In this paper we shall develop a new method for the computation of eigenvalues of singular Sturm-Liouville problems of the Bessel type. This new method is based on the interpolation of a boundary function in Paley-Wiener spaces. Numerical results are provided to illustrate the method.

Let S be a regular semigroup. An inverse subsemigroup S° of S is called an inverse transversal if S° contains a unique inverse of each element of S. An inverse transversal S° of S is called multiplicative if x°xyy° is an idempotent of S° for every x, y∈S, where x° denotes the unique inverse of x∈S in S°. In Section 1, we obtain a necessary and sufficient condition in order for inverse transversals to be multiplicative.

We study a class of second-order nonlinear differential equations on a finite interval with periodic boundary conditions. The nonlinearity in the equations can take negative values and may be unbounded from below. Criteria are established for the existence of non-trivial solutions, positive solutions and negative solutions of the problems under consideration. Applications of our results to related eigenvalue problems are also discussed. Examples are included to illustrate some of the results. Our analysis relies mainly on topological degree theory.

For a large integer $m$, we obtain asymptotic formulae for the number of solutions of certain congruences modulo $m$ with several variables, where the variables belong to special sets of residue classes modulo $m$. In particular, we obtain new information on the exceptional set of the multiplication table problem in the residue ring modulo $m$.

Semiclassical limits of generic multi-parameter quantized coordinate rings A=q(kn) of affine spaces are constructed and related to A, for k an algebraically closed field of characteristic zero and q a multiplicatively antisymmetric matrix whose entries generate a torsion-free subgroup of k×. A semiclassical limit of A is a Poisson algebra structure on the corresponding classical coordinate ring R=(kn), and results of Oh, Park, Shin and the authors are used to construct homeomorphisms from the Poisson-prime and Poisson-primitive spectra of R onto the prime and primitive spectra of~A. The Poisson-primitive spectrum of R is then identified with the space of symplectic cores in kn in the sense of Brown and Gordon, and an example is presented (over ℂ) for which the Poisson-primitive spectrum of R is not homeomorphic to the space of symplectic leaves in kn. Finally, these results are extended from quantum affine spaces to quantum affine toric varieties.

1. The subject-matter of this paper is in some sense known; but we will try to organise, explain and reprove it, and to give examples.

Let L be a nef line bundle on a projective scheme X in positive characteristic. We prove that the augmented base locus of L is equal to the union of the irreducible closed subsets V of X such that L∣V is not big. For a smooth variety in characteristic 0, this was proved by Nakamaye using vanishing theorems.

This paper consists mainly of a review and applications of our old results relating to the title. We discuss how many elliptic fibrations and elliptic fibrations with infinite automorphism groups (or Mordell–Weil groups) an algebraic K3 surface over an algebraically closed field can have. As examples of applications of the same ideas, we also consider K3 surfaces with exotic structures: with a finite number of non-singular rational curves, with a finite number of Enriques involutions, and with naturally arithmetic automorphism groups.

The differential operation known as Lie derivation was introduced by W. Slebodzinski in 1931, and since then it has been used by numerous investigators in applications in pure and applied mathematics and also in physics. A recent monograph by Kentaro Yano (2) devoted to the theory and application of Lie derivatives gives some idea of the wide range of its uses. However, in this monograph, as indeed in other treatments of the subject, the Lie derivative of a tensor field is defined by means of a formula involving partial derivatives of the given tensor field. It is then proved that the Lie derivative is a differential invariant, i.e. it is independent of a transformation from one allowable coordinate system to another. Sometimes some geometrical motivation is given in explanation of the formula, but this is seldom very satisfying.

We study closed Einstein–Weyl structures on compact K-contact, Sasakian and cosymplectic manifolds. In particular we prove that compact Sasakian and cosymplectic manifolds endowed with a closed Einstein–Weyl structure are η-Einstein.

In a recent paper (6) the present author has shown that, for an element a of a Banach algebra A, the condition

for all x∈A and some constant α is equivalent to [x, a]∈Rad a for all x∈A; it turns out that α may be replaced by |α|σ It is the purpose of the present note to investigate a related condition

The purpose of this paper is to study the consequences of an endomorphism near-ring of a finite group being a local near-ring and the existence of such near-rings. As we shall see in Section 2, an endomorphism near-ring of a finite group being local gives us some information about both the structure of the group (Theorem 2.2) and the automorphisms of the group lying in the near-ring (Theorem 2.3). Existence of local endomorphism near-rings of finite groups is considered in Section 3 where we obtain as our main result that any p-group of automorphisms of a p-group containing the inner automorphisms always generates a local endomorphism near-ring. In particular, we get as a corollary that the endomorphism near-ring of a finite group G generated by the inner automorphisms of G is local if and only if G is a p-group. The third section concludes with a discussion of endomorphism near-rings of dihedral 2-groups and generalized quaternion groups.

