extensions of positive definite integral kernels on the heisenberg group abstract we consider positive definite extensions of functions and distributions which are defined on a cylindrical neighborhood and positive definite in the heisenberg group when rotational invariance is assumed we show existence of p d extensions we also give formulas of the classical bochner type for all such extensions they are expanded in terms of an integral transform associated to the sub laplacian we further show that the p d extension problem is closely related to the theory of extensions of hermitian infinitesimal representations in the sense considered earlier by powers and fuglede these are extensions to integrable representations in generally larger so called dilation hilbert spaces

spectrality of a class of infinite convolutions for a finite set d z and an integer b 2 we say that b d is compatible with c z if e 2 idc b d d c c is a hadamard matrix let e 1 e a e a denote the uniformly discrete probability measure on e we prove that the class of infinite convolution moran measure b dk b 1d1 b 2d2 is a spectral measure provided that there is a common c z compatible to all the b dk and c c 0 1 b 1 we also give some examples to illustrate the hypotheses and results in particular the last condition on c is essential

spectral theory of multiple intervals we present a model for spectral theory of families of selfadjoint operators and their corresponding unitary one parameter groups acting in hilbert space the models allow for a scale of complexity indexed by the natural numbers n for each n 2 n we get families of selfadjoint operators indexed by i the unitary matrix group u n and by ii a prescribed set of n non overlapping intervals take to be the complement in r of n fixed closed finite and disjoint intervals and let l 2 be the corresponding hilbert space moreover given b 2 u n then both the lengths of the respective intervals and the gaps between them show up as spectral parameters in our corresponding spectral resolutions within l 2 our models have two advantages one they encompass realistic features from quantum theory from acoustic wave equations and their obstacle scattering as well as from harmonic analysis secondly each choice of the parameters in our models n 2 n b 2 u n and interval configura tion allows for explicit computations and even for closed form formulas computation of spectral resolutions of generalized eigenfunctions in l 2 for the continuous part of spectrum and for scattering coecients our models further allow us to identify embedded point spectrum in the continuum corresponding for example to bound states in scattering to trapped states and to barriers in quantum scattering the possibilities for the discrete atomic part of spectrum includes both periodic and non periodic distributions

differential variational inequalities this paper introduces and studies the class of differential variational inequalities dvis in a finite dimensional euclidean space the dvi provides a powerful modeling paradigm for many applied problems in which dynamics inequalities and discontinuities are present examples of such problems include constrained time dependent physical systems with unilateral constraints differential nash games and hybrid engineering systems with variable structures the dvi unifies several mathematical problem classes that include ordinary differential equations odes with smooth and discontinuous right hand sides differential algebraic equations daes dynamic complementarity systems and evolutionary variational inequalities conditions are presented under which the dvi can be converted either locally or globally to an equivalent ode with a lipschitz continuous right hand function for dvis that cannot be so converted we consider their numerical resolution via an euler time stepping procedure which involves the solution of a sequence of finite dimensional variational inequalities borrowing results from differential inclusions dis with upper semicontinuous closed and convex valued multifunctions we establish the convergence of such a procedure for solving initial value dvis we also present a class of dvis for which the theory of dis is not directly applicable and yet similar convergence can be established finally we extend the method to a boundary value dvi and provide conditions for the convergence of the method the results in this paper pertain exclusively to systems with index not exceeding two and which have absolutely continuous solutions

iterated function systems ruelle operators and invariant projective measures we introduce a fourier based harmonic analysis for a class of discrete dynamical systems which arise from iterated function systems our starting point is the following pair of special features of these systems 1 we assume that a measurable space x comes with a finite to one endomorphism r x x which is onto but not one to one 2 in the case of affine iterated function systems ifss in r d this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets b l in r d of the same cardinality which generate complex hadamard matrices our harmonic analysis for these iterated function systems ifs x is based on a markov process on certain paths the probabilities are determined by a weight function w on x from w we define a transition operator r w acting on functions on x and a corresponding class h of continuous r w harmonic functions the properties of the functions in h are analyzed and they determine the spectral theory of l 2 for affine ifss we establish orthogonal bases in l 2 these bases are generated by paths with infinite repetition of finite words we use this in the last section to analyze tiles in r d

