# All Questions

148,337
questions

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### When is the albanese map an embedding

Let $S$ be a surface defined over a firld $K$, when is the albanese map $S\longrightarrow \text{Alb}(S)$ an embedding?
For curves, for example, with genus at least 2, there is a morphism between the ...

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0
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26
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### An alternative definition for Hilbert cube manifolds

Let $Q=\prod_{i=0}^{+\infty}[-1,1]$ with the product topology be the seperable Hilbert cube and $Q_{n}=\prod_{i=0}^{n-1}[-1,1]$ be the finite dimensional cubes. Recall that an $n$-dimensional ...

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16
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### Completely contractive Banach algebra structure on the dual of a Hopf $C^*$-algebra

Let $(A, \Delta)$ be a Hopf $C^*$-algebra, i.e. $A$ is a $C^*$-algebra, and $\Delta: A \to M(A\otimes A)$ is a non-degenerate $*$-homomorphism that is coassociative:
$$(\iota \otimes \Delta)\Delta = (\...

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49
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### Questions about class field's characteristic

The original version of these photos is Felix Klein's "Development of Mathematics in the 19th Century"
In second photo
In this book, it says class-field and decompose 2 into $(1+i)$ and $(1-...

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22
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### I am passing all input correctly but API is not working for me [closed]

I am correctly providing all the input, but the API is not functioning as expected for me.

1
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0
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20
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### How to get perturbation bounds of singular vectors

Let an adjacency matrix $A={A^\top}\in {\mathbb{R}^{n \times n}}$ (a binary matrix) of a simple undirected graph and its degree matrix $D$ be given.
When adding $Q$ edges into the graph, which is ...

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### how does Huber compute the $var(s_n)/E[s_n]$ and $var(d_n)/E[d_n]$? [migrated]

I cross posted on math stackexchange at the following question because of lack
of response. Perhaps this question was too advanced for math stackexchange,
or perhaps I was asking too many things at ...

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18
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### Perpendicular mapping from one matrix group to a closed matrix subgroup

Let H be a connected closed Lie subgroup of G, which is a connected closed Lie subgroup of GL(n,𝔽), where 𝔽 = ℝ or 𝔽 = ℂ. Assume that G and H are each endowed with the riemannian metric inherited ...

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29
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### Is there is a constant $c$ such that toroidal graphs are minor-$c$-colorable?

A toroidal graph is a graph that can be embedded on a torus. In other words, the graph's vertices can be placed on a torus such that no edges cross.
A minor of graph G is a graph obtained from G by ...

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29
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### Can this question be solved with boolean algebra? (Who owns the fish) [closed]

I’m thinking how to solve the question below with only boolean algebra.
Is it possible?
I cannot solve for 3 days..
Plz help me. It’s homework
The Brit lives in a red house
The Swede keeps dogs
The ...

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16
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### Linear kinetic PDE: Characteristics of the transport operator are given by the flow a Hamiltonian

I am trying to read and understand the article "Hypocoercivity for linear kinetic equations conserving mass." by Dolbeault, Mouhot, Schmeiser. doi: 10.1090/s0002-9947-2015-06012-7 (https://...

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33
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### Computing $G$-theory for a 3-dimensional affine simplicial toric variety

Let $k$ be an algebraically closed field of characteristic zero. Let $\sigma$ be the cone in $\mathbb{R}^3$ generated by $e_1,2e_1+e_2,e_1+2e_2+3e_3$.
Then it is easy to check that $\sigma$ is a 3-...

4
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29
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### Are bounded groups of thin operators on Hilbert space similar to groups of unitaries?

QUESTION. Let $G$ be a group of bounded operators on $\ell^2$, whose elements are all of the form "identity+compact" (sometimes called "thin operators"). Is $G$ always similar, ...

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44
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### Prove that any positive integer can be written as a sum of unique powers of 3s, but allowing negative and positive signs [closed]

Show that any positive integer can be written as a sum of unique powers of 3, but allowing
both positive and negative signs. E.g., $5 = 3^2 − 3 − 1$.

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0
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72
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### On the inequality-integer system

I need to prove this inequality, but I do not have a good background in algebra, if you can guide me:
We have:
$$
p_1 + 2p_2 + \ldots +kp_k \leq q_1 + 2q_2 + \ldots +kq_k+(k+1)q_{k+1}+\ldots+tq_t
$$
...

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1
answer

32
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### Computing the expectation of a quadratic matrix form involving Bernoulli and Gaussian distributed matrices

I am working with two random matrices, $\mathbf{Z}$ and $\mathbf{H}$:
$\mathbf{Z}$ is an $N \times K$ matrix with entries sampled i.i.d. from a Bernoulli distribution: $Z_{nk} \sim \mathrm{Bernoulli}(...

2
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1
answer

36
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### Sufficient conditions for the graph measurability of a multivalued function

I am currently working on a problem related to the measurability of multi-functions in the context of mathematical economics. Specifically, I am searching for sufficient conditions regarding the graph ...

