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1
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1answer
42 views

The magic of the morphisms

Given a set $X$. Let $S\subseteq X$ and consider $(X,S)$ as a very simple mathematical structure, lets call it a spotted set. Given two spotted sets, then a morphism $\alpha ...
1
vote
1answer
45 views

How does this picture called?

Some time ago I saw this in my teacher's room. She called this picture in honor of some scientists (Lagrange,Lie or Liouville, or some other, but I don't remember). Please, name this picture. Thank ...
0
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0answers
11 views

How to define and make the dot product of two continuous matrix?

I was thinking recently that i always learn algebra with discret basis. But in case where the basis is continuous, how can i define a continuous matrix and when it is define how can i do the dot ...
0
votes
0answers
13 views

Can we attach a space with discrete signal?

This question refers to the link https://en.wikipedia.org/wiki/Space_(mathematics) and https://en.wikipedia.org/wiki/Discrete-time_signal. My question is how can we associate a discrete signal with a ...
21
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13answers
307 views

What are some surprising appearances of $e$?

I recently came across the following beautiful and seemingly out-of-the-blue appearance of $e$: $E[\xi]=e$, where $\xi$ is a random variable that is defined as follows. It's the minimum number of ...
0
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1answer
54 views

The role of visualization and intuition in graduate and postgraduate math and developing it

In Visual Complex Analysis's preface, the author gives an analogy with pseudo-deaf musicians and follows the same to mathematics. Mathmatics today, he argues, is mostly build on abstract symbolic ...
-7
votes
0answers
57 views

Can a picture be it's own pie chart? [on hold]

Wow! can vote to close or vote down , but cant say it is correct or incorrect? So this image is going around, but some how I think the visual ratio of what is in the picture is not translated 1-1 to ...
10
votes
1answer
70 views

What is the main purpose of learning about different spaces, like Hilbert, Banach, etc?

I just started to learn about functional analysis and have started to learn about various spaces, like $L^{p}$, Banach, and Hilbert spaces. However, right now my understanding is rather mechanical. ...
2
votes
2answers
35 views

Unsolved problems in graph theory

Is there a good database of unsolved problems in graph theory?
1
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0answers
48 views

How to wite a Statement of Purpose for a Summer Program in Representation Theory. [on hold]

I want to attend a summer research program in Representation theory,$\;$for that I need to write a statement of purpose or simply a write up, so I want to know prerequisites for this course, and what ...
0
votes
2answers
68 views

Non-abelian fundamental group on a path-connected space

I am doing a self-study of algebraic topology, and am having some difficulties comprehending the idea of a non-abelian fundamental group on a path connected space. (See for example Hatcher Exercise ...
1
vote
2answers
57 views

Memorizing Formulas for Differentiation

Once upon a time, I memorized the following formula out of laziness. Let $k(x)=\frac{f(x)^{g(x)}h(x)+i(x)}{j(x)}$. Then $k'(x)$ is as follows. ...
1
vote
0answers
80 views

Do math experts personally care that much about how mathematics is interpreted philosophically? (Platonism vs. formalism, for example) [on hold]

Just wondering about how professional mathematicians feel about philosophically about mathematics: whether philosophy of math matters to them, what their personal views are, etc. I had a few teachers ...
0
votes
5answers
172 views

Why are the symbols of operations written on the left or right of the objects to which they apply? [on hold]

I was wondering why operations, actions and other stuff in mathematics are always defined "on the right" or "on the left". Is that a reflex of our (western) way of writing? For example, japanese is ...
1
vote
1answer
35 views

Is there a measure which allows me to tell how closely something is to an ellipse?

