an extension of a bound for functions in sobolev spaces with applications to m s spline interpolation and smoothing given a function f on a bounded open subset of mathbb r n with a lipschitz continuous boundary we obtain a sobolev bound involving the values of f at finitely many points of overline omega this result improves previous ones due to narcowich et al math comp 74 743 763 2005 and wendland and rieger numer math 101 643 662 2005 we then apply the sobolev bound to derive error estimates for interpolating and smoothing m s splines in the case of smoothing noisy data as well as exact data are considered

synchronization in lattices of coupled oscillators abstract we consider coupled nonlinear oscillators with external periodic forces and the dirichlet boundary conditions we prove that synchronization occurs provided that the coupling is dissipative and the coupling coefficients are sufficiently large the synchronization here is of an obvious type the size of an attractor is comparable to the difference of the subsystems

sobolev bounds on functions with scattered zeros with applications to radial basis function surface fitting in this paper we discuss sobolev bounds on functions that vanish at scattered points in a bounded lipschitz domain that satisfies a uniform interior cone condition the sobolev spaces involved may have fractional as well as integer order we then apply these results to obtain estimates for continuous and discrete least squares surface fits via radial basis functions rbfs these estimates include situations in which the target function does not belong to the native space of the rbf

dynamics of circular arrays of josephson junctions and the discrete sine gordon equation abstract we analyze the damped driven discrete sine gordon equation with periodic boundary conditions and constant forcing analytical and numerical results are presented about the existence stability and bifurcations of traveling waves in this system these results are compared with experimental measurements of the current voltage i v characteristics of a ring of n 8 underdamped josephson junctions we find two types of traveling waves low velocity kinks and high velocity whirling modes the kinks excite small amplitude linear waves intheir wake at certain drive strengths the linear waves phase lock to the kink generating resonant steps in the i v curve steps also occur in the high velocity region due to parametric instabilities of the whirling mode we analyze the onset of these instabilities then numerically study the secondary bifurcations and complex spatiotemporal phenomena that occur past the onset in all cases the measured voltage locations of the resonant steps are in good agreement with the predictions

multiscale approximation and reproducing kernel hilbert space methods we consider reproducing kernels k omega times omega to mathbb r in multiscale series expansion form i e kernels of the form k left boldsymbol x boldsymbol y right sum ell in mathbb n lambda ell sum j in i ell phi ell j left boldsymbol x right phi ell j left boldsymbol y right with weights lambda ell and structurally simple basis functions left phi ell i right here we deal with basis functions such as polynomials or frame systems where for ell in mathbb n the index set i ell is finite or countable we derive relations between approximation properties of spaces based on basis functions phi ell j 1 leq ell leq l j in i ell and spaces spanned by translates of the kernel span k boldsymbol x 1 cdot dots k boldsymbol x n cdot with x n boldsymbol x 1 dots boldsymbol x n subset omega if the truncation index l is appropriately coupled to the discrete set x n an analysis of a numerically feasible approx

deterministic error analysis of support vector regression and related regularized kernel methods we introduce a new technique for the analysis of kernel based regression problems the basic tools are sampling inequalities which apply to all machine learning problems involving penalty terms induced by kernels related to sobolev spaces they lead to explicit deterministic results concerning the worst case behaviour of e and svrs using these we show how to adjust regularization parameters to get best possible approximation orders for regression the results are illustrated by some numerical examples

error estimates for multivariate regression on discretized function spaces in this paper we will discuss the discretization error for the regression setting and derive error bounds relying on the approximation properties of the discretized space furthermore we will point out how the sampling error and the discretization error interact and how they can be balanced appropriately we will present two examples based on tensor product spaces sparse grids hyperbolic crosses which provide a suitable approach in the case of large sample sets in moderate dimensions

on the strongly damped wave equation we prove the existence of the universal attractor for the strongly damped semilinear wave equation in the presence of a quite general nonlinearity of critical growth when the nonlinearity is subcritical we prove the existence of an exponential attractor of optimal regularity having a basin of attraction coinciding with the whole phase space as a byproduct the universal attractor is regular and of finite fractal dimension moreover we carry out a detailed analysis of the asymptotic behavior of the solutions in dependence of the damping coefficient

