ldu decomposition of an extension matrix of the pascal matrix abstract let m n denote a set of all elements of weighted lattice paths with weight in the xy plane from 0 0 to m n such that a vertical step v 0 1 a horizontal step h 1 0 and a diagonal step d 1 1 are endowed with weights and respectively and let m n denote the weight of m n defined by m n p m n ega p where p is the product of the weights of all its steps in p a matrix a a ij is called a lattice path matrix with weight if a ij i j for a triple and of real numbers in this paper we present ldu decomposition of lattice path matrices with weight and related properties for every triple and of real numbers and a necessary and sufficient condition in which the symmetric lattice path matrices are positive definite we also investigate the relationship between the lattice path matrices and generalized pascal matrices

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

on homogenization and scaling limit of some gradient perturbations of a massless free field we study the continuum scaling limit of some statistical mechanical models defined by convex hamiltonians which are gradient perturbations of a massless free field by proving a central limit theorem for these models we show that their long distance behavior is identical to a new homogenized continuum massless free field we shall also obtain some new bounds on the 2 point correlation functions of these models

generalized pascal matrix and recurrence sequences this paper gives a product formula of the generalized pascal matrix n x y from this getting the striking simplicity of the powers of n x y furthermore this paper defines the iegp matrix n x y it is shown that not only can n x y be factorized by special summation matrices but also has the closely relation with generalized fibonacci sequences wn x y an explicit diagonalization of n x y is given

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

matrices determined by a linear recurrence relation among entries abstract a matrix a a ij is called a 7 matrix if its entries satisfy the recurrence relation a i 1 j 1 a i 1 j a ij where are fixed numbers a 7 matrix is completely determined by its first row and first column in this paper we determine the structure of 7 matrices and investigate the sequences represented by columns of infinite 7 matrices

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

irreducible tensor description i a classical gas a classical moderately rarefied simple monatomic gas is considered and it is supposed that its behavior is governed by boltzmann s equation then after a brief review of the fundamental properties of the symmetric traceless tensors the irreducible equations of transfer for the relative symmetric traceless moments of the one point distribution function f are systematically derived and related to those of johnston subsequently grad s expansion of the distribution function in terms of reducible three dimensional hermite polynomials which does not fit in together and with the equations of transfer just mentioned is rigorously transformed into its irreducible counterpart fashioned by mathematical apparatus such as one dimensional laguerre polynomials and ikenberry s tensorial harmonics finally some useful conversion formulas between the relative symmetric traceless moments and the tensorial laguerre ikenberry expansion coefficients of f are deduced and the irreducible variant of grad s moment t

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

renormalization group analysis of spectral problems in quantum field theory abstract in this paper we present a self contained and detailed exposition of the new renormalization group technique proposed in 1 2 its main feature is that the renormalization group transformation acts directly on a space of operators rather than on objects such as a propagator the partition function or correlation functions we apply this renormalization transformation to a hamiltonian describing the physics of an atom interacting with the quantized electromagnetic field and we prove that excited atomic states turn into resonances when the coupling between electrons and field is nonvanishing

the boltzmann equation for persistent scattering models abstract the nonlinear boltzmann equation n l b e for the persistent scattering model of futcher and hoare can be completely solved by a straightforward application of the fourier transform method for general maxwell models it then follows i that the n l b e for this model possesses the bobylev krook wu mode as a special similarity solution ii that all solutions lying inside a certain hilbert space can be given in the form of an expansion in eigenfunctions of the linearized boltzmann equation the coefficients of which can be determined recursively iii that the cornille gervois solutions lying outside the above hilbert space can be constructed not only for the persistent scattering model but also for a general class of maxwell models the above results valid for models with a continuous energy variable are generalized to include the discrete persistent scattering model

