invariant manifolds for impulsive equations and nonuniform polynomial dichotomies for impulsive differential equations in banach spaces we construct stable and unstable invariant manifolds for sufficiently small perturbations of a polynomial dichotomy we also consider the general case of nonuniform polynomial dichotomies moreover we introduce the notions of polynomial lyapunov exponent and of regularity coefficient for a linear impulsive differential equation and we show that when the lyapunov exponent never vanishes the linear equation admits a nonuniform polynomial dichotomy

exit problems associated with affine reflection groups we obtain a formula for the distribution of the first exit time of brownian motion from the alcove of an affine weyl group in most cases the formula is expressed compactly in terms of pfaffians expected exit times are derived in the type widetilde a case the results extend to other markov processes we also give formulas for the real eigenfunctions of the dirichlet and neumann laplacians on alcoves observing that the hot spots conjecture of j rauch is true for alcoves

multivariate rational approximants for multiclass closed queuing networks closed markovian networks of queues with multiclass customers and having a product form equilibrium state probability distribution are useful in the performance evaluation and design of computer and telecommunication systems therefore the efficient computation of the normalizing function the key element of the solution in product form has attracted considerable effort we consider a network that consists of one infinite server is station and one processor sharing ps or fcfs single server station we use multivariate newton pade approximants computed from data for small numbers of customers in each class to estimate the normalizing function for a larger population in the network the effectiveness and tremendous gain in computing time of this procedure are illustrated through various numerical experiments

the homotopy perturbation method for discontinued problems arising in nanotechnology continuum hypothesis on nanoscales is invalid and a differential difference model is considered as an alternative approach to describing discontinued problems this paper applies the homotopy perturbation method to a nonlinear differential difference equation arising in nanotechnology comparison of the approximate solution with the exact one reveals that the method is very effective

schur subalgebras and an application to the symmetric group abstract let k be an infinite field of prime characteristic p and let d r be positive integers of the same parity satisfying a certain congruence condition we prove that the schur algebra s 2 d is isomorphic to a subalgebra of the form es 2 r e where e is a certain idempotent of s 2 r translating this result via ringel duality to the symmetric groups d and r we obtain lattice isomorphisms between specht modules between young modules and between permutation modules here modules labelled by the partitions r k k correspond to modules labelled by d k k this provides a representation theoretical interpretation for part of the fractal structures observed for the decomposition numbers of the symmetric groups corresponding to two part partitions

a solution to an open problem for wave equations with generalized wentzell boundary conditions in this paper we solve an open problem put forward by a favini g r goldstein j a goldstein and s romanelli concerning the mixed problem for wave equations with generalized wentzell boundary conditions as a consequence we also develop the previous wellposedness result regarding the mixed problem for heat equations with generalized wentzell boundary conditions

discrete exact solutions to some nonlinear differential difference equations via the g g expansion method we extended the g g expansion method to two well known nonlinear differential difference equations the discrete nonlinear schrodinger equation and the toda lattice equation for constructing traveling wave solutions discrete soliton and periodic wave solutions with more arbitrary parameters as well as discrete rational wave solutions are revealed it seems that the utilized method can provide highly accurate discrete exact solutions to nddes arising in applied mathematical and physical sciences

on the classification of clifford algebras and their relation to spinors in n dimensions a classification of all the clifford algebras is given in terms of kronecker products of the quaternion and dihedral groups the relationship to spinors in n dimensions is explicitly determined we show that the real clifford algebra in minkowski spacetime is distinct from both the algebra of dirac matrices and the algebra of majorana matrices and cannot be realized by the spinor framework the matrix representations of clifford algebras are discussed and are utilized to give a classification of the real forms of lie algebras we are thus able to relate clifford lie and spinor algebras in an intrinsic geometrical setting

on multivariate lagrange interpolation lagrange interpolation by polynomials in several variables is studied through a finite difference approach we establish an interpolation formula analogous to that of newton and a remainder formula both of them in terms of finite differences we prove that the finite difference admits an integral representation involving simplex spline functions in particular this provides a remainder formula for lagrange interpolation of degree n of a function f which is a sum of integrals of certain n 1 st directional derivatives of f multiplied by simplex spline functions we also provide two algorithms for the computation of lagrange interpolants which use only addition scalar multiplication and point evaluation of polynomials

generic formulas for the values at the singular points of some special monic classical hq ω orthogonal polynomials it is well known that the classical orthogonal polynomials of jacobi bessel laguerre and hermite are solutions of a sturm liouville problem of the type s x y n email protected x y n email protected ny n 0 where s and t are polynomials such that email protected 2 and email protected 1 and l n is a constant independent of x recently based on the hypergeometric character of the solutions of this differential equation w koepf and m masjed jamei a generic formula for the values at the boundary points of monic classical orthogonal polynomials j comput appl math 191 2006 98 105 found a generic formula only in terms of the coefficients of s and t for the values of the classical orthogonal polynomials at the singular points of the above differential hypergeometric equation in this paper we generalize the mentioned result giving the analogous formulas for both the classical q orthogonal polynomials of the q hahn tableau and the classical d w orthogonal polynomials both are special cases of the classical h q w orthogonal polynomials which are solutions of the hypergeometric type difference equation s x h q wh 1 q wy email protected x h q wy email protected ny n 0 where h q w is the difference operator introduced by hahn and s t and l n being as above our approach is algebraic and it does not require hypergeometric functions

