# 17.1: New Page - Mathematics

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17.1: New Page - Mathematics

## Student Classes

New York Math Circle&rsquos goal is to constantly challenge your mind. You&rsquoll get to solve unusual problems and invent your own, apply existing knowledge in new situations, learn famous gems of mathematics, and explore the unknown. The Math Circle will open your eyes and increase your sensitivity to all the mathematics around us. Our main requirement is that you have an open mind and a willingness to work.

NYMC is not a tutoring or test preparation program. Classes focus on material you won&rsquot encounter in the regular curriculum. We&rsquoll help you develop reasoning and problem solving skills, and along the way will help you enjoy, appreciate, and expand your knowledge of mathematics.

To subscribe to announcements about student classes, see our contact page.

Students looking to complete a BS in applied mathematics must complete the required courses in the following areas:

### Foundational Course Requirement

The following foundational courses must be completed before acceptance into the major:

• MTH 161: Calculus IA
• MTH 162: Calculus IIA
• MTH 164: Multidimensional Calculus
• MTH 165: Linear Algebra with Differential Equations
• PHY 121: Mechanics
• PHY 122: Electricity and Magnetism

Equivalent courses can be substituted for the above requirements:

• MTH 171, 172, and 174 for the equivalent MTH 161, 162, and 164
• MTH 173 for MTH 165
• MTH 141-143 for MTH 161-162

AP courses can also be used to satisfy foundational requirements.

### Core Course Requirement

Students must complete the following four courses:

• MTH 235: Linear Algebra*
• MTH 201:  Introduction to Probability
• MTH 265: Functions of a Real Variable I
• MTH 282: Introduction to Complex Variables with Applications

An honors version of a course can always be substituted for the listed course.

* The requirement that MTH 235 be taken can also be satisfied by completing MTH 173. MTH 235 should be taken early in the student's major program.

In addition to the core courses, students must complete five 4-credit advanced courses as follows:

* Any mathematics course numbered 200 or above (excluding core courses) qualifies as an advanced mathematics course.

### Upper Level Writing Requirement

To satisfy the upper level writing requirement, students must pass two  upper level writing courses of a certain type.

## Illustrating Mathematics

This book is for anyone who wishes to illustrate their mathematical ideas, which in our experience means everyone. It is organized by material, rather than by subject area, and purposefully emphasizes the process of creating things, including discussions of failures that occurred along the way. As a result, the reader can learn from the experiences of those who came before, and will be inspired to create their own illustrations.

Topics illustrated within include prime numbers, fractals, the Klein bottle, Borromean rings, tilings, space-filling curves, knot theory, billiards, complex dynamics, algebraic surfaces, groups and prime ideals, the Riemann zeta function, quadratic fields, hyperbolic space, and hyperbolic 3-manifolds. Everyone who opens this book should find a type of mathematics with which they identify.

Each contributor explains the mathematics behind their illustration at an accessible level, so that all readers can appreciate the beauty of both the object itself and the mathematics behind it.

Graduate and undergraduate students and researchers interested in seeing beautiful and thought-provoking illustrations of mathematical ideas and getting ideas for creating one's own.

## [BOOKS] CIE A/AS LEVEL Mathematics books [BOOKS]

Here are some books that I have found online and I think would help every one doing A/AS Level Math.

1. Cambridge International Pure Mathematics 1 (AS LEVEL ONLY)

2012 | ISBN-10: 1444146440 | PDF | 312 pages | 124 MB

This brand new series has been written for the University of Cambridge International Examinations course for AS and A Level Mathematics (9709). This title covers the requirements of P1.

2. Understanding Pure Mathematics (A2 + AS)

1987 | ISBN-10: 0199142432 | PDF | 500 pages | 19 MB

THIS IS THE BOOK THAT I HAVE PERSONALLY USED - THE BEST BOOK FOR THE COMPLETE A LEVEL - HAS EVERY LITTLE THING (Although its old )

A classic single-volume textbook, popular for its direct and straightforward approach. Understanding Pure Mathematics starts by filling the gap between GCSE and A Level and builds on this base for candidates taking either single-subject of double-subject A Level.

3. Pure Mathematics: Complete Advanced Level Mathematics (A2 + AS)

A good book just like the one before.

