lower bounds on the vc dimension of smoothly parameterized function classes we examine the relationship between the vc dimension and the number of parameters of a threshold smoothly parameterized function class we show that the vc dimension of such a function class is at least k if there exists a k dimensional differentiable manifold in the parameter space such that each member of the manifold corresponds to a different decision boundary using this result we are able to obtain lower bounds on the vc dimension proportional to the number of parameters for several thresholded function classes including two layer neural networks with certain smooth activation functions and radial basis functions with a gaussian basis these lower bounds hold even if the magnitudes of the parameters are restricted to be arbitrarily small in valiant s probably approximately correct learning framework this implies that the number of examples necessary for learning these function classes is at least linear in the number of parameters

affine equivalence of quartic homogeneous rotation symmetric boolean functions homogeneous rotation symmetric boolean functions have been extensively studied in recent years because of their applications in cryptography little is known about the basic question of when two such functions in n variables are affine equivalent the simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in 2009 and the first substantial progress on the much more complicated cubic case came in 2010 in this paper we show that much of the work on the cubic case can be extended to the quartic case we also prove an exact formula for the number and sizes of the affine equivalence classes when n is a prime

new formulations for interface problems in polar coordinates in this paper numerical methods are proposed for some interface problems in polar or cartesian coordinates the new methods are based on a formulation that transforms the interface problem with a nonsmooth or discontinuous solution into a problem with a smooth solution the new formulation leads to a simple second order finite difference scheme for the partial differential equation and a new interpolation scheme for the normal derivative of the solution in conjunction with the fast immersed interface method a fast solver has been developed for the interface problems with a piecewise constant but a discontinuous coefficient using the new formulation in a polar coordinate system

polynomial threshold functions and boolean threshold circuits we initiate a comprehensive study of the complexity of computing boolean functions by polynomial threshold functions ptfs on general boolean domains as opposed to domains such as 0 1 n or 1 1 n that enforce multilinearity a typical example of such a general boolean domain for the purpose of our results is 1 2 n we are mainly interested in the length the number of monomials of ptfs with their degree and weight being of secondary interest first we motivate the study of ptfs over the 1 2 n domain by showing their close relation to depth two threshold circuits in particular we show that ptfs of polynomial length and polynomial degree compute exactly the functions computed by polynomial size thr maj circuits we note that known lower bounds for thr maj circuits extend to the likely strictly stronger model of ptfs with no degree restriction we also show that a max plus version of ptfs is related to ac 0 thr circuits we exploit this connection to gain a better understanding of threshold circuits in particular we show that super logarithmic lower bounds for 3 player randomized communication protocols with unbounded error would yield super polynomial size lower bounds for thr thr circuits finally having thus motivated the model we initiate structural studies of ptfs these include relationships between weight and degree of ptfs and a degree lower bound for ptfs of constant length

separating ac 0 from depth 2 majority circuits we construct a function in ac 0 that cannot be computed by a depth 2 majority circuit of size less than exp theta n 1 5 this solves an open problem due to krause and pudlak theoret comput sci 174 1997 pp 137 156 and matches allender s classic result a note on the power of threshold circuits in proceedings of the 30th annual ieee symposium on foundations of computer science focs research triangle park nc 1989 pp 580 584 that ac 0 can be efficiently simulated by depth 3 majority circuits to obtain our result we develop a novel technique for proving lower bounds on communication complexity this technique the degree discrepancy theorem is of independent interest it translates lower bounds on the threshold degree of any boolean function into upper bounds on the discrepancy of a related function upper bounds on the discrepancy in turn immediately imply lower bounds on communication and circuit size in particular we exhibit the first known function in ac 0 with exponentially small discrepancy exp omega n 1 5 thereby establishing the separations sigma 2 cc not subseteq pp cc and pi 2 cc not subseteq pp cc in communication complexity

descartes rule of signs for radial basis function neural networks we establish versions of descartes rule of signs for radial basis function rbf neural networks the rbf rules of signs provide tight bounds for the number of zeros of univariate networks with certain parameter restrictions moreover they can be used to infer that the vapnik chervonenkis vc dimension and pseudodimension of these networks are no more than linear this contrasts with previous work showing that rbf neural networks with two or more input nodes have superlinear vc dimension the rules also give rise to lower bounds for network sizes thus demonstrating the relevance of network parameters for the complexity of computing with rbf neural networks

