Case Of Polynomials Research Articles

The Ronkin number of an exponential sum

AbstractWe give an intrinsic estimate of the number of connected components of the complementary set to the amoeba of an exponential sum with real spectrum improving the result of Forsberg, Passare and Tsikh in the polynomial case and that of Ronkin in the exponential one.

On Humbert Matrix Polynomials of Two Variables

In this paper we introduce Humbert matrix polynomials of two variables. Some hypergeometric matrix representations of the Humbert matrix polynomials of two variables, the double generating matrix functions and expansions of the Humbert matrix polynomials of two variables in series of Hermite polynomials are given. Results of Gegenbauer ma-trix polynomials of two variables follow as particular cases of Humbert matrix polynomials of two variables.

Real root isolation for exp–log–arctan functions

We present a real root isolation procedure for univariate functions obtained by composition and rational operations from exp , log , arctan and real constants. The procedure was first introduced for exp–log functions in Strzeboński (2008). Here we extend the procedure to exp–log–arctan functions, describe computation with elementary constants in detail and discuss the complexity of the root isolation procedure for the general exp–log–arctan case as well as for the special case of sparse polynomials. We discuss implementation of the procedure and present empirical results.

Interlacing properties of zeros of multiple orthogonal polynomials

It is well known that the zeros of orthogonal polynomials interlace. In this paper we study the case of multiple orthogonal polynomials. We recall known results and some recursion relations for multiple orthogonal polynomials. Our main result gives a sufficient condition, based on the coefficients in the recurrence relations, for the interlacing of the zeros of neighboring multiple orthogonal polynomials. We give several examples illustrating our result.

Igusa’s p-adic local zeta function associated to a polynomial mapping and a polynomial integration measure

For p prime, we give an explicit formula for Igusa’s local zeta function associated to a polynomial mapping \({{\bf f} = (f_1, \ldots, f_t) : {\bf Q}_p^{n} \to {\bf Q}_p^{t}}\) , with \({f_1, \ldots, f_t \in {\bf Z}_p[x_1, \ldots, x_n]}\) , and an integration measure on \({{\bf Z}_p^{n}}\) of the form \({|g(x)||dx|}\) , with g another polynomial in Z p [x 1, . . ., x n ]. We treat the special cases of a single polynomial and a monomial ideal separately. The formula is in terms of Newton polyhedra and will be valid for f and g sufficiently non-degenerated over F p with respect to their Newton polyhedra. The formula is based on, and is a generalization of results in Denef and Hoornaert (J Number Theory 89(1):31–64, 2001), Howald et al. (Proc Am Math Soc 135(11):3425–3433, 2007) and Veys and Zúñiga-Galindo (Trans Am Math Soc 360(4):2205–2227, 2008).

Centers for Trigonometric Abel Equations

In this paper we introduce the notion of strongly persistent centers, together with the condition of the annulation of some generalized moments, for Abel differential equations with trigonometric coefficients as a natural candidate to characterize the centers of composition type for these equations. We also recall several related concepts and discuss the differences between the trigonometric and the polynomial cases.

On the correlation functions associated with polynomials of the diffusion operator

AbstractCorrelation functions (CFs) associated with the inverse background‐error correlations (iBECs) represented by polynomials of the diffusion operator D are obtained analytically for the binomial approximations of the Gaussian BEC and in the general case of a quadratic polynomial of D. The respective analytical expressions for one‐, two‐ and three‐dimensional cases have two tuning parameters, which provide enough freedom in adjusting the CFs' shape to experimental data. The polynomial coefficients of the corresponding iBEC operator are obtained in terms of these tuning parameters and may be useful in the design of the BEC models for variational data assimilation. Published in 2011 by John Wiley & Sons Ltd.

Complexity Classifications for Propositional Abduction in Post's Framework

In this article, we investigate the complexity of abduction, a fundamental and important form of non-monotonic reasoning. Given a knowledge base explaining the world's behaviour, it aims at finding an explanation for some observed manifestation. In this article, we consider propositional abduction, where the knowledge base and the manifestation are represented by propositional formulae. The problem of deciding whether there exists an explanation has been shown to be Σ2p-complete in general. We focus on formulae in which the allowed connectives are taken from certain sets of Boolean functions. We consider different variants of the abduction problem in restricting both the manifestations and the hypotheses. For all these variants, we obtain a complexity classification for all possible sets of Boolean functions. In this way, we identify easier cases, namely NP-complete, coNP-complete and polynomial cases. Thus, we get a detailed picture of the complexity of the propositional abduction problem, hence highlighting the sources of intractability. Further, we address the problem of counting the full explanations and prove a trichotomous classification theorem.

