x = 2 is in general false
(since x could be −2).
(⊃ may mean the same as ⇒,[4] or it may have the meaning for
superset given below.)
(subset) A ⊆ B means every element of A is also an element of
B.[5]
subset
is a subset of
⊂
A ⇒ B means if A is true then B is also true; if A is false then
nothing is said about B.
(proper subset) A ⊂ B means A ⊆ B but A ≠ B.
(Some writers use the symbol ⊂ as if it were the same as ⊆.)
(A ∩ B) ⊆ A
ℕ⊂ℚ
ℚ⊂ℝ
119
F. Some List of Math Symbols
⊇
A ⊇ B means every element of B is also an element of A.
superset
(A ∪ B) ⊇ B
is a superset of
⊃
A ⊃ B means A ⊇ B but A ≠ B.
ℝ⊃ℚ
set theory
(Some writers use the symbol ⊃ as if it were the same as ⊇.)
function arrow
→
from ... to
f: X → Y means the function f maps the set X into the set Y.
Let f: ℤ → ℕ ∪ {0} be defined by f(x) := x2.
set theory, type theory
function arrow
↦
f: a ↦ b means the function f maps the element a to the element
maps to
set theory
b.
Let f: x ↦ x + 1 (the successor function).
subtype
is a subtype of
<:
<·
T1 <: T2 means that T1 is a subtype of T2.
If S <: T and T <: U then S <: U (transitivity).
type theory
cover
is covered by
x <• y means that x is covered by y.
{1, 8} <• {1, 3, 8} among the subsets of {1, 2, ..., 10}
ordered by containment.
order theory
entailment
⊧
entails
A ⊧ B means the sentence A entails the sentence B, that is in
every model in which A is true, B is also true.
A ⊧ A ∨ ¬A
model theory
inference
infers;
is derived from
x ⊢ y means y is derivable from x.
A → B ⊢ ¬B → ¬A
p ⊢ n means that p is a partition of n.
(4,3,1,1) ⊢ 9,
propositional logic,
predicate logic
⊢
partition
is a partition of
number theory
bra vector
⟨|
the bra ...;
the dual of ...
⟨φ| means the dual of the vector |φ⟩, a linear functional which
maps a ket |ψ⟩ onto the inner product ⟨φ|ψ⟩.
Dirac notation
ket vector
|⟩
the ket ...;
the vector ...
|φ⟩ means the vector with label φ, which is in a Hilbert space.
A qubit's state can be represented as α|0⟩+ β|1⟩, where α and β are
complex numbers s.t. |α|2 + |β|2 = 1.
Dirac notation
Brackets
120
F. Some List of Math Symbols
Symbol
in
HTML
Symbol
in TeX
Name
Read as
Explanation
Examples
Category
combination;
binomial
means (in the case of n = positive integer) the number of
coefficient
combinations of k elements drawn from a set of n elements.
n choose k
combinatorics (This may also be written as C(n, k), C(n; k), C , nC , or
n k
k
.)
multiset
coefficient
u multichoose
k
(when u is positive integer)
combinatorics means reverse or rising binomial coefficient.
absolute
value;
modulus
|3| = 3
|x| means the distance along the real line (or across the complex
absolute
plane) between x and zero.
value of;
modulus of
|...|
|i|=1
| 3 + 4i | = 5
numbers
Euclidean
norm or
Euclidean
length or
magnitude
|–5| = |5| = 5
For x = (3,−4)
|x| means the (Euclidean) length of vector x.
Euclidean
norm of
geometry
determinant
determinant
|A| means the determinant of the matrix A
of
matrix theory
cardinality
cardinality of; |X| means the cardinality of the set X.
size of;
order of
(# may be used instead as described below.)
|{3, 5, 7, 9}| = 4.
set theory
norm
norm of;
length of
ǁ x ǁ means the norm of the element x of a normed vector space.[6] ǁ x + y ǁ ≤ ǁ x ǁ + ǁ y ǁ
linear algebra
ǁ...ǁ
nearest
integer
function
nearest
integer to
ǁxǁ means the nearest integer to x.
ǁ1ǁ = 1, ǁ1.6ǁ = 2, ǁ−2.4ǁ = −2, ǁ3.49ǁ = 3
(This may also be written [x], ⌊x⌉, nint(x) or Round(x).)
numbers
set brackets
{,}
{:}
{|}
{;}
the set of ... {a,b,c} means the set consisting of a, b, and c.[7]
set theory
ℕ = { 1, 2, 3, ... }
set builder
notation
{x : P(x)} means the set of all x for which P(x) is true.[7] {x | P(x)}
the set of ...
{n ∈ ℕ : n2 < 20} = { 1, 2, 3, 4 }
is the same as {x : P(x)}.
such that
set theory
floor
⌊...⌋
floor;
greatest
integer;
entier
⌊x⌋ means the floor of x, i.e. the largest integer less than or equal
to x.
⌊4⌋ = 4, ⌊2.1⌋ = 2, ⌊2.9⌋ = 2, ⌊−2.6⌋ = −3
(This may also be written [x], floor(x) or int(x).)
numbers
ceiling
⌈...⌉
ceiling
⌈x⌉ means the ceiling of x, i.e. the smallest integer greater than or
equal to x.
⌈4⌉ = 4, ⌈2.1⌉ = 3, ⌈2.9⌉ = 3, ⌈−2.6⌉ = −2
numbers (This may also be written ceil(x) or ceiling(x).)
⌊...⌉
nearest
integer
function
nearest
integer to
⌊x⌉ means the nearest integer to x.
⌊2⌉ = 2, ⌊2.6⌉ = 3, ⌊-3.4⌉ = -3, ⌊4.49⌉ = 4
(This may also be written [x], ||x||, nint(x) or Round(x).)
numbers
[:]
degree of a
field
extension
[,]
[ℂ : ℝ] = 2
the degree of
field theory
[]
[ℚ(√2) : ℚ] = 2
[K : F] means the degree of the extension K : F.
[ℝ : ℚ] = ∞
equivalence
class
[a] means the equivalence class of a, i.e. {x : x ~ a}, where ~ is an Let a ~ b be true iff a ≡ b (mod 5).
equivalence relation.
the
Then [2] = {..., −8, −3, 2, 7, ...}.
equivalence
[a]R means the same, but with R as the equivalence relation.
class of
121
F. Some List of Math Symbols
abstract
algebra
floor
floor;
greatest
integer;
entier
numbers
nearest
integer
function
nearest
integer to
[x] means the floor of x, i.e. the largest integer less than or equal to
x.
[3] = 3, [3.5] = 3, [3.99] = 3, [−3.7] = −4
(This may also be written ⌊x⌋, floor(x) or int(x). Not to be
confused with the nearest integer function, as described below.)
[x] means the nearest integer to x.
(This may also be written ⌊x⌉, ||x||, nint(x) or Round(x). Not to be
confused with the floor function, as described above.)
[2] = 2, [2.6] = 3, [-3.4] = -3, [4.49] = 4
numbers
Iverson
bracket
1 if true, 0
otherwise
[S] maps a true statement S to 1 and a false statement S to 0.
[0=5]=0, [7>0]=1, [2 ∈ {2,3,4}]=1, [5 ∈ {2,3,4}]=0
propositional
logic
[,,]
f[X] means { f(x) : x ∈ X }, the image of the function f under the
set X ⊆ dom(f).
image of ...
under ...
(This may also be written as f(X) if there is no risk of confusing the
everywhere image of f under X with the function application f of X. Another
notation is Im f, the image of f under its domain.)
image
closed
interval
closed
interval
.
0 and 1/2 are in the interval [0,1].
order theory
commutator
the
[g, h] = g−1h−1gh (or ghg−1h−1), if g, h ∈ G (a group).
commutator
of
[a, b] = ab − ba, if a, b ∈ R (a ring or commutative algebra).
group theory,
ring theory
xy = x[x, y] (group theory).
[AB, C] = A[B, C] + [A, C]B (ring theory).
triple scalar
product
the triple
scalar product [a, b, c] = a × b · c, the scalar product of a × b with c.
of
[a, b, c] = [b, c, a] = [c, a, b].
vector
calculus
function
application
of
f(x) means the value of the function f at the element x.
If f(x) := x2, then f(3) = 32 = 9.
set theory
f(X) means { f(x) : x ∈ X }, the image of the function f under the
set X ⊆ dom(f).
image of ...
under ...
(This may also be written as f[X] if there is a risk of confusing the
everywhere image of f under X with the function application f of X. Another
notation is Im f, the image of f under its domain.)
image
precedence
grouping
parentheses
Perform the operations inside the parentheses first.
(8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.
An ordered list (or sequence, or horizontal vector, or row vector)
of values.
(a, b) is an ordered pair (or 2-tuple).
everywhere
()
(,)
tuple
tuple; n-tuple;
ordered
pair/triple/etc;
row vector;
sequence
(Note that the notation (a,b) is ambiguous: it could be an ordered
pair or an open interval. Set theorists and computer scientists
often use angle brackets ⟨ ⟩ instead of parentheses.)
(a, b, c) is an ordered triple (or 3-tuple).
( ) is the empty tuple (or 0-tuple).
everywhere
highest
common
factor
highest
common
(a, b) means the highest common factor of a and b.
factor;
greatest
(This may also be written hcf(a, b) or gcd(a, b).)
common
divisor; hcf;
gcd
(3, 7) = 1 (they are coprime); (15, 25) = 5.
number
theory
(,)
],[
(,]
],]
open interval
.
open interval (Note that the notation (a,b) is ambiguous: it could be an ordered
order theory pair or an open interval. The notation ]a,b[ can be used instead.)
