On sets with rank one in simple homogeneous structures

Ove Ahlman; Vera Koponen

Displaying similar documents to “On sets with rank one in simple homogeneous structures”

Random ε-nets and embeddings in $ℓ_{\infty}^{N}$

Y. Gordon, A. E. Litvak, A. Pajor, N. Tomczak-Jaegermann (2007)

Studia Mathematica

Similarity:

We show that, given an n-dimensional normed space X, a sequence of $N = {(8 / ε)}^{2 n}$ independent random vectors ${(X_{i})}_{i = 1}^{N}$ , uniformly distributed in the unit ball of X*, with high probability forms an ε-net for this unit ball. Thus the random linear map $Γ : ℝ \to ℝ^{N}$ defined by $Γ x = {(⟨ x, X_{i} ⟩)}_{i = 1}^{N}$ embeds X in $ℓ_{\infty}^{N}$ with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into $ℓ_{\infty}^{N}$ with asymptotically best possible relation between N, n, and ε.

Complete convergence theorems for normed row sums from an array of rowwise pairwise negative quadrant dependent random variables with application to the dependent bootstrap

Andrew Rosalsky, Yongfeng Wu (2015)

Applications of Mathematics

Similarity:

Let ${X_{n, j}, 1 \leq j \leq m (n), n \geq 1}$ be an array of rowwise pairwise negative quadrant dependent mean 0 random variables and let $0 < b_{n} \to \infty$ . Conditions are given for $\sum_{j = 1}^{m (n)} X_{n, j} / b_{n} \to 0$ completely and for ${max}_{1 \leq k \leq m (n)} | \sum_{j = 1}^{k} X_{n, j} | / b_{n} \to 0$ completely. As an application of these results, we obtain a complete convergence theorem for the row sums $\sum_{j = 1}^{m (n)} X_{n, j}^{*}$ of the dependent bootstrap samples ${{X_{n, j}^{*}, 1 \leq j \leq m (n)}, n \geq 1}$ arising from a sequence of i.i.d. random variables ${X_{n}, n \geq 1}$ .

About the generating function of a left bounded integer-valued random variable

Charles Delorme, Jean-Marc Rinkel (2008)

Bulletin de la Société Mathématique de France

Similarity:

We give a relation between the sign of the mean of an integer-valued, left bounded, random variable $X$ and the number of zeros of $1 - Φ (z)$ inside the unit disk, where $Φ$ is the generating function of $X$ , under some mild conditions

Cardinalities of DCCC normal spaces with a rank 2-diagonal

Wei-Feng Xuan, Wei-Xue Shi (2016)

Mathematica Bohemica

Similarity:

A topological space $X$ has a rank 2-diagonal if there exists a diagonal sequence on $X$ of rank $2$ , that is, there is a countable family ${𝒰_{n} : n \in ω}$ of open covers of $X$ such that for each $x \in X$ , ${x} = ⋂ {{St}^{2} (x, 𝒰_{n}) : n \in ω}$ . We say that a space $X$ satisfies the Discrete Countable Chain Condition (DCCC for short) if every discrete family of nonempty open subsets of $X$ is countable. We mainly prove that if $X$ is a DCCC normal space with a rank 2-diagonal, then the cardinality of $X$ is at most $𝔠$ . Moreover, we prove that if $X$ is a first...

Positivity of integrated random walks

Vladislav Vysotsky (2014)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

Take a centered random walk $S_{n}$ and consider the sequence of its partial sums $A_{n} : = \sum_{i = 1}^{n} S_{i}$ . Suppose $S_{1}$ is in the domain of normal attraction of an $α$ -stable law with $1 l t; α \leq 2$ . Assuming that $S_{1}$ is either right-exponential (i.e. $ℙ (S_{1} g t; x | S_{1} g t; 0) = e^{- a x}$ for some $a g t; 0$ and all $x g t; 0$ ) or right-continuous (skip free), we prove that $ℙ {A_{1} g t; 0, \dots, A_{N} g t; 0} \sim C_{α} N^{1 / (2 α) - 1 / 2}$ as $N \to \infty$ , where $C_{α} g t; 0$ depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

Type and cotype of operator spaces

Hun Hee Lee (2008)

Studia Mathematica

Similarity:

We consider two operator space versions of type and cotype, namely $S_{p}$ -type, $S_{q}$ -cotype and type (p,H), cotype (q,H) for a homogeneous Hilbertian operator space H and 1 ≤ p ≤ 2 ≤ q ≤ ∞, generalizing “OH-cotype 2” of G. Pisier. We compute type and cotype of some Hilbertian operator spaces and $L_{p}$ spaces, and we investigate the relationship between a homogeneous Hilbertian space H and operator spaces with cotype (2,H). As applications we consider operator space versions of generalized little...

