A Structure Preserving Generalization of Independence,Domination and Irredundance

Abstract

The now famous inequality chain ir≤γ≤i≤β ≤ Γ ≤ IR, where ir and IR denote the lower and upper irredundance numbers of a graph, γ and Γ the lower and upper domination numbers, i the independent domination number and β the independence number of a graph, may be seen as the culmination of a process by which we start with independence (a hereditary property of vertex sets); we characterize maximal independence by domination (a superhereditary property of vertex sets), and then characterize minimal domination by irredundance (again a hereditary property). In this paper we generalize independent, dominating and irredundant sets of a graph G to what we will call s-dominating, s-independent and s-irredundant functions (for s a positive integer), which are functions of the type f : V (G) → N, in such a way that the maximal 1-independent, the minimal 1- dominating and the maximal 1-irredundant functions are the characteristic functions of the maximal independent, the minimal dominating and the maximal irredundant sets of G respectively. In addition, we would want to preserve those properties of and relationships between independence, domination and irredundance needed to extend the inequality chain ir≤γ≤i≤β ≤ Γ ≤ IR to one for s-dominating, s-independent and s-irredundant functions by a process similar to that described above.     

Difference of two sets and estimation of clarke generalized Jacobian via quasidifferential

The notion of difference for two convex compact sets inR ⁿ , proposed by Rubinovet al, is generalized toR ^m×n . A formula of the difference for the two sets, which are convex hulls of a finite number of points, is developed. In the light of this difference, the relation between Clarke generalized Jacobian and quasidifferential, in the sense of Demyanov and Rubinov, for a nonsnooth function, is established. Based on the relation, the method of estimating Clarke generalized Jacobian via quasidifferential for a certain class of functions, is presented.     

Uniform Approximation by Entire Functions That Are All Bounded on a Given Set

For two closed sets F and G in the complex plane C, G C , we solve the following problem Under what conditions on F and G can every function f , continuous on F and analytic in its interior, be uniformly approximated by entire functions, each of which is bounded on G ? February 7, 1995. Date revised: October 31, 1995.     

POSSIBLE RESOLUTIONS FOR A GIVEN HILBERT FUNCTION

ABSTRACT

We consider which sets of graded Betti numbers actually occur among all resolutions of modules with a given Hilbert function.     

Compact,convex sets in R n and a new banach lattice. I.- Theory.

The present paper shows that compact, non-empty convex sets in R ⁿ form a wedge in a well-defined Banach lattice, which turns out to be isometrically Riesz-isomorphic to the continuous functions in S ^n–1, the unit sphere of R ⁿ . Among other results, we obtain Dini-like convergence results for sets, linking order- and norm-convergence.     

Different Bounds on the Different Betti Numbers of Semi-Algebraic Sets

Abstract. A classic result in real algebraic geometry due to Oleinik—Petrovskii, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now no significantly better bounds were known on the individual higher Betti numbers. We prove better bounds on the individual Betti numbers of basic semi-algebraic sets, as well as arrangements of algebraic hypersurfaces. As a corollary we obtain a polynomial bound on the highest Betti numbers of basic semi-algebraic sets defined by quadratic inequalities.     

The complexity of recognizing the preservation of some sets by multivalued functions represented by polynomials

A k-valued logic function is considered for prime numbers k. Polynomial algorithms for recognizing whether k-valued logic functions given in polynomials according to modulo k preserve sets of the forms E(lx ⁱ ) and E(lx ⁱ )\{0}, where E(g(x)) is the range of values of function g(x), are constructed. The number of sets that can be recognized as preserved by the proposed algorithms is estimated.     

Different Bounds on the Different Betti Numbers of Semi-Algebraic Sets

   Abstract. A classic result in real algebraic geometry due to Oleinik—Petrovskii, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now no significantly better bounds were known on the individual higher Betti numbers. We prove better bounds on the individual Betti numbers of basic semi-algebraic sets, as well as arrangements of algebraic hypersurfaces. As a corollary we obtain a polynomial bound on the highest Betti numbers of basic semi-algebraic sets defined by quadratic inequalities.     