some identities on catalan numbers and hypergeometric functions via catalan matrix power abstract in this paper we use the catalan matrix power as a tool for deriving identities involving catalan numbers and hypergeometric functions for that purpose we extend earlier investigated relations between the catalan matrix and the pascal matrix by inserting the catalan matrix power and particulary the squared catalan matrix in those relations we also pay attention to some relations between catalan matrix powers of different degrees which allows us to derive the simplification formula for hypergeometric function 3 f 2 as well as the simplification formula for the product of the catalan number and the hypergeometric function 3 f 2 some identities involving catalan numbers proved by the non matrix approach are also given

green s functions for elliptic and parabolic equations with random coefficients ii this paper is concerned with linear parabolic partial differential equations in divergence form and their discrete analogues it is assumed that the coefficients of the equation are stationary random variables random in both space and time the green s functions for the equations are then random variables regularity properties for expectation values of green s functions are obtained in particular it is shown that the expectation value is a continuously differentiable function in the space variable whose derivatives are bounded by the corresponding derivatives of the green s function for the heat equation similar results are obtained for the related finite difference equations this paper generalises results of a previous paper which considered the case when the coefficients are constant in time but random in space

matrix factorization of the n n shift bell matrix abstract let b n b n k n k 0 be the bell matrix define the n n shift bell matrix p n k by p n k i j b k i 1 k j 1 for i j 1 2 n and k 0 1 2 in this paper matrix factorizations of the n n shift bell matrix and the n n generalized riordan matrix are studied as a result many lower triangular matrices related to bell polynomials can be factorized by the corresponding matrices and some identities are derived from the matrix representations in addition some harmonic number identities are obtained from the riordan array method

symmetric functions and the vandermonde matrix this work deduces the lower and the upper triangular factors of the inverse of the vandermonde matrix using symmetric functions and combinatorial identities the l and u matrices are in turn factored as bidiagonal matrices the elements of the upper triangular matrices in both the vandermonde matrix and its inverse are obtained recursively the particular value x i 1 q qi 1 in the indeterminates of the vandermonde matrix is investigated and it leads to q binomial and q stirling matrices it is also shown that q stirling matrices may be obtained from the pascal matrix

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

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

an optimal variance estimate in stochastic homogenization of discrete elliptic equations we consider a discrete elliptic equation with random coefficients a which to fix ideas are identically distributed and independent from grid point to grid point x in mathbb z d on scales large w r t the grid size i e unity the solution operator is known to behave like the solution operator of a continuous elliptic equation with constant deterministic coefficients these symmetric homogenized coefficients a hom are characterized by xi cdot a hom xi langle left xi nabla phi cdot a xi nabla phi right 0 rangle quad xi in mathbb r d where the random field phi is the unique stationary solution of the corrector problem nabla cdot a xi nabla phi 0 and langle cdot rangle denotes the ensemble average medskip it is known by ergodicity that the above ensemble average of the energy density e xi nabla phi cdot a xi nabla phi which is a stationary random field can be recovered by a system average we quantify this by proving that the variance of a spatial average of e on length scales l is estimated as follows rm var left sum x in mathbb z d eta l x e x right lesssim l d where the averaging function i e sum x in mathbb z d eta l x 1 rm supp eta l subset l l d has to be smooth in the sense that nabla eta l lesssim l 1 in two space dimensions i e d 2 there is a logarithmic correction medskip in other words smooth averages of the energy density e behave like as if e would be independent from grid point to grid point which it is not for d 1 this result is of practical significance since it allows to estimate the error when numerically computing a hom

singularities in the nonisotropic boltzmann equation abstract we consider solutions of the nonlinear boltzmann equation nlbe with nonisotropic singular initial conditions the nlbe is transformed into an integral equation which is solved iteratively the evolution of delta and step singularities in the distribution function during the initial layer is discussed and compared with the isotropic case