4 is not in the interval (4, 18).
(0, +∞) equals the set of positive real numbers.
left-open
interval
half-open
interval;
left-open
interval
.
(−1, 7] and (−∞, −1]
order theory
122
F. Some List of Math Symbols
[,)
[,[
right-open
interval
half-open
interval;
right-open
interval
.
[4, 18) and [1, +∞)
order theory
⟨u,v⟩ means the inner product of u and v, where u and v are
members of an inner product space.
Note that the notation ⟨u, v⟩ may be ambiguous: it could mean the
inner product inner product or the linear span.
The standard inner product between two vectors x = (2, 3) and y = (−1, 5) is:
inner product
of
There are many variants of the notation, such as ⟨u | v⟩ and (u | v), ⟨x, y⟩ = 2 × −1 + 3 × 5 = 13
linear algebra which are described below. For spatial vectors, the dot product
notation, x · y is common. For matrices, the colon notation A : B
may be used. As ⟨ and ⟩ can be hard to type, the more "keyboard
friendly" forms < and > are sometimes seen. These are avoided in
mathematical texts.
for a time series :g(t) (t = 1, 2,...)
average
average of
statistics
⟨⟩
linear span
⟨,⟩
let S be a subset of N for example,
all the element in S.
represents the average of
we can define the structure functions Sq( ):
⟨S⟩ means the span of S ⊆ V. That is, it is the intersection of all
subspaces of V which contain S.
⟨u1, u2, ...⟩ is shorthand for ⟨{u1, u2, ...}⟩.
(linear) span
of;
linear hull of Note that the notation ⟨u, v⟩ may be ambiguous: it could mean the
inner product or the linear span.
linear algebra
.
The span of S may also be written as Sp(S).
subgroup
generated by
a set
the subgroup
generated by
means the smallest subgroup of G (where S ⊆ G, a group)
containing every element of S.
is shorthand for
.
In S3,
and
.
group theory
tuple
tuple; n-tuple; An ordered list (or sequence, or horizontal vector, or row vector)
ordered
of values.
pair/triple/etc;
row vector; (The notation (a,b) is often used as well.)
sequence
is an ordered pair (or 2-tuple).
is an ordered triple (or 3-tuple).
is the empty tuple (or 0-tuple).
everywhere
⟨u | v⟩ means the inner product of u and v, where u and v are
members of an inner product space.[8] (u | v) means the same.
⟨|⟩
(|)
inner product
inner product Another variant of the notation is ⟨u, v⟩ which is described above.
of
For spatial vectors, the dot product notation, x · y is common. For
linear algebra matrices, the colon notation A : B may be used. As ⟨ and ⟩ can be
hard to type, the more "keyboard friendly" forms < and > are
sometimes seen. These are avoided in mathematical texts.
Other non-letter symbols
123
F. Some List of Math Symbols
Symbol
in
HTML
Symbol
in TeX
Name
Explanation
Read as
Examples
Category
convolution
convolution;
convolved with
.
f ∗ g means the convolution of f and g.
functional analysis
complex conjugate z∗ means the complex conjugate of z.
conjugate
complex numbers ( can also be used for the conjugate of z, as described below.)
group of units
*
.
R∗
consists of the set of units of the ring R, along with the operation
the group of units of multiplication.
of
ring theory This may also be written R× as described above, or U(R).
hyperreal numbers
the (set of)
hyperreals
non-standard
analysis
Hodge dual
Hodge dual;
Hodge star
linear algebra
∗R means the set of hyperreal numbers. Other sets can be used in
place of R.
∗v means the Hodge dual of a vector v. If v is a k-vector within an
n-dimensional oriented inner product space, then ∗v is an (n−k)vector.
∗N
If
is the hypernatural numbers.
are the standard basis vectors of
,
proportionality
is proportional to;
y ∝ x means that y = kx for some constant k.
varies as
if y = 2x, then y ∝ x.
everywhere
Karp reduction[9]
∝
is Karp reducible
to;
is polynomial-time A ∝ B means the problem A can be polynomially reduced to the
problem B.
many-one
reducible to
If L1 ∝ L2 and L2 ∈ P, then L1 ∈ P.
computational
complexity theory
set-theoretic
complement
∖
minus;
without;
throw out;
not
A ∖ B means the set that contains all those elements of A that are not
in B.[5]
{1,2,3,4} ∖ {3,4,5,6} = {1,2}
(− can also be used for set-theoretic complement as described
above.)
set theory
conditional event
given
P(A|B) means the probability of the event A occurring given that B
occurs.
if X is a uniformly random day of the year P(X is May 25 | X is in
May) = 1/31
probability
restriction
|
restriction of ... to
f|A means the function f is restricted to the set A, that is, it is the
...;
function with domain A ∩ dom(f) that agrees with f.
restricted to
The function f : R → R defined by f(x) = x2 is not injective, but f|R+ is
injective.
set theory
such that
such that;
so that
| means "such that", see ":" (described below).
S = {(x,y) | 0 < y < f(x)}
The set of (x,y) such that y is greater than 0 and less than f(x).
everywhere
∣
divisor, divides
divides
∤
number theory
exact divisibility
∣∣
exactly divides
number theory
∥
a ∣ b means a divides b.
a ∤ b means a does not divide b.
(The symbol ∣ can be difficult to type, and its negation is rare, so a
regular but slightly shorter vertical bar | character is often used
instead.)
pa ∣∣ n means pa exactly divides n (i.e. pa divides n but pa+1 does
not).
⋕
23 ∣∣ 360.
x ∥ y means x is parallel to y.
x ∦ y means x is not parallel to y.
x ⋕ y means x is equal and parallel to y.
parallel
is parallel to
∦
Since 15 = 3 × 5, it is true that 3 ∣ 15 and 5 ∣ 15.
If l ∥ m and m ⊥ n then l ⊥ n.
geometry (The symbol ∥ can be difficult to type, and its negation is rare, so
two regular but slightly longer vertical bar || characters are often
used instead.)
incomparability
is incomparable to x ∥ y means x is incomparable to y.
order theory
{1,2} ∥ {2,3} under set containment.
cardinality
cardinality of;
size of;
order of
#X means the cardinality of the set X.
#{4, 6, 8} = 3
(|...| may be used instead as described above.)
set theory
#
connected sum
connected sum of;
knot sum of;
A#B is the connected sum of the manifolds A and B. If A and B are
knot composition knots, then this denotes the knot sum, which has a slightly stronger
of
condition.
A#Sm is homeomorphic to A, for any manifold A, and the sphere Sm.
topology, knot
theory
124
F. Some List of Math Symbols
primorial
primorial
n# is product of all prime numbers less than or equal to n.
12# = 2 × 3 × 5 × 7 × 11 = 2310
: means "such that", and is used in proofs and the set-builder
notation (described below).
∃ n ∈ ℕ: n is even.
number theory
such that
such that;
so that
everywhere
field extension
extends;
over
K : F means the field K extends the field F.
ℝ:ℚ
This may also be written as K ≥ F.
field theory
:
inner product of
matrices
A : B means the Frobenius inner product of the matrices A and B.
The general inner product is denoted by ⟨u, v⟩, ⟨u | v⟩ or (u | v), as
described below. For spatial vectors, the dot product notation, x·y is
linear algebra common. See also bra–ket notation.
inner product of
index of a
subgroup
The index of a subgroup H in a group G is the "relative size" of H in
index of subgroup G: equivalently, the number of "copies" (cosets) of H that fill up G
group theory
division
divided by
over
A : B means the division of A with B (dividing A by B)
10 : 2 = 5
everywhere
vertical ellipsis
⋮
vertical ellipsis
Denotes that certain constants and terms are missing out (e.g. for
clarity) and that only the important terms are being listed.
everywhere
wreath product
≀
wreath product of
... by ...
group theory
A ≀ H means the wreath product of the group A by the group H.
This may also be written A wr H.
is isomorphic to the automorphism group of the complete
bipartite graph on (n,n) vertices.
x+4=x-3※
↯
※
downwards zigzag
arrow
Denotes that contradictory statements have been inferred. For
contradiction; this
clarity, the exact point of contradiction can be appended.
contradicts that
everywhere
Statement: Every finite, non-empty, ordered set has a largest element.
Otherwise, let's assume that is a finite, non-empty, ordered set with
no largest element. Then, for some
, there exists an
with
, but then there's also an
with
, and so
on. Thus,
are distinct elements in . ↯ is finite.
exclusive or
⊕
⊻
xor
The statement A ⊕ B is true when either A or B, but not both, are
propositional true. A ⊻ B means the same.
logic, Boolean
algebra
direct sum
The direct sum is a special way of combining several objects into
one general object.
direct sum of
abstract algebra (The bun symbol ⊕, or the coproduct symbol ∐, is used; ⊻ is only
for logic.)
(¬A) ⊕ A is always true, A ⊕ A is always false.
Most commonly, for vector spaces U, V, and W, the following
consequence is used:
U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = {0})
Kulkarni–Nomizu
Derived from the tensor product of two symmetric type (0,2)
product
tensors; it has the algebraic symmetries of the Riemann tensor.
