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Some details about Kirillov-Kostant Poisson bracket
Let $G$ be a finite dimensional Lie group with Lie algebra $\mathfrak{g}$. The Kirillov-Kostant Poisson bracket on $\mathfrak{g}^*$ is defined as $$\{\cdot ,\cdot \} :C^{\infty}(\mathfrak{g}^*)\times ...
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$H^{s,\infty}$ via Triebel Lizorkin spaces
I know that
$F^s_{p,2} = H^{s,p}$
for $p \in (1, \infty)$, but what about $H^{s,1}$ and $H^{s,\infty}$? I know one extension for $F^0_{\infty,2}$ which should be BMO. But how do I get the $H^{s,\infty}...
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ODE satisfied by a special function
Posted on MSE
Context
I would like to estimate the distribution of the difference of two inverse gaussian variables. The convolution doesn't lead to any special functions according to Mathematica . ...
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Proof of the equivalence between Triebel Spaces and Bessel Potential
I've encountered a question regarding the relationship between Triebel-Lizorkin spaces and Bessel potential spaces. Specifically, I understand that
$F^s_{p,2} = H^{s,p}$, for $p \in (1,\infty)$.
...
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Does "epsilon-regular" equal to "cut distance less than epsilon"?
Let $G$ be a bipartite graph (vertex number sufficient large) with bipartition $(U,W)$ and edge density $d$. Does these two statement equal?
$G$ is $\varepsilon$-regular, i.e. $\big|e_G(X,Y)-d|X||Y|\...
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Heuristics for constrained maximal volumes in hypercubes as $n \to \infty$
It can be shown that there is a unique maximal surface of revolution with constant positive Gaussian curvature embedded in $[0,1]^3$ with a pair of antipodal points as cone points which attain the ...
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Compatibility of 2-copulas
An $n$-copula is the joint distribution function of a distribution on $[0,1]^n$ with uniform marginals. A family of 2-copulas $(C_{i,j})_{i<j\leq n}$ is compatible if there exists an $n$-copula $\...
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Projection measure and an integral formula for Lipschitz functions
Let $n\geq m\geq 0$ be integers and put $k=n-m$. Let $A\subset\mathbb{R}^n$ be Borel measurable, we define the projection measure of $A$ as
$$\mu_k(A):=\underset{P_1, \ldots, P_r}{\sup_{A=A_1\sqcup\...
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Position dependent service time in queue
Is there any literature for queuing analysis (waiting time, capacity etc.) of a queue with service time that depends on the position of the customer in the queue?
I have encountered a problem where a ...
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Hamiltonian Circuit Counting and Classification Problem
the Problem Description
background
Consider an undirected complete graph $G_n$ with $n$ vertices, where if the numerical labels of each vertex are consecutive, then the edge weight between them is $1$,...
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Differential-vertex-deletion equation for graph functions $f(x_1,...,x_n;G)$ on $n$ vertices
I encountered a function $f$ defined over a graph $G$ in my research which does not satisfy a deletion–contraction recurrence but an equation of the form
$$\partial_k f(x_1,...,x_k,..x_n;G)=g(x_k, N_{...
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P-equitable graph partition
I am now conducting a research in regards to p equitable graph partition problem. I need to use LP techniques to kernelize this problem parametrized by vertex cover. In basic definition of such ...
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Proof that Component-wise MH algorithm is invariant w.r.t. target measure
consider a standard situation in Bayesian modelling,
given real vector parameter $\theta=(\theta_1,\dotsc,\theta_n)$ and observations $x$ we derive a posterior distribution $\pi$ with posterior ...
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An auxiliary problem while constructing the system of Jordan sets on a plane
Let $\mathfrak{S}$ be a system of rectangles in $R^2$ of the form $[a,b]\times [c,d]$
where $a,b,c, d \in R$, $a<b$, $c<d$.
Let $\mathfrak{A}$ be a system of simple sets based on $\mathfrak{S}$.
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What is the dual of a hyperbolic configuration of points?
Let $C_n$ denote the configuration space of $n$ distinct points in hyperbolic $3$-space $\mathcal{H}$. If $\mathbf{x} := (\mathbf{x}_1, \dots, \mathbf{x}_n) \in C_n$, where $\mathbf{x}_i \in \mathcal{...
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Interpretation of $|V|\,|E|$ for bipartite graphs
Background: the question is motivated by a result in statistical mechanics. I am working with a generalized exclusion process with a fixed number of particles $k$ on a finite graph $G'=(V,E')$, which ...