endomorphisms of the cohomology of complex grassmannians for any complex grassmann manifold c we classify all endomorphisms of the rational cohomology ring of c which are nonzero on dimension 2 some applications of this result are given

existence uniqueness and nonuniqueness of single wave form solutions to josephson junction systems we study the existence and uniqueness of single wave form solutions for series arrays of n josephson junctions coupled through general rlc loads our method recasts the conditions for a single wave form solution as a fixed point equation on a hilbert space we use this technique to prove that single wave form solutions exist when the bias current i is greater than 1 and are essentially unique when either n or i is large we also employ these ideas to give explicit examples of series arrays which posess several distinct single wave form solutions

approximate interpolation with applications to selecting smoothing parameters in this paper we study the global behavior of a function that is known to be small at a given discrete data set such a function might be interpreted as the error function between an unknown function and a given approximant we will show that a small error on the discrete data set leads under mild assumptions automatically to a small error on a larger region we will apply these results to spline smoothing and show that a specific a priori choice of the smoothing parameter is possible and leads to the same approximation order as the classical interpolant this has also a surprising application in stabilizing the interpolation process by splines and positive definite kernels

extension of sampling inequalities to sobolev semi norms of fractional order and derivative data this paper devoted to sampling inequalities provides some extensions of previous results by arcangeli et al numer math 107 2 181 211 2007 j approx theory 161 198 212 2009 given a function u in a suitable sobolev space defined on a domain in mathbb r n sampling inequalities typically yield bounds of integer order sobolev semi norms of u in terms of a higher order sobolev semi norm of u the fill distance d between overline omega and a discrete set a subset overline omega and the values of u on a the extensions established in this paper allow us to bound fractional order semi norms and to incorporate if available values of partial derivatives on the discrete set both the cases of a bounded domain and omega mathbb r n are considered

global behavior in the dynamical equation of j j type on considere l equation de la forme 1 ecos sin asinwt on etudie le cas e 0 2 on montre que chaque trajectoire est asymptotiquement periodique ou quasi periodique

on the homology spectral sequence for topological hochschild homology marcel b6kstedt has computed the homotopy type of the topological hochschild homology of z p using his definition of topological hochschild homology for a functor with smash product here we show that easy conceptual proofs of his main technical result of are possible in the context of the homotopy theory of s algebras as introduced by elmendorf kriz mandell and may we give algebraic arguments based on naturality properties of the topological hochschild homology spectral sequence in the process we demonstrate the utility of the unstable lower notation for the dyer lashof algebra

algebraic k theory of topological k theory we are interested in the arithmetic of ring spectra to make sense of this we must work with structured ring spectra such as s algebras ekmm symmetric ring spectra hss or rings ly we will refer to these as salgebras the commutative objects are then commutative s algebras the category of rings is embedded in the category of s algebras by the eilenberg maclane functor r hr we may therefore view an s algebra as a generalization of a ring in the algebraic sense the added flexibility of s algebras provides room for new examples and constructions which may eventually also shed light upon the category of rings itself in algebraic number theory the arithmetic of the ring of integers in a number field is largely captured by its picard group its unit group and its brauer group these are

a stable method for the evaluation of gaussian radial basis function solutions of interpolation and collocation problems radial basis functions rbfs are a powerful tool for approximating the solution of high dimensional problems they are often referred to as a meshfree method and can be spectrally accurate the best accuracy can often be achieved when the so called shape parameter of the basis functions is small which in turn tends to make the interpolation matrix increasingly ill conditioned to overcome such instability in the numerical method which arises for even the most stable problems one needs to stabilize the method in this paper we present a new stable method for evaluating gaussian radial basis function interpolants based on the eigenfunction expansion for gaussian rbfs this work enhances the ideas proposed in fasshauer and mccourt 2012 by exploiting the properties of the orthogonal eigenfunctions and their zeros we develop our approach in one and two dimensional spaces with the extension to higher dimensions proceeding analogously in the univariate setting the orthogonality of the eigenfunctions and our special collocation locations give rise to easily computable cardinal basis functions the accuracy robustness and computational efficiency of the method are tested by numerically solving several interpolation and boundary value problems in one and two dimensions high accuracy simple implementation and low complexity for high dimensional problems are the advantages of our approach on the down side our method is currently limited to the use of tensor products of unevenly spaced one dimensional data locations

