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Abstract: A graph is polar if the vertex set can be partitioned into and in such a way that induces a complete multipartite graph and induces a disjoint union of cliques (i.e., the complement of a complete multipartite graph). Polar graphs naturally generalize several classes of graphs such as bipartite graphs, cobipartite graphs and split graphs. Recognizing polar graphs is an NP-complete problem in general, and thus it is of interest to restrict the problem to special classes of graphs. Cographs and chordal graphs are among those whose polarity can be recognized in polynomial time. The line-graphs of bipartite graphs are another class of graphs whose polarity has been characterized recently in terms of forbidden subgraphs, but no polynomial time algorithm is given. In this paper, we present an algorithm which decides whether the line-graph of an input bipartite graph is polar and constructs a polar partition when one exists. [ABSTRACT FROM AUTHOR]

Abstract: This paper describes a simple and efficient approach to the Troesch’s problem. In this approach, the hyperbolic nonlinear term in the equation is first converted into polynomial nonlinear terms by variable transformation, and a simple shooting method is then used directly to solve this transformed problem. The calculated results are in excellent agreement with those obtained by other analytical and numerical methods. [Copyright &y& Elsevier]

Abstract: This paper applies the theory of tensor products of functional models to the study of the Stein equation. We analyse the connections between the Sylvester and Stein equations, as well as those between the Anderson–Jury Bezoutian associated with the polynomial Sylvester equation and the -Bezoutian associated with the homogeneous polynomial Stein equation. The functional setting allows us to extend the theory of finite section Toeplitz matrices and their inversion. [Copyright &y& Elsevier]

Abstract: In 2002 Jarque and Villadelprat proved that planar polynomial Hamiltonian systems of degree 4 have no isochronous centers and raised an open question for general planar polynomial Hamiltonian systems of even degree. Recently, it was proved that a planar polynomial Hamiltonian system is non-isochronous if a quantity, denoted by , can be computed such that . As a corollary of this criterion, the open question was answered for those systems with only even degree nonlinearities. In this paper we consider the case of and give a new criterion for non-isochronicity. Applying the new criterion, we also answer the open question for some cases in which some terms of odd degree are included. [Copyright &y& Elsevier]

Abstract: This paper deals with approximating properties and convergence results of the iterates for complex -Stancu polynomials attached to analytic functions on compact disks in the case . Some estimates on the rate of convergence are given. [Copyright &y& Elsevier]

Abstract: It is well-known that an element of the linear group is semisimple if and only if its conjugacy class is Zariski closed. The aim of this paper is to show that the same result holds for the group of complex plane polynomial automorphisms. [Copyright &y& Elsevier]

Abstract: A Bethe tree of levels, , is a rooted tree such that the root vertex has degree , the vertices from level to have degree and the vertices at level are leaves. In this paper, we obtain a recurrence relation for the characteristic polynomial and the Laplacian characteristic polynomial of Bethe trees. As an application, we prove that there are no integral Bethe trees except for the star , and we obtain a recurrence relation for the Laplacian-energy-like invariant of Bethe trees. [Copyright &y& Elsevier]

Abstract: The spectral excess theorem provides a quasi-spectral characterization for a (regular) graph with distinct eigenvalues to be distance-regular graph, in terms of the excess (number of vertices at distance ) of each of its vertices. The original approach, due to Fiol and Garriga in 1997, was obtained by using a local approach, so giving a characterization of the so-called pseudo-distance-regularity around a vertex. In this paper we present a new simple projection method based in a global point of view, and where the mean excess plays an essential role. [Copyright &y& Elsevier]

Abstract: Let be a simple undirected graph with the characteristic polynomial of its Laplacian matrix , . It is well known that for trees the Laplacian coefficient is equal to the Wiener index of , while is equal to the modified hyper-Wiener index of the graph. In this paper, we characterize -vertex trees with given matching number which simultaneously minimize all Laplacian coefficients. The extremal tree is a spur, obtained from the star graph with vertices by attaching a pendant edge to each of certain non-central vertices of . In particular, minimizes the Wiener index, the modified hyper-Wiener index and the recently introduced Incidence energy of trees, defined as , where are the eigenvalues of signless Laplacian matrix . We introduced a general transformation which decreases all Laplacian coefficients simultaneously. In conclusion, we illustrate on examples of Wiener index and Incidence energy that the opposite problem of simultaneously maximizing all Laplacian coefficients has no solution. [Copyright &y& Elsevier]

