the linear algebra of the generalized pascal matrix abstract this paper discusses three kinds of generalized pascal matrix and generalizes the results of r brawer and m pirovino

hilbert coefficients and the gorenstein property of the associated graded ring in this article we investigate the gorenstein property of the associated graded ring ga i of an ideal i in a gorenstein local ring a m of positive dimension we especially concentrate on the case where i is m primary assuming that the rees algebra of i is cohen macaulay we then give necessary and sufficient conditions for the gorensteiness of ga i in terms of the hilbert coefficients of i

a geometrical approach to computing free energy landscapes from short ranged potentials particles interacting with short ranged potentials have attracted increasing interest partly for their ability to model mesoscale systems such as colloids interacting via dna or depletion we consider the free energy landscape of such systems as the range of the potential goes to zero in this limit the landscape is entirely defined by geometrical manifolds plus a single control parameter these manifolds are fundamental objects that do not depend on the details of the interaction potential and provide the starting point from which any quantity characterizing the system equilibrium or nonequilibrium can be computed for arbitrary potentials to consider dynamical quantities we compute the asymptotic limit of the fokker planck equation and show that it becomes restricted to the low dimensional manifolds connected by sticky boundary conditions to illustrate our theory we compute the low dimensional manifolds for identical particles providing a complete description of the lowest energy parts of the landscape including floppy modes with up to 2 internal degrees of freedom the results can be directly tested on colloidal clusters this limit is a unique approach for understanding energy landscapes and our hope is that it can also provide insight into finite range potentials

the numerical rank of krylov matrices abstract in this paper we expose interesting features of krylov matrices and vandermonde matrices let s be a large symmetric matrix of order n let x be a real n vector and let k l denote the related n l krylov matrix the question considered in this paper is how the numerical rank of k l grows as l increases the key for answering this question lies in the close link between k l and the n l vandermonde matrix which is generated by the eigenvalues of s analysis of large vandermonde matrices shows that the numerical rank is expected to remain much smaller than l the proof is based on partition theorems and clustering theorems the basic tool is a new matrix equality the vandermonde pascal toeplitz equality the actual numerical rank of a vandermonde or krylov matrix depends on the distribution of the eigenvalues but often the rank is remarkable small numerical experiments illustrate these points the observation that the numerical rank of krylov matrices stays small has important practical consequences

matrices related to the bell polynomials abstract in this paper we study the matrices related to the partial exponential bell polynomials and those related to the bell polynomials with respect to as a result the factorizations of these matrices are obtained which give unified approaches to the factorizations of many lower triangular matrices moreover some combinatorial identities are also derived from the corresponding matrix representations

tight closure and projective bundles abstract we formulate problems of tight closure theory in terms of projective bundles and subbundles this provides a geometric interpretation of such problems and allows us to apply intersection theory to them this yields new results concerning the tight closure of a primary ideal in a two dimensional graded domain

vandermonde matrices on integer nodes the rectangular case we consider a rectangular vandermonde matrix v on integer nodes using combinatorial identities we give an explicit cholesky factorization of the matrix b v rm t v a factorization of the pseudo inverse of v and asymptotic estimates of the extremal eigenvalues of b we also discuss the numerical properties of the proposed formulas

on connected dominating sets of restricted diameter abstract a connected dominating set cds is commonly used to model a virtual backbone of a wireless network to bound the distance that information must travel through the network we explicitly restrict the diameter of a cds to be no more than s leading to the concept of a dominating s club we prove that for any fixed positive integer s it is np complete to determine if a graph has a dominating s club even when the graph has diameter s 1 as a special case it is np complete to determine if a graph of diameter two has a dominating clique we then propose a compact integer programming formulation for the related minimization problem enhance the approach with variable fixing rules and valid inequalities and present computational results