classes of representable disjoint np pairs for a propositional proof system p we introduce the complexity class dnpp p of all disjoint np pairs for which the disjointness of the pair is efficiently provable in the proof system p we exhibit structural properties of proof systems which make canonical np pairs associated with these proof systems hard or complete for dnpp p moreover we demonstrate that non equivalent proof systems can have equivalent canonical pairs and that depending on the properties of the proof systems different scenarios for dnpp p and the reductions between the canonical pairs exist

computing the region of convergence for power series in many real variables a ratio like test abstract we give an elementary proof that the region of convergence for a power series in many real variables is a star convex domain but not in general a convex domain in doing so we deduce a natural higher dimensional analog of the so called ratio test from univariate power series from the constructive proof of this result we arrive at a method to approximate the region of convergence up to a desired accuracy while most results in the literature are for rather specialized classes of multivariate power series the method devised here is general as far as applications are concerned note that while theorems such as the cauchy kowalevski theorem and its generalizations to many variables grant the existence of a region of convergence for a multivariate taylor series to certain pdes under appropriate restrictions they do not give the actual region of convergence the determination of the maximal region of convergence for such a series solution to a pde is one application of our result

real representations of finite clifford algebras i classification a classification of real matrix irreducible representations of finite dimensional real clifford algebras has been made in contrast to the case of complex representation three distinct types of representations can be obtained which we call normal almost complex and quaternionic the dimension of the latter two cases is twice as large as that of the normal representation a criteria for a given clifford algebra to possess a particular type of the representations is also given with some applications

a nonterminating 8 φ 7 summation for the root system c r using multiple q integrals and a determinant evaluation we establish a nonterminating 8 7 summation for the root system cr we also give some important specializations explicitly

infinite horizon backward stochastic differential equations and elliptic equations in hilbert spaces solutions of semilinear elliptic differential equations in infinite dimensional spaces are obtained by means of forward and backward infinite dimensional stochastic evolution equations the backward equation is considered on an infinite time horizon and a suitable growth condition replaces the final condition elliptic equations are intended in a mild sense suitable also for applications to optimal control we finally notice that due to the lack of smoothing properties the elliptic partial differential equation considered here could not be treated by analytic methods

convergence analysis and adaptive strategy for the certified quadrature over a set defined by inequalities this paper investigates the sufficient conditions for the asymptotic convergence of a generic branch and prune algorithm dedicated to the verified quadrature of a function in several variables quadrature over domains defined by inequalities and adaptive meshing strategies are in the scope of this analysis the framework is instantiated using certified quadrature methods based on taylor models i e taylor approximations with rigorously bounded remainder and reported experiments confirmed the analysis they also show that the performances of the instantiated algorithm are comparable with current methods for certified quadrature

finite geometries and clifford algebras ii the pleasant incidence properties of m dimensional projective geometry enable us upon employing the finite geometry over the field rm f rm 2 to deal nicely with certain commutativity anti commutativity aspects of clifford algebras cl o d in dimensions d 2m 1 1 m 2 3

double centralizing theorems for the alternating groups let v n be the n fold tensor product of a vector space v following i schur we consider the action of the symmetric group sn on v n by permuting coordinates in the super z2 graded case v v0 v1 a sign is added these actions give rise to the corresponding schur algebras s sn v here s sn v is compared with s an v the schur algebra corresponding to the alternating subgroup an sn while in the classical signless case these two schur algebras are the same for n large enough it is proved that in the super case where dim v0 dim v1 s an v is isomorphic to the cross product algebra s an v s sn v z2

general order newton pade approximants for multivariate functions pade approximants are a frequently used tool for the solution of mathematical problems one of the main drawbacks of their use for multivariate functions is the calculation of the derivatives off x 1 x p therefore multivariate newton pade approximants are introduced their computation will only use the value off at some points in sect 1 we shall repeat the univariate newton pade approximation problem which is a rational hermite interpolation problem in sect 2 we sketch some problems that can arise when dealing with multivariate interpolation in sect 3 we define multivariate divided differences and prove some lemmas that will be useful tools for the introduction of multivariate newton pade approximants in sect 4 a numerical example is given in sect 5 together with the proof that forp 1 the classical newton pade approximants for a univariate function are obtained

realization extension and classification of certain physically important groups and algebras an associative algebra of differential forms with division has been constructed the algebra of forms in each different space provides a practical realization of the universal clifford algebra of that space a classification of all such algebras is given in terms of two distinct types of algebras nk and sk the former include the dihedral quaternion and majorana algebras the latter include the complex spinor and dirac algebras the associative product expresses hodge duality as multiplication by a basis element this makes possible the realization of higher order algebras in a calculationally useful algebraic setting the fact that the associative algebras as well as the enveloped lie algebras are precisely those arising in physics suggests that this formalism may be a convenient setting for the formulation of basic physical laws

