Questions tagged [reference-request]

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

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Functions invariant to the permutations of a tuple whose coordinates are probability measures

I have a sequence $(A_t)$ indexed by time $t$. Each $A_t$ is a collection of such elements $(a_{t,k}, A_{t,k})$, i.e., $$A_t = \{(a_{t,1}, A_{t,1}), \ldots, (a_{t,K}, A_{t,K})\}.$$ For all $t$, we ...
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Reference request: the UEA of the LR-algebra of tangent vector fields on a smooth manifold coincides with the derivation ring and the ring of diff ops

Let $\mathcal{M}$ be a smooth real manifold and let $A:= \mathcal{C}\left(\mathcal{M}\right)$ be the real algebra of smooth functions on $\mathcal{M}$. Recall from McConnell, Robson, Noncommutative ...
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15 views

Book recommendations for the mathematics in epidemiology

I would like to educate myself in a proper way about the terminology, concepts and mathematics of epidemiology. So instead of scrambling through separate articles on various websites, I would like to ...
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12 views

Convergence of eigenvalues of random matrix $\mathcal{N}(0, \sigma_n^2)$ with $\lim\limits_{n\to\infty} \sigma_n^2 = 0$

Consider a sequence of random matrices $\{A_n\}_{n\geq 1}$ of dimension $n\times n$ whose entries are i.i.d. Gaussian $\mathcal{N}(0, \sigma_n^2)$ with variance tending to zero, i.e. such that $\lim\...
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Boolean algebras of size equal to non-regular cardinals

In the paper New Proof of a Theorem of Gaifman and Hales by Solovay, he gives a constructive proof of complete Boolean algebras of size larger than $\aleph_\tau$ for any ordinal $\tau$ which are also ...
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(Reference request) Smoothness of desingularized polyharmonic Green kernel on closed manifolds

Let $(M^n,g)$ be a closed Riemannian manifold with Laplace–Beltrami $\Delta_g$. It is known that, for $s>0$, the resolvent $\mathcal G_s$ of $(-\Delta_g)^s$ has integral kernel, say $G_s$, ...
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Geometrical introduction to Caccioppoli sets

Can anyone suggest me a geometrical introduction to Caccioppoli sets, a.k.a sets of finite perimeter. This Encyclopedia of Mathematics article states a definition of the perimeter of a set in terms of ...
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15 views

Name of $\inf_\hat{x} \text{rank}(J(P(\hat{x})))$

Let $P(\bar{x}):\mathbb{C}^n\to\mathbb{C}^n$ be some polynomial map, and let $J_P(\bar{x}) = \left[\begin{smallmatrix}\nabla^T P_1(\bar{x})\\\vdots\\\nabla^T P_n(\bar{x})\end{smallmatrix}\right]$ be ...
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Error of Hermite interpolation

I'm studying Hermite interpolation and I want to know if there is a way of expresing the error. I'm familiar with the result that if $x_0, x_1, \dots x_n$ are distinct points and consider the ...
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51 views

Small abelian categories and module categories.

https://en.wikipedia.org/wiki/Grothendieck%27s_T%C3%B4hoku_paper Wikipedia: "The article "Sur quelques points d'algèbre homologique" by Alexander Grothendieck, now often referred to as ...
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In game theory, the solution is invariant under what changes to the payoff matrices?

Suppose we have a static normal form game described by a collection of payoff matrices $\{A_n\}$. Most commonly, there are two payoff matrices $A,B$, obtained from a payoff table. Does anyone know ...
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What do you mean by bifurcation of the following equations/systems?

I am new to the domain of bifurcation. I encounter the following problems: Analyze the bifurcation of the following equations/systems. Identify the bifurcation values(s), describe the bifurcation and ...
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50 views

Infintely many $n$ such that $an+1$, $bn+1$, $cn+1$ are primes?

Let $a, b, c$ be natural numbers. Is it known that there exist infinitely many $n$ such that $an+1$, $bn+1$, $cn+1$ are primes? or is this problem out of reach of the current knowledge regarding prime ...
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Conditions for a family of vector fields to be bracket generating

In nonlinear control one considers vector fields of the form $$ F(x,u_t)=\dot{x} $$ where $u$ is some space of admissible controls and $F$ is a nonlinear function. Assume that we are on some smooth ...
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A modularity measure for graphs

I wonder if the following number is a good measure for the modularity of a graph, and in which contexts it is possibly used (and under which name). For now let me call it the community index of a ...
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How to prove the existence of weak solutions of parabolic PDEs using Rothe's method?

