Fix integers b > a ≥ 1 with g := gcd(a, b). A set S ⊆ N is {a, b}-multiplicative if ax 6= by for all x, y ∈ S. For all n, we determine an {a, b}-multiplicative set with maximum cardinality in [n], and conclude that the maximum density of an {a, b}-multiplicative set is b b+g . Erdős [2, 3, 4] defined a set S ⊆ N to be multiplicative Sidon1 if ab = cd implies {a, b} = {c, d} for all a, b, c, d ∈...

Let (B t (s); 0 s < 1) be reeecting inhomogeneous Brownian motion with drift t ? s at time s, started with B t (0) = 0. Consider the random graph G(n; n ?1 +tn ?4=3), whose largest components have size of order n 2=3. Normalizing by n ?2=3 , the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of B t (Corollary 2). The dynamics of mergi...

In this paper, we establish the well-posedness for third grade fluid equation perturbed by a multiplicative white noise. This describes motion of non-Newtonian differential type with relevant viscoelastic properties. We are faced strongly nonlinear stochastic partial supplemented Navier slip boundary condition. Taking initial condition in Sobolev space H2, show existence and uniqueness strong (...

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 $ngeq 14$‎.

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...

We study the PageRank mass of principal components in a bow-tie Web Graph, as a function of the damping factor c. It is known that the Web graph can be divided into three principal components: SCC, IN and OUT. The Giant Strongly Connected Component (SCC) contains a large group of pages all having a hyper-link path to each other. The pages in the IN (OUT) component have a path to (from) the SCC,...

Abstract. Let K be a field of characteristic p and G a nonabelian metacyclic finite p-group. We give an explicit list of all metacyclic p-groups G, such that the group algebra KG over a field of characteristic p has a filtered multiplicative K-basis. We also present an example of a non-metacyclic 2-group G, such that the group algebra KG over any field of characteristic 2 has a filtered multipl...

Following classical work by Freidlin [Trans. Amer. Math. Soc. (1988) 305 665–657] and subsequent works by Sowers [Ann. Probab. (1992) 20 504–537] and Peszat [Probab. Theory Related Fields (1994) 98 113–136], we prove large deviation estimates for the small noise limit of systems of stochastic reaction–diffusion equations with globally Lipschitz but unbounded diffusion coefficients, however, ass...