Let $\Omega $ be a bounded open set in $\mathbb R^n$, $n \geq 2$. In a well-known paper {\it Indiana Univ. Math. J.}, 20, 1077--1092 (1971) Moser found the smallest value of $K$ such that $$ \sup \bigg \{\int _{\Omega } \exp \Big (\Big (\frac {\left |f(x)\right |}K\Big )^{n/(n-1)}\Big )\colon f\in W^{1,n}_0(\Omega ),\|\nabla f\|_{L^n}\leq 1\bigg \}<\infty . $$ We extend this result to the situation in which the underlying space $L^n$ is replaced by the generalized Zygmund space $L^n\log ^{n-1}L \log ^{\alpha }\log L$ $(\alpha <n-1)$, the corresponding space of exponential growth then being given by a Young function which behaves like $\exp (\exp (t^{n/(n-1-\alpha )}))$ for large $t$. We also discuss the case of an embedding into triple and other multiple exponential cases.
Assume that $X$, $Y$ are continuous-path martingales taking values in $\mathbb R^\nu $, $\nu \geq 1$, such that $Y$ is differentially subordinate to $X$. The paper contains the proof of the maximal inequality $$ \|\sup _{t\geq 0} |Y_t| \|_1\leq 2\|\sup _{t\geq 0} |X_t| \|_1. $$ The constant $2$ is shown to be the best possible, even in the one-dimensional setting of stochastic integrals with respect to a standard Brownian motion. The proof uses Burkholder's method and rests on the construction of an appropriate special function.
We obtain a sharp upper bound for the spectral radius of a nonnegative matrix. This result is used to present upper bounds for the adjacency spectral radius, the Laplacian spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance Laplacian spectral radius, the distance signless Laplacian spectral radius of an undirected graph or a digraph. These results are new or generalize some known results., Lihua You, Yujie Shu, Xiao-Dong Zhang., and Obsahuje seznam literatury
The theorem about the characterization of a GS-quasigroup by means of a commutative group in which there is an automorphism which satisfies certain conditions, is proved directly.
We study higher local integrability of a weak solution to the steady Stokes problem. We consider the case of a pressure- and shear-rate-dependent viscosity, i.e., the elliptic part of the Stokes problem is assumed to be nonlinear and it depends on p and on the symmetric part of a gradient of u, namely, it is represented by a stress tensor T (Du, p):= v(p, |D|2)D which satisfies r-growth condition with r \in (1, 2]. In order to get the main result, we use Calderón-Zygmund theory and the method which was presented for example in the paper Caffarelli, Peral (1998)., Václav Mácha., and Obsahuje seznam literatury
As said by Mareš and Mesiar, necessity of aggregation of complex real inputs appears almost in any field dealing with observed (measured) real quantities (see the citation below). For aggregation of probability distributions Sklar designed his copulas as early as in 1959. But surprisingly, since that time only a very few literature have appeared dealing with possibility to aggregate several different pairwise dependencies into one multivariate copula. In the present paper this problem is tackled using the well known Iterative Proportional Fitting Procedure. The proposed solution is not an exact mathematical solution of a marginal problem but just its approximation applicable in many practical situations like Monte Carlo sampling. This is why the authors deal not only with the consistent case, when the iterative procedure converges, but also with the inconsistent non-converging case. In the latter situation, the IPF procedure tends to cycle (when combining three pairwise dependencies the procedure creates three convergent subsequences), and thus the authors propose some heuristics yielding a "solution'' of the problem even for inconsistent pairwise dependence relations.
Perez's approximations of probability distributions by dependence structure simplification were introduced in 1970s, much earlier than graphical Markov models. In this paper we will recall these Perez's models, formalize the notion of a compatible system of elementary simplifications and show the necessary and sufficient conditions a system must fulfill to be compatible. For this we will utilize the apparatus of compositional models.
A simple yet powerful procedure for an echo attenuation in signals is introduced. The presented method involves no external reference signal. It is based on comb FIR filtering. To the advantages of the described method belong the simplicity and performance which are beneficial in real time implementations. For illustration, a simulation of the procedure is included. The efficiency of the presented method is demonstrated by a real time implementation on a digital signal processor.
Let $C[0,T]$ denote the space of real-valued continuous functions on the interval $[0,T]$ with an analogue $w_\varphi $ of Wiener measure and for a partition $ 0=t_0< t_1< \cdots < t_n <t_{n+1}= T$ of $[0, T]$, let $X_n\: C[0,T]\to \mathbb R^{n+1}$ and $X_{n+1} \: C [0, T]\to \mathbb R^{n+2}$ be given by $X_n(x) = ( x(t_0), x(t_1), \cdots , x(t_n))$ and $X_{n+1} (x) = ( x(t_0), x(t_1), \cdots , x(t_{n+1}))$, respectively. \endgraf In this paper, using a simple formula for the conditional $w_\varphi $-integral of functions on $C[0, T]$ with the conditioning function $X_{n+1}$, we derive a simple formula for the conditional $w_\varphi $-integral of the functions with the conditioning function $X_n$. As applications of the formula with the function $X_n$, we evaluate the conditional $w_\varphi $-integral of the functions of the form $F_m(x) = \int _0^T (x(t))^m d t$ for $x\in C[0, T]$ and for any positive integer $m$. Moreover, with the conditioning $X_n$, we evaluate the conditional $w_\varphi $-integral of the functions in a Banach algebra $\mathcal S_{w_\varphi }$ which is an analogue of the Cameron and Storvick's Banach algebra $\mathcal S$. Finally, we derive the conditional analytic Feynman $w_\varphi $-integrals of the functions in $\mathcal S_{w_\varphi }$.