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Solve $S_{nm} \sum_{j}x_{nj}\operatorname*{sign}\left( \sum_{k}x_{kj} S_{km} \right) \geq 0$ for $x_{kj}$
Let $S_{nm} \in \{\pm1\}^{N\times M}$ be a matrix of signs. Can we find real numbers $x_{kj} \in \mathbb{R}^{N\times K}$, such that:
$$S_{nm} \sum_{j}x_{nj}\operatorname*{sign}\left( \sum_{k}x_{kj} S_{...
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What are known tightest bounds on packing number over hypothesis class with semi-metric distance?
Let $\mathcal{H}$ denotes a hypothesis class we define the semi-metric on $\mathcal {H}$:
$\|h_1 - h_2 \|_{\mathcal{L}_1} = \underset{x \sim \mathcal{D}}{\mathbb{P}}[h_1(x) \neq h_2(x)]$.
Are there ...
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On the Lipschitz parametrizability of polynomials of fixed Mahler measure
Background
For a polynomial $f(x) = a(x-\alpha_1) \cdots (x - \alpha_n) \in \mathbb{C}[x]$, its Mahler measure is defined to be
$$M(f) = |a| \prod_{i=1}^n \max\{1, |\alpha_i|\}$$
In Lemma 1, Masser-...
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Detecting non-optimality in disjoint unions of perfect matchings
This is a follow-up question to Minimum-weight disjoint union of perfect matchings:
let $G$ be a complete symmetric graph with $2n$ vertices, whose edges are mapped to their weights by $\omega()$ and ...
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Convergence of a series related to counting distinct prime factors
I am here to ask whether the following series is convergent for all real $z$. I am also asking whether this is everywhere real analytic. I conjecture that it is convergent for all real input, or at ...
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What is the lattice point distribution over binary quadratic forms?
Let $f(x,y)=x^2+ny^2$ be the binary quadratic form of interest and consider the lattice points $S=\{ (x,y,f(x,y)) \in \mathbb{N}^3 \}$.
For simplicity, we keep things only on quadrant I of the ...
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On the I-method's energy increment calculation in a paper of Dodson
I am currently reading Dodson's Global Well-posedness for the Defocusing, Quintic Nonlinear Schrödinger Equation in One Dimension for Low Regularity Data article and I am trying to understand Theorem ...
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How to synthetize a controller $\dot{u} = F x + G u$ which stabilizes $\dot{x} = Ax + Bu$?
$\textbf{Introduction}$: I study linear control theory. Among strategies, we begin with vector field $Ax + Bu$, $A \in M_{n^2}(\mathbb{R})$, $B \in M_{n \times m}(\mathbb{R})$, and synthesize a ...
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On the multiplicative group of quotients of polynomial rings
Related to this.
The $p+1$ factorization algorithm works over $\mathbb{Z}/n\mathbb{Z}[x]/f(x)$
and hopes $p+1$ to be smooth.
We are trying to generalize this to multivariate case
and also try to find ...
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Complexity of finding single source paths with capacity constraints and length constraints
Let $G=(V,A)$ be a directed graph with distinguished vertex $s\in V$ and let $c:A\rightarrow{\mathbb N}$ denote arc capacities. For any $t\in V,t\not=s$ we are given two numbers: $C_{t},L_{t}$. Let $...
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What does it mean for a linear algebraic group to act reductively
I was reading this paper by Baues and on page 918 he mention that $S$ acts reductively on the cochain complex and on page 919 again he mention the word "Since $T$ acts reductively on the complex.....
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Kirszbraun-like extension of periodic functions
Let $\Lambda \subset \Lambda' \subset \mathbb{R}^n$ be lattices. Let $f : \Lambda' \rightarrow \mathcal{H}$ be a $a$-Lipschitz function, where $\mathcal{H}$ is a finite-dimensional Hilbert Space. ...
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How to solve recurrences of the form $T(i,n) = \min_{x} f(x) + \max\{T(i,x),T(x,n)\}$?
I am interested in solving a recurrence of the form:
$$T(i,n) = \begin{cases}\min_{x\in\{i+1,\ldots,n-1\}} f(x) + \max\{T(i,x),T(x,n)\} & \mbox{If $i<n-1$}\\ 1 & \mbox{otherwise}\end{cases}\...
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