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0
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0answers
73 views

The poor relation? Time to reassess?

The idea of functions is one of the most important ideas in mathematics. It rules mathematics: one input gives one output. Although, in game theory one study multi-valued functions: one state of the ...
2
votes
1answer
52 views

Research in Mathematics (Formalities vs. Results)

This question is intended to solve an internal dilemma I have been having lately. I assume that most of those who have gone through the process of mathematical research have, at some point, considered ...
1
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2answers
61 views

Books/subjects for proof practice

So I want to practice writing proofs. I've studied general proof-writing but now I want to learn how to apply that to mathematics. From what I understand, the best and most accessible subjects for ...
2
votes
0answers
41 views

Why is the triangle inequality property of a metric space important?

From my understanding, when we use metric spaces, we are trying to measure how "different" certain elements in a metric space are from one another. We all know that a metric space $(S,d)$ satisfies: ...
9
votes
3answers
268 views

How do you go about formalizing a concept?

I am reading Godel Escher Bach. I love it. In the first few chapters, the author shows what a formal system is and gives examples that eventually lead to a typographical formal system of strings that ...
2
votes
1answer
65 views

Breadth vs. Depth in Maths

As a Maths student (beginning undergraduate), I'm often unable to decide whether to spend my time going in more depth and attempt harder and harder problems (which I do to an extent) or whether I ...
2
votes
1answer
113 views

Unsolved/Least Solved IMO Questions

I recently read this article http://blog.mathfights.com/once-upon-a-time-on-imo/ where the author discusses an IMO problem from 2006 that only about 20 participants out of 600 were able to solve. So ...
1
vote
1answer
98 views

How substantial an advance must be to justify publication in pure mathematics?

In order to publish a paper in pure mathematics, specifically functional analysis, do we really need to come out something which improves a lot of existing work in the field? I ask this because I ...
0
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0answers
42 views

Advices on which to choose, uncertainty or certainty?

Currently I am doing my final year project(FYP) on Large-Scale Distributed Storage System, where I apply finite field and coding theory to design new codes which can improve existing codes. I plan to ...
2
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0answers
73 views

A Compact Real Analysis book for a graduate student, who is short of time.

I am a Phd student in Computer Science and I want to focus on Machine Learning, especially on statistical methods. My problem is, I always keep hitting the wall when it comes to studying underlying ...
1
vote
1answer
41 views

Reference request: Measure theory and/or manifolds [duplicate]

I have never encountered measure theory or manifolds yet, despite being close to my third year university level. Any texts for either or both of these subjects would be greatly appreciated.
3
votes
0answers
49 views

What is the difference between $\propto$ and $\sim$

Suppose I have two physical quantities, lets name them $a$ and $b$. I wonder what the difference exactly is between $$a\propto b,\tag{1}$$ and $$a \sim b.\tag{2}$$ I know for eq. 1, that it means ...
0
votes
1answer
43 views

Categorising types of Mathematics

What area of Math do the following fall under? 1) Systems of ODEs and Phase planes 2) Laplace Transforms 3) Fourier series 4) PDEs with grad, div, curl, flux
6
votes
4answers
225 views

Why does the definition of the functional limit involve a limit point?

This may be an odd question, and I'm not sure if these type of questions are at all appreciated in the maths community. But given the definition of the functional limit: Let $f: A \to \mathbb{R}$, ...
3
votes
2answers
63 views

Books for inequality proofs

I was wondering: what books for proving inequalities are used in universities when studying mathematics (undergraduate)? I know there are lots of books, but I would like to know which ones are ...
2
votes
0answers
37 views

Which journals may be interested in papers on justification of the use of notation in mathematics

I will have a paper justifying the use of notation in mathematics. The paper shows how symbolic language is necessary in mathematics and related fields. I have no idea about where to submit such ...
1
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0answers
44 views

Why are empty measurable spaces and empty topological spaces not desirable?