On the real $X$ -ranks of points of $ℙ^{n} (ℝ)$ with respect to a real variety $X \subset ℙ^{n}$

Edoardo Ballico (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

Let $X \subset ℙ^{n}$ be an integral and non-degenerate $m$ -dimensional variety defined over $ℝ$ . For any $P \in ℙ^{n} (ℝ)$ the real $X$ -rank $r_{X, ℝ} (P)$ is the minimal cardinality of $S \subset X (ℝ)$ such that $P \in 〈 S 〉$ . Here we extend to the real case an upper bound for the $X$ -rank due to Landsberg and Teitler.

On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables

Alexander R. Pruss (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let Ω be a countable infinite product $Ω ₁^{ℕ}$ of copies of the same probability space Ω₁, and let Ξₙ be the sequence of the coordinate projection functions from Ω to Ω₁. Let Ψ be a possibly nonmeasurable function from Ω₁ to ℝ, and let Xₙ(ω) = Ψ(Ξₙ(ω)). Then we can think of Xₙ as a sequence of independent but possibly nonmeasurable random variables on Ω. Let Sₙ = X₁ + ⋯ + Xₙ. By the ordinary Strong Law of Large Numbers, we almost surely have $E_{*} [X ₁] \leq l i m i n f S ₙ / n \leq l i m s u p S ₙ / n \leq E * [X ₁]$ , where $E_{*}$ and E* are the lower and upper expectations....

Classifying homogeneous ultrametric spaces up to coarse equivalence

Taras Banakh, Dušan Repovš (2016)

Colloquium Mathematicae

Similarity:

For every metric space X we introduce two cardinal characteristics $c o v^{♭} (X)$ and $c o v^{♯} (X)$ describing the capacity of balls in X. We prove that these cardinal characteristics are invariant under coarse equivalence, and that two ultrametric spaces X,Y are coarsely equivalent if $c o v^{♭} (X) = c o v^{♯} (X) = c o v^{♭} (Y) = c o v^{♯} (Y)$ . This implies that an ultrametric space X is coarsely equivalent to an isometrically homogeneous ultrametric space if and only if $c o v^{♭} (X) = c o v^{♯} (X)$ . Moreover, two isometrically homogeneous ultrametric spaces X,Y are coarsely equivalent if and...

Factorization of CP-rank- $3$ completely positive matrices

Jan Brandts, Michal Křížek (2016)

Czechoslovak Mathematical Journal

Similarity:

A symmetric positive semi-definite matrix $A$ is called completely positive if there exists a matrix $B$ with nonnegative entries such that $A = B B^{⊤}$ . If $B$ is such a matrix with a minimal number $p$ of columns, then $p$ is called the cp-rank of $A$ . In this paper we develop a finite and exact algorithm to factorize any matrix $A$ of cp-rank $3$ . Failure of this algorithm implies that $A$ does not have cp-rank $3$ . Our motivation stems from the question if there exist three nonnegative polynomials of degree at...

Asymptotic behavior of a stochastic combustion growth process

Alejandro Ramírez, Vladas Sidoravicius (2004)

Journal of the European Mathematical Society

Similarity:

We study a continuous time growth process on the $d$ -dimensional hypercubic lattice $𝒵^{d}$ , which admits a phenomenological interpretation as the combustion reaction $A + B \to 2 A$ , where $A$ represents heat particles and $B$ inert particles. This process can be described as an interacting particle system in the following way: at time 0 a simple symmetric continuous time random walk of total jump rate one begins to move from the origin of the hypercubic lattice; then, as soon as any random walk visits a site...

