Some Pólya-Type Irreducibility Criteria for Multivariate Polynomials

We provide irreducibility criteria for multivariate polynomials with coefficients in an arbitrary field that extend a classical result of Pólya for polynomials with integer coefficients. In particular, we provide irreducibility conditions for polynomials of the form f(X)(Y ? f ₁(X))…(Y ? f _n (X)) + g(X), with f, f ₁, ?, f _n , g univariate polynomials over an arbitrary field.     

Scaling coupling of reflecting Brownian motions and the hot spots problem

We introduce a new type of coupling of reflecting Brownian motions in smooth planar domains, called scaling coupling. We apply this to obtain monotonicity properties of antisymmetric second Neumann eigenfunctions of convex planar domains with one line of symmetry. In particular, this gives the proof of the hot spots conjecture for some known types of domains and some new ones.

     

Penalization for Birth and Death Processes

In this paper we study a transient birth and death Markov process penalized by its sojourn time in 0. Under the new probability measure the original process behaves as a recurrent birth and death Markov process. We also show, in a particular case, that an initially recurrent birth and death process behaves as a transient birth and death process after penalization with the event that it can reach zero in infinite time. We illustrate some of our results with the Bessel random walk example.     

High-electric-field limit for the Vlasov–Maxwell–Fokker–Planck system

In this paper we derive the high-electric-field limit of the three-dimensional Vlasov–Maxwell–Fokker–Planck system. We use the relative entropy method which requires the smoothness of the solution of the limit problem. We obtain convergences of the electro-magnetic field, charge and current densities.     

On the number of solutions to systems of Pell equations

We prove that, for positive integers a, b, c and d with c≠d, a>1, b>1, the number of simultaneous solutions in positive integers to ax²−cz²=1, by²−dz²=1 is at most two. This result is the best possible one. We prove a similar result for the system of equations x²−ay²=1, z²−bx²=1.     

Complex symmetric operators and applications II

A bounded linear operator on a complex Hilbert space is called complex symmetric if , where is a conjugation (an isometric, antilinear involution of ). We prove that , where is an auxiliary conjugation commuting with . We consider numerous examples, including the Poincaré-Neumann singular integral (bounded) operator and the Jordan model operator (compressed shift). The decomposition also extends to the class of unbounded -selfadjoint operators, originally introduced by Glazman. In this context, it provides a method for estimating the norms of the resolvents of certain unbounded operators.

     

On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent

We consider the nonlinear eigenvalue problem

in , on , where is a bounded open set in with smooth boundary and , are continuous functions on such that , , and for all . The main result of this paper establishes that any sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland's variational principle.

     

Non-negative hereditary polynomials in a free *−algebra

We prove a non-negative-stellensatz and a null-stellensatz for a class of polynomials called hereditary polynomials in a free *-algebra.Partially supported by NSF, DARPA and Ford Motor Co.Partially supported by NSF grant DMS-0140112Partially supported by NSF grant DMS-0100367     

Some remarks on time-dependent variational problems and their asymptotic behaviour

The subject matter of this paper is the asymptotic behaviour of quasi-static variational inequalities, as regards physical parameters like the friction coefficient, compliance coefficient, etc. By convex duality, the quasi-static problems can be recast into the forms of standard evolution problems, whose study relies on well-known methods. In this framework the stability with respect to small friction coefficients reduces to long time behaviour for evolution problems.     

Viability for Differential Inclusions on Graphs

Let X be a Banach space, I a nonempty, bounded interval and let $K:I\leadsto X$ be a given multi-valued function. We rephrase the concept of tangent set introduced by Cârj? et al. (Trans Amer Math Soc, 2008; Viability, Invariance and Applications. North-Holland Mathematics Studies, vol. 207. Elsevier, 2007) by saying that a bounded set $E\subseteq X$ is right tangent to at if $\liminf_{h\downarrow0}{1}/{h}\,\text{\rm dist}(\xi+hE;K(\tau+h))=0$ . Next, by using a tangency condition expressed in the terms of this concept, we establish several necessary and sufficient conditions for viability referring to differential inclusions on graphs.