on a connection between the pascal vandermonde and stirling matrices ii this article is an extension of the work presented in on a connection between the pascal vandermonde and stirling matrices i applied mathematics and computation 2003 to appear it gives a full matrix t t for which t 1 is exactly the lower triangular stochastic matrix t given in on a connection between the pascal vandermonde and stirling matrices i applied mathematics and computation 2003 to appear a new strategy to generate the elements of the matrix t t 1 is investigated a new conjecture and three combinatorial identities are also given

explicit inverse of a generalized vandermonde matrix in this paper the author gives an explicit closed form expression for the nxn inverse matrix v g k n 1 of the generalized nxn vandermonde matrix v g k n by using the elementary symmetric functions symbolic and numerical results are presented

the average distances in random graphs with given expected degrees abstract random graph theory is used to examine the small world phenomenon any two strangers are connected through a short chain of mutual acquaintances we will show that for certain families of random graphs with given expected degrees the average distance is almost surely of order log n log d where d is the weighted average of the sum of squares of the expected degrees of particular interest are power law random graphs in which the number of vertices of degree k is proportional to 1 k for some fixed exponent for the case of 3 we prove that the average distance of the power law graphs is almost surely of order log n log d however many internet social and citation networks are power law graphs with exponents in the range 2 3 for which the power law random graphs have average distance almost surely of order log log n but have diameter of order log n provided having some mild constraints for the average distance and maximum degree in particular these graphs contain a dense subgraph which we call the core having nc log log n vertices almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core

the average path length for a class of scale free fractal hierarchical lattices rigorous results abstract with the help of the recurrence relations derived from the self similar structure we obtain the closed form solution for the average path length of a class of scale free fractal hierarchical lattices hls with a general parameter q which are simultaneously scale free self similar and disassortative our rigorous solution shows that the average path length of the hls grows logarithmically as d t n t log 2 q 2 in the infinite limit of network size of n t and that they are not small worlds but grow with a power law relationship to the number of nodes

griffiths singularities and algebraic order in the exact solution of an ising model on a fractal modular network we use an exact renormalization group transformation to study the ising model on a complex network composed of tightly knit communities nested hierarchically with the fractal scaling recently discovered in a variety of real world networks varying the ratio k j of intercommunity to intracommunity couplings we obtain an unusual phase diagram at high temperatures or small k j we have a disordered phase with a griffiths singularity in the free energy due to the presence of rare large clusters which we analyze through the yang lee zeros in the complex magnetic field plane as the temperature is lowered true long range order is not seen but there is a transition to algebraic order where pair correlations have power law decay with distance reminiscent of the xy model the transition is infinite order at small k j and becomes second order above a threshold value k j m the existence of such slowly decaying correlations is unexpected in a fat tailed scale free network where correlations longer than nearest neighbor are typically suppressed

notes on particular symmetric polynomials with applications this paper presents some applications using several properties of three important symmetric polynomials elementary symmetric polynomials complete symmetric polynomials and the power sum symmetric polynomials the applications includes a simple proof of el mikkawy conjecture in m e a el mikkawy appl math comput 146 2003 759 769 and a very easy proof of the newton girard formula in addition a generalization of stirling numbers is obtained

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

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

schrodinger processes with unbounded or singular potentials conditional sanov property abstract this paper is concerned with a class of schrodinger processes with creation and killing with unbounded or singular potentials existence of a solution to the corresponding schrodinger s system construction of the schrodinger s bridge associated with it variational characterization and the conditional sanov property

the exact solution of the mean geodesic distance for vicsek fractals vicsek fractals are one of the most interesting classes of fractals and the study of their structural properties is important in this paper the exact formula for the mean geodesic distance of vicsek fractals is found the quantity is computed precisely through the recurrence relations derived from the self similar structure of the fractals considered the obtained exact solution exhibits that the mean geodesic distance approximately increases as a power law function of the number of nodes with the exponent equal to the reciprocal of the fractal dimension the closed form solution is confirmed by extensive numerical calculations

diffusions with singular drift related to wave functions schrodinger equations are equivalent to pairs of mutually time reversed non linear diffusion equations here the associated diffusion processes with singular drift are constructed under assumptions adopted from the theory of schrodinger operators expressed in terms of a local space time sobolev space

