linear estimation of boundary value stochastic processes part ii 1 d smoothing problems this paper addresses the fixed interval smoothing problem for linear two point boundary value stochastic processes of the type introduced by krener 5 as these models are not markovian kalman filtering and associated smoothing algorithms are not applicable the smoothing problem for this class of noncausal processes is solved here by an application of the estimator solution which is developed in part i of this paper 3 via the method of complementary models for an n th order model this approach yields the smoother as a 2 n th order two point boundary value problem it is shown that this smoother can be realized in a stable two filter form which is remarkably similar to two filter smoothers for causal processes in addition expressions for the smoothing error and smoothing error covariance are developed these equations are employed to perform a covariance analysis of estimating the temperature and heat flow in a cooling fin

eichler shimura theory for mock modular forms we use mock modular forms to compute generating functions for the critical values of modular l functions and we answer a generalized form of a question of kohnen and zagier by deriving the extra relation that is satisfied by even periods of weakly holomorphic cusp forms to obtain these results we derive an eichler shimura theory for weakly holomorphic modular forms and mock modular forms this includes two eichler shimura isomorphisms a multiplicity two hecke theory a correspondence between mock modular periods and classical periods and a haberland type formula which expresses petersson s inner product and a related antisymmetric inner product on m k in terms of periods

the f q mock theta function conjecture and partition ranks in 1944 freeman dyson initiated the study of ranks of integer partitions here we solve the classical problem of obtaining formulas for ne n resp no n the number of partitions of n with even resp odd rank thanks to rademacher s celebrated formula for the partition function this problem is equivalent to that of obtaining a formula for the coefficients of the mock theta function f q a problem with its own long history dating to ramanujan s last letter to hardy little was known about this problem until dragonette in 1952 obtained asymptotic results in 1966 g e andrews refined dragonette s results and conjectured an exact formula for the coefficients of f q by constructing a weak maass poincare series whose holomorphic part is q 1f q24 we prove the andrews dragonette conjecture and as a consequence obtain the desired formulas for ne n and no n

minimality and irreducibility of time invariant linear boundary value systems abstract minimality and irreducibility in the class of time invariant linear systems with well posed boundary conditions are introduced and characterized necessary and sufficient conditions for similarity of irreducible or minimal time invariant systems with well posed boundary conditions are given the problem of stable minimality is analysed some of the results are specified for stationary and displacement systems

unicity of minimal rank completions for tri diagonal partial block matrices abstract in this paper we establish necessary and sufficient conditions for a minimal rank completion of a tri diagonal partial block matrix to be unique the case of 4 4 block matrices with four consecutive diagonals is also considered along with its application to the problem of completing a matrix and its inverse

structured quadratic inverse eigenvalue problem i serially linked systems quadratic pencils arising from applications are often inherently structured factors contributing to the structure include the connectivity of elements within the underlying physical system and the mandatory nonnegativity of physical parameters for physical feasibility structural constraints must be respected consequently they impose additional challenges on the inverse eigenvalue problems which intend to construct a structured quadratic pencil from prescribed eigeninformation knowledge of whether a structured quadratic inverse eigenvalue problem is solvable is interesting in both theory and applications however the issue of solvability is problem dependent and has to be addressed structure by structure this paper considers one particular structure where the elements of the physical system if modeled as a mass spring system are serially linked the discussion recasts both undamped and damped problems in a framework of inequality systems that can be adapted for numerical computation some open questions are described

minimal representations of semiseparable kernels and systems with separable boundary conditions abstract the simplest representations of triangular parts of finite rank kernels are analysed criteria for uniqueness up to similarity are given the results are applied to the problem of minimal realization of systems with separable boundary conditions

unique minimal rank extensions of triangular operators the problem to find the minimal rank extensions of the lower triangular part of a finite rank matrix kernel appears in a natural way see 3 in the study of minimal representations of semi separable integral operators or equivalently of minimal realizations of boundary value systems in particular one is interested in those kernels for which the minimal rank extension is unique in this paper we give a full description of this class of so called unique matrix kernels we also define and characterize lower uniqueness for finite and semi infinite block operator matrices to state one of our main results let k t s be an m x n matrix kernel defined on the square a b x a b following 3 we say that k is lower separable if the lower triangular part k of k which is defined by kl t 8 k t s a s t b admits a finite rank extension i e there exists a finite rank matrix kernel h on a b x a b such that k is the lower triangular part of h recall that for an m x n matrix kernel h on a fl x y s the rank of

similarity and reduction for time varying linear systems with well posed boundary conditions the classical theory of similarity and reduction of causal and anti causal systems is generalized and extended to time varying linear systems with well posed boundary conditions two main problems are solved the first is to determine to what extent a system with boundary conditions can be simplified by similarity and reduction the second problem is to find the invariants of such a simplification special attention is given to the time invariant case

linear estimation of boundary value stochastic processes part i the role and construction of complementary models this paper presents a substantial extension of the method of complementary models for minimum variance linear estimation introduced by weinert and desai in their important paper 1 specifically the method of complementary models is extended to solve estimation problems for both discrete and continuous parameter linear boundary value stochastic processes in one and higher dimensions a major contribution of this paper is an application of green s identity in deriving a differential operator representation of the estimator to clarify the development and to illustrate the range of applications of our approach two brief examples are provided one is a 1 d discrete two point boundary value process and the other is a 2 d process governed by poisson s equation on the unit disk

