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0answers
18 views

Chow cover that is an isomorphism in codimension one

Chow's Lemma states that for any proper scheme $X$ over a noetherian scheme $S$, there exists a scheme $X^{\prime}$, projective over $S$, and an $S$ morphism $f \colon X^{\prime} \rightarrow X$ such ...
1
vote
0answers
16 views

Decay estimate on wave equation

In this paper here it is claimed in (1.3) that the solution to the wave equation $$(\partial_t^2 -\Delta )u(t,x)=0$$ with $u(0,x)=0$ and $u_t(0,x)=g(x)$ satifies for some $c>0$ $$\sup_x \vert u(t,x)...
1
vote
0answers
19 views

Convenient vector space and its locally convex structure

I'm trying to understand convenient vector spaces, but I'm unsure about the definition topology on smooth maps. A map $f : E \rightarrow F$ between locally convex vector spaces $E$ and $F$ is called ...
0
votes
0answers
9 views

Total variation convergence of random matrices and convergence of empirical spectral distributions

In the paper https://arxiv.org/pdf/1411.5713.pdf, on page 17, the authors prove in Theorem 7 that the total variation distance between the joint distribution of the entries of certain Wishart matrices ...
2
votes
0answers
15 views

Singularity of reproducing kernel for elliptic operator

Let $(M,g)$ be a smooth compact Riemannian manifold and dimension $2$, $\Gamma$ a smooth vector bundle over $M$, and suppose $L: W^{k,2}(\Gamma)\to W^{k-2,2}(\Gamma)$ is a second order strongly ...
-2
votes
0answers
15 views

PROBABILITY outcomes of two evemts [closed]

if 10 people are in a group and they all have different colours. What is the probability that those2 people in the group are chosen? For example, colours are green, lime, turquoise, black, white, ...
0
votes
0answers
12 views

Bipartedly slice links and their surgeries

A link L in $S^3$ is said to be strongly slice if $L=∂D$,where $D$ is a disjoint union of smoothly and properly embedded disks in $B^4$. A link $L$ in $S^3$ is called bipartedly slice if $L = L_1 \cup ...
-2
votes
0answers
25 views

Complex number multiplication

Goal is to solve this |𝑧+2𝑖/𝑧−1|=1. And the final result that I am supposed to get is 2𝑥+4𝑦+3=0. So I started by converting z into x+bi and i get x +yi + 2i/x+yi-i=1 and then x+i (y+2) / x+i(y-1)....
4
votes
0answers
23 views

What is the relationship between free bicompletion and the Isbell envelope?

Given a small category $\mathbb C$, we can form the free cocompletion $\mathbf y : \mathbb C \to \mathcal P(\mathbb C)$ and the free completion $\mathbf y^\circ : \mathbb C \to \mathcal P^\circ(\...
-1
votes
0answers
26 views

Best way to teach functions, graphing, and transformations

I am a grad student trying to become a high school mathematics teacher and I had a question I wanted to ask to the community. I want to know what your best and most effective way to teach graphs, ...
0
votes
0answers
27 views

Growth rate of exponential sum of $S_j$

Let $X_j$, $1 \leq j \leq n$ be chosen i.i.d. uniform over $[0,2\pi)$. Denote $S_j \triangleq X_1 +X_2+\cdots+X_j$ and suppose that $c_j$, $1 \leq j \leq n$ are some constants such that $|c_j|=1$. I'...
0
votes
0answers
10 views

Dealing with degeneracy in nonlinear programming by “small” perturbations of constraints

CONTEXT: Suppose you have the nonlinear program $$ \begin{aligned} &\min f(x)\\ \text{subject to: }\quad & h_1(x) = 0 \\ &\quad\quad\vdots\\ &h_m(x) = 0 \end{aligned} $$ where $x\in\...
1
vote
4answers
68 views

