Geoffrey Grimmett Abstra t. Known differential inequalities for certain ferromagnetic Potts models with pairinteractions may be extended to Potts models with many-body interactions. As a major application of such differential inequalities, we obtain necessary and sufficient conditions on the set of interactions of such a Potts model in order that its critical point be a strictly monotonic funct...

In this paper, the concepts of positive dependence and linearlypositive quadrant dependence are introduced for fuzzy random variables. Also,an inequality is obtained for partial sums of linearly positive quadrant depen-dent fuzzy random variables. Moreover, a weak law of large numbers is estab-lished for linearly positive quadrant dependent fuzzy random variables. Weextend some well known inequ...

In a celebrated work by Shao [13] several inequalities for negatively associated random variables were proved. In this paper we obtain some maximal inequalities for associated random variables. Also we establish a maximal inequality for demimartingales which generalizes and improves the result of Christofides [4].

Logarithmic Sobolev inequalities first arose in the analysis of elliptic differential operators in infinite dimensions. Many developments and applications can be found in several survey papers [1, 9, 12]. Recently, Diaconis and Saloff-Coste [8] considered logarithmic Sobolev inequalities for Markov chains. The lower bounds for log-Sobolev constants can be used to improve convergence bounds for ...

We consider lower and upper bounds on the difference of differential entropies of a Gaussian random vector and an approximately Gaussian random vector after they are “smoothed” by an arbitrarily distributed random vector of finite power. These bounds are important to establish the optimality of the corner points in the capacity region of Gaussian interference channels. A problematic issue in a ...

Let θ(J) be the order parameter of a (ferromagnetic) Potts or randomcluster process with bond-variables J = (Je : e ∈ K). We discuss differential inequalities of the form ∂θ ∂Je ≤ α(J) ∂θ ∂Jf for all e, f ∈ K. Such inequalities may be established for all random-cluster processes that satisfy the FKG inequality, possibly in the presence of many-body interactions (subject to certain necessary and...

in this paper, we prove the existence and uniqueness results to the random fractional functional differential equations under assumptions more general than the lipschitz type condition. moreover, the distance between exact solution and appropriate solution, and the existence extremal solution of the problem is also considered.