computing the tight closure in dimension two we study computational aspects of the tight closure of a homogeneous primary ideal in a two dimensional normal standard graded domain we show how to use slope criteria for the sheaf of relations for generators of the ideal to compute its tight closure in particular our method gives an algorithm to compute the tight closure of three elements under the condition that we are able to compute the harder narasimhan filtration we apply this to the computation of x a y a z a in k x y z f where f is a homogeneous polynomial

generalized leibniz functional matrices and factorizations of some well known matrices in this paper we introduce the generalized leibniz functional matrices and study some algebraic properties of such matrices to demonstrate applications of these properties we derive several novel factorization forms of some well known matrices such as the complete symmetric polynomial matrix and the elementary symmetric polynomial matrix in addition by applying factorizations of the generalized leibniz functional matrices we redevelop the known results of factorizations of stirling matrices of the first and second kind and the generalized pascal matrix

finding the optimal nets for self folding kirigami three dimensional shells can be synthesized from the spontaneous self folding of two dimensional templates of interconnected panels called nets however some nets are more likely to self fold into the desired shell under random movements the optimal nets are the ones that maximize the number of vertex connections i e vertices that have only two of its faces cut away from each other in the net previous methods for finding such nets are based on random search and thus they do not guarantee the optimal solution here we propose a deterministic procedure we map the connectivity of the shell into a shell graph where the nodes and links of the graph represent the vertices and edges of the shell respectively identifying the nets that maximize the number of vertex connections corresponds to finding the set of maximum leaf spanning trees of the shell graph this method allows us not only to design the self assembly of much larger shell structures but also to apply additional design criteria as a complete catalog of the maximum leaf spanning trees is obtained

an involutory pascal matrix abstract an involutory upper triangular pascal matrix u n is investigated eigenvectors of u n and of u n t are considered a characterization of u n is obtained and it is shown how the results can be extended to matrices over a commutative ring with unity

the algebraic properties of the generalized pascal functional matrices associated with the exponential families abstract in this paper pascal matrices are generalized to functional matrices by using the exponential families and the restricted sequences of polynomials of binomial type some algebraic properties of the pascal functional matrices are presented and proved in addition several important combinatorial identities are obtained

stirling matrix via pascal matrix abstract the pascal type matrices obtained from the stirling numbers of the first kind s n k and of the second kind s n k are studied respectively it is shown that these matrices can be factorized by the pascal matrices also the ldu factorization of a vandermonde matrix of the form v n x x 1 x n 1 for any real number x is obtained furthermore some well known combinatorial identities are obtained from the matrix representation of the stirling numbers and these matrices are generalized in one or two variables

the linear algebra of the k fibonacci matrix abstract for a positive integer k 2 the k fibonacci sequence g k n is defined as g k 1 g k k 2 0 g k k 1 g k k 1 and for n k 2 g k n g k n 1 g k n 2 g k n k the n n k fibonacci matrix f k n f k ij n is defined as for fixed k 2 f k ij g i j 1 i j 1 0 0 i j 1 where gn g k n k 2 also the n by n k symmetric fibonacci matrix q k n q k ij n is defined as q k ij q k ji l 1 k q k i j l i 1 j l 1 k q k i i l g 1 i j where q k ij 0 for j 0 if k 2 then f 2 n is the fibonacci matrix and q 2 n is the symmetric fibonacci matrix the properties of the fibonacci matrix and the symmetric fibonacci matrix are well known in this paper we discuss the linear algebra of the k fibonacci matrix and the symmetric k fibonacci matrix

schroder matrix as inverse of delannoy matrix abstract using riordan arrays we introduce a generalized delannoy matrix by weighted delannoy numbers it turns out that delannoy matrix pascal matrix and fibonacci matrix are all special cases of the generalized delannoy matrices meanwhile schroder matrix and catalan matrix also arise in involving inverses of the generalized delannoy matrices these connections are the focus of our paper the half of generalized delannoy matrix is also considered in addition we obtain a combinatorial interpretation for the generalized fibonacci numbers