on the computational power of depth 2 circuits with threshold and modulo gates abstract we investigate the computational power of depth 2 circuits consisting of mod r gates at the bottom and a threshold gate with arbitrary weights at the top for short threshold mod r circuits and circuits with two levels of mod gates mod p mod q circuits in particular we will show the following results 1 i for all prime numbers p and integers q r it holds that if p divides r but not q then all threshold mod q circuits for mod r have exponentially many nodes 2 ii for all integers r all problems computable by depth 2 and or not circuits of polynomial size have threshold mod r circuits with polynomially many edges 3 iii there is a problem computable by depth 3 and or not circuits of linear size and constant bottom fan in which for all r needs threshold mod r circuits with exponentially many nodes 4 iv for p r different primes and q 2 k positive integers where r does not divide q every mod p k mod q circuit for mod r has exponentially many nodes results i and iii imply the first known exponential lower bounds on the number of nodes of threshold mod r circuits r 2 they are based on a new method for estimating the minimum length of threshold realizations over predefined function bases which in contrast to previous related techniques goldmann et al 1992 bruck and smolensky 1990 kailath et al 1991 goldmann 1993 grolmusz 1993 works even if the weight of the realization is allowed to be unbounded and if the bases are allowed to be nonorthogonal the special importance of result iii consists of the fact that the known spectral theoretically based lower bound methods for threshold xor circuits bruck and smolensky 1990 kailath et al 1991 can provably not be applied to ac 0 functions thus by ii result iii is sharp it gives a partial negative answer to the open question whether there exist simulations of ac 0 circuits by small depth threshold circuits which are more efficient than that given by yao s important result that acc functions have depth 3 threshold circuits of quasipolynomial weight yao 1990 finally we observe that our method works also for mod p mod q circuits if p is a power of a prime iv above see barrington et al 1990 krause and waack 1991 yan and parberry 1994 for related results a preliminary version of this paper appeared in krause and pudlak 1993

fast solvers for 3d poisson equations involving interfaces in a finite or the infinite domain in this paper numerical methods are proposed for poisson equations defined in a finite or infinite domain in three dimensions in the domain there can exists an interface across which the source term the flux and therefore the solution may be discontinuous the existence and uniqueness of the solution are also discussed to deal with the discontinuity in the source term and in the flux the original problem is transformed to a new one with a smooth solution such a transformation can be carried out easily through an extension of the jumps along the normal direction if the interface is expressed as the zero level set of a three dimensional function an auxiliary sphere is used to separate the infinite region into an interior and exterior domain the kelvin s inversion is used to map the exterior domain into an interior domain the two poisson equations defined in the interior and the exterior written in spherical coordinates are solved simultaneously by choosing the mesh size carefully and exploiting the fast fourier transform the resulting finite difference equations can be solved efficiently the approach in dealing with the interface has also been used with the artificial boundary condition technique which truncates the infinite domain numerical results demonstrate second order accuracy of our algorithms

solving partial differential equations on irregular domains with moving interfaces with applications to superconformal electrodeposition in semiconductor manufacturing we present a numerical algorithm for solving partial differential equations on irregular domains with moving interfaces instead of the typical approach of solving in a larger rectangular domain our approach performs most calculations only in the desired domain to do so efficiently we have developed a one sided multigrid method to solve the corresponding large sparse linear systems our focus is on the simulation of the electrodeposition process in semiconductor manufacturing in both two and three dimensions our goal is to track the position of the interface between the metal and the electrolyte as the features are filled and to determine which initial configurations and physical parameters lead to superfilling we begin by motivating the set of equations which model the electrodeposition process building on existing models for superconformal electrodeposition we develop a model which naturally arises from a conservation law form of surface additive evolution we then introduce several numerical algorithms including a conservative material transport level set method and our multigrid method for one sided diffusion equations we then analyze the accuracy of our numerical methods finally we compare our result with experiment over a wide range of physical parameters

universal approximation using radial basis function networks there have been several recent studies concerning feedforward networks and the problem of approximating arbitrary functionals of a finite number of real variables some of these studies deal with cases in which the hidden layer nonlinearity is not a sigmoid this was motivated by successful applications of feedforward networks with nonsigmoidal hidden layer units this paper reports on a related study of radial basis function rbf networks and it is proved that rbf networks having one hidden layer are capable of universal approximation here the emphasis is on the case of typical rbf networks and the results show that a certain class of rbf networks with the same smoothing factor in each kernel node is broad enough for universal approximation

an upper bound on the minimum number of monomials required to separate dichotomies of 1 1 n it is known that any dichotomy of 1 1 n can be learned separated with a higher order neuron polynomial function with 2n inputs monomials in general less than 2n monomials are sufficient to solve a given dichotomy in spite of the efforts to develop algorithms for finding solutions with fewer monomials there have been relatively fewer studies investigating maximum density n the minimum number of monomials that would suffice to separate an arbitrary dichotomy of 1 1 n this article derives a theoretical upper bound for this quantity superseding previously known bounds the main theorem here states that for any binary classification problem in 1 1 n n 1 one can always find a polynomial function solution with 2n 2n 4 or fewer monomials in particular any dichotomy of 1 1 n can be learned by a higher order neuron with a fan in of 2n 2n 4 or less with this result for the first time a deterministic ratio bound independent of n is established as n 2n 0 75 the main theorem is constructive so it provides a deterministic algorithm for achieving the theoretical result the study presented provides the basic mathematical tools and forms the basis for further analyses that may have implications for neural computation mechanisms employed in the cerebral cortex