Help with the convergence of $[\gamma \sqrt[n]{a} + (1-\gamma)\sqrt[n]{b} ]^n$

Let be $a,b> 0$ and $\gamma \in (0,1) $. Set $x_n = (\gamma \sqrt[n]{a} + (1-\gamma)\sqrt[n]{b} )^n$ for each $n\in\mathbb{N}$. My question is if this sequence $(x_n)$ is convergent and, if it so, ...
2
votes
0answers
17 views

References on discrete Sturm-Liouville eigenvectors convergence

Let $ L : u_n \mapsto a_n u_{n + 1} + b_n u_n + a_{n - 1} u_{n -1} = \nabla ( a_n \Delta u_n ) + (b_n + a_n + a_{n - 1}) u_n $ be a discrete Sturm-Liouville operator, with $ \nabla u_n := u_{n + 1} - ...
3
votes
0answers
53 views

Continuity property for Čech cohomology

Suppose we have an inverse system of compact Hausdorff spaces $\lbrace X_i , \varphi_{ij} \rbrace_{i\in I}$ and that each space has a presheaf $\Gamma_i$ assigned to it in such a way that $\Gamma_i(\...
1
vote
0answers
118 views

Is the study of additive functions dead?

I am learning more about the theory of additive functions ($f(nm)=f(m)+f(n)$) and I am struck by how powerful the theorems given are. For example, we have a complete characterization of not only what ...
1
vote
0answers
23 views

Do the solutions of parabolic PDE problems with different initial conditions converge to each other?

Let's say we have a parabolic PDE system: $$ (PDE) \hspace{0.5cm} u_t+f(u)_x=\mu \cdot u_{xx}, $$ where $x \in A \subseteq \mathbb{R}$, $t \in [0,T]$ and $u \in \mathbb{R}^n, \: n\geq 2$. And let's ...
3
votes
0answers
48 views

Inclusion of infinite intersection

Let $E$ be a Banach space, $T:E\rightarrow E$ a continuous bounded nonlinear mapping., and $\{x_n\}_{n\in\mathbb N}$ such that $$x_{n+1}=T(x_n),\:\forall n\in \mathbb{N}^*.$$ Let $$X_n=\overline{\text{...
2
votes
0answers
68 views

Are there any known algebras or vector spaces, where absolute value, modulus or norm is connected to the factors of $\pi$ or $e^{-\gamma}$?

I am currently working on an algebra of divergent integrals and series, and all the elements of that space consist of a regular part (which is a real or complex number) and irregular part (which is ...
2
votes
1answer
38 views

wave equation with vanishing trace at infinity

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Consider the boundary value problem \begin{equation}\label{pf0} \begin{aligned} \begin{cases} \Box u+qu=0\,\quad &\text{on $(0,\infty)\times \...
6
votes
1answer
111 views

What is first-order logic with Dedekind-finite sets of variables?

The usual set up of first-order logic is with an infinite reservoir of variables which we can use in formulas. This is one of the annoying reasons why we need to put $\aleph_0$ into the cardinal ...
2
votes
0answers
27 views

On exactness of associating smooth representation-functor $(\,)^\infty$

Let $G$ be a locally profinite group, e.g. reductive group over $\mathbb{Q}_p$. For a (abstract) representation $(\pi,V)$ of $G$ and $K\subset G$ compact open subgroup denote by $V^K\subset V$ the $\...
1
vote
0answers
65 views

Efficient deterministic algorithm for quadratic residues

In Elliptic Cryptography, one may need the calculation of the quadratic residue over a finite field to generate a random point on the curve. In simple terms, given the equation of the curve as short ...
1
vote
0answers
36 views

Fusion category, formal codegrees and orthogonality

Let $\mathcal{F}$ be the Grothendieck ring of an abelian fusion category. Let $(M_i)$ be its fusion matrices and $(\mathrm{diag}(\lambda_{i,j}))$ their simultaneous diagonalization. The numbers $$c_j:=...
4
votes
1answer
100 views