Roundness is the measure of how closely the shape of an object approaches that of a circle. I am trying to find a similar measure which shows how closely is something to an ellipse. Is there any ...
3
votes
1answer
44 views

What Sort of Discovery Warrants Writing a Paper

I am a high school student who is deeply passionate about mathematics and I have written many different mathematical proofs. I was wondering what sort of discovery warrants writing a mathematical ...
0
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0answers
27 views

Learning outcomes of reading textbooks [on hold]

So I've densely mined this site for the scholarly materials I need most. And have prudently written many of them down, so I can sort them if I see fit. But, are the books always the preferred method ...
-6
votes
1answer
114 views

Why do we teach Calculus in High School instead of, say, programming? [on hold]

I was wondering "Why do we teach Calculus in High School instead of programming?" 'Calculus' only goes up to about partial derivatives, then its called different things like real analysis and other ...
0
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0answers
41 views

Record-holding mathematical proofs [on hold]

Which mathematical theorems admit proofs that are extreme in some sense? Here is what I have in mind: The classification theorem for finite simple groups is the longest proof mathematics has seen. ...
13
votes
4answers
726 views

How to define “being inside of something” in the context of topology?

I'm a Psychologist and Neuroscientist with interest in math and I just started reading about Topology. I have to say it's not easy to grasp the concepts without a practical example, so I'm trying to ...
0
votes
0answers
5 views

Biorthogonal (discrete wavelet) noise bases?

I am slightly interested in discrete wavelet transforms (DWT), but so far I have mostly used already-derived and existing well-known wavelets, such as Daubechies, Cohen-Daubechies-Faveau, Symlets and ...
-2
votes
0answers
60 views

Why are math textbooks that are considered good books so hard to read? Why do authors make their books difficult to read? [on hold]

I've noticed that many books that are difficult to read are considered some of the best. Why does hard to read indicate that it is rigorous? For example: Rudin, Apostol, Lang, Hungerford, Ahlfors, ...
2
votes
0answers
38 views

Why is a projective variety 'the best kind'?

In Hartshorne's AG, he discusses the classification of curves by birational equivalence class says 'based on the idea that a nonsingular projective variety is the best kind..'. What exactly makes a ...
1
vote
0answers
18 views

Infinite series, continued fractions, nested radicals and such - is there a general recursive algorithm theory?

The nature of infinity and irrational numbers (among other things) always gave me trouble. But recently I learned to think about infinite sequences in terms of recursive algorithms and their time ...
0
votes
0answers
32 views

Translation of a book

I am an undergrad studend, my professor told me that a good work I can do is to translate a book on universal algebra (subject which I like a lot) in my language. I'm asking if this kind of work is ...
0
votes
0answers
39 views

how to mathematically represent a matrix of vectors?

My problem is the following: I have a dataset in particular have $4$ dimensions, for didactic reasons I need to represent this dataset as a $m\times n$ matrix array such that the ($i$-th, $j$-th) ...
5
votes
1answer
991 views

Which is the most powerful language, set theory or category theory? [on hold]

As far as I know, mathematics is written based on a language which can be for example set theory or category theory. My concern is about the power of these languages. How can we realize which language ...
1
vote
0answers
26 views

Dimension of a space VS dimension of a function in this space [closed]

A colleague and I ran into a problem when we realised that we had a complete different understanding of dimensions. If we consider this function: $z(x,y)=x^2 + y^2$ Person A believes this function ...
2
votes
1answer
64 views

Possible to do well in Algebra without loving Analysis much? [closed]

Having taken some courses in higher algebra, I realized that what I truly appreciate in mathematics is abstract algebra. But it also appears that I'm not a big fan of real analysis [at least I don't ...
0
votes
1answer
29 views

Is This Mathematical Induction?

Mathematical induction Follows Thus: $1.$ The basis (base case): prove that the statement holds for the first natural number $n$. Usually, $n = 0$ or $n = 1$. $2.$ The inductive step: prove that, ...
5
votes
1answer
232 views

Hillary Clinton's Iowa Caucus Coin Toss Wins and Bayesian Inference

In yesterday's Iowa Caucus, Hillary Clinton beat Bernie Sanders in six out of six tied counties by a coin-toss*. I believe we would have heard the uproar about it by now if this was somehow rigged in ...
5
votes
1answer
124 views

Why is $s$ used for arc length?

Why is $s$ used for arc length? I looked around online, but I can't find a definite answer. Thank you!
5
votes
2answers
57 views

Difference between a limit and accumulation point?

