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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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$\exp: \mathfrak{so}(1,n) \rightarrow SO(1,n)$ is surjective

I am looking for a reference where it is proven that the exponential map described above is surjective. Here, I am denoting by $\mathfrak{so}(1,n)$ the Lie Algebra of the group $SO(1,n)$. So we have ...
2
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0answers
34 views

Gelfand Naimark Segal construction

Let $X$ be a von Neumann algebra. Let $f$ be a faithful semi-finite normal weight on $X$. $$N_f:=\{x \in X : f(x^*x) < \infty \}$$ $$n_f:=\{x \in X : f(x^*x)=0 \}$$ Most of operator algebras ...
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0answers
26 views

Matrices with rows from an arbitrary vector space (and multiplication by number matrices)

Matrices consist of elements of some field. However, if we have a matrix $A\in M_{m,n}(F)$, it is sometimes useful to look at each row as a vector from $F^n$, i.e., we can view the matrix as $$A=\...
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0answers
46 views

What are the search limits of factors of the numbers $10^{2^n}+1$?

For the Fermat numbers, I know such a site which shows how far the factors of $2^{2^n}+1$ have been searched. Is there somewhere a survey for the search limits for factors of the numbers $10^{2^n}+...
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0answers
18 views

Partial pair matching via dynamic programming

Say I have two lists of items $A=(a_1,\ldots,a_n)$ and $B=(b_1,\ldots,b_m)$ where $m<n$. Each pair of items $a_i$ and $b_j$ has a known cost, $c_{ij}>0$. I wish to find the $m$ pairings such ...
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0answers
13 views

On solution methods for min-min optimization problems

Closely related (although not equivalent) to minimax optimization problems is the following: $$\min_{x \in \Omega} \min_{i=1,...,q} f_i (x).$$ Here, $\Omega \subset \Bbb R^n$ and $f_i: \Bbb R^n \to \...
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0answers
9 views

Gradient bounds for stationary Hamilton-Jacobi-Bellman equations

Let $$ u_0\equiv 0\\ -\partial_{t}u=H(x,u,\nabla u,\nabla^2u) $$ be a Hamilton-Jacobi-Bellman equation. Are there lower bounds of the form $$ \max_{x\in B_R}|\lim_{t\to-\infty}\nabla u(t,x)| \leq C $$...
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2answers
30 views

How to identify / measure the acceleration of a chirp?

Which methods do there exist to identify the (angular) acceleration of a chirp function? In other words, given some sampling of the curve for example $\{(t_1,f(t_1)),\cdots(t_n,f(t_n))\}$, can we ...
3
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2answers
56 views

Schrödinger equation involving the Dirac-Delta

I am taking a course on quantum mechanics and I try to understand the time-independent Schrödinger-equation with the Delta-potential: $$\frac{-\hslash^2}{2m}\psi''(x)-V_0\delta(x)\psi(x)=E\psi(x)$$ ...
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1answer
24 views

Representations of non-distributive lattices

So, there are various theorems that show that you can represent a distributive lattice as some sort of lattice of sets (birkhoff, stone, priestley etc). Are there any theorems that provide ...
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0answers
14 views

Degenerate parabolic PDE existence and uniqueness

I have the following PDE $$\partial_t u(t, x) = (ax^2 + b) \partial_{xx}u(t, x) + (cx-d)\partial_x u(t, x) + e u(t,x), (t, x) \in (0, T) \times (-B, B),$$ with $u(t, B) = u(t, -B) = 0$, for all $t \...
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0answers
29 views

Distributions with values in line bundles

Let $M$ be a compact complex manifold, and let $L \rightarrow M$ be a holomorphic line bundle. I'm confused by the notion of a distribution that takes values in $L^{-1}$, and I'm looking for ...
1
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1answer
28 views

Similarity dimension and Hausdorff dimesion

How do we prove the similarity dimension equals the Hausdorff dimension if the self-similar set satisfies the open set condition? Which article contains this proof?
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3answers
66 views

Countable product of metric spaces is metrizable [General Metric]

I know that if we have a countable collection of metric spaces $\{(X_n,\rho_n)\}_{n=1}^{\infty}$ then $X=\Pi^{\infty}_{n=1}X_n$ is a metric space with metric $\rho((x_n)_{n \in \mathbb{N}},(y_n)_{n \...
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0answers
31 views

Bass' formula (homological algebra)

I am studying homological algebra and I want to be familiar with Bass' formula which associate the injective dimension to the depth of module and ring. I am looking for an online version and I can not ...
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0answers
12 views

Which types of topological approaches to regression & denoising of sparsely sampled time-series data exist?