computing elementary symmetric polynomials with a subpolynomial number of multiplications elementary symmetric polynomials s n k are the building blocks of symmetric polynomials in this work we prove that for constant k s s n k modulo composite numbers m p1p2 can be computed using only no 1 multiplications if the coefficients of monomials xi1xi2 xik are allowed to be 1 either mod p1 or mod p2 but not necessarily both to the best of our knowledge no previous result yielded even a sublinear i e n varepsilon 0 varepsilon 1 number of multiplications for similar tasks moreover our algorithm fits in the model of the most restrictive depth 3 arithmetic circuits homogeneous multilinear or the graph model in contrast by a lower bound of nisan and wigderson comput complexity 6 1997 pp 217 234 any homogeneous depth 3 circuit needs size omega n 2k k 2 for computing s n k modulo primes moreover the number of multiplications in our algorithm remains sublinear while k o log log n our results generalize for other nonprime power composite moduli as well the proof uses perfect hashing functions and the famous bbr polynomial of barrington beigel and rudich

large deviations and the propagation of chaos for schrodinger processes schrodinger processes due to schrodinger 1931 the definition of which is given in sect 4 are uniquely characterized by a large deviation principle in terms of the relative entropy with respect to a reference process which is a renormalized diffusion process with creation and killing in applications anapproximate sanov property of a subseta a b is shown wherea a b denotes the set of all probability measures on a path space with prescribed marginal distributions q a qb at finite initial and terminal timesa andb respectively it is shown that there exists the unique markovian modification ofn independent copies of renormalized processes conditioned by the empirical distribution and that the propagation of chaos holds for the system of interacting particles with the schrodinger process as the limiting distribution

analytical solution of average path length for apollonian networks with the help of recursion relations derived from the self similar structure we obtain the solution of average path length overline d t for apollonian networks in contrast to the well known numerical result overline d t ensuremath propto text ln text n t 3 4 j s andrade jr et al phys rev lett 94 018702 2005 our rigorous solution shows that the average path length grows logarithmically as overline d t ensuremath propto text ln text n t in the infinite limit of network size n t the extensive numerical calculations completely agree with our closed form solution

on solving composite power polynomial equations it is well known that a system of power polynomial equations can be reduced to a single variable polynomial equation by exploiting the so called newton s identities in this work by further exploring newton s identities we discover a binomial decomposition rule for composite elementary symmetric polynomials utilizing this decomposition rule we solve three types of systems of composite power polynomial equations by converting each type to single variable polynomial equations that can be solved easily for each type of system we discuss potential applications and characterize the number of nontrivial solutions up to permutations and the complexity of our proposed algorithmic solution

remarks on two symmetric polynomials and some matrices by means of complete symmetric polynomials this paper gives a new proof for the vandermonde determinant formula another alternative proof for this formula is obtained via the collocation matrices it also gives a generalized relationship between the vandermonde the pascal and the stirling matrices a new approach to obtain the explicit inverse of the vandermonde matrix is investigated closed form expressions for some vandermonde related determinants are obtained

diffusion with interactions and collisions between coloured particles and the propagation of chaos one considers a system ofn particles on the real line which are of two different types colours their dynamics is given by a stochastic differential equation with constant diffusion part the drift felt by a particle of either type depends on the empirical measures of type 1 and 2 particles at every instant further a reflection condition is imposed so that particles of different type are not allowed to cross each other the article studies the vlasov mckean limit of the system asn propagation of chaos and an evolution equation for the limiting empirical measures is established from where in particular an equation for the separating front between the two types follos

vertex labeling and routing in expanded apollonian networks we present a family of networks expanded deterministic apollonian networks which are a generalization of the apollonian networks and are simultaneously scale free small world and highly clustered we introduce a labeling of their nodes that allows one to determine the shortest path routing between any two nodes of the network based only on the labels

the limit behavior of elementary symmetric polynomials of i i d random variables when their order tends to infinity let 1 2 be a sequence of i i d random variables and consider the elementary symmetric polynomial s k n of order k k n of the first n elements 1 n of this sequence we are interested in the limit behavior of s k n with an appropriate transformation if k n n a 0 a 1 since k n as n oo the classical methods cannot be applied in this case and new kinds of results appear we solve the problem under some conditions which are satisfied in the generic case the proof is based on the saddlepoint method and a limit theorem for sums of independent random vectors which may have some special interest in itself

on the inversion of the vandermonde matrix abstract the inversion of the vandermonde matrix has received much attention for its role in the solution of some problems of numerical analysis and control theory this work deals with the problem of getting an explicit formula for the generic element of the inverse we derive two algorithms in o n2 and o n3 and compare them with the parker traub and the bjorck pereyra algorithms

universal scaling of distances in complex networks universal scaling of distances between vertices of erd ifmmode mbox h o else h o fi s r enyi random graphs scale free barab asi albert models science collaboration networks biological networks internet autonomous systems and public transport networks are observed a mean distance between two nodes of degrees k i and k j equals to l ij a ensuremath b phantom rule 0 2em 0ex mathrm log k i k j the scaling is valid over several decades a simple theory for the appearance of this scaling is presented parameters a and b depend on the mean value of a node degree k nn calculated for the nearest neighbors and on network clustering coefficients

