We characterise the elements of the (maximum) idempotent-generated subsemigroup of the Kauffman monoid in terms of combinatorial data associated with certain normal forms. We also calculate the smallest size of a generating set and idempotent generating set.

The problem considered is the determination of “lower bounds” of matrix operators on the spaces\ell_p(w) or d(w,p). Under fairly general conditions, the solution is the same for both spaces and is given by the infimum of a certain sequence. Specific cases are considered, with the weighting sequence defined by w_n = 1/n^\alpha . The exact solution is found for the Hilbert operator. For the averaging operator, two different upper bounds are given, and for certain values of p and \alpha it is shown that the smaller of these two bounds is the exact solution.

We examine the solubility of a diagonal, translation invariant, quadratic equation system in arbitrary (dense) subsets $\mathcal{A}$ ⊂ ℤ and show quantitative bounds on the size of $\mathcal{A}$ if there are no non-trivial solutions. We use the circle method and Roth's density increment argument. Due to a restriction theory approach we can deal with equations in s ≥ 7 variables.

The main object of study in this paper is the quantized Weyl algebra which arises from the work of Maltsiniotis [10] on noncommutative differential calculus. This algebra has been studied from the point of view of noncommutative ring theory by various authors including Alev and Dumas [1], the second author [9], Cauchon [3], and Goodearl and Lenagan [5]. In [9], it is shown that has n normal elements zi and, subject to a condition on the parameters, the localization obtained on inverting these elements is simple of Krull and global dimension n. It is easy to show that each of these normal elements generates a height one prime ideal and that these are all the height one prime ideals of . The purpose of this paper is to determine, under a stronger condition on the parameters, all the prime ideals of and to compare the prime spectrum with that of a related algebra . This algebra has more symmetric defining relations than those of but it shares the same simple localization which again is obtained by inverting n normal elements zi. Like the alternative algebra can be regarded as an algebra of skew differential (or difference) operators on the coordinate ring of quantum n-space.

A unit regular semigroup [1, 4] is a regular monoid S such that H1 ∩ A(x) ≠ Ø for every xɛS, where H1, is the group of units and A(x) = {y ɛ S; xyx = x} is the set of associates (or pre-inverses) of x. A uniquely unit regular semigroupis a regular monoid 5 such that |H1 ∩ A(x)| = 1. Here we shall consider a more general situation. Specifically, we consider a regular semigroup S and a subsemigroup T with the property that |T ∩ A(x) = 1 for every x ɛ S. We show that T is necessarily a maximal subgroup Hα for some idempotent α. When Sis orthodox, α is necessarily medial (in the sense that x = xαx for every x ɛ 〈E〉) and αSα is uniquely unit orthodox. When S is orthodox and α is a middle unit (in the sense that xαy = xy for all x, y ɛ S), we obtain a structure theorem which generalises the description given in [2] for uniquely unit orthodox semigroups in terms of a semi-direct product of a band with a identity and a group.

We show that continuous bilinear forms on spaces of continuous functions can be approximated by norm attaining bilinear forms.

We consider a special class of solutions of the BKP hierarchy which we call $\tau$-functions of hypergeometric type. These are series in Schur $Q$-functions over partitions, with coefficients parameterised by a function of one variable $\xi$, where the quantities $\xi(k)$, $k\in\mathbb{Z^+}$, are integrals of motion of the BKP hierarchy. We show that this solution is, at the same time, a infinite soliton solution of a dual BKP hierarchy, where the variables $\xi(k)$ are now related to BKP higher times. In particular, rational solutions of the BKP hierarchy are related to (finite) multi-soliton solution of the dual BKP hierarchy. The momenta of the solitons are given by the parts of partitions in the Schur $Q$-function expansion of the $\tau$-function of hypergeometric type. We also show that the KdV and the NLS soliton $\tau$-functions coinside the BKP $\tau$-functions of hypergeometric type, evaluated at special point of BKP higher time; the variables $\xi$ (which are BKP integrals of motions) being related to KdV and NLS higher times.

It is well known that when the characteristic p(≠ 0) of a field divides the order of a finite group, the group algebra possesses a non-trivial radical and that, if p does not divide the order of the group, the group algebra is semi-simple. A group algebra has a centre, a basis for which consists of the class-sums. The radical may be contained in this centre; we obtain necessary and sufficient conditions for this to happen.

We give an example of a finitely generated commutative semigroup that is not automatic.