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

minimal sign representation of boolean functions algorithms and exact results for low dimensions boolean functions bfs are central in many fields of engineering and mathematics such as cryptography circuit design and combinatorics moreover they provide a simple framework for studying neural computation mechanisms of the brain many representation schemes for bfs exist to satisfy the needs of the domain they are used in in neural computation it is of interest to know how many input lines a neuron would need to represent a given bf a common bf representation to study this is the so called polynomial sign representation where and 1 are associated with true and false respectively the polynomial is treated as a real valued function and evaluated at its parameters and the sign of the polynomial is then taken as the function value the number of input lines for the modeled neuron is exactly the number of terms in the polynomial this letter investigates the minimum number of terms that is the minimum threshold density that is sufficient to represent a given bf and more generally aims to find the maximum over this quantity for all bfs in a given dimension with this work for the first time exact results for four and five variable bfs are obtained and strong bounds for six variable bfs are derived in addition some connections between the sign representation framework and bent functions are derived which are generally studied for their desirable cryptographic properties

accurate numerical methods for micromagnetics simulations with general geometries in current fft based algorithms for micromagnetics simulations the boundary is typically replaced by a staircase approximation along the grid lines either eliminating the incomplete cells or replacing them by complete cells sometimes the magnetizations at the boundary cells are weighted by the volume of the sample in the corresponding cell we show that this leads to large errors in the computed exchange and stray fields one consequence of this is that the predicted switching mechanism depends sensitively on the orientation of the numerical grid we present a boundary corrected algorithm to efficiently and accurately handle the incomplete cells at the boundary we show that this boundary corrected algorithm greatly improves the accuracy in micromagnetics simulations we demonstrate by using a arrott s example of a hexagonal element that the switching mechanism is predicted independently of the grid orientation

on the complexity of computing and learning with multiplicative neural networks in a great variety of neuron models neural inputs are combined using the summing operation we introduce the concept of multiplicative neural networks that contain units that multiply their inputs instead of summing them and thus allow inputs to interact nonlinearly the class of multiplicative neural networks comprises such widely known and well studied network types as higher order networks and product unit networks we investigate the complexity of computing and learning for multiplicative neural networks in particular we derive upper and lower bounds on the vapnik chervonenkis vc dimension and the pseudodimension for various types of networks with multiplicative units as the most general case we consider feedforward networks consisting of product and sigmoidal units showing that their pseudo dimension is bounded from above by a polynomial with the same order of magnitude as the currently best known bound for purely sigmoidal networks moreover we show that this bound holds even when the unit type product or sigmoidal may be learned crucial for these results are calculations of solution set components bounds for new network classes as to lower bounds we construct product unit networks of fixed depth with superlinear vc dimension for sigmoidal networks of higher order we establish polynomial bounds that in contrast to previous results do not involve any restriction of the network order we further consider various classes of higher order units also known as sigma pi units that are characterized by connectivity constraints in terms of these we derive some asymptotically tight bounds multiplication plays an important role in both neural modeling of biological behavior and computing and learning with artificial neural networks we briefly survey research in biology and in applications where multiplication is considered an essential computational element the results we present here provide new tools for assessing the impact of multiplication on the computational power and the learning capabilities of neural networks

product units a computationally powerful and biologically plausible extension to backpropagation networks we introduce a new form of computational unit for feedforward learning networks of the backpropagation type instead of calculating a weighted sum this unit calculates a weighted product where each input is raised to a power determined by a variable weight such a unit can learn an arbitrary polynomial term which would then feed into higher level standard summing units we show how learning operates with product units provide examples to show their efficiency for various types of problems and argue that they naturally extend the family of theoretical feedforward net structures there is a plausible neurobiological interpretation for one interesting configuration of product and summing units

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

immersed interface methods for stokes flow with elastic boundaries or surface tension a second order accurate interface tracking method for the solution of incompressible stokes flow problems with moving interfaces on a uniform cartesian grid is presented the interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids the interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed the stokes equations are then discretized using the second order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper siam j numer anal 31 1994 pp 1019 1044 the resulting velocities are interpolated to the interface to determine the motion of the interface an implicit quasi newton method is developed that allows reasonable time steps to be used

the immersed interface multigrid methods for interface problems new multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic pdes that involve interfaces and discontinuities for elliptic interface problems the multigrid solver developed in this paper works while some other multigrid solvers do not for linear parabolic equations we have developed the second order maximum principle preserving finite difference scheme in this paper we use the crank nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives numerical examples are also presented