zeros of jones polynomials for families of knots and links abstract we calculate jones polynomials v l t for several families of alternating knots and links by computing the tutte polynomials t g x y for the associated graphs g and then obtaining v l t as a special case of the tutte polynomial for each of these families we determine the zeros of the jones polynomial including the accumulation set in the limit of infinitely many crossings a discussion is also given of the calculation of jones polynomials for non alternating links

coding the principal character formula for affine kac moody lie algebras in this paper an algorithm for computing the principal character for affine lie algebras is discussed and presented the principal characters discovered using this program are given and or proven results include level 2 and 3 character formulas in a2n 1 2 and the sole existence of the rogers ramanujan products in a1 1 a2 1 a2 2 c3 1 f4 1 g2 1 a7 2

robust nonuniform dichotomies and parameter dependence abstract we establish the robustness of linear cocycles with an exponential dichotomy under sufficiently small lipschitz perturbations in the sense that the existence of an exponential dichotomy for a given cocycle persists under these perturbations we consider cocycles in banach spaces as well as the general case of nonuniform exponential dichotomies and also the general case of an exponential behavior e c n given by an arbitrary sequence n including the usual exponential behavior n n as a very special case moreover we show that the projections of the exponential dichotomies obtained from the perturbation vary continuously with the parameter and in fact that they are locally lipschitz on finite dimensional parameters

modular invariant of quantum tori the quantum modular invariant j qt of 2r is defined as a discontin uous pgl2 z invariant multi valued map using the distance to the nearest integer function k k for 2 q it is shown that j qt 1 and for quadratic irrationalities pari gp experiments suggest that j qt is a finite set in the case of the golden mean we produce explicit formulas involving weighted versions of the rogers ramanujan functions for the experimental supremum and infimum we then define a universal modular invariant j a mod c as a continuous and single valued map of ultrasolenoids such that 1 the classical modular invariant is a quotient of the re striction of j to a subsolenoid mod cl a mod fibering over the classical moduli space of elliptic curves and 2 the quantum modular invariant is a quotient of the restric tion of j to a subsolenoid mod qt a mod fibering over the moduli space of elliptic curves equipped with a kronecker foliation

an algebraic and geometric approach to non bijective quadratic transformations this work is concerned with an extension of the so called levi civita and kustaanheimo stiefel transformations the extension is achieved along two lines firstly the authors examine the latter transformations in other dimensions than the ones originally considered by on the one hand levi civita and on the other hand by kustaanheimo and stiefel secondly they pass from the compact to the non compact case this leads to quadratic non bijective transformations that they refer to as hurwitz or kustaanheimo stiefel like and quasihurwitz or levi civita like transformations the hurwitz and quasihurwitz transformations are introduced and studied in an algebraic framework which relies on the use of eight dimensional cayley dickson algebras an explicit formulation of the hurwitz and quasihurwitz transformations is also given in terms of clifford algebras the hurwitz transformations are investigated from a geometrical viewpoint indeed they are connected to hopf and pseudohopf fibrations finally some differential aspects of the hurwitz and to a lesser extent quasihurwitz transformations are developed in view of future physical applications

invariant measures and regularity properties of perturbed ornstein uhlenbeck semigroups abstract this paper deals with perturbations of the ornstein uhlenbeck operator on l 2 spaces with respect to a gaussian measure we perturb the generator of the ornstein uhlenbeck semigroup by a certain unbounded non linear drift and show various properties of the perturbed semigroup such as compactness and positivity strong feller property existence and uniqueness of an invariant measure are discussed as well

electromagnetism and the holomorphic properties of spacetime the cauchy riemann equations of holomorphy are extended to fields in higher dimensional spaces in a framework of clifford algebras the equations of holomorphy in minkowski spacetime turn out to be the maxwell equations in vacuum the lorentz gauge condition is a result of the holomorphy sources can be included in an extension of the residue theorem where charges correspond to the residues

trace representation of some generalized cyclotomic sequences of length pq this paper contributes to trace representation of some generalized cyclotomic sequences of length pq p qprime which are defined by ding and helleseth from the relations between these sequences and the legendre sequence we firstly confirm the defining pairs of these sequences of arbitrary order then we obtain their trace representation from which we give their linear complexity using key s method it can be seen that bai et al s conclusion is a special case of our result when the order is two finally an example is given to illustrate the validity of our result

the relationship between finite groups and clifford algebras clifford algebras are traditionally realized in terms of a specific set of representation matrices this paper provides a more effective alternative by giving the finite group associated with each clifford algebra all the representation independent algebraic results which are really direct consequences of the underlying group structure can thus be derived in an easier and more general manner there are five related but distinct classes of finite groups associated with the clifford algebras these groups are constructed from the complex cyclic quaternion and dihedral groups in a way which is discussed here in detail of particular utility is a table which lists the order structure of each group this permits the immediate identification of any clifford algebra in any dimension