Many basic definitions and results in the theory of near-rings can be found in G. Pilz (4). We follow these for the most part, except that we use left near-rings rather than right near-rings. We follow exactly an earlier paper, Meldrum (2), where there are detailed definitions and many results relating to faithful d.g. near-rings. Let R be a d.g. near-ring, distributively generated by the semigroup S, which need not be the semigroup of all distributive elements. Denote such a d.g. near-ring by (R, S). Then (R, +) = Gp < S; > where is a set of defining relations in S. Let (T, U) be a d.g. near-ring. Then a d.g. homomorphism θ from (R, S) to (T, U) is a near-ring homomorphism from R to T which satisfies Sθ ⊆ U. If (G, +) is a group, let T0(G) be the near-ring of all maps from G to itself with pointwise addition and map composition. Let End G be the semigroup of all endomorphisms of G. Then (E(G), End G) is a d.g. near-ring. A d.g. near-ring (R, S) is faithful if there exists a d.g. monomorphism θ:(R, S) → (E(G), End G) for some group G.

A classical theorem due to Young states that the cosine polynomial

$$ C_n(x)=1+\sum_{k=1}^{n}\frac{\cos(kx)}{k} $$

is positive for all $n\geq1$ and $x\in(0,\pi)$. We prove the following refinement. For all $n\geq2$ and $x\in[0,\pi]$ we have

$$ \tfrac{1}{6}+c(\pi-x)^2\leq C_n(x), $$

with the best possible constant factor

$$ c=\min_{0\leq t\lt\pi}\frac{5+6\cos(t)+3\cos(2t)}{6(\pi-t)^2}=0.069\dots. $$

Sequence transformations are extrapolation methods. They are used for the purpose of convergence acceleration. In the scalar case, such algorithms can be obtained by two different approaches which are equivalent. The first one is an elimination approach based on the solution of a system of linear equations and it makes use of determinants. The second approach is based on the notion of annihilation difference operators. In this paper, these two approaches are generalized to the matrix and the vector cases.

For $a\in(0,\tfrac12]$ and $r\in(0,1)$, let $\mu_a(r)$ be the so-called generalized Grötzsch function which appears in Ramanujan's generalized modular equations. In this paper, several sharp inequalities for $\mu_a(r)$ are obtained and a conjecture on $\mu_a(r)$, which was presented by Qiu and Vuorinen in 1999, is proved.

Given a smooth compact surface without focal points and of higher genus, it is shown that its geodesic flow is semi-conjugate to a continuous expansive flow with a local product structure such that the semi-conjugation preserves time parametrization. It is concluded that the geodesic flow has a unique measure of maximal entropy.

The motion of a rigid sphere embedded in an adhering medium and subjected to an external force is analysed exactly in the context of classical elastodynamics. For the limiting case of an incompressible medium it is possible to write down a simple second order differential equation relating displacement of the ball to the external force.

The theory is generalised to the case of a viscoelastic solid and the results obtained are pertinent to recently developed experimental methods for testing the dynamical mechanical properties of very low modulus polymer gels.

In (8) Todd and Coxeter described an algorithm for enumerating the cosets of a finitely generated subgroup of finite index in a finitely presented group. Several authors ((1), (2), (5), (6), (7)) have discussed a modification of the algorithm to give also a presentation of the subgroup in terms of the given generators.

In this paper we consider the existence and multiplicity of positive solutions of a nonlinear elliptic boundary-value problem with nonlinear boundary conditions which arises in population dynamics. While bifurcation problems from lines of trivial solutions are studied, the existence of bifurcation positive solutions from infinity is discussed. The former will be caught by the reduction to a bifurcation equation following the Lyapunov and Schmidt procedure. The latter will be based on a variational argument depending on the corresponding constrained minimization problem.

AMS 2000 Mathematics subject classification: Primary 35J65; 35J20; 35P30; 35B32; 92D25