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1
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1answer
26 views

Is convexity the most general dividing line between “easy” and “hard” optimization problems?

Just got started with Boyd's Convex Optimization. It's great stuff and I see how it directly subsumes the all-important linear programming class of models. However, it seems that if a problem is non-...
0
votes
3answers
177 views

Are there important situations where we study false statements as if they were true?

I know of two situations resulting from asserting that a false mathematical statement is true (by this we assume that the statement has been made to be a mathematical axiom and that it must be true ...
2
votes
0answers
72 views

Big Rudin directly after baby rudin?

I'm a high school student who went through Rudin's Principles of Mathematical analysis a while ago in its entirety, except for the last two chapters. I bought Real and Complex Analysis too, and ...
3
votes
1answer
125 views

Are there any modern mathematicians whose research interest is in “Probability Theory”? [closed]

I have seen professors in universities list "stochastic calculus", "stochastic analysis", "stochastic processes", "stochastic geometry" and "applied probability" as research interests, but are there ...
3
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0answers
59 views

What will happen if evolve metric under Ricci flow on general manifold? [closed]

Because the scalar curvature under Ricci flow evolve by $$ \partial_t R=\Delta R+ 2|Ric|^2 $$ I treat it as heat equation with heat source . So, no matter the heat source is very hot or not , the ...
3
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0answers
66 views

Monads in monoids

This question is almost a duplicate of this one, but not quite. There the person asked about examples and intuition, I am asking about terminology and applications, and I am addressing my question ...
2
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0answers
95 views

“admissible” maps from context

I have been reading Massey's Algebraic Topology and on page 158 came across the following "semi-mystical principle" which he says guides much mathematical research: Whenever we wish to gain ...
-3
votes
2answers
53 views

Redundant proof in Math paper [on hold]

Recently, I read a published math paper and I found that in the excessive argument in the proof of one of its theorem. In fact, in my opinion, the redundant part is not even correct, because it ...
7
votes
1answer
327 views

Corollaries of the Yoneda Lemma in Analysis?

I am looking for some simple examples of how the Yoneda Lemma can be applied in analysis and probability theory and related fields. A simple candidate example that I can think of and somewhat ...
4
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4answers
217 views

Is any real-valued function in physics somehow continuous?

Consider the following well-known function: $$ \operatorname{sinc}(x) = \begin{cases} \sin(x)/x & \text{for } x \ne 0 \\ 1 & \text{for } x =0 \end{cases} $$ In physics, the sinc function has ...
1
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0answers
59 views

What is the Star of David theorem?

I came across a MathWorld entry for the Star of David Theorem, but it doesn't provide much context. I have never heard of this before, can somebody explain its significance and any applications it ...
0
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0answers
59 views

Next book in in learning Analytic Number Theory

I have just finished the book "Tom M. Apostol - Introduction to Analytic Number Theory". My aim is to reach to graduate level to do research, especially on Rationality/Irrationality and Algebraic/...
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votes
2answers
61 views

Is every finite category identifiable with a directed multigraph? (and vice versa?) [duplicate]

What seems implicit in this talk on youtube, is the claim that every directed multigraph (with loops) can be identified with a finite category and vice versa, if we consider the paths of the directed ...
6
votes
1answer
184 views

How to know if you are “tough enough” to study Algebraic Topology [closed]

I am graduating with a BA this summer and I am very interested in topology. I admit it, I never went that deep into topology and all I know is about point-set topology (metric spaces etc.) but from ...
3
votes
3answers
105 views

Is a pure mathematics degree worth it from a financial standpoint? [closed]

I know this isn't a math question but this has been on my mind for quite some time. I am a second year university student who is planning on getting a degree in pure mathematics. I really enjoy the ...
11
votes
1answer
171 views

Lonely theorems [closed]

What are some instances of theorems which are especially unique in mathematics, i.e. for which there are not many other theorems of a similar character? An example I have in mind is Pick's theorem, ...
-2
votes
1answer
23 views

Maximum data storage in a paper sheet? which theory should I look for?