Andy Martin, Kevin Brown, Paul Rigby, Simon Riley, Pure Mathematics: Complete Advanced Level Mathematics
Trans-Atlantic Publications | 1999 | ISBN: 0000 | 336 pages | File type: PDF | 33,4 mb
This title provides numerous exercises, worked examples and clear explanations with questions and diagrams. Colour is used to highlight key mathematical elements and enhance learning. Margin notes provide extra support for key topics and formulas (a key formulas page is also included). Review and Technique exercises Contextual questions Consolidation 'A' and 'B' exercises and Applications and Activities provide a complete range of challenges and exam practice for complete success. Chapter overviews and summaries consolidate understanding.

The Stan Math Library is a C++, reverse-mode automatic differentiation library designed to be usable, extensive and extensible, efficient, scalable, stable, portable, and redistributable in order to facilitate the construction and utilization of algorithms that utilize derivatives.

The Stan Math Library depends on the Intel TBB library which is licensed under the Apache 2.0 license. This dependency implies an additional restriction as compared to the new BSD license alone. The Apache 2.0 license is incompatible with GPL-2 licensed code if distributed as a unitary binary. You may refer to the Apache 2.0 evaluation page on the Stan Math wiki.

Stan Math depends on four libraries:

These are distributed under the lib/ subdirectory. Only these versions of the dependent libraries have been tested with Stan Math.

Documentation for Stan math is available at mc-stan.org/math

The Stan Math Library is a C++ library which depends on the Intel TBB library and requires for some functionality (ordinary differential equations and root solving) the Sundials library. The build system is the make facility, which is used to manage all dependencies.

A simple hello world program using Stan Math is as follows:

If this is in the file /path/to/foo/foo.cpp , then you can compile and run this with something like this, with the /path/to business replaced with actual paths:

The first make command with the math-libs target ensures that all binary dependencies of Stan Math are built and ready to use. The -j4 instructs make to use 4 cores concurrently which should be adapted to your needs. The second make command ensures that the Stan Math sources and all of the dependencies are available to the compiler when building foo .

An example of a real instantiation whenever the path to Stan Math is

The math-libs target has to be called only once, and can be omitted for subsequent compilations.

The standalone makefile ensures that all the required -I include statements are given to the compiler and the necessary libraries are linked:

/stan-dev/math/lib/tbb directory is created by the math-libs makefile target automatically and contains the dynamically loaded Intel TBB library. The flags -Wl,-rpath. instruct the linker to hard-code the path to the dynamically loaded Intel TBB library inside the stan-math directory into the final binary. This way the Intel TBB is found when executing the program.

Note for Windows users: On Windows the -rpath feature as used by Stan Math to hardcode an absolute path to a dynamically loaded library does not work. On Windows the Intel TBB dynamic library tbb.dll is located in the math/lib/tbb directory. The user can choose to copy this file to the same directory of the executable or to add the directory /path/to/math/lib/tbb as absolute path to the system-wide PATH variable.

math supports the new interface of Intel TBB, can be configured to use an external copy of TBB (e.g., with oneTBB or the system TBB library), using the TBB_LIB and TBB_INC environment variables.

To build the development version of math with oneTBB :

For example, installing oneTBB on Linux 64-bit ( x86_64 ) to $HOME directory (change if needed!): Note that you may replace TBB_VERSION=$ with a custom version number if needed ( check available releases here ).

• Set the TBB environment variables (specifically: TBB for the installation prefix, TBB_INC for the directory that includes the header files, and TBB_LIB for the libraries directory).

For example, installing oneTBB on Linux 64-bit ( x86_64 ) to \$HOME directory (change if needed!):

The above example will use the default compiler of the system as determined by make . On Linux this is usually g++ , on MacOS clang++ , and for Windows this is g++ if the RTools for Windows are used. There's nothing special about any of these and they can be changed through the CXX variable of make . The recommended way to set this variable for the Stan Math library is by creating a make/local file within the Stan Math library directory. Defining CXX=g++ in this file will ensure that the GNU C++ compiler is always used, for example. The compiler must be able to fully support C++11 and partially the C++14 standard. The g++ 4.9.3 version part of RTools for Windows currently defines the minimal C++ feature set required by the Stan Math library.