Euler polynomials, Bernoulli polynomials, and Lévyʼs stochastic area formula

In 1951, P. Lévy represented the Euler and Bernoulli numbers in terms of the moments of Lévyʼs stochastic area. Recently the authors extended his result to the case of Eulerian polynomials of types A and B. In this paper, we continue to apply the same method to the Euler and Bernoulli polynomials, and will express these polynomials with the use of Lévyʼs stochastic area. Moreover, a natural problem, arising from such representations, to calculate the expectations of polynomials of the stochastic area and the norm of the Brownian motion will be solved.

Noncommutative biorthogonal polynomials

We define and study biorthogonal sequences of polynomials over noncommutative rings, generalizing previous treatments of biorthogonal polynomials over commutative rings and of orthogonal polynomials over noncommutative rings. We extend known recurrence relations for specific cases of biorthogonal polynomials and prove a general version of Favardʼs theorem.

Relaxed Stability Conditions for Continuous-Time T–S Fuzzy-Control Systems Via Augmented Multi-Indexed Matrix Approach

This paper is concerned with the problem of developing an advanced strategy to reduce the conservatism in stability analysis and control synthesis of continuous-time Takagi-Sugeno (T-S) fuzzy systems. A novel augmented multi-indexed matrix approach is proposed to implement new right-hand-side slack variables technique for the homogenous polynomial setting. Combining with the Finsler lemma with homogenous-matrix Lagrange multipliers, convergent linear-matrix-inequality (LMI) relaxations for stability analysis are proposed by using the generalization of the Polya theorem for the case of positive polynomials with matrix-valued coefficients. A new type of state-feedback controller, namely, the homogeneous polynomially nonquadratic control law (HPNQCL), is developed to conceive less-conservative stabilization conditions. The obtained stability and stabilization conditions are further relaxed by using the proposed right-hand-side slack variables technique. Moreover, the advantages over the existing control schemes are certificated in theory. Three numerical examples are also provided to illustrate the effectiveness of the proposed methods.

Approximating some network design problems with node costs

We study several multi-criteria undirected network design problems with node costs and lengths. All these problems are related to the Multicommodity Buy at Bulk ( MBB) problem in which we are given a graph G = ( V , E ) , demands { d s t : s , t ∈ V } , and a family { c v : v ∈ V } of subadditive cost functions. For every s , t ∈ V we seek to send d s t flow units from s to t , so that ∑ v c v ( f v ) is minimized, where f v is the total amount of flow through v . It is shown in Andrews and Zhang (2002) [2] that with a loss of 2 − ε in the ratio, we may assume that each s t -flow is unsplittable, namely, uses only one path. In the Multicommodity Cost–Distance ( MCD) problem we are also given lengths { ℓ ( v ) : v ∈ V } , and seek a subgraph H of G that minimizes c ( H ) + ∑ s , t ∈ V d s t ⋅ ℓ H ( s , t ) , where ℓ H ( s , t ) is the minimum ℓ -length of an s t -path in H . The approximability of these two problems is equivalent up to a factor 2 − ε [2]. We give an O ( log 3 n ) -approximation algorithm for both problems for the case of the demands polynomial in n . The previously best known approximation ratio for these problems was O ( log 4 n ) (Chekuri et al., 2006, 2007) [5,6]. We also consider the Maximum Covering Tree ( MaxCT) problem which is closely related to MBB: given a graph G = ( V , E ) , costs { c ( v ) : v ∈ V } , profits { p ( v ) : v ∈ V } , and a bound C , find a subtree T of G with c ( T ) ≤ C and p ( T ) maximum. The best known approximation algorithm for MaxCT (Moss and Rabani, 2001) [18] computes a tree T with c ( T ) ≤ 2 C and p ( T ) = Ω ( opt / log n ) . We provide the first nontrivial lower bound on approximation by proving that the problem admits no better than Ω ( 1 / ( log log n ) ) approximation assuming NP ⊈ Quasi(P) . This holds true even if the solution is allowed to violate the budget by a constant ρ , as was done in [18] with ρ = 2 . Our result disproves a conjecture of [18]. Another problem related to MBB is the Shallow Light Steiner Tree ( SLST) problem, in which we are given a graph G = ( V , E ) , costs { c ( v ) : v ∈ V } , lengths { ℓ ( v ) : v ∈ V } , a set U ⊆ V of terminals, and a bound L . The goal is to find a subtree T of G containing U with diam ℓ ( T ) ≤ L and c ( T ) minimum. We give an algorithm that computes a tree T with c ( T ) = O ( log 2 n ) ⋅ opt and diam ℓ ( T ) = O ( log n ) ⋅ L . Previously, a polylogarithmic bicriteria approximation was known only for the case of edge costs and edge lengths.

Zonal polynomials via Stanleyʼs coordinates and free cumulants

We study zonal characters which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power-sum symmetric functions. We show that the zonal characters, just like the characters of the symmetric groups, admit a nice combinatorial description in terms of Stanleyʼs multirectangular coordinates of Young diagrams. We also study the analogue of Kerov polynomials, namely we express the zonal characters as polynomials in free cumulants and we give an explicit combinatorial interpretation of their coefficients. In this way, we prove two recent conjectures of Lassalle for Jack polynomials in the special case of zonal polynomials.

