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Abstract Let M n (n ≥ 3) be a complete noncompact submanifold in a sphere S n + q. In this paper, we prove that the dimension of the space of L p harmonic 1-forms on M n must be finite for certain p > 0 provided the total curvature of M n is finite. This result can be regarded as a generalization of Zhu–Fang's result on L 2 harmonic 1-forms. [ABSTRACT FROM AUTHOR]

This paper deals with an attraction–repulsion chemotaxis system { u t = ∇ ⋅ ( D ( u ) ∇ u ) − χ ∇ ⋅ ( u ∇ v ) + ξ ∇ ⋅ ( u ∇ w ) , x ∈ Ω , t > 0 , τ 1 v t = Δ v + α u − β v , x ∈ Ω , t > 0 , τ 2 w t = Δ w + γ u − δ w , x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R N ( N ≥ 2 ), where parameters τ i ( i = 1 , 2 ) , χ , ξ , α , β , γ and δ are positive, and diffusion coefficient D ( u ) ∈ C 2 ( 0 , + ∞ ) satisfies D ( u ) > 0 for u ≥ 0 , D ( u ) ≥ d u m − 1 with d > 0 and m ≥ 1 for all u > 0 . It is proved that the corresponding initial–boundary value problem possesses a unique global bounded classical solution for m > 2 − 2 N . In particular in the case τ 1 = τ 2 and χ α = ξ γ , the solution is globally bounded if m > 2 − 2 N − N + 2 N 2 − N + 2 . Therefore, due to the inhibition of repulsion to the attraction, the range of m > 2 − 2 N of boundedness is enlarged and the results of [21] is thus extended to the higher dimensional attraction–repulsion chemotaxis system with nonlinear diffusion. [ABSTRACT FROM AUTHOR]

This paper is concerned with the uniform energy decay rates of a structural acoustic model which describes the interactions between the acoustic medium and the elastic structural in an acoustic chamber where one “wall” is flexible and curved. The coupled system considered consists of a variable-coefficient wave equation and a variable-coefficient plate equation defined on the Riemannian manifold with the coupling on the interface between the acoustic medium and the flexible wall. Both the components of the dynamics are subject to nonlinear boundary damping. Furthermore, a nonlinear delay term acting in the boundary feedbacks of the structure component is considered. The Riemannian geometry method is applied to derive the energy estimates of the nonlinear coupled system with variable coefficients. The uniform energy decay rates of the nonlinear variable-coefficient structural acoustic system with delay effects on the Riemannian manifold are quantified by a solution to a constructed nonlinear ODE. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and energy decay estimates of solutions to the variable-coefficient Euler-Bernoulli plate with a delay in localized nonlinear internal feedback. The existence of solutions to the nonlinear system is obtained by using Galerkin method combined with some energy estimates. Furthermore, applying Riemannian geometry method, we establish uniform decay rates of the plate equation with variable coefficients and a nonlinear delay term. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a unified treatment on the linear, differentiable and topological equivalences for time-invariant linear control systems governed by ordinary differential equations (ODEs). Especially, we give a new solution to the topological classification problem for linear control systems by means of an elementary analytic approach, which differs considerably from J.C. Willems’ algebraic method developed in (Willems, 1980) [17]. More precisely, based on P. Brunovsky’s results in (Brunovsky, 1970) [2], we determine all topological invariants for the linear control system. As an interesting by-product, we show that there exist only finitely many topological equivalence classes for any stabilizable system with given numbers of state and control variables, which might be useful for some engineering applications. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the existence and comparison principle of classical solutions for a class of fully nonlinear degenerate parabolic equations. [Copyright &y& Elsevier]

Abstract: We study the exponential stability of the plate equations with potential of second order and indefinite sign damping term. By means of a global Carleman-type estimate and the usual perturbation method, we show that the energy of the system decays exponentially provided that the negative damping is sufficiently small. Both the energy decay rate and the upper bound estimate on the negative damping are given explicitly. [Copyright &y& Elsevier]

Let G be a graph of order n and λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n be the eigenvalues of G . The energy of G is defined as E ( G ) = ∑ k = 1 n | λ k | . A well-known result regarding the energy of graphs is the Coulson integral formula, which defines the relationship between the energy and the characteristic polynomial of graphs. Let μ 1 ≥ μ 2 ≥ ⋯ ≥ μ n = 0 be the Laplacian eigenvalues of G . The general Laplacian energy-like invariant of G , denoted by L E L α ( G ) , is defined as ∑ μ k ≠ 0 μ k α when μ 1 ≠ 0 , and 0 when μ 1 = 0 , where α is a real number. In this study, we give some Coulson-type integral formulas for the general Laplacian energy-like invariant of graphs in the case where α is a rational number. Based on this result, we also give some Coulson-type integral formulas for the general energy and general Laplacian energy of graphs in the case where α is a rational number. We also show that our formulas hold when α is an irrational number where 0 < | α | < 1 , whereas they do not hold when | α | > 1 . [ABSTRACT FROM AUTHOR]

Let G be a simple graph. Its energy is defined as E ( G ) = ∑ k = 1 n | λ k | , where λ 1 , λ 2 , … , λ n are the eigenvalues of G . A well-known result on the energy of graphs is the Coulson integral formula which gives a relationship between the energy and the characteristic polynomial of graphs. Let μ 1 ≥ μ 2 ≥ ⋯ ≥ μ n = 0 be the Laplacian eigenvalues of G . The general Laplacian-energy-like invariant of G , denoted by LEL α ( G ) , is defined as ∑ μ k ≠ 0 μ k α when μ 1 ≠ 0 , and 0 when μ 1 = 0 , where α is a real number. In this paper we give a Coulson-type integral formula for the general Laplacian-energy-like invariant for α = 1 / p with p ∈ Z + \ { 1 } . This implies integral formulas for the Laplacian-energy-like invariant, the normalized incidence energy and the Laplacian incidence energy of graphs. [ABSTRACT FROM AUTHOR]