Note that whenever the compiler is changed, the user usually must clean and rebuild all binary dependencies with the commands:

This ensures that the binary dependencies are created with the new compiler.

## Mathematics

### Department of Mathematics

The Associate in Science in Mathematics for Transfer degree includes curriculum which focuses on the mastery of integration and differentiation and using these techniques to model real-world applications. The Associate in Science in Mathematics for Transfer degree is intended for students who plan to complete a bachelor’s degree in Mathematics or a related field of study offered at various campuses in the California State University system. Students completing this degree are guaranteed admission to the CSU system, but not to a particular campus or major. Students transferring to a CSU campus that accepts this degree will be required to complete no more than 60 units after transfer to earn a bachelor’s degree. This degree may not be the best option for students intending to transfer to a particular CSU campus or to a university or college that is not part of the CSU system. The Associate in Science in Mathematics for Transfer degree also offers the appropriate preparation for students who plan to complete a bachelor’s degree in Mathematics at various campuses in the University of California system. However, students completing this degree are not guaranteed admission to the UC system. In all cases, students should consult with a counselor for more information on university admission and transfer requirements.

###### PROGRAM LEARNING OUTCOMES

Upon satisfactory completion of this award, the student should be prepared to: 1. Successfully complete upper division coursework in mathematics. 2. Master the techniques of integration and differentiation. 3. Use these techniques to model real-world applications.

To earn an Associate in Science for Transfer degree in this major, the student must complete the requirements detailed in the Transfer Model Curriculum pathway. All courses must be completed with a C or better.

###### MATH 30 REPLACES MATH 70

MATH 70 is being offered as MATH 30. Effective Summer 2017, students who previously completed MATH 70 may use MATH 70 in lieu of MATH 30 to satisfy award requirements and prerequisites.

###### WHAT IS STEM?

STEM (Science Technology Engineering & Mathematics) is a curricular pathway that prepares students to take calculus, and to be academically prepared for further study in fields such as medicine, engineering, biology, mathematics, computer programing and some business programs. Students who plan to major in STEM fields, must follow the pathway beginning with MATH 30. Students are encouraged to talk to counselors about academic preparation for these fields of study.

###### NON STEM STUDENTS

Students who do not plan to pursue STEM study or do not need calculus or precalculus for their academic goals may follow the new non-STEM pathway that begins with MATH 29.

###### ACCELERATED FOUNDATIONAL MATH SEQUENCE

MJC is pleased to introduce MATH 9 and MATH 19 as a new, accelerated alternative to the MATH 10/20 sequence. It allows students to complete the entire sequence in one semester instead of two, and for six units as opposed to nine units. Students who would like to progress through the math sequence quickly are encouraged to enroll in these courses.

###### NON-CREDIT MODULES FOR FOUNDATIONAL MATH SKILLS

Math 911-913, Math 921–924, Math 928-929 and Math 988-989 are 4 sequences of non-credit modules designed to allow students to place out of introductory credit math courses. Students who register for 911 are automatically registered for 912 and 913, those who register for 921 are automatically registered for 922 – 924 Students who register for 911 are automatically registered for 912 and 913, those who register for 921 are automatically registered for 921 – 924, those who register for 928 are automatically registered for 928 – 929 and those those who register for 988 are automatically registered for 988 – 989. . Completing the last course in each sequence allows students to move up in the sequence or move into credit courses.

### Mathematics Courses:

Non-Transferable Mathematics Courses (CSU or UC)

 MATH 9 Accelerated Intro to Math MATH 10 Introduction to Mathematics MATH 19 Accelerated Pre-Algebra MATH 20 Pre-Algebra MATH 29 Elementary Algebra for Non-STEM Majors MATH 30 Elementary Algebra Formerly Math 70 - (Before Summer 2017) for STEM Majors MATH 47 Skills for Success in Elementary Algebra MATH 89 Intermediate Algebra for Non STEM majors MATH 90 Intermediate Algebra for STEM Majors

General Education Mathematics Courses (Transfer & Liberal Studies)

#### For those desiring concentrated work in mathematics, the Mathematics Department offers five programs leading to Bachelor degrees.

Bachelor of Science in Actuarial Science | prepares the student to work as an actuary or in applied statistics.