Kulkarni–Nomizu
has components
product
.
tensor algebra
D'Alembertian;
wave operator
□
non-Euclidean
Laplacian
It is the generalisation of the Laplace operator in the sense that it is
the differential operator which is invariant under the isometry group
of the underlying space and it reduces to the Laplace operator if
restricted to time independent functions.
vector calculus
Letter-based symbols
Includes upside-down letters.
Letter modifiers
Also called diacritics.
125
F. Some List of Math Symbols
Symbol
in
HTML
Symbol
in TeX
Name
Explanation
Read as
Examples
Category
mean
overbar;
... bar
(often read as "x bar") is the mean (average value of
.
).
statistics
finite sequence,
tuple
finite sequence,
tuple
means the finite sequence/tuple
.
.
model theory
algebraic closure
a
algebraic closure
of
is the algebraic closure of the field F.
The field of algebraic numbers is sometimes denoted as
algebraic closure of the rational numbers .
because it is the
field theory
complex conjugate
conjugate
means the complex conjugate of z.
.
(z∗
can also be used for the conjugate of z, as described
complex numbers above.)
topological closure
(topological)
closure of
topology
is the topological closure of the set S.
This may also be denoted as cl(S) or Cl(S).
In the space of the real numbers,
real numbers).
(the rational numbers are dense in the
unit vector
hat
â
(pronounced "a hat") is the normalized version of vector ,
having length 1.
geometry
estimator
estimator for
is the estimator or the estimate for the parameter .
The estimator
produces a sample estimate
for the mean .
statistics
derivative
′
... prime;
derivative of
f ′(x) means the derivative of the function f at the point x, i.e.,
the slope of the tangent to f at x.
If f(x) := x2, then f ′(x) = 2x.
(The single-quote character ' is sometimes used instead,
calculus especially in ASCII text.)
derivative
•
... dot;
time derivative of
means the derivative of x with respect to time. That is
.
If x(t) := t2, then
.
calculus
Symbols based on Latin letters
126
F. Some List of Math Symbols
Name
Symbol
in
HTML
Symbol
Read as
Explanation
Category
in TeX
Examples
universal quantification
∀
for all;
for any;
for each;
for every
∀ x: P(x) means P(x) is true for all x.
∀ n ∈ ℕ: n2 ≥ n.
ℂ means {a + b i : a,b ∈ ℝ}.
i = √ −1 ∈ ℂ
predicate logic
ℂ
C
complex numbers
C;
the (set of) complex
numbers
numbers
cardinality of the
continuum
cardinality of the
continuum;
c;
cardinality of the real
numbers
The cardinality of
C).
is denoted by
or by the symbol (a lowercase Fraktur letter
set theory
partial derivative
partial;
d
∂f/∂xi means the partial derivative of f with respect to xi, where f is a function on (x1,
..., xn).
If f(x,y) := x2y, then ∂f/∂x = 2xy,
∂M means the boundary of M
∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}
calculus
∂
boundary
boundary of
topology
degree of a polynomial
∂f means the degree of the polynomial f.
∂(x2 − 1) = 2
degree of
algebra (This may also be written deg f.)
expected value
the value of a random variable one would "expect" to find if one could repeat the
random variable process an infinite number of times and take the average of the
probability theory values obtained
expected value
E
existential quantification
∃
there exists;
there is;
there are
∃ x: P(x) means there is at least one x such that P(x) is true.
∃ n ∈ ℕ: n is even.
predicate logic
uniqueness quantification
∃!
∈
∉
there exists exactly one ∃! x: P(x) means there is exactly one x such that P(x) is true.
predicate logic
set membership
is an element of;
is not an element of
∃! n ∈ ℕ: n + 5 = 2n.
(1/2)−1 ∈ ℕ
a ∈ S means a is an element of the set S;[7] a ∉ S means a is not an element of S.[7]
2−1 ∉ ℕ
everywhere, set theory
set membership
∌
does not contain as an
element
S ∌ e means the same thing as e ∉ S, where S is a set and e is not an element of S.
set theory
such that symbol
such that
∋
mathematical logic
often abbreviated as "s.t."; : and | are also used to abbreviate "such that". The use of ∋ Choose ∋ 2| and 3| . (Here | is used in the
goes back to early mathematical logic and its usage in this sense is declining.
sense of "divides".)
set membership
contains as an element
S ∋ e means the same thing as e ∈ S, where S is a set and e is an element of S.
set theory
ℍ
H
quaternions or
Hamiltonian quaternions
H;
ℍ means {a + b i + c j + d k : a,b,c,d ∈ ℝ}.
the (set of) quaternions
numbers
N means either { 0, 1, 2, 3, ...} or { 1, 2, 3, ...}.
ℕ
natural numbers
the (set of) natural
numbers
N
The choice depends on the area of mathematics being studied; e.g. number theorists
prefer the latter; analysts, set theorists and computer scientists prefer the former. To
avoid confusion, always check an author's definition of N.
ℕ = {|a| : a ∈ ℤ} or ℕ = {|a| > 0: a ∈ ℤ}
numbers
Set theorists often use the notation ω (for least infinite ordinal) to denote the set of
natural numbers (including zero), along with the standard ordering relation ≤.
For two matrices (or vectors) of the same dimensions
product is a matrix of the same dimensions
.
linear algebra In MATLAB this operation is expressed by A.*B.
Hadamard product
o
entrywise product
the Hadamard
with elements given by
function composition
∘
composed with
f ∘ g is the function such that (f ∘ g)(x) = f(g(x)).[10]
set theory
Big O notation
O
if f(x) := 2x, and g(x) := x + 3, then (f ∘ g)(x) =
2(x + 3).
If f(x) = 6x4 − 2x3 + 5 and g(x) = x4, then
big-oh of
The Big O notation describes the limiting behavior of a function, when the argument
Computational tends towards a particular value or infinity.
complexity theory
127
F. Some List of Math Symbols
projective space
P;
the projective space;
the projective line;
the projective plane
ℙ
ℙ means a space with a point at infinity.
topology
probability
ℙ(X) means the probability of the event X occurring.
the probability of
P
probability theory This may also be written as P(X), Pr(X), P[X] or Pr[X].
Given a set S, the power set of S is the set of all subsets of the set S. The power set of
S0 is
Power set
the Power set of
Powerset denoted by P(S).
rational numbers
ℚ
Q
,
If a fair coin is flipped, ℙ(Heads) = ℙ(Tails) =
0.5.
The power set P({0, 1, 2}) is the set of all
subsets of {0, 1, 2}. Hence,
P({0, 1, 2}) = {∅, {0}, {1}, {2}, {0, 1}, {0,
2}, {1, 2}, {0, 1, 2. }}
3.14000... ∈ ℚ
Q;
the (set of) rational
numbers;
the rationals
ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}.
π∉ℚ
numbers
ℝ
R
real numbers
π∈ℝ
R;
the (set of) real numbers; ℝ means the set of real numbers.
the reals
√(−1) ∉ ℝ
numbers
conjugate transpose
†
conjugate transpose;
adjoint;
Hermitian
adjoint/conjugate
/transpose/dagger
A† means the transpose of the complex conjugate of A.[11]
If A = (aij) then A† = (aji).
This may also be written A∗T, AT∗, A∗, AT or AT.
matrix operations
transpose
T
AT means A, but with its rows swapped for columns.
If A = (aij) then AT = (aji).
transpose
matrix operations This may also be written A′, At or Atr.
top element
the top element
⊤
⊤ means the largest element of a lattice.
∀x : x ∨ ⊤ = ⊤
⊤ means the top or universal type; every type in the type system of interest is a
subtype of top.
∀ types T, T <: ⊤
x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y.
If l ⊥ m and m ⊥ n in the plane, then l || n.
W⊥ means the orthogonal complement of W (where W is a subspace of the inner
product space V), the set of all vectors in V orthogonal to every vector in W.
Within
x ⊥ y means x has no factor greater than 1 in common with y.
34 ⊥ 55
A ⊥ B means A is an event whose probability is independent of event B.
If A ⊥ B, then P(A|B) = P(A).
⊥ means the smallest element of a lattice.
∀x : x ∧ ⊥ = ⊥
⊥ means the bottom type (a.k.a. the zero type or empty type); bottom is the subtype
of every type in the type system.
∀ types T, ⊥ <: T
x ⊥ y means that x is comparable to y.
{e, π} ⊥ {1, 2, e, 3, π} under set containment.
A ∪ B means the set of those elements which are either in A, or in B, or in both.[5]
A ⊆ B ⇔ (A ∪ B) = B
lattice theory
top type
the top type; top
type theory
perpendicular
is perpendicular to
geometry
orthogonal complement
orthogonal/
perpendicular
complement of;
perp
,
.
linear algebra
coprime
is coprime to
number theory
⊥
independent
is independent of
probability
bottom element
the bottom element
lattice theory
bottom type
the bottom type;
bot
type theory
comparability
is comparable to
order theory
set-theoretic union
∪
the union of ... or ...;
union
set theory
set-theoretic intersection
∩
intersected with;
intersect
A ∩ B means the set that contains all those elements that A and B have in common.[5] {x ∈ ℝ : x2 = 1} ∩ ℕ = {1}
set theory
logical disjunction or
join in a lattice
∨
or;
max;
join
propositional logic,
lattice theory
∧
logical conjunction or
meet in a lattice
and;
The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement
is false.
n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural
number.
For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), B(x)).
The statement A ∧ B is true if A and B are both true; else it is false.
