the representation of integers as sums of squares we present a uniform method by which we obtain an explicit formula for the number of representations of an integer as the sum of n squares for each n in the range 2 n 8 the final formulas for odd n are new though those for even n are not we also present a formula for the dominant term for the number for an arbitrary n introduction in this paper we are interested in the sum of squares x n

a regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its l norm the i th singular value of a transfer matrix need not be a differentiable function of frequency where its multiplicity is greater than one we show that near a local maximum however the largest singular value has a lipschitz second derivative but need not have a third derivative using this regularity result we give a quadratically convergent algorithm for computing the l norm of a transfer matrix

scale sensitive dimensions uniform convergence and learnability learnability in valiant s pac learning model has been shown to be strongly related to the existence of uniform laws of large numbers these laws define a distribution free convergence property of means to expectations uniformly over classes of random variables classes of real valued functions enjoying such a property are also known as uniform glivenko cantelli classes in this paper we prove through a generalization of sauer s lemma that may be interesting in its own right a new characterization of uniform glivenko cantelli classes our characterization yields dudley gine and zinn s previous characterization as a corollary furthermore it is the first based on a gine and zinn s previous characterization as a corollary furthermore it is the first based on a simple combinatorial quantity generalizing the vapnik chervonenkis dimension we apply this result to obtain the weakest combinatorial condition known to imply pac learnability in the statistical regression or agnostic framework furthermore we find a characterization of learnability in the probabilistic concept model solving an open problem posed by kearns and schapire these results show that the accuracy parameter plays a crucial role in determining the effective complexity of the learner s hypothesis class

existence theorems for positive semidefinite and sign indefinite stabilizing solutions of h riccati equations the existence of both positive semidefinite and sign indefinite stabilizing solutions of h infty type riccati equations is determined via joint eigenvalue and hamiltonian tests these tests determine the inertia properties of the solutions they are dependent on a formula derived in this paper that shows how the solution of game riccati equations can be decomposed into a sum of two well known riccati equations namely the standard filter riccati equation and the bounded real riccati equation other properties of stabilizing solutions such as solution partial order spectral radius monotonicity and rank are also discussed

a gradient method for computing the optimal h sub infinity norm a fast algorithm for computing the optimal h sub infinity norm is presented the algorithm is based on certain explicit computations of gradients examples are given to show that the gradient method is much faster than the bisection method the gradient method can be used without the upper and lower bound computations this could be a substantial additional saving because computing these bounds requires the computation of the h sub infinity norm

coupling of fem and bem in shape optimization in the present paper we consider the numerical solution of shape optimization problems which arise from shape functionals of integral type over a compact region of the unknown shape especially l 2 tracking type functionals the underlying state equation is assumed to satisfy a poisson equation with dirichlet boundary conditions we proof that the shape hessian is not strictly h 1 2 coercive at the optimal domain which implies ill posedness of the optimization problem under consideration since the adjoint state depends directly on the state we propose a coupling of finite element methods fem and boundary element methods bem to realize an efficient first order shape optimization algorithm fem is applied in the compact region while the rest is treated by bem the coupling of fem and bem essentially retains all the structural and computational advantages of treating the free boundary by boundary integral equations

on convergence in elliptic shape optimization the present paper aims at analyzing the existence and convergence of approximate solutions in shape optimization motivated by illustrative examples an abstract setting of the underlying shape optimization problem is suggested taking into account the so called two norm discrepancy a ritz galerkin type method is applied to solve the associated necessary condition existence and convergence of approximate solutions are proved provided that the infinite dimensional shape problem admits a stable second order optimizer the rate of convergence is confirmed by numerical results

mesures et equidistribution sur les espaces de berkovich the proof by ullmo and zhang of bogomolov s conjecture about points of small height in abelian varieties made a crucial use of an equidistribution property for small points in the associated complex abelian variety we study the analogous equidistribution property at p adic places our results can be conveniently stated within the framework of the analytic spaces defined by berkovich the first one is valid in any dimension but is restricted to algebraic metrics the second one is valid for curves but allows for more general metrics in particular to the normalized heights with respect to dynamical systems