I'd love to study a problem: How much information can be stored in a blank paper sheet. with those considerations: "store in a sheet" means writte letters or numbers or equations, with a pen and a ...
20
votes
4answers
862 views

How do I tell if I am able to go to graduate school in math? [closed]

This is my first question on this site, and this question may sound disturbing. My apologies, but I truly need some advice on this. I am a sophomore math major at a fairly good math department (top ...
2
votes
0answers
118 views

Can Local Martingales be characterized only using their FV process and BM?

Prove or Disprove: A process $(X_t)_{t \ge 0}$ is a (continuous) local martingale if and only if it can be represented in the form: $$\int_0^t \xi dB = \large B_{\int_0^t \xi_s^2 ds} $$ where the ...
2
votes
1answer
33 views

How can I remember whether finite or countable cartesian product of countable set is countable

I always forget this result Is cartesian product of countable set countable under finite or countable cartesian products? Is there a good way to remember this? Like a proof sketch where the ...
0
votes
0answers
41 views

How can I study probability?

I want to have a deep understanding of probability. I've tried William Feller's first book on Probability, and E.T Jaynes' Probability theory - the logic of science (which is very different from most ...
2
votes
1answer
81 views

Are there any “default” properties which hold for almost all topological spaces in analysis?

What is the simplest/most general commonly used (type of) topological space in analysis? For instance, every example I can think of in analysis is first-countable. I don't think it can be metric ...
8
votes
2answers
119 views

Should a high schooler be concerned with the abstraction of mathematics? [closed]

I'm currently studying precalculus in high school and have no hands-on experience with advanced mathematics (calculus and beyond). Every time I learn something new, I feel the need to connect it with ...
4
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0answers
84 views

Topology on $\mathcal{C}(X,Y)$ to work with homotopy.

We know that the compact open topology on $\mathcal{C}(X,Y)$ is a good choice for topology on the set of continuous maps, but this seems really efficient, both naively and with respect to existence of ...
3
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0answers
54 views

Is the infinitesimal generator for Lie groups the same as the infinitesimal generator of a Markov semigroup?

Is the infinitesimal generator for Lie groups related to the infinitesimal generator of a Markov semigroup? Or are they totally different concepts? https://en.wikipedia.org/wiki/Lie_group#...
6
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0answers
143 views

Can we characterize all infinite Euclidean-domains having exactly one invertible element?

$\mathbb Z_2$ and $\mathbb Z_2[x]$ are two euclidean-domains having exactly one invertible element ; my question is ; Can we characterize all euclidean domains $D$ having exactly one invertible ...
2
votes
1answer
54 views

How to conceptually visualize the homotopy map?

I hope to be clear in my question, I've been meditating on the definition of Homotopy of two continuous maps and I've come to the following thought: This is the definition I'm adopting: let $f_0, f_1:...
0
votes
1answer
34 views

Book recommendation for engineer turning towards mathematics (Abstract Algebra)

I will be taking abstract algebra course in a month from now. I am first time taking and algebra course and will be sitting with math majors. Can someone suggest me a book suitable for me i.e for ...
1
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0answers
27 views

Is it possible to add up the accuracy rates of 2 predictors?

Weather channel 1 has a 65% accuracy rate of predicting tomorrows weather Weather channel 2 has a 59% accuracy rate of predicting tomorrows weather Is it somehow possible to take into account ...
12
votes
2answers
315 views

Can Path Connectedness be Defined without Using the Unit Interval?

Can path connectedness be defined without using the unit interval or more generally the real numbers? I.e., do we need Dedekind cuts or Cauchy convergence equivalence classes of the rational ...
3
votes
1answer
38 views

When the sheafification map is a presheaf epimorphism

Claim: Suppose $\mathcal{F}$ is a glueable presheaf on a paracompact hausdorff space. Then the sheafication map on global sections $\mathcal{F}(X) \to \tilde{\mathcal{F}}(X)$ is surjective. (Note ...
2
votes
0answers
61 views

Is it worth to be a self taught mathematician? [closed]

The title is self explanatory but I would like to be more precise: With self taught mathematician I mean whether you are enrolled on some college but you don't attend on class, and also whether you ...
2
votes
1answer
22 views

Sublinear functions on a Riemannian manifold

I would like to know if there is any notion of sublinear function or subadditive function for Riemannian manifolds. Thank you!
1
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1answer
31 views

Does there exist a kernel concept for Taylor expansions?

