A uniform estimate for rough paths

It is well known that for two p  -rough paths, if their first ⌊p⌋$\lfloor p\rfloor$ levels of iterated integrals are close in p  -variation sense, then all levels of their iterated integrals are close. In this paper, we prove that a similar result holds for the paths provided the first ⌊p⌋$\lfloor p\rfloor$ terms are close in a ‘uniform’ sense. The estimate is explicit, dimension free, and only involves the p  -variation of two paths and the ‘uniform’ distance between the first ⌊p⌋$\lfloor p\rfloor$ terms. Applications include estimation of the difference of the signatures of two uniformly close paths (Lyons and Xu, 2011 [6]), and convergence rates for Gaussian rough paths (Riedel and Xu, 2012 [7]).     

On Waringʼs problem for systems of skew-symmetric forms

This paper is devoted to a problem of finding the smallest positive integer s(m,n,k)$s(m,n,k)$ such that (m+1)$(m+1)$ generic skew-symmetric (k+1)$(k+1)$-forms in (n+1)$(n+1)$ variables as linear combinations of the same s(m,n,k)$s(m,n,k)$ decomposable skew-symmetric (k+1)$(k+1)$-forms.     

An approximation method for strictly pseudocontractive mappings

Let K$K$ be a closed convex subset of a q$q$-uniformly smooth separable Banach space, T:K→K$T:K\to K$ a strictly pseudocontractive mapping, and f:K→K$f:K\to K$ an L$L$-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1)$t\in (0,1)$, let _xt$x_t$ be the unique fixed point of tf+(1-t)T$\mathit{tf}+(1-t)T$. We prove that if T$T$ has a fixed point, then {_xt}$\{x_t\}$ converges to a fixed point of T$T$ as t$t$ approaches to 0.     

An efficient computation algorithm for a multiserver feedback retrial queue with a large queueing capacity

Kumar et al. consider the M/M/c/N+c feedback queue with constant retrial rate [1]. They provide a solution for the steady state probabilities based on the matrix-geometric method. We show that there exists a more efficient computation method to calculate the steady state probabilities when N+c$N+c$ is large. We prove that the number of zero-eigenvalues of the characteristic matrix polynomial associated with the balance equation is ⌊(N+c+2)/2⌋$\lfloor (N+c+2)/2\rfloor$. As consequence, the remaining eigenvalues inside the unit circle can be computed in a quick manner based on the Sturm sequences. Therefore, the steady state probabilities can be determined in an efficient way.     

Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation

We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical (α<1/2$\alpha <1/2$) dissipation α(−Δ)${(-\mathrm{\Delta })}^{\alpha }$: If a Leray–Hopf weak solution is Hölder continuous θ∈Cδ(R²)$\theta \in C^{\delta }({\mathbb{R}}^2)$ with δ>1−2α$\delta >1-2\alpha$ on the time interval [t₀,t]$[t_0,t]$, then it is actually a classical solution on (t₀,t]$(t_0,t]$.     

Sur la répartition des puissances modulo 1

For almost all x>1$x>1$, (xⁿ)$(x^n)$(n=1,2,…)$(n=1,2,\dots )$ is equidistributed modulo 1, a classical result. What can be said on the exceptional set? It has Hausdorff dimension one. Much more: given an (b_n)$(b_n)$ in [0,1[$[0,1[$ and ε>0$\epsilon >0$, the x  -set such that |xⁿ−b_n|<ε$|x^n-b_n|<\epsilon$ modulo 1 for n   large enough has dimension 1. However, its intersection with an interval [1,X]$[1,X]$ has a dimension <1, depending on ε and X. Some results are given and a question is proposed.     

Distances of group tables and latin squares via equilateral triangle dissections

Denote by gdist(p)$\mathrm{gdist}(p)$ the least non-zero number of cells that have to be changed to get a latin square from the table of addition modulo p  . A conjecture of Drápal, Cavenagh and Wanless states that there exists c>0$c>0$ such that gdist(p)?clog(p)$\mathrm{gdist}(p)?c\log (p)$. In this paper the conjecture is proved for c≈7.21$c\approx 7.21$, and as an intermediate result it is shown that an equilateral triangle of side n   can be non-trivially dissected into at most 5_log2(n)$5{\log }_2(n)$ integer-sided equilateral triangles. The paper also presents some evidence which suggests that gdist(p)/log(p)≈3.56$\mathrm{gdist}(p)/\log (p)\approx 3.56$ for large values of p.     

