In this article, we introduce the notion of the uniquely $I$-clean ring and show that, if $R$ is a ring and $I$ is an ideal of $R$ then $R$ is uniquely $I$-clean if and only if ($R/ I$ is Boolean and idempotents lift uniquely modulo $I$) if and only if (for each $a\in R$ there exists a central idempotent $e\in R$ such that $e- a\in I$ and $I$ is idempotent-free). We examine when ideal extension is uniquely clean relative to an ideal. Also we obtain conditions on a ring $R$ and an ideal $I$ of $R$ under which uniquely $I$-clean rings coincide with uniquely clean rings. Further we prove that a ring $R$ is uniquely nil-clean if and only if ($N(R)$ is an ideal of $R$ and $R$ is uniquely $N(R)$-clean) if and only if $R$ is both uniquely clean and nil-clean if and only if ($R$ is an abelian exchange ring with $J(R)$ nil and every quasiregular element is uniquely clean). We also show that $R$ is a uniquely clean ring such that every prime ideal of $R$ is maximal if and only if $R$ is uniquely nil-clean ring and $N(R)= {\mathrm{Nil} }_{\ast } (R)$.

Let δ be a root system and let V be the Hahn group of real-valued functions on δ Then δ can be order-embedded into P(δ), the root system of prime l-ideals of V. In this note we identify P(δ) in terms of δ without explicit reference to V, up to the convex subgroup structure of the additive groups of real closed η1-fields. In particular, we characterize the minimal prime 1-ideals of V in terms of δ by an ultrafilter construction which generalizes the well-known method when δ is trivially ordered.

Given two m-tuples of commuting spectral operators on a Hilbert space, T = (T1,…, Tm) and S = (S1,…, Sm), an extended version of Henrici perturbation theorem is obtained for the joint approximate spectrum of S under perturbation by T. We also derive an extended version of Bauer-Fike theorem for such tuples of operators. The method used involves Clifford algebra techniques introduced by McIntosh and Pryde.

We study the numbers of involutions and their relation to Frobenius–Schur indicators in the groups $\text{SO}^{\pm }(n,q)$ and $\unicode[STIX]{x1D6FA}^{\pm }(n,q)$. Our point of view for this study comes from two motivations. The first is the conjecture that a finite simple group $G$ is strongly real (all elements are conjugate to their inverses by an involution) if and only if it is totally orthogonal (all Frobenius–Schur indicators are 1), and we observe this holds for all finite simple groups $G$ other than the groups $\unicode[STIX]{x1D6FA}^{\pm }(4m,q)$ with $q$ even. We prove computationally that for small $m$ this statement indeed holds for these groups by equating their character degree sums with the number of involutions. We also prove a result on a certain twisted indicator for the groups $\text{SO}^{\pm }(4m+2,q)$ with $q$ odd. Our second motivation is to continue the work of Fulman, Guralnick, and Stanton on generating functions and asymptotics for involutions in classical groups. We extend their work by finding generating functions for the numbers of involutions in $\text{SO}^{\pm }(n,q)$ and $\unicode[STIX]{x1D6FA}^{\pm }(n,q)$ for all $q$, and we use these to compute the asymptotic behavior for the number of involutions in these groups when $q$ is fixed and $n$ grows.

We show that mono-unary algebras have rank at most two and are thus strongly dualizable. We provide an example of a strong duality for a mono-unary algebra using an alter ego with (partial) operations of arity at most two. This mono-unary algebra has rank two and generates the same quasivariety as an injective, hence rank one, mono-unary algebra.

A construction for balanced ternary designs is given. Based on the designs so obtained, a construction of partially balanced ternary designs is given, which gives balanced ternary designs and series of symmetric balanced ternary designs in special cases.

By using the concept of tame embeddings of chains, a characterization is given of the subobjects of the lattice-ordered groups of order-automorphisms of the chains of rational and real numbers.

The main purpose of this paper is to prove the proposition: “A set of r mutually perspective (m.p.) (s—1)-simplexes have the same [s−2] (say x) of perspectivity, if and only if their centres of perspectivity (c.p.) lie in an [r−2] (say y); there then arises another such set of s m.p. (r—1)-simplexes, having the same rs vertices, which have y as their common [r−2] of perspectivity such that their c.p. lie in x.” The proposition is true in any [k] for k = s−1, s,…, r+s−2 (r ≦ s). The configuration of the proposition in [r+s−2] arises from the incidences of any r+s arbitrary primes therein and is therefore invariant under the symmetric group of permutations of r+s objects, and that in [r+s−3] is self-dual and therefore self- polar for a quadric therein. Some special cases of some interest for r = s are deduced. The treatment is an illustration of the elegance of the Möbius Barycentric Calculus ([15], pp. 136–143; [1], p. 71).

Let K be an algebraically closed field of characteristic zero, and S a nonempty subset of K such that S Q = Ø and card S < card K, where Q is the field of rational numbers. By Zorn's Lemma, there exist subfields F of K which are maximal with respect to the property of being disjoint from S. This paper examines such subfields and investigates the Galois group Gal K/F along with the lattice of intermediate subfields.

