on line learning of smooth functions of a single variable abstract we study the on line learning of classes of functions of a single real variable formed through bounds on various norms of functions derivatives we determine the best bounds obtainable on the worst case sum of squared errors also absolute errors for several such classes we prove upper bounds for these classes of smooth functions for other loss functions and prove upper and lower bounds in terms of the number of trials

a finite difference galerkin formulation for the incompressible navier stokes equations abstract the development of a new computational method for solving the incompressible navier stokes equations in primitive variable form is presented it is found that certain finite difference approximations for these equations can be transformed into an equivalent system which efficiently determines the discrete velocity field and which completely eliminates the pressure two such difference schemes for two dimensional problems are examined and some preliminary numerical results are discussed for the steady driven cavity problem

a quadratic spline based interface quasi reconstruction algorithm for accurate tracking of two phase flows a new quadratic spline based interface quasi reconstruction algorithm is presented which provides an accurate and continuous representation of the interface in a multiphase domain and facilitates the direct estimation of local interfacial curvature the fluid interface in each of the mixed cells is represented by piecewise parabolic curves and an initial discontinuous plic approximation of the interface is progressively converted into a smooth quadratic spline made of these parabolic curves the conversion is achieved by a sequence of predictor corrector operations enforcing function c 0 and derivative c 1 continuity at the cell boundaries using simple analytical expressions for the continuity requirements the efficacy and accuracy of the current algorithm has been demonstrated using standard test cases involving reconstruction of known static interface shapes and dynamically evolving interfaces in prescribed flow situations these benchmark studies illustrate that the present algorithm performs excellently as compared to the other interface reconstruction methods available in literature quadratic rate of error reduction with respect to grid size has been observed in all the cases with curved interface shapes only in situations where the interface geometry is primarily flat the rate of convergence becomes linear with the mesh size the flow algorithm implemented in the current work is designed to accurately balance the pressure gradients with the surface tension force at any location as a consequence it is able to minimize spurious flow currents arising from imperfect normal stress balance at the interface this has been demonstrated through the standard test problem of an inviscid droplet placed in a quiescent medium finally the direct curvature estimation ability of the current algorithm is illustrated through the coupled multiphase flow problem of a deformable air bubble rising through a column of water

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

an unsplit higher order godunov method for scalar conservation laws in multiple dimensions abstract in this paper we develop an unsplit higher order godunov method for scalar conservation laws in two dimensions the method represents an extension for the special case being considered of methods developed by colella and by van leer in our method we begin with a piecewise bilinear representation of the solution on each grid cell a new piecewise bilinear representation at the next time level is then obtained from a two step procedure in the first step a conservative predictor corrector scheme derived from the integral form of the differential equation and characteristic considerations is used to obtain average values of the solution over grid cells at the new time level next these new average values are then used to construct a limited piecewise bilinear profile for each cell at the new time level the resulting method is shown to satisfy a maximum principle for constant coefficient linear advection computational results are presented comparing the new method to colella s method for linear advection the method is also applied to two model problems from porous media flow miscible and immiscible displacement the new scheme provides accurate resolution of sharp fronts without any significant distortion

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

a new family of 5 5 cc 4oc schemes applicable for unsteady navier stokes equations in this paper a new family of implicit compact finite difference schemes for computation of unsteady convection diffusion equation with variable convection coefficient is proposed the schemes which are fourth order accurate in space and second or lower order accurate in time depending on the choice of weighted time average parameter are then applied to unsteady navier stokes system the proposed schemes where transport variable and its first derivatives are carried as the unknowns combine virtues of compact discretization and pade scheme for spatial derivative these schemes which are based on a five point stencil with constant coefficients named as 5 5 constant coefficient 4th order compact 5 5 cc 4oc give rise to a diagonally dominant system of equations and shows higher accuracy and better phase and amplitude error characteristics than some of the standard methods these schemes are capable of using a grid aspect ratio other than unity and are unconditionally stable they efficiently capture both transient and steady solutions of linear and nonlinear convection diffusion equations with dirichlet as well as neumann boundary conditions subsequently the proposed schemes are applied to problems governed by the incompressible navier stokes equations the results obtained are in excellent agreement with analytical and available numerical results in all cases establishing efficiency and accuracy of the proposed schemes

orthogonality from disjoint support in reproducing kernel hilbert spaces we investigate reproducing kernel hilbert spaces rkhs where two functions are orthogonal whenever they have disjoint support necessary and sufficient conditions in terms of feature maps for the reproducing kernel are established we also present concrete examples of finite dimensional rkhs and rkhs with a translation invariant reproducing kernel in particular it is shown that a sobolev space has the orthogonality from disjoint support property if and only if it is of integer index