Abstract: In many realistic situation, a job processed later consumes more time than the same job when it is processed earlier, this phenomenon is known as deteriorated effect. The skills of workers continuously improve when repeating the same or similar tasks, this phenomenon is as the “learning effect” in the literature. However, most studies considering the deteriorated and learning effect ignore the fact that production efficiency can be increased by grouping various parts and products with similar designs and/or production processes. This phenomenon is known “group technology” in the literature. In this paper, we propose a new group scheduling with deteriorated and learning model where the learning effect not only depends on job position, but also depends on the group position; the deteriorated effect depends on its starting time of the job. We then show that the single-machine makespan and the total completion time problems remain polynomial optimal solvable under the proposed model. In addition, we show the maximum lateness have a polynomial optimal solution under certain agreeable restriction. [Copyright &y& Elsevier]

Abstract: Previously we presented a novel approach to program a robot controller based on system identification and robot training techniques. The proposed method works in two stages: first, the programmer demonstrates the desired behaviour to the robot by driving it manually in the target environment. During this run, the sensory perception and the desired velocity commands of the robot are logged. Having thus obtained training data we model the relationship between sensory readings and the motor commands of the robot using ARMAX/NARMAX models and system identification techniques. These produce linear or non-linear polynomials which can be formally analysed, as well as used in place of “traditional robot” control code. In this paper we focus our attention on how the mathematical analysis of NARMAX models can be used to understand the robot’s control actions, to formulate hypotheses and to improve the robot’s behaviour. One main objective behind this approach is to avoid trial-and-error refinement of robot code. Instead, we seek to obtain a reliable design process, where program design decisions are based on the mathematical analysis of the model describing how the robot interacts with its environment to achieve the desired behaviour. We demonstrate this procedure through the analysis of a particular task in mobile robotics: door traversal. [Copyright &y& Elsevier]

Abstract: The well-known formula of Faà di Bruno’s for higher derivatives of a composite function has played an important role in combinatorics. In this paper we generalize the divided difference form of Faà di Bruno’s formula and give an explicit formula for the -th divided difference of a multicomposite function. More generally, we establish the relationships of the Bell polynomials with respect to multicomposite functions. Applying these to multicomposite functions, we obtain some extensions of Faà di Bruno’s formula. [Copyright &y& Elsevier]

Abstract: Zernike Moment (ZM) is an effective region-based shape representation technique. The extracted ZM features should be independent of scale, position and orientation, which can be achieved by ZM-based image normalization. Nevertheless, due to the discretization of digital image and the presence of noise, the normalization is imperfect. Thus, in practice Zernike Moment Invariants (ZMI) cannot perfectly preserve the invariant properties. In this paper, firstly the ZM-based image normalization criteria are derived, and then I theoretically and experimentally evaluate the accuracy of the ZM-based image normalization. Our theoretical and experimental results not only disclose some essential facts, but also have some new findings. The relations between the accuracy of ZM-based image normalization and its influencing factors are established. A creative pseudo-polar coordinate is proposed to cut down the geometrical errors to the greatest extent. Furthermore, I suggest several techniques to improve the accuracy of image normalization. By combining moment-based image normalization with the image regularization theory and the scale-space theory, and several new conclusions are drawn. Our experimental results show that the proposed techniques can preserve the invariance of image normalization and restrain the influence of noise quite effectively. [Copyright &y& Elsevier]