I was to some conference some months ago (about machine learning) and it was one of the first times I heard about topological approaches to do regression / denoising of sparse time series data. It ...
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0answers
21 views

Polar divisor contains no hyperplane when degree $d\geq 3$

Let $X\subset \mathbb P^n=\mathbb P^n_\mathbb C$ $(n>3)$ be a smooth hypersurface of degree $d\geq 3$, defined by a homogeneous polynomial $F$. Let $a=[a_0:\ldots:a_n]\in \mathbb P^n$ be any closed ...
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0answers
34 views

Algebraic de Rham cohomology of the affine plane in characteristic p?

The fact that $H^1_{\mathrm{dR}}(\mathbb{A}^1_k) = \bigoplus_{i=0}^{\infty}k$ for $k$ of positive characteristic exemplifies what can go wrong with de Rham cohomology when the characteristic is not ...
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227 views

Mathematics in Other Languages

The intention of this community wiki is to aid in learning another language in the restricted topic of mathematics. It does matter whether one wants to learn another language for mathematics or use ...
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0answers
19 views

Action of the multiplicative group induced by a grading

I found in several different fonts, (in the first section of this (https://arxiv.org/abs/alg-geom/9405004) paper, or even in this answer (https://mathoverflow.net/questions/212960/intuition-behind-...
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0answers
23 views

Reference with results for common Lie algebras

I frequently need to know facts about classical Lie algebras like $\mathfrak{su}(n)$, $\mathfrak{so}(n)$, $\mathfrak{sl}(2)$... and do not have time to try and compute all of it. I would like to know ...
2
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2answers
56 views

extending a continuous map to quotient spaces.

Let $A\subset B$ and $C\subset D$ metric spaces. Suppose that there is $f: B-A\rightarrow D-C$ a homeomorphism such that for any $x\in A$, and any converging sequence $x_{n}\rightarrow x$ such that $...
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0answers
37 views

Lawvere - Conceptual Mathematics - What is this property of categories called?

In Conceptual Mathematics, Lawvere & Schanuel describe tell us how to create a type of structure (I'm paraphrasing here): A set of names for objects/structural components A set of names for ...
2
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1answer
120 views

Book on Advanced Calculus

I'm an undergraduate student of physics. I have an upcoming course on Advanced Calculus but I do not know which book to follow. So, recommend me an undergraduate level book on Advanced Calculus. Edit:...
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1answer
44 views

Mathematics of war references

I have an upcoming talk in on data science. Now a ways such topics have been hijacked by talks on the use of machine learning. Deviating from the trend, I want to focus of core advancement of ...
2
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0answers
33 views

Is there anything known about $\sup_{s\leq t} \vert B_s \vert - f(s)$, where $B$ is a Brownian motion and $f$ measurable.

If we consider the process $$ Y^f_t := \sup_{s\leq t} \vert B_s \vert - f(s)$$ is there anything known about the distribution or at least the probability $\Bbb P (Y^f_t \leq 0 )$ ? Of particular ...
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0answers
17 views

Book-recommendation: Numerical method for stochastic differential equations

Speaking of numerical stochastic differential equations, the book of Peter Kloeden 1992 Numerical Solution of Stochastic Differential Equations is a quite famous and standard reference. But when I ...
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0answers
22 views

complex infinite series reference needed

I am an undergraduate student and I am looking for some notes and articles about complex infinite series and power series ,and also some problems . Can someone help me please ?
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0answers
24 views

Product of Paracompact spaces being Paracompact

I'm interested in the game-characterization proposed by Telgarsky (paper) of the class of paracompact spaces that preserve paracompactness under cartesian product with another paracompact space. He ...
0
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1answer
33 views

Free presentations of Weyl groups

Can someone please provide me with a reference for the fact that the Weyl group $W$ associated to a root system $\Phi$ can be realised as a Coxeter group? This means that a Weyl group $W$ has a ...
1
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2answers
40 views

Lecture note or article on probability

I am undergraduate student, I am looking for a lecture notes and articles on probability, especially random variables and law of distrubtion ... Can you please help me ?
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0answers
9 views

Reference request on rank properties on union by rank in disjoint set

i am looking for book which has a proof for the following lemmas on union by rank optimization in disjoint set data structure: a tree with rank $k$ has at least $2^k$ vertices all rank k is $<=...
0
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0answers
33 views

De Bruijn–Erdős theorem

I am trying to get a good proof of De Bruijn–Erdős theorem using two of the following: (i) Topology (ii) Propositional Logic. I would be very happy if someone can provide me some info on where can I ...
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0answers
83 views

Reference request about “internal language of categories”

For the last months I've been trying to become familiar with the so-called "internal language of a category". However, I'm still not confident enough when, for instance, I find a subobject (of a given ...
0
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1answer
60 views

Is there a way to find if a fractal has been studied? [closed]

Playing around I've found a fractal with some interesting properties and I would like to know if it has been studied, which is not unlikely, but I haven't been lucky so far. Is there an exhaustive ...
0
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1answer
19 views

Extending a homeomorphism

Let $A\subset B$ be a closed nonempty subspace of $B$ and let $C\subset D$ be a closed nonempty subspace of $D$. All spaces are Hausdorff. Suppose that we have a homeomorphism $f:B-A\rightarrow D-...
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2answers
38 views

If Matrix A, B are similar, is the sum of the diagonals equal?