evaluating higher derivative tensors by forward propagation of univariate taylor series this article considers the problem of evaluating all pure and mixed partial derivatives of some vector function defined by an evaluation procedure the natural approach to evaluating derivative tensors might appear to be their recursive calculation in the usual forward mode of computational differentiation however with the approach presented in this article much simpler data access patterns and similar or lower computational counts can be achieved through propagating a family of univariate taylor series of a suitable degree it is applicable for arbitrary orders of derivatives also it is possible to calculate derivatives only in some directions instead of the full derivative tensor explicit formulas for all tensor entries as well as estimates for the corresponding computational complexities are given

combining hook length formulas and bg ranks for partitions via the littlewood decomposition recently the first author has studied hook length formulas for partitions in a systematic manner in the present paper we show that most of those hook length formulas can be generalized and include more variables via the littlewood decomposition which maps each partition to its t core and t quotient in the case t 2 we ob tain new formulas by combining hook lengths and bg ranks introduced by berkovich and garvan as applications we list several multivari able generalizations of classical and new hook length formulas including the nekrasov okounkov the han carde loubert potechin sanborn the bessenrodt bacher manivel the okada panova and the stanley panova formulas summary

lattices on simplicial partitions which are not simply connected in this paper d 1 pencil lattices on simplicial partitions in r d which are not simply connected are studied it is shown how the fact that a partition is not simply connected can be used to increase the flexibility of a lattice a local modification algorithm is developed also to deal with slight partition topology changes that may appear afterwards a lattice has already been constructed

the exponential mapping in clifford algebras the exponential mapping takes the lie algebra of the lorentz group into the lorentz group each element of the group is defined as a formal power series while the product of two exponential elements usually involves an infinite sum of commutator terms such as in the baker campbell hausdorff formula because of the special arithmetic in clifford algebras many baker campbell hausdorff like formulas and identities can be calculated or summed exactly when they involve only elements from the algebra we calculate exact identities for the baker campbell hausdorff formula and related formulas in the quaternion and dihedral algebras these are useful in treatments of the lorentz group and make possible a truly finite as opposed to infinitesimal description of a transformation group in physics

n dimensional spinors their properties in terms of finite groups a classification scheme is presented for the finite multiplicative group generated by the gamma matrices associated with a given clifford algebra this group reflects the periodicities observed in n dimensional spinors and its representations and other properties are studied thus highlighting the dependence on the number of spacelike and timelike vectors the reality of the representations is examined and tabulated application is made to the imposition of majorana and weyl conditions

a general method of generating and classifying clifford algebras a general construction program is given for the generation of higher order clifford algebras of various signatures together with their faithful representations in terms of pauli type operators the analysis is based upon several isomorphism theorems that provide a simpler understanding of the standard classification scheme of these algebras and an improved signature index notation is suggested that identifies each of the five types of clifford algebras

unitary representations of the 4 1 de sitter group on irreducible representation spaces of the poincare group the construction of the principal continuous series of unitary representations of the simply connected covering group of the 4 1 de sitter group on unitary irreducible representation spaces of the poincare group is presented a unitary irreducible representation space of this covering group of the de sitter group is realized as the direct sum of two irreducible representation spaces of the poincare group possible physical implications are indicated in particular an interpretation of the instantaneous velocity operator in the dirac theory as the spin part of the de sitter boosts is given we obtain a simple method of computing the matrix elements of the generators of the de sitter group in an so 4 basis using the matrix elements of the generators of the four dimensional euclidean group also we obtain explicit expressions for certain matrix elements between the spinor and so 4 basis of the representation space as functions on the coset space so 4 so 3

the complete reduction of clifford algebras the complete reduction of simple clifford algebras is spanned by matrix bases considered as generators of an algebra the basis elements pij satisfy the product relations pijpkl jkpil it follows that subsets of pij with subscript or superscript fixed span minimal left or right ideals the expression of the basis elements pij in the usual multivector basis is derived from isomorphic matrix algebras spinors and hermitian conjugate spinors are defined as members of minimal left and right ideals spinor scalar products are defined these scalar products are in fact scalar multiples of an idempotent

how well can the concept of pade approximant be generalized to the multivariate case what we know about multivariate pade approximation has been developed in the last 25 years in the next sections we compare and discuss many of these results it will become clear that simple properties or requirements such as the uniqueness of the pade approximant and consequently its consistency property can play a crucial role in the development of the multivariate theory a separate section is devoted to a discussion of the convergence properties at the end we include an extensive reference list on the topic

generation of localization in a discrete chain with periodic boundary conditions numerical and analytical results we study the generation of localization in a discrete chain composed of n subsystems and periodic boundary conditions strongly localized motions are studied that is time periodic motions where nearly all of the energy is spatially confined to a single subsystem for varying n numerical results indicate that the strongly localized solutions are generated through a bifurcation from an in phase spatially extended solution however in the limit as n rightarrow infty the bifurcation point tends to infinity and a smooth transition from localization to nonlocalization occurs we then present an analytic technique to complement the numerical results it is based on the matching local asymptotic expansions of a solution branch using pade approximants this leads to global analytic representations of the considered solutions valid over the entire range of the control parameter