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

on prime root of unity sequences with perfect periodic correlation in this paper perfect root of unity sequences prus with entries in alpha p x in complexc x p 1 where p is a prime are studied a lower bound on the number of distinct phase

biorthogonal wavelet approximation for the coupling of fem bem we apply multiscale methods to the coupling of finite and boundary element methods to solve an exterior dirichlet boundary value problem for the two dimensional poisson equation adopting biorthogonal wavelet matrix compression to the boundary terms with n degrees of freedom we show that the resulting compression strategy fits the optimal convergence rate of the coupling galerkin methods while the number of nonzero entries in the corresponding stiffness matrices is considerably smaller than n 2

generalization performance of regularization networks and support vector machines via entropy numbers of compact operators we derive new bounds for the generalization error of kernel machines such as support vector machines and related regularization networks by obtaining new bounds on their covering numbers the proofs make use of a viewpoint that is apparently novel in the field of statistical learning theory the hypothesis class is described in terms of a linear operator mapping from a possibly infinite dimensional unit ball in feature space into a finite dimensional space the covering numbers of the class are then determined via the entropy numbers of the operator these numbers which characterize the degree of compactness of the operator can be bounded in terms of the eigenvalues of an integral operator induced by the kernel function used by the machine as a consequence we are able to theoretically explain the effect of the choice of kernel function on the generalization performance of support vector machines

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

on the transformation formulas of theta series here s is a nondegenerate rational symmetric matrix of signature q s s 3 x iy is a variable in the siegel half space hn of degree n u is a variable in the space cn of all complex n x q matrices runs over the set qq of all rational q x n matrices a is a c valued function on qn that is a finite linear combination of characteristic functions of cosets of qq modulo a z lattice a is a real matrix such that s ta diag lq 1 a and e c exp 2iric for c e c if q 1 s a 1 and a is suitably chosen the series g becomes a classical theta function of jacobi riemann if s is positive definite n 1 u 0 and q is even g is an elliptic modular form of weight q 2 studied by hecke also siegel introduced the series with u 0 in his theory of quadratic forms in si now g satisfies a transformation formula of the following type for every

quadratic diophantine equations and orders in quaternion algebras we first reformulate the main theorem on quadratic diophantine equations given in our recent book in terms of the even clifford group in the quaternary case the group is the multiplicative group of a quaternion algebra and the reformulated theorem connects a class of primitive solutions of a quadratic equation with a class of an order in the algebra in the indefinite case each class is essentially a class of binary hermitian forms in the definite case the result is an analogue of the result of gauss on the sums of three squares

covering numbers for support vector machines support vector sv machines are linear classifiers that use the maximum margin hyperplane in a feature space defined by a kernel function previously the only bounds on the generalization performance of sv machines within valiant s probably approximately correct framework took no account of the kernel used except in its effect on the margin and radius it has been shown that one can bound the relevant covering numbers using tools from functional analysis in this paper we show that the resulting bound can be greatly simplified the new bound involves the eigenvalues of the integral operator induced by the kernel it shows that the effective dimension depends on the rate of decay of these eigenvalues we present an explicit calculation of covering numbers for an sv machine using a gaussian kernel which is significantly better than that implied by previous results

the asymptotic behavior of e entropy of a compact positive operator abstract the purpose of this paper is to apply the methods of real analysis to the investigation of the asymptotic behavior of e entropy of a compact positive operator for sufficiently small e 0 the lower growth order about the e entropy of a compact positive operator is characterized by the proper values associated with this operator moreover about the e entropy per unit time of convolution operators generated by a continuous positive definite and lebesgue integrable function both the upper and the lower growth orders are characterized by the spectral density associated with this function

