We prove that if the “type-II-construct” subsemigroup of a finite semigroup S is regular, then the “type-II” subsemigroup of S is computable (actually in this case, type-II and type-II-construct are equal). This, together with certain older results about pseudo-varieties of finite semigroups, leads to further results:

(1) We get a new proof of Ash's theorem: If the idempotents in a finite semigroup S commute, then S divides a finite inverse semigroup. Equivalently: The pseudo-variety generated by the finite inverse semigroups consists of those finite semigroups whose idempotents commute.

(2) We prove: If the idempotents of a finite semigroup S form a subsemigroup then S divides a finite orthodox semigroup. Equivalently: The pseudo-variety generated by the finite orthodox semigroups consists of those finite semigroups whose idempotents form a subsemigroup.

(3) We prove: The union of all the subgroups of a semigroup S forms a subsemigroup if and only if 5 belongs to the pseudo-variety u * G if and only if Sn belongs to u. Here u denotes the pseudo-variety of finite semigroups which are unions of groups.

For these three classes of semigroups, type-II is equal to type-II construct.

Global wellposedness theorems are established for a class of abstract Cauchy initial value problems and a class of abstract Volterra equations which have a linear semigroup as a convolution kernel. These existence theorems are used to show that a class of nonlinear operators and a class of perturbed linear operators are m-accretive. The m-accretiveness results are used in turn to represent solutions to the differential and integral equations.

Let R be an associative ring with unity 1, N(R) the set of nilpotents, J(R) the Jacobson radical of R and n > 1 be a fixed integer. We prove that R is commutative if and only if it satisfies (xy)n = ynxn for all x, y ∈ R \ N(R) and commutators in R are n(n + 1)-torsion free. Moreover, we extend the same result in the case when x, y ∈ R\J(R).

An asymptotic relation for the expected number of excursions above a boundary g(n) by a random walk Sn, n = 1,2, ‥, N is given in terms of an integral involving g. An integral test is given to determine whether the total excursion time has finite expectation. If some moment assumptions hold then the expectation of the total excursions is finite if and only if .

In axisymmetric irrotational flows of a perfect fluid under gravity there are three basic conserved quantities; axial momentum, energy and a circulation based, radial moment of momentum. This paper adapts these conservation principles to describe cavity collapse adjacent to a rigid boundary in a semi-infinite perfect fluid. They afford a global model accounting for volume change, migration and jet formation; physically the most significant features of bubble collapse close to a rigid boundary.

The existence of maximal arcs of a certain type in symmetric designs is shown to yield semiregular group divisible designs whose duals are also semiregular group divisible. Two infinite families of such group divisible designs are constructed. The group divisible designs in these families are, in general, not symmetric.

In this paper some new results concerning computational analysis are established. The fundamental concepts of interval analysis and fixed point theorems suitable for computational purposes are developed and applied to concrete examples illustrating this new method of proving mathematical statements.

In this paper we introduce the concept of almost nilpotence for Γ-rings, similar to the corresponding concept for rings, as defined by Van Leeuwen and Heyman. An almost mlpotent radical property Α0 is introduced for Γ-rings, and shown to be supernilpotent. If M is a Γ-ring with left and right operator rings L and R respectively, then Α(L)+ = Α0(M) = Α(R)*, where Α(−) denotes the almost nilpotent radical of a ring. If M is a Γ-ring and m, n are positive integers, then Α0(Mm, n) is the almost nilpotent radical of the Γn, m-ring Mm, n.

For a measurable function f on a probability space a basic inequality is ‖f‖p ≤ ‖f‖q where 1 ≤ p < q < ∞ and ‖f‖p denotes the Lp norm of f. The above inequality becomes an equality provided |f| is a constant almost everywhere. We obtain an improvement of the above inequality in all cases that |f| is not a constant almost everywhere.

Let ξ be an irrational number with simple continued fraction expansion ξ = [a0;a1,a2,…,an,…], let pn/qn be its nth convergent and let θn = qn|qnξ − pn|. In this paper a general method is introduced to deduce a series of inequalities involving the triple (θn−1, θn, θn+1).

We generalise some existing results on strict radical classes, and introduce some new notions of hereditariness of radical classes. This is achieved by considering the classes of rings for which a radical class is hereditary and/or strict (in the general sense).

Let F: T → 2x \ {} be a closed-valued multifunction into a separable Banach space X. We define the sets and We prove various convergence theorems for those two sets using the Hausdorff metric and the Kuratowski-Mosco convergence of sets. Then we prove a density theorem of CF and a corresponding convexity theorem for F(·). Finally we study the “differentiability” properties of K(·). Our work extends and improves earlier ones by Artstein, Bridgland, Hermes and Papageorgiou.

Corresponding to each ordered set there is a variety, determined up to equivalence, generated by an algebra whose term operations are all the monotone operations on the ordered set. We produce several characterisations of the finite bounded ordered sets for which the corresponding variety is congruence-distributive. In particular, we find that congruence-distributivity, congruence-modularity, and residual smallness are equivalent for these varieties.

The object of this paper is to prove a conjecture given recently by Nunokawa that if f(z) ∈ Α(p) satisfies Re{1 + zf″(z)/f′(z)} < p + 1/2 in the unit disk U, then f(z) is p-valently starlike in U.

A Skorokhod embedding approach is used to give the other law of the iterated logarithm for square integrable martingales.

For each t ∈ {0,1,2,…}, let Tt be a set of operators in a Banach space. Tt is called a multiple semigroup of operators with respect to some operation ⊗ between sets of operators, if Tt satisfies the semigroup property Tt+s = Tt⊗Ts. Two operations, (∘) and (·), between sets of operators are defined and properties of Tt are studied. Applications to the theory of Controlled Markov Processes are considered.

In this paper we study internal completeness, injectivity and some related notions in the category MBoo of Boolean algebras in the topos MEns of M-sets, for a monoid M.