a coupling interface method for elliptic interface problems we propose a coupling interface method cim under cartesian grid for solving elliptic complex interface problems in arbitrary dimensions where the coefficients the source terms and the solutions may be discontinuous or singular across the interfaces it consists of a first order version cim1 and a second order version cim2 in one dimension the cim1 is derived from a linear approximation on both sides of the interface the method is extended to high dimensions through a dimension by dimension approach to connect information from each dimension a coupled equation for the first order derivatives is derived through the jump conditions in each coordinate direction the resulting stencil uses the standard 5 grid points in two dimensions and 7 grid points in three dimensions similarly the cim2 is derived from a quadratic approximation in each dimension in high dimensions a coupled equation for the principal second order derivatives u x k x k is derived through the jump conditions in each coordinate direction the cross derivatives are approximated by one side interpolation this approach reduces the number of grid points needed for one side interpolation the resulting stencil involves 8 grid points in two dimensions and 12 14 grid points in three dimensions a numerical study for the condition number of the resulting linear system of the cim2 in one dimension has been performed it is shown that the condition number has the same behavior as that of the discrete laplacian independent of the relative location of the interface in a grid cell further we also give a proof of the solvability of the coupling equations provided the curvature k of the interface satisfies kh

neural networks with local receptive fields and superlinear vc dimension local receptive field neurons comprise such well known and widely used unit types as radial basis function rbf neurons and neurons with center surround receptive field we study the vapnik chervonenkis vc dimension of feedforward neural networks with one hidden layer of these units for several variants of local receptive field neurons we show that the vc dimension of these networks is superlinear in particular we establish the bound w log k for any reasonably sized network with w parameters and k hidden nodes this bound is shown to hold for discrete center surround receptive field neurons which are physiologically relevant models of cells in the mammalian visual system for neurons computing a difference of gaussians which are popular in computational vision and for standard rbf neurons a major alternative to sigmoidal neurons in artificial neural networks the resutlt for rbf neural networks is of particular interest since it answers a question that has been open for several years the results also give rise to lower bounds for networks with fixed input dimension regarding constants all bounds are larger than those known thus far for similar architectures with sigmoidal neurons the superlinear lower bounds contrast with linear upper bounds for single local receptive field neurons also derived here

the adelic zeta function associated with the space of binary cubic forms with coefficients in a function field in this paper we study the adelic zeta function associated with the prehomogeneous vector space of binary cubic forms defined over a function field we establish its rationality find its poles and residues and a simple functional equation that this zeta function satisfies introduction the subject of integral equivalence of binary cubic forms although not as popular or well developed as the theory of integral equivalence of binary quadratic forms has received a fair amount of attention the first paper on this topic 9 written by eisenstein appeared in 1844 later on several other first rate mathematicians tried their hand at the subject among others the list includes arndt cayley hermite and davenport the classical method of attacking this problem is to draw upon a one to one discriminant preserving relation between binary cubic and binary quadratic forms which assigns to a cubic form its hessian and then use the reduction theory for binary quadratic forms to obtain results on class numbers of binary cubic forms the best of the classical results are due to davenport 5 who showed that e h ir n 2 36x o x15 16 nax e h ir n v2 12x o x 5116 nax where h ir n is the class number of irreducible binary cubic forms of discriminant n the credit for a new surge of interest in the study of integral equivalence of binary cubic forms must indisputably go to shintani for it was his 1972 paper 25 that put this study on modern footing in that paper shintani introduced a certain zeta function which as it turns out decomposes into a linear combination of products of dirichlet series whose coefficients are weighted class numbers of binary cubic forms and certain mellin transforms by studying analytic properties of the zeta function shintani was able to improve the error term in davenport s result to o x576 it is of interest to note that the zeta function and the dirichlet series it received by the editors september 3 1985 1980 mathematics subject classification 1985 revision primary 12a70 secondary iocio 12a90 1987 american mathematical society 0002 9947 87 1 00 25 per page

a general lower bound on the number of examples needed for learning abstract we prove a lower bound of 1 ln 1 vcdim c on the number of random examples required for distribution free learning of a concept class c where vcdim c is the vapnik chervonenkis dimension and and are the accuracy and confidence parameters this improves the previous best lower bound of 1 ln 1 vcdim c and comes close to the known general upper bound of o 1 ln 1 vcdim c ln 1 for consistent algorithms we show that for many interesting concept classes including k cnf and k dnf our bound is actually tight to within a constant factor

a multigrid ghost point level set method for incompressible navier stokes equations on moving domains with curved boundaries abstract in this paper we present a numerical approach to solve the navier stokes equations on moving domains with second order accuracy the space discretization is based on the ghost point method which falls under the category of unfitted boundary methods since the mesh does not adapt to the moving boundary the equations are advanced in time by using crank nicholson the momentum and continuity equations are solved simultaneously for the velocity and the pressure by adopting a proper multigrid approach to avoid the checkerboard instability for the pressure a staggered grid is adopted where velocities are defined at the sides of the cell and the pressure is defined at the centre the lack of uniqueness for the pressure is circumvented by the inclusion of an additional scalar unknown representing the average divergence of the velocity and an additional equation to set the average pressure to zero several tests are performed to simulate the motion of an incompressible fluid around a moving object as well as the lid driven cavity tests around steady objects the object is implicitly defined by a level set approach that allows a natural computation of geometrical properties such as distance from the boundary normal directions and curvature different shapes are tested circle ellipse and flower numerical results show the second order accuracy for the velocity and the divergence that decays to zero with second order and the efficiency of the multigrid that is comparable with the tests available in literature for rectangular domains without objects showing that the presence of a complex shaped object does not degrade the performance

