Given any group G, its tensor square G ⊗ G is defined by the followingpresentation(see [[3]):

where g,g′,h,h′ range independently over G, and gh=ghg−1. In what follows, gg′ ⊗ gh is often written in the abbreviated form g(g′ ⊗ h).

In this paper we classify all infinite metacyclic groups up to isomorphism and determine their non-abelian tensor squares. As an application we compute various other functors, among them are the exterior square, the symmetric product, and the second homology group for these groups. We show that an infinite non-abelian metacyclic group is capable if and only if it is isomorphic to the infinite dihedral group

Suppose λ is a positive number, and let , x∈Rd, denote the d-dimensional Gaussian. Basic theory of cardinal interpolation asserts the existence of a unique function , x∈Rd, satisfying the interpolatory conditions , k∈Zd, and decaying exponentially for large argument. In particular, the Gaussian cardinal-interpolation operator , given by , x∈Rd, , is a well-defīned linear map from ℓ2(Zd) into L2(Rd). It is shown here that its associated operator-norm is , implying, in particular, that is contractive. Some sidelights are also presented.

Let $R$ be a commutative ring with unity and let $G$ be a group. The group ring $RG$ has a natural involution that maps each element $g\in G$ to its inverse. We denote by $RG^-$ the set of skew symmetric elements under this involution. We study necessary and sufficient conditions for $RG^-$ to be commutative.

In this paper, we determine all the factorials that are a sum of at most three Fibonacci numbers.

We prove results on geodesic metric spaces which guarantee that some spaces are not hyperbolic in the Gromov sense. We use these theorems in order to study the hyperbolicity of Riemann surfaces. We obtain a criterion on the genus of a surface which implies non-hyperbolicity. We also include a characterization of the hyperbolicity of a Riemann surface $S^*$ obtained by deleting a closed set from one original surface $S$. In the particular case when the closed set is a union of continua and isolated points, the results clarify the role of punctures and funnels (and other more general ends) in the hyperbolicity of Riemann surfaces.

1. V. Bernstein, N. Levinson, R. P. Boas, Jr., and others have investigated under what conditions on the sequence

for all functions f(z) regular and of suitably restricted growth in the halfplane x≥0. (For references see [1].)

In this note the restrictions on f(z) are that f(z) is regular, and not identically zero in continuous in and that for some positive B

We consider classes of subsets of [0, 1], originally introduced by Falconer, that are closed under countable intersections, and such that every set in the class has Hausdorff dimension at least s. We provide a Frostman-type lemma to determine if a limsup set is in such a class. Suppose that E = lim sup En ⊂ [0, 1], and that μn are probability measures with support in En. If there exists a constant C such that

for all n, then, under suitable conditions on the limit measure of the sequence (μn), we prove that the set E is in the class .

As an application we prove that, for α > 1 and almost all λ ∈ (½, 1), the set

where and ak ∈ {0, 1}}, belongs to the class . This improves one of our previously published results.

Consider a sufficiently smooth simple closed convex plane curve enclosing the origin, expanding linearly with time. The root mean square of the discrepancy (number of lattice points minus area) from time t = M to t = M + 1 is almost as small as the root mean square discrepancy from time t = 0 to t = M, so the discrepancy has no memory.

Nevanlinna's second main theorem is a far-reaching generalization of Picard's theorem concerning the value distribution of an arbitrary meromorphic function f. The theorem takes the form of an inequality containing a ramification term in which the zeros and poles of the derivative f′ appear. We show that a similar result holds for special subfields of meromorphic functions where the derivative is replaced by a more general linear operator, such as higher-order differential operators and differential-difference operators. We subsequently derive generalizations of Picard's theorem and the defect relations.

This paper takes a new look at ideals generated by 2×2 minors of 2×3 matrices whose entries are powers of three elements not necessarily forming a regular sequence. A special case of this is the ideals determining monomial curves in three-dimensional space, which were studied by Herzog. In the broader context studied here, these ideals are identified as Northcott ideals in the sense of Vasconcelos, and so their liaison properties are displayed. It is shown that they are set-theoretically complete intersections, revisiting the work of Bresinsky and of Valla. Even when the three elements are taken to be variables in a polynomial ring in three variables over a field, this point of view gives a larger class of ideals than just the defining ideals of monomial curves. We then characterize when the ideals in this larger class are prime, we show that they are usually radical and, using the theory of multiplicities, we give upper bounds on the number of their minimal prime ideals, one of these primes being a uniquely determined prime ideal of definition of a monomial curve. Finally, we provide examples of characteristic-dependent minimal prime and primary structures for these ideals.