For functions A(x) and B(x), A(x) ∧ B(x) is used to mean min(A(x), B(x)).
n < 4 ∧ n > 2 ⇔ n = 3 when n is a natural
number.
128
F. Some List of Math Symbols
min;
meet
propositional logic,
lattice theory
wedge product
wedge product;
exterior product
exterior algebra
u ∧ v means the wedge product of any multivectors u and v. In three-dimensional
Euclidean space the wedge product and the cross product of two vectors are each
other's Hodge dual.
exponentiation
a ^ b means a raised to the power of b
... (raised) to the power
of ...
(a ^ b is more commonly written ab. The symbol ^ is generally used in programming
everywhere languages where ease of typing and use of plain ASCII text is preferred.)
multiplication
2^3 = 23 = 8
3 × 4 means the multiplication of 3 by 4.
times;
multiplied by
(The symbol * is generally used in programming languages, where ease of typing and
arithmetic use of ASCII text is preferred.)
7 × 8 = 56
Cartesian product
×
the Cartesian product of
... and ...;
X × Y means the set of all ordered pairs with the first element of each pair selected
the direct product of ... from X and the second element selected from Y.
and ...
{1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}
set theory
cross product
cross
(1,2,5) × (3,4,−1) =
(−22, 16, − 2)
u × v means the cross product of vectors u and v
linear algebra
group of units
R× consists of the set of units of the ring R, along with the operation of multiplication.
the group of units of
ring theory This may also be written R∗ as described below, or U(R).
⊗
tensor product, tensor
product of modules
tensor product of
means the tensor product of V and U.[12]
modules V and U over the ring R.
means the tensor product of {1, 2, 3, 4} ⊗ {1, 1, 2} =
{{1, 1, 2}, {2, 2, 4}, {3, 3, 6}, {4, 4, 8}}
linear algebra
semidirect product
N ⋊φ H is the semidirect product of N (a normal subgroup) and H (a subgroup), with
⋉
the semidirect product of
respect to φ. Also, if G = N ⋊φ H, then G is said to split over N.
⋊
semijoin
group theory
the semijoin of
relational algebra
(⋊ may also be written the other way round, as ⋉, or as ×.)
R ⋉ S is the semijoin of the relations R and S, the set of all tuples in R for which there R
is a tuple in S that is equal on their common attribute names.
S=
a1,..,an(R
S)
natural join
⋈
the natural join of
relational algebra
ℤ
R ⋈ S is the natural join of the relations R and S, the set of all combinations of tuples
in R and S that are equal on their common attribute names.
ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...}.
integers
ℤ+ or ℤ> means {1, 2, 3, ...} .
numbers ℤ* or ℤ≥ means {0, 1, 2, 3, ...} .
the (set of) integers
Z
ℤn
ℤp
Zn
Zp
integers mod n
ℤ = {p, −p : p ∈ ℕ ∪ {0}}
ℤn means {[0], [1], [2], ...[n−1]} with addition and multiplication modulo n.
the (set of) integers
modulo n
Note that any letter may be used instead of n, such as p. To avoid confusion with
numbers p-adic numbers, use ℤ/pℤ or ℤ/(p) instead.
ℤ3 = {[0], [1], [2]}
p-adic integers
the (set of) p-adic
integers
Note that any letter may be used instead of p, such as n or l.
numbers
Symbols based on Hebrew or Greek letters
129
F. Some List of Math Symbols
Symbol
in
HTML
Symbol
Name
Explanation
Read as
in TeX
Examples
Category
aleph number
ℵ
aleph
set theory
beth number
ℶ
beth
set theory
ℵα represents an infinite cardinality (specifically, the α-th one, where α
|ℕ| = ℵ0, which is called aleph-null.
is an ordinal).
ℶα represents an infinite cardinality (similar to ℵ, but ℶ does not
necessarily index all of the numbers indexed by ℵ. ).
Dirac delta
function
δ(x)
Dirac delta of
hyperfunction
Kronecker
delta
δij
Kronecker
delta of
δ
hyperfunction
Functional
derivative
Functional
derivative of
Differential
operators
∆
symmetric
difference
symmetric
difference
⊖
A ∆ B (or A ⊖ B) means the set of elements in exactly one of A or B.
{1,5,6,8} ∆ {2,5,8} = {1,2,6}
(Not to be confused with delta, Δ, described below.)
{3,4,5,6} ⊖ {1,2,5,6} = {1,2,3,4}
set theory
delta
Δx means a (non-infinitesimal) change in x.
delta;
change in
is the gradient of a straight line.
(If the change becomes infinitesimal, δ and even d are used instead.
calculus Not to be confused with the symmetric difference, written ∆, above.)
Δ
Laplacian
Laplace
operator
The Laplace operator is a second order differential operator in
n-dimensional Euclidean space
If ƒ is a twice-differentiable real-valued function, then the Laplacian of ƒ
is defined by
∇f (x1, ..., xn) is the vector of partial derivatives (∂f / ∂x1, ..., ∂f / ∂xn).
If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)
vector
calculus
gradient
del;
nabla;
gradient of
vector
calculus
divergence
∇
del dot;
divergence of
If
, then
If
, then
.
vector
calculus
curl
curl of
vector
calculus
.
Pi
pi;
Used in various formulas involving circles; π is equivalent to the
3.1415926...; amount of area a circle would take up in a square of equal width with
≈355÷113 an area of 4 square units, roughly 3.14159. It is also the ratio of the
mathematical circumference to the diameter of a circle.
A = πR2 = 314.16 → R = 10
constant
projection
Projection of
π
restricts
to the
attribute set.
relational
algebra
Homotopy
group
the nth
Homotopy
group of
consists of homotopy equivalence classes of base point
preserving maps from an n-dimensional sphere (with base point) into
the pointed space X.
Homotopy
theory
product
product over
... from ... to ...
of
means
.
arithmetic
∏
Cartesian
product
the Cartesian
product of;
the direct
product of
means the set of all (n+1)-tuples
(y0, ..., yn).
set theory
coproduct
∐
A general construction which subsumes the disjoint union of sets and
coproduct of topological spaces, the free product of groups, and the direct sum of
over ... from ... modules and vector spaces. The coproduct of a family of objects is
130
F. Some List of Math Symbols
to ... of
σ
essentially the "least specific" object to which each object in the family
category
admits a morphism.
theory
selection
The selection
selects all those tuples in for which holds
Selection of between the and the attribute. The selection
selects all
relational those tuples in for which holds between the attribute and the
value
.
algebra
summation
∑
sum over ...
from ... to ...
of
means
.
arithmetic
∅
{}
empty set
the empty set ∅ means the set with no elements.[7] { } means the same.
set theory
{n ∈ ℕ : 1 < n2 < 4} = ∅
Variations
In mathematics written in Arabic, some symbols may be reversed to make right-to-left writing and reading easier.[13]
See also
Greek letters used in mathematics, science, and engineering
Diacritic
ISO 31-11 (Mathematical signs and symbols for use in physical sciences and technology)
Latin letters used in mathematics
List of mathematical abbreviations
List of mathematical symbols by subject
Mathematical Alphanumeric Symbols (Unicode block)
Mathematical constants and functions
Mathematical notation
Mathematical operators and symbols in Unicode
Notation in probability and statistics
Physical constants
Table of logic symbols
Table of mathematical symbols by introduction date
Typographical conventions in mathematical formulae
References
1. "Math is Fun website".
2. Rónyai, Lajos (1998), Algoritmusok(Algorithms), TYPOTEX, ISBN 963-9132-16-0
3. Deb, K.; Pratap, A.; Agarwal, S.; Meyarivan, T. (2002). "A fast and elitist multiobjective genetic algorithm: NSGA-II". IEEE Transactions on Evolutionary Computation 6 (2): 182.
doi:10.1109/4235.996017.
4. Copi, Irving M.; Cohen, Carl (1990) [1953], "Chapter 8.3: Conditional Statements and Material Implication", Introduction to Logic (8th ed.), New York: Macmillan Publishers (United States),
pp. 268–269, ISBN 0-02-325035-6, LCCN 89037742
5. Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 4, ISBN 0-412-60610-0
6. Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press, p. 66, ISBN 0-521-63503-9, OCLC 43641333
7. Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 3, ISBN 0-412-60610-0
8. Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press, p. 62, ISBN 0-521-63503-9, OCLC 43641333
9. Berman, Kenneth A; Paul, Jerome L. (2005), Algorithms: Sequential, Parallel, and Distributed, Boston: Course Technology, p. 822, ISBN 0-534-42057-5
10. Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 5, ISBN 0-412-60610-0
11. Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press, pp. 69–70, ISBN 0-521-63503-9, OCLC 43641333
12. Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press, pp. 71–72, ISBN 0-521-63503-9, OCLC 43641333
13. M. Benatia, A. Lazrik, and K. Sami, "Arabic mathematical symbols in Unicode (http://www.ucam.ac.ma/fssm/rydarab/doc/expose/unicodeme.pdf)", 27th Internationalization and Unicode Conference,
2005.
External links
The complete set of mathematics Unicode characters (http://krestavilis.com/math.php)
Jeff Miller: Earliest Uses of Various Mathematical Symbols (http://jeff560.tripod.com/mathsym.html)
Numericana: Scientific Symbols and Icons (http://www.numericana.com/answer/symbol.htm)
GIF and PNG Images for Math Symbols (http://us.metamath.org/symbols/symbols.html)
Mathematical Symbols in Unicode (http://tlt.psu.edu/suggestions/international/bylanguage/math.html#browsers)
Using Greek and special characters from Symbol font in HTML (http://www.alanwood.net/demos/symbol.html)
Unicode Math Symbols (http://mathsymbols.net/) - a quick form for using unicode math symbols.
