# All Questions

102,879 questions

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### 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 $\{\...

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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 ...

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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 ...

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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 $(\mathcal D(A),A)$;
$E_n$ ...

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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 ...

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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 $\...

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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 ...

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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 ...

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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+...

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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} = \...

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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 ...

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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 ...

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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 $\bar G$...

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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 ...

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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 & \...

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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 ...

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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| ...

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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_{\...

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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\ |\ \...

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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 ...

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**1**answer

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 $\sqrt{2}$ ...

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**1**answer

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 ...

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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 ...

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**1**answer

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 ...

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**1**answer

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{ ...

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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 $H_i$ ...

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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 ...

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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 ...

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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 $k$-...

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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 ...

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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 ...

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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 ...

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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 ...

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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}(...

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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 ...

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**2**answers

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 ...

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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 ...

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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&.&...

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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(...

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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{...

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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 ...

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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 ...

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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 ...

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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 $X$ ...

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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 ...

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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})+\...

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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 ...

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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**

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**1**answer

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. ...

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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), ...