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49
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### Why can we find the volume of the n-ball through the Gaussian integral? [closed]

I already know how to derive the n-ball, but I don’t know why we are able to use the Gaussian integral to derive it. Answers would be appreciated.

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70
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### Would proving that every positive integer > 1 gets mapped to 1 and only 1 unique point in the 2D plane be useful for proving Collatz Conjecture? [closed]

Before diving into the topic of the Collatz Conjecture, I'd like to briefly share my background, not as a means of boasting, but to provide some context to my perspective. I studied computer science ...

5
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23
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### Connectivity of the space of transverse vector fields

Suppose we have a smooth, closed manifold $M$ of dimension $n$ and connectivity $k$. What can we say about the connectivity of the space of all tangent vector fields on $M$ that are transverse to the ...

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84
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### Initial conditions to falsify Rowland's conjecture

Based on the Rowland's paper (A natural prime-generating recurrence), is there any theorem to show that for which initial condition $a(1) = k$ the conjecture can be falsified?
For example, for $k$ ...

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48
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### Restrictions of affinoid Functions from wide open neighbourhoods

Let $X=\operatorname{Sp}(A)$ be an affinoid $K$-space, where $K$ is a p-adic field. Suppose that $X$ lies in the interior of another affinoid $K$-space $X'=\operatorname{Sp}(B)$. Recall that this ...

2
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57
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### Bounding the degree of the Weierstrass polynomial of a product of a holomorphic function and a polynomial

In brief. For a fixed holomorphic function $v$, I want to bound the degree $q$ of the Weierstrass polynomial $Q$ in the Weierstrass decomposition $v^TP = uQ$, in terms of the degree $p$ of the ...

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29
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### Hardness of an optimization problem when some variables are fixed

Given a general optimization problem, I would like to know what we can say about the hardness of the problem when a subset of its variables are fixed.
With the two (related) examples, it is clear that ...

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69
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### Criteria on $f$ such that $\begin{bmatrix} 1 & f(t)\\ 0 & 1 \end{bmatrix}\mathbb Z^2$ is equidistributed on the circle (periodic orbit)

Consider $\left(\begin{bmatrix} 1 & t \\ 0 & 1 \end{bmatrix}\mathbb Z^2 \right)_{t\in \mathbb R} \cong S^1$. Let $\mu$ denote the rotation invariant Haar measure $m$ on this orbit. I wonder if ...

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63
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### Gluing faces of n-cube

Assuming $C_n$ be the $n$-cube, the intersection of $C_n$ with a supporting hyperplane $H(P, v)$ is called a face or more precisely a $d$-face if the dimension is $d$.
Let $f_0$ and $f_1$ be faces ...

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84
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### Proj of Weil divisors and fibrations

Let $X$ be a normal, $\mathbb{Q}$-factorial complex projective variety, and let $A$, $B$ be two Weil divisors on $X$. From my understanding, the associated sheaves $\mathcal{O}_X(A)$, $\mathcal{O}_X(B)...

2
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31
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### Size of set of positive integers no sum of two distinct elements giving square

Question: find the size of a maximal subset $A$ of $[n]=\{1,\cdots,n\}$ satisfying that for any distinct elements $x,y\in A$, $x+y$ is not a perfect square.
Consider a graph with $n$ vertices: $x$ and ...

8
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113
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### Sheaf of compact Hausdorff spaces but not a condensed anima

Consider the site $\mathbf{CHaus}$ of compact Hausdorff spaces together with the finitely jointly surjective families of maps as coverings. Restriction induces an equivalence of categories $$ \mathbf{...

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1
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98
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### A non-example of a graded Frobenius algebra

Take the class of finite dimensional graded algebras $A = \sum_i A_i$ satisfying $|A_n| = 1$ where $A_n \neq 0$ and $A_m = 0$, for all $m > n$. What is an example in this class that is not ...

2
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77
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### Approximating $L^p$ functions by eigenfunctions of Laplacian

I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932.
In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...

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### please help me solve the problem from the textbook. I can't understand her in any way [closed]

task text:
The fishing net has the shape of a rectangle of size 245.0 x 350.0 cells. Inside the net there is a rectangular hole of size 100.0 x 155.0 cells (the boundary of the hole is whole). What is ...

1
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1
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112
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### On Dirac/ Clifford matrices

Let $(\eta^{\mu\nu})=\operatorname{diag}(+1,-1,-1,-1)$.
The Dirac matrices $\gamma^\mu$, $\mu=0,1,2,3$ satisfy by definition
$$\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}\tag{1}\label{1}$$
where $\{A,B\}=...

2
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0
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### A possible generalization of the Frobenius theorem

Motivation: We can reinterprete the Frobenious theorem as follows: Vector fields tangent to the leaves of a foliation forms a Lie algebra. In fact the flux of these vector fields do not pass throught ...

