This tag is for questions where the poster seeks references (books, articles, etc.) about a particular subject. Please do not use this as the only tag for a question.

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17 views

Some terminology and reference questions on singular values

Let $T: V \rightarrow W$ be an operator between to inner product spaces. Then singular values $s_1 \leq s_2 .... \leq s_n$ of $T$ are square roots of eigenvalues of $T^*T$ where $T^*$ is the conjugate ...
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0answers
18 views

branched cover along a closed curve in the $3$-sphere

Let $c$ be a closed embedded smooth curve in the $3$-sphere $\mathbb S^3$. I was told that $\mathbb S^3$ admits a two fold branched cover $X(c)$, branched along $c$, which corresponds to the ...
7
votes
2answers
172 views

A planar Brownian motion has area zero

I'm looking for proofs of Paul Lévy's theorem that a planar Brownian motion has Lebesgue measure $0$. I know of only two proofs: one is in Lévy's original paper (Théorème 12, p. 532) and the other is ...
3
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4answers
495 views

Objects are finite sets, arrows are matrices. How is this a category?

I just started to read this book on category theory. How is this example below a category? I have difficulty imagining what this construct really is. Could someone please illuminate me ? I ...
5
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1answer
105 views

Is it possible to extend a $C^1$-function smoothly from any Lipschitz domain?

If $\Omega$ is a cube in $\mathbb{R}^n$ and $f\in C^1(\overline\Omega)$. By reflection one can extend such a function to all of $\mathbb{R}^n$ and the extenstion is in $C^1(\mathbb{R}^n)$. If ...
3
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0answers
37 views

Logic in closed symmetric monoidal categories; reference request.

Suppose we want an algebraic theory $T$ to be interpretable in any closed symmetric monoidal category $\mathbf{C}.$ I am thinking in particular of the case where $\mathbf{C}$ is the category of models ...
1
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2answers
121 views

Why do people stick with Riemann-Integration when dealing with differential geometry?

I asked a question yesterday that is, "Is there an introductory differential geometry text using Lebesgue integration?" Then, i got an answer that "since we are dealing with differential geometry we ...
2
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1answer
19 views

how to prove that this weak solution is subharmonic?

My question is about this article http://hal.inria.fr/docs/00/12/87/60/PDF/fbpLaplacian.pdf. My question is : Consider a smooth, bounded and convex domain $K$ in $R^n$ such that $K\subset \{ x_1 = ...
3
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0answers
59 views

Grothendieck's manuscript on differential manifolds

I have a Japanese book on Grothendieck's life and his mathematical works. The author writes that Grothendieck wrote manuscripts(over 250 pages) on "the category of manifolds" and "differential ...
3
votes
1answer
34 views

Integration by parts in Bochner Lebesgue spaces.

Does there exist an analogous of integration by parts for expressions such as: $$\int_0^T {\langle u(t),v(t) \rangle }\, \mathrm{d}t,$$ where $u,v\in L^2([0,T];H)$, for some Hilbert space $H$? If so, ...
3
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0answers
73 views

Mathematical YouTube channels?

So I'm wondering if anybody knows any good math/science related YouTube channels? As for the math channels, I'm currently subscribed to Numberphile, and that is about it. I know few other channels, ...
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0answers
51 views

is there an introductory differential geometry text using Lebesgue integration?

Is there an introductory differential geometry text using Lebesgue integration? Every differential geometry text I saw introduces the theory using Riemann integration. (Even Spivak) Would someone ...
0
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0answers
14 views

Recommendation for Order Theory texts

Order Theory, Lattice Theory, or any directly relatable subject is not taught at my university, and quick searches don't give much clue into good textbooks for the subjects. My question is: what are ...
2
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0answers
49 views

handle moves: proof

In several 4-manifold textbooks, when handle moves (creation, cancellation, sliding) are discussed, they are explained using very helpful drawings. However, I would like to know if there is a ...
2
votes
1answer
52 views

Solutions for $x^2+y^2=2007$ and reference request.

I have never formally studied number theory, it is not a part of my course work, and what I have learnt is reading Wikipedia or the answers here. This question was on a test and I tried to use ...
3
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1answer
77 views

Recommended books/articles for learning set theory

What is the recommended reading for thoroughly learning set theory? I'm currently studying Kunen's book [1]. But what then, and in what order? One needs to learn large cardinals, inner models and ...
2
votes
1answer
20 views

Is the von neumann algebra of locally compact amenable group hyperfinite?