maximum principle preserving schemes for interface problems with discontinuous coefficients new finite difference methods using cartesian grids are developed for elliptic interface problems with variable discontinuous coefficients singular sources and nonsmooth or even discontinuous solutions the new finite difference schemes are constructed to satisfy the sign property of the discrete maximum principle using quadratic optimization techniques the methods are shown to converge under certain conditions using comparison functions the coefficient matrix of the resulting linear system of equations is an m matrix and is coupled with a multigrid solver numerical examples are also provided to show the efficiency of the proposed methods

a second order cartesian method for the simulation of electropermeabilization cell models in this paper we present a new finite differences method to simulate electropermeabilization models like the model of neu and krassowska or the recent model of kavian et al these models are based on the evolution of the electric potential in a cell embedded in a conducting medium the main feature lies in the transmission of the voltage potential across the cell membrane the jump of the potential is proportional to the normal flux thanks to the well known kirchoff law an adapted scheme is thus necessary to accurately simulate the voltage potential in the whole cell notably at the membrane separating the cell from the outer medium we present a second order finite differences scheme in the spirit of the method introduced by cisternino and weynans for elliptic problems with immersed interfaces this is a cartesian grid method based on the accurate discretization of the fluxes at the interface through the use of additional interface unknowns the main novelty of our present work lies in the fact that the jump of the potential is proportional to the flux and therefore is not explicitly known the original use of interface unknowns makes it possible to discretize the transmission conditions with enough accuracy to obtain a second order spatial convergence we prove the second order spatial convergence in the stationary linear one dimensional case and the first order temporal convergence for the dynamical non linear model in one dimension we then perform numerical experiments in two dimensions that corroborate these results

fast marching methods fast marching methods are numerical schemes for computing solutions to the nonlinear eikonal equation and related static hamilton jacobi equations based on entropy satisfying upwind schemes and fast sorting techniques they yield consistent accurate and highly efficient algorithms they are optimal in the sense that the computational complexity of the algorithms is o n log n where n is the total number of points in the domain the schemes are of use in a variety of applications including problems in shape offsetting computing distances from complex curves and surfaces shape from shading photolithographic development computing first arrivals in seismic travel times construction of shortest geodesics on surfaces optimal path planning around obstacles and visibility and reflection calculations in this paper we review the development of these techniques including the theoretical and numerical underpinnings provide details of the computational schemes including higher order versions and demonstrate the techniques in a collection of different areas

immersed interface methods for neumann and related problems in two and three dimensions we develop and apply a finite difference method to discretize the laplacian operator with neumann boundary conditions on an irregular domain using a regular cartesian grid the method is an extension of the immersed interface method developed by leveque and li siam j numer anal 31 1994 1019 1044 with careful selection of stencils the method is second order accurate and produces a matrix that is stable diagonally semidominant the method is illustrated on several two dimensional problems and one three dimensional problem

high order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources this paper introduces a novel high order interface scheme the matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and singular sources on cartesian grids by appropriate use of auxiliary line and or fictitious points physical jump conditions are enforced at the interface unlike other existing interface schemes the proposed method disassociates the enforcement of physical jump conditions from the discretization of the differential equation under study to construct higher order interface schemes the proposed mib method bypasses the major challenge of implementing high order jump conditions by repeatedly enforcing the lowest order jump conditions the proposed mib method is of arbitrarily high order in principle in treating straight regular interfaces we construct mib schemes up to 16th order for more general elliptic problems with curved irregular interfaces and boundary up to 6th order mib schemes have been demonstrated by employing the standard high order finite difference schemes to discretize the laplacian the present mib method automatically reduces to the standard central difference scheme when the interface is absent the immersed interface method iim is regenerated for a comparison study of the proposed method the robustness of the mib method is verified against the large magnitude of the jump discontinuity across the interface the nature of high efficiency and low memory requirement of the mib method is extensively validated via solving various elliptic immersed interface problems in two and three dimensions ee dimensions

