In this paper we give a new bound for the solutionsx of the title equation, provided that k \ge 8. This bound is polynomial in d. Moreover, under the same condition, a similar bound for the number of solutions in (x, k, y, l) is given.

Let $(\Omega,\, \Sigma,\,\Prob)$ be a nonatomic probability space and let $\F=(\F_n)_{n{\in}\Z}$ be a filtration. If $f=(\,f_n)_{n{\in}\Z}$ is a uniformly integrable $\F$-martingale, let $\A_{\F}f=(\A_{\F}f_n)_{n{\in}\Z}$ denote the martingale defined by $\A_{\F}f_n =\E[|\,f_{\infty}||\F_n]\; (n \,{\in}\, \Z)$, where $f_{\infty}=\lim_n f_n$ a.s. Let $X$ be a Banach function space over $\Omega$. We give a necessary and sufficient condition for $X$ to have the property that $S(\,\hspace*{.2pt}f\hspace*{.3pt}) \,{\in}\, X$ if and only if $S(\A_{\F}f) \,{\in}\, X$, where $S(\,\hspace*{.2pt}f\hspace*{.3pt})$ stands for the square function of $f=(\,f_n)$.

We calculate the height of quantum determinantal ideals in the algebra of quantum matrices, and also the Gelfand-Kirillov, Krull and Classical Krull dimensions of the corresponding quantum determinantal factors.

With much sadness we note the death of John Leech, on 28 September 1992. Perhaps best known for his discovery of the “Leech Lattice” (which provides the best known sphere-packing in 24 dimensions), John will also be remembered for his contributions to the use of computers in mathematics, and to computational algebra in particular.

Let G be a compact Lie group and X a G-CW complex. We are interested in the calculation of the Borel cohomology of X

where EG is a universal free G-space and we use on the right hand side cellular cohomology. For an introduction to G-CW complexes see Matumoto [4] and for a good exposition on Borel cohomology see for instance torn Dieck [2], We want to replace X with an ordinary CW complex Y in order to find an ordinary CW structure on the Borel construction EG ΧGY so we can use cellular chains to compute the Borel cohomology of X. For every compact Lie group one has an extension

where G0 is the identity component, so for our case G0 is isomorphic to the circle group . We are dealing with the case in which π0(G) is isomorphic to C2, the cyclic group of order

Given a probability space (X, ℱ, μ) and a σ-algebra A ⊂ ℱ, arguably the most powerful tool in gaining information about an ℱ-measurable function f from restricted knowledge of -measurability is that of the conditional expectation E(f | ); written throughout the remainder of this note. Two properties of conditional expectation that may be exploited to gain information, but which also limit conditional expectation's use are the following.

If k denotes a number field and εm is the product of an elliptic curve ε with itself m times over k, then for each prime π where ε has non-degenerate reduction, the zeta factor ζ(επ'S) can be expressed as

Where |π| denotes the norm of π. It is a consequence of a conjecture of Tate [16] that if ε does not have complex multiplications, then the numbers are distributed according to the density function

that is, the density of the set of primes π such that – is

We shall extend results of Samuel [19] and Griffin [8, 9] about conditions which generalise the notion of valuation domain in a field. Let U be a commutative ring with identity, R a subring of U and L an R-submodule of U. The conditions we study have in common the property (EV), that the submodules L:x (x ∈ U) form a chain. We pay particular attention to the strongest of the conditions, viz, that L be a Manis valuation (MV) subring, i.e. having a prime ideal P such that (L, P) is a maximal pair in U (see [19], [16] and e.g. [4]). Such P is unique, being the union of all L:x such that x ∉ L, which we call P+(L) the centre of L. This set P+ plays a key role in the study of all our valuation conditions.

One of the concepts introduced in [2] is that of a hyperbornological space, an idea which effectively replaces that of a bornological space when semiconvex spaces are being considered. In Section 2 of the present paper, it is shown how the topology of such a space may be described in terms of bounded pseudometrices. This is used in Section 3 to tackle the problem of when a product of separated hyperbornological spaces has the same property. It is shown that, as in the classical case of bornological spaces, this problem is equivalent to one in measure theory.

Normal-convex embeddings are introduced for inverse semigroups, generalizing the group-theoretic concept, due to Papakyriakopoulos [4]. It is shown that every E-unitary inverse semigroup admits a normal-convex embedding into a semidirect product of a semilattice by a group, a stronger version of a result by O'Carroll [3]. A general embedding result for inverse semigroups is also obtained.

