In [6] the question of the existence of perfect e-codes in the infinite family of distance-transitive graphs Ok was considered. It was pointed out that it is difficult to rule out completely any particular value of e because of the difficulty of working with the sphere packing condition. For e= 1, 2, 3 it can be seen from the results of [6] that the condition given by the generalisation of Lloyd's theorem is satisfied for infinitely many values of k. We shall show that this is not the case for e=4 and we shall prove that there are no perfect 4-codes in Ok.

In 1956, Jacobson asked whether the intersection of the powers of the Jacobson radical, J(R), of a right Noetherian ring R, must always be zero [4, p. 200]. His question was answered in the negative by I. N. Herstein [3], who noted that , where Z(2) denotes the ring of rational numbers with denominator prime to 2, affords a counterexample. In contrast, the ring , though similar in appearance to R1, satisfies . (Here, k denotes a field.)

The present paper incorporates a preliminary study of a new generalization of several known polynomial systems belonging to (or providing extensions of) the families of the classical Jacobi, Hermite and Laguerre polynomials. It is shown how suitable specializations will yield a number of known or new results in the theory of the special functions considered.

We consider a nonlinear Dirichlet problem driven by the p(ċ)-Laplacian. Using variational methods based on the critical point theory, together with suitable truncation techniques and the use of upper-lower solutions and of critical groups, we show that the problem has at least three nontrivial solutions, two of which have constant sign (one positive, the other negative). The hypotheses on the nonlinearity incorporates in our framework of analysis, both coercive and noncoercive problems.

In this paper, we deal with a tree-shaped network of strings with a fixed root node. By imposing velocity feedback controllers on all vertices except the root node, we show that the spectrum of the system operator consists of all isolated eigenvalues of finite multiplicity and is distributed in a strip parallel to the imaginary axis under certain conditions. Moreover, we prove that there exists a sequence of eigenvectors and generalised eigenvectors that forms a Riesz basis with parentheses, and that the imaginary axis is not an asymptote of the spectrum. Thereby, we deduce that the system is exponentially stable.

The present paper deals with relations between flat overrings and quotient rings. Weare mainly concerned with Richman's results [10] on flat overrings and withthose of Davis [2], Gilmer [3], Gilmer and Heinzer [4], Gilmer and Ohm [5], and Mott [8], on rings with the QR propertyand with the property (#) defined in Section 1. Some of their results are generalized, and it is shown that certain theorems, which at first glance seem to have nothing in common, are in fact particular cases of a single more general theorem.

The structure of groups which have at most two isomorphism classes of derived subgroups ($\mathfrak{D}$2-groups) is investigated. A complete description of $\mathfrak{D}$2-groups is obtained in the case where the derived subgroup is finite: the solution leads an interesting number theoretic problem. In addition, detailed information is obtained about soluble $\mathfrak{D}$2-groups, especially those with finite rank, where algebraic number fields play an important role. Also, detailed structural information about insoluble $\mathfrak{D}$2-groups is found, and the locally free $\mathfrak{D}$2-groups are characterized.

Given a complete hereditary cotorsion pair $(\mathcal{X}, \mathcal{Y})$, we introduce the concept of $(\mathcal{X}, \mathcal{X} \cap \mathcal{Y})$-Gorenstein projective modules and study its stability properties. As applications, we first get two model structures related to Gorenstein flat modules over a right coherent ring. Secondly, for any non-negative integer n, we construct a cofibrantly generated model structure on Mod(R) in which the class of fibrant objects are the modules of Gorenstein injective dimension ≤ n over a left Noetherian ring R. Similarly, if R is a left coherent ring in which all flat left R-modules have finite projective dimension, then there is a cofibrantly generated model structure on Mod(R) such that the cofibrant objects are the modules of Gorenstein projective dimension ≤ n. These structures have their analogous in the category of chain complexes.

In this paper we study the dependence of the first eigenvalue of the infinity Laplace with respect to the domain. We prove that this first eigenvalue is continuous under some weak convergence conditions which are fulfilled when a sequence of domains converges in Hausdorff distance. Moreover, it is Lipschitz continuous but not differentiable when we consider deformations obtained via a vector field. Our results are illustrated with simple examples.

Let ω be a non-empty set, ℱ a Boolean σ-algebra of subsets of Ω, k a natural number, and let m:ℱ→ℝk be a non-atomic vector measure. Then, by the celebrated theorem of Liapounov [11], the range m[3F] = {m(A): A ε ℱ3F} of m is a compact convex subset of ℝk. This theorem has been generalized in a number of ways. For example Kingman and Robertson [8] and Knowles [9] have shown that, under appropriate conditions, results in the same spirit can be proved for measures taking their values in infinite-dimensional vector spaces. Another type of generalization was obtained by Dvoretsky, Wald and Wolfowitz [6,7]. What they do is to take m as above together with a natural number n≥ 1. They then consider the set Knof all vectors

where (A1 A2,…, An) is an ordered ℱ-measurable partition of Ω (i.e. a partition whose terms A, all belong to ℱ). They prove in [6] that Kn is a compact convex subset of ℝnk and moreover that Kn is equal to the set of all vectors of the form

where (ϕ1, ϕ2…, ϕn) is an ℱ-measurable partition of unity; i.e. it is an n-tuple of non-negative ϕr on Ω such that

Liapounov's theorem can be obtained as a corollary of this result by taking n= 2.

