We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a full L1 setting requiring no growth assumptions on the nonline...

Markov Random Fields (MRF) are commonly used in computer vision and maching learning applications to model interactions of interdependant variables. Finding the Maximum Aposteriori (MAP) solution of an MRF is in general intractable, and one has to resort to approximate solutions. We review some of the recent literature on convex relaxations for MAP estimation. Our starting point is to notice th...

for a graph $g$ with edge set $e(g)$, the multiplicative sum zagreb index of $g$ is defined as$pi^*(g)=pi_{uvin e(g)}[d_g(u)+d_g(v)]$, where $d_g(v)$ is the degree of vertex $v$ in $g$.in this paper, we first introduce some graph transformations that decreasethis index. in application, we identify the fourteen class of trees, with the first through fourteenth smallest multiplicative sum zagreb ...

‎for a graph $g$ with edge set $e(g)$‎, ‎the multiplicative second zagreb index of $g$ is defined as‎ ‎$pi_2(g)=pi_{uvin e(g)}[d_g(u)d_g(v)]$‎, ‎where $d_g(v)$ is the degree of vertex $v$ in $g$‎. ‎in this paper‎, ‎we identify the eighth class of trees‎, ‎with the first through eighth smallest multiplicative second zagreb indeces among all trees of order $ngeq 14$‎.

We give a formula of the Riemann-Hurwitz type for classes defined by multiplicative sequences as corollaries of the Chern number formula for ramified coverings. 1. 1.1. In the classical results on Riemann surfaces, we have the Riemann-Hurwitz formula which relates the topological Euler numbers of the covering space and base space. The difference of the Euler numbers can be expressed by local re...

To analyze the stability of Itô stochastic differential equations with multiplicative noise, we introduce the stochastic logarithmic norm. The logarithmic norm was originally introduced by G. Dahlquist in 1958 as a tool to study the growth of solutions to ordinary differential equations and for estimating the error growth in discretization methods for their approximate solutions. We extend the ...

We consider the long time limit theorems for the solutions of a discrete wave equation with a weak stochastic forcing. The multiplicative noise conserves the energy, and in the unpinned case also conserves the momentum. We obtain a time-inhomogeneous OrnsteinUhlenbeck equation for the limit wave function that holds both for square integrable and statistically homogeneous initial data. The limit...

Leptin’s theorem asserts that a locally compact group is amenable if and only if its Fourier algebra has a bounded (by one) approximate identity. In the language of locally compact quantum groups—in the sense of J. Kustermans and S. Vaes—, it states that a locally compact group is amenable if and only if its quantum group dual is co-amenable. It is an open problem whether this is true for gener...