The definition of a $\sigma$-field $\mathscr{F}$ on a set $X$ (or $\sigma$-ring) requires $\mathscr{F}$ to be a non-empty subset of $\mathscr{P}(X)$. Why is this convention taken? What is the issue ...
8
votes
1answer
132 views

Modern Research in Algebraic Topology

What are some of the main directions and trends in modern (let's say within the last ~10 years) algebraic topology? What are some major open problems or recent results? In a more specific direction, ...
1
vote
2answers
60 views

Question about the limit definitions of derivative and definite integral

Actually this question may look simple and basic, but it is about something which bugs me for a long time, since when I took my first calculus classes. The limit $\lim_{x \to a}f(x) = L$ is defined ...
8
votes
3answers
253 views

Resources for Integrals?

I want to learn to solve integrals of some type, probably definite integrals with results involving various constants such as Catalan's, Euler-Mascheroni,Golden-ratio etc. and involving various ...
3
votes
0answers
83 views

Why is there no collection about all mathematical theorems and open questions?

I really would like to have a giant math collection which is sorted according to the Mathematics Subject Classification, but with more then 3 levels, and it should contain all known theorems and also ...
1
vote
3answers
98 views

Recommendation on Category theory textbook [duplicate]

I had posted a question about category theory some months ago, and I got answered that there are two ways to study Category Theory. One is to treat Category Theory as a logic system independent from ...
2
votes
2answers
70 views

$f(\alpha x)-f(\beta x) \to 0$ faster than $\frac{1}{\log x}$ for all functions $f$

I was solving the following problem (# 19-46 in Spivak's Calculus, 4th ed.): Suppose that $\dfrac{f(x)}{x}$ is integrable on every interval $[a,b]$ for $0<a<b$, and that $\lim_{x\to ...
3
votes
1answer
55 views

What are Goldbach conjecture for other algebra structures, matrix, polynomial, algebraic number, etc?

All: what are Goldbach conjectures for other algebraic structures, such as: matrix, polynomial ring, algebraic number, vectors, and other algebraic structures ? In other word, for other algebraic ...
1
vote
1answer
22 views

Quantifiers and Mathematical Modelling

When we are in the context of pure mathematics, quantifiers are everywhere. When we are in the context of mathematical modelling, quantifiers usually disappear. For instance, in statistics we often ...
18
votes
4answers
3k views

Why is an image called an “image”?

Given a function $f : A \to B$, the image, denoted by $\operatorname{Im}f$ is the set of all $f(x)$ where $x \in A$. Why do we call this set the image? When was it first used, and what motivated its ...
1
vote
2answers
47 views

Minimum prerequisites for Basic Complex Analysis by J. Marsden, M. Hoffman

I want to self study Basic Complex Analysis by J. Marsden, M. Hoffman but I don't know much real analysis and not very interested in learning real analysis. I know single and multivariable calculus, ...
4
votes
1answer
112 views

How to read this in English?

I am a teacher of English for IT. Please, help me! I do not know how to read $s(n), y(n), e(n)$ in English. We are studying signal processing and the way a filter operates. The sentence is: The ...
4
votes
5answers
84 views

$\mathbb{C}$ and $\mathbb{R}^{2}$

If a complex number is a pair of real numbers, then why we need to introduce the new symbol $\mathbb{C}$ for complexes instead of using $\mathbb{R}^{2}$? What are the subtle differences involved?
8
votes
1answer
107 views

Why are Banach manifolds not so popular?

Why are Banach and Frechet manifolds studied not even remotely as much as Euclidean manifolds? I assume like many other mathematical subjects, theory of manifolds has been developed much more than the ...
5
votes
2answers
162 views

Why should the generalization of a 'sequence' be called a 'net'?

The title says it all, really. Reading through Reed & Simon's book on functional analysis, I have now reached the chapter on topological spaces, and the notion of a net is introduced there to ...
8
votes
0answers
67 views

Why is the Leibniz rule a sufficient ingredient in the construction of the tangent space?

This is a very soft question, but I am wondering if anyone can shed light on why it is that the product rule (and linearity) provide exactly the right requirement for the space of derivations to be ...
0
votes
1answer
48 views

How to pronounce $\sim$ and $\overline a$ in equivalence relations?