feedback particle filter the feedback particle filter introduced in this paper is a new approach to approximate nonlinear filtering motivated by techniques from mean field game theory the filter is defined by an ensemble of controlled stochastic systems the particles each particle evolves under feedback control based on its own state and features of the empirical distribution of the ensemble the feedback control law is obtained as the solution to an optimal control problem in which the optimization criterion is the kullback leibler divergence between the actual posterior and the common posterior of any particle the following conclusions are obtained for diffusions with continuous observations 1 the optimal control solution is exact the two posteriors match exactly provided they are initialized with identical priors 2 the optimal filter admits an innovation error based gain feedback structure 3 the optimal feedback gain is obtained via a solution of an euler lagrange boundary value problem the feedback gain equals the kalman gain in the linear gaussian case numerical algorithms are introduced and implemented in two general examples and a neuroscience application involving coupled oscillators in some cases it is found that the filter exhibits significantly lower variance when compared to the bootstrap particle filter

central limit theorem for sequential monte carlo methods and its application to bayesian inference the term sequential monte carlo methods or equivalently particle filters refers to a general class of iterative algorithms that performs monte carlo approximations of a given sequence of distributions of interest t we establish in this paper a central limit theorem for the monte carlo estimates produced by these computational methods this result holds under minimal assumptions on the distributions t and applies in a general framework which encompasses most of the sequential monte carlo methods that have been considered in the literature including the resample move algorithm of gilks and berzuini j r stat soc ser b stat methodol 63 2001 127 146 and the residual resampling scheme the corresponding asymptotic variances provide a convenient measurement of the precision of a given particle filter we study in particular in some typical examples of bayesian applications whether and at which rate these asymptotic variances diverge in time in order to assess the long term reliability of the considered algorithm

nonuniform multiresolution analyses and spectral pairs abstract a generalization of the notion of multiresolution analysis based on the theory of spectral pairs is considered in contrast to the standard setting the associated subspacev0ofl2 r has as an orthonormal basis a collection of translates of the scaling function of the form x where 0 r n 2z n 1 is an integer andris an odd integer with 1 r 2n 1 such thatrandnare relatively prime andzis the set of all integers furthermore the corresponding dilation factor is 2n the case wheren 1 corresponding to the usual definition of a multiresolution analysis with dilation factor 2 a necessary and sufficient condition for the existence of associated wavelets which is always satisfied whenn 1 or 2 is obtained and is shown to always hold if the fourier transform of is a constant multiple of the characteristic function of a set

analysis of orthogonality and of orbits in affine iterated function systems we introduce a duality for affine iterated function systems aifs which is naturally motivated by group duality in the context of traditional harmonic analysis our affine systems yield fractals defined by iteration of contractive affine mappings we build a duality for such systems by scaling in two directions fractals in the small by contractive iterations and fractals in the large by recursion involving iteration of an expansive matrix by a fractal in the small we mean a compact attractor x supporting hutchinson s canonical measure and we ask when is a spectral measure i e when the hilbert space l 2 mu has an orthonormal basis onb of exponentials e lambda lambda in lambda we further introduce a fourier duality using a matched pair of such affine systems using next certain extreme cycles and positive powers of the expansive matrix we build fractals in the large which are modeled on lacunary fourier series and which serve as spectra for x our two main results offer simple geometric conditions allowing us to decide when the fractal in the large is a spectrum for x our results in turn are illustrated with concrete sierpinski like fractals in dimensions 2 and 3

spectral sets and tiles on vector space over local fields abstract in this article we prove that a set which is of positive and finite haar measure in k d is a spectral set with quasi lattice as a spectrum if and only if it tiles k d by the dual quasi lattice as a translation set moreover we obtain some results connected with the dual spectral set conjecture on vector space over local fields

