We obtain optimal L2-lower bounds for nonzero solutions to – ΔΨ + VΔ = EΨ in Rn, n ≥ 2, E ∈ R where V is a measurable complex-valued potential with V(x) = 0(|x|-c) as |x|→∞, for some ε∈ R. We show that if 3δ = max{0, 1 – 2ε} and exp (τ|x|1+δ)Ψ ∈ L2(Rn)for all τ > 0, then Ψ; has compact support. This result is new for 0 < ε ½ and generalizes similar results obtained by Meshkov for = 0, and by Froese, Herbst, M. Hoffmann-Ostenhof, and T. Hoffmann-Ostenhof for both ε≤O and ε≥½. These L2-lower bounds are well known to be optimal for ε ≥ ½ while for ε < ½ this last is only known for ε = O in view of an example of Meshkov. We generalize Meshkov's example for ε< ½ and thus show that for complex-valued potentials our result is optimal for all ε ∈ R.

This paper is concerned with two notions of cohomological triviality for Banach algebras, weak amenability and cyclic amenability. The first is defined within Hochschild cohomology and the latter within cyclic cohomology. Our main result is that where ℱ is a Banach algebraic free product of two Banach algebras and ℬ. It follows that cyclic amenability is preserved under the formation of free products.

The generalised zeta-function ζ(s, α) is defined by

where α>0 and Res>l. Clearly, ζ(s, 1)=, where ζ(s) denotes the Riemann zeta-function. In this paper we consider a general class of Dirichlet series satisfying a functional equation similar to that of ζ(s). If ø(s) is such a series, we analogously define ø(s, α). We shall derive a representation for ø(s, α) which will be valid in the entire complex s-plane. From this representation we determine some simple properties of ø(s, α).

In recent years, the problem of embedding the projective spaces in Euclidean spaces was studied very much, by different methods. Usually, the negative results on the embedding problem are proved by using suitable homotopy invariants. The best known example of such homotopy invariants is given by the Stiefel–Whitney classes.

In this paper we obtain a sharp height estimate concerning compact hypersurfaces immersed into warped product spaces with some constant higher-order mean curvature and whose boundary is contained in a slice. We apply these results to draw topological conclusions at the end of the paper.

Let μ λ be the Bernoulli convolution associated with λ ∈ (0, 1). The well-known result of Jorgensen and Pedersen shows that if λ = 1/(2k) for some k ∈ ℕ, then μ 1/(2k) is a spectral measure with spectrum Γ(1/(2k)). The recent research on the spectrality of μ λ shows that μ λ is a spectral measure only if λ = 1/(2k) for some k ∈ ℕ. Moreover, for certain odd integer p, the multiple set pΓ(1/(2k)) is also a spectrum for μ 1/(2k). This is surprising because some spectra for the measure μ 1/(2k) are thinning. In this paper we mainly characterize the number p that has the above property. By applying the properties of congruences and the order of elements in the finite group, we obtain several conditions on p such that pΓ(1/(2k)) is a spectrum for μ 1/(2k).

In a recent paper, J. L. Synge gives an interesting derivation of the conservation equations Tij,j = 0 satisfied by the energy tensor Tij of a continuous medium. Previous to the appearance of this paper, these equations were generally obtained by assuming the classical equations of motion and continuity, after which it was necessary to appeal to the Principle of Equivalence. It then follows that the path of a free particle is a geodesic. Synge however starts with the hypothesis that the path of a particle between collisions is a geodesic and that the proper mass is constant. The conservation equations are then deduced exactly from the law of conservation of momentum for collisions.

This paper concludes the investigation of axially symmetric stress distributions in elastic solids containing penny-shaped cracks, commenced in previous papers (1), (2), by considering the stress distribution in a circular beam containing a crack opened by internal pressure or by uniform tension. The method of analysis, developed in the previous papers, is to first seek a representation of the displacement at a point of the beam as a sum of two terms, one of which is a representation of the displacement due to the crack in an otherwise unbounded infinite solid whilst the second is a general representation of the displacement in an undamaged beam, and then to show that this representation satisfies the conditions on the crack and the curved surface of the beam provided an unknown function occurring in it is the solution of a certain Fredholm integral equation. This equation holds whatever the ratio of the radius of the crack to that of the beam, but is most readily solved by iteration when this ratio is small, this solution being a perturbation on that for a crack in an infinite solid.

If the group of inner automorphisms of a semigroup S of transformations of a finite n-element set contains an isomorphic copy of the alternating group Altn, then S is an Sn-normal semigroup and all the automorphisms of S are inner.

We study the existence, uniqueness and qualitative properties of global solutions of abstract differential equations with state-dependent delay. Results on the existence of almost periodic-type solutions (including, periodic, almost periodic, asymptotically almost periodic and almost automorphic solutions) are proved. Some examples of partial differential equations with state-dependent delay arising in population dynamics are presented.