This paper shows how to construct Galois field extensions of Hilbertian fields with a given group out of some subclass (called ‘semiabelian groups’ by Matzat [2]) of all soluble groups as Galois group. This is done in a fairly explicit way by constructing polynomials whose Galois groups are universal in the sense that every group in the above subclass is obtained as a quotient of some of them.

In a recent paper [1] we showed that there is a (1,) -correspondence between the homomorphisms of an inverse semigroup S and its normal subsemigroups. The normal subsemigroup of S corresponding to and determining the homomorphism μ of S is the inverse image under μ of the set of idempotents of Sμ and is called the kernel of the homomorphism μ. The inverse image of each idempotent of Sμ is itself an inverse semigroup [1], and each such inverse semigroup is said to be a component of the normal subsemigroup determined by μ.

We construct, for q a root of unity of odd order, an embedding of the projective special linear group PSL(n) into the group of bi-Galois objects over uq(sl(n))*, the coordinate algebra of the Frobenius–Lusztig kernel of SL(n), which is shown to be an isomorphism at n=2.

In an earlier paper [5] of the author bisimple weakly inverse semigroups with partial identities were studied. The aim of this paper is to extend the results to a wider class of semigroups, viz: bisimple weakly inverse semigroups with partial right unitoids. It is found that an ℛ-class of weakly inverse semigroup is a right skew groupoid R = (R, P), where P is a right skew semigroup [5], P⊆R, and R is a partial semigroup satisfying certain conditions. When S is a bisimple weakly inverse semigroup with E the set of partial right unitoids, it can be shown that the ℛ-class R = (R, P) containing E, which is a right skew groupoid, satisfies the following:

(i) for any a, b ∈ R, there exists c ∈ R such that Pa ∩ Pb = Pc;

(ii) for any a ∈ R, there exists a left identity e of R such that (Pa ∩ P)e = Pa ∩ P.

Let K be a totally real algebraic number field of degree n > 1, and let OK be the maximal order. We denote by гk, the Hilbert modular group SL2(OK) associated with K. On the extent of the weight of an automorphy factor for гK, some restrictions are imposed, not as in the elliptic modular case. Maass [5] showed that the weight is integral for K = ℚ(√5). It was shown by Christian [1] that for any Hilbert modular group it is a rational number with the bounded denominator depending on the group.

In this paper, we provide an explicit upper bound for the Loxton–Kedlaya rank of the maximal abelian extension of ℚ.

Recently Ovsienko and Tabachnikov considered extensions of Somos and Gale-Robinson sequences, defined over the algebra of dual numbers. Ovsienko used the same idea to construct so-called shadow sequences derived from other nonlinear recurrence relations exhibiting the Laurent phenomenon, with the original motivation being the hope that these examples should lead to an appropriate notion of a cluster superalgebra, incorporating Grassmann variables. Here, we present various explicit expressions for the shadow of Somos-4 sequences and describe the solution of a general Somos-4 recurrence defined over the $\mathbb{C}$ -algebra of dual numbers from several different viewpoints: analytic formulae in terms of elliptic functions, linear difference equations, and Hankel determinants.

In this paper we construct examples which answer three questions in the general area of noncommutative Noetherian local rings and rings of finite global dimension. The examples are formed in the same basic way, beginning with a commutative polynomial ring A over a field k and a k-derivation δ of A, taking the skew polynomial ring R = A[x;δ] and localizing at a prime ideal of the form IR, where I is a prime ideal of A invariant under δ. The localization is possible by a result of Sigurdsson [13].

Several people, including Wallace [4] and Passman [3], have studied the Jacobson radical of the group algebra F[G] where F is a field and G is a multiplicative group. In [4], for instance, Wallace proves that if G is an abelian group with Sylow p-subgroup P and if F is a field of characteristic p, then the Jacobson radical of F[G] equals the right ideal generated by the radical of F[P]. In this paper we shall study group algebras over arbitrary commutative rings. By a reduction to the case of a semi-simple commutative ring, we obtain Theorem 1 whose corollary contains a generalization of Wallace's theorem. Theorem 2, on the other hand, uses the first theorem to obtain results related to the main theorem of [3].

§1. Introductory. The formula

where w is zero or a positive integer and | ζ | > 1, was given by F. E. Neumann “Crelle's Journal, XXXVII (1848), p. 24”. In § 2 of this paper some related formulae are given; the extension to the case when n is not integral is dealt with in § 3; while in § 4 the corresponding formulae for the Associated Legendre Functions when the sum of the degree and the order is a positive integer are established.