Several formulas are developed which can be used to determine constant solutions and the possible periods of periodic solutions (if any) of autonomous homogeneous matrix Riccati differential equations. These formulas are used to analyse some 2 × 2 cases, as well as to discuss the existence of periodic solutions under weak periodic forcing.

Let Ω⊂ℝnbe a bounded open domain and T = ∂Ω. It β is a maximal monotone graph in ℝ×ℝ with 0ϵβ(0), and f: ℝ×Ω→ℝ is measurable with t→ f(t,.) S2-almost periodic as a function ℝ→L2(Ω), we consider the nonlinear hyperbolic equation

We show that:

• (i) if ゲ is strictly increasing and (1) has a solution ω on ℝ with [ω, Əω/Ət] almost periodic: , for any solution of (1) there exists with u(t,.)–ω(t,.)—ξin

• (ii) if β is single valued and everywhere defined, the existence of ω as above implies that, for every solution of (1), there exists Ϛ(t, x) with ә2Ϛ/әt2–0△Ϛ = in ℝ×Ω and u(t,.)–ω(t,.)—0 in as t → +∞

• (iii) if β–1 is uniformly continuous and ゲ satisfies some growth assumption (depending on N), for every f as above, there exists ω solution of (1) on ℝ with [ω, Əω/Ət] almost periodic: ℝ → .

We discuss smooth changes of eigenvalues under perturbation of the boundary value problems given in the title. The simple eigenvalue criterion is developed in the setting of Banach spaces, so very general perturbations of both the differential equation and the boundary conditions are allowed. Further, we need no assumptions about self-adjointness of the original or perturbed problems. The discussion is concluded with the application of the simple eigenvalue criterion to two examples.

Existence and uniqueness theorems are proved for the solution of a Dirichlet-Neumann-Third mixed boundary value problem for the Helmholtz equation in ℝ3. The proofs make use of an equivalent system of two integral equations of the second kind.

In three recent papers by Cavaretta et al., progress has been made in understanding the structure of bi-infinite totally positive matrices which have a block Toeplitz structure. The motivation for these papers came from certain problems of infinite spline interpolation where total positivity played an important role.

In this paper, we re-examine a class of infinite spline interpolation problems. We derive new results concerning the associated infinite matrices (periodic B-spline collocation matrices) which go beyond consequences of the general theory. Among other things, we identify the dimension of the null space of these matrices as the width of the largest band of strictly positive elements.

Let Γ be a graph with n points, and let G be the group of automorphisms of Γ. An orbit of G on which G acts as an elementary abelian 2-group is said to be exceptional. It is shown that the number of simple eigenvalues of Γ is at most (5n+4t)/9, where t is the number of points of Γ lying in exceptional orbits of G.

This is a study of the family of power series where Σ αnZn has unit radius of convergence and the εn are independent random variables taking the values ±1 with equal probability. It is shown that if

then almost all these power series take every complex value infinitely often in the unit disk.

In §§1 and 2, we consider mainly a system of reaction-diffusion equations with general diffusion matrix and we establish the stabilization of all solutions at t →∞. The interest of this problem derives from two separate facts. First, the sets that are useful for localizing the asymptotics cease to be invariant as soon as the diffusion matrix is not a multiple of the identity. Second, the set of equilibria is connected. In §3, we establish uniform L§ bounds for the solutions of a class of parabolic systems. The unifying feature in the problems considered is the lack of any conventional maximum principles.

We prove that the selfadjoint elliptic differential equation (1) has rectangular nodal domains if the quadratic form of the equation takes on negative values. The existence of nodal domains is closely connected with the position of the smallest point of the spectrum of the corresponding selfadjoint operator (Friedrichs extension). If the smallest point of a second order selfadjoint differential operator with Dirichlet boundary conditions is an eigenvalue, then this eigenvalue is strictly increasing when the (possibly unbounded) domain, where the coefficients of the differential operator are denned, is shrinking (Theorem 4).

We consider a common abstraction of de Morgan algebras and Stone algebras which we call an MS-algebra. The variety of MS-algebras is easily described by adjoining only three simple equations to the axioms for a bounded distributive lattice. We first investigate the elementary properties of these algebras, then we characterise the least congruence which collapses all the elements of an ideal, and those ideals which are congruence kernels. We introduce a congruence which is similar to the Glivenko congruence in a p-algebra and show that the location of this congruence in the lattice of congruences is closely related to the subdirect irreducibility of the algebra. Finally, we give a complete description of the subdirectly irreducible MS-algebras.

Liouville theorems are obtained for general elliptic PDE-systems ℒU(x): = Σlɑl≦21Aα(x) əαU(x)=0 essentially under the assumption that ℒ satisfies a coerciveness estimate over .

This paper shows that if S is a sphere in a Banach space and f: S → S is an α-contraction, then f has a fixed point. The paper generalizes a result of R. D. Nussbaum which holds for α-contractions only. The proof uses the Browder nonrepulsive fixed point theorem and is motivated by recent work of M. Martelli and G. D. Cooperman.

This paper is concerned with the existence of solutions of a two point boundary value problem for neutral functional differential equations. We consider the problem

where M and N are n × n matrices. This is examined by using the “shooting method”. Also, an example is given to illustrate how our result can be applied to yield the existence of solutions of a periodic boundary value problem.

We give a coordinate-free algebraic interpretation for singularity indices of quasihomogeneous functions and their unfoldings, as first exploited systematically by Berry in statistical optics. We also make some observations on geometric interpretation, on formulae for indices, and on the difficulties of rigorous treatment for non-quasihomogeneous functions.

Let H be a separable Hilbert space and let CL(H) be the semigroup of continuous, linear maps from H to H. Let E+ be the idempotents of CL(H). Let Ker ɑ and Im ɑ be the null-space and range, respectively, of an element ɑ of CL(H) and let St ɑ be the subspace {x∊H: xɑ = x} of H. It is shown that 〈E+〉 = I∪F∪{i}, where

and ι is the identity map. From the proof it is clear that I and F both form subsemigroups of 〈E+〉 and that the depth of I is 3. It is also shown that the depths of F and 〈E+〉 are infinite.