All Questions

0
votes
0answers
12 views

Name for mappings that are “not quite projections”

Is there a known name for the following definition? Consider topological spaces $X$, $Y$ and $f: X \rightarrow Y$ a continuous mapping. Then, $f$ is an "almost projection" if there is a topological ...
1
vote
0answers
38 views

Uniqueness of limits and compactness implies closure

It is not difficult to prove that in a Hausdorff topological space every compact set is closed, and almost trivial that if in a topological space X every compact set is closed then X is T1. As ...
-1
votes
0answers
68 views

Specific examples of an algebraic closure of a finite field [on hold]

I'm struggling to understand the concept of algebraic closure for finite fields. Are there specific examples I can use to get an intuitive understanding? What sorts of elements do the algebraic ...
3
votes
0answers
45 views

Theory of surfaces in $\mathbb{R}^3$ as level sets

Is there a book that treats the classical theory of surfaces in $\mathbb{R}^3$ from the point of view of level sets of a function? I seem to remember someone telling me that such a book exists, but I ...
0
votes
0answers
28 views

$m-$cycles in $S_n$ modulo an equivalence relation

Let $A$ be the set of all $m-$cycles in $S_n$. Define an equivalence relation $i$ in $A$ by $\sigma_1$ is related to $\sigma_2$ by $i$ if $\sigma_1$ is a power of $\sigma_2$ or viz., then the number ...
1
vote
0answers
34 views

Non-zero homomorphism from a module to its ground ring

Let $c_1,\dots,c_k$ be some non-zero complex numbers and $M$ be the abelian subgroup generated by $c_1,\dots,c_k$ (i.e. all $\mathbb{Z}$-linear combinations of $c_1\dots,c_k$). Suppose further that $\...
0
votes
0answers
72 views

The prime number matrix sieve [on hold]

I have derived the following theorem: An odd positive integer $N=6n−1$ is a prime iff neither of two diophantine equations $6x^2+(6x−1)y=n$ $6x^2+(6x+1)y=n$ has a solution. An odd positive ...
2
votes
0answers
34 views

Oscillation operator of a function

Call a function from $[0, 1]$ to itself a box function. Given any box function $f$, define its oscillation function Of as $$Of(x) = \lim _{d \to 0} \sup _{y, z \in B_d (x)} |f(y) - f(z)|$$ Then $Of(x)...
0
votes
0answers
17 views

Optimal transport between Gaussian mixtures and their centers

I have a question about bounding the Wasserstein loss between a continuous Gaussian mixture and a discrete uniform distribution of its centers. In particular, let $P=\frac 1 k \sum_{i=1}^k \mathcal{N}(...
4
votes
1answer
79 views

Smooth structure on the space of sections of a fiber bundle and gauge group

Let $\xi$ be a fiber bundle $F\hookrightarrow E\to B$ (where every space is smooth, T2 and second countable), let $\Gamma(\xi)$ be the space of smooth sections. We can complete $\Gamma(\xi)$ with ...
1
vote
1answer
61 views

Does $f$ have the same minimiser as $\|\nabla f \|$ for $f$ strictly convex?

This question is migrated from MathStackExchange where it seemed to be too hard. I wonder does anyone here have any ideas? Suppose $f: K \to \mathbb R$ is $\mathcal C^2$ and strictly convex on some ...
1
vote
0answers
47 views

Given a set of vertices of the Boolean cube, what is the equivalent union with least number of faces?

For example. Suppose you are compacting binary using the symbol X. Like a pattern that expands to (0,1). Example. BINARY SET 0000 0010 1000 1010 MINIMAL ...
0
votes
0answers
48 views

Give an example of a paracompact space whose has finite cohomological dimension

Borel-Atiyah-Quillen-Hsiang localization theorem is stated for compact Lie group actions on compact spaces or paracompact spaces of finite cohomological dimension. In fact, Satya Deo generalized the ...
2
votes
0answers
66 views

Bounds on the L^1 norm of a discrete Fourier spectrum

I am dealing with a function $f$ of the form \begin{equation} f(t):=\sum_{k=1}^Na_ke^{\mathrm{i}\phi_k t} \end{equation} and I have a promise that \begin{equation} 0\leq f(t)\leq C\;\;\;\text{for all}...
3
votes
1answer
196 views

does recursive (decidable) languages closed under division (Quotient) with any language?