outline capturing using rational functions with the help of genetic algorithm a technique to capture outline of planar objects using two rational cubic functions for approximating the boundary curve is presented in this paper genetic algorithm is utilized to find suitable values of parameters in the description of rational functions the proposed devised scheme eventually helps construct optimal curve to contours of images of generic shapes

on rank selection probabilities the concept of rank selection probabilities for stack filters has previously been introduced if the probabilities are known then the output distribution for i i d input follows easily there is another expression for the output distribution of a stack filter using certain quantities called a sub i the authors utilize the different forms of output distribution to derive a simple connection between the quantities a sub i and the rank selection probabilities this also leads to a new fast method for the computation of rank selection probabilities

the linear algebra of the pascal matrix abstract pascal s triangle can be represented as a square matrix in two basically different ways as a lower triangular matrix pn or as a full symmetric matrix qn it has been found that the pn pnt is the cholesky factorization of qn pn can be factorized by special summation matrices it can be shown that the inverses of these matrices are the operators which perform the gaussian elimination steps for calculating cholesky s factorization by applying linear algebra we produce combinatorial identities and an existence theorem for diophantine equation systems finally an explicit formula for the sum of the kth powers is given

an exact algorithm for the maximum leaf spanning tree problem given an undirected graph with n vertices the maximum leaf spanning tree problem is to find a spanning tree with as many leaves as possible when parameterized in the number of leaves k this problem can be solved in time o 4kpoly n using a simple branching algorithm introduced by a subset of the authors kneis et al 2008 16 daligault et al 2010 6 improved the branching and obtained a running time of o 3 72kpoly n in this paper we study the problem from an exponential time viewpoint where it is equivalent to the connected dominating set problem here fomin grandoni and kratsch showed how to break the 2n barrier and proposed an o 1 9407n time algorithm fomin et al 2008 11 based on some useful properties of kneis et al 2008 16 and daligault et al 2010 6 we present a branching algorithm whose running time of o 1 8966n has been analyzed using the measure and conquer technique finally we provide a lower bound of 1 4422n for the worst case running time of our algorithm

an efficient method for fitting large data sets using t splines data fitting is a fundamental tool in scientific research and engineering applications generally there are two ingredients in solving data fitting problems one is the fitting representation and the other is the fitting method nowadays the fitting of larger and larger quantities of data sets requires more compact fitting representation and faster fitting methods the t spline is a recently invented spline representation whose control mesh t mesh allows a row of control points to terminate thus reducing the number of superfluous control points in the b spline representation significantly this property makes t splines more compact than b splines in fitting large data sets however the adaptivity of the t spline causes the coefficient matrix of a least squares fitting linear system to lose its block structure thus when fitting large data sets with t splines by iterative methods only point iterative methods can be used and the iteration speeds of typically employed point iterative methods are r

generalized leibniz functional matrices and divided difference form of the lagrange burmann formula abstract in this paper we derive two versions of divided difference form of the lagrange burmann expansion formula the first version expresses all coefficients by a form of a determinant by means of the factorizations of the generalized leibniz functional matrices with respect to a sequence we obtain the second version with coefficients represented by a finite sum some new matrix inversions are also investigated

looking out for stable syzygy bundles abstract we study slope stability properties of syzygy bundles on a projective space p n given by ideal generators of a homogeneous primary ideal in particular we give a combinatorial criterion for a monomial ideal to have a semistable syzygy bundle restriction theorems for semistable bundles yield the same stability results on the generic complete intersection curve from this we deduce a numerical formula for the tight closure of an ideal generated by monomials or by generic homogeneous elements in a generic two dimensional complete intersection ring