Abstract: This paper studies the forward kinematics of a symmetrical 6-6 Stewart platform, in which both the base and the mobile platform are hexagons and the joint centers satisfy some conditions. A concise algebraic elimination algorithm to solve the closed-form forward kinematics of the Stewart platform is presented. Based on the presented algebraic method, an interesting result that comes out of our study is that the forward kinematics problem is reduced to solve a univariate polynomial equation of degree at most 14. The 14th degree univariate polynomial is derived from the determinant of the 15×15 Sylvester’s matrix, which is relatively small in size, without factoring out or deriving the greatest common divisor. The algorithm is comparatively concise and requires fairly less computation time. The result is verified by a numerical example. [Copyright &y& Elsevier]

Abstract: The theme of the present paper is to introduce and study two different versions of tensor products of functional models, one over the underlying field and the other over the corresponding algebra of polynomials, as well as related models based on the Kronecker product of polynomial matrices. In the process, we study the Sylvester equation and its reduction to a polynomial matrix equation. We analyse the relation between the two tensor products and use this to elucidate the role of the Anderson-Jury generalized Bezoutians in this context. [Copyright &y& Elsevier]

Abstract: Let denote a field and let denote a vector space over with finite positive dimension. We consider a pair of linear transformations and that satisfy the following conditions: (i) each of is diagonalizable; (ii) there exists an ordering of the eigenspaces of such that for , where and ; (iii) there exists an ordering of the eigenspaces of such that for , where and ; (iv) there is no subspace of such that , , , . We call such a pair a tridiagonal pair on . It is known that and for the dimensions of , , , coincide; we denote this common dimension by . In this paper we prove that for . It is already known that if is algebraically closed. [Copyright &y& Elsevier]

Abstract: In this paper we present an implementation of the well known “fast multipole” method (FMM) for the efficient calculation of dipole fields. The main advantage of the present implementation is simplicity—we believe that a major reason for the lack of use of FMMs is their complexity. One of the simplifications is the use of polynomials in the Cartesian coordinates rather than spherical harmonics. We have implemented it in the context of an arbitrary hierarchical system of cells—no periodic mesh is required, as it is for FFT (fast Fourier transform) methods. The implementation is in terms of recursive functions. Results are given for application to micromagnetic simulation. Complete source code is provided for an open-source implementation of this method, as well as an installer for the resulting program. [Copyright &y& Elsevier]

Abstract: Spline quasi-interpolants are practical and effective approximation operators. In this paper, we construct QIs with optimal approximation orders and small infinity norms called near-best discrete quasi-interpolants which are based on -splines, i.e. B-splines with octagonal supports on the uniform four-directional mesh of the plane. These quasi-interpolants are exact on some space of polynomials and they minimize an upper bound of their infinity norms depending on a finite number of free parameters. We show that this problem has always a solution, in general nonunique. Concrete examples of such quasi-interpolants are given in the last section. [Copyright &y& Elsevier]

Abstract: Thirty years ago, G. N. de Oliveira has proposed the following completion problems: Describe the possible characteristic polynomials of , where and are square submatrices, when some of the blocks are fixed and the others vary. Several of these problems remain unsolved. This paper gives the solution, over the field of real numbers, of Oliveira’s problem where the blocks are fixed and the others vary. [Copyright &y& Elsevier]

Abstract: In this paper, we shall follow a companion matrix approach to study the relationship between zeros of a wide range of pairs of complex polynomials, for example, a polynomial and its polar derivative or Sz.-Nagy’s generalized derivative. We shall introduce some new companion matrices and obtain a generalization of the Weinstein–Aronszajn Formula which will then be used to prove some inequalities similar to Sendov conjecture and Schoenberg conjecture and to study the distribution of equilibrium points of logarithmic potentials for finitely many discrete charges. Our method can also be used to produce, in an easy and systematic way, a lot of identities relating the sums of powers of zeros of a polynomial to that of the other polynomial. [Copyright &y& Elsevier]

Abstract: A twisted link is a generalization of a virtual link, which is related to a link diagram on a closed, possibly non-orientable surface. In this paper we generalize the Miyazawa polynomial invariant of a virtual link to an invariant of a twisted link in two formulae one of which is introduced by A. Ishii and the other by the author. [Copyright &y& Elsevier]