I need to prove or disprove that R languages are closed under divison. I have managed to prove thet CFL are't closed under division. I read in wikipedia that RE languages are closed, but I didn't find ...
1
vote
0answers
25 views

Decomposition into irreducible of a representation of the wreath product $S_d \wr S_m$ (2)

This is a question following Decomposition into irreducible of a representation of the wreath product $S_d\wr S_n$ I call: $$ R_m= \bigl( F^{\widetilde{\otimes n-m}} \boxtimes S^{\widetilde{\otimes m}...
7
votes
0answers
75 views

What does the classifying space of a topological monoid classify?

The classifying space $BG$ of a topological group $G$ classifies principal $G$ bundles. I have come to appreciate this. I hope the following question is appropriate for MathOverflow: What does the ...
-1
votes
0answers
37 views

How to understand the sign periodicity of any conditionally convergent series? [on hold]

Given any conditionally convergent series $\sum_{n\geq1} a_n$. I wonder if there is a "standard way/method" to "investigate/estimate" the sign periodicity of $a_n$ by some explicit functions $f(x)$ i....
-4
votes
0answers
43 views

what are the math books worth to read published in 2018? [on hold]

Holidays are approaching fast and books are always great gifts. By math books, I mean undergraduate textbooks, large audience math books, specific field books. The only request is that they have ...
3
votes
1answer
58 views

Globalizing Feigin--Frenkel duality

Let $\mathfrak{g}$ be a semisimple Lie algebra, $\mathfrak{g}^L$ be its Langlands dual. Feigin--Frenkel duality says $$ W^k(\mathfrak{g})=W^{k_L}(\mathfrak{g}^L) $$ if $r'(k+h^{'})(k_L+h'_L)=1$, where ...
2
votes
0answers
34 views

A question related to (random) matrix factorization

Let $\mathbf{B}$ be an $m\times r$ binary matrix taking values in $\{0,1\}$, and assume that $m\geq r$. I am wondering what can be said about the recovery of $\mathbf{B}$ from $\mathbf{B}\mathbf{B}^T$....
4
votes
0answers
50 views

measure of generic reals in forcing extensions

It is well-known that if $V[G]$ is a generic extension by adding a Cohen real, then the set $\{r \in V[G]: r$ is Cohen generic over $V\}$ has measure zero. On the other hand, if $V[G]$ is a generic ...
2
votes
2answers
89 views

Examples of group $G=N \rtimes H$ where $N$ and $H$ are as below

I am searching for examples of connected locally compact group $G = N \rtimes H$, where $N$ is a simply connected nilpotent non-abelian Lie group, $H$ is linear reductive and $H$ operates on $N$ ...
1
vote
0answers
87 views

SL(2,R) invariant which are not SL(2,C) invariants

Consider four points, $\sigma_i$ i=1,2,3,4 on the line $\mathrm{Im}(z) = 0$ in the complex plane $\mathbb{C}$. Does it exist a rational function of these four points which is $\mathrm{SL}(2,\mathbb{R})...
1
vote
0answers
43 views

Simplicity Criterion for Verma module

In Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$, $\lambda\in\mathfrak{h}^*$ is antidominant if $\langle \lambda + \rho, \alpha^\lor\rangle \not\in \mathbb{Z}^{>0}$ ...
5
votes
1answer
94 views

Kato's Euler System for Isogenous Elliptic Curves

Let $E,E^\prime$ be elliptic curves over $\mathbb{Q}$ and also suppose they are $p$-isogenous. How are the Euler systems corresponding to the two isogenous elliptic curves related, if at all?
1
vote
0answers
51 views

Is the set of fixed points of hyperbolic elements from $\mathrm{PSL}_{2}(A)$ a group?