fourier asymptotics of fractal measures abstract a measure on r n will be called locally uniformly dimensional if br x cr for all r 1 and all x where br x denotes the ball of radius r about x for l 2 d the measure d is in i so d is well defined we show it is locally in l2 and sup r 1 r n b r y f d 2 d c f 2 under additional hypotheses we show that lim r r n b r y f d 2 2 d is comparable in size to 2 2 a number of other related results are established the special case when is an integer and is the surface measure on a c1 manifold was treated by s agmon and l hormander j analyse math 30 1976 1 38

a high accuracy method for solving odes with discontinuous right hand side ordinary differential equations with discontinuities in the state variables require a differential inclusion formulation to guarantee existence 8 from this formulation a high accuracy method for solving such initial value problems is developed which can give any order of accuracy and tracks the discontinuities the method uses an active set approach and determines appropriate active sets from solutions to linear complementarity problems convergence results are established under some non degeneracy assumptions the method has been implemented and results compare favourably with previously published methods 7 21

optimal adaptive solution of piecewise regular systems of ivps with unknown switching hypersurface we study the solution of piecewise regular systems of ivps with a switching hypersurface defined by an unknown event function our aim is to design an algorithm that is not based on heuristic steps it does not use information on the event function and preserves the order known for regular systems the algorithm is based on the adaptive mesh which differs from the equidistant mesh only by a single point we provide a rigorous that is independent of heuristic arguments analysis of its error and cost

keller s cube tiling conjecture is false in high dimensions o h keller conjectured in 1930 that in any tiling of r n by unit n cubes there exist two of them having a complete facet in common 0 perron proved this conjecture for n 6 we show that for all n 10 there exists a tiling of r n by unit n cubes such that no two n cubes have a complete facet in common

how to avoid the curse of dimensionality scalability of particle filters with and without importance weights particle filters are a popular and flexible class of numerical algorithms to solve a large class of nonlinear filtering problems however standard particle filters with importance weights have bee

on characterizations of spectra and tilings abstract in a recent paper lagarias reeds and wang established a characterization of spectra and tilings that can be used to prove a conjecture of jorgensen and pedersen by keller s criterion different techniques to prove these facts have also been developed by kolountzakis iosevich and pedersen the primary aim of this paper is to present an elementary method of describing certain characterizations of spectra and tilings to illustrate this method we first give a simple proof of this characterization we then use the method to derive some characteristic results connected with the dual fuglede s spectral set conjecture the results here extend several known conclusions in a simple manner

endomorphisms of b h ii finitely correlated states on on we identify sets of conjugacy classes of ergodic endomorphisms of b h where h is a fixed separable hilbert space they correspond to certain equivalence classes of pure states on the cuntz algebras onwherenis the powers index these states called finitely correlated states and strongly asymptotically shift invariant states are defined and characterized the subsets of these states defining shifts will in general be identified in a later work but here an interesting cross section for the conjugacy classes of shifts called diagonalizable shifts is introduced and studied

commuting self adjoint extensions of symmetric operators defined from the partial derivatives we consider the problem of finding commuting self adjoint extensions of the partial derivatives 1 i xj j 1 d with domain cc where the self adjointness is defined relative to l2 and is a given open subset of rd the measure on is lebesgue measure on rd restricted to the problem originates with segal and fuglede and is difficult in general in this paper we provide a representation theoretic answer in the special case when i 2 and i is an open interval we then apply the results to the case when is a d cube id and we describe possible subsets rd such that e id is an orthonormal basis in l2 id

wavelets associated with nonuniform multiresolution analyses and one dimensional spectral pairs a generalization of mallat s classic multiresolution analysis mra based on the theory of spectral pairs was considered in two papers by gabardo and nashed in this nonstandard setting the translation set is no longer a subgroup or a translate of a subgroup of r but is a spectrum associated with a one dimensional spectral pair in this paper we continue the study based on this nonstandard setting and give the characterization for nonuniform wavelets associated with a nonuniform mra these characterizations are consistent with both the known necessary and sufficient conditions for the existence of nonuniform mra wavelets and the known characterization for standard dyadic wavelets associated with an mra