a computation of tight closure in diagonal hypersurfaces the aim of this paper is to settle a question about the tight closure of 2 2 2 w x 3 3 3 the ideal x y z in the ring r s k x y z r x q y q z where k is a field of prime characteristic p 3 lower case letters denote the images of the corresponding variables m mcdermott has studied the tight closure of various irreducible ideals in r and has established that 2 2 2 w x xyz g x y z when p 200 see mc the general case however existed as a classic example of the difficulty involved in tight closure w x 2 2 2 computations see also hu example 1 2 we show that xyz g x y z in arbitrary prime characteristic p and furthermore establish that xyz g 2 2 2 f x y z whenever r is not f pure i e when p 2 mod 3 we move on to generalize these results to the diagonal hypersurfaces r s w x n n k x x r x q qx 1 n 1 n these issues relate to the question whether the tight closure i of an ideal i agrees with its plus closure iqs irql r where r is a domain over a field of characteristic p and rq is the integral closure of r in an algebraic closure of its fraction field in this setting we may think of the frobenius closure of i as i f s ir l r where r is the extension of r obtained by adjoining pth roots of all nonzero elements of r for e g n it is not difficult to see that iq i and equality in general is a formidable open question it should be mentioned that in the case when i is an ideal generated by part of a system of parameters the equality is a result

groups of generalized pascal matrices abstract we present a new approach to the study of generalized pascal matrices that yields general results about group structures and explicit formulas for powers and inverses of the pascal matrices shows how the groups of pascal matrices are related to groups of 2 2 matrices and clarifies the way in which symmetries and duality relate to the algebraic aspects of the theory we consider bilaterally infinite generalized pascal matrices obtained as the matrix representations of certain linear operators on spaces of formal laurent series their column generating functions are simple rational functions closely related to the linear fractional maps of the complex plane we obtain explicit expressions for the inverses and the powers of the pascal matrices that generalize most of the analogous results in the literature

a composite iterative procedure with fast convergence rate for the progressive iteration approximation of curves abstract the progressive iterative approximation pia is an iterative method for solving the linear system of equations corresponding to the interpolation problem in this paper a composite iterative procedure for the progressive and iteration approximation is presented the proposed method generates a sequence of matrices based on the schulz iterative method which it uses in the adjusting vectors the present method develops the progressive and iterative approximation for least square fitting lspia given in deng and lin 2014 and analyzes its convergence along with the estimation of error bounds a comparative study of experimental results embosses various advantages of the proposed technique such as faster convergence rate and low approximation errors

tight closure plus closure and frobenius closure in cubical cones we consider tight closure plus closure and frobenius closure in the rings r k x y z x3 y3 z3 where k is a field of characteristic p and p 74 3 we use a z3 grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular ring k x y we show that frobenius closure is the same as tight closure in certain classes of ideals when p 2 mod 3 since if c ir n r c 1 we conclude that ir n r 1 for these ideals using injective modules over the ring r the union of all peth roots of elements of r we reduce the question of whether if i for 23 graded ideals to the case of 23 graded irreducible modules we classify the irreducible m primary z3 graded ideals we then show that jf 1 for most irreducible rn primary z3 graded ideals in k x y z x3 y3 z3 where k is a field of characteristic p and p 2 mod 3 hence 1 irmf n r for these ideals in this paper we discuss the conjecture that i ir n r where r denotes the integral closure of a domain r of characteristic p in an algebraic closure of its fraction field and i denotes the tight closure of i the ring r is characterized by the property that it is a domain integral over r and every monic polynomial with coefficients in r factors into monic linear factors this characterization can be used to prove the following property of r if w is a multiplicatively closed set of r then w lr w 1r aside from providing a much more concrete description of tight closure proving that i ir n r would solve the localization problem for tight closure it is known that i ir n r for parameter ideals sml and for rings in which every ideal of the normalization is tightly closed also for those ideals i of an excellent local domain r such that r i has finite phantom projective dimension it is known that i ir n r ab however the conjecture is open even for two dimensional normal gorenstein domains in particular the conjecture is open for the cubical cone k x y z x3 y3 z3 where k is a field of characteristic p and p 5l 3 and more generally for rings of the form k x y z f x y z where f is a homogeneous cubic polynomial we consider tight closure plus closure and frobenius closure in the rings r k x y z x3 y3 z3 where k is a field of characteristic p and p 5l 3 in section 1 we use a z3 grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular rings k x y in section 2 we show that the frobenius closure of an ideal i denoted if is the same as the tight closure in certain classes of ideals when p 2 mod 3 since if c ir n r c i received by the editors august 27 1997 1991 mathematics subject classification primary 13a35 13a02 13hi10