the complexity of weighted boolean csp with mixed signs we give a complexity dichotomy for the problem of computing the partition function of a weighted boolean constraint satisfaction problem such a problem is parameterized by a set c of rational valued functions which generalize constraints each function assigns a weight to every assignment to a set of boolean variables our dichotomy extends previous work in which the weight functions were restricted to being non negative we represent a weight function as a product of the form 1 sg where the polynomial s determines the sign of the weight and the non negative function g determines its magnitude we show that the problem of computing the partition function the sum of the weights of all possible variable assignments is in polynomial time if either every function in c can be defined by a pure affine magnitude with a quadratic sign polynomial or every function can be defined by a magnitude of product type with a linear sign polynomial in all other cases computing the partition function is fp p complete

the ablowitz ladik lattice system by means of the extended g g expansion method abstract we analyzed the ablowitz ladik lattice system by using the extended g g expansion method further discrete soliton and periodic wave solutions with more arbitrary parameters are obtained we observed that some previously known results can be recovered by assigning special values to the arbitrary parameters

algebras with three anticommuting elements i spinors and quaternions a general construction of alternative algebras with three anticommuting elements and a unit is given as an exhaustive result over the real and complex fields we obtain the clifford algebras h quaternions n1 dihedral clifford algebra which is related to real 2 spinors and s1 algebra of pauli matrices which is related to complex 2 spinors what is important is that the algebras n1 and s1 possess inverses everywhere except on a region akin to the light cone of the minkowski space while the quaternion algebra h has inverses everywhere except at the zero element we discuss the reasons why the three algebras n1 h and s1 are so difficult to distinguish in the representation space of 2 2 complex matrices

dirac general covariance and tetrads i clifford and lie bundles and torsion the isomorphic map of clifford and lie bundles to arbitrary coordinate atlases using a global orthonormal tetrad field on a parallelizable space time is used to construct a fully covariant dirac spinor theory the klein gordon equation exhibits a natural spin torsion coupling of the einstein cartan form the torsion coming from the tetrad field the tetrad connection coefficients are explicitly derived in addition to their relationship to the usual levi civita coefficients various topological conditions for vanishing torsion are given the dirac and adjoint dirac equations are obtained from a simple lagrangian and the structure of the adjoint equation is discussed

new combinatorial interpretations of ramanujan s partition congruences mod 5 7 and 11 soit p n le nombre de partitions non restreintes de n on donne de nouvelles interpretations combinatoires des congruences p 5n 4 p 7n 5 et p 11n 6 0 mod 5 7 et 11 respectivement

multivariate polynomial interpolation in newton forms techniques of univariate newton interpolating polynomials are extended to multivariate data points by different generalizations and practical algorithms the newton basis format with divided diffe

a taylor series method for the solution of the linear initial boundary value problems for partial differential equations in this paper a numerical method for solving the linear initial problems for partial differential equations with constant coefficients and analytic initial conditions in two and three independent variables is presented the technique is based upon the taylor series expansion properties and the operational matrices for partial derivatives for the taylor series in two and three variables are first presented these properties are then used to reduce the solution of partial differential equations in two and three independent variables to a system of algebraic equations the procedure can be extended to linear partial differential equations with more independent variables the taylor series may not converge if the solution is not analytic in the whole domain however the present method can be applied to boundary value problems for linear partial differential equations when the solution is analytic in the interior of the domain and also on some open subsets for each distinct part of the boundary the method is computationally very attractive and applications are demonstrated through illustrative examples

a de giorgi nash type theorem for time fractional diffusion equations we study the regularity of weak solutions to linear time fractional diffusion equations in divergence form of arbitrary time order alpha in 0 1 the coefficients are merely assumed to be bounded and measurable and they satisfy a uniform parabolicity condition our main result is a de giorgi nash type theorem which gives an interior holder estimate for bounded weak solutions in terms of the data and the l infty bound of the solution the proof relies on new a priori estimates for time fractional problems and uses de giorgi s technique and the method of non local growth lemmas which has been introduced recently in the context of nonlocal elliptic equations involving operators like the fractional laplacian

directions for computing truncated multivariate taylor series efficient recurrence relations for computing arbitrary order taylor coefficients for any univariate function can be directly applied to a function of n variables by fixing a direction in r n after a sequence of directions the multivariate taylor coefficients or partial derivatives can be reconstructed or interpolated the sequence of univariate calculations is more efficient than multivariate methods although previous work indicates a space cost for this savings and significant cost for the reconstruction we completely eliminate this space cost and develop a much more efficient algorithm to perform the reconstruction by appropriate choice of directions the reconstruction reduces to a sequence of lagrange polynomial interpolation problems in r n 1 for which a divided difference algorithm computes the coefficients of a newton form another algorithm collects like terms from the newton form and returns the desired multivariate coefficients

some inequalities for the q beta and the q gamma functions via some q integral inequalities some new inequalities for the q gamma the q beta and the q analogue of the psi functions are established via some q integral inequalities