a new approximate maximal margin classification algorithm a new incremental learning algorithm is described which approximates the maximal margin hyperplane w r t norm p 2 for a set of linearly separable data our algorithm called alma p approximate large margin algorithm w r t norm p takes o p 1 2 2 corrections to separate the data with p norm margin larger than 1 where g is the normalized p norm margin of the data alma p avoids quadratic or higher order programming methods it is very easy to implement and is as fast as on line algorithms such as rosenblatt s perceptron algorithm we performed extensive experiments on both real world and artificial datasets we compared alma 2 i e alma p with p 2 to standard support vector machines svm and to two incremental algorithms the perceptron algorithm and li and long s romma the accuracy levels achieved by alma 2 are superior to those achieved by the perceptron algorithm and romma but slightly inferior to svm s on the other hand alma 2 is quite faster and easier to implement than standard svm training algorithms when learning sparse target vectors alma p with p 2 largely outperforms perceptron like algorithms such as alma 2

efficient implementation of thinc scheme in this paper we improve the multi dimensional thinc tangent of hyperbola for interface capturing scheme f xiao y honma t kono a simple algebraic interface capturing scheme using hyperbolic tangent function int j numer meth fluid 48 2005 1023 the thinc scheme is a vof volume of fluid type method in the original thinc scheme one dimensional thinc scheme was straightforwardly used for multi dimensional cases in this paper we propose the wlic weighed line interface calculation method to extend the thinc scheme to multi dimension in the wlic method the interface is reconstructed by taking an average of line interfaces along x y and z coordinates with the weights calculated from surface normal the wlic method can extend the thinc scheme to multi dimension while maintaining simplicity of implementation and achieve a higher accuracy than the original thinc scheme the wlic method can readily extend the thinc scheme to three dimension

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

a high order discontinuous galerkin method for 2d incompressible flows in this paper we introduce a high order discontinuous galerkin method for two dimensional incompressible flow in the vorticity stream function formulation the momentum equation is treated explicitly utilizing the efficiency of the discontinuous galerkin method the stream function is obtained by a standard poisson solver using continuous finite elements there is a natural matching between these two finite element spaces since the normal component of the velocity field is continuous across element boundaries this allows for a correct upwinding gluing in the discontinuous galerkin framework while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability the method is efficient for inviscid or high reynolds number flows optimal error estimates are proved and verified by numerical experiments

a stable and convergent scheme for viscoelastic flow in contraction channels we present a new algorithm to simulate unsteady viscoelastic flows in abrupt contraction channels in our approach we split the viscoelastic terms of the oldroyd b constitutive equation using duhamel s formula and discretize the resulting pdes using a semi implicit finite difference method based on a lax wendroff method for hyperbolic terms in particular we leave a small residual elastic term in the viscous limit by design to make the hyperbolic piece well posed a projection method is used to impose the incompressibility constraint we are able to compute the full range of unsteady elastic flows in an abrupt contraction channel from the viscous limit to the elastic limit in a stable and convergent manner we demonstrate the range of our method for unsteady flow of a maxwell fluid with and without viscosity in planar contraction channels we also demonstrate stable and convergent results for benchmark high weissenberg number problems at we 1 and we 10

unitary representations of the 4 1 de sitter group on unitary irreducible representation spaces of the poincare group equivalence with their realizations as induced representations in a previous work we have constructed realizations of the principal continuous series of unitary irreducible representations of the simply connected covering group of the 4 1 de sitter group on unitary irreducible representation spaces of the simply connected covering group of the poincare group in this work we demonstrate the equivalence of the representations constructed in the previous work with their realizations as induced representations

overlapping schwarz methods for a singularly perturbed semilinear elliptic problem and their parallel implementation domain decomposition algorithms based on the schwarz alternating method are applied to the numerical solution of a singularly perturbed semilinear elliptic problem the emphasis is given to the description of the algorithms multidomain schwarz alternating method and two level schwarz method as well as their computer realization on a distributed memory multiprocessor computer numerical experiments related to a model problem illustrate the performance of the algorithms on parallel environment as well as their behavior with respect to the critical parameters such as the perturbation parameter the amount of overlapping and sizes of subdomains application of the two level schwarz method to a model semiconductor problem is also discussed

discrete exterior calculus discretization of incompressible navier stokes equations over surface simplicial meshes a conservative discretization of incompressible navier stokes equations is developed based on discrete exterior calculus dec a distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product the governing equations are first rewritten using the exterior calculus notation replacing vector calculus differential operators by the exterior derivative hodge star and wedge product operators the discretization is then carried out by substituting with the corresponding discrete operators based on the dec framework numerical experiments for flows over surfaces reveal a second order accuracy for the developed scheme when using structured triangular meshes and first order accuracy for otherwise unstructured meshes by construction the method is conservative in that both mass and vorticity are conserved up to machine precision the relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step

