Let G be a group and let K be an algebraically closed field of characteristic p>0. The twisted group algebra Kt(G) of G over K is defined as follows: let G have elements a, b, c, … and let Kt(G) be a vector space over K with basis elements , …; a multiplication is defined on this basis of Kt(G) and extended by linearity to Kt(G) by letting

where α(x, y) is a non-zero element of K, subject to the condition that

which is both necessary and sufficient for associativity. If, for all x, y ∈ G, α{x, y) is the identity of K then Kt(G) is the usual group algebra K(G) of G over K. We denote the Jacobson radical of Kt(G) by JKt(G). We are interested in the relationship between JKt(G) and JKt(H) where H is a normal subgroup of G. In § 2 we show, among other results, that if certain centralising conditions are satisfied and if JK(H) is locally nilpotent then JK(H)K(G) is also locally nilpotent and thus contained in JK(G). It is observed that in the absence of some centralising conditions these conclusions are false. We show, in particular, that if H and G/C(H) are locally finite, C(H) being the centraliser of H, and if G/H has no non-trivial elements of order p, then JK(G) coincides with the locally nilpotent ideal JK(H)K(G). The latter, and probably more significant, part of this paper is concerned with particular types of groups. We introduce the notion of a restricted SN-group and show that if G is such a group and ifG has no non-trivial elements of order p then JKt(G) = {0}. It is also shown that if G is polycyclic then JKt(G) is nilpotent.

If the elements of a symmetric matrix lie in the real field it is well known that the roots of its characteristic equation are real. This implies that the discriminant of that equation (i.e. the product of the squared differences of the roots) is a polynomial in the elements which is non-negative and the same must be true for the leading coefficients of all the other Sturm functions associated with the characteristic equation. One would expect that it should be possible to express them as a sum of squares. Conversely, such an expression would establish the reality of the roots.

The problem of determining the distribution of stress in the neighbourhood of a Griffith crack, denned by ∣x∣ ≦ 1, y = 0, which is subject to an internal pressure varying along the length of the crack has been solved by Sneddon and Elliott (3) and by England and Green (1). In the former paper a Fourier cosine method is used to arrive at a solution while in the latter paper the problem is reduced to an Abel integral equation by making use of integral representations of the complex potentials given in Green and Zerna (2). Neither paper deals with the calculation of the stress intensity factor

which is extremely important to workers in fracture mechanics.

Let ξ be an irrational number with simple continued fraction expansion ξ= [a0;a1,a2,…], Pn/qn be its nth convergent, . The following two theorems were proved by Müller [9] and rediscovered by Bagemihl and McLaughlin [1]:

Theorem 1.For n>1,

We study homogeneous Kähler structures on a non-compact Hermitian symmetric space and their lifts to homogeneous Sasakian structures on the total space of a principal line bundle over it, and we analyse the case of the complex hyperbolic space.

This paper studies questions connected with when the Rees algebra of an ideal or the formring of an ideal is Gorenstein. The main results are for ideals of small analytic deviation, and for m-primary ideals of a regular local ring (R, m). The general point proved is that the Gorenstein property forces (and is sometimes equivalent to) lowering the reduction number of the ideal by one from the value predicted if one only assumes the Rees algebra or formring is Cohen–Macaulay.

In (2), Holcombe investigated near-rings of zero-preserving mappings of a group Γ which commute with the elements of a semigroup S of endomorphisms of Γ and examined the question: under what conditions do near-rings of this type have near-rings of right quotients which are 2-primitive with minimum condition on right ideals? In the first part of this paper (§2) we investigate further properties of near-rings of this type. The second part of the paper (§3) deals with those near-rings which have semisimple near-rings of right quotients. Our results here are analogous to those of Goldie (1); in particular, with a suitable definition of finite rank we prove that a near-ring which has a semisimple near-ring of right quotients has finite rank

In various problems in harmonic analysis (4, 5, 6), I have required an estimate for the integral for the hypergeometric function

in the case where a>c−b>0, b>0, and 0 ≦ x < 1 (the integral is then unbounded as x→l−). Although there are innumerable identities for hypergeometric functions, few inequalities for these functions seem to be known, and in estimating the integral (1) I employed ad hoc arguments that made no use of the theory of hypergeometric functions. The estimates obtained were adequate for my purposes, but were far from sharp, and the object of this note is to show that, by assembling a few known facts concerning hypergeometric functions, we can obtain sharp inequalities for the integral (1). We also give (in §4) several related inequalities that can be obtained by transforming the integral.

Consider the ordinary linear matrix differential system

ψ(x) is a scalar mapping, X and A(x) are n by n matrices. Both belong to C1([a,∞)) for some integer l. The stability and asymptotic behaviour of its solutions have been subject to much investigation. See Bellman [2], Levinson [24], Hartman and Wintner [20], Devinatz [9], Fedoryuk [11], Harris and Lutz [16,17,18] and Cassell [30]. The special interest in eigenvalue problems and in the deficiency index problem stimulated a continued interest in asymptotic integration. See e.g. Naimark [36], Eastham and Grundniewicz [10] and [8,9]. Harris and Lutz [16,17,18] succeeded in explaining how to derive many known theorems in asymptotic integration by repeatedly using certain “(1 + Q)” linear transformations.