twists of the iwasawa tate zeta function the intent of this paper is to develop the theory of a simple variation of the adelic zeta function first described by tate 14 and iwasawa 9 in independent efforts to simplify hecke s analysis of his l functions attached to gr6bencharakters our twists of the iwasawa tate zeta function arise in the framework of a more general theory of zeta functions associated to representations of algebraic groups many of the basic problems of this general theory have already been solved by yukie and the author see 20 and 21 1 but still await appearance in final written form thus it seems worthwhile to give here a general statement of these problems before proceeding to the very special case considered in this paper let k be a global field i e algebraic number field or function field of a curve defined over a finite field let n and a denote the ring of adeles and the group of ideles of k respectively let co be a quasicharacter of the idele group trivial on the embedded group k of nonzero elements of k the group f2 f2 k of all such quasicharacters co has a natural riemann surface structure let l i denote the usual idele norm on a and for s tl2 let co be the principal quasicharacter co t ltl let re co be the real part of co f2 defined as the unique real number a such that l o i co for any reductive linear algebraic group g defined over k let g denote the adelizat ion of g over k as defined in 16 and let gk be the group of k rational points of g embedded as a discrete group in ga for simplicity we shall fix g as g1 now although the theory to be described below can also be effected for any reductive algebraic group with at least one nontrivial k rational character let p g g1 v be an irreducible k rational representation of g in the finite dimensional vector space v of dimension m let i be the identity matrix in g by schur s lemma there is an integer d such that p ti tdp i for all scalars t we shall always assume that d 0 let dg be any convenient left invariant measure on the quotient ga gk let v be the schwartz bruhat space of functions on v for the time being let vk be any gk invariant subset of vk the zeta function associated to the representation p may now be defined

a note on immersed interface method for three dimensional elliptic equations abstract the immersed interface method proposed by leveque and li 1 is extended to three dimensional elliptic equations of the form x u x x u x f x we study the situation in which there is an irregular interface surface s contained in the solution domain across which and f may be discontinuous or even singular as a result the solution u will usually be nonsmooth or even discontinuous a finite difference approach with a uniform cartesian grid is used in the discretization local truncation error analysis is performed to estimate the accuracy of the numerical solution

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

classification of cubic n 4 resilient boolean functions carlet and charpin classified the set of cubic n 4 resilient boolean functions into four different types with respect to the walsh spectrum and the dimension of the linear space based on the classification of rm 3 6 rm 1 6 we have completed this classification of cubic n 4 resilient boolean functions by deriving the corresponding algebraic normal form anf and autocorrelation spectrum for each of the four types at the same time we have solved an open problem by proving that all plateaued cubic n 4 resilient boolean functions have dimension of the linear space equal either to n 5 or n 6

a fast iterative algorithm for elliptic interface problems a fast second order accurate iterative method is proposed for the elliptic equation grad cdot beta x y grad u f x y in a rectangular region omega in two space dimensions we assume that there is an irregular interface across which the coefficient beta the solution u and its derivatives and or the source term f may have jumps we are especially interested in the cases where the coefficients beta are piecewise constant and the jump in beta is large the interface may or may not align with an underlying cartesian grid the idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by leveque and li siam j numer anal 4 1994 pp 1019 1044 in order to take advantage of fast poisson solvers on a rectangular region an intermediate unknown function the jump in the normal derivative across the interface is introduced our discretization is equivalent to using a second order difference scheme for a corresponding poisson equation in the region and a second order discretization for a neumann like interface condition thus second order accuracy is guaranteed a gmres iteration is employed to solve the schur complement system derived from the discretization a new weighted least squares method is also proposed to approximate interface quantities from a grid function numerical experiments are provided and analyzed the number of iterations in solving the schur complement system appears to be independent of both the jump in the coefficient and the mesh size

on xor lemmas for the weight of polynomial threshold functions abstract a multilinear polynomial p is said to sign represent a boolean function f 1 1 n 1 1 if f x s g n p x for all x 1 1 n in this paper we consider the length and weight of polynomials sign representing boolean functions of the form k f the xor of k copies of f on disjoint sets of variables firstly we show that for an infinite family of functions f a naive construction does not yield a shortest polynomial sign representing k f more precisely we give a construction of polynomials sign representing k and n whose length is strictly smaller than the k th power of the minimum length of a polynomial sign representing and n for every k 2 and n 2 except for k n 2 previously such polynomials were known only for n 2 sezener and oztop 2015 a similar result for the weight is also provided secondly we introduce a parameter v f of a boolean function f and show that the k th root of the minimum weight of a polynomial sign representing k f converges between v f and v f 2 as k goes to infinity

polynomial threshold functions ac 0 functions and spectral norms this paper examines the class of polynomial threshold functions using harmonic analysis and applies the results to derive lower bounds related to ac 0 functions a boolean function is polynomial threshold if it can be represented as the sign of a sparse polynomial one that consists of a polynomial number of terms the main result of this paper is that the class of polynomial threshold functions can be characterized using their spectral representation in particular it is proved that an n variable boolean function whose l 1 spectral norm is bounded by a polynomial in n is a polynomial threshold function while boolean function whose l infty 1 spectral norm is not bounded by a polynomial in n is not a polynomial threshold function j bruck siam j discrete math 3 1990 pp 168 177 the motivation is that the characterization of polynomial threshold functions can be applied to obtain upper and lower bounds on the complexity of computing with networks of linear threshold elements in