What is the exact difference between a limit point and an accumulation point? An accumulation point of a set is a point, every neighborhood of which has infinitely many points of the set. ...
1
vote
2answers
48 views

Where do I have to use Chain Rule of differentiation?

I have come across many examples of chain rule of differentiation while studying physics (eg. finding velocity of SHM,differentiating Kinetic Energy with respect to time etc.).But,I feel I lack the ...
0
votes
0answers
16 views

Research experience/summer schools for UK undergraduates.

I'm an undergraduate student in my 2nd year in UK who has a strong interest in entering the academia in the future, and I have been planning for my future career since the start of this year. I ...
9
votes
1answer
64 views

Categorification of algebra structures

This might be a bit of a soft question. Take a $\mathbb{C}$-linear category. Form the complex vector space spanned by its objecs modulo exact sequences. This construction is, as far as I know, the ...
32
votes
9answers
2k views

Definition of “well defined” in mathematics

I have encountered this term "well defined" in many places in maths like well-defined set, well-defined function, well-defined group, etc. What are the contexts in which we can talk about well ...
3
votes
1answer
47 views

The Mathematics of Finite State Automata

I am a final year undergraduate mathematics student preparing to undertake my BSc-HONS project, provisionally titled for the time being, "Finite State Automata and Regular Languages". Having had a ...
1
vote
2answers
44 views

What are some interesting uses for/motivations of projective spaces?

I have trouble motivating myself to think about real projective spaces, for instance. Are there any cool results about them? Are there any motivating examples?
-1
votes
3answers
45 views

Textbooks for Groups & Rings [closed]

Please I need suggestions on the best textbooks to help me comprehend this Groups and Rings and relate it with the rudimentary aspect of Set Theory
1
vote
0answers
37 views

Why graph factorization problems emphasize $k$-regular graphs

When we study graph factorization a lot of emphasis is placed on $k$-factorable graphs, i.e. graphs in which the factors are all $k$-regular. Why is such a factorization more important then say ...
1
vote
1answer
98 views

What to do as a teenager interested in mathematics? [closed]

Suppose that I am a teenager going to school who is interested in mathematics. Suppose that I don't want to learn mathematics as it is taught at the university because later, I have enough time for ...
4
votes
0answers
48 views

Riemann Surface Tennis [closed]

So a few months ago I read Brave New World by Aldous Huxley and found the authors insights and style of humor remarkable. It describes a utopian society where people are conditioned to be enslaved by ...
1
vote
0answers
41 views

Ideas for approaching set theory when you've already studied higher abstractions?

I've come to acknowledge (or so I think). That many of the concepts e.g. in real analysis (like, say, continuity), actually don't (in modern times) boil down just real analysis. But rather, e.g. set ...
0
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0answers
3 views

Convergence in constraint propagation

Introduction To the best of my knowledge, constraint propagation can be thought of (in a very heuristic sense) as a class of algorithms that solve a sort of generalized Sudoku problem. Some initial ...
0
votes
0answers
18 views

When studying 2D gabor functions why is a gaussian called elliptical?

Consider $$G(x,y)=\frac{1}{2\pi\sigma\beta}e^{-\pi\left[\frac{(x-x_0)^2}{\sigma^2}+\frac{(y-y_0)^2}{\beta^2}\right]}e^{i[\xi_0x+\nu_0y]}.$$ This is the product of a complex plane wave and what this ...
3
votes
1answer
26 views

Why is Logistic Distribution called logistic?

What is logistic about Logistic Distribution, in a common sense way? What is the lexical rationale of the name, not just pure math definition?
1
vote
0answers
23 views

Where does the name of the hypergeometric distribution come from?

I understand what it does and how to get there, but why is it called hypergeometric? All the other distributions I know of have rather self-explanatory names like "binomial" or "exponential", or are ...
0
votes
0answers
8 views

Books for large deviations and maximum entropy principle?

What are some good introductory resources for learning more about the theory of large deviations and the maximum entropy principle?
2
votes
2answers
46 views

Rigorous Probability/Statistics Book reference?

Im wondering if anyone could recommend a book (or a few books) about statistics/probability for someone at the advanced undergraduate level who has taken some real analysis (at the level of baby ...