All Questions
152,895
questions
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4
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The mean of iterations for choosing a unique set
We need to choose a set of unique numbers (with replacement) out of a group of n unique numbers. The set we choose should end up being in the size of exactly 10% of the group(n/10). What would be the ...
0
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0
answers
18
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The relay use of invariant set theory
For a dynamical system, set $A$ is an invariant set with a function $V_1$, whose derivative is semi negative definite on $A$, and the region where the derivative is $0$ is the set $B$, which is also ...
1
vote
0
answers
11
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Failure of Tomiyama's property ($F$) for reduced group $C*$-algebras
Are there known examples of discrete groups such that the minimal tensor product of their reduced group $C^\ast$-algebras does not have Tomiyama's property ($F$)?
Such groups must necessarily be non-...
1
vote
0
answers
45
views
Grothendieck group and an almost localization
Let $T$ be a small triangulated category and let $S\subset T$ be a full triangulated subcategory. We denote this embedding by $I: S\rightarrow T$.
Let $F: T\rightarrow S$ be a triangulated functor ...
-4
votes
0
answers
61
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Link between different notions of dimensions [closed]
I was self studying dimensions in algebraic geometry. There are many notions of dimensions define algebraically, like projective dimension, injective dimension, homological dimension, cohomological ...
0
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0
answers
35
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Must "$p$ divides $X$" and "$q$ divides $X$" be independent if $\omega\left(X\left(n\right)\right)\sim N\left(\log \log n, \log \log n\right)$?
Let $p$ and $q$ denote distinct primes.
For a uniform variable $N\left(n\right)$ on $\lbrace 1,\ldots, n\rbrace$, the events $\lbrace N\left(n\right) \text{ is divisible by } p\rbrace$ and $\lbrace N\...
1
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0
answers
51
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Reference for cocommutative coalgebras
I'm looking for references on cocommutative coalgebras where I can see them as kind of infinitesimal spaces. I'm trying to understand this post Why do Lie algebras pop up, from a categorical point of ...
2
votes
0
answers
22
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Covering base sets $X$ with a subset family satisfying a "partial covering property"
Let $X$ be an $n$ element base set. Suppose I have a subset family $\mathscr{F} \subset 2^X$ satisfying the following property:
(*) For any subset $Y \subset X$, we can find an element $F \in \mathscr{...
-5
votes
0
answers
88
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Mellin Transform and Analytic parts of the Zeta function [closed]
Suppose I had this transform
\begin{equation}
\zeta(s) = \dfrac{1}{\Gamma(s)}\int_{0}^{\infty}{\dfrac{x^{s-1}}{e^x-1}dx}\tag{1}
\end{equation}
for $0\le\Re(s)\le 1$. If we write $$\Gamma(s) = \alpha_1(...
0
votes
0
answers
22
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Boundary behavior of bilateral Laplace transform
Of all the examples I know of (bilateral) Laplace transforms $F$ defined on their maximal vertical strips $V_{a,b}=\{ z \in \mathbf{C} : a < Re(z) < b \}$ with $-\infty < a < b \leq + \...
1
vote
0
answers
85
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The Lebesgue covering dimension of the Cosmic String interval topology
Take the spacetime $(M,g)$ that satistfies Einstein's Field Equations exactly where $g$ is locally:
$$g=-(cdt-a d\phi)^2 + d \rho^2 + \kappa^2 \rho^2 d \phi^2 +dz^2 \ , \ \ \kappa>0 , \ a\in \...
-1
votes
0
answers
20
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Find a conditional for factorizing the sum of a set of gaussian integer-valued matrices
In my research project, we're exploring the decomposition of Gaussian integer-valued square matrices as a cross-product of other Gaussian integer matrices (GIM) with the same dimension. One of the ...
0
votes
0
answers
64
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CAT(k) spaces and law of cosines
Let $M^n_{\kappa}$ be a model space with curvature $\kappa>0$. The sides are $a,b,c$, and the angle at the vertex opposite to the side of $c$ is $\gamma$. $c$ is known.