a new multiscale finite element method for high contrast elliptic interface problems we introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes and without resolving the interfaces a typical application would be the modelling of flow in a porous medium containing a number of inclusions of low or high permeability embedded in a matrix of high respectively low permeability our method is h 1 conforming with degrees of freedom at the nodes of a triangular mesh and requiring the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface but which use standard linear approximation otherwise a key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems under moderate assumptions we prove that our methods have optimal convergence rate of o h in the energy norm and o h 2 in the l 2 norm where h is the coarse mesh diameter and the hidden constants in these estimates are independent of the contrast i e ratio of largest to smallest value of the pde coefficient for standard elements the best estimate in the energy norm would be o h 1 2 e with a hidden constant which in general depends on the contrast the new interior boundary conditions depend not only on the contrast of the coefficients but also on the angles of intersection of the interface with the element edges

lower bound on vc dimension by local shattering we show that the vc dimension of a smoothly parameterized function class is not less than the dimension of any manifold in the parameter space as long as distinct parameter values induce distinct decision boundaries a similar theorem was published recently and used to introduce lower bounds on vc dimension for several cases lee bartlett williamson 1995 this theorem is not correct but our theorem could replace it for those cases and many other practical ones

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

a classification of the cosets of the reed muller code r 1 6 the weight distribution of a coset of a reed muller code m 1 m is invariant under a large transformation group consisting of all affine rearrangements of a vector space with dimension m we discuss a general algorithm that produces an ordered list of orbit representatives for this group action as a byproduct the procedure finds the order of the symmetry group of a coset with m 6 we can implement the algorithm on a computer and find that there are 150357 equivalence classes these classes produce 2082 distinct weight distributions their symmetry groups have 122 different orders

finite element methods and their convergence for elliptic and parabolic interface problems in this paper we consider the finite element methods for solving second order elliptic and parabolic interface problems in two dimensional convex polygonal domains nearly the same optimal l 2 norm and energy norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth though the regularities of the solutions are low on the whole domain the assumptions on the finite element triangulation are reasonable and practical

a novel concept for the design of immersed interface methods the objective of this paper is to present a novel robust high order accurate immersed interface method iim for advection diffusion type equations in contrast to other immersed methods that were designed for consistency and accuracy with a posteriori check of the numerical stability we combine local taylor series expansion at irregular grid points with a local stability constraint as part of the design process stability investigations of the iim are employed to demonstrate that the local stability constraint is sufficient for obtaining a globally stable method as long as the neumann number is less than its limiting value one of the key aspects of this iim is that the irregular finite difference stencils can be isolated from the rest of the computational domain to validate our novel immersed interface approach two dimensional and three dimensional test cases for model equations are presented in addition this method is applied to the incompressible navier stokes equations to conduct stability investigations of a boundary layer flow over a rough surface and for investigations of pulsatile stenotic flows stability investigations of wall bounded flows are challenging for immersed methods because the near wall accuracy is important for correctly capturing the characteristics of the hydrodynamic instability mechanisms in particular regarding the wave relation between the wave velocity components close to the wall

on the capabilities of higher order neurons a radial basis function approach higher order neurons with k monomials in n variables are shown to have vapnik chervonenkis vc dimension at least nk 1 this result supersedes the previously known lower bound obtained via k term monotone disjunctive normal form dnf formulas moreover it implies that the vc dimension of higher order neurons with k monomials is strictly larger than the vc dimension of k term monotone dnf the result is achieved by introducing an exponential approach that employs gaussian radial basis function neural networks for obtaining classifications of points in terms of higher order neurons

computing affine equivalence classes of boolean functions by group isomorphism affine equivalence classification of boolean functions has significant applications in logic synthesis and cryptography previous studies for classification have been limited by the large set of boolean functions and the complex operations on the affine group although there are many research on affine equivalence classification for parts of boolean functions in recent years there are very few results for the entire set of boolean functions the best existing result has been achieved by harrison with 15768919 affine equivalence classes for 6 variable boolean functions this paper presents a concise formula for affine equivalence classification of the entire set of boolean functions as well as a formula for affine classification of boolean functions with distinct on set size respectively the method outlined in this paper greatly simplifies the affine group s action by constructing an isomorphism mapping from the affine group to a permutation group by this method we can compute the affine equivalence classes for up to 10 variables experiment results indicate that our scheme for calculating the affine equivalence classes for more than 6 variables is a significant advancement over previous published methods