Given a subring $A$ of $\mathbb{R}$, we can consider the set $\mathrm{PSL}_{2}(A)$ of elements in $\mathrm{PSL}_{2}(\mathbb{R})$ with entries in $A$ and the determinant of associated matrix is equal ...
0
votes
0answers
30 views

Minimizing a diophantine functions / quadratic form

Note - not a mathematician here, pardon me potential terminology misuse. I need minimize the following diophantine function, in integer coefficients a,b and variables x, y. I won't always have a ...
8
votes
1answer
206 views

Are there infinitely many real multiplication fields of abelian surfaces over $\mathbb Q$?

Do there exist infinitely many real quadratic fields $F$ such that there is an abelian surface $A$ over $\mathbb Q$ whose ring of endomorphisms, tensored with $\mathbb Q$, is $F$? Do there exist ...
1
vote
0answers
21 views

A $\mathbb C$-linear map from $M(p-1,\mathbb C)$ to $\mathbb C^\hat G$, where $p$ is an odd prime and $G=\mathbb Z/(p) ^\times$

Let $p$ be an odd prime and $G=(\mathbb Z/(p))^\times=\{1,2,...,p-1\}$ i.e. $G$ is a cyclic group of order $p-1$. Let $\hat G:=\{\chi:G \to \mathbb C^\times : \chi $ is a group homomorphism $\}$. For ...
13
votes
0answers
80 views

Can we always shift two disjoint convex bodies a little bit to decrease the volume of their convex hull?

Let $K,L\subset\mathbb R^d$ be two disjoint compact convex sets with non-empty interiors. Can $x=0$ be a point of local minimum for the function $F(x)=\text{vol}_d(\text{conv(K,L+x))}$? I was asked ...
2
votes
1answer
43 views

Spectral density of $D + XX^T$

Let $D$ be a fixed diagonal matrix with real entries, and $X$ a random $m\times n$ matrix. More precisely, the entries $X_{ij}$ are real independent and identically distributed. It can be shown that ...
14
votes
1answer
226 views

Conceptual explanation for curious linear-algebra fact in characteristic $2$

All matrices and vectors in this post have entries in the field $\mathbb{F}_2$. Fix some $n \geq 1$. For an $n \times n$ matrix $X$, write $X_0$ for the column vector whose entries are the diagonal ...
0
votes
0answers
51 views

What is the consistency strength of this kind of reflection principle?

If $\psi$ is a predicate that is definable in $FOL(\in,=)$ by a formula from parameters in $V$, then if $\psi$ hold of the class $\small ORD$ of all ordinals in $V$, then the class of all cardinals in ...
6
votes
1answer
149 views

Remark 12.8.8 in Arinkin--Gaitsgory

I can not understand Remark 12.8.8 in the preprint "SINGULAR SUPPORT OF COHERENT SHEAVES AND THE GEOMETRIC LANGLANDS CONJECTURE". I am somewhat embarrased by the degree of my confusion, hopefully ...
10
votes
1answer
414 views

Is the p-adic Lindemann-Weierstrass Conjecture still open?

The p-adic Lindemann-Weierstrass Conjecture: Let $\alpha_{1},\ldots,\alpha_{N}\in\overline{\mathbb{Q}_{p}}$ be $p$-adic algebraic numbers satisfying $\left|\alpha_{n}\right|_{p}<p^{-\frac{1}{p-1}}...
2
votes
0answers
59 views

Existence of a “generic enough” lattice point interior to a lattice triangle

Let $T$ be a lattice triangle in $\Bbb R^2$ (i.e. the convex hull of three noncolinear points in $\Bbb Z^2$), and assume it has at least one interior lattice point. Is it always possible to find a ...
7
votes
1answer
238 views

Is there a conceptual reason why the notion of “quasicoherent sheaf” is independent of the choice of topology?