Atiyah-Singer for Riemannian and Kaehler manifolds

I am trying to understand the proof of the Atiyah--Singer index theorem, and would like to see how it works for two "simple" examples. Could somebody direct me to a proof for the special ...
7
votes
1answer
237 views

Are generators defined in Tohoku paper equivalent to that defined in Wikipedia (Which I believe is a more widely used definition)

As I was reading Grothendieck's Tohoku paper(translated by M.L.Barr and M.Barr), I found that the definition of a generator in the category differs from that defined in wikipedia. Let $\mathbf{C}$ be ...
2
votes
0answers
51 views

Weak sequential continuity vs strong continuity

Let $E$ be a Banach space, $T:E\rightarrow E$ a non-linear operator. $T$ is said to be Weakly Sequentially Continuous (shortly W.S.C) on $E,$ if for every $\left(x_{n}\right)_{n}\subseteq E$ with $x_{...
-5
votes
0answers
30 views

Prove that 3(a3+b3+c3)≥a2+b2+c2 if a+b+c=1 and a,b,c are non-negative reals [closed]

This question is designed my me,and is good question.I want the answer in 2 days ,please solve it
-6
votes
0answers
19 views

Prove CI is a bisector ACB [closed]

Given triangle ABC. A circle with center O goes through point B and C, but doesn't go through A. Draw the bisector of BCA, cutting AO at L. On BL, take a point F such that OCL = FCL. KF cuts BO at I. ...
5
votes
2answers
205 views

Which singular homology classes can be represented by embedded manifolds?

Given a connected CW-complex $X$ I'm interested in if a given homology class $\sigma \in H_n(X)$ can be represented by a manifold meaning if there is a map $f : M^n \to X$ from a oriented manifold $M$ ...
1
vote
0answers
13 views

Stratification which makes the defining functions isotrivial

Let $0\in X\subset\mathbb{C}^N$ be a germ of complex space and $0\in Z\subset X$ be a closed analytic subset (globally) defined by holomorphic functions $f_1,\dots,f_r$. Is there a complex analytic ...
0
votes
1answer
146 views

Integer solution of optimal transport

Let us consider two vectors $\mathbf{a}=(a_1,...,a_n)$ and $\mathbf{b}=(b_1,...,b_m)$ so that each quantity is an integer $a_i,b_j \in \mathbb{N}$. It represents for example supply and demand. Let $\...
4
votes
1answer
115 views

Zariski's main theorem for non-representable morphisms?

Let $f:\mathcal{X}\to \mathcal{Y}$ be a separated quasi-finite map of qcqs Deligne-Mumford stacks. Is there a version of Zariski's main theorem that makes sense in this context? Rydh proved a ...
2
votes
0answers
67 views

Show that these vectors are linearly independent almost surely

So I'm doing research in control theory and I have been stuck with this problem for a while. Let me explain my issue, then my proposal, and finally my concrete question. x_1,x_2\in\mathbb{R} Problem: ...
1
vote
2answers
56 views

Distribution of pre-images of the divisor function $\sigma$

If $A\subseteq\mathbb{N}$ is a subset of the positive integers, we let $$\mu^+(A) = \lim\sup_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}$$ be the upper density of $A$. For $n\in\mathbb{N}$ we let $\...
-2
votes
0answers
12 views

Transforming large size kernel into equivalent small size kernels [closed]

I have a problem where I want to convert kernel of large size,i.e, 64x64 into the number of 3x3 size kernels such that convolving them with the same image results in the equivalent output. (It does ...
2
votes
0answers
21 views

Changing a little assumptions in famous paper Vanishing viscosity solutions of nonlinear hyperbolic systems?

