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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Is this statement about roots of polynomials well-known?

Here is the statement : Let $P$ be a non constant polynomial of $\mathbb{C}[X]$ which has at least two distinct roots. If $P''$ divides $P$ hence all the roots of $P$ belong to the same (complex) ...
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1 vote
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Elementary theory of the double category of spans

Is there an existing axiomization of an "Elementary theory of the double category of spans?" Among other nice things a category could be then defined as a monad inside the theory of spans. I ...
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6 votes
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For which values of $q$ is $\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^q}{|x-y|^{N+sq}}dxdy$ is finite?

Let $u=u(r)$ be radially symmetric nonnegative decreasing function. Let $s\in (0, 1)$ and $p, q\in\mathbb{R}$ such that $1<q<p$ and $ps<N$ with $N\in\mathbb{N}, N\ge 2$. Assume that $u\in D^{...
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Prove comparison principle for $u_t + f(u)_x = k u_{xx} + g(t,x)$ with $g \in L^\infty(0,T; L^2(\mathbb R))$

Let us consider $$u_t + f(u)_x = k u_{xx} + g(t,x),$$ with $k>0$, $g \in L^\infty(0,T; L^2(\mathbb R))$ and $f \in W^{1,\infty}(\mathbb R)$ and $f \not \equiv 0$ (possibly, we can also add the ...
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3 votes
0 answers
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Topos-theoretic proof of the consistency of CH

I hope this is not a duplicate, if so please let me know. So in MacLane Moerdijk‘s book Sheaves in Geometry and Logic it is shown that there is a boolean, two-valued topos with nno and choice, in ...
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-1 votes
0 answers
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Book for real anlaysis/measure theory course?

Does exist a book (possibly with solutions) about closed, open space, compact spaces and convergence in $L^1(\mathbb R^n)$. I would like solutions only to know if I am doing good or not. Thanks
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0 votes
1 answer
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Can reparameterization make Cramer-Rao bounds tight?

Given a family of distributions parametrized by $\theta$ for which Cramer-Rao bounds on variance for a (biased or unbiased) estimator of $\theta$ exist, these bounds may be unattainable. In the proof ...
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0 answers
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Conceptual problems in measure theory.

I am studying measure theory as a graduate student. I have studied the theory up to integration. Now I want to test my concepts by doing some problems as the best way to learn a topic is to do ...
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-1 votes
1 answer
44 views

Sylow 2-subgroup of Suzuki Group $Sz(8)$

I need to find the isomorphism class containing the Sylow 2-subgroup of the Suzuki group $Sz(8)$. Can anyone give a reference?
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0 answers
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Character Tables of large groups

I need to export LaTex files of character tables of various groups to write a project report. I usually use this site: https://people.maths.bris.ac.uk/~matyd/GroupNames/index.html However they do not ...
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1 vote
1 answer
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Reference request - textbook on linear algebra

I'm looking for a textbook on linear algebra that is on a level higher than that of a first-read text, but isn't of the machine-gun-definition-theorem-proof-corollary type. I used Lay's text as an ...
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Group cohomology reference

I'm interested in studying group cohomology (for discrete groups). Are there accessible (lecture) notes that give a nice overview of the basics for group cohomology and develop the categorical ...
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2 votes
1 answer
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Cocomplete abelian category with enough injectives has exact coproducts

In this post it is claimed that for any (cocomplete) abelian category with enough injectives, the coproduct functor is exact, that is for a family of short exact sequences $0 \to A_i \to B_i \to C_i \...
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0 votes
1 answer
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Embedding $X \ni x \mapsto \delta_x \in P_2(X)$ is totally convex

I am looking for a reference to a proof of the following result: Let $X$ be a compact, connected, smooth Riemannian manifold. Then, the embedding $$X \ni x \mapsto \delta_x \in P_2(X) $$ has totally ...
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0 answers
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When is a Markov FBSDE solution differentiable in the starting point of the Markov process?

Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, $\mu,\sigma\colon [0;T]\times\mathbb R \to \mathbb R$ ...
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1 vote
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Uniqueness of the ternary Golay codes

In [Van Lint - Introduction to Coding Theory] the uniqueness of the binary Golay codes is shown quite easily. In essence, the proof boils down to the fact that there is only one 2-$(11,5,2)$-design up ...
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0 votes
1 answer
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References for me to understand current approaches to settle $P$ vs $NP$ [closed]

I am an undergrad student that likes to study approaches to settle $P$ vs $NP$. I know that there is GCT method, and another way is to attack it by logic equivalent. I am a double major student in CS ...
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0 answers
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Books about operator algebra and number theory

Does anyone know books that covers both operator algebras and number theory. Actually, a number theory books that has operator algebraic approaches.
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0 votes
0 answers
11 views