Bachelor of Science in Mathematics | prepares the student as a mathematician for industry or graduate work.

Bachelor of Science in Applied Mathematics | prepares the student as an applied mathematician for industry or graduate work.

Bachelor of Arts in Secondary Teaching | prepares the student to teach at the high school level. Contact Mark Oursland [[email protected]] for more information.

Bachelor of Arts in Middle-level Mathematics Teaching | prepares the student to teach at the middle school level. Contact Peter Klosterman [email protected]] for more information.

The Mathematics Department is located in Samuelson Hall.

## 17.1: New Page - Mathematics

Section 1: Introduction: Why bother?

Good mathematical writing, like good mathematics thinking, is a skill which must be practiced and developed for optimal performance. The purpose of this paper is to provide assistance for young mathematicians writing their first paper. The aim is not only to aid in the development of a well written paper, but also to help students begin to think about mathematical writing.

I am greatly indebted to a wonderful booklet, "How to Write Mathematics," which provided much of the substance of this essay. I will reference many direct quotations, especially from the section written by Paul Halmos, but I suspect that nearly everything idea in this paper has it origin in my reading of the booklet. It is available from the American Mathematical Society, and serious students of mathematical writing should consult this booklet themselves. Most of the other ideas originated in my own frustrations with bad mathematical writing. Although studying mathematics from bad mathematical writing is not the best way to learn good writing, it can provide excellent examples of procedures to be avoided. Thus, one activity of the active mathematical reader is to note the places at which a sample of written mathematics becomes unclear, and to avoid making the same mistakes his own writing.

Mathematical communication, both written and spoken, is the filter through which your mathematical work is viewed. If the creative aspect of mathematics is compared to the act of composing a piece of music, then the art of writing may be viewed as conducting a performance of that same piece. As a mathematician, you have the privilege of conducting a performance of your own composition! Doing a good job of conducting is just as important to the listeners as composing a good piece. If you do mathematics purely for your own pleasure, then there is no reason to write about it. If you hope to share the beauty of the mathematics you have done, then it is not sufficient to simply write you must strive to write well.

This essay will begin with general ideas about mathematical writing. The purpose is to help the student develop an outline for the paper. The next section will describe the difference between "formal" and "informal" parts of a paper, and give guidelines for each one. Section four will discuss the writing of an individual proof. The essay will conclude with a section containing specific recommendations to consider as you write and rewrite the paper.

Section 2. Before you write: Structuring the paper

The purpose of nearly all writing is to communicate. In order to communicate well, you must consider both what you want to communicate, and to whom you hope to communicate it. This is no less true for mathematical writing than for any other form of writing. The primary goal of mathematical writing is to assert, using carefully constructed logical deductions, the truth of a mathematical statement. Careful mathematical readers do not assume that your work is well-founded they must be convinced. This is your first goal in mathematical writing.

However, convincing the reader of the simple truth of your work is not sufficient. When you write about your own mathematical research, you will have another goal, which includes these two you want your reader to appreciate the beauty of the mathematics you have done, and to understand its importance. If the whole of mathematics, or even the subfield in which you are working, is thought of as a large painting, then your research will necessarily constitute a relatively minuscule portion of the entire work. Its beauty is seen not only in the examination of the specific region which you have painted (although this is important), but also by observing the way in which your own work 'fits' in the picture as a whole.

These two goals--to convince your reader of the truth of your deductions, and to allow your audience to see the beauty of your work in relation to the whole of mathematics--will be critical as you develop the outline for your paper. At times you may think of yourself as a travel guide, leading the reader through territory charted only by you.

A successful mathematical writer will lay out for her readers two logical maps, one which displays the connections between her own work and the wide world of mathematics, and another which reveals the internal logical structure of her own work.

• Does your result strengthen a previous result by giving a more precise characterization of something?
• Have you proved a stronger result of an old theorem by weakening the hypotheses or by strengthening the conclusions?
• Have you proven the equivalence of two definitions?
• Is it a classification theorem of structures which were previously defined but not understood?
• Does is connect two previously unrelated aspects of mathematics?
• Does it apply a new method to an old problem?
• Does it provide a new proof for an old theorem?
• Is it a special case of a larger question?

