the determinant of a tree s neighborhood matrix abstract let n be an n n neighborhood matrix for a tree or forest we show that det n is bounded by the nth fibonacci number we obtain a simple elegant algorithm to compute det n that operates directly on the forest and uses o n space and o n arithmetic operations

operads and knot spaces let em denote the space of embeddings of the interval i 1 1 in the cube im with endpoints and tangent vectors at those endpoints fixed on opposite faces of the cube equipped with a homotopy through immersions to the unknot see definition 5 1 by proposition 5 17 em is homotopy equivalent to emb i im x i lmm l im in 28 mcclure and smith define a cosimplicial object o associated to an operad with multiplication 0 whose homotopy invariant totalization we denote tot see definition 2 17 and definition 2 5 below let cm denote the rath kontsevich operad introduced in 22 whose entries are compactified configuration spaces and which is weakly equivalent to the little ra disks operad 37 see definition 4 1 and theorem 4 5

algebraic graph theory without orientation abstract let g be an undirected graph with vertices v 1 v 2 v and edges e1 e2 e let m be the matrix whose ijth entry is 1 if ej is a link incident with vi 2 if ej is a loop at vi and 0 otherwise the matrix obtained by orienting the edges of a loopless graph g i e changing one of the 1 s to a 1 in each column of m has been studied extensively in the literature the purpose of this paper is to explore the substructures of g and the vector spaces associated with the matrix m without imposing such an orientation we describe explicitly bases for the kernel and range of the linear transformation from r to r defined by m our main results are determinantal formulas using the unoriented laplacian matrix mmt to count certain spanning substructures of g these formulas may be viewed as generalizations of the matrix tree theorem the point of view adopted in this paper also gives rise to a matroid structure on the edges of g analogous to the cycle matroid and its dual in this setting the analogue of a spanning forest can have components with one odd cycle and the analogue of an edge cut has the property that its removal creates a new bipartite component

on characterizations of classical polynomials it is well known that the classical families of jacobi laguerre hermite and bessel polynomials are characterized as eigenvectors of a second order linear differential operator with polynomial coefficients rodrigues formula etc in this paper we present a unified study of the classical discrete polynomials and q polynomials of the q hahn tableau by using the difference calculus on linear type lattices we obtain in a straightforward way several characterization theorems for the classical discrete and q polynomials of the q hahn tableau finally a detailed discussion of a characterization by marcellan et al is presented

lie coalgebras and rational homotopy theory ii hopf invariants we give a new definitive resolution of the homotopy periods problem modulo torsion this problem was addressed most famously by sullivan on the first page and then section 11 of his seminal paper 24 there sullivan starts to give explicit integrals over a sphere which determine whether two maps from that sphere are equivalent in the quotient of the homotopy groups of a simply connected space modulo their torsion subgroups these integrals are a direct generalization of whitehead s formula for hopf invariant 25 and they encode linking numbers more integrals along these lines were given by haefliger 10 and some framework was developed by novikov 17 before sullivan s work boardman and steer 3 gave a set of linking number invariants defined for maps to a suspension finally hain in his thesis 11 uses chen integrals as his model for the bar construction to give one resolution of the homotopy period question but produces integrals over products of spheres with simplices rather than over spheres themselves and an auxiliary formalism is required to address redundancy we give a complete set of integrals we can use integer valued cochains if we wish over the sphere which distinguish maps modulo torsion proceeding from a simple use of the bar construction we also connect these integrals to quillen functors in the framework of differential graded lie coalgebras as developed in 23 on the formal side we are then able to evaluate hopf invariants on whitehead products and are able to understand the naturality of these maps in the long exact sequence of a fibration for applications we can show for example that the homotopy groups of homogeneous spaces are controlled by classical linking numbers we proceed in two steps first using the classical bar complex to define integer valued homotopy functionals which coincide with evaluation of the cohomology of x on the looping of a map from s to x one basic apparently new observation is that calculations in the classical bar complex yield the hopf invariant formula of whitehead as well as those of haefliger and sullivan we establish basic properties and give examples in the second part we use the harrison complex on commutative cochains the standard cohomology theory for commutative algebras and thus must switch to rational coefficients using our graph coalgebraic presentation we show that a product coproduct formula established geometrically in the bar complex descends to the duality predicted by koszul moore theory in summary we find that rational homotopy groups are controlled by linking phenomena and that this linking phenomena are best described in the language of lie coalgebras one direction we plan to pursue further is the use of hopf invariants to realize koszul moore duality isomorphisms in general a second direction we plan to pursue is that of spaces which are not simply connected where the graph coalgebra models seem relevant even for k 1 spaces