Reference for time dependent traces

Consider the spaces $H^{1/2}(0,T; L^2(\partial\Omega))$, or $L^2(0,T;H^{3/2}(\Omega))$ and what not. I'm interested in a reference book illustrating the meaning, properties of these spaces (so, ...
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1 vote
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+50

Absolute value of functions in $H^1(\mathbb R^n)$

Let $f$ be a function in the atomic Hardy space $H^1_{at}(\mathbb R^n)$. That is, there exists a sequence of atoms $a_j$ satisfying supp $a_j \subset B_j$ for some ball $B_j$, $\int a_j dx = 0$, $\...
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Vladimir Zorich an ideal substitute for Baby Rudin?

I am currently ploughing through Zorich's Analysis Volume I, after which I plan on reading the Volume II. Truth be told, I am truly enjoying this read. A little background: I am a first year math ...
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-2 votes
0 answers
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Is there a solution manual for GMT Graph Theory ( Adrian Bondy, U.S.R. Murty)? [closed]

I am looking for an offical solutions manual for Graph theory. There is S&M for another Graph theory with Apps. However it seems no S&M for GMT Graph theory(the yellow front), considering ...
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0 votes
0 answers
18 views

Books for learning Hyperbolic Dynamical Systems and differentiable manifolds

I am looking for some books/lectures that cover Hyperbolic Dynamical systems and supplemental materials that cover the very basics of differentiable manifolds, enough to understand everything relevant ...
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0 votes
0 answers
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Let $f:\Bbb N^2\to\Bbb R:(m,n)\mapsto a^mb^n$, with $0<a<1<b$. Is $\operatorname{im}\! f$ dense in $\Bbb R_{>0}$?

Given real $a$ and $b$ with $0<a<1<b$, can every positive real number be arbitrarily well approximated by a number of the form $a^mb^n$ ($m,n\in\Bbb N$), provided that $a^mb^n=1$ only when $m=...
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1 vote
1 answer
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Am I learning math wrong?

This may be not really related to math. I'm currently learning differentiation, and no matter how many math problems I do, it seems that I always get it wrong in the exam, either having calculated ...
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1 vote
1 answer
41 views

Dual of $L \log L(\mathbb{R})$

Consider the space$$L\log L(\mathbb{R})=\left \{f\in L^1(\mathbb{R}):\int \limits _\mathbb{R}|f(x)|\log ^+|f(x)|\,dx<\infty \right \}.$$Is it known what its dual and predual spaces are? Also any ...
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Attempting to restate the question of whether the collatz conjecture has a nontrivial cycle as a combinatorics problem

It occurs to me that the question about whether non-trivial cycles exist for the collatz conjecture can be restated as these two questions (details on how this relates to the collatz conjecture can be ...
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1 vote
0 answers
22 views

Reference request for books on Sieve Theory

For various reasons, I am hoping to study sieving methods some during this summer. My general goal would be to read a book on the topic, complete relatively large amounts of questions in my own, and ...
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3 votes
0 answers
28 views

Multidimentional real analysis.Reference request.

I wanted a rigorous self contained book on multivariable calculus. I found someone recommending this book "Multidimensional real analysis" by Duistermaat,Kolk on mSE but couldn't find any ...
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4 votes
2 answers
84 views

Build a "rich" first-order logic within a given category

I would like to know a mathematical framework with an internal logic where isomorphic objects can be considered equal. For example, consider the rationals $\mathbb{Q}$. With this set we can construct ...
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1 vote
0 answers
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Reference request: Bounded operators are not a Hilbert space

I believe that the following is true: Let $X$ and $Y$ be normed spaces, both of dimension at least $2$. Then, the space of bounded linear operators $L(X,Y)$ is not a Hilbert space. Is there a nice ...
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-1 votes
0 answers
28 views

Where can I find matrix topology problems?

I am an NBHM aspirant. I am currently studying MSc in Mathematics. In NBHM, I have seen questions from topology of the space of matrices for example compactness, connectedness, openness, closedness ...
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3 votes
2 answers
93 views

Pseudo-periodicity of analytic self-maps of the upper half-plane

I have a couple of questions, in increasing order of softness: Consider an analytic map of the upper-half plane into itself $f: \mathbb{H}\to\mathbb{H}$. When this function is $1$-periodic, i.e., $f(z+...
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0 votes
0 answers
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Is there a probabilistic concept or theory for infinitesimal logarithmic product interpretation of integral?

If we have a number of independent events in probability, we can calculate it's likelihood : $$\prod_{\forall i} p_{i}$$ We can also consider ( where $H$ is the Heaviside step function ) $$\int L(t) H(...
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1 vote
0 answers
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When is an SDE solution differentiable in its starting value?

Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, $\mu,\sigma\colon [0;T]\times\mathbb R \to \mathbb R$ ...
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  • 495
0 votes
1 answer
17 views

Name for triangle with vertices on and sides perpendicular to acute triangle $ABC$

Given an acute triangle $\Delta ABC$ there is a unique triangle $\Delta XYZ$ where $X$ lies on $\overline{AB}$, $Y$ on $\overline{BC}$ and $Z$ on $\overline{AC}$ such that $XY \perp AB$, $YZ \perp BC$,...
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0 votes
1 answer
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Ratio of products of line segments

The points $A,B,C,D$ are collinear. The point $P$ sits off the line, and $\angle{APB}=\angle{CPD}=\theta.$ I'd like to show that if the points $P,A,D$ are fixed, the ratio $\dfrac{AB\cdot AC}{DC\...
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2 votes
1 answer
49 views

Is There a Notion of Diagram in Multicategories and/or Operads?

In ordinary category theory there is a notion of a diagram in a category $\mathsf{C}$ which is usually described as a functor $F: \mathsf{J \to C}$ where $\mathsf{J}$ is some small category. Based on ...
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3 votes
1 answer
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Is the complement of a set with vanishing $(d-2)-$dimensional Hausdorff measure simply connected?

In the same vein as this question, I want to ask whether $\mathbb{R}^d\setminus A$ is simply connected if $A\subseteq \mathbb{R}^d$ has vanishing $(d-2)-$dimensional Hausdorff measure, i.e. ${\cal H}^{...
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3 votes
0 answers
40 views

If $A$ and $B$ have the same measure, are they isomorphic?

During a discussion, the following result was mentioned. Theorem. Let $A$ and $B$ two Borel-measurable subsets of $\mathbb R$ with the same (positive and finite) Lebesgue measure. Then, there exists a ...
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0 votes
1 answer
34 views

Reference request for studying product measure.

I am a graduate student of Mathematics.I am self-studying measure theory.I have already completed measure,integration,convergence theorem etc.I am yet to study product measures but I am not finding ...
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2 votes
0 answers
36 views

Alternative reference request for Kato's Perturbation Theory

I am studying Analytic Perturbation Theory from Kato's book.But sometimes I find it really difficult to follow.(e.g. In Chapter 2 , he directly talks about algebraic singularities without defining it ...
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0 votes
0 answers
38 views

What are the classical mathematics textbooks? [closed]

I have some classical books such as Richard Courant Differential and Integral Calculus both volumes I and II (English version). I also have Hardy’s Pure Mathematics. Apostol Introduction to ...
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5 votes
1 answer
60 views

Reference request for perfection of schemes over finite fields

I am currently reading a paper from 2021 which uses "perfection" of schemes over finite fields. If $X$ is such a scheme over $\mathbb F_q$, the associated perfection is denoted by $X^{\...
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  • 4,579
2 votes
1 answer
29 views

What kinds of graphs are known to exhibit sharp threshold for bernoulli percolation?

What kinds of graphs are known to exhibit sharp threshold for independent bernoulli percolation? Here, sharp threshold stands for exponential decay of the probability of the cluster range below the ...
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  • 663
5 votes
1 answer
112 views

Formalizing Natural Languages

I've been interested in the subject of metalanguages and how (if) we can formalize them. Most metalanguages I've encountered use some variation of a natural language (such as English, German or French)...
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-1 votes
0 answers
32 views

Tor functor using tensor product of free resolutions

I was studying these notes by Huneke: https://home.adelphi.edu/~bstone/commalg-notes/commalg-2/algebra-notes-II.pdf He considerar $M$ and $N$ $R$-modules over a commutative ring $R$. Then, he says ...
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2 votes
1 answer
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What is this concept called (differentiating a matrix, NOT talking about Jacobians)

In my differential geometry course, we try to prove that the set of orthogonal matrices is a smooth-submanifold. The submersion we use is $f(A) = A^TA - I$, where $I$ is the identity. For $f$ to be a ...
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1 vote
0 answers
20 views

Prove that if $f\in C^{r}$ then the map $g(u,x):=(df)_u(x)$ is of class $C^{r-1}$

Let $X,Y$ be two finite dimensional $\mathbb{R}$-banach spaces and $U\subseteq X$ an open subset. Suppose that $f:U\to Y$ is of class $C^{r}$ with $r\geq 1$. Show that the map $g:U\times X\to Y$ given ...
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-2 votes
0 answers
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Is there any solutions manual to N Piskunov - Differential and Integral Calculus [closed]

Just as the questions says, is there any solutions manual or repository of solutions of the book N Piskunov - Differential and Integral Calculus.
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