Consider the corresponding ...
-1
votes
0
answers
62
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Reference request - logarithmic average
Consider a set $A\subseteq\mathbb{N}$. Consider an arithmetic function $a(n):\mathbb{N}\to\mathbb{C}$. I am looking for notation which describes the following:\begin{equation}\frac{\sum_{n\in A}\frac{...
0
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0
answers
24
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Maximizing the integral of a transformation that depends on a neighborhood of values of the original function
I'm not an expert in analysis whatsoever, so I might be posing a well-established question, or even an unanswerable one.
We are working with non-negative real functions over a sufficiently nice region ...
9
votes
1
answer
237
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Identifying two definitions of orientation on a vector space
Let $V$ be an $n$-dimensional real vector space. Here are two definitions of an orientation on $V$:
A generator of the $1$-dimensional vector space $\wedge^n V$, up to multiplication by positive ...
4
votes
1
answer
124
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Multiplication factors in folding root systems and Lie algebras by automorphisms
When Stembridge, in the paper Folding by automorphisms, considers folding by automorphism $\sigma$ he considers the root system generated by for each orbit $J$.
$$\sum_{i \in J} \alpha_i .$$
Whereas ...
2
votes
0
answers
80
views
Limit involving the fractional part and the Fibonacci numbers
Helo,
Let $F(n)$ be the $n$th Fibonacci number, if $\left\{ x\right\}$ denotes the fractional part of $x$, how proving
$$\lim_{n\rightarrow\infty}\frac{1}{2n}\sum_{k=1}^{2n}\left\{ \frac{F(2n)}{F(k)}\...
1
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0
answers
80
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Does anyone have a good example of an injective resolution?
I'm learning about injective resolutions and derived functor sheaf cohomology, and it seems that every source on injective resolutions gives no examples. I feel like just one good example would make ...
0
votes
1
answer
179
views
Decomposition of identity
Fix an integer $n$ and consider a finite numbers $m$ of subsets $ S_i \subset [n]$ such that $$ \bigcup_{i = 1}^m S_i = [n].$$ Do we have a necessary and sufficient condition on the subsets $S_i$ so ...
-2
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0
answers
67
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What happens if we restrict inputs in Separation and Replacement axioms to definable sets?
If we replace the axiom of Foundation by
Foundation schema: if $\varphi(x)$ is a formula in which "$x$" occurs free and only free, and in which "$y$" doesn't occur, whose free ...
1
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0
answers
29
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Structural description of a particular set motivated by graph reconstruction
$\DeclareMathOperator\Coh{Coh}\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\Aut{Aut}$In this post, I asked a question regarding a particular function $\psi$ whose construction is motivated by the ...
4
votes
1
answer
134
views
Dehn surgery on $RP^2 \times S^1$
A standard example of Dehn surgery is obtaining $S^3$ from $S^2 \times S^1$. Consider a unknot $L$ wrapping the non-trivial cycle $S^1$ in $S^2 \times S^1$. We drill out a tubular neighborhood $T_{L} $...
1
vote
0
answers
19
views
Finding the point within a convex n-gon that minimizes the largest angle subtended there by an edge of the n-gon
This post records a variant to the question asked in this post: Finding the point within a convex n-gon that maximizes the least angle subtended there by an edge of the n-gon
Given a convex n-gon, ...
0
votes
0
answers
22
views
Comparison between the expected values of the inverse of the CDF of binomial-distributed random variables
Let us denote with $F(x;j,\mu)$ the cdf of a Binomial distributed random variable with $j$ trial with success probability $\mu$ considered in $x$, and let $f(x;j,\mu)$ be the pmf. Defining $0\leq \...
4
votes
0
answers
76
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A "lax Boardman-Vogt tensor product," or what object represents duoidal categories?
Let me preface this by saying I'm not sure what the fundamental examples should be, and perhaps that's part of my question.
