All Questions

0
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0answers
3 views

Closed convex cone - equivalence of definition via closure and via infinite sums

I have a set $P$ of points in a Banach space. Consider the following two cones: The closure of the set of all (finite) nonnegative linear combinations of $P$. (I.e., the topological closure of $\{\...
1
vote
0answers
26 views

Geometric interpretation of j-invariants of supersingular elliptic curves

In the classical theory of Complex Multiplication, one considers elliptic curves with an endomorphism ring larger than the integers. In this theory, it's possible to determine the j-invariants of all ...
1
vote
0answers
26 views

Do i.i.d Sums Concentrate Any Faster Than Martingales?

Suppose $X_1,X_2, \ldots, X_N \in \mathbb R^d$ are random variables with each $\|X_n\|_2 \le 1/2$ (this choice of the constant simplifies later formulae). The simplest concentration inequality I ...
1
vote
0answers
14 views

Trotter-Kato approximation theorem for transition semigroups of Feller processes

Let $E$ be a locally compact separable metric space; $(T(t))_{t\ge0}$ be a strongly continuous contraction semigroup on $C_0(E)$ with generator $C_0(E)$; $D$ be a core of $(\mathcal D(A),A)$; $E_n$ ...
1
vote
0answers
15 views

Which complete orthocomplemented lattices arise as the lattice of 'regular opens' in a closure space?

Every complete Boolean algebra arises as the lattice of regular open sets in some topological space, namely given a complete Boolean algebra $B$, the corresponding Stone space $S(B)$ will be ...
2
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0answers
26 views

Integers with a Hamiltonian Square Path

Let $n>1$ be an integer and set $[n]=\{1,\ldots,n\}$. We say that $n$ has a "Hamiltonian Square Path" if there is a bijection $\varphi:[n]\to[n]$ such that for all $k\in [n-1]$ we have that $\...
2
votes
0answers
62 views

Kazhdan–Lusztig polynomials in terms of Ext groups

Let $P_{x,w}$ be the Kazhdan–Lusztig polynomial, $\rho$ be the half sum of positive roots in $\Phi^+$, $M_x$ be the Verma module with highest weight $x\cdot(-2\rho)$ and $L_w$ be the simple highest ...
0
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0answers
63 views

Reading list in dynamical systems

So I’ve managed to gather from various sources, a plethora of books in dynamical systems. Now I’m wondering which of them to read, and in what order. So far these are the books I’ve found/been ...
0
votes
0answers
25 views

Solve recursion with characteristic polynomial [on hold]

The equation $a_n=7\cdot a_{n-1} -7\cdot a_{n-2}+175\cdot a_{n-3}+450\cdot a_{n-4}+(5+13\cdot n)\cdot9^n$ where $a_0=148, a_1=144, a_2=-55, a_3=-61$ I assume that solution will look like that $a^s_n+...
0
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0answers
10 views

Continuity of a constrained parameterized convex optimization problem

Consider the parameterized optimization problem: \begin{align} \boldsymbol{s}(p)= &\arg \min_{ \boldsymbol{x}} \quad g( \boldsymbol{x})\\ \text{s.t. } & \boldsymbol{A}(p) \textbf{x} = \...
3
votes
0answers
37 views

Analog of cellular approximation theorem for $CW_0$-complexes ($CW_\mathcal P$-complexes)

$CW_0$-complexes are analogs of $CW$-complexes, in which the "building blocks" are the rational disks $D^{n+1}_0$ whose boundaries are given by $\partial D^{n+1}_0= S^n_0$, where $S^n_0$ is a ...
1
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0answers
13 views

Convergence acceleration of a series by using optimal parameters

One of the ways of accelerating the convergence of a series is by transforming into a faster series using optimal parameters. Examples of this approach can be found in this paper. I obtained a ...
1
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0answers
7 views

Orthogonal similarity of adjacency matrices of graphs which are cospectral, have cospectral complements and have a common equitable partition

