Some generalized hypergeometric d-orthogonal polynomial sets

In this paper, we characterize the d-orthogonal polynomial sets given by their explicit expressions in a specific basis. As application, we consider the generalized hypergeometric case to characterize d-orthogonal polynomial sets of Laguerre type, Meixner type, Meixner-Pollaczek type, Krawtchouk type, continuous dual Hahn type, and dual Hahn type. For d=1, we obtain a unification of some characterization theorems in the orthogonal polynomials theory.     

d-Orthogonality of Humbert and Jacobi type polynomials

In this paper, we treat three questions related to the d-orthogonality of the Humbert polynomials. The first one consists to determinate the explicit expression of the d-dimensional functional vector for which the d-orthogonality holds. The second one is the investigation of the components of Humbert polynomial sequence. That allows us to introduce, as far as we know, new d-orthogonal polynomials generalizing the classical Jacobi ones. The third one consists to solve a characterization problem related to a generalized hypergeometric representation of the Humbert polynomials.     

On the zeros of d-symmetric d-orthogonal polynomials

In this paper, we give some properties of the zeros of d-symmetric d-orthogonal polynomials and we localize these zeros on (d+1) rays emanating from the origin. We apply the obtained results to some known polynomials. In particular, we partially solve the conjecture about the zeros of the Humbert polynomials stated by Milovanovi? and Dordevi? [G.V. Milovanovi?, G.B. Dordevi?, On some properties of Humbert's polynomials, II, Ser. Math. Inform. 6 (1991) 23-30]. A study of the eigenvalues of a particular banded Hessenberg matrix is done.     

d-Symmetric d-orthogonal polynomials of Brenke type

In this paper, we characterize the d-symmetric d-orthogonal polynomials of Brenke type. We obtain two new families of polynomials and the moments of the corresponding d-dimensional vectors of linear functionals. Weights, providing integral representations for these moments, were also given.     

Uniform asymptotics of some q-orthogonal polynomials

Uniform asymptotic formulas are obtained for the Stieltjes-Wigert polynomial, the q⁻¹-Hermite polynomial and the q-Laguerre polynomial as the degree of the polynomial tends to infinity. In these formulas, the q-Airy polynomial, defined by truncating the q-Airy function, plays a significant role. While the standard Airy function, used frequently in the uniform asymptotic formulas for classical orthogonal polynomials, behaves like the exponential function on one side and the trigonometric functions on the other side of an extreme zero, the q-Airy polynomial behaves like the q-Airy function on one side and the q-Theta function on the other side. The last two special functions are involved in the local asymptotic formulas of the q-orthogonal polynomials. It seems therefore reasonable to expect that the q-Airy polynomial will play an important role in the asymptotic theory of the q-orthogonal polynomials.     

Connection and inversion coefficients for basic hypergeometric polynomials

In this paper, we give a closed-form expression of the inversion and the connection coefficients for general basic hypergeometric polynomial sets using some known inverse relations. We derive expansion formulas corresponding to all the families within the q-Askey scheme and we connect some d-orthogonal basic hypergeometric polynomials.     

The centroid of the zeroes of a polynomial via certain Laguerre-type operators

In this paper, we present several properties of the centroid of the zeroes of a polynomial. As an illustration, we apply these results to the d-orthogonal polynomials. Finally, we provide the relationship between different centroids of a general monic polynomial and its image under a certain Laguerre–type operator.     

Counting solutions to binomial complete intersections

We study the problem of counting the total number of affine solutions of a system of n binomials in n   variables over an algebraically closed field of characteristic zero. We show that we may decide in polynomial time if that number is finite. We give a combinatorial formula for computing the total number of affine solutions (with or without multiplicity) from which we deduce that this counting problem is #P$\#P$-complete. We discuss special cases in which this formula may be computed in polynomial time; in particular, this is true for generic exponent vectors.     