DeTeXify handwritten symbol recognition (http://detexify.kirelabs.org/classify.html) — doodle a symbol in the box, and the program will tell you what its name is
Handbook for Spoken Mathematics (http://web.efzg.hr/dok/MAT/vkojic/Larrys_speakeasy.pdf) — pronunciation guide to many commonly used symbols
Some Unicode charts of mathematical operators:
Index of Unicode symbols (http://www.unicode.org/charts/#symbols)
Range 2100–214F: Unicode Letterlike Symbols (http://www.unicode.org/charts/PDF/U2100.pdf)
Range 2190–21FF: Unicode Arrows (http://www.unicode.org/charts/PDF/U2190.pdf)
Range 2200–22FF: Unicode Mathematical Operators (http://www.unicode.org/charts/PDF/U2200.pdf)
Range 27C0–27EF: Unicode Miscellaneous Mathematical Symbols–A (http://www.unicode.org/charts/PDF/U27C0.pdf)
Range 2980–29FF: Unicode Miscellaneous Mathematical Symbols–B (http://www.unicode.org/charts/PDF/U2980.pdf)
Range 2A00–2AFF: Unicode Supplementary Mathematical Operators (http://www.unicode.org/charts/PDF/U2A00.pdf)
Some Unicode cross-references:
Short list of commonly used LaTeX symbols (http://www.artofproblemsolving.com/Wiki/index.php/LaTeX:Symbols) and Comprehensive LaTeX Symbol List
(http://mirrors.med.harvard.edu/ctan/info/symbols/comprehensive/)
MathML Characters (http://www.robinlionheart.com/stds/html4/entities-mathml) - sorts out Unicode, HTML and MathML/TeX names on one page
Unicode values and MathML names (http://www.w3.org/TR/REC-MathML/chap6/bycodes.html)
Unicode values and Postscript names (http://svn.ghostscript.com/ghostscript/branches/gs-db/Resource/Decoding/Unicode) from the source code for Ghostscript
Retrieved from "https://en.wikipedia.org/w/index.php?title=List_of_mathematical_symbols&oldid=730409871"
Categories: Mathematical notation Mathematics-related lists Mathematical symbols Mathematical tables Mathematical logic Lists of symbols
This page was last modified on 18 July 2016, at 21:19.
Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy.
Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
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Appendix G
DISLAYING MATH EXPRESSIONS
132
G. Dislaying Math Expressions
Help:Displaying a formula
From Wikipedia, the free encyclopedia
"WP:MATH" and "WP:MATHS" redirect here. For the WikiProject on mathematics, see Wikipedia:WikiProject
Mathematics. For Wikipedia's mathematics style manual, see Wikipedia:Manual of Style/Mathematics. For the
mathematics reference desk, see Wikipedia:Reference desk/Mathematics.
MediaWiki renders mathematical equations using a combination of html markup and a variant of LaTeX.
The version of LaTeX used is a subset of AMS-LaTeX markup, a superset of LaTeX markup which is in turn a superset of TeX
markup, for mathematical formulae. Only a limited part of the full TeX language is supported; see below for details.[a]
By default SVG images with non-visible MathML are generated. The older PNG images can be set via user preferences.[b] On
some browsers like Firefox, it is possible to use MathML for display via extensions; see the main extension page at
mw:Extension:Math for details. Client side MathJax is no longer supported.
Contents
1 Basics
1.1
1.2
1.3
1.4
1.5
LaTeX commands
Special characters
Spaces
LaTeX environments
Rendering
1.5.1 Force-rerendering of formulas
2 TeX vs HTML
2.1 Pros of HTML
2.2 Pros of TeX
2.3 Using MathML
3 Formatting using TeX
3.1 Functions, symbols, special characters
3.1.1 Accents/diacritics
3.1.2 Standard numerical functions
3.1.3 Bounds
3.1.4 Projections
3.1.5 Differentials and derivatives
3.1.6 Letter-like symbols or constants
3.1.7 Modular arithmetic
3.1.8 Radicals
3.1.9 Operators
3.1.10 Sets
3.1.11 Relations
3.1.12 Geometric
3.1.13 Logic
3.1.14 Arrows
3.1.15 Special
3.1.16 Unsorted (new stuff)
3.2 Larger expressions
3.2.1 Subscripts, superscripts, integrals
3.3 Display attribute
3.3.1 Inline
3.3.1.1 Example
3.3.1.2 Technical implementation
3.3.2 Block
3.3.2.1 Example
3.3.2.2 Technical implementation
3.3.3 Not specified
3.3.3.1 Example
3.3.4 Fractions, matrices, multilines
3.3.5 Parenthesizing big expressions, brackets, bars
3.3.6 Equation numbering
3.4 Alphabets and typefaces
3.4.1 Mixed text faces
3.5 Color
3.6 Formatting issues
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G. Dislaying Math Expressions
4
5
6
7
8
3.6.1 Spacing
3.6.2 Alignment with normal text flow
3.7 Commutative diagrams
3.7.1 Diagrams in TeX
3.7.2 Convert to SVG
3.7.3 Upload the file
3.7.4 Examples
3.8 Unimplemented elements and workarounds
3.8.1 \oiint and \oiiint
3.8.1.1 \oiint and \oiiint as PNG images
3.8.1.1.1 Examples
3.8.1.2 Oriented \oiint and \oiiint as PNG images
3.8.2 \overarc
3.8.3 \dddot
3.9 Syntax to avoid
3.9.1 Percentages
3.9.2 \textrm
3.9.3 Unicode characters
Chemistry
4.1 Molecular and Condensed formula
4.2 Bonds
4.3 Charges
4.4 Addition Compounds and Stoichiometric Numbers
4.5 (Italic) Math
4.6 Oxidation States
4.7 Greek characters
4.8 Isotopes
4.9 States
4.10 Precipitate
4.11 Reaction Arrows
4.12 Further Examples Using Ordinary LaTeX tags
Examples of implemented TeX formulas
5.1 Quadratic polynomial
5.2 Quadratic formula
5.3 Tall parentheses and fractions
5.4 Integrals
5.5 Matrices and determinants
5.6 Summation
5.7 Differential equation
5.8 Complex numbers
5.9 Limits
5.10 Integral equation
5.11 Example
5.12 Continuation and cases
5.13 Prefixed subscript
5.14 Fraction and small fraction
5.15 Area of a quadrilateral
5.16 Volume of a sphere-stand
5.17 Multiple equations
See also
References
7.1 Footnotes
7.2 Citations
External links
Basics
Math markup goes inside . Chemistry markup goes inside or
.... All these tags use TeX.
The TeX code has to be put literally: MediaWiki templates, predefined templates, and parameters cannot be used within math
tags: pairs of double braces are ignored and "#" gives an error message. However, math tags work in the then and else part of
#if, etc. See m:Template:Demo of attempt to use parameters within TeX (backlinks edit (https://meta.wikimedia.org
/wiki/Template:Demo_of_attempt_to_use_parameters_within_TeX?action=edit)) for more information.
134
G. Dislaying Math Expressions
LaTeX commands
LaTeX commands are case-sensitive, and take one of the following two formats:
They start with a backslash \ and then have a name consisting of letters only. Command names are terminated by a
space, a number or any other "non-letter".
They consist of a backslash \ and exactly one non-letter.
Some commands need an argument, which has to be given between curly braces { } after the command name. Some
commands support optional parameters, which are added after the command name in square brackets []. The general syntax is:
\commandname[option1,option2,...]{argument1}{argument2}...
Special characters
The following symbols are reserved characters that either have a special meaning under LaTeX or are unavailable in all the
fonts. If you enter them directly in your text, they will normally not render, but rather do things you did not intend.
# $ % ^ & _ { } ~ \
These characters can be entered by adding a prefix backslash or using special sequences:
\# \$ \% ^\wedge \& \_ \{ \} \sim \backslash
yielding
.
The backslash character \ can not be entered by adding another backslash in front of it (\\); this sequence is used for line
breaking. For introducing a backslash in math mode, you can use \backslash instead which gives .
The command \tilde produces a tilde which is placed over the next letter. For example, \tilde{a} gives . To produce just a
tilda character ~, use \tilde{} which gives , placing a ~ over an empty box. Alternatively \sim produces , a large centred ~
which may be more appropriate in some situations.
The command \hat produces a hat over the next character, for example \hat{o} produces . For a stretchable version use
giving
. The wedge \wedge is normally used as a mathematical operator
the best equivalent to the ascii caret ^ character.
\widehat{abc}
the sequence ^\wedge produces
Spaces
"Whitespace" characters, such as blank or tab, are treated uniformly as "space" by LaTeX. Several consecutive whitespace
characters are treated as one "space". See below for commands that produces spaces of different size.
LaTeX environments
Environments in LaTeX have a role that is quite similar to commands, but they usually have effect on a wider part of formula.
Their syntax is:
\begin{environmentname}
text to be influenced
\end{environmentname}
Environments supported by Wikipedia include matrix, align, etc. See below.
Rendering
By default, the PNG images are rendered black on white, with a transparent
background. On darker backgrounds, the characters may show white edges. To
remove these, match the PNG background color with the background color of the
page using \pagecolor. However, black text on a dark background is hard to read
and should be avoided altogether where possible.
