The asymptotic behaviour of L2-solutions of one-body Schrödinger equations (–δ+V–E)ψ = 0 in ΩR = {x ∊ Rn||x|>R} is investigated. We show, for example, that if V tends to zero in a certain sense for |x|→∞, then either |x|γ exp for some γ>0 or ψ has compact support. Related results are given for potentials tending to infinity for |x|→∞.

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.

The existence of a smallest inverse congruence on an orthodox semigroup is known. It is also known that a regular semigroup S is locally inverse and orthodox if and only if there exists a local isomorphism from S onto an inverse semigroup T.

In this paper, we show the existence of a smallest R-unipotent congruence ρ on an orthodox semigroup S and give its expression in the case where S is also left quasinormal. Finally, we prove that a regular semigroup S is left quasinormal and orthodox if and only if there exists a local isomorphism from S onto an R-unipotent semigroup T.

In this paper, the existence of global smooth solutions and the formation of singularities of solutions for strictly hyperbolic systems with general eigenvalues are discussed for the Cauchy problem with essentially periodic small initial data or nonperiodic initial data. A result of Klainerman and Majda is thus extended to the general case.

The exterior Dirichlet and Neumann problems can be treated very satisfactorily by using a fundamental solution which is modified by adding radiating spherical wave functions. It has been shown [3], that the coefficients of these added terms can be chosen to ensure that the associated boundary integral equation formulation of the problem was uniquely solvable and, in addition, that the modified Green's function was a least squares best approximation to the exact Green's function for the problem. Here we show that the coefficients can be chosen to ensure not only unique solvability but also minimization of the norm of the modified integral operator. This leads to a constructive method of solution. The theory is illustrated when the boundary is a sphere and when it is a perturbation of a sphere.

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.

A theorem of Rudin states that if B is the open unit ball in ℂN, N > 1, if 0<ρ < 1, if is the family of all complex lines in ℂN at a distance ρ from the origin and if f ∈ C(∂B) is such that for every Λ∈ the function f|Λ∂B has a continuous extension to Λ ∩ B which is holomorphic in Λ ∩ B, then f has a continuous extension to B which is holomorphic in B. In this paper we show that when N = 2, the theorem still holds if is replaced by a considerably smaller family.

The operators Δhf ≡ f(x) on function spaces and Δxn ≡ xn+1–xn on sequence spaces replace derivatives to yield analogues of the Kolmogorov inequality. Estimates for best constants are given for many spaces and for a few the best constants are actually given.

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.

Regular solutions of the forced nonlinear wave equation uu + L4u + LΦLu = r are studied in Hilbert spaces. L is a linear, positive, selfadjoint operator and the nonlinear nucleus Φ(u) = f(|u|2)u is generated by a C1-function f, such that LΦ(Lu) = f(|Lu|2)L2u. If the initial value data u(0) = ϕ and u1(0) = ψ belong to the domain D(Lk+4) and D(Lk+2), respectively, and if rεD(Lk), then there is a (global) solution u(t) such that u ε D(Lk+4), ut ε D(Lk+2) and uuε D(Lk) for all times t. The abstract result is applied to examples in nonlinear elasticity theory.

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).

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.

A subalgebra M of a Lie algebra L is called modular in L if M is a modular element in the lattice of the subalgebras of L. Our aim is to study the finite-dimensional Lie algebras all of whose maximal subalgebras are modular. We characterize these algebras over any field of characteristic zero.

There is a residual subset of embeddings of an m-manifold, M in Rm+1 (m ≦ 6), for which the induced Maxwell subset on the sphere Sm is a stratified subset. We define and study two different stratifications of this subset and their extensions to the whole Sm: the Gauss stratification and the core stratification. We also find relations between the Euler numbers of the strata of the core stratification and the “exposed” singularities of the Gauss map on M.

Integral transformations T of the formal form

where G is a Meijer-G function are studied on the spaces ℐμ,r of functions f such that

Results are obtained on the boundedness, representation and range of the transformations, and on when they have an inverse of the same formal form.

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 → .

In this paper entire solutions of differential equations with polynomial coefficients are considered and bounds on the maximum modulus and the index are obtained, when the equation is of second order and the coefficients are of second degree.