In the sense of kernel which one can use when weighing coefficients together in Discrete Fourier Transform (Dirichlet and Fejér kernels maybe most well known). Here are some examples of such kernels. ...
1
vote
2answers
90 views

When does circular reasoning go wrong?

Consider the following erroneous usage of L'hopital's rule: $$\lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{D_h(f(x+h) - f(x))}{D_h(h)} = \lim_{h \to 0} \frac{f'(x+h)}{1} = f'(x) \...
3
votes
1answer
51 views

What is the difference between operator theory and functional analysis?

In my undergrad mind they are the same subject because functional analysis studies functional spaces like Banach and Hilbert spaces. Operators are function, so shouldn't they be the same subject? ...
0
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2answers
79 views

Need Guidance for engineer taking rigorous analysis course for first time

I will be taking analysis course in a month from now. Topics are given below. I am doing engineering and had been through calculus courses but nothing like sort of analysis before. Many of my friends ...
5
votes
6answers
630 views

How to Axiomize the Notion of “Continuous Space”?

EDIT (to clear up controversy and misunderstandings caused by my poor wording): Historically, Riesz's efforts to try and make rigorous a notion of a "continuous space" (as opposed to "discrete ones") ...
4
votes
1answer
174 views

Deceptively simple math conjectures [closed]

Why is it that some mathematical problems with seemingly simple statements end up soliciting extremely complicated and groundbreaking proofs or remain unsolved for extended periods of time? (Ex. ...
0
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0answers
62 views

When is it true that $V \subseteq \overline V \subseteq U$ will hold for open sets?

Let $(X, \mathfrak{T})$ be a topological space Let $V, U \in \mathfrak{T}$, and suppose that $V \subseteq U$ Then when it is true that $V \subseteq \overline V \subseteq U$, where $\overline V$ is ...
3
votes
2answers
140 views

Is the class of imaginable objects which cannot exist a worthwhile thing to talk about?

Clearly, in mathematics, there are certain objects which we can imagine, yet which have no real meaning, or do not exist. The real number whose square is negative, the quantity represented by ${1 \...
1
vote
2answers
69 views

Self Study of number theory

I have always wanted to learn about number theory. There is actually no one here who can teach me and it's also not in my regular school syllabus, but the greatness of number theory attracts me ...
4
votes
1answer
53 views

Predicting next headache

I am thinking of making a model or whatever name i don't know. Idea is this that I suffers from headache, say twice in a month. What i want to do is that i want to make some kind of system in which i ...
0
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0answers
27 views

The best known bounds for spectral radius of a graph

There are many bounds for the spectral radius of graphs in terms of no. of vertices, maximum degree, chromatic number etc. I wish to know till date what are the best lower and upper bound for the ...
0
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0answers
34 views

What are some other operators like infinite sums and products? [duplicate]

I've heard of the sigma and capital pi notations (hasn't everyone?), but I know there are some other ones, like the ones signified E, F and K. What are these? Are there any more related to infinite ...
0
votes
1answer
33 views

Recommendation in orthogonal polynomials

I am searching for a good book in orthogonal polynomials (for good I mean detailed and with all the basic results one has to know before doing some research in that branch ) for beginners ( nothing at ...
4
votes
2answers
88 views

On the HoTT Cauchy Reals

In the Homotopy Type Theory Book there is a construction given of a kind of Cauchy reals via higher inductive type and the authors remarked, that this construction is preferred to other notions (reals ...
17
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4answers
651 views

Tough integrals that can be easily beaten by using simple techniques

This question is just idle curiosity. Today I find that an integral problem can be easily evaluated by using simple techniques like my answer to evaluate \begin{equation} \int_0^{\pi/2}\frac{\cos{x}}{...
1
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0answers
90 views

Why are symplectic manifolds and Riemannian manifolds so different?

They seem superficially similar in some ways 1) both have a given two tensor on the tangent space 2) if we choose a hamiltonian function for the symplectic manifold then both give a canonical ...
4
votes
3answers
221 views

Why study non-T1 topological spaces?

I can understand (somewhat) why one would want to study non-Hausdorff topologies, since for example the Zariski topology is so important to algebraists, and the weak topology generated by lower ...