Abstract: In this paper, we consider two single-machine rescheduling problems with linear deteriorating jobs under disruption. By a deteriorating jobs, we mean that the actual processing time of the job is an increasing function of its starting time. The two problems correspond to two different increasing linear function. Rescheduling means a set of original jobs has already been scheduled to minimize some classical objective, then a new set of jobs arrives and creates a disruption. We consider the rescheduling problem to minimize the total completion time under a limit of the disruption from the original scheduling. For each problem, we consider two versions. For each version, the polynomial algorithms are proposed, respectively. [Copyright &y& Elsevier]

Abstract: Almost difference families (ADFs) were introduced by Ding and Yin as a useful generalization of almost difference sets (ADSs), and a number of infinite classes of almost difference families had been constructed. Suppose is a prime power. To construct combinatorial designs in , one often needs to find an element , such that some polynomials in of degree one or two satisfying certain conditions. Weil''s theorem on character sum estimates is very useful to do this. In this paper, a general bound for finding such is given. By using this bound and computer searching, some known results on almost difference families by Ding and Yin are improved. [Copyright &y& Elsevier]

Abstract: In this paper we deal from an algorithmic perspective with different questions regarding monochromatic and properly edge-colored s-t paths/trails on edge-colored graphs. Given a c-edge-colored graph without properly edge-colored closed trails, we present a polynomial time procedure for the determination of properly edge-colored s-t trails visiting all vertices of a prescribed number of times. As an immediate consequence, we polynomially solve the Hamiltonian path (resp., Eulerian trail) problem for this particular class of graphs. In addition, we prove that to check whether contains 2 properly edge-colored s-t paths/trails with length at most is NP-complete in the strong sense. Finally, we prove that, if is a general c-edge-colored graph, to find 2 monochromatic vertex disjoint s-t paths with different colors is NP-complete. [Copyright &y& Elsevier]

Abstract: Given an undirected edge-weighted network in which one edge capacity is allowed to vary, we propose in this paper a polynomial algorithm that needs only two 2-cut-trees computations to provide the all pairs maximum 2-route flow values. [Copyright &y& Elsevier]

Abstract: The paper is devoted to demonstrating a randomized algorithm for determining a dominating set in a given graph having a maximum degree of five. The algorithm follows the Las Vegas technique. Furthermore, the concept of a 2-separated collection of subsets of vertices in graphs is used. The suggested algorithm is based on a condition of the upper bound of the cardinality of a local dominating set. If the condition is not satisfied, then the algorithm halts with an appropriate message. Otherwise, the algorithm determines the dominating set. The given algorithm is considered a polynomial-time approximation one. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the explicit solutions of two matrix equations, namely, the Yakubovich matrix equation and Sylvester matrix equations and . For this purpose, we make use of Kronecker map and Sylvester sum as well as the concept of coefficients of characteristic polynomial of the matrix . Some lemmas and theorems are stated and proved where explicit and parametric solutions are obtained. The proposed methods are illustrated by numerical examples. The results obtained show that the methods are very neat and efficient. [Copyright &y& Elsevier]

Abstract: In this paper, integrability and generalized complex resonant center condition of degenerate resonant singular point for a class of complex polynomial differential system were studied. The concept of generalized singular point quantity of degenerate resonant singular point was proposed and the construction of that was studied. Two methods of computing generalized singular point quantities were given. Furthermore, the sufficient and necessary condition of integrability of degenerate resonant singular point was discussed for the first time. [Copyright &y& Elsevier]

Abstract: In this paper we classify the centers and isochronous centers for a class of polynomial differential systems in of degree that in complex notation can be written as where is an arbitrary even positive integer and . Note that if we obtain a special case of quartic polynomial differential systems which can have a center at the origin. [Copyright &y& Elsevier]