Let $G$ be a discrete group and $\mathcal{L}(G)$ the associated von Neumann algebra. It is well known that $G$ is amenable if and only if $\mathcal{L}(G)$ is hyperfinite. Does there exist a ...
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0answers
24 views

Reference Request: Simplex dynamics

I want to know if there's any general method to investigate linear system restrained on standard simplex. It's hard for me to start with such a system because if I directly regard it as general ...
3
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0answers
26 views

Can the Milnor number be used to resolve curve singularities?

Let $f(x,y)\in \mathbb{C}[x,y]$ define a curve $C$ which is singular at the origin. By successively blowing-up the origin, we can resolve the singularities of $C$. Of course to make sure this process ...
0
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0answers
35 views

Finding the mathematical formula base on this C++ code

The code below is the solution to a ACM regional programming contest problem: Using credit cards for your purchases is convenient, but they have high interest rates if you do not pay your balance in ...
1
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1answer
30 views

Parabolic Cusp of an Action on the Upper Half Plane

This is a basic definition question. Parabolic bundles are used in certain counting arguments in my research area. I asked my advisor for a reference on these, and he directed me to the paper of Mehta ...
0
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0answers
30 views

Reference for power series

I would need some references for power series, Taylors series of elementary functions, derivation and integration of power series, convergence of sequences of functions and series of functions. The ...
6
votes
1answer
63 views

Gowers' proof of Szemerdi's theorem

Are there any good books or other resources (expository notes) which explains Gowers' proof of Szemerdi's theorem in detail?
0
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1answer
92 views

Reference request: Partition of unity…

I was looking for some material that could help me understand a real analysis course (1st year undergraduate). My teacher treated the following topics: Partition of unity Existence of regular ...
5
votes
2answers
64 views

Open problems involving p-adic numbers

I am in my final year of my undergraduate degree, and I'm doing a project on p-adic numbers, and in particular, trying to find Galois groups of simple extensions of $\mathbb{Q}_p$ (this is a Galois ...
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0answers
24 views

Is there a program like ALEKS for mathematical logic?

ALEKS (http://www.aleks.com/) is a good way of learning procedural math, because it is very systematic and forces you to master the dependencies of a kind of problem before working on that kind of ...
7
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1answer
108 views

Commutator subgroup - or?

If $G$ is a group and $X, Y \subseteq G$ then the commutator subgroup of $G$ is defined as $[G, G] = \langle [x, y] \mid x, y\in G \rangle$, where $[x, y] = x^{-1}y^{-1}xy$ and the group generated by ...
2
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0answers
50 views

What would be a good book to self study basic algebra?

I am asking about basic algebra so that I can tie it into learning about number theory and set theory. before I tackle Geometry, and College Algebra/Analytical Geometry. I will top it off with ...
0
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0answers
28 views

Regulairty of eigenfunctions of singular integral equations

Can you provide a proof or a reference, to study from, for the following problem: Let $\Gamma$ be a real analytic rectifiable closed curve in the plane, $ds$ is the arc-length , and kernel $K(z,w)$ ...
1
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1answer
33 views

Poincare lemma on current

The Poincare lemma on current states that: If $U$ is a star-shaped open set in $\mathbb R^n$ and $T$ is a $k$-current on $U$ such that $dT=0$, then there is a $k-1$-current $S$ on $U$ such that $dS = ...
2
votes
2answers
56 views

Historic proof of the area of a circle

The area of a circle radius $R$ is $\pi R^2$ which is quite easy to prove with integral calculus. Consider a ring of radius $\mathrm{d}r$ at a distance $r$ from the centre. This ring has area $2\pi r ...
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5answers
65 views

A book to inspire people about math

I'm looking for a book for someone who only knows high-school level math that will show him what math really is and how amazing it is. Do you know a book that could do that?
1
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1answer
27 views

Introduction to functional interpretations

Any good recomendations for an introduction to functional interpretations? I understand this is a little vague but i haven't had much contact with the area. I am particularly interested in the ...
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0answers
23 views

Learning resources for Probability Distributions/Models

I've a good background in basic probability. I need to learn and get a good grip on the probability distributions and stochastic processes, counting processes, and other related topics. I am already ...
2
votes
1answer
62 views

Mumford-Oda - Algebraic Geometry II . There will be a complete book?