Higher-order matching polynomials and d-orthogonality

We show combinatorially that the higher-order matching polynomials of several families of graphs are d-orthogonal polynomials. The matching polynomial of a graph is a generating function for coverings of a graph by disjoint edges; the higher-order matching polynomial corresponds to coverings by paths. Several families of classical orthogonal polynomials—the Chebyshev, Hermite, and Laguerre polynomials—can be interpreted as matching polynomials of paths, cycles, complete graphs, and complete bipartite graphs. The notion of d-orthogonality is a generalization of the usual idea of orthogonality for polynomials and we use sign-reversing involutions to show that the higher-order Chebyshev (first and second kinds), Hermite, and Laguerre polynomials are d-orthogonal. We also investigate the moments and find generating functions of those polynomials.     

Derivations and local derivations on Freudenthal almost f-algebras

Let A be an Archimedean f  -algebra and let N(A)$N(A)$ be the set of all nilpotent elements of A. Colville et al. [4] proved that a positive linear map d:A→A$d:A\to A$ is a derivation if and only if d(A)⊂N(A)$d(A)\subset N(A)$ and d(A²)={0}$d(A^2)=\{0\}$, where A²$A^2$ is the set of all products ab in A.     

Coboundaries, Flows, and Tutte Polynomials of Matrices

Using matroid duality and the critical problem, we show that certain evaluations of the Tutte polynomial of a matroid represented as a matrix over a finite field GF(q) can be interpreted as weighted sums over pairs f , g of functions defined from the ground set to GF(q) whose difference f – g is the restriction of a linear functional on the column space of the matrix. Similar interpretations are given for the characteristic polynomial evaluated at q. These interpretations extend and elaborate interpretations for Tutte and chromatic polynomials of graphs due to Goodall and Matiyasevich. Received July 14, 2006     

Minimax inequalities and fixed points of expansive set-valued mappings with noncompact and nonconvex domains and ranges in topological spaces

Some minimax inequalities involving two bifunctions with noncompact and nonconvex domains are first proved in finite continuous topological spaces (in short, FC$FC$-spaces) without convexity structure. As applications some new Fan–Browder type fixed point theorems for expansive set-valued maps with noncompact and nonconvex domains and ranges are obtained in general topological spaces. These results generalize some known results in the recent literature.     

Kirchberger-type theorems for separation by convex domains

We say that a convex set K in ? ^d strictly separates the set A from the set B if A ? int(K) and B ? cl K = ø. The well-known Theorem of Kirchberger states the following. If A and B are finite sets in ? ^d with the property that for every T ? A?B of cardinality at most d + 2, there is a half space strictly separating T ? A and T ? B, then there is a half space strictly separating A and B. In short, we say that the strict separation number of the family of half spaces in ? ^d is d + 2.In this note we investigate the problem of strict separation of two finite sets by the family of positive homothetic (resp., similar) copies of a closed, convex set. We prove Kirchberger-type theorems for the family of positive homothets of planar convex sets and for the family of homothets of certain polyhedral sets. Moreover, we provide examples that show that, for certain convex sets, the family of positive homothets (resp., the family of similar copies) has a large strict separation number, in some cases, infinity. Finally, we examine how our results translate to the setting of non-strict separation.     

A variant smoothing Newton method for P0-NCP based on a new smoothing function

In this paper, we present a new one-step smoothing Newton method proposed for solving the non-linear complementarity problem with _P0$P_0$-function based on a new smoothing NCP$NCP$-function. We adopt a variant merit function. Our algorithm needs only to solve one linear system of equations and perform one line search per iteration. It shows that any accumulation point of the iteration sequence generated by our algorithm is a solution of _P0-NCP$P_0\text{-}NCP$. Furthermore, under the assumption that the solution set is non-empty and bounded, we can guarantee at least one accumulation point of the generated sequence. Numerical experiments show the feasibility and efficiency of the algorithm.     

Solutions to degenerate complex Hessian equations

Let (X,ω)$(X,\omega )$ be an n-dimensional compact Kähler manifold and fix an integer m   such that 1?m?n$1?m?n$. We study degenerate complex Hessian equations of the form (ω+dd^cφ)^m∧ω^n−m=F(x,φ)ωⁿ${(\omega +d{d}^c\phi )}^m\wedge {\omega }^{n-m}=F(x,\phi ){\omega }^n$. Under some natural conditions on F, this equation has a unique continuous solution. When X is homogeneous and ω is invariant under the Lie group action, we further show that the solution is Hölder continuous.     

