Recently A. D. Sands [7] solved a problem posed in [6], and characterised the semisimple classes of associative rings as classes being regular, coinductive and closed under extensions. It is the purpose of this note to prove the same assertion for alternative rings. This result is perhaps not surprising, nevertheless its proof cannot be considered an easy one, and it requires a technique of dealing with ideals of ideals. In addition, semisimple classes of hereditary radicals and those of supernilpotent radicals will be characterised as easy consequences of our theorem.

In this note we show that if we have an exact sequence of AH algebras (AH stands for “approximately homogeneous”) 0 → I → A → B → 0, then A has the ideal property (i.e., any ideal is generated by its projections) if and only if I and B have the ideal property. Also, we prove that an extension of two AT algebras (AT stands for “approximately circle”) with the ideal property is an AT algebra with the ideal property if and only if the extension is quasidiagonal.

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).

If $A$ is a stable basis algebra of rank $n$, then the set $S_{n-1}$ of endomorphisms of rank at most $n-1$ is a subsemigroup of the endomorphism monoid of $A$. This paper gives a number of necessary and sufficient conditions for $S_{n-1}$ to be generated by idempotents. These conditions are satisfied by finitely generated free modules over Euclidean domains and by free left $T$-sets of finite rank, where $T$ is cancellative monoid in which every finitely generated left ideal is principal.

The purpose of this paper is to describe , the bordism module of unitary T-manifolds, where T denotes the circle group S1. We give both an algebraic and a geometric description. The algebraic result is

where I = (i(1), i(2),…i(2n)) runs through all finite ordered 2n-tuples (n≧0) of non-negative integers which satisfy the conditions (a) i(l) + i(2n)≠0 and (b) if i(2n)≠0 then i(2n)=≠. The isomorphism is also described geometrically and this leads to geometric generators of .

With differential equations in the neighbourhood of an irregular singular point, it sometimes happens that formal solutions may converge. For example, this occurs for Bessel's equation at∞ when the parameter is half of an odd integer. In addition, there are some classical theorems of Perron and Lettenmeyer which give sufficient conditions for the existence of linearly independent analytic solutions at (generally) an irregular singular point. Using the principle of reduction of order, such a solution may be used to transform the differential equation into one whose coefficient matrix is triangularly blocked with an (n – 1) and 1-block on the diagonal. The solutions of the given differential equation can thus be obtained by solving a lower dimensional differential equation plus quadrature.

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

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.

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.

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.

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.

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.

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.

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 the one-dimensional operator,

on 0<x<∞ with . The coefficients p, V1 and V2 are assumed to be real, locally Lebesgue integrable functions; c1 and c2 are positive numbers. The operator L acts in the Hilbert space H of all equivalence classes of complex vector-value functions such that . L has domain D(L) consisting of ally∈H such that y is locally absolutely continuous and Ly∈H; thus in the language of differential operators L is a maximal operator. Associated with L is the minimal operator L0 defined as the closure of where is the restriction of L to the functions with compact support in (0,∞).

Let R be a semiprime ring with the extended centroid C and Q the maximal right ring of quotients of R. Set [y, x]1 = [y, x] = yx − xy for x, y ∈ Q and inductively [y, x]k = [[y, x]k−1, x] for k > 1. Suppose that f : R → Q is an additive map satisfying [f(x), x]n = 0 for all x ∈ R, where n is a fixed positive integer. Then it can be shown that there exist λ ∈ C and an additive map μ : R → C such that f(x) = λx + μ(x) for all x ∈ R. This gives the affirmative answer to the unsolved problem of such functional identities initiated by Brešar in 1996.

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.

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 .

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

Consider the nonautonomous delay logistic difference equation

where (pn)n≧0 is a sequence of nonnegative numbers, (ln)n≧0 is a sequence of positive integers with limn→∞(n−ln) = ∞ and K is a positive constant. Only solutions which are positive for n≧0 are considered. We established a sharp condition under which all solutions of (E0) are oscillatory about the equilibrium point K. Also we obtained sufficient conditions for the existence of a solution of (E0) which is nonoscillatory about K.