# All Questions

98,519 questions

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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**

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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**

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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**

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**0**answers

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 ...

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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 ...

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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 $\...

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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**

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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)...

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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

**1**answer

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 ...

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**1**answer

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 ...

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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 ...

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**0**answers

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 ...

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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

**1**answer

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 ...

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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}...

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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 ...

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**0**answers

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**

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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

**1**answer

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**

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**0**answers

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 $\mathbf{B}\mathbf{B}^T$....

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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

**2**answers

88 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 $N$ ...

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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})...

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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

**1**answer

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?

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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 ...

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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

**1**answer

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 ...

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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 ...

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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

**1**answer

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**

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**1**answer

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 ...

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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

**1**answer

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**

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**1**answer

413 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}}...

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**0**answers

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 ...

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**1**answer

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 $\...

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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}...

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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 $\...

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votes

**1**answer

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

**1**answer

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**

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**1**answer

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 ...

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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**

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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 ...

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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

**3**answers

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**

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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

**3**answers

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 $S \cap (S+t)$ ...

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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) =...