a christoffel darboux formula and a favard s theorem for orthogonal laurent polynomials on the unit circle let f k z k 0 be the family of orthonormal laurent polynomials on the unit circle which spans d in the ordering induced by p n e n 1 2 from the three term recurrence relation satisfied by f k z k 0 we deduce a christoffel darboux formula particular examples are considered and a favard type theorem is proved a connection with the ordering induced by p n e n 2 is also established

stabilizing quantum states by constructive design of open quantum dynamics based on recent work on the asymptotic behavior of controlled quantum markovian dynamics we show that any generic quantum state can be stabilized by devising constructively a simple lindblad gks generator that can achieve global asymptotic stability at the desired state the applicability of such result is demonstrated by designing a direct feedback strategy that achieves global stabilization of a qubit state encoded in a noise protected subspace

applications of recurrence relations for the characteristic polynomials of bethe trees a bethe tree of k levels b k d is a rooted tree such that the root vertex has degree d the vertices from level 2 to k 1 have degree d 1 and the vertices at level k 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 b 2 n 2 and we obtain a recurrence relation for the laplacian energy like invariant of bethe trees

on a class of matrix orthogonal polynomials on the real line abstract it is shown how to construct matrix orthogonal polynomials on the real line provided we have a sequence of scalar orthogonal polynomials over an algebraic harmonic curve we consider the moment matrix associated with a distribution on and related to a basis of the linear space of polynomials obtained through the polynomial which defines the curve this particular moment matrix can be regarded as a hankel block matrix and thus matrix orthogonal polynomials appear in a natural way

on the favard theorem and its extensions in this paper we present a survey on the favard theorem and its extensions c 2001 elsevier science b v all rights reserved

irreducible decomposition of powers of edge ideals abstract in this paper by using some tools from graph theory mainly the theorem on ear decomposition of factor critical graphs given by l lovasz and the canonical decomposition of a graph given by edmonds and gallai we can describe each irreducible component of powers of edge ideals of a graph as an application we use our results to persistence of irreducible components to bound dstab and to count the number of irreducible components of powers of edge ideals of a graph

laguerre domain identification of continuous linear time delay systems from impulse response data an expression for the laguerre spectrum of the impulse response of a linear continuous time invariant system with input or output delay is derived a discrete state space description of the time delay system in the laguerre shift operator is obtained opening up for the use of conventional identification techniques a method for laguerre domain identification of continuous time delay systems from impulse response data is then proposed linear time invariant systems resulting from cascading finite dimensional dynamics with pure time delays are considered subspace identification is utilized for estimation of finite dimensional dynamics an application to blind identification of a mathematical model of an endocrine system with pulsatile regulation is also provided

a class of matrix orthogonal polynomials on the unit circle abstract the study of matrix orthogonal polynomials on the unit circle is connected with problems of applied mathematics such as filtering theory prediction theory and scattering theory in this paper some families of matrix orthogonal polynomials on the unit circle are given explicitly the decomposition a z z a z z h 1 a z where is the linear space of complex algebraic polynomials is the main idea for this approach