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### Finite (schema) axiomatizability of representable cylindric algebras

If we know that the class of all representable cylindric algebras of dimension $\alpha$ (for any ordinal number $\alpha>2$) is NOT finitely (schema) axiomatizable*, then does it (perhaps trivially) ...

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53
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### Does a symmetric monoidal functor between cartesian monoidal categories automatically preserve products?

Suppose $\cal A, B$ are cartesian monoidal categories, that is, categories equipped with a choice of finite products (including the nullary one, $1$). Suppose, moreover, that $F: \cal A \to B$ is a ...

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0
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### Can a holomorphic disk in the closure of a pseudoconvex domain be partially contained in the boundary?

A holomorphic disk is the image of an injective holomorphic map $f:\mathbb D \to \mathbb C^n$ from the unit disk $\mathbb D \subset \mathbb C$ to $\mathbb C^n$.
Let $\Omega$ be a pseudoconvex domain ...

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### Positivity of the Fourier transform: prove or disprove that $\operatorname{Re}(\overline{\widehat{u}}(\xi) \widehat{F\circ u}(\xi))\geq0$

Let $F:[0,\infty) \to[0,\infty)$ be increasing, $C^1$ and $L-$Lipschitz with $F(0)=0$. Let $u\in L^1 (\Bbb R^d)$, $u\geq0$ so that $F\circ u\in L^1 (\Bbb R^d)$
I would like to prove (or disprove) ...

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### Large-deviation inequalities for a class of simple random multivariate polynomials

Let $N$ be a large positive integer and let $[N] := \{1,2,\ldots,N\}$. For any $k$, let $K_{N,k}$ denote the collection of $k$-element subsets of $[N]$. Let $x=(x_1,\ldots,x_N)$ be a uniformly random ...

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### Checking for an inscribed square within a Jordan curve [closed]

Is there a method to find or prove the existence of a inscribed square within any arbitrary piecewise Jordan curve?

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75
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### Not a twin prime pair test using $\gcd$ only

Let $m$ be an odd positive integer such that $m=2k+1$, $k\in\mathbb{N}$.
Let $v$ be a vector of $n$ positive integers. Let $v(i)$ be the $i$-th element of the vector. Then we start with $v(i)=m(i+1)-2$...

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### set connectedness with two equivalent definition. how to prove they are equal? [migrated]

I've seen two different and perhaps equivalent definitions of connected sets.
set $E$ is connected when.
$\nexists{C, D}$ such that both open and $C \cap D = \emptyset$ & $E = C \cup D$
$\nexists{...

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1
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85
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### Strong law of large numbers for a sequence of random variables in different probability spaces

Is it known whether the following version of the strong law of large numbers holds?
For each $k\in\mathbb{N}$, let $\Omega_k$ be a finite set and $\mu_k$ be a probability measure on $\Omega_k$. Let $(...

-4
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0
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77
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### Structures in classes of primes [closed]

I am convinced that there exist structures in certain classes of primes. For structures I mean a very complex machinery that makes that classes of numbers prime.
Are known examples of such classes of ...

2
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0
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33
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### Klyachko-type inequalities for shifted Schur structure constants

Klyachko (+Knutson-Tao) provide a set of inequalities that are necessary and sufficient for the Littlewood-Richardson coefficient $c^{\lambda}_{\mu\nu}$ to be non-zero.
Is there a similar result for ...

18
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4
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1k
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### Why study finite topological spaces?

In rereading Thurston's essay On Proof and Progress in Mathematics I ran across this passage:
… this means that some concepts that I use freely and naturally in
my personal thinking are foreign to ...

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0
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22
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### A question about almost realcompactification

R. G. Woods made the following definition in the article ''A Tychonoff Almost Realcompactification'':
Let $a_{1}X$ denote the set {$p\in \beta X:$ there exists an ultrafilter $\mathcal{A}$ on $\...

0
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0
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57
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### Markov process with time varying transition kernels

I cross post this question from StackExchange as it may be more appropriate.
I am interested in studying the evolution of a variable $\alpha_t\in [0,1]$ governed by the following stochastic dynamical ...

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0
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94
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### Is this class of $p$-groups large?

Call a $p$-group $G$ good if for each subgroups $H, H_1, H_2\subseteq G$ for which $H_1\subseteq H$, $H_2\subseteq H$, $|H_1| = |H_2| = |H|/p$, $H_1\not= H_2$, $H'\not=\{e\}$ holds we have that there ...

1
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0
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77
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### Is the Frobenius semisimple on the de-Rham cohomology?

Suppose $K$ is a unramified finite extension of $\mathbb Q_p$, and $X$ is a projective smooth curve defined over $K$. By $p$-adic Hodge theory we know $D_{cris}(H_{et}^i(X,\mathbb Q_p))=H_{dR}^i(X)$. ...