optimal control of systems with discontinuous differential equations in this paper we discuss the problem of verifying and computing optimal controls of systems whose dynamics is governed by differential systems with a discontinuous right hand side in our work we are motivated by optimal control of mechanical systems with coulomb friction which exhibit such a right hand side notwithstanding the impressive development of nonsmooth and set valued analysis these systems have not been closely studied either computationally or analytically we show that even when the solution crosses and does not stay on the discontinuity differentiating the results of a simulation gives gradients that have errors of a size independent of the stepsize this means that the strategy of optimize the discretization will usually fail for problems of this kind we approximate the discontinuous right hand side for the differential equations or inclusions by a smooth right hand side for these smoothed approximations we show that the resulting gradients approach the true gradients provided that the start and end points of the trajectory do not lie on the discontinuity and that euler s method is used where the step size is sufficiently small in comparison with the smoothing parameter numerical results are presented for a crude model of car racing that involves coulomb friction and slip showing that this approach is practical and can handle problems of moderate complexity

spectral theory for borel sets in rn of finite measure abstract let be a pair of subsets in r n such that has finite positive lebesgue measure the pair is called a spectral pair if the functions e i x for form an orthogonal total family in l 2 i e when restricted to our present work continues earlier papers on simultaneous diagonalization of vector fields for the special case when is also assumed open however the setting here is more general and the results are stated in the measure theoretic category in considering fundamental domains for lattices for example we allow distinct lattice point translates of the domain to overlap on sets of measure zero rather than the traditional condition of empty overlap two classes of examples of spectral pairs are known fundamental domains and simple factors the latter may be identified in terms of a certain finite group action we show that every spectral pair can be factored in the sense of cartesian products as a product of two pairs of lower dimension where the first factor is a fundamental domain and the second a simple factor the dimensions add a simple factor yields in a natural way a finite covering of an associated compact torus but not every such covering corresponds to a spectral pair we give a complete spectral characterization of the finite coverings and identify those which correspond to simple factors the latter identification is phrased in terms of generators and relations for certain c algebras which are analogous to but different from algebras studied earlier by cuntz and arveson

on spectral cantor measures a probability measure in rd is called a spectral measure if it has an orthonormal basis consisting of exponentials in this paper we study spectral cantor measures we establish a large class of such measures and give a necessary and sufficient condition on the spectrum of a spectral cantor measure these results extend the studies by jorgensen and pedersen j anal math 75 1998 185 228 and strichartz j d analyse math 81 2000 209 238

tiling sets and spectral sets over finite fields abstract we study tiling and spectral sets in vector spaces over prime fields the classical fuglede conjecture in locally compact abelian groups says that a set is spectral if and only if it tiles by translation this conjecture was disproved by t tao in euclidean spaces of dimensions 5 and higher using constructions over prime fields in vector spaces over finite fields of prime order and lifting them to the euclidean setting over prime fields when the dimension of the vector space is less than or equal to 2 it has recently been proven that the fuglede conjecture holds see 6 in this paper we study this question in higher dimensions over prime fields and provide some results and counterexamples in particular we prove the existence of spectral sets which do not tile in z p 5 for all odd primes p and z p 4 for all odd primes p such that p 3 mod 4 although counterexamples in low dimensional groups over cyclic rings z n were previously known they were usually for non prime n or a small sporadic set of primes p rather than general constructions this paper is a result of a research experience for undergraduates program ran at the university of rochester during the summer of 2015 by a iosevich j pakianathan and g petridis

orthogonal exponentials translations and bohr completions abstract we are concerned with an harmonic analysis in hilbert spaces l 2 where is a probability measure on r n the unifying question is the presence of families of orthogonal complex exponentials e x exp 2 i x in l 2 this question in turn is connected to the existence of a natural embedding of l 2 into an l 2 space of bohr almost periodic functions on r n in particular we explore when l 2 contains an orthogonal basis of e functions for in a suitable discrete subset in r n i e when the measure is spectral we give a new characterization of finite spectral sets in terms of the existence of a group of local translation we also consider measures that arise as fixed points in the sense of hutchinson of iterated function systems ifss and we specialize to the case when the function system in the ifs consists of affine and contractive mappings in r n we show in this case that if is then assumed spectral then its partitions induced by the ifs at hand have zero overlap measured in this solves part of the aba wang conjecture as an application of the new non overlap result we solve the spectral pair problem for bernoulli convolutions advancing in this way a theorem of ka sing lau in addition we present a new perspective on spectral measures and orthogonal fourier exponentials via the bohr compactification

