A few steps are made towards a representation theory of embeddability among uncountable graphs. A class of graphs is defined by forbidding some countable configurations which are related to the graph's end-structure. Using uncountable combinatorics, a representation theorem for embeddability in this class is proved, which asserts the existence of a surjective homomorphism from the relation of embeddability over isomorphism types of regular cardinality λ>[Nfr ]1 onto set inclusion over all subsets of reals or cardinality λ or less. As corollaries we obtain: (1) the complexity of the class in every regular uncountable λ>[Nfr ]1 is at least

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(2) a characterisation of graphs in the class for which some invariant of the graph has to be inherited by one of fewer than λ subgraphs whose union covers G. Some continuity properties of the homomorphism in the representation theorem are explored and are used to extend the first corollary to all singular cardinals below the first fixed point of second order. The first corollary shows that, contrary to what Shelah has shown for the class of all graphs, the relations of embeddability in the class under discussion is not independent of negations of the generalised continuum hypothesis.

The paper studies projective stationary sets. The Projective Stationary Reflection Principle is the statement that every projective stationary set contains an increasing continuous ∈-chain of length ω1. It is shown that, if Martin's Maximum holds, then the Projective Stationary Reflection Principle holds. Also, this principle is equivalent to the Strong Reflection Principle. The paper shows that the saturation of the nonstationary ideal on ω1 is equivalent to a certain kind of reflection.

We compare the classical semidirect product V[midast ] W of pseudovarieties of semigroups with the recently found ‘regular’ semidirect product [Vscr ] [midast ]e [Wscr ] of e-pseudovarieties of regular semigroups. Applications of our findings include a substantial class of additions to the list of pseudovarieties generated by classes of regular semigroups and for which the membership problem is solved. For example, we compute the pseudovarieties generated by all locally inverse and locally orthodox semigroups, respectively, thereby answering a question raised by Almeida and Kaďourek.

A family [Fscr ] of functions from ℝn to ℝ is k-point separating if, for every k-subset S of ℝn, there is a function f∈[Fscr ] such that f is one-to-one on S. The paper shows that, if the functions are required to be linear (or smooth), then a minimum k-point separating family [Fscr ] has cardinality n(k−1). In the linear case, this result is extended to a larger class of fields including all infinite fields as well as some finite fields (depending on k and n). Also, some partial results are obtained for continuous functions on ℝn, including the case when k is infinite. The proof of the main result is based on graph theoretic results that have some interest in their own right. Say that a graph is an n-tree if it is a union of n edge-disjoint spanning trees. It is shown that every graph with k[ges ]2 vertices and n(k−1) edges has a non-trivial subgraph which is an n-tree. A determinantal criterion is also established for a graph with k vertices and n(k−1) edges to be an n-tree.

Let u be a solution of the differential equation u″+Ru=0, where R is rational. Newton's method of finding the zeros of u consists of iterating the function f(z) =z−u(z)/u′(z). With suitable hypotheses on R and u, it is shown that the iterates of f converge on an open dense subset of the plane if they converge for the zeros of R. The proof is based on the iteration theory of meromorphic functions, and in particular on the result that, if the family of K-quasiconformal deformations of a meromorphic function f depends on only finitely many parameters, then every cycle of Baker domains of f contains a singularity of f−1. This result, together with classical results of Hille concerning the asymptotic behaviour of solutions of the above differential equations, is also used to study their value distribution. For example, it is shown that, if R is a rational function which satisfies R(z)∼amzm as z→∞ and has only k distinct zeros where k<(m+2)/2, then δ(0, u)[les ]k/(m+2−k)<1.

The paper shows that the distinction between tangential sequences and tangential curves, which is known to be relevant for the boundary behaviour of harmonic functions in Euclidean half spaces, is also meaningful and relevant in the discrete setting of the potential theory on a tree associated to a very regular, nearest neighbour random walk. In particular, the paper first defines the class of tangential curves on a tree, and then proves that, for any family of tangential curves which have the same order of tangency and are associated (and converging) to points in the boundary, there is a bounded harmonic function which, for (almost) every point in the boundary, has positive oscillation along the corresponding curve. This result does not hold if, in place of tangential curves, tangential sequences are admitted.

The tree is not assumed to be homogeneous. In particular, the results do not depend on the existence of isometries.

In this paper, we investigate two problems concerning a class of higher order uniformly elliptic operators with measurable coefficients. In our first result, we prove a stability estimate for the resolvent of such an operator under perturbation of its principal coefficients in Lp spaces. Our second result is a regularity theorem concerning the smoothing properties of the resolvent, and it provides a sufficient condition under which the first result can be applied.

The paper derives a formula for the second variation of the displacement function for polynomial perturbations of Hamiltonian systems with elliptic or hyperelliptic Hamiltonians H(x, y)=½y2−U(x) in terms of the coefficients of the perturbation. As an application, the conjecture stated by C. Chicone that a specific cubic system appearing in a deformation of singularity with two zero eigenvalues has at most two limit cycles is proved.

As is well known, the classical Titchmarsh–Weyl m-function for second order differential operators admits a generalisation to considerably larger classes of differential equations. In the applications, however, the use of the general M-matrix often turns out to be rather cumbersome. The paper interprets the m-function in terms of Hilbert space notions, and shows that, in a sense that is made precise, the classical m-function can be recovered as a part of the more complicated one. Applications of this trick lead to results on spectral multiplicity and on stability properties of the spectrum.