On uniqueness of distribution of a random variable whose independent copies span a subspace in $L_{p}$

S. Astashkin, F. Sukochev, D. Zanin (2015)

Studia Mathematica

Similarity:

Let 1 ≤ p < 2 and let $L_{p} = L_{p} [0, 1]$ be the classical $L_{p}$ -space of all (classes of) p-integrable functions on [0,1]. It is known that a sequence of independent copies of a mean zero random variable $f \in L_{p}$ spans in $L_{p}$ a subspace isomorphic to some Orlicz sequence space $l_{M}$ . We give precise connections between M and f and establish conditions under which the distribution of a random variable $f \in L_{p}$ whose independent copies span $l_{M}$ in $L_{p}$ is essentially unique.

Giant component and vacant set for random walk on a discrete torus

Itai Benjamini, Alain-Sol Sznitman (2008)

Journal of the European Mathematical Society

Similarity:

We consider random walk on a discrete torus $E$ of side-length $N$ , in sufficiently high dimension $d$ . We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time $u N^{d}$ . We show that when $u$ is chosen small, as $N$ tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const $log N$ . Moreover, this connected component occupies a...

The 4-string braid group $B_{4}$ has property RD and exponential mesoscopic rank

Sylvain Barré, Mikaël Pichot (2011)

Bulletin de la Société Mathématique de France

Similarity:

We prove that the braid group $B_{4}$ on 4 strings, its central quotient $B_{4} / 〈 z 〉$ , and the automorphism group $Aut (F_{2})$ of the free group $F_{2}$ on 2 generators, have the property RD of Haagerup–Jolissaint. We also prove that the braid group $B_{4}$ is a group of intermediate mesoscopic rank (of dimension 3). More precisely, we show that the above three groups have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.

Coherent randomness tests and computing the $K$ -trivial sets

Laurent Bienvenu, Noam Greenberg, Antonín Kučera, André Nies, Dan Turetsky (2016)

Journal of the European Mathematical Society

Similarity:

We introduce Oberwolfach randomness, a notion within Demuth’s framework of statistical tests with moving components; here the components’ movement has to be coherent across levels. We show that a ML-random set computes all $K$ -trivial sets if and only if it is not Oberwolfach random, and indeed that there is a $K$ -trivial set which is not computable from any Oberwolfach random set. We show that Oberwolfach random sets satisfy effective versions of almost-everywhere theorems of analysis,...

Simultaneous stabilization in $A_{ℝ} ()$

Raymond Mortini, Brett D. Wick (2009)

Studia Mathematica

Similarity:

We study the problem of simultaneous stabilization for the algebra $A_{ℝ} ()$ . Invertible pairs $(f_{j}, g_{j})$ , j = 1,..., n, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair (α,β) of elements such that $α f_{j} + β g_{j}$ is invertible in this algebra for j = 1,..., n. For n = 2, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since $A_{ℝ} ()$ has stable rank two, we are faced here with a different...

On the geometry of proportional quotients of $l ₁^{m}$

Piotr Mankiewicz, Stanisław J. Szarek (2003)

Studia Mathematica

Similarity:

We compare various constructions of random proportional quotients of $l ₁^{m}$ (i.e., with the dimension of the quotient roughly equal to a fixed proportion of m as m → ∞) and show that several of those constructions are equivalent. As a consequence of our approach we conclude that the most natural “geometric” models possess a number of asymptotically extremal properties, some of which were hitherto not known for any model.

Unique decomposition for a polynomial of low rank

Edoardo Ballico, Alessandra Bernardi (2013)

Annales Polonici Mathematici

Similarity:

Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the sth secant variety of the d-uple Veronese embedding of $ℙ^{m}$ into $ℙ^{\binom{m + d}{d} - 1}$ but that its minimal decomposition as a sum of dth powers of linear forms requires more than s summands. We show that if s ≤ d then F can be uniquely written as $F = M ₁^{d} + \dots + M_{t}^{d} + Q$ , where $M ₁, . . ., M_{t}$ are linear forms with t ≤ (d-1)/2, and Q is a binary form such that $Q = \sum_{i = 1}^{q} l_{i}^{d - d_{i}} m_{i}$ with $l_{i}$ ’s linear forms...