We continue our investigation of the general notion of universal enveloping algebra introduced in [A. Ardizzoni, A Milnor–Moore type theorem for primitively generated braided Bialgebras, J. Algebra 327(1) (2011), 337–365]. Namely, we study a universal enveloping algebra when it is of Poincaré–Birkhoff–Witt (PBW) type, meaning that a suitable PBW-type theorem holds. We discuss the problem of finding a basis for a universal enveloping algebra of PBW type: as an application, we recover the PBW basis both of an ordinary universal enveloping algebra and of a restricted enveloping algebra. We prove that a universal enveloping algebra is of PBW type if and only if it is cosymmetric. We characterise braided bialgebra liftings of Nichols algebras as universal enveloping algebras of PBW type.

In recent years, a number of special integral equations of the first kind was discussed by several authors (see [l]–[4], [6], [7], [9]–[18]). The kernels of these integral equations are special functions of the hypergeometric family, and it was necessary to restrict the parameters appearing in these functions to secure convergence of the integrals. If these restrictions are removed, the integral fails to converge but it may possess a finite part (in Hadamard's sense), and the question arises whether the methods used in the restricted case will alsoapply in the new situation. Indeed, one could pose the moregeneral problem of Volterra integral equations involving finite parts of divergent integrals [19]

A point x in real Hilbert space is represented by an infinite sequence (x1, x2, x3, …) of real numbers such that

is convergent. The unit “sphere“ S consists of all points × for which ‖x‖ ≤ 1. The sphere of radius a and centre y is denoted by Sa(y) and consists of all points × for which ‖x−y‖ ≤ a.

A ring $R$ satisfies the dual of the isomorphism theorem if $R/Ra\cong \mathtt{l}(a)$ for every element $a\in R.$ We call these rings left morphic, and say that $R$ is left P-morphic if, in addition, every left ideal is principal. In this paper we characterize the left and right P-morphic rings and show that they form a Morita invariant class. We also characterize the semiperfect left P-morphic rings as the finite direct products of matrix rings over left uniserial rings of finite composition length. J. Clark has an example of a commutative, uniserial ring with exactly one non-principal ideal. We show that Clark's example is (left) morphic and obtain a non-commutative analogue.

We prove that every infinite subset of the dual of a compact, connected group contains an infinite, central, weighted I0 set. This yields a new proof of the fact that the duals of such groups admit infinite central p-Sidon sets for each p > 1. We also establish the existence of infinite, weighted I0 sets in the duals of many compact, abelian hypergroups.

As a special case of fully invariant subgroups, strongly invariant subgroups are introduced and studied for Abelian groups.

Let be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation in with two independent approximation functions; that is if a certain sum converges then the set of all points (x,y) on the curve which satisfy simultaneously the inequalities ||qx|| < ψ1(q) and ||qy|| < ψ2(q) infinitely often has induced measure 0. This completes the metric theory for the Lebesgue case. Further, for multiplicative approximation ||qx|| ||qy|| < ψ(q) we establish a Hausdorff measure convergence result for the same class of curves, the first such result for a general class of manifolds in this particular setup.

For any permutation σ on N = {1,2,…,n}: (i) Green’s relations are characterized in the centralizer C(σ) of σ (relative to the semigroup PTn of partial transformations on N); and (ii) a criterion is given for C(σ) to be a regular semigroup (inverse semigroup, union of groups).

1. A discussion of the Erdélyi-Kober operator

and its adjoint operator Kη,α together with an account of some of their applications can be found in the book by Sneddon [5].

The study of pure-injectivity is accessed from an alternative point of view. A module M is called pure-subinjective relative to a module N if for every pure extension K of N, every homomorphism N → M can be extended to a homomorphism K → M. The pure-subinjectivity domain of the module M is defined to be the class of modules N such that M is N-pure-subinjective. Basic properties of the notion of pure-subinjectivity are investigated. We obtain characterizations for various types of rings and modules, including absolutely pure (or, FP-injective) modules, von Neumann regular rings and (pure-) semisimple rings in terms of pure-subinjectivity domains. We also consider cotorsion modules, endomorphism rings of certain modules, and, for a module N, (pure) quotients of N-pure-subinjective modules.

In this paper we give new information about the conjugacy vector of the group , the Sylow p-subgroup of GL(n, q) consisting of the upper unitriangular matrices. The first two components of this vector are given in [4]. Here, we obtain the third component, that is, the number of conjugacy classes whose centralizer has qn+l elements. Besides, we give the whole set of numbers which compose this vector: