a combinatorial identity involving gamma function and pochhammer symbol in this paper we establish an identity that relates pochhammer symbol and the ratio of gamma functions the identity is derived using the mellin series representation for the solution of a general algebraic equation

deflation of conjugate gradients with applications to boundary value problems a method for improving the convergence of the standard conjugate gradient method is given this method involves the use of auxiliary subspaces it is shown how such subspaces may be constructed for boundary value problems and an analysis of convergence for second order problems is presented

a block qmr method for computing multiple simultaneous solutions to complex symmetric systems the solution of complex symmetric indefinite systems of equations where multiple solutions are required is considered the quasi minimum residual qmr method ideally suited for these matrices is generalized using the block lanczos algorithm to solve multiple solutions simultaneously this modification alone is shown through numerical examples involving large sparse matrices from finite element discretization of maxwell s equations to accelerate the convergence by a factor almost as great as the number of simultaneous solutions a natural convergence criterion for this method is presented that is shown to be as effective as and easier to compute than the usual equation residual finally a numerical comparison of the classical incomplete cholesky and a variant of the ilu t preconditioners is given showing superior performance by the latter

computation of galerkin double surface integrals in the 3 d boundary element method the galerkin boundary element method bem also known as the method of moments is a powerful method for solving the laplace equation in three dimensions there are advantages to galerkin formulations for integral equations as they treat problems associated with kernel singularity and lead to symmetric and better conditioned matrices however the galerkin method requires the computation of double surface integral over pairs of triangles there are many semianalytical methods to treat these integrals which all have some issues and are discussed in this paper novel methods inspired by the treatment of these kernels in the fast multipole method are presented for computing all the integrals that arise in the galerkin formulation to any accuracy integrals involving completely geometrically separated triangles are nonsingular and are computed using a technique based on spherical harmonics and multipole expansions and translations which require the integration of polynomial functions over the triangles integrals involving cases where the triangles have common vertices edges or are coincident are treated via scaling and symmetry arguments combined with automatic recursive geometric decomposition of the integrals the methods are validated and example results are presented

an inequality for ratios of gamma functions abstract let x denote euler s gamma function the following inequality is proved for y 0 and x 1 we have x y 1 y 1 1 x x y 2 y 1 1 x 1 x y x y 1 the inequality is reversed if 0 x 1 this resolves an open problem of guo and qi b n guo f qi inequalities and monotonicity for the ratio of gamma functions taiwanese j math 7 2003 239 247

null structure and almost optimal local well posedness of the maxwell dirac system we uncover the full null structure of the maxwell dirac system in lorenz gauge this structure which cannot be seen in the individual compo nent equations but only when considering the system as a whole is expressed in terms of tri and quadrilinear integral forms with cancellations measured by the angles between spatial frequencies in the 3d case we prove frequency localized l 2 space time estimates for these integral forms at the scale invariant regularity up to a logarithmic loss hence we obtain almost optimal local well posedness of the system by iteration

almost optimal local well posedness for the 3 1 dimensional maxwell klein gordon equations this paper contains a detailed study of the local in time regularity properties of the maxwell klein gordon mkg equations the mkg equations represent a physical model for the interaction of a spin 0 particle sometimes referred to as a higgs or scalar field with the classical equations of electrodynamics i e maxwell s equations in terms of the calculus of variations the mkg system consists of the euler lagrange equations associated to the density l 4 f f 1 2 d d where f a a is the electro magnetic field associated to the set a of real valued potentials on the minkowski space d 1a is the corresponding covariant derivative with respect to those potentials and r c is a complex scalar field in terms of the field f the euler lagrange equations read f d 1a d d 0 1b

well posedness for a quadratic derivative nonlinear schrodinger system at the critical regularity abstract we consider the cauchy problem for a quadratic derivative nonlinear schrodinger system in sobolev space h s in this paper under the null condition and the mass resonance relation we prove large data local well posedness if the space dimension d 3 and s s c d 2 and s s c or d 1 and s 0 where s c is the scaling critical regularity moreover under the same assumptions we also prove small data global well posedness and scattering in h s if d 3 and s s c our proof is based on a contraction argument using the u p and v p type function spaces

the lanczos algorithm with partial reorthogonalization the lanczos algorithm is becoming accepted as a powerful tool for finding the cigenvalues and for solving linear systems of equations any practical implementation of the algorithm suffers however from roundoff errors which usually cause the lanczos vectors to lose their mutual orthogonality in order to maintain some level of orthogonality full reorthogonalization fro and selective orthogonalization so have been used in the past as a remedy here partial reorthogonalization pro is proposed as a new method for maintain ing semiorthogonality among the lanczos vectors pro is based on a simple recurrence which allows us to monitor the loss of orthogonality among the lanczos vectors directly wvithout computing the inner products based on the information from the recurrence reorthogonalizations occur only when necessary thus substantial savings are made as compared to fro in some numerical examples we apply the lanczos algorithm with pro to the solution of large symmetric systems of linear equations and show that it is a robust and efficient algorithm for maintaining semiorthogonality among the lanczos vectors the results obtained compare favorably with the conjugate gradient method

extending substructure based iterative solvers to multiple load and repeated analyses direct solvers currently dominate commercial finite element structural software but do not scale well in the fine granularity regime targeted by emerging parallel processors substructure based iterative solvers often called also domain decomposition algorithms lend themselves better to parallel processing but must overcome several obstacles before earning their place in general purpose structural analysis programs one such obstacle is the solution of systems with many or repeated right hand sides such systems arise for example in multiple load static analyses and in implicit linear dynamics computations direct solvers are well suited for these problems because after the system matrix has been factored the multiple or repeated solutions can be obtained through relatively inexpensive forward and backward substitutions on the other hand iterative solvers in general are ill suited for these problems because they often must restart from scratch for every different right hand side in this paper we present a methodology for extending the range of applications of domain decomposition methods to problems with multiple or repeated right hand sides basically we formulate the overall problem as a series of minimization problems over k orthogonal and supplementary subspaces and tailor the preconditioned conjugate gradient algorithm to solve them efficiently the resulting solution method is scalable whereas direct factorization schemes and forward and backward substitution algorithms are not we illustrate the proposed methodology with the solution of static and dynamic structural problems and highlight its potential to outperform forward and backward substitutions on parallel computers as an example we show that for a linear structural dynamics problem with 11640 degrees of freedom every time step beyond time step 15 is solved in a single iteration and consumes 1 0 second on a 32 processor ipsc 860 system for the same problem and the same parallel processor a pair of forward backward substitutions at each step consumes 15 0 seconds

local well posedness of the yang mills equation in the temporal gauge below the energy norm abstract we show that the yang mills equation in three dimensions in the temporal gauge is locally well posed in hs for s 3 4 if the hs norm is sufficiently small the temporal gauge is slightly less convenient technically than the more popular coulomb gauge but has the advantage of uniqueness even for large initial data and does not require solving a nonlinear elliptic problem to handle the temporal gauge correctly we project the connection into curl and divergence free components and develop some new bilinear estimates of xs b type which can handle integration in the time direction

local well posedness for low regularity data for the higher dimensional maxwell klein gordon system in lorenz gauge the cauchy problem for the maxwell klein gordon equations in the lorenz gauge in n space dimensions n 4 is shown to be locally well posed for low regularity large data the result relies on the null structure for the main bilinear terms which was shown to be present not only in the coulomb gauge but also in the lorenz gauge by selberg and tesfahun who proved global well posedness for finite energy data in three space dimensions this null structure is combined with product estimates for wave sobolev spaces crucial for the improvement are the solution spaces introduced by klainerman selberg the cauchy problem for the maxwell klein gordon equations in the lorenz gauge in n space dimensions n 4 is shown to be locally well posed for low regularity large data the result relies on the null structure for the main bilinear terms which was shown to be present not only in the coulomb gauge but also in the lorenz gauge by selberg and tesfahun who proved global well posedness for finite energy data in three space dimensions this null structure is combined with product estimates for wave sobolev spaces crucial for the improvement are the solution spaces introduced by klainerman selberg

multilinear weighted convolution of l 2 functions and applications to nonlinear dispersive equations the x s b spaces as used by beals bourgain kenig ponce vega klainerman machedon and others are fundamental tools to study the low regularity behavior of nonlinear dispersive equations it is of particular interest to obtain bilinear or multilinear estimates involving these spaces by plancherel s theorem and duality these estimates reduce to estimating a weighted convolution integral in terms of the l 2 norms of the component functions in this paper we systematically study weighted convolution estimates on l 2 as a consequence we obtain sharp bilinear estimates for the kdv wave and schrodinger x s b spaces

global small solutions to the critical radial dirac equation with potential abstract we solve globally a radial cubic dirac equation perturbed with a small potential with data of small critical norm h 1 the main tool is a new endpoint estimate of the perturbed dirac flow for a class of radial type initial data