On the symmetry function of a convex set

We attempt a broad exploration of properties and connections between the symmetry function of a convex set S ${S \subset\mathbb{R}^n}We attempt a broad exploration of properties and connections between the symmetry function of a convex set S and other arenas of convexity including convex functions, convex geometry, probability theory on convex sets, and computational complexity. Given a point , let sym(x,S) denote the symmetry value of x in S: , which essentially measures how symmetric S is about the point x, and define x ^* is called a symmetry point of S if x ^* achieves the above maximum. The set S is a symmetric set if sym (S)=1. There are many important properties of symmetric convex sets; herein we explore how these properties extend as a function of sym (S) and/or sym (x,S). By accounting for the role of the symmetry function, we reduce the dependence of many mathematical results on the strong assumption that S is symmetric, and we are able to capture and otherwise quantify many of the ways that the symmetry function influences properties of convex sets and functions. The results in this paper include functional properties of sym (x,S), relations with several convex geometry quantities such as volume, distance, and cross-ratio distance, as well as set approximation results, including a refinement of the L?wner-John rounding theorems, and applications of symmetry to probability theory on convex sets. We provide a characterization of symmetry points x ^* for general convex sets. Finally, in the polyhedral case, we show how to efficiently compute sym(S) and a symmetry point x ^* using linear programming. The paper also contains discussions of open questions as well as unproved conjectures regarding the symmetry function and its connection to other areas of convexity theory. Dedicated to Clovis Gonzaga on the occasion of his 60th birthday.     

Measurable chromatic number of geometric graphs and sets without some distances in euclidean space

LetH be a set of positive real numbers. We define the geometric graphG _H as follows: the vertex set isR ⁿ (or the unit circleS ¹) andx, y are joined if their distance belongs toH. We define the measurable chromatic number of geometric graphs as the minimum number of classes in a measurable partition into independent sets. In this paper we investigate the difference between the notions of the ordinary and measurable chromatic numbers. We also prove upper and lower bounds on the Lebesgue upper density of independent sets.     

Approximation and Complexity II: Iterated Integration

We introduce two classes of real analytic functions W \subset U on an interval. Starting with rational functions to construct functions in W we allow the application of three types of operations: addition, integration, and multiplication by a polynomial with rational coefficients. In a similar way, to construct functions in U we allow integration, addition, and multiplication of functions already constructed in U and multiplication by rational numbers. Thus, U is a subring of the ring of Pfaffian functions [7]. Two lower bounds on the L _∈fty -norm are proved on a function f from W (or from U , respectively) in terms of the complexity of constructing f .     

An Extended Faber Operator

   Abstract. One of the basic tools in the theory of polynomial approximation in the uniform norm on compact plane sets is the Faber operator. Usually, the Faber operator is viewed as an operator acting on functions in the disk algebra, that is, functions which are holomorphic in the open unit disk D and continuous on D. We consider an extended Faber operator acting on arbitrary functions continuous on ; D.     

On directional convexity

Motivated by problems from calculus of variations and partial differential equations, we investigate geometric properties of D-convexity. A function f: R ^d → R is called D-convex, where D is a set of vectors in R ^d, if its restriction to each line parallel to a nonzero v ∈ D is convex. The D-convex hull of a compact set A ⊂ R ^d, denoted by co^D(A), is the intersection of the zero sets of all nonnegative D-convex functions that are zero on A. It also equals the zero set of the D-convex envelope of the distance function of A. We give an example of an n-point set A ⊂ R ² where the D-convex envelope of the distance function is exponentially close to zero at points lying relatively far from co ^D(A), showing that the definition of the D-convex hull can be very nonrobust. For separate convexity in R ³ (where D is the orthonormal basis of R ³), we construct arbitrarily large finite sets A with co ^D(A) ≠ A whose proper subsets are all equal to their D-convex hull. This implies the existence of analogous sets for rank-one convexity and for quasiconvexity on 3 × 3 (or larger) matrices. This research was supported by Charles University Grants No. 158/99 and 159/99.     