Let M be a real hypersurface of the complex projective space Pn(C). The Ricci tensor S of M is recurrent if there exists a 1-form $\alpha$ such that $\nablaS=S\otimes\alpha;$. In this paper we show that there are no real hypersurfaces with recurrent Ricci tensor of Pn(C) under the condition that $\xi$ is a principal curvature vector.

1991 Mathematics Subject Classification 53C40 (53C25).

Let E be a Banach space such that its dual E* is separable. We show that there exists a hypercyclic bounded operator T on E such that its adjoint T* is also hypercyclic on E*. We also exhibit a new kind of dual hypercyclic operator. Thus answers affirmatively two of the questions raised by Henrik Petersson in a recent paper.

Let R be a semiprime ring (possibly without 1). The symmetric ring of quotients of R is defined as the set of equivalence classes of essentially defined double centralizers (ƒ, g) on R; see [1], [8]. So, by definition, ƒ is a left R-module homomorphism from an essential ideal I of R into R, g is a right R-module homomorphism from an essential ideal J of R into R, and they satisfy the balanced condition ƒ(x)y = xg(y) for x ∈ Iand y ∈ J. This ring was used by Kharchenko in his investigations on the Galois theory of semiprime rings [4] and it is also a useful tool for the study of crossed products of prime rings [7]. We denote the symmetric ring of quotients of a semiprime ring R by Q(R).

Given a contact form $\eta$, there is a one-to-one correspondence between the Riemannian structures $(\eta,g)$ and the CR-structures $(\eta,L)$. It is interesting to study the interaction between the two associated structures. We approach the geometry of contact Riemannian manifolds in connection with their associated CR-structures. In this context, for a contact Riemannian manifold $(M;\eta,g)$ we consider the Jacobi-type operator $R_{\dot\gamma}=R(\cdot,\dot\gamma)\dot\gamma$ along a self-parallel curve $\gamma$ with respect to the (generalized) Tanaka connection $\hatbnabla$.This work was financially supported by Chonnam National University in the program, 2001.

S(X) is the semigroup of all continuous self maps of the topological space X and for any semigroup S, Cong(S) will denote the complete lattice of congruences on S. Cong(S) has a zero Z and a unit U. Specifically, Z = {(a, a):a ∈ S} and U = S × S. Evidently, Z and U are distinct if S has at least two elements. By a proper congruence on S we mean any congruence which differs from each of these. Since S(X) has more than one element when X is nondegenerate, we will assume without further mention that the spaces we discuss in this paper have more than one point. We observed in [4] that there are a number of topological spaces X such that S(X) has a largest proper congruence, that is, Cong(S(X)) has a unique dual atom which is greater than every other proper congruence on S(X). On the other hand, we also found out in [5] that it is also common for S(X) to fail to have a largest proper congruence. We will see that the situation is quite different at the other end of the spectrum in that it is rather rare for S(X) not to have a smallest proper congruence. In other words, for most spaces X, Cong(S(X)) has a unique atom which is smaller than every other proper congruence.

Let K be a compact Hausdorff space, X a Banach space and C(K, X) the Banach space of all continuous functions f: K → X endowed with the supremum norm. In this paper we study weakly precompact operators defined on C(K, X).

Let d > 0 be a squarefree integer and denote by h = h(−d) the class number of the imaginary quadratic field . It is well known (see e.g. [25]) that for a given positive integer N there are only finitely many squarefree d's for which h(−d) = N. In [45], Saradha and Srinivasan and in [28] Le and Zhu considered the equation in the title and solved it completely under the assumption h(−d) = 1 apart from the case d ≡ 7 (mod 8) in which case y was supposed to be odd. We investigate the title equation in unknown integers (x, y, l, n) with x ≥ 1, y ≥ 1, n ≥ 3, l ≥ 0 and gcd(x, y) = 1. The purpose of this paper is to extend the above result of Saradha and Srinivasan to the case h(−d) ∈ {2, 3}.

In this paper, we show that Weyl's theorem holds for operators having the single valued extension property and quasisimilarity preserves Weyl's theorem for these operators under some assumptions for spectral subsets, respectively.

We discuss practical methods for computing the space of solutions to an arbitrary homogeneous linear system of partial differential equations with constant coefficients. These rest on the Fundamental Principle of Ehrenpreis–Palamodov from the 1960s. We develop this further using recent advances in computational commutative algebra.

We show that Diestel's theorem on weak compactness of subsets of L1,(μ, X) can be derived as a simple corollary of James's theorem. It is a pleasure to acknowledge several stimulating conserversations with Dave Emmons and the remarks of an anonymous referee. Errors are, of course, solely mine.