Recently we introduced the notion of an MS-algebra as a common abstraction of a de Morgan algebra and a Stone algebra [2]. Precisely, an MS-algebra is an algebra 〈L; ∧, ∨ ∘, 0, 1〉 of type 〈2, 2, 1, 0, 0〉 such that 〈L; ∧, ∨, 0, 1〉 is a distributive lattice with least element 0 and greatest element 1, and x → x∘ is a unary operation such that x ≤ x∘∘, (x ∧ y)∘ = x∘ ∨ y∘ and 1∘ = 0. It follows that ∘ is a dual endomorphism of L and that L∘∘ = {x∘∘ x ∊ L} is a subalgebra of L that is called the skeleton of L and that belongs to M, the class of de Morgan algebras. Clearly, theclass MS of MS-algebras is equational. All the subvarieties of MS were described in [3]. The lattice Λ (MS) of subvarieties of MS has 20 elements (see Fig. 1) and its non-trivial part (we exclude T, the class of one-element algebras) splits into the prime filter generated by M, that is [M, M1], the prime ideal generated by S, that is [B, S], and the interval [K, K2 ∨ K3].

The formula to be proved is

In [3] P. E. Conner showed that no abelian group with rank greater than 2 can act freely on Sn×Sn, the product of two spheres. G. Lewis [6] studied free actions of p-groups on Sn×Sn, when n is odd, n≢−l(p) and p is an odd prime. He showed that any p-group which has such an action must be abelian.

We describe two congruences α and γ contained in ℒ on an arbitrary orthodox semigroup. Let S be a right unipotent semigroup. We show that (i) α is an inverse semigroup congruence and γ is the finest fundamental inverse semigroup congruence on S, (ii) S is a union of groups if and only if ℒ on S and (iii) S is a band of groups if and only if ℒ on S.

In this note we formally solve the following dual integral equations:

where h is a constant and the Fourier cosine transform of u–1 φ(u) is assumed to exist. These dual equations arise in a crack problem in elasticity theory.

Throughout this note all rings considered will be commutative and noetherian and will have non-zero identity elements. A will always denote such a ring and the category of all A-modules and all A-homomorphisms will be denoted by .

The first formula to be proved is

where R(z)>0.

Let K be a totally real number field of degree nover ℚ and let c be an integral ideal of a maximal order of K. Given a nonnegative integer j and a Hecke character on the group of ideles of K, let denote the space of Hilbert cusp forms of holomorphic type on ℋn of weight j, level c and character ψ where ℋn is the n-th power of the Poincaré upper half plane ℋ.Let g be an element of , where 1 is the trivial character. If u ∈ Sk(c, ψ), then the product gu is an element of Sk+l (c, ψ), and therefore we can consider the linear map sending u to gu. Let be the adjoint of the linear map Φg with respect to the Petersson inner product.

The torsion of beams of L-cross-section was studied for the first time, from a mathematical standpoint, by Kotter [1]. He solved the problem in the case of an L-section both arms of which are infinite. Some time later, Trefftz [2], in his work on the torsion of beams of polygonal cross-section, applied his method also to an infinite L-section. In 1934, Seth [3] solved the case of a beam of an L-section with only one infinite arm. In 1949, Arutyanyan [4] solved the torsion problem of an L-section that has both arms finite, but of equal length, reducing the problem to that of solving an infinite system of equations.

We shall solve the equation

where 0 <a<b, and f(x) is a continuous function on the interval (a, b).

Let A be a complex Banach algebra with unit e of norm one. We show that A can be represented on a compact Hausdorff space ω which arises entirely out of the algebraic and norm structures of A. This space induces an order structure on A that is preserved by the representation. In the commutative case, ω is the spectrum of A, and we have a generalization of Gelfand's representation theorem for commutative complex Banach algebras with unit. Various aspects of this representation are illustrated by considering algebras of n × n complex matrices.

For a single space curve (that is, a smooth curve embedded in ℝ3) much geometrical information is contained in the dual and the focal set of the curve. These are both (singular) surfaces in ℝ3, the dual being a model of the set of all tangent planes to the curve, and the focal set being the locus of centres of spheres having at least 3-point contact with the curve. The local structures of the dual and the focal set are (for a generic curve) determined by viewing them as (respectively) the discriminant of a family derived from the height functions on the curve, and the bifurcation set of the family of distance-squared functions on the curve. For details of this see for example [6, pp. 123–8].

In [2], K. D. Magill, Jr. has proved that every automorphism of the semigroup (with respect to composition) of all real-valued differentiable functions of a real variable is inner. The purpose of this paper is to generalize this fact to arbitrary finite-dimensional real Banach spaces.