matrix solutions of wave equations and clifford algebras we are extending the formation of matrix solutions un for linear and nonlinear wave equations by construction of unitary anti hermitian anti commuting matrices up to the eighth order we use clifford algebras c 0 n with periodicity in modulo 8 to construct coupled matrix solutions we also propose to use the matrix solutions for describing the intrinsic rotations of particles

taylor meshless method for solving non linear partial differential equations a true meshless integration free method based on taylor series named taylor meshless method tmm has been proposed recently to solve partial differential equations pdes where the shape functions are high degree polynomials and the discretization concerns only the boundary with high computational efficiency and exponential convergence the tmm has been confirmed to be very robust in linear problems including large scale cases in this paper the tmm and the automatic differentiation ad are combined with the newton method to solve non linear elliptic pdes where the ad is used to compute shape functions in a fast manner the numerical results illustrate that the proposed algorithm is very accurate and efficient in solving non linear elliptic problems

some solutions of the yang mills equations in dimension greater than four the yang mills equations for gauge fields of an arbitrary semisimple lie group g in euclidean spaces rn with dimension n 4 is considered for gauge fields aa we introduce the ansatze which reduce the yang mills equations in rn to a set of nonlinear differential matrix equations rouhani ward equations this permits us to write out the explicit form of cosmological plane wave and other types of solutions

fast algorithms for toeplitz and hankel matrices abstract the paper gives a self contained survey of fast algorithms for solving linear systems of equations with toeplitz or hankel coefficient matrices it is written in the style of a textbook algorithms of levinson type and schur type are discussed their connections with triangular factorizations pade recursions and lanczos methods are demonstrated in the case in which the matrices possess additional symmetry properties split algorithms are designed and their relations to butterfly factorizations are developed

the asymptotic bounds of solutions of linear delay systems abstract our aim is to obtain conditions under which every solutionyof the vector functional differential equation formula can be estimated in terms of a solution of a scalar functional equation formula wherea x is a suitable real positive function andbis a suitable real negative constant moreover ifb x is diagonal matrix we derive more precise estimates ofyin terms of a solution of a vector functional equation formula where the elements ofa x are formed by the absolute values of the corresponding elements ofa x these asymptotic results enable us to obtain sharp estimates of solutions of certain linear delay systems

the lorentz group and the thomas precession ii exact results for the product of two boosts the product of two lorentz boosts in different directions is equal to the product of a pure boost and a spatial rotation to second order the resulting boost is simply the sum of the individual boosts and the rotation is responsible for the thomas precession here the resulting boost and the rotation correction are calculated exactly the calculation of higher order corrections to the sum of two finite lorentz boosts is not usually done and here previous exact results are critically compared the complete expression is to the best of the author s knowledge a new result

sparse interpolation of multivariate rational functions consider the black box interpolation of a sparse n variate rational function f where is the maximum number of terms in either numerator or denominator when numerator and denominator are at most of degree d then the number of possible terms in f is o dn and explodes exponentially as the number of variables increases the complexity of our sparse rational interpolation algorithm does not depend exponentially on n anymore it still depends on d because we densely interpolate univariate auxiliary rational functions of the same degree we remove the exponent n and introduce the sparsity in the complexity by reconstructing the auxiliary function s coefficients via sparse multivariate interpolation the approach is new and builds on the normalization of the rational function s representation our method can be combined with probabilistic and deterministic components from sparse polynomial black box interpolation to suit either an exact or a finite precision computational environment the latter is illustrated with several examples running from exact finite field arithmetic to noisy floating point evaluations in general the performance of our sparse rational black box interpolation depends on the choice of the employed sparse polynomial black box interpolation if the early termination ben or tiwari algorithm is used our method achieves rational interpolation in o d black box evaluations and thus is sensitive to the sparsity of the multivariate f

grassmann calculus pseudoclassical mechanics and geometric algebra a reformulation of grassmann calculus is presented in terms of geometric algebra a unified language for physics based on clifford algebra in this reformulation grassmann generators are replaced by vectors so that every product of generators has a natural geometric interpretation the calculus introduced by berezin the method of second quantization academic new york 1966 is shown to be unnecessary amounting to no more than an algebraic contraction this approach is not only conceptually clearer but it is also computationally more efficient as demonstrated by treatments of the grauss integral and the grassmann fourier transform the reformulation is applied to pseudoclassical mechanics ann phys 104 336 1977 where it is shown to lead to a new concept the multivector lagrangian to illustrate this idea the three dimensional fermi oscillator is reformulated and solved and its symmetry properties discussed as a result a new and highly compact formula for generating super lie algebras is revealed the paper ends with a discussion of quantization outlining a new approach to fermionic path integrals