Let $G$ and $H$ be two undirected graphs of the same order (i.e., they have the same number of vertices). Denote by $A_G$ and $A_H$ the corresponding adjacency matrices. Furthermore, denote by $\bar G$...
1
vote
1answer
18 views

Probability of 4 Points being in Convex/Deltoid Configuration when their MST has 2 resp. 3 Leaf Nodes

Question: what is the probability that four distinct points in general position in the Euclidean plane are in convex configuration, depending on the number of leaf nodes in their Minimum ...
2
votes
0answers
53 views

Ramification divisor with base change

Let's work over $\mathbb{C}$. Consider the following commutative diagram \begin{array}{llllllllllll} E_1& \xrightarrow{f} &E_2\\ \downarrow{\pi} &&\downarrow{\pi}\\ P_1 & \...
0
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0answers
31 views

A comprehensive list of random walk inequalities?

I am interested in finding a comprehensive list of all noticeable random walk inequalities. ie. $S_n = \sum_{k\leq n} X_i$ for i.i.d symmetric $X_i$ I can only seem to find books/papers that list ...
1
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0answers
23 views

Shifts-induced group of a toroidal cube

Consider $[n]^d$ -- a $d$-dimensional toroidal cube with side length $n$ divided into $n^d$ unit cubes. Define $k$-shift as a following permutation type on unit cubes: choose $S \subset [n]$ with $|S| ...
0
votes
0answers
176 views

A maximization problem

Consider the minimization problem described this paper. Let $f_{\lambda}$ be the minimizer. As a part of extending my work, I am able to show the following facts $$\lim_\limits{\lambda \to 0}\|f_{\...
3
votes
0answers
90 views

Infinite Order Automorphisms of Planar Polynomials

Let $R_n$ be the integral polynomial ring $\mathbb{Z}[x_1,x_2,...,x_n]$, let $A_n$ be the group of ring automorphisms $\mathrm{Aut}(R_n)$, and for $f\in R_n$ let $\mathrm{Aut}(f)=\{\alpha\in A_n\ |\ \...
2
votes
0answers
28 views

Explicit, small resolving sets for Hamming graphs

Definition. Let $G = (V;E)$ be a finite, undirected graph. $R = \{r_1, \ldots, r_k \} \subseteq V$. $R$ resolves $G$ if $$ V \to [0, \infty]^k, v \mapsto (d_G(v,r_1), \ldots, d_G(v, r_k)) $$ is ...
2
votes
1answer
39 views

Diagonal shortcuts to minimize all-pairs shortest-paths in grid graph

Augment the grid graph $G$ on lattice points $[1,n]^2$, which connects each point to its four distance-$1$ vertical and horizontal neighbors. Augment $G$ to $G'$ by adding in one of the two $\sqrt{2}$ ...
3
votes
1answer
248 views

Quantum Field Theory: completing the “A Bridge between Mathematicians and Physicists” series

I decided to read the series "A Bridge between Mathematicians and Physicists" written by Eberhard Zeidler. But when I read the preface of the first book I realized that at first this series should be ...
0
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0answers
49 views

Fractional and Exterior Calculus

I am interested in fractional calculus and was wondering is there an equivalent/analagous fractional exterior calculus or fractional theory of differential forms? If so can someone point me out to ...
5
votes
1answer
104 views

Maxima of Brownian motion

It is well-known that Brownian motion attains infinitely many maxima in each time interval $[0,T]$ a.s.. From a physics perspective it seems reasonable that when the disorder of the path of a ...
2
votes
1answer
149 views

A simple proof for a case where: $\mathbf{L}_\mu \models ZF^-$?

I am looking for a simple proof (no fine structure, please) of the following: Let $\lambda$ be a limit ordinal, and $\mu < \lambda$, infinite: If $\mathbf{L}_\lambda \models \texttt{"}\mu \mbox{ ...
0
votes
0answers
7 views

Can a bramble of maximal order be efficiently found from a tree decomposition of minimal width?

