We introduce the notion of a strong representation of a semigroup in the monoid of endomorphisms of any mathematical structure, and use this concept to provide a theoretical description of the automorphism group of any semigroup. As an application of our general theorems, we extend to semigroups a well-known result concerning automorphisms of groups, and we determine the automorphism groups of certain transformation semigroups and of the fundamental inverse semigroups.

We establish new sampling representations for linear integral transforms associated with arbitrary general Birkhoff regular boundary value problems. The new approach is developed in connection with the analytical properties of Green’s function, and does not require the root functions to be a basis or complete. Unlike most of the known sampling expansions associated with eigenvalue problems, the results obtained are, generally speaking, of Hermite interpolation type.

We describe a scheme for constructing explicitly solvable arbitrage-free models for stock price. This is used to study a model similar to one introduced by Cox and Ross, where the volatility of the stock is proportional to the square root of the stock price. We derive a formula for the value of a European call option based on this model and give a procedure for estimating parameters and for testing the validity of the model.

This paper deals with the interpolation of given real boundary values into a bounded domain in Euclidean n-space, under a prescribed gradient bound. It is well known that there exist an upper solution (an inf-convolution) and a lower solution (a sup-convolution) to this problem, provided that a certain compatibility condition is satisfied. If the upper and lower solutions coincide somewhere in the domain, then several interesting consequences follow. They are considered here. Basically, the upper and lower solutions must be regular wherever they coincide.

For a class of convex curves in ℝd we prove that the corresponding maximal operator and Hilbert transform are of weak type Llog L. The point of interest here is that this class admits curves which are infinitely flat at the origin. We also prove an analogous weak type result for a class of nonconvex hypersurfaces.

A one-dimensional tiling is a bi-infinite string on a finite alphabet, and its tiling semigroup is an inverse semigroup whose elements are marked finite substrings of the tiling. We characterize the structure of these semigroups in the periodic case, in which the tiling is obtained by repetition of a fixed primitive word.

In this paper using the Quillen–Barr–Beck (co-)homology theory of universal algebras we define (co-)homology groups for commutative algebras with divided powers in prime characteristic. In particular, we determine for A a commutative 𝔽p-algebra with divided powers, the category of Beck A-modules and the group of Beck derivations. We construct the abelianization functor and we define (co-)homology. Moreover, we determine the cohomology in low dimensions and we interpret the first cohomology in terms of extensions.

A p-group is called powerful if every commutator is a product of pth powers when p is odd and a product of fourth powers when p=2. In the group algebra of a group G of p-power order over a finite field of characteristic p, the group of normalized units is always a p-group. We prove that it is never powerful except, of course, when G is abelian.

We give a qualitative description of the set 𝒪G(H) of overgroups in G of primitive subgroups H of finite alternating and symmetric groups G, and particularly of the maximal overgroups. We then show that certain weak restrictions on the lattice 𝒪G(H) impose strong restrictions on H and its overgroup lattice.

The best known results about the distribution of prime numbers in short intervals imply that all intervals [n,n+H]⊂[N,2N] contain the expected number of primes for all H≥N7/12, and almost all intervals [n,n+H]⊂[N,2N] contain the expected number of primes for all H≥N1/6. As a natural generalization, this paper is concerned with the distribution of prime numbers in intervals of type [nα,nα+H] with α>1.

Let G be a finite group and let δ(G) be the number of prime order subgroups of G. We determine the groups G with the property δ(G)≥∣G∣/2−1, extending earlier work of C. T. C. Wall, and we use our classification to obtain new results on the generation of near-rings by units of prime order.

We analyze the automorphism group for the norm closed quiver algebras 𝒯+(Q). We begin by focusing on two normal subgroups of the automorphism group which are characterized by their actions on the maximal ideal space of 𝒯+(Q). To further discuss arbitrary automorphisms we factor automorphism through subalgebras for which the automorphism group can be better understood. This allows us to classify a large number of noninner automorphisms. We suggest a candidate for the group of inner automorphisms.

We give a graph-theoretic definition for the number of ends of Cayley digraphs for finitely generated semigroups and monoids. For semigroups and monoids, left Cayley digraphs can be very different from right Cayley digraphs. In either case, the number of ends for the Cayley digraph does not depend upon which finite set of generators is used for the semigroup or monoid. For natural numbers m and n, we exhibit finitely generated monoids for which the left Cayley digraphs have m ends while the right Cayley digraphs have n ends. For direct products and for many other semidirect products of a pair of finitely generated infinite monoids, the right Cayley digraph of the semidirect product has only one end. A finitely generated subsemigroup of a free semigroup has either one end or else has infinitely many ends.

We define asymptotically p-flat and innerly asymptotically p-flat sets in Banach spaces in terms of uniform weak* Kadec–Klee asymptotic smoothness, and use these concepts to characterize weakly compactly generated (Asplund) spaces that are c0(ω1)-generated or ℓp(ω1)-generated, where p∈(1,∞). In particular, we show that every subspace of c0(ω1) is c0(ω1)-generated and every subspace of ℓp(ω1) is ℓp(ω1)-generated for every p∈(1,∞). As a byproduct of the technology of projectional resolutions of the identity we get an alternative proof of Rosenthal’s theorem on fixing c0(ω1).

We study an important class of Finsler metrics, namely, Randers metrics. We classify Randers metrics of scalar flag curvature whose S-curvatures are isotropic. This class of Randers metrics contains all projectively flat Randers metrics with isotropic S-curvature and Randers metrics of constant flag curvature.

The duality properties of the integration map associated with a vector measure m are used to obtain a representation of the (pre)dual space of the space Lp(m) of p-integrable functions (where 1<p<∞) with respect to the measure m. For this, we provide suitable topologies for the tensor product of the space of q-integrable functions with respect to m (where p and q are conjugate real numbers) and the dual of the Banach space where m takes its values. Our main result asserts that under the assumption of compactness of the unit ball with respect to a particular topology, the space Lp(m) can be written as the dual of a suitable normed space.

In this note, we provide an example of a Banach space X for which that is not isomorphic to any Hilbert space, where denotes the infimum of all von Neumann–Jordan constants for equivalent norms of X.

We prove that the relative class number of a nonabelian normal CM-field of degree 2pq (where p and q are two distinct odd primes) is always greater than four. Not only does this result solve the class number one problem for the nonabelian normal CM-fields of degree 42, but it has also been used elsewhere to solve the class number one problem for the nonabelian normal CM-fields of degree 84.

This paper continues the investigation of semigroup constructions motivated by applications in data mining. We give a complete description of the error-correcting capabilities of a large family of clusterers based on Rees matrix semigroups well known in semigroup theory. This result strengthens and complements previous formulas recently obtained in the literature. Examples show that our theorems do not generalize to other classes of semigroups.