A criterion on the semisimple Brauer algebras

In this paper, we give a necessary and sufficient condition for a Brauer algebra to be semisimple.     

On the stability of the Serrin problem

We investigate stability issues concerning the radial symmetry of solutions to Serrin's overdetermined problems. In particular, we show that, if u is a solution to Δu=n in a smooth domain Ω⊂Rn, u=0 on ∂Ω and |Du| is “close” to 1 on ∂Ω, then Ω is “close” to the union of a certain number of disjoint unitary balls.     

A characterization of the simplex

In the vein of Rado's beautiful characterization of matroids by the functioning of the Greedy Algorithm we prove that our persistent frustrations in obtaining analogues and generalizations of Shapiro's Lemma had a good reason.Essentially, only the classical case of the power set can work.     

Global solvability for first order real linear partial differential operators

F. Treves, in [17], using a notion of convexity of sets with respect to operators due to B. Malgrange and a theorem of C. Harvey, characterized globally solvable linear partial differential operators on C^∞(X), for an open subset X of Rn.Let P=L+c be a linear partial differential operator with real coefficients on a C^∞ manifold X, where L is a vector field and c is a function. If L has no critical points, J. Duistermaat and L. Hörmander, in [2], proved five equivalent conditions for global solvability of P on C^∞(X).Based on Harvey-Treves's result we prove sufficient conditions for the global solvability of P on C^∞(X), in the spirit of geometrical Duistermaat-Hörmander's characterizations, when L is zero at precisely one point. For this case, additional non-resonance type conditions on the value of c at the equilibrium point are necessary.     

A criterion on asymptotic stability for partially equicontinuous Markov operators

In this paper, we prove a slight, but practically useful, generalisation of a criterion on asymptotic stability for Markov e-chains by T. Szarek, which is based on the so-called lower bound technique, developed by A. Lasota and J. York. Simultaneously, we present an alternative proof of this theorem using an asymptotic coupling method introduced by M. Hairer. Our main result is illustrated by an application to random iterated function systems, which are not contracting on average.     

Evolution Problems involving non-stationary Operators between two Banach Spaces II. Ancillary results

This paper is a direct continuation of the first paper in this series[Quaestiones Math. 8 (1985), no. 2,], which deals Kith the evolution problem [B(t)u(t)]' = A(t)u(t), Bu(0) = y. whereas in the first paper existence and uniqueness results were presented, this paper contains the ancillary results formulated in the first paper. These results describe the properties of B(t)-evolutions with non-stationary generating pairs, as well as of the solution operator of [B(t)u(t)]' = Au, Bu(0) = y.

The notation of the first paper is assumed throughout and is used without further note. Citation of results from the first paper is made by equation number. The numbering of this paper follows that of the first paper in sequence.     

含有快速增长非线性恢复力的梁方程的行波解

Abstract. On studying traveling waves on a nonlinearly suspended bridge,the following partial differential equation has been considered:     

Evolution Problems involving non-stationary Operators between two Banach Spaces I. Existence and Uniqueness Theorems

Synopsis

(for ‘Evolution Problems involving non-stationary Operators between two Banach Spaces I-II)

In this series of two papers the initial-value problem [B(t)u(t)' = A(t)u(t), Bu(0) = y, with A = A(t) and B = B(t) time-varying operators from one Banach space X to another Banach space Y, and y an arbitrary element of Y, is considered. By making use of the theory of B-evolutions and by integrating certain temporally inhomogeneous equations, a unique solution is obtained for any y in Y. The solution is formulated explicitly in terms of a certain solution operator which involves the B(t)-evolution generated by the closed pair >A(t),B(t)< of operators. Certain properties of the solution operator are also studied. The well-known results, obtained by making use of semigroup theory, for the evolution problem [u(t)]' = A(t)u(t), u(0) = u₀, where A is a closed operator in a Banach space with dense domain, may also be derived from our results.     

Pointwise Green's function estimates toward stability for multidimensional fourth-order viscous shock fronts

For the case of multidimensional viscous conservation laws with fourth-order smoothing only, we develop detailed pointwise estimates on the Green's function for the linear fourth-order convection equation that arises upon linearization of the conservation law about a viscous planar wave solution. As in previous analyses in the case of second-order smoothing, our estimates are sufficient to establish that spectral stability implies nonlinear stability, though the full development of this result will be considered in a companion paper.     

Hörmander form and uniqueness for the Cauchy problem

We give a new geometric proof to Hörmander's uniqueness theorem in the Cauchy problem for systems of differential equations (possibly with multiple characteristics).     

Well-posedness of higher-order Camassa-Holm equations

We consider higher-order Camassa-Holm equations describing exponential curves of the manifold of smooth orientation-preserving diffeomorphisms of the unit circle in the plane. We establish the existence of global weak solutions. We also present some invariant spaces under the action of the equation. Moreover, we prove a “weak equals strong” uniqueness result.     

Cauchy problem for p-evolution equations

We consider p-evolution equations with real characteristics. We give a condition, on the lower order terms, that is sufficient for well-posedness of the Cauchy problem in Sobolev spaces.     

H 1-perturbations of Smooth Solutions for a Weakly Dissipative Hyperelastic-rod Wave Equation

We consider a weakly dissipative hyperelastic-rod wave equation (or weakly dissipative Camassa–Holm equation) describing nonlinear dispersive dissipative waves in compressible hyperelastic rods. We fix a smooth solution and establish the existence of a strongly continuous semigroup of global weak solutions for any initial perturbation from In particular, the supersonic solitary shock waves [8] are included in the analysis. Dedicated to the memory of Professor Aldo Cossu The research of K.H. Karlsen is supported by an Outstanding Young Investigators Award from the Research Council of Norway. The current address of G.M. Coclite is Department of Mathematics, University of Bari, Via E. Orabona 4, 70125 Bari, Italy     

Global stability of solutions with discontinuous initial data containing vacuum states for the isentropic Euler equations

The global stability of Lipschitz continuous solutions with discontinuous initial data is established in a broad class of entropy solutions in containing vacuum states. In particular, the uniqueness of Lipschitz solutions with discontinuous initial data is obtained in the broad class of entropy solutions in .     

A one-round,two-prover,zero-knowledge protocol for NP

The model of zero-knowledge multi-prover interactive proofs was introduced by Ben-Or, Goldwasser, Kilian and Wigderson in [4]. A major open problem associated with this model is whether NP problems can be proven by one-round, two-prover, zero-knowledge protocols with exponentially small error probability (e.g. via parallel executions). A positive answer was claimed by Fortnow, Rompel and Sipser in [12], but its proof was later shown to be flawed by Fortnow who demonstrated that the probability of cheating inn independent parallel rounds can be much higher than the probability of cheating inn independent sequential rounds (with exponential ratio between them). In this paper we solve this problem: We show a new one-round two-prover interactive proof for Graph Hamiltonicity, we prove that it is complete, sound and perfect zeroknowledge, and thus every problem in NP has a one-round two-prover interactive proof which is perfectly zero knowledge under no cryptographic assumptions. The main difficulty is in proving the soundness of our parallel protocol namely, proving that the probability of cheating in this one-round protocol is upper bounded by some exponentially low threshold. We prove that this probability is at most 1/2 ^n/9 (wheren is the number of parallel rounds), by translating the soundness problem into some extremal combinatorial problem, and then solving this new problem.     

Local well-posedness and stability of peakons for a generalized Dullin-Gottwald-Holm equation

We establish the local well-posedness for a generalized Dullin-Gottwald-Holm equation by using Kato’s theory. Furthermore, the orbital stability of the peaked solitary waves is also proved.     

Local well-posedness and regularity criterion for the density dependent incompressible Euler equations

In this paper, local well-posedness for the density dependent incompressible Euler equations is established in Besov spaces. We also obtain a blow-up criterion for the corresponding solution.     

A blow-up criterion for 3D compressible viscoelastic flow with large initial data

In this paper, we consider a Cauchy problem for the three-dimensional compressible viscoelastic flow with large initial data. We establish a blow-up criterion for the strong solutions in terms of the gradient of velocity only, which is similar to the Beale-Kato-Majda criterion for ideal incompressible flow (cf. Beale et al. (1984) [20]) and the blow-up criterion for the compressible Navier-Stokes equations (cf. Huang et al. (2011) [21]).