solution of dense systems of linear equations in the discrete dipole approximation the discrete dipole approximation dda is a method for calculating the scattering of light by an irregular particle the dda has been used for example in calculations of optical properties of cosmic dust in this method the particle is approximated by interacting electromagnetic dipoles computationally the dda method includes the solution of large dense systems of linear equations where the coefficient matrix is complex symmetric in this work the linear systems of equations are solved by various iterative methods qmr was found to be the best iterative method in this application it converged in only a few more iterations than the full generalized minimal residual gmres method when the discretization of the particle was refined the number of iterations remained constant even without preconditioning the matrix vector product in the iterative methods can be computed with the fast fourier transform or the fast multipole algorithm these algorithms make it feasible to solve dense linear systems of

finite energy solutions of the yang mills equations in mathbb r 3 1 yang mills equations in r3 1 is well posed in the energy norm this means that for an appropriate gauge condition we construct local unique solutions in a time interval which depends only on the size of the energy norm of the data since the energy norm is left invariant by the yang mills flow the local solution is automatically extended to the entire space time thus our results which settle a problem stated in str imply the well known fundamental regularity result of eardley and moncrief e m that result proved in the temporal gauge requires a higher degree of smoothness for the data our main result proved also in the temporal gauge allows us to extend the concept of solutions to arbitrary finite energy initial data the solutions are automatically unique in the class of solutions obtained by our procedure moreover the global regularity proof given by eardley and moncrief depends in an essential way on the specific properties of the fundamental solution of the wave equation in the flat minkowski space time r3 1 namely the strong huygens principle indeed due mainly to this fact their proof does not seem to extend to general curved space times we have reasons to hope that the very different approach we take here will resolve this difficulty the basic ingredients of our method are 1 the introduction of appropriate local coulomb gauges adapted to the causal structure of the equations 2 an appropriate method of localizing the new space time estimates for

an iterative scheme for dense complex symmetric linear systems in acoustics boundary element computations we consider the indirect variational boundary integral formulation for the helmholtz equation in acoustics applications this formulation leads after discretization to a dense complex symmetric system of linear equations which are typically solved by direct elimination methods using the bunch kaufman pivoting strategy for matrices in the symmetric packed storage format as implemented in the lapack library anderson et al lapack users guide siam philadelphia 1995 in this paper we describe an iterative solution procedure which uses an adaptation of the orthogonal factorization method gill and murray in numerical methods for constrained optimization academic press new york 1974 pp 29 65 in conjunction with the qmr iteration freund siam j sci statist comput 13 1992 pp 425 448 this iterative procedure which achieves rapid convergence by exploiting the structure of the matrix generated by the discretization is competitive with the lapack direct elimination approach for small but difficult industrial applications on which it has been tested so far and for larger problems the iterative procedure is expected to provide significant advantages in terms of robustness computational cost and parallelizability

on the lanczos method for solving symmetric linear systems with several right hand sides this paper analyzes a few methods based on the lanczos algorithm for solving large sparse symmetric linear systems with several right hand sides the methods examined are suitable for the situation when the right sides are not too different from one another as is often the case in time dependent or parameter dependent problems we propose a theoretical error bound for the approximation obtained from a projection process onto a krylov subspace generated from processing a previous right hand side

global well posedness of the maxwell dirac system in two space dimensions in recent work grunrock and pecher proved that the dirac klein gordon system in 2d is globally well posed in the charge class data in l 2 for the spinor and in a suitable sobolev space for the scalar field here we obtain the analogous result for the full maxwell dirac system in 2d making use of the null structure of the system found in earlier joint work with damiano foschi we first prove local well posedness in the charge class to extend the solutions globally we build on an idea due to colliander holmer and tzirakis for this we rely on the fact that md is charge subcritical in two space dimensions and make use of the null structure of the maxwell part

a new asymptotic series for the gamma function the famous stirling s formula says that s 1 2 s s e se s 2 s e se s 12s in this paper we obtain a novel convergent asymptotic series of s and proved that s is increasing for s 0

projection techniques for iterative solution of ax b with successive right hand sides we present two projection techniques for computing approximate solutions to linear systems of the form ax n b n for a sequence n 1 2 e g such as arises from time discretization of a partial differential equation the inexpensive approximate solutions can be used as initial guesses for iterative solution of the system resulting in significantly reduced computational expense examples of two and three dimensional incompressible navier stokes calculations are presented in which x represents the pressure and a is a discrete poisson operator in flows containing significant dynamic activity these projection techniques lead to as much as a two fold reduction in solution time