composition of interactions in relativistic quantum theory a scheme of composition of interactions which is the generalisation of sokolov s method 1975 of packing operators is proposed this scheme can be used in all forms of relativistic quantum dynamics in contrast to the formulation of coester and polyzou 1982 the author makes no use of the properties of multichannel wave operators but uses only the properties of the poincare group unitary representations a solution to the problem of composition of interactions in the instant form is given and it is shown that this solution agrees with that obtained by coester and polyzou

physically motivated domain decomposition for singularly perturbed equations a domain decomposition algorithm suitable for the efficient and accurate solution of a parabolic reaction convection diffusion equation with small parameter multiplying the diffusion term is presented convergence is established via maximum principle arguments the equation arises in the modeling of laminar transonic flow decomposition into subdomains is accomplished via singular perturbation analysis which dictates regions where certain reduced equations may be solved in place of the full equation effectively preconditioning the problem this paper concentrates on the theoretical basis of the method establishing local and global a priori error bounds

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

su 2 su 2 shift operators and representations of so 5 su 2 su 2 shift operators analogous to the su 2 shift operators developed and used by the author for the classification and analysis of representations of lie algebras in an su 2 or so 3 basis are obtained for the su 2 su 2 lie algebra in the case where one has an additional set of operators forming an irreducible four dimensional tensor representation of su 2 su 2 the shift operators obtained are used to treat the representations of so 5 in an su 2 su 2 basis

generalized semi inner products with applications to regularized learning abstract we propose a definition of generalized semi inner products g s i p by relating them to duality mappings from a normed vector space to its dual space a characterization for all g s i p satisfying this definition is obtained we then study the riesz representation of continuous linear functionals via g s i p as applications we establish a representer theorem and characterization equation for the minimizer of a regularized learning from finite or infinite samples in banach spaces of functions

reproducing kernel banach spaces for machine learning we introduce the notion of reproducing kernel banach spaces rkbs and study special semi inner product rkbs by making use of semi inner products and the duality mapping properties of an rkbs and its reproducing kernel are investigated as applications we develop in the framework of rkbs standard learning schemes including minimal norm interpolation regularization network support vector machines and kernel principal component analysis in particular existence uniqueness and representer theorems are established

asymptotic analysis of penalized likelihood and related estimators a general approach to the first order asymptotic analysis ofpenalized likelihood and related estimators is described the method gives expansions for the systematic and random error asymptotic convergence rates in a family of spectral norms are obtained the theory applies to a broad range of function estimation prob erns including non paxametric dellsity hazard and generalized regression curve estimation some examples are provided ams 1980 subject classifications primary 62 g05 secondary 62j05 41 a35 41 a25 47 a53 45 llo 45 m05

the problem of interactions in de sitter invariant theories the structure of representations describing systems of free particles in the theory with the invariance group is investigated the property of the particles to be free means as usual that the representation describing a many particle system is the tensor product of the corresponding single particle representations i e no interaction is introduced it is shown that the mass operator contains only continuous spectrum in the interval and such representations are unitarily equivalent to ones describing interactions gravitational electromagnetic etc this means that there are no bound states in the theory and the hilbert space of the many particle system contains a subspace of states with the following property the action of free representation operators on these states is manifested in the form of different interactions possible consequences of the results are discussed

some group theoretical aspects of the so0 1 4 invariant theory some properties of unitary representations of the de sitter group so0 1 4 are considered in the case of spin zero and positivity of the mass operator these representations involve a decomposition which ensures the fulfilment of the cluster separability condition and allows for the conventional interpretation of the de sitter analogues of momentum and boost operators the decomposition of the tensor product of representations belonging to the principal series contains the representations of the principal series with all masses under natural assumptions concerning the interaction operators there are no bound states in the theory which is due to the universal de sitter antigravity

refinement of reproducing kernels we continue our recent study on constructing a refinement kernel for a given kernel so that the reproducing kernel hilbert space associated with the refinement kernel contains that with the original kernel as a subspace to motivate this study we first develop a refinement kernel method for learning which gives an efficient algorithm for updating a learning predictor several characterizations of refinement kernels are then presented it is shown that a nontrivial refinement kernel for a given kernel always exists if the input space has an infinite cardinal number refinement kernels for translation invariant kernels and hilbert schmidt kernels are investigated various concrete examples are provided

two level parallel schwarz methods for singularly perturbed semilinear elliptic problems abstract two level domain decomposition methods based on the schwarz alternating procedure are proposed for the numerical solution of singularly perturbed semilinear elliptic problems convergence properties of the two level schwarz methods are established numerical experiments for a test problem illustrate the behaviour of the methods with respect to the critical parameters such as the perturbation parameter the amount of overlapping and sizes of subdomains application of one of the two level schwarz methods to a model semiconductor problem is also discussed