Let $X$ be a scheme and $\mathcal S$ a site which is a full subcategory of the category $Aff/X$ of affine schemes with a map to $X$. If I understand correctly, the category $QCoh^\mathcal S(X)$ of $\...
3
votes
0answers
48 views

Jacobians of pointed curves

Let $Y$ be an algebraic curve of genus $g \geq 1$ defined over a number field $K$. If $Y$ has a $K$-point, then one can define the Abel-Jacobi map which embeds $Y$ into its Jacobian variety $\text{Jac}...
1
vote
0answers
30 views

Slightly finer topology vs a quasi-component

Let $(X,\tau)$ be a topological space, and let $Q$ be a quasi-component of $X$. Let $S$ be a subset of $X\setminus Q$. Then is $Q$ necessarily a quasi-component of $X$ in the topology generated by $\...
0
votes
1answer
55 views

Differentiating Riemannian logarithmic map

Let $(M,g)$ be a Riemannian manifold, geodesically complete, and assume logarithms are well defined and smooth. Let $c: I\to M $ be a smooth path in $M$, and $x\in M$. Can we say something about $$\...
1
vote
1answer
64 views

Relative weight lattice

Let $G$ be a reductive group over an algebraically closed field $k$. Let $T$ be a maximal torus, $B$ be a Borel subgroup and $I_G$ is the set of simple roots. Let $P$ be a parabolic subgroup, $M$ be ...
9
votes
1answer
209 views

Map of Grassmannians associated with a Veronese embedding

I'm quite sure this should be classically known, however I am not an expert on the topic and I was unable to find a precise reference in the huge literature concerning Veronese embeddings and ...
0
votes
0answers
31 views

Energy estimates involving test functions for weak solutions of PDE problems

I was reading an article on Arxiv.org about Navier-Stokes system ([Breit]) and I stumble on this sentence on the second page: "A weak (in the PDE sense) solution satisfying the energy inequality ...
2
votes
0answers
56 views

Understanding an equality in the paper “Class groups, totally positive units, and squares”

This is an excerpt from the paper "Class groups, totally positive units, and squares" (page 36). I am struggling to understand the last equality $|K^{(1)}_{2}:K|=|\overline{O}_K^{+}|$, the bar ...
1
vote
0answers
74 views

Espace étalé for derived category

It is known that for a sheaf $\mathcal{F}$ on $X$, we can associate $X_\mathcal{F}$, the étalé space of $\mathcal{F}$ over $X$ such that section of $X_\mathcal{F}$ coincides with section of $\mathcal{...
10
votes
3answers
225 views

Minimizing geodesics in incomplete Riemannian manifolds

Let $(M, g)$ be a Riemannian manifold, not necessarily complete. Let $x$ be a point in $M$, and let $r>0$ be such that the exponential map $\operatorname{exp}_x$ is defined on an open ball $B=B(0,r)...
-4
votes
0answers
103 views

On $\det\big[x+\big(\frac{i^2-\frac{p-1}2!\,j}p\big)\big]_{1\le i,j\le(p-1)/2}$ for primes $p\equiv 3\pmod 4$

Motivated by Question 302323 and Question 317509, I have formulated the followng conjecture on the basis of my computation. Conjecture. For any prime $p\equiv3\pmod4$, there is a positive integer $...
3
votes
3answers
233 views

Intersection of $\{2^a 3^b 5^c 7^d\}$ and its translates

Let $S$ be the set of positive integers of the form $2^a3^b 5^c 7^d$. I need information about the cardinality of the intersection of $S$ and its translates. In particular, is $S \cap (S+t)$ ...
1
vote
0answers
68 views

A computing shortcut to $Dedekind Number(n)$?

OEIS A132581 gives a functional extension of Dedekind numbers. $F(n)$ is the number of antichains in the first $n$ elements of "the infinite boolean lattice". And $\operatorname{DedekindNumber}(e) =...

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