A representation of weak effect algebras

Weak effect algebras were introduced by the author as a generalization of effect algebras and pseudoeffect algebras. It was shown that having a basic algebra, we can restrict its binary operation to orthogonal elements only and what we get is just a weak effect algebra. However, the converse construction is impossible due to the fact that the underlying poset of a basic algebra is a lattice which need not be true for weak effect algebras. Hence, we found a weaker structure than a basic algebra which can serve as a representation of a weak effect algebra.     

Algebras in the Positive Cone of po-Groups

We systemize a number of algebras that are especially known in the field of quantum structures and that in particular arise from the positive cones of partially ordered groups. Generalized effect algebras, generalized difference posets, cone algebras, commutative BCK-algebras with the relative cancellation property, and positive minimal clans are included in the text.All these structures are conveniently characterizable as special cases of generalized pseudoeffect algebras, which we introduced in a previous paper. We establish the exact relations between all mentioned structures, thereby adding new structures whenever necessary to make the scheme of order complete.Generalized pseudoeffect algebras were under certain conditions proved to be representable by means of a po-group. From this fact, we will easily establish representation theorems for all of the structures included in discussion.     

Maximally Clustered Elements and Schubert Varieties

We introduce and study a class of “maximally clustered” elements for simply laced Coxeter groups. Such elements include as a special case the freely braided elements of Green and the author, which in turn constitute a superset of the iji-avoiding elements of Fan. We show that any reduced expression for a maximally clustered element is short-braid equivalent to a “contracted” expression, which can be characterized in terms of certain subwords called “braid clusters”. We establish some properties of contracted reduced expressions and apply these to the study of Schubert varieties in the simply laced setting. Specifically, we give a smoothness criterion for Schubert varieties indexed by maximally clustered elements. Received December 30, 2005     

On ∗-minimal ∗-ideals and ∗-biideals in involution rings

For semiprime involution rings, we determine some ∗-minimal ∗-ideals using idempotent elements. Nevertheless, ∗-minimal ∗-biideals are characterized by idempotent elements. Moreover, the involutive version of a theorem due to Steinfeld, which investigates a semiprime involution ring A if A=SocA, is given. Finally, semiprime involution rings having no proper nonzero ∗-biideals are characterized.     

On ∈-Derivations and Local ∈-Derivations

In this paper, we describe ∈-derivations in certain graded algebras by their actions on elements satisfying some special conditions. One of the main results is applied to local ∈-derivations on some certain graded algebras.     

W-Gleitgleitkinematik

The following gives some starting elements of a theory, which the author calls glide-glide-kinematics and which goes far beyond the kinematics of helical motion in ℝ ⁿ , which was studied up to now. We are mainly concerned here with “angle-preservering glide-glide-kinematics”, generalising (and giving as well some idea of) the distance-preservering glide-glide-kinematics, which we develop in a forthcoming book ([5]) in detail.      

Common universal restrictions of power series

In this short paper, we study the existence of common universal series for uncountable families of specific linear operators. In particular we deal with some derived forms of Seleznev’s theorem and we obtain common universal elements in the space of formal power series in several complex variables.     

Bochner-Weitzenböck formulas and curvature actions on Riemannian manifolds

Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give formulas in the enveloping algebra that induce not only identities for higher Casimir elements but also all Bochner-Weitzenböck formulas for gradients. As applications, we give some vanishing theorems.

     

The structure of blocks

In this paper, we focus on the structure of p-blocks with defect group satisfying some special condition. These special conditions include: two elements of the defect group are conjugate to each other in defect D if and only if they are conjugate to each other in G; the number of conjugacy classes whose p-part is contained in P by conjugacy is not larger than ∣P∣     

Finiteness conditions in groups of tree automorphisms

For any prime p and for every natural n>2, we construct an n-generated (n−2)-finite residually finite p-group of unbounded period such that the growth function of periods of its elements grows slower than any number of logarithms does. This gives the answer to Grigorchuk's question. Also constructed are residually finite but not locally finite p-groups, in which every finite set of elements whose order does not exceed a fixed one generate a finite subgroup. In the class of residually finite groups, some of the finiteness conditions specified by Shunkov are generalized and separated. In particular, question 8.66 from the Kourovka Notebook is settled and some progress is made toward solving problems 9.77 and 9.78. To the Light Memory of My Teacher Yuri I. Merzlyakov Translated fromAlgebra i Logika, Vol. 37, No. 5, pp. 568–605, September–October, 1998.     

Design principles and error analysis for reduced-shear plate-bending finite elements

Summary. In this paper, we consider the problem of designing plate-bending elements which are free of shear locking. This phenomenon is known to afflict several elements for the Reissner-Mindlin plate model when the thickness of the plate is small, due to the inability of the approximating subspaces to satisfy the Kirchhoff constraint. To avoid locking, a “reduction operator” is often applied to the stress, to modify the variational formulation and reduce the effect of this constraint. We investigate the conditions required on such reduction operators to ensure that the approximability and consistency errors are of the right order. A set of sufficient conditions is presented, under which optimal errors can be obtained – these are derived directly, without transforming the problem via a Hemholtz decomposition, or considering it as a mixed method. Our analysis explicitly takes into account boundary layers and their resolution, and we prove, via an asymptotic analysis, that convergence of the finite element approximations will occur uniformly as , even on quasiuniform meshes. The analysis is carried out in the case of a free boundary, where the boundary layer is known to be strong. We also propose and analyze a simple post-processing scheme for the shear stress. Our general theory is used to analyze the well-known MITC elements for the Reissner-Mindlin plate. As we show, the theory makes it possible to analyze both straight and curved elements. We also analyze some other elements. Received June 19, 1995     

On the inertia sets of some symmetric sign patterns

A matrix whose entries consist of elements from the set {+, −, 0} is a sign pattern matrix. Using a linear algebra theoretical approach we generalize of some recent results due to Hall, Li and others involving the inertia of symmetric tridiagonal sign matrices.     