essential prime divisors and projectively equivalent ideals let z be an ideal in a noetherian ring r we concern ourselves with the essential prime divisors of z an interesting subset of ass r p for all large n we first take z br with b a regular element of r we show that there is a ring t with r c te r such that t is a finite r module and the essential primes of bt are exactly the prime divisors of bt we next consider an arbitrary ideal z and apply our principal arguments to the element u in the kees ring of z we thereby deduce that there is an idcal j projectively equivalent to z such that the set of essential primes of z equals the set u ass r j over ii 1 2 3 notation let z be an ideal in a noetherian ring r we will use w or

a bracket power characterization of analytic spread one ideals the main theorem characterizes in terms of bracket powers analytic spread one ideals in local rings specifically let bl bg x be r egular nonunits in a local noetherian ring r m and assume that i c xr a the integral closure of xr where i b1 bg x r then the main result shows that for all but finitely many units u1 ug in r that are non congruent modulo m and for all large integers n and k it holds that jin ii for j 1 i k and j not divisible by char r m where i i is the j th bracket power bi uix i bg ugx i xi r of i bi ulx bg ugx x r and conversely if there exist positive integers g n and k n g such that i has a basis 31 39 x such that ilkn 3i k 3gk xk n r then i has analytic spread one

an extension of the generalized pascal matrix and its algebraic properties abstract the extended generalized pascal matrix can be represented in two different ways as a lower triangular matrix n x y or as a symmetric n x y these matrices generalize p n x q n x and r n x which are defined by zhang and by call and velleman a product formula for n x y has been found which generalizes the result of call and velleman it is shown that not only can n x y be factorized by special summation but also n x y as q n xy s t y 1 x or n x y p n t y x finally the inverse of n x y and the values of det n x y det n 1 x y det n x y and det n 1 x y are given

positive data modeling using spline function a rational cubic function with two parameters has been constructed to visualize the positive data the main focus of the work is the representation of the data in such a way that its view looks smooth and attractive in the first step simple data dependent constraints are derived on the parameters in the description of the rational cubic function to visualize the shape of positive data then it is extended to a rational bi cubic partially blended functions coons patches and derived constraints on parameters to visualize the shape of positive surface data the developed scheme is locally positive and economical the approximation order of rational cubic spline function is oh i 3

extended symmetric pascal matrices via hypergeometric functions in this paper we give a matrix representation of the hypergeometric functions of the type 2f 1 a b c x as a result we obtain a connection between the hypergeometric functions the legendre polynomials and the delannoy numbers moreover it is shown that each entry of p n x y p n x y t can be represented by the hypergeometric functions where p n x y i j x i jy i j 2i 1j 1 is the extended generalized pascal matrix which is defined by zhang and liu linear algebra appl 271 1998 169

regenerator location problem polyhedral study and effective branch and cut algorithms abstract in this paper we study the regenerator location problem rlp this problem arises in optical networks where an optical signal can only travel a certain maximum distance called the optical reach before its quality deteriorates needing regenerations by regenerators deployed at network nodes the rlp is to determine a minimal number of network nodes for regenerator placement such that for each node pair there exists a path of which no sub path without internal regenerators has a length greater than the optical reach starting with a set covering formulation involving an exponential number of constraints reported and studied in rahman 2012 and aneja 2012 we study the facial structure of the polytope arising from this formulation significantly extending known results making use of these polyhedral results we present a new branch and cut b c solution approach to solve the rlp to optimality we present a series of computational experiments to evaluate two versions of the proposed b c approach over 400 benchmark rlp instances we first compare them with an available exact method for the rlp in the literature because of the equivalence among the rlp the minimum connected dominating set problem mcdsp and the maximum leaf spanning tree problem mlstp we further compare our approaches with other available exact algorithms using 47 benchmark mcdsp mlstp instances