discontinuous bubble scheme for elliptic problems with jumps in the solution abstract we propose a new numerical method to solve an elliptic problem with jumps both in the solution and derivative along an interface by considering a suitable function which has the same jumps as the solution we transform the problem into one without jumps then we apply the immersed finite element method in which we allow uniform meshes so that the interface may cut through elements to discretize the problem as introduced in 1 2 3 some convenient way of approximating the jumps of the solution by piecewise linear functions is suggested our method can also handle the case when the interface passes through grid points we believe this paper presents the first resolution of such cases numerical experiments for various problems show second order convergence in l2 and first order in h1 norms moreover the convergence order is very robust for all problems tested

spectral method for matching exterior and interior elliptic problems a spectral method is described for solving coupled elliptic problems on an interior and an exterior domain the method is formulated and tested on the two dimensional interior poisson and exterior laplace problems whose solutions and their normal derivatives are required to be continuous across the interface a complete basis of homogeneous solutions for the interior and exterior regions corresponding to all possible dirichlet boundary values at the interface are calculated in a preprocessing step this basis is used to construct the influence matrix which serves to transform the coupled boundary conditions into conditions on the interior problem chebyshev approximations are used to represent both the interior solutions and the boundary values a standard chebyshev spectral method is used to calculate the interior solutions the exterior harmonic solutions are calculated as the convolution of the free space green s function with a surface density this surface density is itself the solution to an integral equation which has an analytic solution when the boundary values are given as a chebyshev expansion properties of chebyshev approximations insure that the basis of exterior harmonic functions represents the external near boundary solutions uniformly the method is tested by calculating the electrostatic potential resulting from charge distributions in a rectangle the resulting influence matrix is well conditioned and solutions converge exponentially as the resolution is increased the generalization of this approach to three dimensional problems is discussed in particular the magnetohydrodynamic equations in a finite cylindrical domain surrounded by a vacuum

three dimensional matched interface and boundary mib method for treating geometric singularities this paper reports the three dimensional 3d generalization of our previous 2d higher order matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and non smooth interfaces new mib algorithms that make use of two sets of interface jump conditions are proposed to remove the critical acute angle constraint of our earlier mib scheme for treating interfaces with sharp geometric singularities such as sharp edges sharp wedges and sharp tips the resulting 3d mib schemes are of second order accuracy for arbitrarily complex interfaces with sharp geometric singularities of fourth order accuracy for complex interfaces with moderate geometric singularities and of sixth order accuracy for curved smooth interfaces a systematical procedure is introduced to make the mib matrix optimally symmetric and banded by appropriately choosing auxiliary grid points consequently the new mib linear algebraic equations can be solved with fewer number of iterations the proposed mib method makes use of cartesian grids standard finite difference schemes lowest order interface jump conditions and fictitious values the interface jump conditions are enforced at each intersecting point of the interface and mesh lines to overcome the staircase phenomena in finite difference approximation while a pair of fictitious values are determined along a mesh at a time an iterative procedure is proposed to determine all the required fictitious values for higher order schemes by repeatedly using the lowest order jump conditions a variety of mib techniques are developed to overcome geometric constraints the essential strategy of the mib method is to locally reduce a 2d or a 3d interface problem into 1d like ones the proposed mib method is extensively validated in terms of the order of accuracy the speed of convergence the number of iterations and cpu time numerical experiments are carried out to complex interfaces including the molecular surfaces of a protein a missile interface and van der waals surfaces of intersecting spheres

the sign rank of ac 0 the sign rank of a matrix a a ij with pm1 entries is the least rank of a real matrix b b ij with a ij b ij 0 for all i j we obtain the first exponential lower bound on the sign rank of a function in mathsf ac 0 namely let f x y bigwedge i 1 dots m bigvee j 1 dots m 2 x ij wedge y ij we show that the matrix f x y x y has sign rank exp omega m this in particular implies that sigma 2 cc not subseteq mathsf upp cc which solves a longstanding open problem in communication complexity posed by babai frankl and simon proceedings of the 27th symposium on foundations of computer science focs 1986 pp 337 347 our result additionally implies a lower bound in learning theory specifically let phi 1 dots phi r 0 1 n to mathbb r be functions such that every dnf formula f 0 1 n to 1 1 of polynomial size has the representation f equiv mathrm sgn a 1 phi 1 dots a r phi r for some reals a 1 dots a r we prove that then r geqslant exp omega n 1 3 which essentially matches an upper bound of exp tilde o n 1 3 due to klivans and servedio j comput system sci 68 2004 pp 303 318 finally our work yields the first exponential lower bound on the size of threshold of majority circuits computing a function in mathsf ac 0 this substantially generalizes and strengthens the results of krause and pudlak theoret comput sci 174 1997 pp 137 156

vc dimension and uniform learnability of sparse polynomials and rational functions the authors prove upper and lower bounds on the vc dimension of sparse univariate polynomials over reals and apply these results to prove uniform learnability of sparse polynomials and rational functions as an application the solution to the open problem of vapnik in estimation of dependences based on empirical data springer verlag berlin 1982 on computational approximation of the regression in a class of polynomials used in the theory of empirical data dependences is given