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

solution to the nonlinear boltzmann equation for maxwell models for nonisotropic initial conditions the known solution to the spatially homogeneous nonlinear boltzmann equation for maxwell models in a series of laguerre polynomials is extended to include nonisotropic initial conditions existence proofs for a class of solutions are supplied the equations for the generalized nonisotropic laguerre moments are derived in explicit form for two and three dimensional models further it is shown that the ordinary moments satisfy the same set of equations as the hermite polynomial moments

the additive structure of elliptic homogenization one of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions since the latter are nonlocal functions of the former in this paper we address this problem in a new way in the context of linear elliptic equations in divergence form by showing that certain quantities associated to the energy density of solutions are essentially additive as a result we are able to prove quantitative estimates on the weak convergence of the gradients fluxes and energy densities of the first order correctors under blow down which are optimal in both scaling and stochastic integrability the proof of the additivity is a bootstrap argument completing the program initiated in armstrong et al commun math phys 347 2 315 361 2016 using the regularity theory recently developed for stochastic homogenization we reduce the error in additivity as we pass to larger and larger length scales in the second part of the paper we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables in particular we prove that the first order correctors converge in the large scale limit to a variant of the gaussian free field

quantification of ergodicity in stochastic homogenization optimal bounds via spectral gap on glauber dynamics we study quantitatively the effective large scale behavior of discrete elliptic equations on the lattice mathbb z d with random coefficients the theory of stochastic homogenization relates the random stationary and ergodic field of coefficients with a deterministic matrix of effective coefficients this is done via the corrector problem which can be viewed as a highly degenerate elliptic equation on the infinite dimensional space of admissible coefficient fields in this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a spectral gap estimate w r t a glauber dynamics on coefficient fields as it is the case for independent and identically distributed coefficients as a main result we prove an optimal decay in time of the semigroup associated with the corrector problem i e of the generator of the process called random environment as seen from the particle as a corollary we recover existence of stationary correctors in dimensions d 2 and prove new optimal estimates for regularized versions of the corrector in dimensions d ge 2 we also give a self contained proof of a new estimate on the gradient of the parabolic variable coefficient green s function which is a crucial analytic ingredient in our approach as an application of these results we prove the first and optimal estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients

exact solutions of the nonlinear boltzmann equation a review is given of research activities since 1976 on the nonlinear boltzmann equation and related equations of boltzmann type in which several rediscoveries have been made and several conjectures have been disproved subjects are i the bkw solution of the boltzmann equation for maxwell molecules first discovered by krupp in 1967 and the krook wu conjecture concerning the universal significance of the bkw solution for the large v t behavior of the velocity distribution functionf v t ii moment equations and polynomial expansions off v t iii model boltzmann equation for a spatially uniform system of very hard particles that can be solved in closed form for general initial conditions iv for maxwell and non maxwell type molecules there exist solutions of the nonlinear boltzmann equation with algebraic decrease at connections with nonuniqueness and violation of conservation laws v conjectured super h theorem and the bkw solution vi exactly soluble one dimensional boltzmann equation with spatial dependence

generalized pascal matrices generate classes closed under multiplication abstract classes of matrices which are the hadamard product of a fixed lower triangular generating matrix p and any toeplitz matrix are studied these classes are generalizations of the special case when a toeplitz matrix is generated by the vector of powers and p is either also toeplitz or a pascal triangle the matrix p which lead to the class being closed under matrix multiplication is fully characterized explicit formulae for inverses are derived and commutativity of products within each class proven examples using lower triangular matrices with binomial coefficients are given

annealed estimates on the green function we consider a random uniformly elliptic coefficient field a x on the d dimensional integer lattice mathbb z d we are interested in the spatial decay of the quenched elliptic green function g a x y next to stationarity we assume that the spatial correlation of the coefficient field decays sufficiently fast to the effect that a logarithmic sobolev inequality holds for the ensemble langle cdot rangle we prove that all stochastic moments of the first and second mixed derivatives of the green function that is langle nabla x g x y p rangle and langle nabla x nabla y g x y p rangle have the same decay rates in x y gg 1 as for the constant coefficient green function respectively this result relies on and substantially extends the one by delmotte and deuschel probab theory relat fields 133 358 390 2005 which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of g that is langle nabla x g x y 2 rangle and langle nabla x nabla y g x y rangle as an application we are able to obtain optimal estimates on the random part of the homogenization error even for large ellipticity contrast