On the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/GFEM/PUFEM)

AbstractBy using the strain smoothing technique proposed by Chen et al. (Comput. Mech. 2000; 25:137–156) for meshless methods in the context of the finite element method (FEM), Liu et al. (Comput. Mech. 2007; 39(6):859–877) developed the Smoothed FEM (SFEM). Although the SFEM is not yet well understood mathematically, numerical experiments point to potentially useful features of this particularly simple modification of the FEM. To date, the SFEM has only been investigated for bilinear and Wachspress approximations and is limited to linear reproducing conditions. The goal of this paper is to extend the strain smoothing to higher order elements and to investigate numerically in which condition strain smoothing is beneficial to accuracy and convergence of enriched finite element approximations. We focus on three widely used enrichment schemes, namely: (a) weak discontinuities; (b) strong discontinuities; (c) near‐tip linear elastic fracture mechanics functions. The main conclusion is that strain smoothing in enriched approximation is only beneficial when the enrichment functions are polynomial (cases (a) and (b)), but that non‐polynomial enrichment of type (c) lead to inferior methods compared to the standard enriched FEM (e.g. XFEM). Copyright © 2011 John Wiley & Sons, Ltd.

Complexity reduction of C-Algorithm

The C-Algorithm introduced in [5] is designed to determine isochronous centers for Lienard-type differential systems, in the general real analytic case. However, it has a large complexity that prevents computations, even in the quartic polynomial case. The main result of this paper is an efficient algorithmic implementation of C-Algorithm, called ReCA (Reduced C-Algorithm). Moreover, an adapted version of it is proposed in the rational case. It is called RCA (Rational C-Algorithm) and is widely used in [1,2] to find many new examples of isochronous centers for the Liénard type equation.

On Decompositions of Real Polynomials Using Mathematical Programming Methods

We present a procedure that gives us an SOS (sum of squares) decomposition of a given real polynomial in variables, if there exists such decomposition. For the case of real polynomials in non-commutative variables we extend this procedure to obtain a sum of hermitian squares SOHS) decomposition whenever there exists any. This extended procedure is the main scientific contribution of the paper.

Orthogonal Polynomials of Compact Simple Lie Groups

Recursive algebraic construction of two infinite families of polynomials in n variables is proposed as a uniform method applicable to every semisimple Lie group of rank n. Its result recognizes Chebyshev polynomials of the first and second kind as the special case of the simple group of type A1. The obtained not Laurent‐type polynomials are equivalent to the partial cases of the Macdonald symmetric polynomials. Recurrence relations are shown for the Lie groups of types A1, A2, A3, C2, C3, G2, and B3 together with lowest polynomials.

Gravitational attraction and potential of spherical shell with radially dependent density

Solutions to the direct problem in gravimetric interpretation are well-known for wide class of source bodies with constant density contrast. On the other hand, sources with non-uniform density can lead to relatively complicated formalisms. This is probably why analytical solutions for this type of sources are rather rare although utilization of these bodies can sometimes be very effective in gravity modeling. I demonstrate an analytical solution to that problem for a spherical shell with radial polynomial density distribution, and illustrate this result when applied to a special case of 5th degree polynomial. As a practical example, attraction of the normal atmosphere is calculated.

The Minimal Design Problem on Dynamic Polynomial Combinants

AbstractThe theory of dynamic polynomial combinants is linked to the linear part of the Dynamic Determinantal Assignment Problems, which provides the unifying description of the pole and zero dynamic assignment problems in Linear Systems. The fundamentals of the theory of dynamic polynomial combinants have been recently developed by examining issues of their representation, parameterization of dynamic polynomial combinants according to the notions of order and degree and spectral assignment. Central to this study is the link of dynamic combinants to the theory of “Generalised Resultants“, which provide the matrix representation of the dynamic combinants. The paper considers the case of coprime set polynomials for which spectral assignability is always feasible and provides a complete characterisation of all assignable combinants with order above and below the Sylvester order. A complete parameterization of combinants and respective Generalised Resultants is given and this leads naturally to the characterisation of the minimal degree and order combinant for which spectrum assignability may be achieved, referred to as the “Dynamic Combinant Minimal Design“ (DCMD) problem. Such solutions provide low bounds for the respective Dynamic Assignment control problems.

Quasiconvexity: the quadratic case revisited, and some consequences for fourth-degree polynomials

We provide an alternative proof for the well-known quadratic case for which quasiconvexity and rank-one convexity are equivalent, which does not make use of Plancherel formula. Some consequences of the same ideas for the case of 4th degree homogeneous polynomials are shown.