We prove that the module categories of Noether algebras (i.e., algebras module finite over a noetherian center) and affine noetherian PI algebras over a field enjoy the following product property: whenever a direct product \prod _(n \in ℕ) M_n of finitely generated indecomposable modules M_n is a direct sum of finitely generated objects, there are repeats among the isomorphism types of the M_n. The rings with this property satisfy the pure semisimplicity conjecture which stipulates that vanishing one-sided pure global dimension entails finite representation type.

Let the locally finite group G be the product of two locally nilpotent subgroups A and B, and assume that H is a subgroup of G belonging to a group classℱ . The question is considered whether there exists a subgroup X of G containing H which belongs to ℱ and satisfies $X=(X\capA)(X\capB)$. Under various assumptions on G and ℱ, necessary and sufficient conditions for the existence of such a subgroup X are obtained.

We generalize the notion of summable Szlenk index from a Banach space to an arbitrary weak*-compact set. We prove that a weak*-compact set has summable Szlenk index if and only if its weak*-closed, absolutely convex hull does. As a consequence, we offer a new, short proof of a result from Draga and Kochanek [J. Funct. Anal. 271 (2016), 642–671] regarding the behavior of summability of the Szlenk index under c0 direct sums. We also use this result to prove that the injective tensor product of two Banach spaces has summable Szlenk index if both spaces do, which answers a question from Draga and Kochanek [Proc. Amer. Math. Soc. 145 (2017), 1685–1698]. As a final consequence of this result, we prove that a separable Banach space has summable Szlenk index if and only if it embeds into a Banach space with an asymptotic c0 finite dimensional decomposition, which generalizes a result from Odell et al. [Q. J. Math. 59, (2008), 85–122]. We also introduce an ideal norm $\mathfrak{s}$ on the class $\mathfrak{S}$ of operators with summable Szlenk index and prove that $(\mathfrak{S}, \mathfrak{s})$ is a Banach ideal. For 1 ⩽ p ⩽ ∞, we prove precise results regarding the summability of the Szlenk index of an ℓp direct sum of a collection of operators.

We introduce a generalization ${\rm{\pounds}}_d^{(\alpha)}(X)$ of the finite polylogarithms ${\rm{\pounds}}_d^{(0)}(X) = {{\rm{\pounds}}_d}(X) = \sum\nolimits_{k = 1}^{p - 1} {X^k}/{k^d}$ , in characteristic p, which depends on a parameter α. The special case ${\rm{\pounds}}_1^{(\alpha)}(X)$ was previously investigated by the authors as the inverse, in an appropriate sense, of a parametrized generalization of the truncated exponential which is instrumental in a grading switching technique for nonassociative algebras. Here, we extend such generalization to ${\rm{\pounds}}_d^{(\alpha)}(X)$ in a natural manner and study some properties satisfied by those polynomials. In particular, we find how the polynomials ${\rm{\pounds}}_d^{(\alpha)}(X)$ are related to the powers of ${\rm{\pounds}}_1^{(\alpha)}(X)$ and derive some consequences.

The aim of this paper is to prove that the Ricci curvature ${\rm Ric}_M$ of a complete hypersurface $M^n$, $n\,{\ge}\,3$, of the Euclidean sphere $\mathbb{S}^{n+1}$, with two distinct principal curvatures of multiplicity 1 and $n-1$, satisfies $\sup {\rm Ric}_M\,{\ge}\,\inf\, f(H)$, for a function\, $f$ depending only on $n$ and the mean curvature $H$. Supposing in addition that $M^n$ is compact, we will show that the equality occurs if and only if $H$ is constant and $M^n$ is isometric to a Clifford torus $S^{n-1}(r) \times S^1(\sqrt{1-r^2})$.

Let G be a group and A a set equipped with a collection of finitary operations. We study cellular automata $$\tau :{A^G} \to {A^G}$$ that preserve the operations AG of induced componentwise from the operations of A. We show τ that is an endomorphism of AG if and only if its local function is a homomorphism. When A is entropic (i.e. all finitary operations are homomorphisms), we establish that the set EndCA(G;A), consisting of all such endomorphic cellular automata, is isomorphic to the direct limit of Hom(AS, A), where S runs among all finite subsets of G. In particular, when A is an R-module, we show that EndCA(G;A) is isomorphic to the group algebra $${\rm{End}}(A)[G]$$ . Moreover, when A is a finite Boolean algebra, we establish that the number of endomorphic cellular automata over AG admitting a memory set S is precisely $${(k|S|)^k}$$ , where k is the number of atoms of A.

We prove that the profinite completion of a profinite projective group is projective.