In the Theory of Potential the term Green's Function, used in a slightly different sense by Maxwell, now denotes a function associated with a closed surface S, with the following properties:—

(i) In the interior of S, it satisfies ∇2V = 0.

(ii) At the boundary of S, it vanishes.

(iii) In the interior of S, it is finite and continuous, as also its first and second derivatives, except at the point (x1, y1,z1).

In Part I it has been shown that, given a contact transformation, two equations

can be derived which lead to the compatible differential equations

It will be shown in the present communication that the necessary and sufficient condition that (1.3), (1.4) should be compatible is that

regarded as an equation in the non-commutative variables q, p which themselves satisfy the condition

We shall call functions satisfying this condition conjugate functions. From this point of view the method employed by Professor Whittaker in his original paper, involving the use of a contact transformation,, was really a particular method of generating conjugate functions. This powerful method may be supplemented and extended by the other methods developed in the following pages.

One of the most fundamental problems in the study of Lagrangian submanifolds from a Riemannian geometric point of view is the classification of Lagrangian immersions of real-space forms into complex-space forms. In this article, we solve this problem for the most basic case; namely, we classify Lagrangian surfaces of constant curvature in the complex Euclidean plane $\mathbb{C}^2$. Our main result states that there exist 19 families of Lagrangian surfaces of constant curvature in $\mathbb{C}^2$. Twelve of the 19 families are obtained via Legendre curves. Conversely, Lagrangian surfaces of constant curvature in $\mathbb{C}^2$ can be obtained locally from the 19 families.

Kirchhoff's kinematic hypothesis that leads to an approximate two-dimensional theory of bending of elastic plates consists in assuming that the displacements have the form [1]

In general, the Dirichlet and Neumann problems for the equilibrium equations obtained on the basis of (1.1) cannot be solved by the boundary integral equation method both inside and outside a bounded domain because the corresponding matrix of fundamental solutions does not vanish at infinity [2]. However, as we show in this paper, the method is still applicable if the asymptotic behaviour of the solution is suitably restricted.

It has been proved by CAYLEY that if x11, x12, x21 … are independent variables, x = det (xik), ξ = det (ξik), (i, k = 1, … n) where ξik =∂/∂xik then by formal derivation ξxα = α(α + 1)…(α + n − 1)xα−1. This is a special case of the formula

where m=1,…,n and with i = i1,..im; k= k1,…km and xi,… is the algebracial complement of i = i1,..im; k = k1,…km, in .

Let v = (a1 …, an) be a real n-tuple and be the numbers a1 …, an arranged in decreasing order. Let denote the sum of m greatest components of v and the sum of m smallest components of v, i.e.,

We prove the following: Assume that , where p is an odd positive integer, g(ζ is a transcendental entire function with order of growth less than 1, and set A(z) = B(ezz). Then for every solution , the exponent of convergence of the zero-sequence is infinite, and, in fact, the stronger conclusion holds. We also give an example to show that if the order of growth of g(ζ) equals 1 (or, in fact, equals an arbitrary positive integer), this conclusion doesn't hold.

Throughout this paper H will denote a complex separable Hilbert space and L(H) denotes the algebra of all bounded linear operators on H. If T lies in L(H), its spectrum σ(T) is the set of all complex numbers z such zI–T is not invertible in L(H) and its compression spectrum σcomp(T) is the set of all complex numbers z such that the range (zI-T)(H) is not dense in H ([3, p. 240]). This paper is concerned with the Sturm–Liouville operator problem

where λ is a complex parameter and X(t), Q, Ei, Fi for i = l,2, and t∈[0,a], are bounded operators in L(H). For the scalar case, the classical Sturm-Liouville theory yields a complete solution of the problem, see [4], and [7]. For the finite-dimensional case, second order operator differential equations are important in the theory of damped oscillatory systems and vibrational systems ([2, 6]). Infinite-dimensional differential equations occur frequently in the theory of stochastic processes, the degradation of polymers, infinite ladder network theory in engineering [1, 17], denumerable Markov chains, and moment problems [10, 20]. Sturm-Liouville operator problems have been studied by several authors and with several techniques ([12, 13, 14, 15, 16]).

The well-known algebraic concept of tensor product exists for any variety of algebras.The tensor product of groups and of rings have been studied extensively. For other varieties, such as the variety of semigroups, the tensor product has been investigated more recently (5). In this paper we investigate the tensor product of distributive lattices.