It is necessary that you explicitly consider this question of placement in the structure of mathematics, because it will linger in your readers' minds until you answer it. Failure to address this very question will leave the reader feeling quite dissatisfied.

• Are your results concentrated in one dramatic theorem?
• Or do you have several theorems which are related, but equally significant?
• Have you found important counterexamples?
• Is your research purely theoretical mathematics, in the theorem-proof sense, or does your research involve several different types of activity, for example, modeling a problem on the computer, proving a theorem, and then doing physical experiments related to your work?
• Is your work a clear (although small) step toward the solution of a classic problem, or is it a new problem?

Since your reader does not know what you will be proving until after he has read your paper, advising him beforehand about what he will read, just as the travel agent prepares his customer, will allow him to enjoy the trip more, and to understand more of the things you lead him to.

To honestly and deliberately explain where your work fits into the big picture of mathematical research may require a great deal of humility. You will likely despair that your accomplishments seem rather small. Do not fret! Mathematics has been accumulating for thousands of years, based on the work of thousands (or millions) of practitioners. It has been said that even the best mathematicians rarely have more than one really outstanding idea during their lifetimes. It would be truly surprising if yours were to come as a high school student!

Section 3. Formal and Informal Exposition

Once you have a basic outline for your paper, you should consider "the formal or logical structure consisting of definitions, theorems, and proofs, and the complementary informal or introductory material consisting of motivations, analogies, examples, and metamathematical explanations. This division of the material should be conspicuously maintained in any mathematical presentation, because the nature of the subject requires above all else that the logical structure be clear." (p.1) These two types of material work in parallel to enable your reader to understand your work both logically and cognitively (which are often quite different--how many of you believed that integrals could be calculated using antiderivatives before you could prove the Fundamental Theorem of Calculus?) "Since the formal structure does not depend on the informal, the author can write up the former in complete detail before adding any of the latter." (p. 2)

Thus, the next stage in the writing process may be to develop an outline of the logical structure of your paper. Several questions may help: To begin, what exactly have you proven? What are the lemmas (your own or others) on which these theorems stand. Which are the corollaries of these theorems? In deciding which results to call lemmas, which theorems, and which corollaries, ask yourself which are the central ideas. Which ones follow naturally from others, and which ones are the real work horses of the paper? The structure of writing requires that your hypotheses and deductions must conform to a linear order. However, few research papers actually have a linear structure, in which lemmas become more and more complicated, one on top of another, until one theorem is proven, followed by a sequence of increasingly complex corollaries. On the contrary, most proofs could be modeled with very complicated graphs, in which several basic hypotheses combine with a few well known theorems in a complex way. There may be several seemingly independent lines of reasoning which converge at the final step. It goes without saying that any assertion should follow the lemmas and theorems on which it depends. However, there may be many linear orders which satisfy this requirement. In view of this difficulty, it is your responsibility to, first, understand this structure, and, second, to arrange the necessarily linear structure of your writing to reflect the structure of the work as well as possible. The exact way in which this will proceed depends, of course, on the specific situation.

One technique to assist you in revealing the complex logical structure of your paper is a proper naming of results. By naming your results appropriately (lemmas as underpinnings, theorems as the real substance, and corollaries as the finishing work), you will create a certain sense of parallelness among your lemmas, and help your reader to appreciate, without having struggled through the research with you, which are the really critical ideas, and which they can skim through more quickly.

Another technique for developing a concise logical outline stems from a warning by Paul Halmos, in HTWM, never to repeat a proof:

If several steps in the proof of Theorem 2 bear a very close resemblance to parts of the proof of Theorem 1, that's a signal that something may be less than completely understood. Other symptoms of the same disease are: 'by the same technique (or method, or device, or trick) as in the proof of Theorem 1. ', or, brutally, 'see the proof of Theorem 1'. When that happens the chances are very good that there is a lemma that is worth finding, formulating, and proving, a lemma from which both Theorem 1 and Theorem 2 are more easily and more clearly deduced. (p. 35)

These issues of structure should be well thought through BEFORE you begin to write your paper, although the process of writing itself which surely help you better understand the structure.