Rough approximations of vague sets in fuzzy approximation space

The combination of the rough set theory, vague set theory and fuzzy set theory is a novel research direction in dealing with incomplete and imprecise information. This paper mainly concerns the problem of how to construct rough approximations of a vague set in fuzzy approximation space. Firstly, the β-operator and its complement operator are introduced, and some new properties are examined. Secondly, the approximation operators are constructed based on β-(complement) operator. Meantime, λ-lower (upper) approximation is firstly proposed, and then some properties of two types of approximation operators are studied. Afterwards, for two different kinds of approximation operators, we introduce two roughness measure methods of the same vague set and discuss a property. Finally, an example is given to illustrate how to calculate the rough approximations and roughness measure of a vague set using the β-(complement) product between two fuzzy matrixes. The results show that the proposed rough approximations and roughness measure of a vague set in fuzzy environment are reasonable.     

Symbolic-numeric Gaussian cubature rules

It is well known that Gaussian cubature rules are related to multivariate orthogonal polynomials. The cubature rules found in the literature use common zeroes of some linearly independent set of products of basically univariate polynomials. We show how a new family of multivariate orthogonal polynomials, so-called spherical orthogonal polynomials, leads to symbolic-numeric Gaussian cubature rules in a very natural way. They can be used for the integration of multivariate functions that in addition may depend on a vector of parameters and they are exact for multivariate parameterized polynomials. Purely numeric Gaussian cubature rules for the exact integration of multivariate polynomials can also be obtained.We illustrate their use for the symbolic-numeric solution of the partial differential equations satisfied by the Appell function F₂, which arises frequently in various physical and chemical applications. The advantage of a symbolic-numeric formula over a purely numeric one is that one obtains a continuous extension, in terms of the parameters, of the numeric solution. The number of symbolic-numeric nodes in our Gaussian cubature rules is minimal, namely m for the exact integration of a polynomial of homogeneous degree 2m−1.In Section 1 we describe how the symbolic-numeric rules are constructed, in any dimension and for any order. In Sections 2, 3 and 4 we explicit them on different domains and for different weight functions. An illustration of the new formulas is given in Section 5 and we show in Section 6 how numeric cubature rules can be derived for the exact integration of multivariate polynomials. From Section 7 it is clear that there is a connection between our symbolic-numeric cubature rules and numeric cubature formulae with a minimal (or small) number of nodes.     

The d-precoloring problem for k-degenerate graphs

In this paper we deal with the d-PRECOLORING EXTENSION (d-PREXT) problem in various classes of graphs. The d-PREXT problem is the special case of PRECOLORING EXTENSION problem where, for a fixed constant d, input instances are restricted to contain at most d precolored vertices for every available color. The goal is to decide if there exists an extension of given precoloring using only available colors or to find it.We present a linear time algorithm for both, the decision and the search version of d-PREXT, in the following cases: (i) restricted to the class of k-degenerate graphs (hence also planar graphs) and with sufficiently large set S of available colors, and (ii) restricted to the class of partial k-trees (without any size restriction on S). We also study the following problem related to d-PREXT: given an instance of the d-PREXT problem which is extendable by colors of S, what is the minimum number of colors of S sufficient to use for precolorless vertices over all such extensions? We establish lower and upper bounds on this value for k-degenerate graphs and its various subclasses (e.g., planar graphs, outerplanar graphs) and prove tight results for the class of trees.     

On distinct distances among points in general position and other related problems

A set of points in the plane is said to be in general position if no three of them are collinear and no four of them are cocircular. If a point set determines only distinct vectors, it is called parallelogram free. We show that there exist n-element point sets in the plane in general position, and parallelogram free, that determine only O(n ²/√log n) distinct distances. This answers a question of Erd?s, Hickerson and Pach. We then revisit an old problem of Erd?s: given any n points in the plane (or in d dimensions), how many of them can one select so that the distances which are determined are all distinct? — and provide (make explicit) some new bounds in one and two dimensions. Other related distance problems are also discussed.