schrodinger processes and large deviations for a large system of independent diffusing particles each of which is killed at a certain space time dependent rate the conditional distribution of surviving trajectories in a bounded time interval is computed given the approximate form of the initial and final empirical distribution of surviving particles this generalizes a result for the brownian case without killing which was first obtained by schrodinger sitzungsber preuss akad wiss phys math kl 1931 144

degree dependent intervertex separation in complex networks we study the mean length l k of the shortest paths between a vertex of degree k and other vertices in growing networks where correlations are essential in a number of deterministic scale free networks we observe a power law correction to a logarithmic dependence l k a ln n k gamma 1 2 ck gamma 1 n in a wide range of network sizes here n is the number of vertices in the network gamma is the degree distribution exponent and the coefficients a and c depend on a network we compare this law with a corresponding l k dependence obtained for random scale free networks growing through the preferential attachment mechanism in stochastic and deterministic growing trees with an exponential degree distribution we observe a linear dependence on degree l k approximately a ln n ck we compare our findings for growing networks with those for uncorrelated graphs

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

factorial stirling matrix and related combinatorial sequences abstract in this paper some relationships between the stirling matrix the vandermonde matrix the benoulli numbers and the eulerian numbers are studied from a matrix representation of k s n k which will be called the factorial stirling matrix where s n k are the stirling numbers of the second kind as a consequence a number of interesting and useful identities are obtained

transformations of diffusion and schrodinger processes a transformation by means of a new type of multiplicative functionals is given which is a generalization of doob s space time harmonic transformation in the case of arbitrary non harmonic function t x which may vanish on a subset of a b x d the transformation induces an additional singular drift term like in the case of doob s space time harmonic transformation to handle the transformation an integral equation of singular perturbations and a diffusion equation with singular potentials are discussed and the feynman kac theorem is established for a class of singular potentials the transformation is applied to schrodinger processes which are defined following an idea of e schrodinger 1931

percolation in hierarchical scale free nets we study the percolation phase transition in hierarchical scale free nets depending on the method of construction the nets can be fractal or small world the diameter grows either algebraically or logarithmically with the net size assortative or disassortative a measure of the tendency of like degree nodes to be connected to one another or possess various degrees of clustering the percolation phase transition can be analyzed exactly in all these cases due to the self similar structure of the hierarchical nets we find different types of criticality illustrating the crucial effect of other structural properties aside from the scale free degree distribution of the nets

a family of symmetric polynomials of the eigenvalues of a matrix in this paper we consider a family of symmetric polynomials of the eigenvalues of a complex matrix a and find an explicit expression of each member of the family as a polynomial of the entries of a with positive coefficients in the case of a nonnegative matrix one immediately obtains a family of inequalities involving matrix eigenvalues and diagonal entries equivalent forms of some of the obtained results as well as connections with known results and specific applications are also presented in the concluding part of the paper we provide comments and conjecture further inequalities related with nonnegative matrices

inverted berezinskii kosterlitz thouless singularity and high temperature algebraic order in an ising model on a scale free hierarchical lattice small world network we have obtained exact results for the ising model on a hierarchical lattice incorporating three key features characterizing many real world networks a scale free degree distribution a high clustering coefficient and the small world effect by varying the probability p of long range bonds the entire spectrum from an unclustered non small world network to a highly clustered small world system is studied using the self similar structure of the network we obtain analytic expressions for the degree distribution p k and clustering coefficient c for all p as well as the average path length ensuremath ell for p 0 and 1 the ferromagnetic ising model on this network is studied through an exact renormalization group transformation of the quenched bond probability distribution using up to 562 phantom rule 0 2em 0ex 500 renormalized probability bins to represent the distribution for pl0 494 we find power law critical behavior of the magnetization and susceptibility with critical exponents continuously varying with p and exponential decay of correlations away from t c for p ensuremath ge 0 494 in fact where the network exhibits small world character the critical behavior radically changes we find a highly unusual phase transition namely an inverted berezinskii kosterlitz thouless singularity between a low temperature phase with nonzero magnetization and finite correlation length and a high temperature phase with zero magnetization and infinite correlation length with power law decay of correlations throughout the phase approaching t c from below the magnetization and the susceptibility respectively exhibit the singularities of mathrm exp ensuremath c sqrt t c ensuremath t and mathrm exp d sqrt t c ensuremath t with c and d positive constants with long range bond strengths decaying with distance we see a phase transition with power law critical singularities for all p and evaluate an unusually narrow critical region and important corrections to power law behavior that depend on the exponent characterizing the decay of long range interactions