finiteness of physics and its possible consequences the modular analog of representations of the so 1 4 algebra for a system of two spinless particles is considered in the framework of approach proposed by the author earlier in which physical systems are described by the elements of a linear space over a finite field and operators of physical quantities by linear operators in this space the eigenvalues of the free mass operator and corresponding eigenvectors are found it is shown that if the finite field under consideration satisfies some properties then the particles necessarily interact with each other the corresponding interaction is universal since it is fully defined by the masses of particles and characteristics of the finite field the formula for the kernel of the interaction operator is derived but since the kernel is defined by the sum over the galois field and this sum cannot be calculated explicitly then it cannot be determined how far the above interaction manifests itself in the usual conditions

iterative solution of elliptic problems by approximate factorization an iterative method for the numerical solution of singularly perturbed second order linear elliptic problems is presented it is a defect correction iteration in which the approximate operator is the product of two first order operators which is readily inverted numerically the approximate operator is generated by formal asymptotic factorization of the original operator hence this is a quasi analytic defect correction iteration quad both its continuous and discrete versions are analyzed in one dimension the scheme is extended to a variety of two dimensional operators and it is analyzed for a model advection diffusion equation numerical calculations show the effectiveness of the scheme over a wide range of values of the small parameter

comparison of parallel implementation of some multi level schwarz methods for singularly perturbed parabolic problems abstract parallel multi level algorithms combining a time discretization and an overlapping domain decomposition technique are applied to the numerical solution of singularly perturbed parabolic problems two methods based on the schwarz alternating procedure are considered a two level method with auxiliary correcting subproblems as well as a three level method with auxiliary predicting and correcting subproblems moreover modifications of the methods using time extrapolation on subdomain interfaces are investigated the emphasis is given to the description of the algorithms as well as their computer realization on a distributed memory multiprocessor computer numerical experiments illustrate the performance of the algorithms on parallel environment and their behaviour with respect to the critical parameters such as the perturbation parameter and the size of the auxiliary subdomains

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

gerris a tree based adaptive solver for the incompressible euler equations in complex geometries an adaptive mesh projection method for the time dependent incompressible euler equations is presented the domain is spatially discretised using quad octrees and a multilevel poisson solver is used to obtain the pressure complex solid boundaries are represented using a volume of fluid approach second order convergence in space and time is demonstrated on regular statically and dynamically refined grids the quad octree discretisation proves to be very flexible and allows accurate and efficient tracking of flow features the source code of the method implementation is freely available

improving the speed and accuracy of projection type incompressible flow solvers abstract superseding so called first generation incompressible flow solvers of the projection type based on taylor galerkin advection second order pressure damping and element based data structures the current second generation solvers based on high order upwind advection fourth order pressure damping and edge based data structures have now been in use for half a decade and have proven remarkably robust and efficient for many large scale problems in order to achieve higher accuracy and speed these solvers have recently been enhanced in a variety of ways a substepping for advection b implicit treatment of advective terms via sgs and gmres lu sgs iterative solvers c fully implicit time accurate advancement of pressure and velocities and d linelet preconditioning for the pressure poisson equation the combined effect of these third generation improvements leads to speedups of the order of o 1 5 1 10 with similar or even better temporal accuracy as demonstrated on a variety of academic and industrial problems

when is there a representer theorem vector versus matrix regularizers we consider a general class of regularization methods which learn a vector of parameters on the basis of linear measurements it is well known that if the regularizer is a nondecreasing function of the l2 norm then the learned vector is a linear combination of the input data this result known as the representer theorem lies at the basis of kernel based methods in machine learning in this paper we prove the necessity of the above condition in the case of differentiable regularizers we further extend our analysis to regularization methods which learn a matrix a problem which is motivated by the application to multi task learning in this context we study a more general representer theorem which holds for a larger class of regularizers we provide a necessary and sufficient condition characterizing this class of matrix regularizers and we highlight some concrete examples of practical importance our analysis uses basic principles from matrix theory especially the useful notion of matrix nondecreasing functions

modular representations as a possible basis of finite physics the approach in which physical systems are described by the elements of a linear space over a finite field and operators of physical quantities by linear operators in this space is discussed the mathematical formulation of the correspondence between such a description and the conventional one is given for a case when the characteristic of the finite field is sufficiently large the above correspondence is considered in the examples of finite analogs of representations of so 2 3 so 1 4 and osp 1 4 algebras finite analogs of representations of infinite dimensional algebras are briefly discussed

an iterative method for systems of nonlinear hyperbolic equations abstract an iterative algorithm for the efficient solution of systems of nonlinear hyperbolic equations is presented parallelism is evident at several levels in the formation of the iteration the equations are decoupled thereby providing large grain parallelism parallelism may also be exploited within the solves for each equation convergence of the iteration is established via a bounding function argument experimental results in two dimensions are presented