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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.
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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 ...
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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
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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
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2answers
65 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 ...
1
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2answers
61 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 ...
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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 ...
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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 ...
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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 ...
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3answers
345 views

Applications of model theory to analysis

Some of the more organic theories considered in model theory (other than set theory, which, from what I've seen, seems to be quite distinct from "mainstream" model theory) are those which arise from ...
8
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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, ...
8
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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 ...
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2answers
71 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 ...
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1answer
128 views

Suggestion about Algebraic Topology talk

following the content of the title I am writing here to ask some suggestions concerning a talk I will be presenting at my university in a week or two. The main topic I chose is the fundamental ...
5
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1answer
220 views

What kind of matrices are non-diagonalizable?

I'm trying to build an intuitive geometric picture about diagonalization. Let me show what I got so far. Eigenvector of some linear operator signifies a direction in which operator just ''works'' ...
3
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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 ...
8
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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 ...
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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 ...
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2answers
1k views

Simplest examples of real world situations that can be elegantly represented with complex numbers

Mathematics could be defined as the study of formally defined abstractions. These abstractions may or may not be useful for describing real world phenomenon. Indeed, Physics could be defined as the ...
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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
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0answers
261 views

The status of $\mathbb{R}$ in homotopy theory.

The definition of a path as a continuous map $I \rightarrow X$ is a completely natural one. But this raises two questions in my mind. First, what properties of the interval give rise to useful ...
3
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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 ...
18
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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 ...
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1answer
39 views

Prerequisites for Hilbert Cohn-Vossen's Geometry and the Imagination?

I've not read this book(not really),but I would like to know how much is assumed by the reader. can I recommend this to the layperson? Also ,any more recent similar books? I already know of Courant ...
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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 ...
2
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1answer
82 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 ...
6
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2answers
163 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 ...
91
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10answers
7k views

Are mathematical articles on Wikipedia reliable?

I know that Wikipedia gets a bad rap, and it seems like some teachers of mine have nothing better to do in class than harp on about the Great Academic Pastime of calling Wikipedia untrustworthy, but ...
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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 ...
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7answers
543 views

Is it possible to alternate the law of mathematics?

I am freelance writer. Recently I have been planning a science fiction - just planning, nothing solid yet - and I was wondering would it be possible for some other universes that have different set of ...
2
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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 ...
2
votes
2answers
80 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 ...
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0answers
29 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 ...
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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 ...
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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 ...
4
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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. ...
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4answers
256 views

Confused about differentiation

I'm new to calculus and have been taught that $\displaystyle \frac{dy}{dx}$ is the rate of change of y with respect to x. Does $\displaystyle \frac{dy}{dx}$ show how much the variable y changes as x ...
627
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25answers
43k views

Can I use my powers for good?

I hesitate to ask this question, but I read a lot of the career advice from mathOverflow and math.stackexchange, and I couldn't find anything similar. Four years after the PhD, I am pretty sure that ...
1
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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 ...
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6answers
414 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? ...
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 \\ ...
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1answer
418 views

Linear algebra and Multivariable calculus prerequisites for Stochastic Calculus

Which topics are considered "graduate-level" for the following subjects: Linear algebra Multivariable calculus On Internet, it is said that you need "graduate level" Linear algebra and ...
2
votes
1answer
257 views

Is it worth pursuing a statistics minor? I want to go to pure math grad school.

My school offers a minor in "quantitative methods" from the psychology department. I've taken one class from this minor (Intro to Stats) and will most likely get an A. That said, the classes in this ...
2
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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 ...
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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 ...
4
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3answers
124 views

(translated)Russian mathematics books?

Most russian mathematician(generally) are known to do and teach mathematics in a very original manner,they do in a very intuitive yet rigorous way, with/through wonderful connection to physics. ...
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2answers
158 views

Theorems with one-line proofs [closed]

Inspired by this very concise answer, which proves that $$\sin^2(\theta)+\cos^2(\theta) \equiv 1 $$ as follows: $f(\theta)=\cos^2\theta+\sin^2\theta \quad;$ then it's simple to see that ...
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2answers
57 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
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2answers
303 views

Largest number that can be written using N characters

I remember reading in RPF's biographical book, "Surely you're joking Mr. Feynman" that he used to have timed contest about writing down the biggest number using standard symbols. Instead of the time ...