In this paper, we consider a stochastic lattice differential equation with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and multiplicative white noise at each node. We prove the existence of a compact global random attractor which, pulled back, attracts tempered random bounded sets.

In this paper, we present all the Leibniz 2-cocycles of the centerless twisted Schrödinger-Virasoro algebra ℒ, which determine the second Leibniz cohomology group of ℒ.

We prove a Bernstein type theorem for constant mean curvature hypersurfaces in ℝⁿ⁺¹ under certain growth conditions for n ⩽ 3. Our result extends the case when M is a minimal hypersurface in the same condition.

In this paper, we prove that a group G is isomorphic to M, where M is a simple K₄-group, if and only if the following hold: (1) |G| = |M|, (2) nse(G) = nse(M).

In this paper, we construct families of irreducible representations for a class of quantum groups U_q(ƒ_m(K)). First, we give a natural construction of irreducible weight representations for U_q(ƒ_m(K)) using methods in spectral theory developed by Rosenberg. Second, we study the Whittaker model for the center of U_q(ƒ_m(K)). As a result, the structure of Whittaker representations is determined, and all irreducible Whittaker representations are explicitly constructed. Finally, we prove that the annihilator of a Whittaker representation is centrally generated.

In this paper, we study the maximal prime subgraphs and their corresponding structure for any undirected graph. We introduce the notion of junction trees and investigate their structural characteristics, including junction properties, induced-subtree properties, running-intersection properties and maximum-weight spanning tree properties. Furthermore, the characters of leaves and edges on junction trees are discussed.

In this paper, we study a posteriori error estimates of the edge stabilization Galerkin method for the constrained optimal control problem governed by convection-dominated diffusion equations. The residual-type a posteriori error estimators yield both upper and lower bounds for control u measured in L²-norm and for state y and costate p measured in energy norm. Two numerical examples are presented to illustrate the effectiveness of the error estimators provided in this paper.

In this paper, we derive non-exponential asymptotic forms for solutions of defective renewal equations. These include as special cases asymptotics for compound geometric distribution and the convolution of a compound geometric distribution with a distribution function. As applications of these results, we study the Gerber-Shiu discounted penalty function in the classical risk model and the reliability of a two-unit cold standby system in reliability theory.

Let M be a simple group whose order is less than 10⁸. In this paper, we prove that if G is a finite group with the same order and degree pattern as M, then the following statements hold: (a) If M ≠ A₁₀, U₄(2), then G ≅ M; (b) If M = A₁₀, then G ≅ A₁₀ or J₂ × ℤ₃; (c) If M = U₄(2), then G is isomorphic to a 2-Frobenius group or U₄(2). In particular, all simple groups whose orders are less than 10⁸ but A₁₀ and U₄(2) are OD-characterizable. As a consequence of this result, we can give a positive answer to a conjecture put forward by W. J. Shi and J. X. Bi in 1990 [Lecture Notes in Mathematics, Vol. 1456, 171–180].