Abstract: By introducing a bivariate matrix-valued linear functional on the scalar polynomial space, a general two-dimensional (2-D) matrix Padé-type approximant () in the inner product space is defined in this paper. The coefficients of its denominator polynomials are determined by taking the direct inner product of matrices. The remainder formula is developed and an algorithm for the numerator polynomials is presented when the generating polynomials are given in advance. By means of the Hankel-like coefficient matrix, a determinantal expression of is presented. Moreover, to avoid the computation of the determinants, two efficient recursive algorithms are proposed. At the end the method of is applied to partial realization problems of 2-D linear systems. [Copyright &y& Elsevier]

Abstract: An -labeling of a graph is an assignment from the vertex set to the set of nonnegative integers such that if and are adjacent and if and are at distance 2 for all and in . A --labeling is an -labeling , and the -labeling problem asks the minimum , which we denote by , among all possible -labelings. It is known that this problem is NP-hard even for graphs of treewidth 2. Tree is one of a few classes for which the problem is polynomially solvable, but still only an time algorithm for a tree has been known so far, where is the maximum degree of and . In this paper, we first show that an existent necessary condition for is also sufficient for a tree with , which leads to a linear time algorithm for computing under this condition. We then show that can be computed in time for any tree . Combining these, we finally obtain an time algorithm, which substantially improves upon previously known results. [Copyright &y& Elsevier]

Abstract: A word is a fixed point of a nontrivial morphism if and is not the identity on the alphabet of . The paper presents the first polynomial algorithm deciding whether a given finite word is such a fixed point. The algorithm also constructs the corresponding morphism, which has the smallest possible number of non-erased letters. [Copyright &y& Elsevier]

Abstract: In contrast to the minimization of deterministic finite automata (DFA’s), the task of constructing a minimal nondeterministic finite automaton (NFA) for a given NFA is PSPACE-complete. Moreover, there are no polynomial approximation algorithms with a constant approximation ratio for estimating the number of states of minimal NFA’s. Since one is unable to efficiently estimate the size of a minimal NFA in an efficient way, one should ask at least for developing mathematical proof methods that help to prove good lower bounds on the size of a minimal NFA for a given regular language. Here we consider the robust and most successful lower bound proof technique that is based on communication complexity. In this paper it is proved that even a strong generalization of this method fails for some concrete regular languages. “To fail” is considered here in a very strong sense. There is an exponential gap between the size of a minimal NFA and the achievable lower bound for a specific sequence of regular languages. The generalization of the concept of communication protocols is also strong here. It is shown that cutting the input word into pieces for a size of a minimal nondeterministic finite automaton and investigating the necessary communication transfer between these pieces as parties of a multiparty protocol does not suffice to get good lower bounds on the size of minimal nondeterministic automata. It seems that for some regular languages one cannot really abstract from the automata model that cuts the input words into particular symbols of the alphabet and reads them one by one using its input head. [Copyright &y& Elsevier]

Abstract: It is well known that by adding some extra zeros to a Daubechies low-pass wavelet filter, one gets new orthonormal wavelet basis with better regularity property. In this paper, we give a detailed study of this procedure in the general case of a wavelet filter associated with any integer dilation factor . Moreover, we describe an algorithm for the construction of symmetric scaling functions with dilation factor . Finally, we provide the reader with some numerical examples that illustrate the results of this work. [Copyright &y& Elsevier]

Abstract: The main purpose of this paper is to analyze new determinantal expressions which define the subresultant sequence of two polynomials, when the coefficients of such polynomials depend on parameters. [Copyright &y& Elsevier]

Abstract: Let be a nonempty closed set with . For and , define by and We call a super-stationary set if holds for any infinite subset of . Denoting the derived set (i.e. the set of accumulating points) of and with , it is known [T. Kamae, Uniform set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)] that for any nonempty closed subset of such that there exists an infinite subset of with , there exists an infinite subset such that is a super-stationary set. Moreover, if for any infinite subset of , then the maximal pattern complexity [T. Kamae, Uniform set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)] is . Thus, the uniform complexity functions are realized by the super-stationary sets [T. Kamae, Uniform set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)]. We call a super-subword of if there exists with such that . Let be the set of having no super-subword . Denote where . In this paper, we prove that the class of super-stationary sets other than coincides with the class of for nonempty finite sets . Moreover, it also coincides with the class of for nonempty finite sets , where is the set of minimal covers of . Using these expressions, we can calculate the complexity of super-stationary sets and prove that the complexity function of a super-stationary set in is either or a polynomial function of for large . We also discuss the word problems related to the super-subwords. [Copyright &y& Elsevier]