The Boardman-Vogt tensor product of $\infty$-operads $\mathcal{O}$ and $\...
3
votes
0
answers
24
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Coherence of the graphical language for pivotal categories
Throughout I follow A survey of graphical languages for monoidal categories, Peter Selinger, arXiv.
A pivotal category is a monoidal category where each object $A$ has a dual $A^*$, together with a ...
0
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0
answers
27
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Does this "linear-approximated" version of Graph Counting Lemma hold?
Let $0\leq d\ll\varepsilon,\frac{1}{e},\frac{1}{v}\leq 1.$ Let $G$ be a $n$-vertices graph ($n$ is sufficient large, $1/n\ll d$) and for any $A,B\subseteq V(G)$, the edge density $d(A,B)\geq d.$ Then ...
-5
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0
answers
35
views
Chain Rule of Quadratic Matrix? [closed]
Hi I want to find an easy way to get the derivative of this function expressed in matrix format. What is results of df/dy? Thanks a ton!!
χ and y are vectors
Λ and Ω are square matrices
f(y)=(χ-Ωy)'Λ(...
3
votes
1
answer
112
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(Derived category of) sheaves over an infinite union
The short version of my question is:
Suppose $X$ is a (reasonably nice) topological space such that $X = \bigcup_{n \ge 1} X_n$ for an increasing sequence of (closed) subspaces $X_1 \subset X_2 \...
0
votes
0
answers
20
views
Time complexity of Magma's `NormEquation` for quadratic extensions of $2$-adic fields
Note: This is similar to, but easier than, a previous question I asked here. It is a different question! I'm hoping this one might get an answer because it concerns a standard algorithm, whereas the ...
2
votes
2
answers
221
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Naturality of Lie bracket - alternate proof
Let $M$ and $N$ be smooth manifolds, and let $F: M \to N$ be a smooth map. Let $X$ and $Y$ be vector fields on $M$, and let $\tilde{X}$ and $\tilde{Y}$ be vector fields on $N$. We say that $X$ and $\...
0
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0
answers
31
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Vertices of hyperbolic quadrilateral with given angles
In a previous post (link to previous post), I received help from user dan_fulea on constructing a hyperbolic triangle with given angles. Now, I am attempting to extend this method to quadrilaterals in ...
0
votes
0
answers
59
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Convolution of $\mathscr{F}\{ \log \}(x) * \mu$ with compactly supported measure $\mu$
As I read in this post the Fourier transform of $\psi(\lambda) = \log{|\lambda|}$ must be interpreted in distributional sense and it is given by:
$$\mathscr{F}\{\psi\}(x)=-2\pi \gamma \delta(x)-\pi \...
10
votes
1
answer
350
views
Long chains of amorphous cardinalities
An amorphous set is an infinite set that cannot be partitioned into 2 infinite subsets. An amorphous cardinality is the cardinality of an amorphous set. Working in $\sf ZF$, it is consistent that ...
0
votes
0
answers
25
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Are there 4-connected planar non-hamilton multi-graphs?
Tutte proved the famous result: Every planar 4-connected graph has a hamiltonian cycle. But I read in Section 111.6.5 on book Eulerian Graphs and Related Topics that the author Herbert Fleischner ...
2
votes
0
answers
76
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Artin-Schreier theorem for rings (a little different)
Motivation:
Let me recall the well-known Artin-Schreier theorem (AST) for fields in a non-formal way; if $L$ is an algebraically closed field, and $K \subset L$ a subfield not 'much smaller' than $L$, ...
0
votes
1
answer
142
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Check an equation on the Heisenberg group $H_1$
The Heisenberg group $H_1$ is the set $\mathbb C\times \mathbb R$ endowed with the group law
$$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right); \quad \forall z,w \in \mathbb C\,...
