Numerical ranges of the powers of an operator

The numerical range W(A) of a bounded linear operator A on a Hilbert space is the collection of complex numbers of the form (Av,v) with v ranging over the unit vectors in the Hilbert space. In terms of the location of W(A), inclusion regions are obtained for W(Ak) for positive integers k, and also for negative integers k if A−1 exists. Related inequalities on the numerical radius and the Crawford number are deduced.     

The utilization of total mass to determine the switching points in the symmetric boundary control problem with a linear reaction term

The authors study the problem , and u(0,t)=u(1,t)=ψ(t), where ψ(t)=u₀ for t2k<t<t2k+1 and ψ(t)=0 for , with t₀=0 and the sequence tk is determined by the equations , for , and , for k=2,4,6,… and where 0<m<M. Note that the switching points , are unknown. Existence and uniqueness are demonstrated. Theoretical estimates of the tk and tk+1−tk are obtained and numerical verifications of the estimates are presented. The case of ux(0,t)=ux(1,t)=ψ(t) is also considered and analyzed.     

Construction of a certain circular unit and its applications

For an abelian number field k, let CS(k) be the group of circular units of k defined by Sinnott, and CW(k) be that suggested by Washington. In this paper, we construct an element in CW(k) for a real subfield k of conductor . We will see that the order of in the factor group CW(k)/CS(k) can be very large. As an application, we derive some information about the class number of k for special cases.     

Doubly nonlinear periodic problems with unbounded operators

The solvability of the evolution system v′(t)+B(t)u(t)∋f(t), v(t)∈A(t)u(t), 0<t<T, with the periodic condition v(0)=v(T) is investigated in the case where are bounded, possibly degenerate, subdifferentials and are unbounded subdifferentials.     

Homogeneous polynomials as Lyapunov functions in the stability research of solutions of difference equations

The linear autonomous system of difference equations x(n+1)=Ax(n) is considered, where is a real nonsingular k×k matrix. In this paper it has been proved that if W(x) is any homogeneous polynomial of m-th degree in x, then there exists a unique homogeneous polynomial V(x) of m-th degree such that ΔV=V(Ax)-V(x)=W(x) if and only if where are the eigenvalues of the matrix A. The theorem on the instability has also been proved.     

Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces

Let be the space of solutions to the parabolic equation having finite norm. We characterize nonnegative Radon measures μ on having the property , 1≤p≤q<∞, whenever . Meanwhile, denoting by v(t,x) the solution of the above equation with Cauchy data v₀(x), we characterize nonnegative Radon measures μ on satisfying , β∈(0,n), p∈[1,n/β], q∈(0,∞). Moreover, we obtain the decay of v(t,x), an isocapacitary inequality and a trace inequality.     

Design of RLS Wiener estimators from randomly delayed observations in linear discrete-time stochastic systems

This paper presents the design of a new recursive least-squares (RLS) Wiener filter and fixed-point smoother based on randomly delayed observed values by one sampling time in linear discrete-time wide-sense stationary stochastic systems. The mixed observed value y(k) consists of the past observed value by one sampling time with the probability p(k) and of the current observed value at time k with the probability 1 − p(k). It is assumed that the delayed measurements are characterized by Bernoulli random variables. The observation is given as the sum of the signal z(k) and the white observation noise v(k). The RLS Wiener estimators explicitly require the following information: (a) the system matrix for the state vector; (b) the observation matrix; (c) the variance of the state vector; (d) the delayed probability p(k); (e) the variance of white observation noise v(k).     

Infinitely many homoclinic solutions for second order Hamiltonian systems

In this paper we study the existence of infinitely many homoclinic solutions for second order Hamiltonian systems , , where L(t) is unnecessarily positive definite for all t∈R, and W(t,u) is of subquadratic growth as |u|→∞.     

Arithmetical properties of wendt's determinant

Wendt's determinant of order n is the circulant determinant W_n whose (i,j)-th entry is the binomial coefficient , for 1?i,j?n, where n is a positive integer. We establish some congruence relations satisfied by these rational integers. Thus, if p is a prime number and k a positive integer, then and . If q is another prime, distinct from p, and h any positive integer, then . Furthermore, if p is odd, then . In particular, if p?5, then . Also, if m and n are relatively prime positive integers, then W_mW_n divides W_mn.     

Spatial asymptotic behavior of homeomorphic global flows for non-Lipschitz SDEs

Let x→?s,t(x) be a -valued stochastic homeomorphic flow produced by non-Lipschitz stochastic differential equation , where W=(W¹,W²,…) is an infinite sequence of independent standard Brownian motions. We first give some estimates of modulus of continuity of {?s,t(⋅)}, then prove that the flow ?s,t(x), when x nears infinity, grows slower than for some constant c>0 and integrable random variable Z via lemma of Garsia-Rodemich-Rumsey Lemma (abbreviated as GRR Lemma) improved by Arnold and Imkeller [L. Arnold, P. Imkeller, Stratonovich calculus with spatial parameters and anticipative problems in multiplicative ergodic theory, Stochastic Process. Appl. 62 (1996) 19-54] and moment estimates for one- and two-point motions.     

Continuous dependence results for inhomogeneous ill-posed problems in Banach space

We apply semigroup theory and other operator-theoretic methods to prove Hölder-continuous dependence on modeling for the inhomogeneous ill-posed Cauchy problem in Banach space. The inhomogeneous ill-posed Cauchy problem is given by , u(0)=χ, 0?t<T; where −A is the infinitesimal generator of a holomorphic semigroup on a Banach space X, χ∈X, and . For a suitable function f, the approximate problem is given by , v(0)=χ. Under certain stabilizing conditions, we prove that for a related norm, where and M are computable constants independent of β, 0<β<1, and ω(t) is a harmonic function. These results extend earlier work of Ames and Hughes on the homogeneous ill-posed problem.     

Distance irredundance and connected domination numbers of a graph

Let k be a positive integer and G be a connected graph. This paper considers the relations among four graph theoretical parameters: the k-domination number γ_k(G), the connected k-domination number ; the k-independent domination number and the k-irredundance number ir_k(G). The authors prove that if an ir_k-set X is a k-independent set of G, then , and that for k?2, if ir_k(G)=1, if ir_k(G) is odd, and if ir_k(G) is even, which generalize some known results.     

Hecke group algebras as quotients of affine Hecke algebras at level 0

The Hecke group algebra of a finite Coxeter group , as introduced by the first and last authors, is obtained from by gluing appropriately its 0-Hecke algebra and its group algebra. In this paper, we give an equivalent alternative construction in the case when is the finite Weyl group associated to an affine Weyl group W. Namely, we prove that, for q not a root of unity of small order, is the natural quotient of the affine Hecke algebra H(W)(q) through its level 0 representation.The proof relies on the following core combinatorial result: at level 0 the 0-Hecke algebra H(W)(0) acts transitively on . Equivalently, in type A, a word written on a circle can be both sorted and antisorted by elementary bubble sort operators. We further show that the level 0 representation is a calibrated principal series representation M(t) for a suitable choice of character t, so that the quotient factors (non-trivially) through the principal central specialization. This explains in particular the similarities between the representation theory of the 0-Hecke algebra and that of the affine Hecke algebra H(W)(q) at this specialization.