Abstract: For a graph , we denote by the adjoint polynomial of . Let denote the minimum real root of . In this paper, we characterize all the connected graphs with . [Copyright &y& Elsevier]

Abstract: In this paper, we characterize symmetric Sobolev bilinear forms defined on , where is the space of polynomials. More specifically we show that symmetric Sobolev bilinear forms, like symmetric matrices, can be re-written with a diagonal representation. As an application, we introduce the notion of a ghost matrix, extending some classic work of Stieltjes. [Copyright &y& Elsevier]

Abstract: Let be the chromatic polynomial of a graph . A graph is chromatically unique if for any graph , implies H is isomorphic to . For integers , , denote by the complete -partite graph that has partite sets of size and one partite set of size . Let be the set of graphs obtained from by adding a set of edges to the partite set of size such that is bipartite. If , denote the only graph in by . In this paper, we shall prove that for and , each graph is chromatically unique if and only if is a chromatically unique graph that has no cut-vertex. As a direct consequence, the graph is chromatically unique for and . [Copyright &y& Elsevier]

Abstract: This paper revisits the ‘branchwidth territories’ of Kloks, Kratochvíl and Müller [T. Kloks, J. Kratochvíl, H. Müller, New branchwidth territories, in: 16th Ann. Symp. on Theoretical Aspect of Computer Science, STACS, in: Lecture Notes in Computer Science, vol. 1563, 1999, pp. 173–183] to provide a simpler proof, and a faster algorithm for computing the branchwidth of an interval graph. We also generalize the algorithm to the class of chordal graphs, albeit at the expense of exponential running time. Compliance with the ternary constraint of the branchwidth definition is facilitated by a simple new tool called -troikas: three sets of size at most each are a -troika of set , if any two have union . We give a straightforward algorithm, computing branchwidth for an interval graph on edges, vertices and maximal cliques. We also prove a conjecture of Mazoit [F. Mazoit, A general scheme for deciding the branchwidth, Technical Report RR2004-34, LIP — École Normale Supérieure de Lyon, 2004. http://www.ens-lyon.fr/LIP/Pub/Rapports/RR/RR2004/RR2004-34.pdf], by showing that branchwidth can be computed in polynomial time for a chordal graph given with a clique tree having a polynomial number of subtrees. [Copyright &y& Elsevier]

Abstract: Graph searching was introduced by Parson [T. Parson, Pursuit-evasion in a graph, in: Theory and Applications of Graphs, in: Lecture Notes in Mathematics, Springer-Verlag, 1976, pp. 426–441]: given a “contaminated” graph (e.g., a network containing a hostile intruder), the search number of the graph is the minimum number of searchers needed to “clear” the graph (or to capture the intruder). A search strategy is connected if, at every step of the strategy, the set of cleared edges induces a connected subgraph. The connected search number of a graph is the minimum such that there exists a connected search strategy for the graph using at most searchers. This paper is concerned with the ratio between the connected search number and the search number. We prove that, for any chordal graph of treewidth , . More precisely, we propose a polynomial-time algorithm that, given any chordal graph , computes a connected search strategy for using at most searchers. Our main tool is the notion of connected tree-decomposition. We show that, for any connected graph of chordality , there exists a connected search strategy using at most searchers where is an optimal tree-decomposition of . [Copyright &y& Elsevier]