The colors, as well as font sizes and types, are independent of browser settings or CSS. Font sizes and types will often deviate
from what HTML renders. Vertical alignment with the surrounding text can also be a problem; a work-around is described in
the "Alignment with normal text flow" section below. The css selector of the images is img.tex.
135
G. Dislaying Math Expressions
The alt text of the PNG images, which is displayed to visually impaired and other readers who cannot see the images, and is
also used when the text is selected and copied, defaults to the wikitext that produced the image, excluding the . You can override this by explicitly specifying an alt attribute for the math element. For example, generates an image
whose alt text is "Square root of pi". This should not
be confused with the title attribute that produces popup text when the hovering over the PNG image, for example generates an image whose popup text is "pi".
Apart from function and operator names, as is customary in mathematics, variables and letters are in italics; digits are not. For
other text, (like variable labels) to avoid being rendered in italics like variables, use \text, \mbox, or \mathrm. You can also
define new function names using \operatorname{...}. For example, \text{abc} gives
. \operatorname{...} provides
spacing before and after the operator name when appropriate, as when a\operatorname{\sn}b is rendered as
(with
space to the left and right of "sn") and a\operatorname{\sn}(b+c) as
(with space to the left and not to the right).
Latex does not have full support for Unicode characters and not all characters render. Most Latin characters with accents
render correctly. However some do not, in particular those that include multiple diacritics (e.g. with Latin letters used in
Vietnamese) or that cannot be precomposed into a single character (such as the uppercase Latin letter W with ring), or that use
other diacritics (like the ogonek or the double grave accent, used in Central European languages like Polish, or the horn
attached above some vowels in Vietnamese), or other modified letter forms (used in IPA notations, or African languages, or in
medieval texts), some digram ligatures (like IJ in Dutch), or Latin letters borrowed from Greek, or small capitals, as well as
superscripts and subscript letters. For example, \text{ð} or \mbox{ð}, and \text{þ} or \mbox{þ} (used in Icelandic) will give
errors.
Force-rerendering of formulas
MediaWiki stores rendered formulas in a cache so that the images of those formulas do not need to be created each time the
page is opened by a user. To force the rerendering of all formulas of a page, you must open it with the getter variables
action=purge&mathpurge=true. Imagine for example there is a wrong rendered formula in the article Integral. To force the
rerendering of this formula you need to open the URL https://en.wikipedia.org/w/index.php?title=Integral&action=purge&
mathpurge=true . Afterwards you need to bypass your browser cache so that the new created images of the formulas are
actually downloaded. See also mw:Extension:Math#Purging pages that contain equations for more details.
TeX vs HTML
Main page: Wikipedia:Rendering math
Before using TeX markup for producing special characters, it should be noted that, as this comparison table shows, sometimes
similar results can be achieved in HTML using Template:Math. See also Help:Special characters.
TeX syntax
TeX rendering
HTML syntax
HTML rendering
\alpha
{{math|''α''}}
α
f(x) = x^2
{{math|''f''(''x'') {{=}} ''x''2}}
f(x) = x2
\sqrt{2}
{{math|{{radical|2}}}}
√2
\sqrt{1-e^2}
{{math|{{radical|1 − ''e''2}}}}
√ 1 − e2
The codes on the left produce the symbols on the right, but the latter can also be put directly in the wikitext, except for ‘=’.
136
G. Dislaying Math Expressions
HTML syntax
Rendering
α β γ δ ε ζ
η θ ι κ λ μ ν
ξ ο π ρ σ ς
τ υ φ χ ψ ω
αβγδεζ
ηθικλμν
ξοπρσς
τυφχψω
Γ Δ Θ Λ Ξ Π
Σ Φ Ψ Ω
ΓΔΘΛΞΠ
ΣΦΨΩ
∫ ∑ ∏ √ − ± ∞
≈ ∝ = ≡ ≠ ≤ ≥
× · ⋅ ÷ ∂ ′ ″
∇ ‰ ° ∴ ∅
∫∑∏√−±∞
≈∝=≡≠≤≥
×·⋅÷∂′″
∇‰°∴∅
∈ ∉ ∩ ∪ ⊂ ⊃ ⊆ ⊇
¬ ∧ ∨ ∃ ∀
⇒ ⇔ → ↔ ↑ ↓
ℵ - – —
∈∉∩∪⊂⊃⊆⊇
¬∧∨∃∀
⇒⇔→↔↑↓
ℵ-–—
The project has settled on using both HTML and TeX because each has advantages in some situations.
Pros of HTML
1. Formulas in HTML behave more like regular text. In-line HTML formulae always align properly with the rest of the
HTML text and, to some degree, can be copied-and-pasted (this is not a problem if TeX is rendered using MathJax, and
the alignment should not be a problem for PNG rendering once bug 32694 is fixed).
2. The formula’s background and font size match the rest of HTML contents (this can be fixed on TeX formulas by using
the commands \pagecolor and \definecolor) and the appearance respects CSS and browser settings while the
typeface is conveniently altered to help you identify formulae.
3. Pages using HTML code for formulae will load faster and they will create less clutter on your hard disk.
4. Formulae typeset with HTML code will be accessible to client-side script links (a.k.a. scriptlets).
5. The display of a formula entered using mathematical templates can be conveniently altered by modifying the templates
involved; this modification will affect all relevant formulae without any manual intervention.
6. The HTML code, if entered diligently, will contain all semantic information to transform the equation back to TeX or
any other code as needed. It can even contain differences TeX does not normally catch, e.g. {{math|''i''}} for the
imaginary unit and {{math|i}} for an arbitrary index variable.
7. Unlike generated bitmaps, HTML is not sensitive to dots per inch variances between viewing platforms.
Pros of TeX
1. TeX is semantically more precise than HTML.
1. In TeX, "x" means "mathematical variable " ", whereas in HTML "x" is generic and somewhat ambiguous.
2. On the other hand, if you encode the same formula as "{{math|x}}", you get the same visual result x
and no information is lost. This requires diligence and more typing that could make the formula harder to
understand as you type it. However, since there are far more readers than editors, this effort is worth considering if
no other rendering options are available (such as MathJax, which is available to logged-in users as a preferences
opt-in).
2. One consequence of point 1 is that TeX code can be transformed into HTML, but not vice versa.[1] This means that on
the server side we can always transform a formula, based on its complexity and location within the text, user
preferences, type of browser, etc. Therefore, where possible, all the benefits of HTML can be retained, together with the
benefits of TeX. It is true that the current situation is not ideal, but that is not a good reason to drop information or
contents. It is more a reason to help improve the situation.
3. Another consequence of point 1 is that TeX can be converted to MathML (e.g. by MathJax) for browsers which support
it, thus keeping its semantics and allowing the rendering to be better suited for the reader’s graphic device.
4. TeX is the preferred text formatting language of most professional mathematicians, scientists, and engineers. It is easier
to persuade them to contribute if they can write in TeX.
5. TeX has been specifically designed for typesetting formulae, so input is easier and more natural if you are accustomed to
it, and output is more aesthetically pleasing if you focus on a single formula rather than on the whole containing page.
6. Once a formula is done correctly in TeX, it will render reliably, whereas the success of HTML formulae is somewhat
dependent on browsers or versions of browsers. Another aspect of this dependency is fonts: the serif font used for
rendering formulae is browser-dependent and it may be missing some important glyphs. While the browser is generally
capable to substitute a matching glyph from a different font family, it need not be the case for combined glyphs (compare
137
G. Dislaying Math Expressions
"a̅" and " a̅").
7. When writing in TeX, editors need not worry about whether this or that version of this or that browser supports this or
that HTML entity. The burden of these decisions is put on the software. This does not hold for HTML formulae, which
can easily end up being rendered wrongly or differently from the editor’s intentions on a different browser.[2]
8. TeX formulae, by default, render larger and are usually more readable than HTML formulae and are not dependent on
client-side browser resources, such as fonts, and so the results are more reliably WYSIWYG.
9. While TeX does not assist you in finding HTML codes or Unicode values (which you can obtain by viewing the HTML
source in your browser), copying and pasting from a TeX PNG image in Wikipedia into simple text will return the
LaTeX source.
^ Unless your wikitext follows the style of point 1.2
^ The entity support problem is not limited to mathematical formulae though; it can be easily solved by using the corresponding characters
instead of entities, as the character repertoire links do, except for cases where the corresponding glyphs are visually indiscernible (e.g.
– for ‘–’ and − for ‘−’).
In some cases it may be the best choice to use neither TeX nor the HTML substitutes, but instead the simple ASCII symbols of
a standard keyboard (see hereafter, for an example).
Using MathML
The default MathML/SVG renderer option, selectable through My Preferences - Appearance - Math generate hidden MathML
code. This code can be used by screen readers and other assistive technology. To actually display the MathML in Firefox the
Native MathML (https://addons.mozilla.org/en-US/firefox/addon/native-mathml/) extension and the MathML fonts
(https://developer.mozilla.org/en-US/docs/Mozilla/MathML_Project/Fonts) must be installed. Details on using MathML in
other systems can be found at mw:Extension:Math.