We consider the iterates of the heat operator

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on Rn+1={(X, t); X =(x1, x2, …, xn) ∈Rn, t∈R}. Let Ω⊂Rn+1 be a domain, and let m[ges ]1 be an integer. A lower semi-continuous and locally integrable function u on Ω is called a poly-supertemperature of degree m if

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If u and −u are both poly-supertemperatures of degree m, then u is called a poly-temperature of degree m. Since H is hypoelliptic, every poly-temperature belongs to C∞(Ω), and hence

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For the case m=1, we simply call the functions the supertemperature and the temperature.

In this paper, we characterise a poly-temperature and a poly-supertemperature on a strip

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by an integral mean on a hyperplane. To state our result precisely, we define a mean A[·, ·]. This plays an essential role in our argument.

Let Ω be a fixed open cube in ℝn. For r∈[1, ∞) and α∈[0, ∞) we define

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where Q is a cube in ℝn (with sides parallel to the coordinate axes) and χQ stands for the characteristic function of the cube Q.

A well-known result of Gehring [5] states that if

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for some p∈(1, ∞) and c∈(0, ∞), then there exist q∈(p, ∞) and C=C(p, q, n, c)∈(0, ∞) such that

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for all cubes Q⊂Ω, where [mid ]Q[mid ] denotes the n-dimensional Lebesgue measure of Q. In particular, a function f∈L1(Ω) satisfying (1.1) belongs to Lq(Ω).

In [9] it was shown that Gehring's result is a particular case of a more general principle from the real method of interpolation. Roughly speaking, this principle states that if a certain reversed inequality between K-functionals holds at one point of an interpolation scale, then it holds at other nearby points of this scale. Using an extension of Holmstedt's reiteration formulae of [4] and results of [8] on weighted inequalities for monotone functions, we prove here two variants of this principle involving extrapolation spaces of an ordered pair of (quasi-) Banach spaces. As an application we prove the following Gehring-type lemmas.

This paper defines the n-fold central Haagerup tensor product [otimes ]ni=1[Lscr ]h[Rscr ] of a von Neumann algebra [Rscr ], and shows that the map θz[ratio ][otimes ]ni =1[Lscr ]h[Rscr ]→CB([Rscr ]n−1, [Rscr ]) given by

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is an isometry.

The paper proves several weighted imbedding theorems for domains with fractal boundaries. The weights considered are distances to the boundary to certain powers, and the domains are so-called s-John domains. The paper also proves, in the general setting, that the existence of an imbedding implies compactness of the imbedding for lower exponents. Moreover, following Maz'ya, the paper reformulates the imbedding theorems in the language of local isoperimetric and capacity estimates.

The paper shows that, if the operator T[ratio ]A(γ)→B(γ) is compact for almost every γ∈Γ, then T[ratio ]F(Ā)→F([Bmacron]) is compact when F([Xmacron])=(X)SK or F([Xmacron])=(X)SJ is the interpolation functor constructed for infinite families of Banach spaces and S satisfies certain conditions.

We characterize the minimal isometric dilation of a non-commutative contractive sequence of operators as a universal object for certain diagrams of completely positive maps. A non-spatial construction of the minimal isometric dilation is also given, using Hilbert modules over C*-algebras.

It is shown that the non-commutative disc algebras [Ascr ]n (n[ges ]2) are the universal algebras generated by contractive sequences of operators and the identity, and C*(S1, …, Sn) (n[ges ]2), the extension through compact operators of the Cuntz algebra [Oscr ]n, is the universal C*-algebra generated by a contractive sequence of isometries. It is also shown that the algebras [Ascr ]n and C*(S1, …, Sn) are completely isometrically isomorphic to some free operator algebras considered by D. Blecher. In particular, the universal operator algebra of a row (respectively column) contraction is identified with a subalgebra of C*(S1, …, Sn). The internal characterization of the matrix norm on a universal algebra leads to some factorization theorems.

Suppose that K is a closed, total cone in a real Banach space X, that A[ratio ]X→X is a bounded linear operator which maps K into itself, and that A′ denotes the Banach space adjoint of A. Assume that r, the spectral radius of A, is positive, and that there exist x0≠0 and m[ges ]1 with Am(x0) =rmx0 (or, more generally, that there exist x0∉(−K) and m[ges ]1 with Am(x0) [ges ]rmx0). If, in addition, A satisfies some hypotheses of a type used in mean ergodic theorems, it is proved that there exist u∈K−{0} and θ∈K′−{0} with A(u)=ru, A′(θ)=rθ and θ(u)>0. The support boundary of K is used to discuss the algebraic simplicity of the eigenvalue r. The relation of the support boundary to H. Schaefer's ideas of quasi-interior elements of K and irreducible operators A is treated, and it is noted that, if dim(X)>1, then there exists an x∈K−{0} which is not a quasi-interior point. The motivation for the results is recent work of Toland, who considered the case in which X is a Hilbert space and A is self-adjoint; the theorems in the paper generalize several of Toland's propositions.

The discrete spectrum of the Schrödinger operator is studied for a system of three identical particles with short-range interactions in a homogeneous magnetic field. All the two-particle subsystems are supposed to be unstable. Finiteness of the discrete spectrum is established under some assumptions about the solutions of the corresponding two-particle Schrödinger equation.