Online there is the draft of a book written by Mumford and Oda that should be the continuation of "Algebraix Geometry I complex projetive varieties" (Mumford,1976). Do you know if and when this book ...
0
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0answers
28 views

The role of verifiable computing in the formalization of mathematics

I've been thinking about this for a while, and it seems to me that mathematics "works" because (in principle) we can to check proofs very quickly, even though the discovery of that proof may have ...
1
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1answer
49 views

Which of these things is not like the others?

What's in a name? Well quite a lot, if you're confused enough. I have an engineering-style mathematics education, based on good old hand waving and learning bits and pieces from all over the place. I ...
3
votes
1answer
57 views

How does integration over $\delta^{(n)}(x)$ work?

For a math paper I need to be able to evaluate $\int_{-a}^{a}\delta^{(n)}(x)\ f(x)\ dx$ for differentiable $f$. I know that it is 'supposed' to equal $(-1)^nf^{(n)}(0)$: $$\int_{-a}^a\delta^{(n)}(x)\ ...
2
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1answer
26 views

Reference Request - Series Solutions to Differential Equations

I am looking for a text that gives a good exposition of power series solutions to second order equations with variable coefficients. My course I'm guessing focuses mainly on this section. My knowledge ...
2
votes
1answer
128 views

Where does this probability problem come from?

A long time ago, a friend gave me a probability problem. Here is rough reconstruction. A spaceship is lost in deep ($3$-d) space. Its home planet is $X$ meters away. Every second, the spaceship ...
2
votes
2answers
30 views

Various “sizes” of 0-measured sets

I am looking for a formalization of an intuitive concept of size, in cases simple measure is too coarse. It will be easier for me to give an example. Let $\mu$ be the Lebesgue measure on the unit ...
0
votes
1answer
41 views

A question about an asymptotic formula

I've been told that the asymptotic formula $\pi(x+y)-\pi(x)\sim y/\ln x$ holds for $y\ge x^{1/2+\varepsilon}$ if Riemann's hypothesis is true, but I was unable to find a journal reference for this. ...
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3answers
94 views

University-level books focusing on intuition?

I help some students with difficulties in Mathematics and Physics (especially math, physics, and engineering majors). While in high school they usually don't study, or are not interested, etc., in ...
2
votes
1answer
27 views

Primes between consecutive cubes

I am looking at Dudley's proof of the existence of Mill's constant. It starts out as follows The proof depends on the following theorem: there is an integer $A$ such that if $n>A$, then there ...
0
votes
4answers
57 views

Rigour and Formal Reference for Ergodic Theory

I am not even a beginner to Ergodic Theory, but I want to start to read about it. I am coming from a math background and for me its quite important that the definitions to be stated and the formalism ...
0
votes
0answers
31 views

Intuition analysis-deconstruction-reconstruction.

The following question is a refinement of this question, which caused a lot of people to give answers that were missing the point entirely, probably because the question was not clear. Being human, ...
6
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1answer
61 views

Cohomological Whitehead theorem

Let $X$ and $Y$ be CW complexes (resp. Kan complexes) and let $f : X \to Y$ be a continuous map (resp. morphism of simplicial sets). The following seems to be a folklore result: Theorem. The ...
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0answers
26 views

A game of repeatedly taking the union of sets

Suppose that I have the set family $\mathcal{A}_1 = \{A_1, \dots, A_k\}$. We play the following game, which consists of a sequence of iterations. In the first iteration, we choose an arbitrary ...
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2answers
33 views

Looking for good literature on Markov Chains with explicit calculations

I am currently starting my thesis on Markov Chains and am looking for good books and papers that include explicit calculations. I have taken a small course on Markov Chains so the subject is not ...
6
votes
3answers
171 views

Categorical introduction to Algebra and Topology

I am self-studying Mathematics in my free time. At the moment I am reading books on Algebra and on Category theory. More exactly, I started working through the book $\textit{Algebra}$ by Serge Lang. I ...