structure of the global attractor for a second order strongly damped lattice system in this paper we consider the dynamical behavior of a second order strongly damped lattice system where the coupled operator is nonnegative definite symmetric firstly we prove the existence of a global attractor and give an upper bound of hausdorff dimension of the global attractor which keeps bounded for large strongly damping then we show that when the damping term is linear and the damping is suitable large the system has an unbounded one dimensional global attractor which is a restricted horizontal curve

sheared algebra maps and operation bialgebras for mod 2 homology and cohomology the mod 2 steenrod algebraa and dyer lashof al gebra r have both striking similarities and dierences arising from their common origins in lower indexed algebraic operations these algebraic operations and their relations generate a bigraded bialgebrak whose module actions are equivalent to but quite dif ferent from those ofa andr the exact relationships emerge as sheared algebra bijections which also illuminate the role of the cohomology ofk as a bialgebra k has a particularly attractive and potentially useful structure providing a bridge between those ofa andr and suggesting possible applications to the miller spectral sequence and thea structure of dickson algebras

discrete rotating waves in a ring of coupled mechanical oscillators with strong damping by establishing the monotonicity in a ring of coupled oscillators with strong damping we show the global stability of the discrete rotating wave whose existence is demonstrated by applying the schauder fixed point theorem

stable evaluation of gaussian radial basis function interpolants we provide a new way to compute and evaluate gaussian radial basis function interpolants in a stable way with a special focus on small values of the shape parameter i e for flat kernels this work is motivated by the fundamental ideas proposed earlier by bengt fornberg and his coworkers however following mercer s theorem an l 2 mathbb r d rho orthonormal expansion of the gaussian kernel allows us to come up with an algorithm that is simpler than the one proposed by fornberg larsson and flyer and that is applicable in arbitrary space dimensions d in addition to obtaining an accurate approximation of the radial basis function interpolant using many terms in the series expansion of the kernel we also propose and investigate a highly accurate least squares approximation based on early truncation of the kernel expansion

dynamics in a chain of overdamped pendula driven by constant torques this paper studies the dynamical behavior of a chain of overdamped pendula driven by constant torques with nearest neighbor coupling the coupling constant k is assumed to be 0 independent of n it is shown that when the system does not have equilibrium points the global attractor of this system is a one dimensional closed curve so no matter what input frequencies omega j are used the existence uniqueness and global stability of a limit cycle of second kind are proved therefore any solution will be frequency locked in the long time limit on the other hand if there are equilibrium points in the system any solution is bounded and converges to an equilibrium point

single waveform solutions for linear arrays of josephson junctions abstract in this paper we study the existence of single waveform solutions to system of equations governing the dynamics of series arrays of identical current biased josephson junctions which are coupled through a shared lrc load we use a homotopy argument and degree theory to prove the existence of single waveform solutions which are either in phase running solution discrete rotating waves or clustered discrete rotating waves in which the junctions are grouped into clusters of equal size which form a discrete rotating wave

on learning with integral operators a large number of learning algorithms for example spectral clustering kernel principal components analysis and many manifold methods are based on estimating eigenvalues and eigenfunctions of operators defined by a similarity function or a kernel given empirical data thus for the analysis of algorithms it is an important problem to be able to assess the quality of such approximations the contribution of our paper is two fold 1 we use a technique based on a concentration inequality for hilbert spaces to provide new much simplified proofs for a number of results in spectral approximation 2 using these methods we provide several new results for estimating spectral properties of the graph laplacian operator extending and strengthening results from von luxburg et al 2008

topological hochschild homology of connective complex k theory let ku be the connective complex k theory spectrum completed at an odd prime p we present a computation of the mod p 1 homotopy algebra of the topological hochschild homology spectrum of ku