kernel methods for learning languages this paper studies a novel paradigm for learning formal languages from positive and negative examples which consists of mapping strings to an appropriate high dimensional feature space and learning a separating hyperplane in that space such mappings can often be represented flexibly with string kernels with the additional benefit of computational efficiency the paradigm inspected can thus be viewed as that of using kernel methods for learning languages we initiate the study of the linear separability of automata and languages by examining the rich class of piecewise testable languages we introduce a subsequence feature mapping to a hilbert space and prove that piecewise testable languages are linearly separable in that space the proof makes use of word combinatorial results relating to subsequences we also show that the positive definite symmetric kernel associated to this embedding is a rational kernel and show that it can be computed in quadratic time using general purpose weighted automata algorithms our examination of the linear separability of piecewise testable languages leads us to study the general problem of separability with other finite regular covers we show that all languages linearly separable under a regular finite cover embedding a generalization of the subsequence embedding we use are regular we give a general analysis of the use of support vector machines in combination with kernels to determine a separating hyperplane for languages and study the corresponding learning guarantees our analysis includes several additional linear separability results in abstract settings and partial characterizations for the linear separability of the family of all regular languages

a recursive computation scheme for multivariate rational interpolants we derive here a recursive computation scheme for the rational interpolation method introduced in 7 explicit formulas for these multivariate rational interpolants are repeated in 2 while the recursive algorithm is described in 3 a number of interesting special cases such as the univariate rational interpolation problem and the multivariate pade approximants introduced in 6 and 10 are dealt with in 4 for some of these rational approximants other recursive schemes were described previously finally 5 contains the numerical results the multivariate rational interpolants described here are compared with multivariate polynomial interpolants interpolating branched continued fractions introduced by cuyt and verdonk 8 interpolating branched continued fractions introduced by siemaszko 12 and several multivariate pade approximants 6 3 10 besides the fact that our multivariate rational interpolants allow a large degree of freedom in the choice for the numerator and denominator in ord

pade approximation for stochastic discrete event systems we show that pade approximation can be effectively used for approximation of performance functions in discrete event systems the method is 1 obtaining the maclaurin coefficients of the performance function and 2 finding a pade approximant from the maclaurin coefficients and use it to approximate the function we use the method with the expected number of renewals in a random interval gi g 1 systems and inventory systems the results are very good

wavelets of multiplicity r a multiresolution approximation km m z of l2 r is of multi plicity r 0 if there are r functions x r whose translates form a riesz basis for v in the general theory we derive necessary and sufficient conditions for the translates of x r y x y r to form a riesz basis for v the resulting reconstruction and decomposition sequences lead to the construction of dual bases for v0 and its orthogonal complement w0 in vx the general theory is applied in the construction of spline wavelets with mul tiple knots algorithms for the construction of these wavelets for some special cases are given

approximation of solutions of polynomial partial differential equations in two independent variables abstract a numerical method for solving polynomial partial differential equations in two independent variables defined in the paper is presented the technique is based on polynomial approximation properties and the operational matrices for partial derivatives for a polynomial in two variables are presented first these properties are then used to reduce the solution of partial differential equations in two independent variables to a system of algebraic equations five illustrative examples are presented to prove the effectiveness of the present method results show that the numerical scheme is very convenient for solving polynomial partial differential equations

quasidiagonality of direct sums of weighted shifts let h be a separable hilbert space a bounded linear operator a defined on h is said to be quasidiagonal if there exists a sequence p n of projections of finite rank such that p n i strongly and ap n p n a 0 as n we give a necessary and sufficient condition for a finite direct sum of weighted shifts to be quasidiagonal the condition is stated using a markedgraph a graph with a 0 or attached to its vertices that can be associated with the direct sum

the x ray transform and its inversion for the series expansion basis functions in three dimensional tomography a formula for the three dimensional x ray transform and an expression for its inversion have been derived the spherical harmonics and the rotation matrices in terms of euler angles are used in the generalized fourier series expansion some properties of the transformed fourier components are discussed

an improved sectional anosov closing lemma we prove that all nonwandering points of a sectional anosov flow on a compact 3 manifold can be approximated by periodic points or by points for which the omega limit set is a singularity this improves the closing lemma in morales mich math j 56 1 29 53 2008 we also describe a sectional anosov flow for which the recurrent points are not dense in the nonwandering set

n point pade approximants to real valued stieltjes series with nonzero radii of convergence abstract by employing special continued fractions to two stieltjes series with nonzero radii of convergence we extend the inequalities for one point pade approximants reported by baker 1975 corollory 17 1 to the case of two point pade approximants we prove that some convergents of the continued fractions form a monotone sequences of upper and lower bounds converging uniformly to stieltjes function x1 x on compact subsets of r where r is a radius of convergence of an expansion of x f1 x at x 0 for an illustration of theoretical results we provide nontrivial numerical examples as an application to real physical problems second order pade approximants bounds on the effective conductivity of a square array of cylinders are evaluated

clifford algebra notes on the spinor metric and lorentz poincare and conformal groups a particular normalization for the set of basis elements i of the complex clifford algebrasc p q is motivated and defined by demanding that the physical bispinor densities i i be real this condition referred to here as dirac normalization also necessitates the introduction of the spinor metric and the solution of the metric conditions is given for arbitrary p q when n p q is even the metric is unique and when n is odd there are two distinct metrics then the dirac normalization preserving automorphism group of the basis is explored this is also the group of transformations leaving the spinor metric invariant in particular the physically important cases of the lorentz poincare and conformal groups are sought as subgroups of the automorphism group as expected it is found that the lorentz group is always contained in the automorphism group however it is found that the poincare and conformal groups are contained only in the cases where n is even and q is odd furthermore when n is odd these groups may be found in the full isomorphism group but only for one of the two possible spinor metrics possible physical implications of these results are discussed