Formatting using TeX
Functions, symbols, special characters
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G. Dislaying Math Expressions
Accents/diacritics
\dot{a}, \ddot{a}, \acute{a}, \grave{a}
\check{a}, \breve{a}, \tilde{a}, \bar{a}
\hat{a}, \widehat{a}, \vec{a}
Standard numerical functions
\exp_a b = a^b, \exp b = e^b, 10^m
\ln c, \lg d = \log e, \log_{10} f
\sin a, \cos b, \tan c, \cot d, \sec e, \csc f
\arcsin h, \arccos i, \arctan j
\sinh k, \cosh l, \tanh m, \coth n
\operatorname{sh}\,k, \operatorname{ch}\,l, \operatorname{th}\,m,
\operatorname{coth}\,n
\operatorname{argsh}\,o, \operatorname{argch}\,p,
\operatorname{argth}\,q
\sgn r, \left\vert s \right\vert
\min(x,y), \max(x,y)
Bounds
\min x, \max y, \inf s, \sup t
\lim u, \liminf v, \limsup w
\dim p, \deg q, \det m, \ker\phi
Projections
\Pr j, \hom l, \lVert z \rVert, \arg z
Differentials and derivatives
dt, \operatorname{d}\!t, \partial t, \nabla\psi
dy/dx, \operatorname{d}\!y/\operatorname{d}\!x, {dy \over dx},
{\operatorname{d}\!y\over\operatorname{d}\!x}, {\partial^2\over\partial
x_1\partial x_2}y
\prime, \backprime, f^\prime, f', f'', f^{(3)}, \dot y, \ddot y
Letter-like symbols or constants
\infty, \aleph, \complement, \backepsilon, \eth, \Finv, \hbar
\Im, \imath, \jmath, \Bbbk, \ell, \mho, \wp, \Re, \circledS
Modular arithmetic
s_k \equiv 0 \pmod{m}
a\,\bmod\,b
\gcd(m, n), \operatorname{lcm}(m, n)
\mid, \nmid, \shortmid, \nshortmid
Radicals
139
G. Dislaying Math Expressions
For a little more semantics on these symbols, see the brief TeX Cookbook (http://www.math.upenn.edu/tex-stuff
/cookbook.pdf).
Larger expressions
Subscripts, superscripts, integrals
140
G. Dislaying Math Expressions
Feature
Syntax
Superscript
a^2
Subscript
a_2
How it looks rendered
10^{30} a^{2+2}
Grouping
a_{i,j} b_{f'}
Combining sub & super without and with
horizontal separation
x_2^3
Super super
10^{10^{8}}
{x_2}^3
\sideset{_1^2}{_3^4}\prod_a^b
Preceding and/or additional sub & super
{}_1^2\!\Omega_3^4
\overset{\alpha}{\omega}
\underset{\alpha}{\omega}
Stacking
\overset{\alpha}{\underset{\gamma}{\omega}}
\stackrel{\alpha}{\omega}
x', y'', f', f''
Derivatives
x^\prime, y^{\prime\prime}
Derivative dots
\dot{x}, \ddot{x}
\hat a \ \bar b \ \vec c
Underlines, overlines, vectors
\overrightarrow{a b} \ \overleftarrow{c d}
\ \widehat{d e f}
\overline{g h i} \ \underline{j k l}
Arc (workaround)
Arrows
\overset{\frown} {AB}
A \xleftarrow{n+\mu-1} B
\xrightarrow[T]{n\pm i-1} C
Overbraces
\overbrace{ 1+2+\cdots+100 }^{5050}
Underbraces
\underbrace{ a+b+\cdots+z }_{26}
Sum
\sum_{k=1}^N k^2
Sum (force \textstyle)
\textstyle \sum_{k=1}^N k^2
Sum in a fraction (default \textstyle)
\frac{\sum_{k=1}^N k^2}{a}
Sum in a fraction (force \displaystyle)
\frac{\displaystyle \sum_{k=1}^N k^2}{a}
Sum in a fraction (alternative limits style)
\frac{\sum\limits^{^N}_{k=1} k^2}{a}
Product
\prod_{i=1}^N x_i
Product (force \textstyle)
\textstyle \prod_{i=1}^N x_i
Coproduct
\coprod_{i=1}^N x_i
Coproduct (force \textstyle)
\textstyle \coprod_{i=1}^N x_i
141
G. Dislaying Math Expressions
Limit
\lim_{n \to \infty}x_n
Limit (force \textstyle)
\textstyle \lim_{n \to \infty}x_n
Integral
\int\limits_{1}^{3}\frac{e^3/x}{x^2}\, dx
Integral (alternative limits style)
\int_{1}^{3}\frac{e^3/x}{x^2}\, dx
Integral (force \textstyle)
\textstyle \int\limits_{-N}^{N} e^x\, dx
Integral (force \textstyle, alternative limits
\textstyle \int_{-N}^{N} e^x\, dx
style)
Double integral
\iint\limits_D \, dx\,dy
Triple integral
\iiint\limits_E \, dx\,dy\,dz
Quadruple integral
\iiiint\limits_F \, dx\,dy\,dz\,dt
Line or path integral
\int_{(x,y)\in C} x^3\, dx + 4y^2\, dy
Closed line or path integral
\oint_{(x,y)\in C} x^3\, dx + 4y^2\, dy
Intersections
\bigcap_{i=_1}^n E_i
Unions
\bigcup_{i=_1}^n E_i
Display attribute
The and play with the
vertical-align argument until you get it right; however, how it looks may depend on the browser and the browser settings.
Also note that if you rely on this workaround, if/when the rendering on the server gets fixed in future releases, as a result of
this extra manual offset your formulae will suddenly be aligned incorrectly. So use it sparingly, if at all.
Commutative diagrams
To make a commutative diagram, there are three steps:
1. write the diagram in TeX
2. convert to SVG
3. upload the file to Wikimedia Commons
Diagrams in TeX
Xy-pic (http://www.tug.org/applications/Xy-pic/) (online manual (http://tex.loria.fr
/graph-pack/doc-xypic/xyguide-html/xyguide-html.html)) is the most powerful and
general-purpose diagram package in TeX. Diagrams created using it can be found at
Commons: Category:Xy-pic diagrams.
A sample commutative diagram,
created in the manner described
Simpler packages include:
AMS's amscd (http://www.dante.de/CTAN//help/Catalogue/entries/amscd.html)
Paul Taylor's diagrams (http://www.ctan.org/tex-archive/macros/generic/diagrams/taylor/)
François Borceux Diagrams (http://www.ctan.org/tex-archive/help/Catalogue/entries/borceux.html)
The following is a template for Xy-pic, together with a hack to increase the margins in dvips, so that the diagram is not
truncated by over-eager cropping (suggested in TUGboat: TUGboat, Volume 17 1996, No. 3 (http://www.tug.org/TUGboat
/Articles/tb17-3/tb52rahtz.pdf)):
\documentclass{amsart}
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G. Dislaying Math Expressions
\usepackage[all, ps, dvips]{xy}
% Loading the XY-Pic package
% Using postscript driver for smoother curves
\usepackage{color} % For invisible frame
\begin{document}
\thispagestyle{empty} % No page numbers
\SelectTips{eu}{} % Euler arrowheads (tips)
\setlength{\fboxsep}{0pt} % Frame box margin
{\color{white}\framebox{{\color{black}$$ % Frame for margin
\xymatrix{
%%% Diagram goes here %%%
}
$$}}} % end math, end frame
\end{document}
Convert to SVG
Once you have produced your diagram in LaTeX (or TeX), you can convert it to an SVG file using the following sequence of
commands:
pdflatex file.tex
pdfcrop --clip file.pdf tmp.pdf
pdf2svg tmp.pdf file.svg
rm tmp.pdf
The pdfcrop (http://pdfcrop.sourceforge.net) and pdf2svg (http://www.cityinthesky.co.uk/opensource/pdf2svg) utilities are
needed for this procedure. You can alternatively use pdf2svg (http://www.pdftron.com/pdf2svg/) from PDFTron for the last
step.
If you do not have pdfTeX (which is unlikely) you can use the following commands to replace the first step (TeX → PDF):
latex file.tex
dvipdfm file.dvi
In general, you will not be able to get anywhere with diagrams without TeX and Ghostscript, and the inkscape program is a
useful tool for creating or modifying your diagrams by hand. There is also a utility pstoedit which supports direct conversion
from Postscript files to many vector graphics formats, but it requires a non-free plugin to convert to SVG, and regardless of the
format, this editor has not been successful in using it to convert diagrams with diagonal arrows from TeX-created files.
These programs are:
a working TeX distribution, such as TeX Live
Ghostscript
pstoedit
Inkscape
Upload the file
See also: commons:Commons:First steps/Upload form
See also: Help:Contents/Images and media
As the diagram is your own work, upload it to Wikimedia Commons, so that all projects (notably, all languages) can use it
without having to copy it to their language's Wiki. (If you've previously uploaded a file to somewhere other than Commons, to
Commons.)
Check size
Before uploading, check that the default size of the image is neither too large nor too small by opening in an SVG
application and viewing at default size (100% scaling), otherwise adjust the -y option to dvips.
Name
Make sure the file has a meaningful name.
Upload
Login to Wikimedia Commons, then upload the file (http://commons.wikimedia.org/w/index.php?title=Special:Upload&
uselang=ownwork); for the Summary, give a brief description.