sliding motion on discontinuity surfaces of high co dimension a construction for selecting a filippov vector field in this paper we consider the issue of sliding motion in filippov systems on the intersection of two or more surfaces to this end we propose an extension of the filippov sliding vector field on manifolds of co dimension p with p 2 our model passes through the use of a multivalued sign function reformulation to justify our proposal we will restrict to cases where the sliding manifold is attractive for the case of co dimension p 2 we will distinguish between two types of attractive sliding manifold node like and spiral like the case of node like attractive manifold will be further extended to the case of p 3 finally we compare our model to other existing methodologies on some examples

harmonic analysis on constant curvature surfaces with point singularities let rn2 denote the n fold covering space of the plane with the origin deleted 0 n parametrized by polar coordinates r with periodic of period 2 n and equipped with the flat riemannian metric dr2 r2 d 2 there is a natural harmonic analysis with the fourier transform given by f c 12 0 f r j v r eiv r dr d fourier inversion formula f c 0 f r j v r eiv d dv and plancherel formula f 22 2 0 f v 2 d dv in the case n and analogous expressions with v restricted to the values kn when n is finite this harmonic analysis realizes the joint spectral decompositions of the self adjoint operators i and where is the dirichlet or friedrichs extension of the laplace beltrami operator defined on d this paper gives a detailed development of this harmonic analysis including the effect of basic differentiation and multiplication operators on the fourier transform the theory of l2 sobolev spaces hk and the distinctions between h2 the domain of and the space of functions f with both f and f in l2 in the process it is necessary to study certain generalizations of the hilbert transform that are related to the theory of hankel transforms two applications of this harmonic analysis are given to the study of a natural radon transform or x ray transform on rn2 and to the solution of the heat equation on rn2 the latter also gives information about the heat equation in a sector of the plane with dirichlet or neumann boundary conditions and about brownian motion in the plane taking into account the winding number of a brownian path with respect to the origin analogous problems for spaces of non zero constant curvature are discussed and in this context some higher dimensional spaces arise naturally

tiling the line with translates of one tile a region t is a closed subset of the real line of positive finite lebesgue measure which has a boundary of measure zero call a region t a tile if bbb r can be tiled by measure disjoint translates of t for a bounded tile all tilings of bbb r with its translates are periodic and there are finitely many translation equivalence classes of such tilings the main result of the paper is that for any tiling of bbb r by a bounded tile any two tiles in the tiling differ by a rational multiple of the minimal period of the tiling from it we deduce a structure theorem characterizing such tiles in terms of complementing sets for finite cyclic groups

a filippov sliding vector field on an attracting co dimension 2 discontinuity surface and a limited loss of attractivity analysis we consider sliding motion in the sense of filippov on a discontinuity surface of co dimension 2 we characterize and restrict to the case of being attractive through sliding in this situation we show that a certain filippov sliding vector field ff suggested in alexander and seidman 1998 2 di bernardo et al 2008 6 dieci and lopez 2011 10 exists and is unique we also propose a characterization of first order exit conditions clarify its relation to generic co dimension 1 losses of attractivity for and examine what happens to the dynamics on for the aforementioned vector field ff examples illustrate our results

switching relay and complementarity systems a tutorial on their well posedness and relationships this paper reviews the relationships between switching systems defined from a partition of the state space into convex cells and relay or complementarity dynamical systems which are other classes of discontinuous systems well posedness results i e results on the existence and the uniqueness of solutions for different classes of relay and complementarity systems which are also switching systems are presented the reverse issue when can a switching system be rewritten equivalently as a relay or a complementarity system is also tackled many examples from mechanics circuits and biology illustrate the developments throughout the paper the paper focuses on systems with continuous solutions i e with no state jumps convexity is the central property