powerlike decreasing solutions of the boltzmann equation for a maxwell gas we study the homogeneous isotropic nonlinear boltzmann equation for a maxwellian interaction we show that solutions decreasing like inverse powers of the energy are physically acceptable both in the linearized and the quadratic problem because all moments may not exist we introduce a generalized generating function and a finite differential system for generalized sonine moments is derived these new solutions may lead to small relaxation rates and justify in most cases the linear approximation

riordan matrices in the reciprocation of quadratic polynomials we iterate contractive one degree polynomials with coefficients in the ring k x of formal power series to calculate the reciprocal in k x of a quadratic polynomial doing this we meet the structure of riordan array we interpret certain changes of variable as a riordan array we finish the paper by using our techniques to find new ways to get known formulas for the sum of powers of natural numbers involving stirling and eulerian numbers

on estimating the derivatives of symmetric diffusions in stationary random environment with applications to ϕ interface model we consider diffusions on d or random walks on d in a random environment which is stationary in space and in time and with symmetric and uniformly elliptic coefficients we show existence and holder continuity of second space derivatives and time derivatives for the annealed kernels of such diffusions and give estimates for these derivatives in the case of random walks these estimates are applied to the ginzburg landau interface model

on the averaged green s function of an elliptic equation with random coefficients we consider a divergence form elliptic difference operator on the lattice mathbb z d with a coefficient matrix that is an i i d perturbation of the identity matrix recently bourgain introduced novel techniques from harmonic analysis to prove the convergence of the feshbach schur perturbation series related to the averaged green s function of this model our main contribution is a refinement of bourgain s approach which improves the key decay rate from 2d epsilon to 3d epsilon the optimal decay rate is conjectured to be 3d as an application we derive estimates on higher derivatives of the averaged green s function which go beyond the second derivatives considered by delmotte deuschel and related works

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

a fast eigenvalue algorithm for pascal matrices abstract we present an algorithm that can find all the eigenvalues of an n n symmetric pascal matrices in o n 2 log n operations we take advantage of real symmetry and the pascal structure our scheme consists of an o n 2 log n lanczos tridiagonalization procedure and an o n qr diagonalization method and the fast fourier transform fft

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

on a homogenization problem we study the average green s function of stochastic uniformly elliptic operators of divergence form on zd mathbb z d when the randomness is independent and has small variance we prove regularity of the fourier transform of the self energy the proof relies on the schur complement formula and the analysis of singular integral operators combined with a steinhaus system

lipschitz regularity for elliptic equations with random coefficients we develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients the main result is a large scale l type estimate for the gradient of a solution the estimate is proved with optimal stochastic integrability under a one parameter family of mixing assumptions allowing for very weak mixing with non integrable correlations to very strong mixing for example finite range of dependence we also prove a quenched l2 estimate for the error in homogenization of dirichlet problems the approach is based on subadditive arguments which rely on a variational formulation of general quasilinear divergence form equations

inverting linear combinations of identity and generalized catalan matrices abstract we introduce the notion of the generalized catalan matrix as a kind of lower triangular toeplitz matrix whose nonzero elements involve the generalized catalan numbers inverse of the linear combination of the pascal matrix with the identity matrix is computed in aggarwala and lamoureux 2002 1 in this paper continuing this idea we invert various linear combinations of the generalized catalan matrix with the identity matrix a simple and efficient approach to invert the pascal matrix plus one in terms of the hadamard product of the pascal matrix and appropriate lower triangular toeplitz matrices is considered in yang and liu 2006 14 we derive representations for inverses of linear combinations of the generalized catalan matrix and the identity matrix in terms of the hadamard product which includes the generalized catalan matrix and appropriate lower triangular toeplitz matrix