algebras with three anticommuting elements ii two algebras over a singular field the clifford algebra generated by the elements 1 with 2 1 is an abelian ring of dimension two with properties analogous to the complex field c the ring has a string of singular inverses and may be regarded as a singular field which circumvents both the fundamental theorem of algebra and the frobenius theorem we construct two associative algebras of dimension four over the clifford algebra 1 and the biquaternions of clifford 2 and demonstrate that both algebras possess inverses everywhere except on a singular region akin to the light cone of the minkowski space matrix representations are discussed as well as the importance of the algebras 1 and 2 in the description of physical vector fields

introduction to automatic differentiation and matlab object oriented programming an introduction to both automatic differentiation and object oriented programming can enrich a numerical analysis course that typically incorporates numerical differentiation and basic matlab computation automatic differentiation consists of exact algorithms on floating point arguments this implementation overloads standard elementary operators and functions in matlab with a derivative rule in addition to the function value for example sin u will also compute cos u ast u prime where u and u prime are numerical values these methods are mostly one line programs that operate on a class of value and derivative objects providing a simple example of object oriented programming in matlab using the new as of release 2008a class definition structure the resulting powerful tool computes derivative values and multivariable gradients and is applied to newton s method for root finding in both single and multivariable settings to compute higher order derivatives of a single variable function another class of series objects keeps taylor polynomial coefficients up to some order overloading multiplication on series objects is a combination discrete convolution of coefficients this idea leads to algorithms for other operations and functions on series objects a survey of more advanced topics in automatic differentiation includes an introduction to the reverse mode our implementation is forward mode and considerations in arbitrary order multivariable series computation

partition identities involving gaps and weights we obtain interesting new identities connecting the famous partition functions of euler gauss lebesgue rogers ramanujan and others by attaching weights to the gaps between parts the weights are in general multiplicative some identities involve powers of 2 as weights and yield combinatorial information about some remarkable partition congruences modulo powers of 2

central limit theorem for products of random matrices using the semigroup product formula of p chernoff a central limit theorem is derived for products of random matrices applications are presented for representations of solutions to linear systems of stochastic differential equations and to the corresponding partial differential evolution equations included is a discussion of stochastic semigroups and a stochastic version of the lie trotter product formula

on the q analogues of some transformations of nearly poised hypergeometric series a number of transformation formulas for very well poised basic hypergeometric series have been obtained which in the limit q 1 approach the known transformation formulas for nearly poised ordinary hypergeometric series

on modular homology in the boolean algebra iii abstract let be a set ra ring of characteristicp 0 and denote bymkther module withk element subsets of as basis theset inclusion map mk mk 1is the homomorphism which associates to ak element subset the sum 1 2 kof all its k 1 element subsets i in this paper we study the chain formula arising from we introduce the notion ofp exactness for a sequence if is infinite we show that isp exact for all prime characteristicsp 0 this result can be extended to various submodules and quotient modules and we give general constructions arising from permutation groups with a finitary section two particular applications are the following the orbit module sequence of such a permutation group on isp exact for every primep and we give a formula for thep rank of the orbit inclusion matrix if the group has finitely many orbits onk element subsets

exponents of nonnegative matrix pairs the notions of primitivity and exponent of a square nonnegative matrix a are classical a is primitive provided there is a nonnegative integer k such that a k is entrywise positive and in the case a is primitive the exponent of a is the smallest such k fornasini and valcher have extended the notion of primitivity to pairs a b of square nonnegative matrices of the same order the pair a b is primitive provided there exist nonnegative integers h and k such that the sum of all products formed by words consisting of ha s and kb s is entrywise positive this paper defines the exponent of a nonnegative matrix pair to be the smallest value of h k over all such h and k it is then shown that the largest exponent of a primitive pair of n by n nonnegative matrices lies in the interval n 3 5n 2 2 3n 3 2n 2 2n 2 in addition the exponent of a pair of nonnegative matrices is related to properties of an associated twodimensional dynamical system 2001 elsevier science inc all rights reserved

on null controllability of linear systems in banach spaces abstract our aim here is to give characterizations for the null controllability of linear systems in general banach spaces the starting point of this paper is the work of pengnian chen and huashu qin systems control lett 45 2002 155 which solves the problem of exact controllability in fact the results of systems control lett 45 2002 155 say that in case of setting in general banach spaces there are the same characterizations of exact controllability as in the case of reflexive banach spaces an open problem raised in systems control lett 45 2002 155 is the validity of a similar result for null controllability we give here a positive answer to this problem however it seems to us that the technique of systems control lett 45 2002 155 does not work for null controllability and consequently our approach is completely different