The question that I hope to find some answer here is: do the results from Bianchini, Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, 2005 paper still apply if we change a ...
1
vote
0answers
53 views

Constructive way to optimally cover a compact subset of Euclidean space

Let, $(X,d)$ be a simply connected compact subset of $\mathbb{R}^d$ with non-empty interiorn, let $d$ denote the Euclidean metric, and let $\varepsilon>0$. Is there a way to iteratively select ...
4
votes
1answer
236 views

What is the meaning of the $L$-group?

Langlands' functoriality conjecture predicts that to a suitable homomorphism of $L$-groups $$ \psi : ^LG \to ^LH $$ there should be a transfer of automorphic representations from $G$ to $H$. For the ...
3
votes
1answer
146 views

Finite sets and relations with boolean matrix

I'm reading about the category $\mathbf{FinRel}$ on the $n$Lab and it said: "$\mathbf{FinRel}$ is equivalent to $\mathrm{Mat(Bool)}$", without giving any explanation. Does anyone know how to ...
-2
votes
0answers
43 views

Computing the Cube root of a decimal number manually [closed]

In general, it will be difficult to compute the cube root of a decimal number manually? More the precision of a decimal number, More the Harder? Examples : 48.70 72.14 93.65 156.3451
6
votes
2answers
233 views

Do filtered colimits commute with finite limits in the category of pointed sets?

It seems to be the case that filtered colimits commute with finite limits in the category Set (for instance, this is shown in Why do filtered colimits commute with finite limits?), but does the same ...
14
votes
4answers
2k views

Has any open/difficult problem in ordinary mathematics been solved only/mostly by appeal to set theory?

We know that many (if not all) mathematical notions can be reduced to the talk of sets and set-membership. But it nevertheless sounds like a grueling task (if at all possible) to actually get advanced ...
9
votes
0answers
147 views

Finiteness of $\pi_n(Top/O)$

For $n>4$ one may identify $\pi_n(Top/O)$ via smoothing theory as concordance class of smooth structures on $S^n$ where two structures are concordant if they bound a smooth structure on the product ...
0
votes
0answers
52 views

L_q matrix inequality

The following arose out of studying $\ell_q$ Lewis weights. Let $P$ be a real $n \times n$ orthogonal projection matrix (i.e., $P$ is symmetric and $P^2 = P$) and let $W$ be the diagonal matrix ...
4
votes
0answers
54 views

Maximal Thurston--Bennequin number of boundary knot classes in contact handlebodies

Let $H$ be a contact handlebody. In other words, $H$ is a small regular neighborhood of a Legendrian graph in a contact $3$-manifold (wlog $\mathbb R^3$). Equivalently, $H=(\Sigma\times[0,1],dt+\...
4
votes
1answer
46 views

Mass distributions for high dimensional simplex and cross polytope

In this question, it is shown that the radial mass distribution of an $n$-cube (i.e. the probability density for the distance from a point selected uniformly from within an $n$-cube to the cube's ...
5
votes
1answer
121 views

An elementary inequality for three complex numbers

The following problem arose in asymptotic analysis of difference equations. Numerical maximization suggests that for all nonzero complex numbers $a,b,c$ we have $$h\big(r(a,b,c),r(b,c,a),r(c,a,b)\big)...
0
votes
0answers
64 views

Transcendence à la Liouville

Let $\alpha\in\overline{\mathbb Q}^*$. How to prove that for a non ultimately constant sequence $(\varepsilon_n)_n$ with $\varepsilon_n\in\{0,1\}$, the number $\sum_{n\ge0}\frac{\varepsilon_n}{\alpha^{...
1
vote
1answer
68 views

If $\theta_n \sim N(\theta,1/n)$, can we find the rate of convergence of $p_{\theta_n X}(x) $ to $p_{\theta X}(x)$?

Let $X$ be a random variable. Let $\theta$ be a constant, and let $\theta_n \sim N(\theta,1/n)$ so that $\theta_n$ converges towards $\theta$ as $n$ gets large. Define $p_{\theta X}$ and $p_{\theta_n ...

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