Let A = [aij] be an n-th order irreducible non-negative matrix. As is very well-known, the matrix A has a positive characteristic root ρ (provided that n>l), which is simple and maximal in the sense that every characteristic root λ satisfies |λ| ≤ρ, and the characteristic vector x belonging to ρ may be chosen positive. These results, originally due to Frobenius, have been proved by Wielandt (4) by means of a strikingly simple basic idea. Recently, a variant of Wielandt's proof has been given by Householder (2).

We complete a classification of the groups of endotrivial modules for the modular group algebras of symmetric groups and alternating groups. We show that, for n ≥ p2, the torsion subgroup of the group of endotrivial modules for the symmetric groups is generated by the sign representation. The torsion subgroup is trivial for the alternating groups. The torsion-free part of the group is free abelian of rank 1 if n ≥ p2 + p and has rank 2 if p2 ≤ n < p2 + p. This completes the work begun earlier by Carlson, Mazza and Nakano.

In 1990, Howie and McFadden showed that every proper two-sided ideal of the full transformation monoid $T_n$, the set of all maps from an $n$-set to itself under composition, has a generating set, consisting of idempotents, that is no larger than any other generating set. This fact is a direct consequence of the same property holding in an associated finite $0$-simple semigroup. We show a correspondence between finite $0$-simple semigroups that have this property and bipartite graphs that satisfy a condition that is similar to, but slightly stronger than, Hall's condition. The results are applied in order to recover the above result for the full transformation monoid and to prove the analogous result for the proper two-sided ideals of the monoid of endomorphisms of a finite vector space.

The diffraction of a simple harmonic wave train by a straightedged semi-infinite screen was originally discussed by Sommerfeld in 1895. The analysis is of a recondite character, involving the use of multivalued functions and Riemann surfaces (1). An alternative formulation of the problem is as an inhomogeneous Wiener-Hop integral equation, the solution of which also involves considerable difficulties (2). It is the purpose of this note to show that following Friedlander (3) it is possible by the use of parabolic co-ordinates to solve the problem by elementary methods. The method can be applied either to the case of sound or that of electromagnetism, the results being formally identical.

If we write the product

in the form

the coefficients have been called by Professor Nielsen (following Thiele) the Stirling Numbers of the First Species, because James Stirling, in his Methodus Differentialis (1730), was the first writer to draw attention to their use, and furnished a small table of their initial values.

This paper suggests a new approach to questions of rationality of 3-folds based on category theory. Following work by Ballard et al., we enhance constructions of Kuznetsov by introducing Noether–Lefschetz spectra: an interplay between Orlov spectra and Hochschild homology. The main goal of this paper is to suggest a series of interesting examples where the above techniques might apply. We start by constructing a sextic double solid X with 35 nodes and torsion in H3(X, ℤ). This is a novelty: after the classical example of Artin and Mumford, this is the second example of a Fano 3-fold with a torsion in the third integer homology group. In particular, X is non-rational. We consider other examples as well: V10 with 10 singular points, and the double covering of a quadric ramified in an octic with 20 nodal singular points. After analysing the geometry of their Landau–Ginzburg models, we suggest a general non-rationality picture based on homological mirror symmetry and category theory.

A graph in which each line is designated as either positive or negative is called a signed graph S. The sign of a cycle in S is a product of the signs of its lines. A signed graph in which every cycle is positive is called balanced. This concept was introduced by Harary in (3) and the following characterisation of balanced signed graphs was given.

I begin by recalling the well known definitions for summability by the methods of Cesaro and Riesz.

The series Σan is said to be summable (C, K), k> – 1, to the sum s if, as n → ∞

where

and is defined formally by the relation

For a valued field (K, v), let Kv denote the residue field of v and Gv its value group. One way of extending a valuation v defined on a field K to a simple transcendental extension K(x) is to choose any α in K and any μ in a totally ordered Abelian group containing Gv, and define a valuation w on K[x] by w(Σici(x – α)i) = mini (v(ci) + iμ). Clearly either Gv is a subgroup of finite index in Gw = Gv + ℤμ or Gw/Gv is not a torsion group. It can be easily shown that K(x)w is a simple transcendental extension of Kv in the former case. Conversely it is well known that for an algebraically closed field K with a valuation v, if w is an extension of v to K(x) such that either K(x)w is not algebraic over Kv or Gw/Gv is not a torsion group, then w is of the type described above. The present paper deals with the converse problem for any field K. It determines explicitly all such valuations w together with their residue fields and value groups.

The Hecke algebra of a Hecke pair (G, H) is studied using the Schlichting completion (Ḡ, ), which is a Hecke pair whose Hecke algebra is isomorphic to and which is topologized so that is a compact open subgroup of Ḡ. In particular, the representation theory and C*-completions of are addressed in terms of the projection using both Fell's and Rieffel's imprimitivity theorems and the identity . An extended analysis of the case where H is contained in a normal subgroup of G (and in particular the case where G is a semi-direct product) is carried out, and several specific examples are analysed using this approach.