spectrum is periodic for n intervals abstract in this paper we study spectral sets which are unions of finitely many intervals in r we show that any spectrum associated with such a spectral set is periodic with the period an integral multiple of the measure of as a consequence we get a structure theorem for such spectral sets and observe that the generic case is that of the equal interval case

spectral sets whose spectrum is a lattice with a base abstract let l be a finite subset of euclidean d dimensional space r d and let s be a lattice in r d such that l l l l l s 0 a classification of all lebesgue measurable subsets of r d for which the functions e x exp i 2 x x l s l s l l s s form an orthogonal basis for l 2 is given in terms of geometric and combinatorial data the data is derived from l and s

fourier frequencies in affine iterated function systems abstract we examine two questions regarding fourier frequencies for a class of iterated function systems ifs these are iteration limits arising from a fixed finite families of affine and contractive mappings in r d and the ifs refers to such a finite system of transformations or functions the iteration limits are pairs x where x is a compact subset of r d the support of and the measure is a probability measure determined uniquely by the initial ifs mappings and a certain strong invariance axiom the two questions we study are 1 existence of an orthogonal fourier basis in the hilbert space l 2 x and 2 explicit constructions of fourier bases from the given data defining the ifs

translation representations and scattering by two intervals studying unitary one parameter groups in hilbert space u t h we show that a model for obstacle scattering can be built up to unitary equivalence with the use of translation representations for l2 functions in the complement of two finite and disjoint intervals the model encompasses a family of systems u t h for each we obtain a detailed spectral representation and we compute the scattering operator and scattering matrix we illustrate our results in the lax phillips model where u t h represents an acoustic wave equation in an exterior domain and in quantum tunneling for dynamics of quantum states

a survey of convergence results on particle filtering methods for practitioners optimal filtering problems are ubiquitous in signal processing and related fields except for a restricted class of models the optimal filter does not admit a closed form expression particle filtering methods are a set of flexible and powerful sequential monte carlo methods designed to solve the optimal filtering problem numerically the posterior distribution of the state is approximated by a large set of dirac delta masses samples particles that evolve randomly in time according to the dynamics of the model and the observations the particles are interacting thus classical limit theorems relying on statistically independent samples do not apply in this paper our aim is to present a survey of convergence results on this class of methods to make them accessible to practitioners

sliding motion in filippov differential systems theoretical results and a computational approach in this work we discuss some theoretical and numerical aspects of solving differential equations with discontinuous right hand sides of filippov type in particular i we propose second order corrections to the theory of filippov ii we provide a systematic and nonambiguous way to define the vector field on the intersection of several surfaces of discontinuity and iii we propose and implement a numerical method to approximate a trajectory of systems with discontinuous right hand sides and illustrate its performance on a few examples

a duality for endomorphisms of von neumann algebras we consider endomorphisms of von neumann algebras let m be a von neumann algebra represented on a hilbert space h and let m be the corresponding commutant let end m be given and suppose m has a cyclic vector in h such that the corresponding state leaves invariant then there is a dual completely positive mapping on m which we find and describe each of the two and has an associated spectral group and we show that the group for is contained in that for we consider the following three restrictions on i is a shift on m ii is strongly ergodic and iii is ergodic we give spectral theoretic conditions on using the two groups described above to fall into each of the three classes we also show that the two groups are conjugacy invariants and we discuss the case of cocycle conjugacy

spectral theory of finite volume domains in rn abstract let be an arbitrary open subset of r n of finite positive measure and assume the existence of a subset r n such that the exponential functions e exp i 1x1 nxn 1 n form an orthonormal basis for l 2 with normalized measure assume 0 and define subgroups k and a of r n by k 0 r n 2 z a a r n ua m u a m where ut is the unitary representation of r n on l 2 given by ute eit e t r n and where m is the multiplication algebra of l on l2 assume that a is discrete then there is a discrete subgroup d a of dimension n a fundamental domain d for d and finite sets of representers r r r each containing 0 r for a k in k0 and r for a k in a such that is disjoint union of translates of d a r a d neglecting null sets and r d0 if r is a set of representers for d a in d then r k is a translation set for i e r n direct sum neglecting null sets the case a r n corresponds to d d0 and k this last case corresponds in turn to a function theoretic assumption of forelli