Re-nnd generalized inverses

In this paper, we give some necessary and sufficient conditions for the existence of Re-nnd and nonnegative definite {1,3}$\{1,3\}$- and {1,4}$\{1,4\}$-inverses of a matrix A∈C^n×n$A\in {\mathbb{C}}^{n\times n}$ and completely described these sets. Moreover, we prove that the existence of nonnegative definite {1,3}$\{1,3\}$-inverse of a matrix A   is equivalent with the existence of its nonnegative definite {1,2,3}$\{1,2,3\}$-inverse and present the necessary and sufficient conditions for the existence of Re-nnd {1,3,4}$\{1,3,4\}$-inverse of A.     

On the nonexistence of a relation between σ-left-porosity and σ-right-porosity

Given an arbitrarily weak notion of left-〈f〉$〈f〉$-porosity and an arbitrarily strong notion of right-〈g〉$〈g〉$-porosity, we construct an example of closed subset of R$\mathbb{R}$ which is not σ  -left-〈f〉$〈f〉$-porous and is right-〈g〉$〈g〉$-porous. We also briefly summarize the relations between three different definitions of porosity controlled by a function; we then observe that our construction gives the example for any combination of these definitions of left-porosity and right-porosity.     

Subconstituents of orthogonal graphs of odd characteristic – continued

The subconstituents of the orthogonal graph O(2ν+δ,q)$O(2\nu +\delta ,q)$, where ν?2$\nu ?2$ and δ∈{1,2}$\delta \in \{1,2\}$, over a finite field of odd characteristic are shown to be quasi-strongly regular. Furthermore, the first subconstituent is shown to be co-edge regular, and when ν?3$\nu ?3$ its automorphism group is determined. The second subconstituent is shown to be edge regular, and when ν?2$\nu ?2$ its automorphism group is determined. Their parameters and chromatic numbers are also determined.     

Diagonals and numerical ranges of direct sums of matrices

For any n-by-n matrix A  , we consider the maximum number k=k(A)$k=k(A)$ for which there is a k-by-k compression of A   with all its diagonal entries in the boundary ∂W(A)$\partial W(A)$ of the numerical range W(A)$W(A)$ of A. If A   is a normal or a quadratic matrix, then the exact value of k(A)$k(A)$ can be computed. For a matrix A   of the form B⊕C$B\oplus C$, we show that k(A)=2$k(A)=2$ if and only if the numerical range of one summand, say, B is contained in the interior of the numerical range of the other summand C   and k(C)=2$k(C)=2$. For an irreducible matrix A  , we can determine exactly when the value of k(A)$k(A)$ equals the size of A  . These are then applied to determine k(A)$k(A)$ for a reducible matrix A   of size 4 in terms of the shape of W(A)$W(A)$.     

Ewens sampling formulae with and without selection

We shall first consider the random Dirichlet partitioning of the interval into n   fragments at temperature θ>0.$\theta >0.$ Using calculus for Dirichlet integrals, pre-asymptotic versions of the Ewens sampling formulae from finite Dirichlet partitions follow up. From these preliminaries, straightforward proofs of the usual sampling formulae from random proportions with Poisson–Dirichlet (PD)(γ)$(\mathrm{PD})(\gamma )$ distribution can be obtained, while considering the Kingman limit n↗∞$n\nearrow \infty$, θ↘0$\theta \searrow 0$, with nθ=γ>0$n\theta =\gamma >0$.     

Dimension spectra of random subfractals of self-similar fractals

The dimension of a point x   in Euclidean space (meaning the constructive Hausdorff dimension of the singleton set {x}$\{x\}$) is the algorithmic information density of x  . Roughly speaking, this is the least real number dim(x)$\dim (x)$ such that r×dim(x)$r\times \dim (x)$ bits suffice to specify x   on a general-purpose computer with arbitrarily high precision 2^−r$2^{-r}$. The dimension spectrum of a set X   in Euclidean space is the subset of [0,n]$[0,n]$ consisting of the dimensions of all points in X.     

A converse of Loewner–Heinz inequality and applications to operator means

Let f(t)$f(t)$ be an operator monotone function. Then A?B$A?B$ implies f(A)?f(B)$f(A)?f(B)$, but the converse implication is not true. Let A?B$A?B$ be the geometric mean of A,B?0$A,B?0$. If A?B$A?B$, then B⁻¹?A?I$B^{-1}?A?I$; the converse implication is not true either. We will show that if f(λB+I)⁻¹?f(λA+I)?I$f{(\lambda B+I)}^{-1}?f(\lambda A+I)?I$ for all sufficiently small λ>0$\lambda >0$, then f(λA+I)?f(λB+I)$f(\lambda A+I)?f(\lambda B+I)$ and A?B$A?B$. Moreover, we extend it to multi-variable matrices means.