In analogy with the maximal tensor product of C*-algebras, we define the “maximal” tensor product E1⊗μE2 of two operator spaces E1 and E2 and we show that it can be identified completely isometrically with the sum of the two Haagerup tensor products: E1⊗hE2 + E2⊗hE1. We also study the extension to more than two factors. Let E be an n-dimensional operator space. As an application, we show that the equality E*⊗μE = E*⊗min E holds isometrically iff E = Rn or E = Cn (the row or column n-dimensional Hilbert spaces). Moreover, we show that if an operator space E is such that, for any operator space F, we have F ⊗min E = F⊗μ E isomorphically, then E is completely isomorphic to either a row or a column Hilbert space.

In the present paper we deal with a quasilinear problem involving a singular term. By combining truncation techniques with variational methods, we prove the existence of three weak solutions. As far as we know, this is the first contribution in this direction in the high-dimensional case.

Let R be a noetherian prime PI ring and let P be a prime ideal of R. There is a set of prime ideals, the fundamental prime ideals, associated with the injective hull of R/P and denoted by Fund(P). The set Fund(P) is finite, by a result of Miiller. In this paper we give a natural description of Fund(P) in terms of the trace ring of R which strengthens Miiller's result by establishing a uniform bound for the size of Fund(P) for all primes P in the ring.

Let R be a commutative ring and let q be an ideal in R. Let En(R) be the subgroup of GLn(R) generated by the elementary matrices and let En(q) be the normal subgroup of En(R) generated by the q-elenientary matrices. The order of a subgroup S of GLn(R) is the ideal q0 in R generated by xij, xii−xjj, where (xij)∈S, with 1≦i, j≦n and i≠j. The subgroup S is called a standard subgroup if En(q0)≦S. An almost-normal subgroup of GLn(R) is a non-normal subgroup which is normalized by En(R).

By analogy with the concept of “inverse semi-group” in semi-group theory, in this paper we introduce the concept of “generalized near-field” in near-rings. A near-ring N is called a generalized near-field (GNF) if for each a ε N there exists a unique b ε N such that a = aba and b = bab, that is (N, ·) is an inverse semi-group. Surprisingly, this concept in rings coincides with that of “strong regularity”. But this is not true in the case of near-rings. Every GNF is strongly regular, but in general the converse is not true.

The treatment of the Fourier Series, that is, of the series which proceeds according to sines and cosines of multiples of the variable, is in most English text-books very unsatisfactory; in many cases it shows almost no advance upon that of Poisson and, even where a more or less accurate reproduction of Dirichlet's investigations is given, there is no attempt at indicating the advantages it possesses over the so-called proof of Poisson. Nor is the uniformity of the convergence of the series so much as mentioned, not to say discussed. I have therefore thought it might be useful to give a fairly complete outline of the historical development of the series so far as the materials at my disposal allow. I do not think that any important contribution to the theory is omitted, but, as I indicate at one or two places, there are some memoirs to which I have not had access and which I only know at second hand.

We give an explicit formula for the projective dynamics of planar homogeneous polynomial differential systems in terms of natural local invariants and we establish explicit algebraic connections (syzygies) between these invariants (leading to restrictions on possible global dynamics). We discuss multidimensional generalizations together with applications to the existence of first integrals and bounded solutions.

We study systems which in characteristic coordinates have the form

where A is a k × k diagonal matrix with distinct real eigenvalues. The nonlinearity F is assumed to be asymptotically homogeneous in the sense, that it is a sum of two terms, one positively homogeneous of degree one in u and a second which is sublinear in u and vanishes when u = 0. In this case, F(t, x, u(t)) is meaningful provided that u(t) is a Radon measure, and, for Radon measure initial data there is a unique solution (Theorem 2.1).

The main result asserts that if μn is a sequence of initial data such that, in characteristic coordinates, the positive and negative parts of each component, , converge weakly to μ±, then the solutions coverge weakly and the limit has an interesting description given by a nonlinear superposition principle.

Simple weak converge of the initial data does not imply weak convergence of the solutions.

The asymptotic behaviour of inhomogeneous power-law type functionals is undertaken via De Giorgi’s Γ-convergence. Our results generalize recent work dealing with the asymptotic behaviour of power-law functionals acting on fields belonging to variable exponent Lebesgue and Sobolev spaces to the Orlicz–Sobolev setting.

In this paper we review the classification of isolated quotient singularities over the field of complex numbers due to Zassenhaus, Vincent and Wolf. As an application, we describe Gorenstein isolated quotient singularities over ℂ, generalizing a result of Kurano and Nishi.

In this paper we shall introduce a certain class of operators from a Banach lattice X into a Banach space B (see Definition 1) which is closely related to p-absolutely summing operators defined by Pietsch [8].

These operators, called positive p-summing, have already been considered in [9] in the case p = 1 (there they are called cone absolutely summing, c.a.s.) and in [1] by the author who found this space to be the space of boundary values of harmonic B-valued functions in .

Here we shall use these spaces and the space of majorizing operators to characterize the space of bounded p-variation measures and to endow the tensor product with a norm in order to get as its completion in this norm.