fourier duality for fractal measures with affine scales for a family of fractal measures wend an explicit fourier d uality the measures in the pair have compact support in r d and they both have the same matrix scaling but the two use dierent translation vectors one by a subset b in r d and the other by a related subset l among other things we show that there is then a pair of innite disc rete sets l and b in r d such that the l fourier exponentials are orthogonal in l 2 b and the b fourier exponentials are orthogonal in l 2 l these sets of orthogonal frequencies are typically lac unary and they will be obtained by scaling in the large the nature of our duality is explored below both in higher dimensions and for examples on the real line our duality pairs do not always yield orthonormal fourier bases in the respective l 2 hilbert spaces but depending on the geometry of certainnite orbit s we show that they do in some cases we further show that there are new and surprising scaling symmetries of relevance for the ergodic theory of these ane fractal measures

stability of the linear complementarity problem at a solution point in this paper we study the behavior of a solution of the linear complementarity problem when data are perturbed we give characterizations of strong stability of the linear complementarity problem at a solution in the case of stability we give sufficient and necessary conditions

lipschitz algebras and derivations ii exterior differentiation abstract basic aspects of differential geometry can be extended to various non classical settings lipschitz manifolds rectifiable sets sub riemannian manifolds banach manifolds wiener space etc although the constructions differ in each of these cases one can define a module of measurable 1 forms and a first order exterior derivative we give a general construction which applies to any metric space equipped with a finite measure and produces the desired result in all of the above cases it agrees with cheeger s construction when the latter is defined it also applies to an important class of dirichlet spaces where however the known first order differential calculus in general differs from ours although the two are related

multivariable feedback particle filter this paper presents the multivariable extension of the feedback particle filter fpf algorithm for the nonlinear filtering problem in continuous time the fpf is a control oriented approach to particle filtering the approach does not require importance sampling or resampling and offers significant variance improvements in particular the algorithm can be applied to systems that are not stable this paper describes new representations and algorithms for the fpf in the general multivariable nonlinear non gaussian setting theory surrounding the fpf is improved exactness of the fpf is established in the general setting as well as well posedness of the associated boundary value problem to obtain the filter gain a galerkin finite element algorithm is proposed for approximation of the gain its performance is illustrated in numerical experiments

harmonic analysis and fractal limit measures induced by representations of a certain c algebra we describe a class of measurable subsets in rd such that l2 has an orthogonal basis of frequencies e x ei2 x x indexed by rd we show that such spectral pairs have a self similarity which may be used to generate associated fractal measures typically with cantor set support the hilbert space l2 does not have a total set of orthogonal frequencies but a harmonic analysis of may be built instead from a natural representation of the cuntz c algebra which is constructed from a pair of lattices supporting the given spectral pair we show conversely that such a pair may be reconstructed from a certain cuntz representation given to act on l2

unitary groups and spectral sets abstract we study spectral theory for bounded borel subsets of r and in particular finite unions of intervals for hilbert space we take l 2 of the union of the intervals this yields a boundary value problem arising from the minimal operator d 1 2 i d d x with domain consisting of c functions vanishing at the endpoints we offer a detailed interplay between geometric configurations of unions of intervals and a spectral theory for the corresponding self adjoint extensions of d and for the associated unitary groups of local translations while motivated by scattering theory and quantum graphs our present focus is on the fuglede spectral pair problem stated more generally this problem asks for a determination of those bounded borel sets in r k such that l 2 has an orthogonal basis of fourier frequencies spectrum i e a total set of orthogonal complex exponentials restricted to in the general case we characterize borel sets having this spectral property in terms of a unitary representation of r acting by local translations the case of k 1 is of special interest hence the interval configurations we give a characterization of those geometric interval configurations which allow fourier spectra directly in terms of the self adjoint extensions of the minimal operator d this allows for a direct and explicit interplay between geometry and spectra