eigenvalues of hadamard powers of large symmetric pascal matrices abstract let s n be the positive real symmetric matrix of order n with i j entry equal to i j 2 j 1 and let x be a positive real number eigenvalues of the hadamard or entry wise power s n x are considered in particular for k a positive integer it is shown that both the perron root and the trace of s n k are approximately equal to 4 k 4 k 1 2 n 2 n 1 k

the cyclic groups via the pascal matrices and the generalized pascal matrices abstract in this paper given a positive integer m we consider the multiplicative order of upper and lower triangular matrices and symmetric matrices derived from pascal s triangle when read modulo m we also consider the corresponding problems when the binomial coefficients i j are replaced by x i j i j and x i j i j where x is an integer satisfying certain divisibility hypotheses we also offer some open conjectures on these orders

generalized pascal functional matrix and its applications in this paper we introduce the generalized pascal functional matrix and show that the existing variations of pascal matrices are special cases of this generalization we study some algebraic properties of such generalized pascal functional matrices in addition we demonstrate a direct application of these properties by deriving several novel combinatorial identities and a nontraditional approach for lu decompositions of some well known matrices such as symmetric pascal matrices

matrix approach to polynomials 2 abstract we continue our study of the structure initiated in t arponen a matrix approach to polynomials linear algebra appl 359 2003 181 196 our main emphasis is exploring further structure into the formalism introduced in t arponen a matrix approach to polynomials linear algebra appl 359 2003 181 196 this formalism reveals beautiful interplay between certain elementary operators and provides tools for example for checking handling and generalizing combinatorial identities as we show in examples in addition to that we discover a group structure among vandermonde matrices a fascinating diophantine equation and new proofs and generalizations of recently found related results

enumerating minimal connected dominating sets in graphs of bounded chordality enumerating objects of specified type is one of the principal tasks in algorithmics in graph algorithms one often enumerates vertex subsets satisfying a certain property we study the enumeration of all minimal connected dominating sets of an input graph from various graph classes of bounded chordality we establish enumeration algorithms as well as lower and upper bounds for the maximum number of minimal connected dominating sets in such graphs in particular we present algorithms to enumerate all minimal connected dominating sets of chordal graphs in time o 1 7159 n of split graphs in time o 1 3803 n and of at free strongly chordal and distance hereditary graphs in time o 3 n 3 where n is the number of vertices of the input graph our algorithms imply corresponding upper bounds for the number of minimal connected dominating sets for these graph classes

algorithmic design of self folding polyhedra self assembly has emerged as a paradigm for highly parallel fabrication of complex three dimensional structures however there are few principles that guide a priori design yield and defect tolerance of self assembling structures we examine with experiment and theory the geometric principles that underlie self folding of submillimeter scale higher polyhedra from two dimensional nets in particular we computationally search for nets within a large set of possibilities and then test these nets experimentally our main findings are that i compactness is a simple and effective design principle for maximizing the yield of self folding polyhedra and ii shortest paths from 2d nets to 3d polyhedra in the configuration space are important for rationalizing experimentally observed folding pathways our work provides a model problem amenable to experimental and theoretical analysis of design principles and pathways in self assembly

capturing planar shapes by approximating their outlines a non deterministic evolutionary approach for approximating the outlines of planar shapes has been developed non uniform rational b splines nurbs have been utilized as an underlying approximation curve scheme simulated annealing heuristic is used as an evolutionary methodology in addition to independent studies of the optimization of weight and knot parameters of the nurbs a separate scheme has also been developed for the optimization of weights and knots simultaneously the optimized nurbs models have been fitted over the contour data of the planar shapes for the ultimate and automatic output the output results are visually pleasing with respect to the threshold provided by the user a web based system has also been developed for the effective and worldwide utilization the objective of this system is to provide the facility to visualize the output to the whole world through internet by providing the freedom to the user for various desired input parameters setting in the algorithm designed

