In this paper, we consider the linear and circular consecutive $k$-out-of-$n$ systems consisting of arbitrarily dependent components. Under the condition that at least $n-r+1$ components ($r\le n$) of the system are working at time $t$, we study the reliability properties of the residual lifetime of such systems. Also, we present some stochastic ordering properties of residual lifetime of consecutive $k$-out-of-$n$ systems. In the following, we investigate the inactivity time of the component with lifetime...

We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order $\alpha$ with drift and diffusion coefficients $b$, $\sigma$. When $\alpha \in (1,2)$, we investigate pathwise uniqueness for this equation. When $\alpha \in (0,1)$, we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether $\alpha \in (0,1)$ or $\alpha \in (1,2)$ and on whether the driving stable process is symmetric or not. Our...

We consider a family of nonlinear stochastic heat equations of the form ${\partial }_{t}u=ℒu+\sigma \left(u\right)\dot{W}$, where $\dot{W}$ denotes space–time white noise, $ℒ$ the generator of a symmetric Lévy process on $𝐑$, and $\sigma$ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure $u_0$. Tight a priori bounds on the moments of the solution are also obtained. In the particular case that $ℒf=c{f}^{\text{'}\text{'}}$ for some $cgt;0$, we prove that if $u_0$ is a finite measure of compact support, then the...

For lower-semicontinuous and convex stochastic processes $Z_n$ and nonnegative random variables ${ϵ}_n$ we investigate the pertaining random sets $A(Z_n,{ϵ}_n)$ of all ${ϵ}_n$-approximating minimizers of $Z_n$. It is shown that, if the finite dimensional distributions of the $Z_n$ converge to some $Z$ and if the ${ϵ}_n$ converge in probability to some constant $c$, then the $A(Z_n,{ϵ}_n)$ converge in distribution to $A(Z,c)$ in the hyperspace of Vietoris. As a simple corollary we obtain an extension of several argmin-theorems in the literature. In particular,...

For $X,Y\in {𝐌}_{n,m}$, it is said that $X$ is majorized by $Y$ (and it is denoted by $X{\prec }_{gt}Y$) if there exists a tridiagonal g-doubly stochastic matrix $A$ such that $X=AY$. In this paper, the linear preservers and strong linear preservers of ${\prec }_{gt}$ are characterized on ${𝐌}_{n,m}$.

An $m\times n$ matrix $R$ with nonnegative entries is called row stochastic if the sum of entries on every row of $R$ is 1. Let ${𝐌}_{m,n}$ be the set of all $m\times n$ real matrices. For $A,B\in {𝐌}_{m,n}$, we say that $A$ is row Hadamard majorized by $B$ (denoted by $A{\prec }_{RH}B)$ if there exists an $m\times n$ row stochastic matrix $R$ such that $A=R\circ B$, where $X\circ Y$ is the Hadamard product (entrywise product) of matrices $X,Y\in {𝐌}_{m,n}$. In this paper, we consider the concept of row Hadamard majorization as a relation on ${𝐌}_{m,n}$ and characterize the structure of all linear operators $T:{𝐌}_{m,n}\to {𝐌}_{m,n}$ preserving...

We consider the stochastic differential equation $X_t=X₀+{\int }_0^t(A_s+B_s{X}_s)ds+{\int }_0^t{C}_{s}d{Y}_s$, where $A_t$, $B_t$, $C_t$ are nonrandom continuous functions of t, X₀ is an initial random variable, $Y=(Y_t,t\ge 0)$ is a Gaussian process and X₀, Y are independent. We give the form of the solution ($X_t$) to (0.1) and then basing on the results of Plucińska [Teor. Veroyatnost. i Primenen. 25 (1980)] we prove that ($X_t$) is a quasi-diffusion proces.

We consider “nonconventional” averaging setup in the form $\frac{\mathrm{d}{X}^{\epsilon }\left(t\right)}{\mathrm{d}t}=\epsilon B(X^{\epsilon }\left(t\right)$, $𝛯\left(q_1\left(t\right)\right),𝛯\left(q_2\left(t\right)\right),...,𝛯\left(q_{\ell }\left(t\right)\right))$ where $𝛯\left(t\right)$, $t\ge 0$ is either a stochastic process or a dynamical system with sufficiently fast mixing while $q_j\left(t\right)={\alpha }_{j}t$, ${\alpha }_{1}lt;{\alpha }_{2}lt;\cdots lt;{\alpha }_k$ and $q_j$, $j=k+1,...,\ell$ grow faster than linearly. We show that the properly normalized error term in the “nonconventional” averaging principle is asymptotically Gaussian.

