Densely algebraic bounds for the exponential function

An upper bound for that implies the inequality between the arithmetic and geometric means is generalized with the introduction of a new parameter . The new upper bound is smoothly and densely algebraic in , and valid for for arbitrarily large positive provided that () is sufficiently close to . The range of its validity for negative is investigated through the study of a certain family of quadrinomials.

     

On disjunctions of algebraic sets in completely simple semigroups

A semigroup S is called an equational domain if any finite union of algebraic sets over S is algebraic. We give some necessary and su?cient conditions for a completely simple semigroup to be an equational domain.     

On the asymptotic magnitude of subsets of Euclidean space

Magnitude is a canonical invariant of finite metric spaces which has its origins in category theory; it is analogous to cardinality of finite sets. Here, by approximating certain compact subsets of Euclidean space with finite subsets, the magnitudes of line segments, circles and Cantor sets are defined and calculated. It is observed that asymptotically these satisfy the inclusion-exclusion principle, relating them to intrinsic volumes of polyconvex sets.     

Extended symmetric spaces and θ-twisted involution graphs

Abstract

For a Weyl group G and an automorphism θ of order 2, the set of involutions and θ-twisted involutions can be generated by considering actions by basis elements, creating a poset structure on the elements. Haas and Helminck showed that there is a relationship between these sets and their Bruhat posets. We extend that result by considering other bases and automorphisms. We show for G = S_n, θ an involution, and any basis consisting of transpositions, the extended symmetric space is generated by a similar algorithm. Moreover, there is an isomorphism of the poset graphs for certain bases and θ.     

Fuzzy sets and type 2 under algebraic product and algebraic sum

The concept of fuzzy sets of type 2 has been proposed by L.A. Zadeh as an extension of ordinary fuzzy sets. A fuzzy set of type 2 can be defined by a fuzzy membership function, the grade (or fuzzy grade) of which is taken to be a fuzzy set in the unit interval [0, 1] rather than a point in [0, 1].This paper investigates the algebraic properties of fuzzy grades (that is, fuzzy sets of type 2) under the operations of algebraic product and algebraic sum which can be defined by using the concept of the extension principle and shows that fuzzy grades under these operations do not form such algebraic structures as a lattice and a semiring. Moreover, the properties of fuzzy grades are also discussed in the case where algebraic product and algebraic sum are combined with the well-known operations of join and meet for fuzzy grades and it is shown that normal convex fuzzy grades form a lattice ordered semigroup under join, meet and algebraic product.     

Zero sets of bivariate Hermite polynomials

We establish various properties for the zero sets of three families of bivariate Hermite polynomials. Special emphasis is given to those bivariate orthogonal polynomials introduced by Hermite by means of a Rodrigues type formula related to a general positive definite quadratic form. For this family we prove that the zero set of the polynomial of total degree n+m$n+m$ consists of exactly n+m$n+m$ disjoint branches and possesses n+m$n+m$ asymptotes. A natural extension of the notion of interlacing is introduced and it is proved that the zero sets of the family under discussion obey this property. The results show that the properties of the zero sets, considered as affine algebraic curves in R²${\mathbb{R}}^2$, are completely different for the three families analyzed.     

Smoothness spaces of higher order on lower dimensional subsets of the Euclidean space

We study Sobolev type spaces defined in terms of sharp maximal functions on Ahlfors regular subsets of and the relation between these spaces and traces of classical Sobolev spaces.     

The Taikov functional in the space of algebraic polynomials on the multidimensional Euclidean sphere

We discuss three related extremal problems on the set of algebraic polynomials of given degree n on the unit sphere $ \mathbb{S}^{m - 1} $ of Euclidean space ? ^m of dimension m ≥ 2. (1) The norm of the functional F(h) = FhP _n = ∫_?(h) P _n (x)dx, which is equal to the integral over the spherical cap ?(h) of angular radius arccos h, ?1 < h < 1, on the set with the norm of the space L( $ \mathbb{S}^{m - 1} $ ) of summable functions on the sphere. (2) The best approximation in L _∞( $ \mathbb{S}^{m - 1} $ ) of the characteristic function χ _h of the cap ?(h) by the subspace of functions from L _∞( $ \mathbb{S}^{m - 1} $ ) that are orthogonal to the space of polynomials . (3) The best approximation in the space L( $ \mathbb{S}^{m - 1} $ ) of the function χ _h by the space of polynomials . We present the solution of all three problems for the value h = t(n,m) which is the largest root of the polynomial in a single variable of degree n + 1 least deviating from zero in the space L ₁ ^? on the interval (?1, 1) with ultraspheric weight ?(t) = (1 ? t ²) ^α , α = (m ? 3)/2.     