Abstract: A clique-transversal of a graph is a subset of vertices that meets all the cliques of . A clique-independent set is a collection of pairwise vertex-disjoint cliques. A graph is clique-perfect if the sizes of a minimum clique-transversal and a maximum clique-independent set are equal for every induced subgraph of . The list of minimal forbidden induced subgraphs for the class of clique-perfect graphs is not known. Another open question concerning clique-perfect graphs is the complexity of the recognition problem. Recently we were able to characterize clique-perfect graphs by a restricted list of forbidden induced subgraphs when the graph belongs to two different subclasses of claw-free graphs. These characterizations lead to polynomial time recognition of clique-perfect graphs in these classes of graphs. In this paper we solve the characterization problem in two new classes of graphs: diamond-free and Helly circular-arc () graphs. This last characterization leads to a polynomial time recognition algorithm for clique-perfect graphs. [Copyright &y& Elsevier]

Abstract: A Directed Path Family is a family of subsets of some finite ground set whose members can be realized as arc sets of simple directed paths in some directed graph. In this paper we show that recognizing whether a given family is a Directed Path family is an NP-Complete problem, even when all members in the family have at most two elements. If instead of a family of subsets, we are given a collection of words from some finite alphabet, then deciding whether there exists a directed graph such that each word in the language is the set of arcs of some path in , is a polynomial-time solvable problem. [Copyright &y& Elsevier]

Abstract: We prove that an analytic function f on the unit ball with Hadamard gaps, that is, (the homogeneous polynomial expansion of f) satisfying for all , belongs to the space if and only if . Moreover, we show that the following asymptotic relation holds . Also we prove that if and only if . These results confirm two conjectures from the following recent paper [S. Stević, On Bloch-type functions with Hadamard gaps, Abstr. Appl. Anal. 2007 (2007) 8 pages (Article ID 39176)]. [Copyright &y& Elsevier]

Abstract: In this paper, the authors investigate the question of when a partition of is an -core and also a -core when and are not relatively prime. A characterization of all such -cores is given, as well as a generating function dependent upon the polynomial generating functions for -cores when and are relatively prime. Furthermore, characterizations and generating functions are given for -cores which are self-conjugate and also for -cores. [Copyright &y& Elsevier]

Abstract: Throughout the last decades, several results have been published in the area of the so-called Matrix Completion Problems. In this paper, we survey several results in this field. In particular, we describe the possible eigenvalues, the characteristic polynomial, the invariant polynomials, or the number of nontrivial invariant polynomials of a square matrix, over a field, when some of its entries are prescribed and the others vary. Finally, we present our contribution, generalizing some of the previous cases, to an matrix partitioned into blocks, with entries in a field, when some of its blocks are prescribed and the others vary. [Copyright &y& Elsevier]

Abstract: It is well-known that when a polynomial whose coefficients are continuous functions of a parameter loses its degree then some of its zeros must vanish at infinity. In this paper, we consider such a situation: we examine how roots of a complex polynomial tend to infinity as some of its coefficients, including the leading one, tend to zero. We show, among other things, that in such a situation the unbounded paths traced by the roots of the polynomial have asymptotes; we also obtain their formulas. Some examples are presented to complete and illustrate the results. [Copyright &y& Elsevier]

Abstract: In this paper, we study exponential partial Bell polynomials and Sheffer sequences. Two new characterizations of Sheffer sequences are presented, which indicate the relations between Sheffer sequences and Riordan arrays. Several general identities involving Bell polynomials and Sheffer sequences are established, which reduce to some elegant identities for associated sequences and cross sequences. [Copyright &y& Elsevier]

Abstract: This paper deals with the maximum triangle packing problem. For this problem, Hassin and Rubinstein gave a randomized polynomial-time approximation algorithm that achieves an expected ratio of for any constant . By modifying their algorithm, we obtain a new randomized polynomial-time approximation algorithm for the problem which achieves an expected ratio of for any constant . [Copyright &y& Elsevier]

Abstract: Precondition plays a critical role in the numerical methods for large and sparse linear systems. It is also true for nonlinear algebraic systems. In this paper incomplete Gröbner basis (IGB) is proposed as a preconditioner of homotopy methods for polynomial systems of equations, which transforms a deficient system into a system with the same finite solutions, but smaller degree. The reduced system can thus be solved faster. Numerical results show the efficiency of the preconditioner. [Copyright &y& Elsevier]