A convolution operator, bounded on $L^q({ℝ}^n)$, is bounded on $L^p({ℝ}^n)$, with the same operator norm, if $p$ and $q$ are conjugate exponents. It is well known that this fact is false if we replace ${ℝ}^n$ with a general non-commutative locally compact group $G$. In this paper we give a simple construction of a convolution operator on a suitable compact group $G$, wich is bounded on $L^q(G)$ for every $q\in [2,\mathrm{\infty })$ and is unbounded on $L^p(G)$ if $p\in [1,2)$.

For $X,Y\in {𝐌}_{n,m}$ it is said that $X$ is gut-majorized by $Y$, and we write $X{\prec }_{\mathrm{gut}}Y$, if there exists an $n$-by-$n$ upper triangular g-row stochastic matrix $R$ such that $X=RY$. Define the relation ${\sim }_{\mathrm{gut}}$ as follows. $X{\sim }_{\mathrm{gut}}Y$ if $X$ is gut-majorized by $Y$ and $Y$ is gut-majorized by $X$. The (strong) linear preservers of ${\prec }_{\mathrm{gut}}$ on ${ℝ}^n$ and strong linear preservers of this relation on ${𝐌}_{n,m}$ have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of ${\sim }_{\mathrm{gut}}$ on ${ℝ}^n$ and ${𝐌}_{n,m}$.

Let $\mu \in {\left({ℝ}^d\right)}^{\text{'}}$ be an analytic functional and let $T_{\mu }$ be the corresponding convolution operator on Sato’s space $\left({ℝ}^d\right)$ of hyperfunctions. We show that $T_{\mu }$ is surjective iff $T_{\mu }$ admits an elementary solution in $\left({ℝ}^d\right)$ iff the Fourier transform μ̂ satisfies Kawai’s slowly decreasing condition (S). We also show that there are $0\ne \mu \in {\left({ℝ}^d\right)}^{\text{'}}$ such that $T_{\mu }$ is not surjective on $\left({ℝ}^d\right)$.

For the convolution operators $A_a^{\alpha }$ with symbols ${a\left(\right|\xi \left|\right)\left|\xi \right|}^{-\alpha }expi\left|\xi \right|$, 0 ≤ Re α < n, $a\left(\right|\xi \left|\right)\in L_{\infty }$, we construct integral representations and give the exact description of the set of pairs (1/p,1/q) for which the operators are bounded from $L_p$ to $L_q$.

We consider one-parameter (C₀)-semigroups of operators in the space ${}^{\text{'}}(ℝⁿ;{ℂ}^m)$ with infinitesimal generator of the form ${(G*)|}_{{}^{\text{'}}(ℝⁿ;{ℂ}^m)}$ where G is an $M_{m\times m}$-valued rapidly decreasing distribution on ℝⁿ. It is proved that the Petrovskiĭ condition for forward evolution ensures not only the existence and uniqueness of the above semigroup but also its nice behaviour after restriction to whichever of the function spaces $(ℝⁿ;{ℂ}^m)$, ${}_{L^p}(ℝⁿ;{ℂ}^m)$, p ∈ [1,∞], ${(}_a)(ℝⁿ;{ℂ}^m)$, a ∈ ]0,∞[, or the spaces ${}_{L^q}^{\text{'}}(ℝⁿ;{ℂ}^m)$, q ∈ ]1,∞], of bounded distributions.

A $0/1$-simplex is the convex hull of $n+1$ affinely independent vertices of the unit $n$-cube $I^n$. It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute $0/1$-simplices in $I^n$ can be represented by $0/1$-matrices $P$ of size $n\times n$ whose Gramians $G=P^{\top }P$ have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part $D$ of the transposed inverse $P^{-\top }$ of $P$ is doubly stochastic and has the...