Let $G$ be a connected graph with vertices $V(G)$. A bramble of $G$ is a set of connected subgraphs $H_1,\ldots,H_n$ such that for each $i$ and $j$, $H_i$ touches $H_j$; that is, either $H_i$ ...
1
vote
0answers
50 views

Joint drunkard walks

The drunkard walk is a game where two players have $a$ and $b$ dollars, respectively, and they play a series of fair games (both risking one dollar in each game) until one of them goes broke. My ...
0
votes
0answers
22 views

Computation of Wasserstein Profile Function

I am trying to compute the value of the following function $R:\mathbb{R}^m\mapsto\mathbb{R}$ $$R_n(\theta)=\sup_{\lambda\in\mathbb{R}^m}\left\{-\frac{1}{n}\sum_{i=1}^{n}\sup_{x\in\mathbb{R}^m}\lbrace ...
0
votes
0answers
69 views

Do disjoint manifolds get separated by embedded disks in higher Euclidean space?

Let $A,B$ two disjoint $p$ and $q$ manifolds embedded in $R^n$. Can we find always a PL-map $f:R^n \longrightarrow R^k$ such that $f(A)$ and $f(B)$ are contained in two separate disjoint embedded $k$-...
0
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0answers
44 views

When volume ratio of concentric balls decays faster than Gaussian?

Let $\mu$ be a finite measure on $\mathbb{R}^n$. Let $B_1$ to be the unit Euclidean ball centered at 0 in $\mathbb{R}^n$. Therefore, for any $t>0$ and $\theta\in\mathbb{R}^n$, $tB_1+\theta$ is the ...
-1
votes
0answers
31 views

Under what conditions is this family normal?

Let $\mathcal{S} = \{s \in \mathbb{C}\,\mid\,|\Im(s)| < 1\}$ be a strip of the complex plane. Let $q(s,z)$ be a holomorphic function on $\mathcal{S} \times \mathbb{C}$. Letting $\mathcal{K}$ be a ...
7
votes
2answers
407 views

Roots of $x^n-x^{n-1}-\cdots-x-1$

It is easy to see that $f(x)=x^n-x^{n-1}-\cdots-x-1$ has only one positive root $\alpha$ which lies in the interval $(1,2)$. But it is claimed that this root is a Pisot number, i.e., the other roots ...
3
votes
0answers
96 views

Properties of rings of global functions of open subschemes

It is known (although maybe not so well) that there are nice algebraic varieties whose ring of global functions is not finitely generated over the ground field. One can find examples on the web, but ...
1
vote
0answers
24 views

Describing hull of vertex intersections of two convex bounded polytopes?

We have two convex bounded polytopes $P_1$ and $P_2$ where a. $P_2\subseteq P_1$ b. $\mathcal{V}(P_2)\cap\mathcal{V}(P_1)\neq\emptyset$. Is there a name for the polytope $P=\mbox{Conv}(\mathcal{V}(...
0
votes
0answers
75 views

Stable bundles on Curves

Suppose E is an ample vector bundle on a curve. Is there a simple way to manufacture a (slope)stable vector bundle out of it, without destroying its ampleness? Sorry if this question is too vague or ...
4
votes
2answers
447 views

What is a really good book for complex variables? [on hold]

I'm an engineering student but I self-study pure mathematics. I am looking for a Complex Variables Introduction book (to study before complex analysis). I have the Brown and Churchill book but I was ...
-4
votes
0answers
105 views

Would this definition of True(F, x) select all true sentences of F? [on hold]

Would this definition of True(F, x) select all true sentences of F? Introduction to Mathematical logic Sixth edition Elliott Mendelson (2015):28 A wf C is said to be a consequence in S of a set Γ of ...
-2
votes
0answers
44 views

In search of a proof for the Mayank-Israel determinants

Let the $a_i$'s be some indeterminates. In this MO post, Mayank introduced the matrix $$M= \begin{pmatrix}2a_1&a_2&a_3&.&.\\a_2&2a_2&a_3&.&.\\a_3&a_3&2a_3&.&...
-1
votes
0answers
53 views

Divisibility Properties of Pisano Periods

Let $(F_n)$ the Fibonacci sequence and $\pi(m)$ the Pisano period of $m$ (i.e., the smallest period of $F_n \pmod{m}$). There are many proved results about $\pi(m)$. For example, it is known that $\pi(...
1
vote
0answers
50 views

How to obtain mathematical expectation with the vector as random variable?