Now that we have discussed the formal structure, we turn to the informal structure. The formal structure contains the formal definitions, theorem-proof format, and rigorous logic which is the language of 'pure' mathematics. The informal structure complements the formal and runs in parallel. It uses less rigorous, (but no less accurate!) language, and plays an important part in elucidating both the mathematical location of the work, as we discussed above, and in presenting to the reader a more cognitive presentation of the work. For although mathematicians write in the language of logic, very few actually think in the language of logic (although we do think logically), and so to understand your work, they will be immensely aided by subtle demonstration of why something is true, and how you came to prove such a theorem. Outlining, before you write, what you hope to communicate in these informal sections will, most likely, lead to more effective communication.

Before you begin to write, you must also consider notation. The selection of notation is a critical part of writing a research paper. In effect, you are inventing a language which your readers must learn in order to understand your paper. Good notation firstly allows the reader to forget that he is learning a new language, and secondly provides a framework in which the essentials of your proof are clearly understood. Bad notation, on the other hand, is disastrous and may deter the reader from even reading your paper. In most cases, it is wise to follow convention. Using epsilon for a prime integer, or x(f) for a function, is certainly possible, but almost never a good idea.

Section 4: Writing a Proof

The first step in writing a good proof comes with the statement of the theorem. A well-worded theorem will make writing the proof much easier. The statement of the theorem should, first of all, contain exactly the right hypotheses. Of course, all the necessary hypotheses must be included. On the other hand, extraneous assumptions will simply distract from the point of the theorem, and should be eliminated when possible.

When writing a proof, as when writing an entire paper, you must put down, in a linear order, a set of hypotheses and deductions which are probably not linear in form. I suggest that, before you write you map out the hypotheses and the deductions, and attempt to order the statements in a way which will cause the least confusion to the reader.

In HTWM, Halmos offers several important recommendations about writing proofs:

1. Write the proof forward

A familiar trick of bad teaching is to begin a proof by saying: "Given e, let d be e/2". This is the traditional backward proof-writing of classical analysis. It has the advantage of being easily verifiable by a machine (as opposed to understandable by a human being), and it has the dubious advantage that something at the end comes out to be less than e. The way to make the human reader's task less demanding is obvious: write the proof forward. Start, as the author always starts, by putting something less than e, and then do what needs to be done--multiply by 3M2 + 7 at the right time and divide by 24 later, etc., etc.--till you end up with what you end up with. Neither arrangement is elegant, but the forward one is graspable and rememberable. (p. 43)

2. Avoid unnecessary notation. Consider:

a proof that consists of a long chain of expressions separated by equal signs. Such a proof is easy to write. The author starts from the first equation, makes a natural substitution to get the second, collects terms, permutes, inserts and immediately cancels an inspired factor, and by steps such as these proceeds till he gets the last equation. This is, once again, coding, and the reader is forced not only to learn as he goes, but, at the same time, to decode as he goes. The double effort is needless. By spending another ten minutes writing a carefully worded paragraph, the author can save each of his readers half an hour and a lot of confusion. The paragraph should be a recipe for action, to replace the unhelpful code that merely reports the results of the act and leaves the reader to guess how they were obtained. The paragraph would say something like this: "For the proof, first substitute p for q, the collect terms, permute the factors, and, finally, insert and cancel a factor r. (p. 42-43)

Section 5. Specific Recommendations

As in any form of communication, there are certain stylistic practice which will make your writing more or less understandable. These may be best checked and corrected after writing the first draft. Many of these ideas are from HTWM, and are more fully justified there.

## Research Group: Pure Mathematics

The Pure Mathematics Group has a strong tradition of making important contributions to research in algebra, analysis, geometry and topology.

### Group Overview

We are internationally recognised as research leaders in several fields, including geometric group theory, non commutative geometry and analysis, algebraic topology and applied topology.

The Pure Mathematics group is a vibrant, dynamic team of 30 academics and postgraduate students. We collaborate with researchers around the world and have strong contacts with groups in the UK, the EU, North America and East Asia. The Centre for Geometry, Topology, and Applications has recently been created to provide a focal point for a large part of this activity. We run multiple seminars and a colloquium in order to share our latest research results and hear of new research by external speakers.

We welcome applications for PhD study in one of our areas of expertise. Please have a look at our research areas and projects, and do not hesitate to contact our staff for more information. More information on how to apply can be found via this link. For external sources of fellowships and funding, please see this page.