The authors investigate max min μF(x) F(y)for two standard distance functions F in R2, where μ denotes the area of {x ∈ R2; F(x)≤ 1}, the maximum is over all (geometric) lattices Λ in R2 and the minimum is over all positive points x ∈ Λ and y ∈ Λ* (the polar lattice of Λ). An application is given to a problem on fractional parts.

In this paper we determine the lower bound on |z| = r < 1 for the functional Re{αp(z) + β zp′(z)/p(z)}, α ≧0, β ≧ 0, over the class Pk (A, B). By means of this result, sharp bounds for |F(z)|, |F',(z)| in the family and the radius of convexity for are obtained. Furthermore, we establish the radius of starlikness of order β, 0 ≦ β < 1, for the functions F(z) = λf(Z) + (1-λ) zf′ (Z), |z| < 1, where ∞ < λ <1, and .

An alternative approach is proposed to the basic definitions of the lassical lambda calculus. A proof is sketched of the equivalence of the approach with the classical case. The new formulation simplifies some aspects of the syntactic theory of the lambda calculus. In particular it provides a justification for omitting in syntactic theory discussion of changes of bound variable.

In this note a characterization of semigroups with atomistic consruence lattices, given for weakly reductive semigroups, is generalized to arbitrary semigroups. Also, it is shown that there is a complete congruence on the congruence lattice of such a semigroup that decomposes it into a disjoint union of intervals of the partition lattice.

In this paper we consider the Plateau problem for surfaces of annular type bounded by a pair of convex, non-compact curves in parallel planes. We prove that for certain symmetric boundaries there are solutions to the non-compact Plateau problems (Theorem B). Except for boundaries consisting of a pair of parallel straight lines, these are the first known examples.

Let F(x) be a distribution function and denote by its characteristic function and by its moment of order k H. Milicer-Grużewska [2]has derived the following theorem: Suppose that F(x) has moments of all orders and that they satisfy the relations. where while C is a positive constant. Then f(t) is an entire function of order not exceeding two. The proof given by H. Milicer-Gruzewska is rather complicated, moreover the restriction (1.2) seems to be artificial and motivated only by the particular method of proof. We note that condition (1.1) means that the moments of F (x) are majorated by the moments of a normal distribution and we use this remark to generalize the problem.

We discuss the projective geometry defined in terms of the hollow factor modules of a given module. In particular, we derive an explicit expression for the division ring obtained in coordinatizing such a projective geometry.

John Conway's analysis in 1968 of the automorphism group of the Leech lattice and his discovery of three sporadic simple groups led to the immediate speculation that other Z-lattices might have interesting automorphism groups which give rise to (possibly new) finite simple groups. (The classification theorem for the finite simple groups has since told us that no new finite simple groups can arise in this or any other way.) For example in 1973, M. Broué and M. Enguehard constructed, in every dimension 2n, an even lattice (unimodular if n is odd) whose automorphism group is related to the simple Chevalley group of type Dn. This family of integral lattices received attention and acclaim in the subsequent literature. What escaped the attention of this literature, however, was the fact that these lattices had been discovered years earlier. Indeed in 1959, E. S. Barnes and G. E. Wall gave a uniform construction for a large class of positive definite Z-lattices in dimensions 2n which include those of Broué and Enguehard as special cases. The present article introduces an abstracted and generalized version of the construction of Barnes and Wall. In addition, there are some new observations about Barnes-Wall lattices. In particular, it is shown how to associate to each such lattice a continuous, piecewise linear graph in the plane from which all the important properties of the lattice, for example, its minimum, whether it is integral, unimodular, even, or perfect can be read off directly.

Let N be a complete connected Riemannian manifold with sectional curvatures bounded from below. Let M be a complete simply connected Riemannian manifold with sectional curvatures KM(σ)≤ −a2 (a ≥ 0) and with dimension < 2 dim N. Suppose that N is isometrically immersed in M and that its image lies in a closed ball of radius ρ. Then sup(KN(σ) − KM(σ)) ≥ μ2(aρ)/ρ2 where the function μ is defined by μ(x) = x coth x for x > 0, μ(0) = 1 and the supremum is taken over all sections tangent to N.

On the basis of physical principles having a very general nature, A. Landé [1] has demonstrated that the mathematical structure of quantum mechanics can be derived without having recourse to the introduction of special assumptions of an ad hoc type (such as commutation rules governing canonical observables) which are not immediately suggested by our knowledge of the physical world, but which have simply originated as rules which mathematical physicists have discovered by past experience to yield conclusions in conformity with experiment.

Let S be an inverse semigroup and let π:S→T be a surjective homomorphism with kernel K. We show how to obtain a presentation for K from a presentation for S, and vice versa. We then investigate the relationship between the properties of S, K and T, focusing mainly on finiteness conditions. In particular we consider finite presentability, solubility of the word problem, residual finiteness, and the homological finiteness property FPn. Our results extend several classical results from combinatorial group theory concerning group extensions to inverse semigroups. Examples are also provided that highlight the differences with the special case of groups.