I am reading a textbook on PDE (Introduction to Elliptic and Parabolic Partial Differential Equations by Z. Wu, J. Yin and C. Wang, written in Chinese) where a proof of the existence of weak solutions ...
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Reference request: characteristic numbers of manifolds

I am wondering if there is a good resource out there that computes characteristic numbers of common manifolds. For example, the $\hat{A}$-genus of the tori $T^n$ or spheres $S^n$ for various $n$. It ...
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“Standard” pairs and group actions

In Aschbacher's paper about symmetries of symmetric block designs, there is mention of "standard $(G,\pi)$ pairs" and "standard permutation groups": If $G$ is a permutation group ...
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Bracewell - The Fourier Transform and its Applications [closed]

Anybody has an errata for "The Fourier Transform and its Applications" from Ronald Bracewell?
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1answer
23 views

Does these convergences hold true? (In unbounded domains, too?)

Let $\Omega$ be an open bounded subset of $\mathbb{R}^N$, $n\ge 2$. Let $(u_n)_n\subset W_0^{1, p}(\Omega)$ be a bounded sequence in $W_0^{1, p}(\Omega)$. It implies, in general, that $$u_n\...
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32 views

Which of the following is not the correct quadratic equation?

If $3/2$ and $4$ are the two roots of a quadratic equation, then which one of the following is not the correct quadratic equation? (A) $2x^2-11x+6=0$, (B) $6x^2-33x+18=0$, (C) $-10x^2-55x-30=0$, (D) $...
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53 views

What is the name of this theorem in euclidean geometry

the theorem is this: Given the $\triangle ABC$; let $O$ be it's incenter, let $D,E$ and $F$ be the points of contact of it's incircle with sides $BC,AC$ and $AB$ respectively and let point $G$ be the ...
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29 views

Do generic smooth functions have no degenerate points?

In this question, $\mathbb{N}$ includes $0$. Let $M$ be a compact $D$-dimensional $C^\infty$-smooth manifold, and fix a finite atlas $\,\mathcal{U}=\{\phi_j \,\colon U_j \to \mathbb{R}^D\}_{j=1,\ldots,...
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44 views

Can one interpret the derivative of $f(x)=x^2$ as a way to predict a value under which $f(x)$ cannot fall at a given point? and $f'$as a minimum rate?

Note : my questions is not about the application of derivatives to determine the max / min values of a function ; also, t is not about linear approximation ( though it is related to this topic); what ...
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58 views

Are there tessellations of the plane with curved $C^1$ shapes?

Does there exist a $C^1$ curve $\alpha:\mathbb{S}^1 \to\mathbb{R}^2$ such that the plane can be tessellated by a union of congruent shapes, each has boundary identical to $\text{Image}(\alpha)$ up to ...
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1answer
60 views

Good introduction to free groups and free products

In my undergraduate research project, I am going to study a paper on free products in division rings. To do this, however, I, of course, need to learn about free groups and free products. Right now, ...
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20 views

Sums of squares and positivity on the half line. (Related to Hilbert Theorem 1888)

How do I prove that $f(x) \geq 0$ for all $x\geq 0$ if and only if $f$ can be decomposed as $f(x)= P_1(x) + x P_2(x)$, where $P_1$ an $P_2$ are sums of squares. Does this theorem have a name? Is ...
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Are there any good textbooks in recent years about the math of neural networks?

I'm looking into doing neural networks for my mathematics degree's student seminar, but I'm having a hard time finding a textbook to use that is more recent as this field is growing quickly, as well ...
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26 views

Name of a notion in Linear algebra/Matrices

Let $A$ be a symmetric matrices. Is there any name for the number $$(\text{sum of all off diagonal entries})/2+(\text{ sum of all diagonal entries})?$$ For example, If $A = \begin{bmatrix}a_{11}&...
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Modal Heyting Algebras

Is there a standard way to add modal operators over a Heyting algebra -- as it was done e.g. by Johnstone and Tarski for Boolean algebras? Does this provide a semantics to some version of ...
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17 views

Best resources to learn the foundations of math for computer science [duplicate]

So I decided that this year I would go all out and learn a bunch which I have been interested in for a long time but never fully tried to get into and am trying to take on discrete math, abstract ...
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1answer
66 views
+50

Reference request for IIT JAM.

I am a BSc $3$rd year student.I am aspiring for IIT JAM exam in Mathematics.So I am looking for a problem book that contains sample problems similar to JAM exam questions.Can anyone provide me a link ...
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13 views

Mathematical Definition of White Noise

I wish to understand more about Gaussian white noise through a formal math definition if possible and how its related to the mean and variance (if we are speaking in the time-continuous domain) or if ...
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1answer
45 views

Ascoli Arzelà in Infinite dimension

In which book can I find the Ascoli Arzelà theorem for the space $C([0,T];H)$ where $H$ is a generic Hilbert space?
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81 views

The hairy ball theorem, from Brouwer's fixed point.

EDIT : The question is now the following. I know this statement of the hairy ball theorem : Theorem : Let $n \geq 3$ be an odd number, and $f:\mathbb{S}^{n-1} \rightarrow \mathbb{R}^n$ be a continuous ...
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27 views

Changing the Riemannian metric does not affect the $Spin^c$-structure

It is well known that changing the Riemannian metric on a manifold does not change the $Spin$-structure. I suppose the same should be true for $Spin^c$-structures, but I am unable to prove it or to ...
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1answer
39 views

Book recommendation/reference request on a gentle introduction to cyclotomic polynomials

Can anyone recommend a book or reference material (written in English) that offers a gentle introduction to cyclotomic polynomials? The book does not have to be too comprehensive. I plan to apply the ...
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19 views

Reference request for the free loop space

I've been trying to read about the free loop space $LX:= C(\mathbb{S}^1, X)$ for a topological space $X$, that is, the space of continuous maps between $\mathbb{S}^1$ and $X$, however the only ...
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47 views

Riemannian textbook reference request on umbilic points and totally umbilic hypersurfaces

I'm doing a project exploring Riemannian immersions and submersions. In O'Neil's book, Semi-Riemannian Geometry, he mentions umbilic points and totally umbilic hypersurfaces and proves some ...
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Time Complexity of Ordinal Exponentiation (ordinals below $\epsilon_0$ in CNF)

Based on this question, I was thinking about the following: Suppose that we have finite alphabet $\Sigma=\{w,+,\times,exp,(,)\}$. Consider the set $A\subset \Sigma^*$ such that a finite string $s \in ...
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14 views

Schwarz reflection principle for general linear elliptic PDEs

Suppose we have a harmonic function $u$ inside the half ball $B_+ \subset \mathbb R^n$ such that $u \in C(\bar B_+)$ and $u$ is $0$ when restricted to the flat part of $\partial B_+$. Then the Schwarz ...
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+50

Synthetic geometry theorems that relate lengths to areas

Does anyone know any synthetic geometry theorems (so, no algebra at all) or sources with synthetic geometry theorems, that relate lengths to areas? My only reference currently is Euclid's Elements, ...
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1answer
30 views

Definition of $*$-endomorphisms on a $C^*$-algebra

I'm reading a paper which talks about $*$-endomorphisms on a $C^*$-algebra without defining the notion. I browsed through various books on $C^*$-algebras but could not find a definition. I only found ...
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19 views

Reference on how discrete sums of multiple variables work

I am looking for a reference which goes into dealing with concepts like the 'discrete fubini theorem' (see here), how to visualize the index set of a sum and related concepts.
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2answers
52 views

Reference for analytification of schemes/varieties

It seems that algebraic geometers form some of their intuition by thinking about the analytification of a scheme. E.g. Alex Youcis mentions that this is often the "correct" topology to think ...
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1answer
74 views

What's the nicest proof of the formula for the sum of the $k$-th powers of the first natural numbers?

Do you know of a text where I can find a nicely motivated proof of the formula for $1^{k}+2^{k}+\cdots+n^{k}$? At the very beginning of page 68 of Professor H. S. Wilf's generatingfunctionology, one ...
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41 views

Reference Request: Eager for more after Chapter on Modules in Lang's Algebra

After about six months of study (over the course of about 16 months total) I’m coming to the end of Ch. 3 (on modules) in Lang’s Algebra (third edition) and I don’t know if it’s the high I’m feeling ...
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20 views

Give a reference of a text,pdf,anything , where geodesic passing by a certain point is calculated.

I'm looking for a text , pdf , any reference , where it calculated in details geodesic passing by a certain point , for example on a surface of revolution. Something in the type , a geodesic passing ...
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1answer
32 views

Reference request for improper integrals.

I am an undergrad student.We have a course on improper integrals of first and second kind along with beta and gamma function.I am looking for a suitable text that describes the problems and theory in ...
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1answer
45 views

In a C*-algebra , if $a\leq b$, then $a^{\alpha}\leq b^{\alpha}$, for $0<\alpha\leq1$ [closed]

I need help to find a reference to prove the the following theorem: Theorem 6.9. Let $A$ be a $C^*$ algebra. If $a,b\in A_+$ and $a\leq b$ then $a^\alpha\leq b^\alpha$ for $0\leq \alpha\leq 1$. On ...

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