In analogy with earlier work on the forward-backward case, we consider an explicit construction of the forward-forward double stochastic product integral ${\prod }^{\to \to }(1+dr)$ with generator $dr=\lambda (d{A}^{†}\otimes dA-dA\otimes d{A}^{†})$. The method of construction is to approximate the product integral by a discrete double product ${\prod }_{(j,k)\in {\mathbb{N}}_m\times \mathbb{N}ₙ}^{\to \to }\Gamma \left(R_{m,n}^{(j,k)}\right)=\Gamma \left({\prod }_{(j,k)\in {\mathbb{N}}_m\times \mathbb{N}ₙ}^{\to \to }\left(R_{m,n}^{(j,k)}\right)\right)$ of second quantised rotations $R_{m,n}^{(j,k)}$ in different planes using the embedding of ${ℂ}^m\oplus ℂⁿ$ into L²(ℝ) ⊕ L²(ℝ) in which the standard orthonormal bases of ${ℂ}^m$ and ℂⁿ are mapped to the orthonormal sets consisting of normalised indicator...

Let ${𝕄}_{n,m}$ be the set of all $n\times m$ real or complex matrices. For $A,B\in {𝕄}_{n,m}$, we say that $A$ is row-sum majorized by $B$ (written as $A{\prec }^{\mathrm{rs}}B$) if $R\left(A\right)\prec R\left(B\right)$, where $R\left(A\right)$ is the row sum vector of $A$ and $\prec$ is the classical majorization on ${ℝ}^n$. In the present paper, the structure of all linear operators $T:{𝕄}_{n,m}\to {𝕄}_{n,m}$ preserving or strongly preserving row-sum majorization is characterized. Also we consider the concepts of even and circulant majorization on ${ℝ}^n$ and then find the linear preservers of row-sum majorization of these relations on ${𝕄}_{n,m}$. ...

We consider $L_G\left(s\right)$ to be the $L$-function attached to a particular automorphic form $G$ on $GL\left(6\right)$. We establish an upper bound for the mean square estimate on the critical line of Rankin-Selberg $L$-function $L_{G\times G}\left(s\right)$. As an application of this result, we give an asymptotic formula for the discrete sum of coefficients of $L_{G\times G}\left(s\right)$.

Let ${\alpha }_i,{\beta }_i>0$, 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let $t•x=(t^{\alpha ₁}x₁,...,t^{\alpha ₙ}xₙ)$, $t\circ x=(t^{\beta ₁}x₁,...,t^{\beta ₙ}xₙ)$ and $\left|\right|x\left|\right|={\sum }_{i=1}^n{\left|x_i\right|}^{1/{\alpha }_i}$. Let φ₁,...,φₙ be real functions in $C^{\infty }(ℝⁿ-0)$ such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on ${ℝ}^{2n}$ given by $\mu \left(E\right)={\int }_{ℝⁿ}{\chi }_E{(x,\phi \left(x\right))\left|\right|x\left|\right|}^{\gamma -\alpha }dx$, where $\alpha ={\sum }_{i=1}^n{\alpha }_i$ and dx denotes the Lebesgue measure on ℝⁿ. Let $T_{\mu }f=\mu \ast f$ and let $\left|\right|T_{\mu }{\left|\right|}_{p,q}$ be the operator norm of $T_{\mu }$ from $L^p\left({ℝ}^{2n}\right)$ into $L^q\left({ℝ}^{2n}\right)$, where the $L^p$ spaces are taken with respect to the Lebesgue measure. The type set $E_{\mu }$ is defined by $E_{\mu }=(1/p,1/q):\left|\right|T_{\mu }{\left|\right|}_{p,q}<\infty ,1\le p,q\le \infty$. In the case ${\alpha }_i\ne {\beta }_k$ for 1 ≤ i,k ≤ n we characterize the...

An important result on submajorization, which goes back to Hardy, Littlewood and Pólya, states that b ⪯ a if and only if there is a doubly stochastic matrix A such that b = Aa. We prove that under monotonicity assumptions on the vectors a and b the matrix A may be chosen monotone. This result is then applied to show that $(\widetilde{L^p},L^{\infty })$ is a Calderón couple for 1 ≤ p < ∞, where $\widetilde{L^p}$ is the Köthe dual of the Cesàro space $Ce{s}_{p^{\text{'}}}$ (or equivalently the down space $L_{\downarrow }^{p^{\text{'}}}$). In particular, $(\widetilde{L¹},L^{\infty })$ is a Calderón couple, which...