locating the eigenvalues of trees abstract e give an o n method that computes for any tree t and interval how many eigenvalues of t lie within the interval our method is based on sylvester s law of inertia we use our algorithm to show that the nonzero eigenvalues of a caterpillar are simple it follows that caterpillars having b back nodes where b 2 1 are not integral we also show that among the regular caterpillars c b k formed by adjoining k legs to each of b back nodes all positive roots are in the interval k 1 k 2 and c b k is not integral if b 2

combinatorial characterization of the null spaces of symmetric h matrices abstract we characterize the structure of null spaces of symmetric diagonally dominant sdd matrices and symmetric h matrices with non negative diagonal entries we show that the structure of the null space of a symmetric sdd matrix or h matrix a depends on the structure of the connected components of its underlying graph each connected component contributes at most one vector to the null space this paper provides a combinatorial characterization of the rank of each connected component and a combinatorial characterization of a null vector if one exists for sdd matrices we also present an efficient combinatorial algorithm for constructing an orthonormal basis for the null space the paper also shows a close connection between gain graphs and h matrices which extends known results regarding the connection between undirected graphs and laplacian matrices and between signed graphs and sdd matrices we show how to exploit these combinatorial algorithms to reliably solve certain singular linear systems in finite precision arithmetic

laguerre series approximation of infinite dimensional systems laguerre fourier approximations of stable systems are shown to exhibit many desirable properties for various classes of infinite dimensional systems specifically time domain supremum and l1 norm convergence results and frequency domain h norm convergence results are given for laguerre fourier approximations it is also shown that the theory of laguerre polynomials solves explicitly the problem of determining laguerre fourier approximations for a large class of delay systems furthermore it is believed that these results are important for the study of orthonormal series identification as a general technique for identification of infinite dimensional systems

the determinant of a unicyclic graph s neighborhood matrix abstract let g be a unicyclic graph with n vertices and the unique cycle c a g and n g its adjacency matrix and neighborhood matrix respectively and a scalar we obtain an algorithm for computing the determinant of in a g which uses o n space and o n arithmetic operations for n where min dg x x v c applications include computing the determinants of a g and n g and computing the characteristic polynomial of a g

parameters estimation of systems with delayed and structured entries this paper deals with on line identification of continuous time systems with structured entries such entries which may consist of inputs perturbations or piecewise polynomial time varying parameters can be defined as signals that can be easily annihilated the proposed cancellation method allows to obtain non asymptotic estimators for the unknown coefficients application to delayed and switching hybrid systems are proposed numerical simulations with noisy data but also experimental results on a delay process are provided

new perspectives on self linking abstract we initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot which is the induced map on the compactified n point configuration space sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots we compute the homotopy types of the first three mapping space models showing that the third model gives rise to an integer valued invariant we realize this invariant in two ways in terms of collinearities of three or four points on the knot and give some explicit computations we show this invariant coincides with the second coefficient of the conway polynomial thus giving a new geometric definition of the simplest finite type invariant finally using this geometric definition we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot

expansions of one density via polynomials orthogonal with respect to the other abstract we expand the chebyshev polynomials and some of its linear combination in linear combinations of the q hermite the rogers q utraspherical and the al salam chihara polynomials and vice versa we use these expansions to obtain expansions of some densities including q normal and some related to it in infinite series constructed of the products of the other density times polynomials orthogonal to it allowing deeper analysis and discovering new properties on the way we find an easy proof of expansion of the poisson mehler kernel as well as its reciprocal we also formulate simple rule relating one set of orthogonal polynomials to the other given the properties of the ratio of the respective densities of measures orthogonalizing these polynomials sets

the symmetric meixner pollaczek polynomials with real parameter abstract the symmetric meixner pollaczek polynomials p n x 2 2 for 0 are well studied polynomials these are polynomials orthogonal on the real line with respect to a continuous positive real measure for 0 p n x 2 2 are also polynomials however they are not orthogonal on the real line with respect to any real measure this paper defines a non standard inner product with respect to which the polynomials p n x 2 2 for 0 become orthogonal polynomials it examines the major properties of the polynomials p n x 2 2 for 0 which are also shared by the polynomials p n x 2 2 for 0