Given Matrix $A \sim B$ , can the sum of the diagonals be equal? E.g $$ A = \begin{bmatrix} 0 & 2 & -3 \\ -1 & 3 & -3 \\ 1 & -2 & a \\ \end{bmatrix},\qquad\qquad B = \begin{...
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1answer
79 views

Set $\int_0^1{x^n\sqrt{1-x^2}}dx\quad (n=0,1,2,…)$. Prove that the sequence $(a_n)$ is monotonically decreasing; [closed]

Set $a_{n}$=$\int_0^1{x^n\sqrt{1-x^2}}dx \quad (n=0,1,2,....)$ $ 1.$ Prove that the sequence $(a_n)$ is monotonically decreasing; $2.$ Prove $a_{n}$ = $\frac{(n-1)}{(n+2)}a_{n-2}\quad (n=2,3,…)$ ...
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1answer
14 views

how to split a separable algebra?

I'm trying to factor ideals in a function field (more precisely, ideals in a maximal order of a function field), and I've come across a step in the published Buchman-Lenstra algorithm which works in ...
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0answers
25 views

Poincare's Inequality

Assume $\Omega \subset L_d$, for some $d > 0$. Then, for all $u \in W^{1,q}_0(\Omega)$ $1 \leq q \leq \infty$ , $\left \|u \right \|_p \leq (d/2)\left \| \nabla u \right \|_p$. Prove that the ...
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0answers
40 views

Group structure of an elliptic curve over a finite field

I'd be interested in a short proof of and a reference (first source, not a textbook as e.g. one of Silverman's monographs) for the following result: Let $q$ be a prime power and $C/F_q$ be an ...
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0answers
26 views

Primitive of a function for all but countable many points

Let $f$ be a real-valued function with $D(f) = D = <a,b>$. In our calculus course we introduce such definitions: Definition 1. $F$ is an exact primitive of $f$ iff $D(F) = D$, $F$ is ...
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1answer
45 views

Find the area between the graph $y=e^{-x}\sin x, x \geq 0$ and the $x$-axis. Calculate the area of ​the area.

Find the area between the graph $y=e^{-x}\sin x, x \geq 0$ and the $x$-axis. Calculate the area of ​​the area. answer: The part of the red line is marked out, how do you think of it? If you don’t ...
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0answers
92 views

Book recommendation to study topics of Linear Programming for self study

I need some reference book suitable for self-study with many solved examples and solutions preferably for exercise questions for following study. Need basic honours undergraduate level text , ...
2
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1answer
59 views

Axioms of Propositional Logic with as few negation axioms as possible

Could you direct me to an axiom system for propositional logic over the connectives $\land$, $\lor$, and $\lnot$ with as few axioms over negation as reasonably possible? I've done a fair bit of ...
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2answers
167 views
+100

Image of $T:E \rightarrow \alpha E + (1-\alpha) E$ where $\alpha>1$.

Fix a real number $\alpha>1$ and an integer $n \geq 1$. Let $T_\alpha$ be the mapping defined on the set $\mathcal{E}$ of closed convex subsets of $\mathbb{R}^n$ by \begin{equation*} T_\alpha(E) = \...
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0answers
14 views

Table of structure constants for closed 3-manifolds (Thurston geometries)

I was wondering if there exist a tabulation of the structure constants of the eight types of geometries for 3-manifolds according to Thurston. Those constants involved in the equation $$[e_i,e_j] = c^...
2
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1answer
76 views

An application of the pigeon hole principle

I'm interested in a reference (preferably the original source) of the following result which can be proven using the pigeon hole principle: Let $K$ be a subset of $\{1,...,2n\}$ with $|K| \geq n+1$. ...
2
votes
2answers
47 views

Natural Extensions of the $p$-Adic Norm to Higher Dimensions

There are two classes of completions for $\Bbb{Q}$, we get $\Bbb{R}$ by considering Cauchy sequences with respect to the standard Euclidean metric, and we get the $p$-adic numbers, $\Bbb{Q}_p$ when ...
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0answers
16 views

Hessian of solutions to heat equation in bounded domain

Consider the solution $$u(t,x) = \sum_{i=1}^\infty e^{-\lambda_i t} \hat{g}_i \phi_i(x)$$ of the Heat equation with initial data $g$ and homogeneous Dirichlet or Neumann boundary data (where we ...