product eigenvalue problems many eigenvalue problems are most naturally viewed as product eigenvalue problems the eigenvalues of a matrix a are wanted but a is not given explicitly instead it is presented as a product of several factors a a k a k 1 cdots a 1 usually more accurate results are obtained by working with the factors rather than forming a explicitly for example if we want eigenvalues vectors of b t b it is better to work directly with b and not compute the product the intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems for all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic gr algorithm applied to a related cyclic matrix

symmetric polynomials pascal matrices and stirling matrices we consider lower triangular matrices consisting of symmetric polynomials and we show how to factorize and invert them since binomial coefficients and stirling numbers can be represented in terms of symmetric polynomials these results contain factorizations and inverses of pascal and stirling matrices as special cases this work generalizes that of several other authors on pascal and stirling matrices

totally positive bases and progressive iteration approximation in this paper we study the progressive iteration approximation property of a curve tensor product surface generated by blending a given data point set and a set of basis functions the curve tensor product surface has the progressive iteration approximation property as long as the basis is totally positive and the corresponding collocation matrix is nonsingular thus the b spline and nurbs curve surface have the progressive iteration approximation property and bezier curve surface also has the property if the corresponding collocation matrix is nonsingular

a measure conquer approach for the analysis of exact algorithms for more than 40 years branch reduce exponential time backtracking algorithms have been among the most common tools used for finding exact solutions of np hard problems despite that the way to analyze such recursive algorithms is still far from producing tight worst case running time bounds motivated by this we use an approach that we call measure conquer as an attempt to step beyond such limitations the approach is based on the careful design of a nonstandard measure of the subproblem size this measure is then used to lower bound the progress made by the algorithm at each branching step the idea is that a smarter measure may capture behaviors of the algorithm that a standard measure might not be able to exploit and hence lead to a significantly better worst case time analysis in order to show the potentialities of measure conquer we consider two well studied np hard problems minimum dominating set and maximum independent set for the first problem we consider the current best algorithm and prove thanks to a better measure a much tighter running time bound for it for the second problem we describe a new simple algorithm and show that its running time is competitive with the current best time bounds achieved with far more complicated algorithms and standard analysis our examples show that a good choice of the measure made in the very first stages of exact algorithms design can have a tremendous impact on the running time bounds achievable

pascal k eliminated functional matrix and its property abstract in this paper we shall first define the pascal k eliminated functional matrix for two variables over an arbitrary field f 1 8 then using the previous results we obtain several interesting combinatorial identities we also investigate the relationship between these matrices and cesaro matrices 9 finally we give an affirmative answer to an open problem raised in 1 for minimal polynomial of pascal k eliminated functional matrix over the field z p

catalan matrix and related combinatorial identities we introduce the notion of the catalan matrix c n x whose non zero elements are expressions which contain the catalan numbers arranged into a lower triangular toeplitz matrix inverse of the catalan matrix is derived correlations between the matrix c n x and the generalized pascal matrix are considered some combinatorial identities involving catalan numbers binomial coefficients and the generalized hypergeometric function are derived using these correlations moreover an additional explicit representation of the catalan number as well as an explicit representation of the sum of the first m catalan numbers are given

triangular powers of integers from determinants of binomial coefficient matrices abstract formulas are obtained for the determinants of certain matrices whose entries are zero and either binomial coefficients or their negatives a consequence is that for all integers n 2 and k 2 there exists an n 1 k 1 n 1 k 1 matrix m n k whose entries are the alternating binomial coefficients 1 j 1 jn and zeros such that det m n k ktn 1 where tn 1 is the n l th triangular number further if we form the infinite matrix download download full size image whose kth row is 0k 1k 2k then each of the above mentioned determinants is up to sign the determinant of an n n submatrix a of download download full size image obtained by selecting the initial n columns and some choice of n rows of download download full size image the matrices m n k and others that we will consider also have the unexpected property that det m n k det m n k where m denotes the matrix obtained from m by replacing each entry with its absolute value