1
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0
answers
28
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Wellposedness of SDE with switching diffusion
Let $b:\mathbb R\to [-1,1]$ and $a_1, a_2:\mathbb R\to [1,2]$ be Lipschitz functions. Consider the stochastic differential equation (SDE) as follows :
$$dX_t = b(X_t)dt + a(X_t)dW_t,$$
where $(W_t)_t$ ...
0
votes
0
answers
91
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Calculate special integrals [migrated]
Prove:$$\int_0^1\frac{1-\cos x}{x} \, dx-\int_1^{+\infty}\frac{\cos x}{x} \, dx=\gamma, \\ \gamma=\lim_{n\to\infty}(1+1/2+\cdots+1/n-\ln n)$$
I try to use the Taylor expansion of $1-\cos x$ and ...
0
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0
answers
76
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Relation between nullspace and row-equivalence of matrices over $\mathbb{Z}$ and $\frac{\mathbb{Z}}{n \mathbb{Z}}$?
Two matrices $D$ and $E$ over a field have the same nullspace if only if they are row-equivalent. Is the same true if those matrices are over the ring of integers ($\mathbb{Z}$) or integers mod a ...
2
votes
1
answer
162
views
Functions with asymmetrically decreasing Fourier transform?
$\def\ii{{\rm i}}\def\bbR{\mathbb R}\def\bbC{\mathbb C}\def\bbNo{\mathbb N_0}\def\Fou{\mathscr F}$Specifically, I would like to have a compactly supported continuous function $f=u+\ii\,v:\bbR\to\bbC$ ...
0
votes
0
answers
34
views
Lipschitz approximation of a probability measure with finite $1$-st moment by the ones with finite $p$-th moment
For $p \in [1, \infty)$, let $\mathcal P_p (\mathbb{R^d})$ be the space of Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. We endow $\mathcal P_p (\mathbb{R^d})$ with the ...
6
votes
0
answers
133
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Do precipitous ideals "always" come from collapsing?
It's well-known that if $\kappa$ is a measurable cardinal, then there is a poset $\mathbb{P}$ that forces $\kappa$ to carry a precipitous ideal.
Suppose that $\omega_1$ carries a preciptous ideal $I$.
...
1
vote
0
answers
120
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An open ended question: The dual of a covering map? Is this a real thing?
Reposted from this Reddit post as I didn't get good answers there:
So I've been reading about the Galois theory of covering maps and been staring at this equation for way too long:
$$\left| \pi_1(X,...
1
vote
0
answers
31
views
Hyperplane arrangements and tropical linear spaces
I have been trying to understand Chapter 5.4 of this Brief Introduction to Tropical Geometry, but I am struggling because of my lack of mathematical background. I will ask a few questions after giving ...
1
vote
0
answers
35
views
How can we calculate the Euler-lagrange equations?
In this paper https://arxiv.org/pdf/1907.09605.pdf \
let $\Omega \subset \mathbb{R}^n$ with $n \geq 1$ be a bounded Lipschitz domain with boundary $\partial \Omega$, $f: \Omega \rightarrow \mathbb{R}$ ...
1
vote
0
answers
32
views
Density of zero modes
Let $(M,g)$ be a compact smooth Riemannian manifold with a smooth boundary. Let $\{(\lambda_k,\phi_k)\}_{k\in\mathbb N}$ be the spectral data on $(M,g)$, namely an orthonormal basis for $L^2(M)$ ...
0
votes
0
answers
37
views
Extension of a type A Springer fibre
Given a decomposition $p=(p_1,\dots,p_n)$ of $n$, one can associate its corresponding
partial flag variety $$\mathcal{B}_p=\{F=(0=F_0\subset F_1\subset \dots \subset F_n=\mathbb{C}^n) \mid \dim F_i/F_{...
5
votes
1
answer
119
views
The action of the Grothendieck group on higher K-theory groups
Let $(C,\otimes)$ be a monoidal (non symmetric) Waldhausen category. In particular, under these conditions,
$K_{0}(C)$ is a ring and $K_{i}(C)$ are $K_{0}(C)$-bimodule for any $i\in \mathbb{Z}$.
...