Matroid Enumeration for Incidence Geometry

Matroids are combinatorial abstractions for point configurations and hyperplane arrangements, which are fundamental objects in discrete geometry. Matroids merely encode incidence information of geometric configurations such as collinearity or coplanarity, but they are still enough to describe many problems in discrete geometry, which are called incidence problems. We investigate two kinds of incidence problem, the points–lines–planes conjecture and the so-called Sylvester–Gallai type problems derived from the Sylvester–Gallai theorem, by developing a new algorithm for the enumeration of non-isomorphic matroids. We confirm the conjectures of Welsh–Seymour on ≤11 points in ℝ³ and that of Motzkin on ≤12 lines in ℝ², extending previous results. With respect to matroids, this algorithm succeeds to enumerate a complete list of the isomorph-free rank 4 matroids on 10 elements. When geometric configurations corresponding to specific matroids are of interest in some incidence problems, they should be analyzed on oriented matroids. Using an encoding of oriented matroid axioms as a boolean satisfiability (SAT) problem, we also enumerate oriented matroids from the matroids of rank 3 on n≤12 elements and rank 4 on n≤9 elements. We further list several new minimal non-orientable matroids.     

On the set of grouplikes of a coring

We focus our attention to the set Gr(■) of grouplike elements of a coring ■ over a ring A.We do some observations on the actions of the groups U(A) and Aut(■) of units of A and of automorphisms of corings of ■,respectively,on Gr(■),and on the subset Gal(■) of all Galois grouplike elements.Among them,we give conditions on ■ under which Gal(■) is a group,in such a way that there is an exact sequence of groups {1} → U(Ag) → U(A) → Gal(■) → {1},where Ag is the subalgebra of coinvariants for some g ∈ Gal(■).     

Irregular locus of the commuting variety of reductive symmetric lie algebras and rigid pairs

The aim of this paper is to describe the irregular locus of the commuting variety of a reductive symmetric Lie algebra. More precisely, we want to enlighten a remark of V. L. Popov. In one of his papers, the irregular locus of the commuting variety of any reductive Lie algebra is described and its codimension is computed. This provides a bound for the codimension of the singular locus of this commuting variety. V. L. Popov also suggests that his arguments and methods are suitable for obtaining analogous results in the symmetric setting. We show that some difficulties arise in this case and we obtain some results on the irregular locus of the component of maximal dimension of the “symmetric commuting variety”. As a by-product, we study some pairs of commuting elements specific to the symmetric case, that we call rigid pairs. These pairs allow us to find all symmetric Lie algebras whose commuting variety is reducible.     

On a vector q‐d algorithm

Using the framework provided by Clifford algebras, we consider a non‐commutative quotient‐difference algorithm for obtaining the elements of a continued fraction corresponding to a given vector‐valued power series. We demonstrate that these elements are ratios of vectors, which may be calculated with the aid of a cross rule using only vector operations. For vector‐valued meromorphic functions we derive the asymptotic behaviour of these vectors, and hence of the continued fraction elements themselves. The behaviour of these elements is similar to that in the scalar case, while the vectors are linked with the residues of the given function. In the particular case of vector power series arising from matrix iteration the new algorithm amounts to a generalisation of the power method to sub‐dominant eigenvalues, and their eigenvectors. This revised version was published online in June 2006 with corrections to the Cover Date.     

Binary relations as lattice isomorphisms

Summary In 1963, Zaretskiį established a one-to-one correspondence between the setB _X of binary relations on a set X and the set of triples of the form (W, ϕ, V) where W and V are certain lattices and ϕ: W→V is an isomorphism. We provide a multiplication for these triples making the Zaretskiį correspondence a semigroup isomorphism. In addition, we consider faithful representations ofB _X by pairs of partial transformations and also as the translational hull of its rectangular relations. Using these triples, we study idempotents, regular and completely regular elements and relationsH-equivalent to some relations with familiar properties such as reflexivity, transitivity, etc. Entrata in Redazione il 14 aprile 1998.     

Sufficient conditions for realizability of Boolean functions by asymptotically optimal circuits with the unreliability 2<Emphasis Type="Italic">ɛ</Emphasis>

We consider realizations of Boolean functions by circuits composed of unreliable functional elements in some complete finite basis B. We assume that all elements independently of each other with the probability ɛ (ɛ ∈ (0; 1/2)) are subjected to inverse failures at the output.     

Some new closures on orders

For each of the relations “less than or equal to”, “less than”, “covered by”, and “covered by or equal to”, we characterize finite orders (also called posets) with the property that the pair of Galois closure operators induced by the relation in question coincides with the pair of closure operators introduced and applied in our previous paper in 2007. We also consider the “less than or equal to” relation between the set of join-irreducible elements and the set of meet-irreducible elements, and we show that the above-mentioned pairs of closure operators coincide for finite modular lattices.