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Link between different notions of dimentions

I was self studying dimensions in algebraic geometry. There are many notions of dimensions define algebracally, like projective dimension, infective dimension, homological dimension, cohomological ...
KAK's user avatar
  • 319
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0 answers
18 views

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\...
The Substitute's user avatar
0 votes
0 answers
19 views

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 ...
Leandro Lorenzetti's user avatar
1 vote
0 answers
16 views

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{...
abacaba's user avatar
  • 354
-3 votes
0 answers
66 views

Analytic parts of the Zeta function

please ask developer to remove this thanks lol
free_lancer's user avatar
0 votes
0 answers
17 views

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 + \...
proofromthebook's user avatar
1 vote
0 answers
29 views

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 \...
Bastam Tajik's user avatar
-1 votes
0 answers
18 views

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 ...
IV-301's user avatar
  • 1
0 votes
0 answers
60 views

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 ...
Nelson's user avatar
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-2 votes
0 answers
52 views

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{...
alidixon222's user avatar
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0 answers
23 views

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 ...
Juan Meleiro's user avatar
7 votes
1 answer
163 views

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 ...
Jean's user avatar
  • 73
2 votes
1 answer
86 views

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 ...
Smith's user avatar
  • 33
2 votes
0 answers
70 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)}\...
 Babar's user avatar
  • 31
1 vote
0 answers
66 views

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 ...
A. Kriegman's user avatar
0 votes
1 answer
152 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 ...
Anthony's user avatar
  • 31
-2 votes
0 answers
36 views

What happens if we restrict parameters 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, and if $\varphi(...
Zuhair Al-Johar's user avatar
1 vote
0 answers
26 views

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 ...
Joseph Zambrano's user avatar
4 votes
1 answer
109 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} $...
Topology_Dummy_Boy's user avatar
1 vote
0 answers
14 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, ...
Nandakumar R's user avatar
  • 5,473
0 votes
0 answers
19 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 \...
Marco Max Fiandri's user avatar
4 votes
0 answers
64 views

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 $\...
Reuben Stern's user avatar
2 votes
0 answers
22 views

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 ...
Léo S.'s user avatar
  • 121
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0 answers
25 views

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 ...
bc a's user avatar
  • 41
-5 votes
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(...
tteclinc's user avatar
3 votes
1 answer
106 views

(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 \...
jessetvogel's user avatar
0 votes
0 answers
19 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 ...
Sebastian Monnet's user avatar
1 vote
2 answers
150 views

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 $\...
Zhang Yuhan's user avatar
0 votes
0 answers
30 views

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 ...
Rowing0914's user avatar
0 votes
0 answers
59 views

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 \...
Grandes Jorasses's user avatar
9 votes
1 answer
299 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 ...
Ynir Paz's user avatar
  • 193
0 votes
0 answers
25 views

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 ...
L.C. Zhang's user avatar
  • 1,605
2 votes
0 answers
75 views

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 $L$, ...
Maty Mangoo's user avatar
0 votes
1 answer
140 views

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\,...
Z. Alfata's user avatar
  • 640
1 vote
0 answers
28 views

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 $(W_t)_t$ ...
GJC20's user avatar
  • 1,220
0 votes
0 answers
88 views

Calculate special integrals

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 ...
MathNoob's user avatar
0 votes
0 answers
73 views

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 ...
José's user avatar
  • 209
2 votes
1 answer
160 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 $f=u+\ii\,v:\bbR\to\bbC$ ...
TaQ's user avatar
  • 3,348
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 ...
Akira's user avatar
  • 815
6 votes
0 answers
130 views

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 $I$. ...
Toby Meadows's user avatar
  • 1,111
1 vote
0 answers
113 views

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,...
Tetrahedron's user avatar
1 vote
0 answers
28 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 ...
mijucik's user avatar
  • 187
1 vote
0 answers
33 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 $\partial \Omega$, $f: \Omega \rightarrow \mathbb{R}$ ...
Mohamed's user avatar
  • 11
1 vote
0 answers
31 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 $L^2(M)$ ...
Ali's user avatar
  • 4,077
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_{...
Filip's user avatar
  • 1,617
5 votes
1 answer
114 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 $i\in \mathbb{Z}$. ...
cellular's user avatar
  • 983
5 votes
1 answer
72 views

Does the first fundamental representation of $\frak{sp}_n$ generates all the other fundamental representations

Let $\mathfrak{sp_n}$ be the symplectic Lie algebra, that is, the $C_n$ complex simple Lie algebra. Is it true that the first fundamental, which is to say the vector space, representation $V_1$ of $\...
Zoltan Fleishman's user avatar
3 votes
1 answer
89 views

When the fundamental group of subgraph of groups embeds?

Given a connected graph of groups $\mathcal G$ (where edge maps are embeddings), by a subgraph we mean a graph of groups obtain by omitting some vertices, some edges, and replacing the remaining ...
tomasz's user avatar
  • 1,214
2 votes
0 answers
50 views

Quasi-isomorphisms of P-algebras

In the paper "Homotopy algebras are homotopy algebras" from Markl a notion of strong homotopy morphism between strong homotopy P-algebras is defined. The author restricts to the case where $...
groupoid's user avatar
  • 217
1 vote
0 answers
99 views

Homotopical interpretation of Langlands correspondence

Recently I began learning about homotopy theory, I am very far from being familiar with all the basic notions and constructions, however I heard of the notion of topological modular forms. I also ...
kindasorta's user avatar
  • 1,473

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