Lattices of algebraic subsets

This survey article tackles different aspects of lattices of algebraic subsets, with the emphasis on the following: the theory of quasivarieties, general lattice theory and the theory of closure spaces with the anti-exchange axiom.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived August 24, 2002; accepted in final form October 2, 2003.     

Zimmermann type cancellation in the free Faà di Bruno algebra

Haiman and Schmitt showed that one can use the antipode SF of the colored Faà di Bruno Hopf algebra F to compute the (compositional) inverse of a multivariable formal power series. It is shown that the antipode SH of an algebraically free analogue H of F may be used to invert non-commutative power series. Whereas F is the incidence Hopf algebra of the colored partitions of finite colored sets, H is the incidence Hopf algebra of the colored interval partitions of finite totally ordered colored sets. Haiman and Schmitt showed that the monomials in the geometric series for SF are labeled by trees. By contrast, the non-commuting monomials of SH are labeled by colored planar trees. The order of the factors in each summand is determined by the breadth first ordering on the vertices of the planar tree. Finally there is a parallel to Haiman and Schmitt's reduced tree formula for the antipode, in which one uses reduced planar trees and the depth first ordering on the vertices. The reduced planar tree formula is proved by recursion, and again by an unusual cancellation technique. The one variable case of H has also been considered by Brouder, Frabetti, and Krattenthaler, who point out its relation to Foissy's free analogue of the Connes-Kreimer Hopf algebra.     

On algebraic connectivity augmentation

Suppose that G is an undirected graph, and that H is a spanning subgraph of G^c whose edges induce a subgraph on p vertices. We consider the expression α(G∪H)-α(G), where α denotes the algebraic connectivity. Specifically, we provide upper and lower bounds on α(G∪H)-α(G) in terms of p, and characterise the corresponding equality cases. We also discuss the density of the expression α(G∪H)-α(G) in the interval [0,p]. A bound on α(G∪H)-α(G) is provided in a special case, and several examples are considered.     

Maximum algebraic connectivity augmentation is NP-hard

The algebraic connectivity of a graph, which is the second-smallest eigenvalue of the Laplacian of the graph, is a measure of connectivity. We show that the problem of adding a specified number of edges to an input graph to maximize the algebraic connectivity of the augmented graph is NP-hard.     

Pure homology of algebraic varieties

We show that for a complete complex algebraic variety the pure term of the weight filtration in homology coincides with the image of intersection homology. Therefore pure homology is topologically invariant. To obtain slightly more general results we introduce image homology for noncomplete varieties.     

Sparse and slender subsets of monoids

We generalize the notions of sparse and slender sets for an arbitrary monoid and characterize the unambiguous rational sets which are sparse or slender.     

Finitary algebraic logic II

This is a supplement to the paper “Finitary Algebraic Logic” [1]. It includes corrections for several errors and some additional results. MSC: 03G15, 03G25.     

Rep-tiling Euclidean space

Summary Arep-tiling is a self replicating, lattice tiling ofR ⁿ .Lattice tiling means a tiling by translates of a single compact tile by the points of a lattice, andself-replicating means that there is a non-singular linear mapø: R ⁿ Rⁿ such that, for eachT , the imageø(T) is, in turn, tiled by . This topic has recently come under investigation, not only because of its recreational appeal, but because of its application to the theory of wavelets and to computer addressing. The paper presents an exposition of some recent results on rep-tiling, including a construction of essentially all rep-tilings of Euclidean space. The construction is based on radix representation of points of a lattice. One particular radix representation, called thegeneralized balanced ternary, is singled out as an example because of its relevance to the field of computer vision.