unimodular equivalence of graphs abstract let g be a simple graph with n vertices and m edges denote by q the vertex edge incidence matrix corresponding to some orientation of the edges of g define l g qqt and k g qtq then l g is the so called laplacian matrix and k g its edge version viewed as integer matrices the smith normal form of l g is complicated but the smith normal form of k g for connected graphs is always i n 2 n 0 m n 1 an edge version of the matrix tree theorem is used in the proof there is an application to digraph flows with constant resultants over abelian groups finally if l g and l h are congruent then g and h have the same chromatic polynomial

periodical solutions in a pulse modulated model of endocrine regulation with time delay a hybrid mathematical model of endocrine regulation obtained by augmenting the classical continuous smith model with a pulse modulated feedback to describe episodic pulsatile secretion is considered conditions for existence and local orbital stability of periodical solutions with m impulses in the least period m cycles are derived an important implication of the performed analysis is that the nonlinear dynamics of the pulse modulated system and not the delay itself cause the sustained closed loop oscillations furthermore simulation and bifurcation analysis indicate that increasing the time delay in the system in hand typically but not always leads to less complex dynamic pattern in the closed loop system by giving rise to stable cycles of lower periodicity

alternation points and bivariate lagrange interpolation abstract given m 1 strictly decreasing numbers h 0 h 1 h m we give an algorithm to construct a corresponding finite sequence of orthogonal polynomials p 0 p 1 p m such that p 0 1 p j has degree j and p m j h n 1 n p j h n for all j n 0 1 m using these polynomials we construct bivariate lagrange polynomials and cubature formulas for nodes that are points in r 2 where the coordinates are taken from given finite decreasing sequences of the same length and where the indices have the same or opposite parity

eigenvalue location in threshold graphs abstract let g be a threshold graph of order n with adjacency matrix a we present an o n algorithm for constructing a diagonal matrix congruent to b x a x i for any x an application using sylvester s law of inertia can determine how many eigenvalues lie in an interval allowing efficient approximation we prove that eigenvalues of threshold graphs other than 0 or 1 are simple we give the spectrum for threshold graphs g k j represented by 0 k 1 j when n 3 g n n 3 n 3 has the minimum eigenvalue min n among threshold graphs of order n and a formula for min n is given there is one more graph if n 2 mod 3

orthogonal matrix polynomials and higher order recurrence relations abstract it is well known that orthogonal polynomials on the real line satisfy a three term recurrence relation and conversely every system of polynomials satisfying a three term recurrence relation is orthogonal with respect to some positive borel measure on the real line we extend this result and show that every system of polynomials satisfying some 2 n 1 term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients are n n matrices we apply this result to polynomials orthogonal with respect to a discrete sobolev inner product and other inner products in the linear space of polynomials as an application we give a short proof of krein s characterization of orthogonal polynomials with a spectrum having a finite number of accumulation points

on the minors of an incidence matrix and its smith normal form abstract consider the vertex edge incidence matrix of an arbitrary undirected loopless graph we completely determine the possible minors of such a matrix these depend on the maximum number of vertex disjoint odd cycles i e the odd tulgeity of the graph the problem of determining this number is shown to be np hard turning to maximal minors we determine the rank of the incidence matrix this depends on the number of components of the graph containing no odd cycle we then determine the maximum and minimum absolute values of the maximal minors of the incidence matrix as well as its smith normal form these results are used to obtain sufficient conditions for relaxing the integrality constraints in integer linear programming problems related to undirected graphs finally we give a sufficient condition for a system of equations whose coefficient matrix is an incidence matrix to admit an integer solution

linear time lup decomposition of forest like matrices a square matrix m has an lup decomposition if it can be written as lup where l is a unit lower triangular matrix u is an upper triangular matrix and p is a permutation matrix we present a linear time algorithm for finding an lup decomposition for a nonsingular neighborhood matrix of a tree we also identify a more general class of matrices we call forest like for which the algorithm can be modified