I just was thinking about the basic statement shown below relating equivalence relations and partitions. My question specifically is, how to pronounce this statement, $$ \overline a = \{x \in S: x ...
2
votes
1answer
81 views

Analogy of ideals with Normal subgroups in groups.

I've started with Ideals in ring theory but still not comfortable with the analogy it has with normal subgroups in group theory.Like we can visualize normal subgroups as Is there some good intutive ...
2
votes
1answer
55 views

What exactly is integration?

Consider the function $y=2x$. The graph of this function is here. Next, Consider $\int 2x dx=x^2 + c$. Here is the graph : http://www.wolframalpha.com/input/?i=plot+y%3Dx^2+from+-2+to+2. My ...
0
votes
0answers
27 views

Can I start apostol (vol 2) with no background in multivariable calculus?

I searched a lot in web and almost everyone says if you want to read spivak or apostol, you should first read an introductory book on calculus like stewart. I didn't read stewart but I studied single ...
2
votes
2answers
79 views

What is the use of scheme theory?

I should preface this by saying that my background in Algebraic Geometry is (more or less) the content of Vakil's notes up through Chapter 4 (i.e. through the definition of a scheme and several ...
0
votes
0answers
21 views

Problem supplement for Advanced Calculus (Loomis and Sternberg)

There are too many problems in Loomis and Sternberg's Advanced Calculus for them to be useful. Can someone recommend a collection of problems to supplement this book? A short list of its best problems ...
1
vote
0answers
69 views

Learning math vs problem solving

Ok so I am about to start my final year in high school we will be learning calculus this year, but I already know single- and multi-variable calculus and linear algebra so I want to spend my final ...
1
vote
0answers
41 views

Working on strengths vs. weaknesses as an undergraduate student

I realize there's a lot of general advice on this site and elsewhere about studying mathematics, but I couldn't find anything about this, so I've decided to ask: as an undergraduate student, I'm ...
4
votes
2answers
65 views

Visualizing Lie algebra of SO(3)

Let $SO(3)$ be the Lie group of 3D rotations. Rotation about z-axis by an angle $\phi$ is represented in standard basis by this matrix: $$ \begin{pmatrix} \cos \phi & -\sin\phi & 0 \\ ...
1
vote
2answers
52 views

Proof - Inverse of linear function is linear

This is my first proof related to linear functions. It refers to the linear-algebra-$\textit{linear}$ (not the calculus-$\textit{linear}$). Please comment. Theorem The inverse of a linear bijection ...
2
votes
0answers
120 views

Is mathematics invented or discovered? [closed]

A google search yields millions of results, most of which are made by laymen who have nothing to do with math and it's "just another article" for the authors, so I assume here with so much passion in ...
4
votes
2answers
124 views

What is a geometric structure?

Every elementary book on abstract algebra usually begins with giving a definition of algebraic structures; generally speaking one or several functions on cartesian product of a point-set to the set. ...
1
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0answers
82 views

Any other mathematicians like Galois in recent history?

Any other mathematicians like Galois in recent history ? For "someone like Galois", I mean someone who developed a completely new theory all by himself, solved a big problem and the theory has big ...
-1
votes
2answers
47 views

Proof Techniques ( Soft Question )

I've been googling around for books of methods of mathematical proofing, and I haven't had much luck finding anything reputable in book form. I do recall running by a few in a university library ( I ...
9
votes
6answers
412 views

Reference request for undergraduate complex analysis.

I am a second year student studying electrical engineering. I self-study pure mathematics and want to pursue a career as a mathematician. What are some prerequisites for studying complex analysis? ...
2
votes
3answers
55 views

how to fairly select a leader

I recently came across a rather practical problem: A large group (around 30 people) wanted to elect a new leader (someone who is not part of the group) of 4 possible candidates. Each of the ...
18
votes
6answers
2k views

What is the motivation for quaternions?

I know imaginary numbers solve $x^2 +1=0$, but what is the motivation for quaternions?
0
votes
2answers
55 views

Problem book for abstract linear algebra

Kindly suggest a good book for abstract linear algebra other than finite dimensional vector space by P R Halmos