Highly Nonlinear Resilient Functions Through Disjoint Codes in Projective Spaces

Functions which map n-bits to m-bits are important cryptographic sub-primitives in the design of additive stream ciphers. We construct highly nonlinear t-resilient such functions ((n, m, t) functions) by using a class of binary disjoint codes, a construction which was introduced in IEEE Trans. Inform. Theory, Vol. IT-49 (2) (2003). Our main contribution concerns the generation of suitable sets of such disjoint codes. We propose a deterministic method for finding disjoint codes of length ν m by considering the points of PG ). We then obtain some lower bounds on the number of disjoint codes, by fixing some parameters. Through these sets, we deduce in certain cases the existence of resilient functions with very high nonlinearity values. We show how, thanks to our method, the degree and the differential properties of (n, m, t) functions can be improved.Communicated by: J.D. Key     

A closed formula for the generating function of p-Bernoulli numbers

Abstract

In this paper, using geometric polynomials, we obtain a generating function of p-Bernoulli numbers in terms of harmonic numbers. As consequences of this generating function, we derive closed formulas for the finite summation of Bernoulli and harmonic numbers involving Stirling numbers of the second kind. We also give a relationship between the p-Bernoulli numbers and the generalized Bernoulli polynomials.     

Generating functions of the (<Emphasis Type="Italic">h,q</Emphasis>) extension of twisted Euler polynomials and numbers

By using p-adic q-deformed fermionic integral on ℤ _p , we construct new generating functions of the twisted (h, q)-Euler numbers and polynomials attached to a Dirichlet character χ. By applying Mellin transformation and derivative operator to these functions, we define twisted (h, q)-extension of zeta functions and l-functions, which interpolate the twisted (h, q)-extension of Euler numbers at negative integers. Moreover, we construct the partially twisted (h, q)-zeta function. We give some relations between the partially twisted (h, q)-zeta function and twisted (h, q)-extension of Euler numbers.      

Generating functions of the incomplete Fibonacci and Lucas numbers

For the incomplete Fibonacci and incomplete Lucas numbers, which were introduced and studied recently by P. Filliponi [Rend. Circ. Math. Palermo (2)45 (1996), 37–56], the authors derive two classes of generating functions in terms of the familiar Fibonacci and Lucas numbers, respectively.     

A survey of bounds for classical Ramsey numbers

This paper is a survey of the methods used for determining exact values and bounds for the classical Ramsey numbers in the case that the sets being colored are two-element sets. Results concerning the asymptotic behavior of the Ramsey functions R(k,l) and R_m(k) are also given.     

Order Embeddings with Irrational Codomain: Debreu Properties of Real Subsets

The objective of this paper is to investigate the role of the set of irrational numbers as the codomain of order-preserving functions defined on topological totally preordered sets. We will show that although the set of irrational numbers does not satisfy the Debreu property it is still nonetheless true that any lower (respectively, upper) semicontinuous total preorder representable by a real-valued strictly isotone function (semicontinuous or not) also admits a representation by means of a lower (respectively, upper) semicontinuous strictly isotone function that takes values in the set of irrational numbers. These results are obtained by means of a direct construction. Moreover, they can be related to Cantor’s characterization of the real line to obtain much more general results on the semicontinuous Debreu properties of a wide family of subsets of the real line.      

Quasi-convex Functions in Carnot Groups

In this paper, the authors introduce the concept of h-quasiconvex functions on Carnot groups G. It is shown that the notions of h-quasiconvex functions and h-convex sets are equivalent and the L∞estimates of first derivatives of h-quasiconvex functions are given. For a Carnot group G of step two, it is proved that h-quasiconvex functions are locally bounded from above. Furthermore, the authors obtain that h-convex functions are locally Lipschitz continuous and that an h-convex function is twice differentiable almost everywhere.