Now go to the image page and add a description, including the source code, using this template:
{{Information
|description =
{{en|1= Description [[:en:Link to WP page|topic]]
}}
|source = {{own}}, created as per:
[[:en:Help:Displaying a formula#Commutative diagrams]];
source code below.
|date = The Creation Date, like 1999-12-31
|author = [[User:YourUserName|Your Real Name]]
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G. Dislaying Math Expressions
|permission = {{self|PD-self (or other license)
|author = [[User:YourUserName|Your Real Name]]}}
}}
==TeX source==
% TeX source here
[[Category:Commutative diagrams]]
[[Category:Xy-pic diagrams]]
[[Category:Images with LaTeX source code]]
Source code
Include the source code in the image page, in the Source section of the {{Information}} template, so that the diagram
can be edited in future.
Include the complete .tex file, not just the fragment, so future editors do not need to reconstruct a compilable file.
You may optionally make the source code section collapsible, using the {{cot}}/{{cob}} templates.
(Don't include it in the Summary section, which is just supposed to be a summary.)
License
The most common license for commutative diagrams is PD-self; some use PD-ineligible, especially for simple
diagrams, or other licenses. Please do not use the GFDL (http://www.gnu.org/copyleft/fdl.html), as it requires the entire
text of the GFDL to be attached to any document that uses the diagram.
Description
If possible, link to a Wikipedia page relevant to the diagram. (The 1= is necessary if you use nest templates within the
description, and harmless otherwise.)
Category
Include [[Category:Commutative diagrams]], so that it appears in commons:Category:Commutative diagrams. There
are also subcategories, which you may choose to use.
Include image
Now include the image on the original page via [[File:Diagram.svg]]
Examples
A sample conforming diagram is commons:Image:PSU-PU.svg.
Unimplemented elements and workarounds
\oiint
and \oiiint
Elements which are not yet implemented are \oiint, namely a two-fold integral \iint (
) with a circular curve through the
centre of the two integrals, and similarly \oiiint, a circular curve through three integrals. In contrast, \oint ( ) exists for the
single dimension (integration over a curved line within a plane or any space with higher dimension).
These elements appear in many contexts: \oiint denotes a surface integral over the closed 2d boundary of a 3d region (which
occurs in much of 3d vector calculus and physical applications – like Maxwell's equations), likewise \oiiint denotes
integration over the closed 3d boundary (surface volume) of a 4d region, and they would be strong candidates for the next TeX
version. As such there are a lot of workarounds in the present version.
\oiint
\oiint
and \oiiint using currently implemented symbols
looks like:
, which uses \iint along with \subset and \supset (overdrawn after
backspacing):
\iint\limits_{S}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset \mathbf D \cdot
\mathrm{d}\mathbf A
, which uses \int twice (with some backward kerning) along with \bigcirc
(also overdrawn after backpacing) to produce a more consistent circle:
\int\!\!\!\!\int_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;
\bigcirc\,\,\mathbf D\cdot\mathrm{d}\mathbf A
\oiiint
(should also be preferably more tightly kerned) looks more or less like:
153
G. Dislaying Math Expressions
which uses three \int symbols (with more backward kerning) with \subset
and \supset (overdrawn after backspacing):
\int\!\!\!\!\!\int\!\!\!\!\!\int_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\;\;\;\subset\!\supset \mathbf D\;\cdot\mathrm{d}\mathbf A
, which uses three \int symbols (with more backward kerning) along with
\bigcirc
(also overdrawn after backspacing):
\int\!\!\!\!\!\int\!\!\!\!\!\int_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\;\;\;\bigcirc\,\,\mathbf D\;\cdot\mathrm{d}\mathbf A
However, since no standardisation exists as yet, any workaround like this (which uses many \! symbols for backspacing)
should be avoided, if possible. See below for a possibility using PNG image enforcement.
Note that \iint (the double integral) and \iiint (the triple integral) are still not kerned as they should preferably be, and are
currently rendered as if they were successive \int symbols ; this is not a major problem for reading the formulas, even if the
integral symbols before the last one do not have bounds, so it's best to avoid backspacing "hacks" as they may be inconsistent
with a possible future better implementation of integrals symbols (with more precisely computed kerning positions).
\oiint
and \oiiint as PNG images
These symbols are available as PNG images which are also integrated into two templates, {{oiint}} and {{oiiint}}, which take
care of the formatting around the symbols.
The templates have three parameters:
preintegral
the text or formula immediately before the integral
intsubscpt
the subscript below the integral
integrand
the text or formula immediately after the integral
Examples
Stokes' theorem: {{oiint | intsubscpt = | integrand= }}
Ampère's law + correction: {{oiint | preintegral= | intsubscpt = | integrand = }}
Continuity of 4-momentum flux (in general relativity):[1]{{oiiint | preintegral= |
intsubscpt = | integrand = }}
Oriented \oiint and \oiiint as PNG images
Some variants of \oiint and \oiiint have arrows on them to indicate the sense of integration, such as a line integral around a
closed curve in the clockwise sense, and higher dimensional analogues. These are not implemented in TeX on Wikipedia
either, although the template {{intorient}} is available - see link for details.
\overarc
\overarc
is not yet implemented to display the arc notation. However, there exists a workaround: use \overset{\frown}{AB},
which gives
154
G. Dislaying Math Expressions
\dddot
is not implemented in the TexVC renderer but does work in MathJax. For a workaround use \overset{...}{x}, which
gives .
\dddot
Syntax to avoid
The texvc processor accepts some non-standard syntax. These should be avoided as the MathJax based renderers do not
support these syntax.
Percentages
Texvc accepts % for representing percentages. This causes an error with MathJax and should be replaced with \% in all
renderers.
\textrm
In texvc spaces need to be represented inside the \textrm environment using \, \ and normal spaces are ignored i.e.
\textrm{A\,B C} would render as A BC. In mathjax \textrm is an alias for \text which is renders its argument as normal text,
hence \textrm{A\,B C} renders as A\,B C. To ensure compatibility between versions alway use the \text environment:
\text{A B C}.
Unicode characters
Non-ASCII Unicode characters like π work in MathML, and MathJax but not in texvc so should be avoided.
Chemistry
There are three ways to render chemical sum formulae as used in chemical equations:
...
{{chem}}
X
is short for
(where X is a chemical sum formula)
Technically, is a math tag with the extension mhchem enabled, according to the MathJax
documentation (http://mathjax.readthedocs.org/en/latest/tex.html#mhchem).
Note, that the commands \cee and \cf are disabled, because they are marked as deprecated in the mhchem LaTeX package
documentation (http://www.ctan.org/pkg/mhchem).
Please note that there are still major issues (https://phabricator.wikimedia.org/T140217) with mhchem support in MediaWiki.
Molecular and Condensed formula
mhchem
Markup
{{chem}}
Equivalent
HTML
Renders as
H2OSb2O3(NH4)2S
Bonds
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G. Dislaying Math Expressions
Equivalent
{{chem}}
mhchem
Markup
and
HTML
Renders as
C6H5-CHOA-B=C{\equiv}D
Charges
mhchem
Markup
{{chem}}
Equivalent
HTML
Renders as
H+NO3-CrO4^2-AgCl2-[AgCl2]-Y^{99}+Y^{99+}
Addition Compounds and Stoichiometric Numbers
mhchem
Markup
{{chem}}
Renders as
MgSO4.7H2OKCr(SO4)2*12H2O{CaSO4.1/2H2O} + 1\!1/2H2O -> CaSO4.2H2O{25/2O2} + C8H18 -> {8CO2} + 9H2O
(Italic) Math
156
G. Dislaying Math Expressions
Markup
mhchem
{{chem}}
{C_\mathit{x}H_\mathit{y}} + \mathit{z}O2 -> {\mathit{x}CO2} + \frac{\mathit{y}}{2}H2O
Renders
as
Markup
{{chem|C|''x''|H|''y''}} + ''z''{{chem|O|2}} → ''x''{{chem|C|O|2}} + {{frac|''y''|2}}
{{chem|H|2|O}}
Renders
as
CxHy + zO2 → xCO2 + y⁄2H2O
Oxidation States
Markup
mhchem
{{chem}}
Fe^{II}Fe^{III}2O4
Renders
as
Markup
{{chem|Fe|II|Fe|III|2|O|4}}
Renders
as
FeIIFeIII2O4
with
...
Greek characters
Equivalent
{{chem}}
mhchem
Markup
and
HTML
Renders as
\mu-Cl[Fe(\eta^5-C5H5)2]
Isotopes
Equivalent
{{chem}}
mhchem
Markup
and
HTML
Renders as
^{227}_{90}Th+^0_{-1}n-
States
States Subscripting is not IUPAC recommendation.
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G. Dislaying Math Expressions
mhchem
{{chem}}
Markup
Renders as
H2_{(aq)}CO3^{2-}{(aq)}
Precipitate
Markup
mhchem
{Ba^2+} + SO4^{2-} -> BaSO4 v
Renders
as
Markup
{{chem}}
{{chem|Ba|2+}} + {{chem|S|O|4|2-}} → {{chem|Ba|S|O|4}}↓
Renders Ba
as
Markup
Equivalent HTML
2+
+ SO2−
4 → BaSO4↓
Ba2+ + SO42- → BaSO4↓
Renders Ba2+ + SO42- → BaSO4↓
as
Reaction Arrows
Markup
Renders as
A ->[x] BA ->[\text{text
above}][\text{text
below}] BA ->[\ce{+H2O}]
B
Comparison of arrow symbols
Markup
Renders as
-><=>
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G. Dislaying Math Expressions
<->
Further Examples Using Ordinary LaTeX tags
To align the equations or color them, use
.