h sub infinity control by state feedback and fast algorithms for the computation of optimal h sub infinity norms the suboptimality of some parameter for h sub infinity optimization by dynamic state feedback is characterized in terms of the solvability of riccati inequalities this is done without restricting the finite zero structure of the plant if there are no system zeros on the imaginary axis the h sub infinity problem can be treated in a complete and satisfactory way explicit characterizations optimum to be achieved are provided and a closed formula for the optimal value is derived in terms of the h sub infinity norm of some fixed transfer matrix if the optimum is not attained any sequence of controllers of bounded size which is constructed to approach the infimal norm must necessarily be high gain a globally and quadratically convergent algorithm to compute the optimal value is proposed this algorithm is generalized to the h sub infinity optimization problem by measurement feedback

numerical solution of the discrete time periodic riccati equation in this paper we present a method for the computation of the periodic nonnegative definite stabilizing solution of the periodic riccati equation this method simultaneously triangularizes by orthogonal equivalences a sequence of matrices associated with a cyclic pencil formulation related to the euler lagrange difference equations in doing so it is possible to extract a basis for the stable deflating subspace of the extended pencil from which the riccati solution is obtained this algorithm is an extension of the standard qz algorithm and retains its attractive features such as quadratic convergence and small relative backward error a method to compute the optimal feedback controller gains for linear discrete time periodic systems is dealt with

coupling of finite and boundary element methods for an elastoplastic interface problem a class of transmission problems is considered in which a nonlinear variational problem in one domain is coupled with a linear elliptic problem in a second domain a typical example is a problem from three dimensional elasticity theory where an elastoplastic material is embedded into a linear elastic material the nonlinear problem is given in variational form with a strictly convex functional the linear elliptic problem is described by boundary integral equations on the coupling boundary the typical saddle point structure of such problems is analyzed galerkin approximations are studied which consist of a finite element approximation in the first domain coupled with a boundary element method on the coupling boundary the convergence of the galerkin approximation is based on the saddle point structure which is shown to hold for the exact as well as the discretized problems

classification of integer valued symmetric forms we classify all bf z valued symmetric forms that are reduced in the sense that they cannot be represented nontrivially by another bf z valued symmetric form we determine the possible values of the discriminant and signature of each such form and also the number of genera of the forms with the given discriminant and signature

inhomogeneous quadratic forms and triangular numbers we prove an explicit formula for the number of representations of an integer as the sum of n triangular numbers for each n in the range 2 n 8 as special cases of a more general formula applicable to an inhomogeneous quadratic form over a totally real number field the formula can be derived by calculating explicitly the fourier coefficients of a certain eisenstein series that appears in the siegel weil formula for an inhomogeneous quadratic form

zeta functions and eisenstein series on classical groups we construct an euler product from the hecke eigenvalues of an automorphic form on a classical group and prove its analytic continuation to the whole complex plane when the group is a unitary group over a cm field and the eigenform is holomorphic we also prove analytic continuation of an eisenstein series on another unitary group containing the group just mentioned defined with such an eigenform as an application of our methods we prove an explicit class number formula for a totally definite hermitian form over a cm field

on the siegel weil formula for quaternionic unitary groups we extend the siegel weil formula to all quaternion dual pairs applications include the classification problem of skew hermitian forms over a quaternion algebra over a number field and a product formula for the weighted average of the representation numbers of a skew hermitian form by another skew hermitian form

on the game riccati equations arising in h_ infty control problems in the state space approach to h infty optimal control feasible closed loop gains gamma are characterized via a pair of game riccati equations depending on gamma this paper is concerned with the properties of these equations as gamma varies the most general problem is considered d 11 neq 0 and the variations of the riccati solutions are thoroughly analyzed insight is gained into the behavior near the optimum and into the dependence on gamma of the suboptimality conditions in addition concavity is established for a criterion that synthesizes the three conditions x geq 0 y geq 0 and rho xy gamma 2 this suggests a numerically reliable newton scheme for the computation of the optimal gamma most results presented here are extensions of earlier contributions the main concern is to provide a complete and synthetic overview as well as results and formulas tailored to the development of numerically sound algorithms

a preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems this paper provides a preconditioned iterative technique for the solution of saddle point problems these problems typically arise in the numerical approximation of partial differential equations by lagrange multiplier techniques and or mixed methods the saddle point problem is reformulated as a symmetric positive definite system which is then solved by conjugate gradient iteration applications to the equations of elasticity and stokes are discussed and the results of numerical experiments are given

on the computation of the optimal h norms for two feedback control problems abstract numerical computation of the optimal h norms for the state feedback and the output feedback control problems is considered in 1989 doyle et al derived for a given parameter r 0 necessary and sufficient conditions under which an admissible output feedback controller exists as well as conditions for the existence of a state feedback controller we explore numerical methods to find the infimum of parameter r that satisfies those conditions

confluent hypergeometric functions on tube domains our problem specialized to the case of the siegel upper half space h of degree m concerns the fourier expansion of a series s z fl det z a det a a l here z is a variable on h l is a lattice in the space v of all real symmetric matrices of size m and fl ec it can be shown that this is convergent if re f l m and also that if re m 2 and r e f l m 2 it has a fourier expansion of the form

multilevel preconditioning this paper is concerned with multilevel techniques for preconditioning linear systems arising from galerkin methods for elliptic boundary value problems a general estimate is derived which is based on the characterization of besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions the result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers in particular our analysis indicates how to realize even uniformly bounded condition numbers for example the general results are used to show that the bramble pasciak xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform furthermore they are applied to a general multivariate setting of refinable shift invariant spaces in particular covering those induced by various types of wavelets

on an exact mass formula of shimura in a series of recent papers g shimura obtained an exact formula for the mass of a maximal lattice in a quadratic or hermitian space over a totally real number field using bruhat tits theory we obtain a quick and more conceptual proof of his formula when the form is totally definite

a fast algorithm to computer the h norm of a transfer function matrix abstract a fast algorithm is presented to compute the h norm of a transfer function matrix based on the relation between the singular values of the transfer function matrix and the eigenvalues of a related hamiltonian matrix the norm is computed with guaranteed accuracy a very simple method to compute a rather close lower bound on the h norm is given

optimization of a variable mouth acoustic horn by using boundary shape optimization on the end part of a semi infinite waveguide for acoustic waves we design transmission efficient interfacial devices without imposing an upper bound on the mou

the importance of convexity in learning with squared loss we show that if the closure of a function class f under the metric induced by some probability distribution is not convex then the sample complexity for agnostically learning f with squared loss using only hypotheses in f is spl omega ln 1 spl delta spl epsiv sup 2 where 1 spl delta is the probability of success and spl epsiv is the required accuracy in comparison if the class f is convex and has finite pseudodimension then the sample complexity is o 1 spl epsiv ln 1 spl epsiv ln 1 b if a nonconvex class f has finite pseudodimension then the sample complexity for agnostically learning the closure of the convex hull of f is o 1 spl epsiv 1 spl epsiv ln 1 spl epsiv ln 1 spl delta hence for agnostic learning learning the convex hull provides better approximation capabilities with little sample complexity penalty

regularized principal manifolds many settings of unsupervised learning can be viewed as quantization problems the minimization of the expected quantization error subject to some restrictions this allows the use of tools such as regularization from the theory of supervised risk minimization for unsupervised learning this setting turns out to be closely related to principal curves the generative topographic map and robust coding we explore this connection in two ways 1 we propose an algorithm for finding principal manifolds that can be regularized in a variety of ways and 2 we derive uniform convergence bounds and hence bounds on the learning rates of the algorithm in particular we give bounds on the covering numbers which allows us to obtain nearly optimal learning rates for certain types of regularization operators experimental results demonstrate the feasibility of the approach