In my study, I wish to get the mathematical expectation for the term below. The vector $\boldsymbol{z} \in \mathcal{C}^{N\times1}$ and $\boldsymbol z \sim \mathcal{CN}\left(\boldsymbol{0},\boldsymbol{...
-1
votes
0answers
73 views

homotopy VS isotopy classes of embeddings

Let $X$ any compact set in $R^n$, not necessarily a manifold. Let $f,g:X \longrightarrow M^k $ be two homotopic PL embeddings of $X$. When $X$ is an $m$-manifold then the two embeddings are also ...
20
votes
0answers
442 views

Are we better in computing integrals than mathematicians of 19th century?

When I started to learn mathematics, I was fascinating by legendary «Демидович»: problems in mathematical analysis. Fifteen years later, when I open chapters about integrals, I see a long list of ...
0
votes
0answers
62 views

Elementary constraints for the solutions of $z^{n-2}y(y+z)=x^n$?

Related to FLT and this question. For natural $n > 4 $ define the curve $C_n : z^{n-2}y(y+z)=x^n$. $C_n$ has the trivial points with $x=0$ for all $n$. The answer in the linked question shows ...
7
votes
0answers
109 views

Can a non-trivial algebraic variety carry a vector bundle whose total space is affine space?

Suppose $X$ is an algebraic variety over $\mathbb{C}$, and let $Y\to X$ be an algebraic vector bundle. Suppose $Y$ is algebraically isomorphic to $\mathbb{C}^n$ for some $n$. Does it follow that $X$ ...
0
votes
0answers
21 views

Obtaining a lower bound on the expectation using the Sudakov-Fernique inequality

In my work I wish to obtain a lower bound for the term below, independent of the vector $x$. Here the expectation is taken over $h$, a standard random Gaussian vector of length $n$. The vector $x$ is ...
-1
votes
0answers
60 views

Solution existence theorems of polynomial system of equations

Consider the following system $a_i(1-\sum\limits_{z=1}^N(x_z\theta_z))=x_i(1-\theta_i)$ $b_i(1-\sum\limits_{z=1}^N(y_z\phi_z))=y_i(1-\phi_i)$ where $\theta_i=q^u(\sum\limits_{k=i+1}^N(y_kq_{ik})+\...
1
vote
0answers
79 views

Generalised CRT - How to compute the cokernel?

Let $R$ be a commutative ring of dimension one with minimal prime ideals $P_1,\ldots,P_n$. We have the canonical injective map $$\phi_n: R/(P_1 \cap \ldots \cap P_n) \to \prod_{i=1}^n R/P_i.$$ My ...
4
votes
0answers
35 views

Symmetric and anti-symmetric parts of the covariant derivative of a connection

The following is an excerpt from Sharpe's Differential Geometry - Cartan's Generalization of Klein's Erlangen Program. Now we come to the question of higher derivatives. As usual in modern ...
11
votes
1answer
253 views

Realizing cohomology classes by submanifolds

In "Quelques propriétés globales des variétés différentiables", Thom gives conditions for a class in singular homology of a compact manifold to be realized by a smooth oriented submanifold (see e.g. ...
0
votes
0answers
84 views

A typo in Jacquet-Langlands

On pages 516-517 of "Automorphic forms on $GL(2)$", there is a list of expressions contributing to Selberg trace formula. In (IV), should there be an additional $\frac{1}{2}$ multiple? In (VII), ...

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