On bilinear forms based on the resolvent of large random matrices

Walid Hachem, Philippe Loubaton, Jamal Najim, Pascal Vallet (2013)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

Consider a $N \times n$ non-centered matrix $𝛴_{n}$ with a separable variance profile: $𝛴_{n} = \frac{D_{n}^{1 / 2} X_{n} {\tilde{D}}_{n}^{1 / 2}}{\sqrt{n}} + A_{n} .$ Matrices $D_{n}$ and ${\tilde{D}}_{n}$ are non-negative deterministic diagonal, while matrix $A_{n}$ is deterministic, and $X_{n}$ is a random matrix with complex independent and identically distributed random variables, each with mean zero and variance one. Denote by $Q_{n} (z)$ the resolvent associated to $𝛴_{n} 𝛴_{n}^{*}$ , i.e. $Q_{n} (z) = {(𝛴_{n} 𝛴_{n}^{*} - z I_{N})}^{- 1} .$ Given two sequences of deterministic vectors $(u_{n})$ and $(v_{n})$ with bounded Euclidean norms, we study the limiting behavior of the random bilinear form:...

Geometrically strictly semistable laws as the limit laws

Marek T. Malinowski (2007)

Discussiones Mathematicae Probability and Statistics

Similarity:

A random variable X is geometrically infinitely divisible iff for every p ∈ (0,1) there exists random variable $X_{p}$ such that $X \overset{d}{=} \sum_{k = 1}^{T (p)} X_{p, k}$ , where $X_{p, k}$ ’s are i.i.d. copies of $X_{p}$ , and random variable T(p) independent of $X_{p, 1}, X_{p, 2}, . . .$ has geometric distribution with the parameter p. In the paper we give some new characterization of geometrically infinitely divisible distribution. The main results concern geometrically strictly semistable distributions which form a subset of geometrically infinitely divisible distributions....

The spread of a catalytic branching random walk

Philippe Carmona, Yueyun Hu (2014)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

We consider a catalytic branching random walk on $ℤ$ that branches at the origin only. In the supercritical regime we establish a law of large number for the maximal position $M_{n}$ : For some constant $α$ , $\frac{M_{n}}{n} \to α$ almost surely on the set of infinite number of visits of the origin. Then we determine all possible limiting laws for $M_{n} - α n$ as $n$ goes to infinity.

Quasi-polynomial mixing of the 2D stochastic Ising model with “plus” boundary up to criticality

Eyal Lubetzky, Fabio Martinelli, Allan Sly, Fabio Lucio Toninelli (2013)

Journal of the European Mathematical Society

Similarity:

We considerably improve upon the recent result of [37] on the mixing time of Glauber dynamics for the 2D Ising model in a box of side $L$ at low temperature and with random boundary conditions whose distribution $P$ stochastically dominates the extremal plus phase. An important special case is when $P$ is concentrated on the homogeneous all-plus configuration, where the mixing time $T_{M I X}$ is conjectured to be polynomial in $L$ . In [37] it was shown that for a large enough inverse-temperature $β$ and...

A continuum $X$ such that $C (X)$ is not continuously homogeneous

Alejandro Illanes (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

A metric continuum $X$ is said to be continuously homogeneous provided that for every two points $p, q \in X$ there exists a continuous surjective function $f : X \to X$ such that $f (p) = q$ . Answering a question by W.J. Charatonik and Z. Garncarek, in this paper we show a continuum $X$ such that the hyperspace of subcontinua of $X$ , $C (X)$ , is not continuously homogeneous.

The absolute continuity of the invariant measure of random iterated function systems with overlaps

Balázs Bárány, Tomas Persson (2010)

Fundamenta Mathematicae

Similarity:

We consider iterated function systems on the interval with random perturbation. Let $Y_{ε}$ be uniformly distributed in [1-ε,1+ ε] and let $f_{i} \in C^{1 + α}$ be contractions with fixpoints $a_{i}$ . We consider the iterated function system $Y_{ε} f_{i} + a_{i} (1 - Y_{ε}) ⁿ_{i = 1}$ , where each of the maps is chosen with probability $p_{i}$ . It is shown that the invariant density is in L² and its L² norm does not grow faster than 1/√ε as ε vanishes. The proof relies on defining a piecewise hyperbolic dynamical system on the cube with an SRB-measure whose projection...