a class of gaussian processes with fractional spectral measures abstract we study a family of stationary increment gaussian processes indexed by time these processes are determined by certain measures generalized spectral measures and our focus here is on the case when the measure is a singular measure we characterize the processes arising from when is in one of the classes of affine selfsimilar measures our analysis makes use of kondratiev white noise spaces with the use of a priori estimates and the wick calculus we extend and sharpen see theorem 7 1 earlier computations of ito stochastic integration developed for the special case of stationary increment processes having absolutely continuous measures we further obtain an associated ito formula see theorem 8 1

spectral theory of commuting self adjoint partial differential operators let denote a connected and open subset of rn the existence of n commuting self adjoint operators h1 hn on l2 such that each hj is an extension of i xj acting on cc is shown to be equivalent to the existence of a measure on rn such that f tf the fourier transform of f is unitary from l2 onto it is shown that the support of can be chosen as a subgroup of rn iff h1 hn can be chosen such that the unitary groups generated by h1 hn act multiplicatively on l2 this happens iff after correction by a null set forms a system of representatives for the quotient of rn by some subgroup i e iff is essentially a fundamental domain

harmonic analysis of fractal measures induced by representations of a certain c algebra we describe a class of measurable subsets s in md such that l2 q has an orthogonal basis of frequencies ex x el2nx x x e 2 indexed by x e a c ir we show that such spectral pairs i2 a have a self similarity which may be used to generate associated fractal measures p with cantor set support the hubert space l2 fi does not have a total set of orthogonal frequencies but a harmonic analysis of ii may be built instead from a natural representation of the cuntz c algebra which is constructed from a pair of lattices supporting the given spectral pair i2 a we show conversely that such a pair may be reconstructed from a certain cuntz representation given to act on l2 i

optimal solution of a class of non autonomous initial value problems with unknown singularities in this paper we are interested in a rigorous analysis of the solution of a class of initial value problems with singularities we consider scalar non autonomous ivps with separated variables and one unknown singularity in each variable which leads to four unknown events in the two dimensional space many algorithms proposed in the literature for singular ivps are practically oriented they do not avoid heuristic arguments and are often checked for efficiency by numerical experiments we design an adaptive algorithm with no heuristic steps for solving the considered problems and provide rigorous bounds on the error we show that in spite of the presence of singularities the algorithm preserves the optimal error known for the regular case lower bounds on the error of any algorithm in the case of a larger number of singularities are also discussed

when does a bernoulli convolution admit a spectrum abstract in this paper we solve a long standing problem on bernoulli convolutions in particular we show that the bernoulli convolution with contraction rate 0 1 admits a spectrum if and only if is the reciprocal of an even integer

compact open spectral sets in qp abstract in this article we prove that a compact open set in the field q p of p adic numbers is a spectral set if and only if it tiles q p by translation and also if and only if it is p homogeneous which is easy to check we also characterize spectral sets in z p n z p 2 prime n 1 integer by tiling property and also by homogeneity moreover we construct a class of singular spectral measures in q p some of which are self similar measures

spectral sets and factorizations of finite abelian groups aspectral setis a subset ofrnwith lebesgue measure 0 such that there exists a set of exponential functions which form an orthogonal basis ofl2 the spectral set conjecture of b fuglede states that a set 0 is a spectral set if and only if tilesrnby translation we study sets which tilernusing a rational periodic tile set s zn a where a 1 n1 z 1 nn zis finite we characterize geometrically bounded measurable sets that tilernwith such a tile set certain tile sets s have the property that every bounded measurable set which tilesrnwith s is a spectral set with a fixed spectrum s we call sa universal spectrum for such s we give a necessary and sufficient condition for a rational periodic set to be a universal spectrum for s which is expressed in terms of factorizationsa b gwhereg zn1 znn anda a modzn in dimensionn 1 we show that s has a universal spectrum whenevern1is the order of a good group in the sense of hajos and for various other sets s

on fourier transforms of functions supported on sets of finite lebesgue measure soit g un groupe abelien localement compact et son groupe dual soient m et m les mesures de haar sur g et on demontre soit f l 1 r n et f sa transformee de fourier et soit a x r n f x 0 et b r n f 0 alors m a et m b f 0 a e m ici m et m denotent la mesure de lebesgue sur r n resp r n