construction and classification of minimal representations of semi separable kernels abstract an algorithm to construct a minimal lower separable representation out of a lower separable representation of a kernel of an integral operator is given the minimal lower separable representations of a kernel are classified the number of non similar minimal lower separable representations of a kernel is discussed

zagier type dualities and lifting maps for harmonic maass jacobi forms abstract the real analytic jacobi forms of zwegers phd thesis play an important role in the study of mock theta functions and related topics but have not been part of a rigorous theory yet in this paper we introduce harmonic maass jacobi forms which include the classical jacobi forms as well as zwegers functions as examples maass jacobi poincare series also provide prime examples we compute their fourier expansions which yield zagier type dualities and also yield a lift to skew holomorphic jacobi poincare series finally we link harmonic maass jacobi forms to different kinds of automorphic forms via a commutative diagram

on state models of linear time variant discrete time systems a decoupled full dimension form is introduced for state models in this form the dimension of the state vector is time invariant which is equal to the dimension of the state space at each time instant therefore the dimension of the state vector is minimized from the time variant point of view and all the state variables are independent moreover the state vector in a state model of the decoupled full dimension form is partitioned into two decoupled subvectors each of which alone forms a subsystem one of the subsystems is an fir system whose state vector is dependent only on inputs at finitely past instants in addition this subsystem is completely reachable the other subsystem is an iir system whose state space is equal to the force free solution set of the state vector of the subsystem it is proved that every state model in the conventional form can be mapped by a linear one to one map into an equivalent state model in the decoupled full dimension form in comparison with a state model in the

minimal rank completions for block matrices abstract this paper deals with the following problem of completion of block matrices let aij i j be given matrices find additional matrices aij such that the completion a aij ni j 1 has lowest possible rank the structure of the set of all minimal rank completions is studied special attention is paid to the toeplitz case together with its connection to the partial realization problem

inverse eigenvalue problem for a modified vibrating system let lambda i 1 n be the eigenvalues of a simply connected mass spring system suppose that a simple oscillator of mass m and stiffness k is attached to the system and let mu i 1 n

stability stochastic stationarity and generalized lyapunov equations for two point boundary value descriptor systems the authors introduce the concept of internal stability for two point boundary value descriptor systems tpbvdss since tpbvdss are defined only over a finite interval the concept of stability is not easy to formulate the definition which is used consists of requiring that as the length of the interval of definition increases the effect of boundary conditions on states located close to the center of the interval should go to zero stochastic tpbvdss are studied and the property of stochastic stationarity is characterized in terms of a generalized lyapunov equation satisfied by the variance of the boundary vector a second generalized lyapunov equation satisfied by the state variance of a stochastically stationary tpbvds is also introduced and the existence and uniqueness of positive definite solutions to this equation are then used to characterize the property of internal stability

harmonic maass forms of weight 1 the object of this paper is to initiate a study of the fourier coefficients of a weight one harmonic maass form and relate them to the complex galois representation associated to a weight one newform which is the form s image under a certain differential operator in this paper our focus will be on weight one dihedral newforms of prime level p 3 mod 4 in this case we give properties of the fourier coefficients that are similar to and sometimes reduce to cases of stark s conjectures on derivatives of l functions we also give a new modular interpretation of certain products of differences of singular moduli studied by gross and zagier finally we provide some numerical evidence that the fourier coefficients of a mock modular form whose shadow is exotic are similarly related to the associated complex galois representation

transfer functions and operator theory abstract the notion of a transfer function from systems theory serves as an important tool in dealing with certain problems in operator and matrix theory this is illustrated with a variety of material taken from existing literature a striking feature of the results is their high degree of explicitness

inverse problems for multidimensional vibrating systems many vibrating systems involve interactions only between neighbouring parts of the system frequently such systems are analysed by supposing that the mass distribution is lumped at the generalized coordinates undamped systems of this type involve a block tri diagonal stiffness matrix k and a block diagonal inertia matrix m the inverse problem is to reconstruct k and m from frequency response data the known reconstruction of a jacobi matrix using the forsythe algorithm is generalized so that a block jacobi matrix can be reconstructed from a certain spectral function this analysis is used to construct the dynamic stiffness matrix a l 1 kl text t m ll text t from the frequency response a is determined to within a block diagonal orthogonal matrix it is shown that k and m can be separated by using special information about their forms a vibrating lattice composed of rods and masses is used as an example

inverse vibration problem for un damped 3 dimensional multi story shear building models various researchers have contributed to the identification of the mass and stiffness matrices of two dimensional 2 d shear building structural models for a given set of vibratory frequencies the suggested methods are based on the specific characteristics of the jacobi matrices i e symmetric tri diagonal and semi positive definite matrices however in case of three dimensional 3 d structural models those methods are no longer applicable since their stiffness matrices are not tri diagonal in this paper the inverse problem for a special class of vibratory structural systems i e 3 d shear building models is investigated a practical algorithm is proposed for solving the inverse eigenvalue problem for un damped 3 d shear buildings the problem is addressed in two steps first using the target frequencies a so called normalized eigenvector matrix which is a banded matrix containing the information related to the frequencies and mode shapes of the target structural system is determined in this regard similar to the solution of inverse problem for 2 d shear building structural models in which an auxiliary structure is constructed by adding constraints or springs to the original system three auxiliary structures are proposed to solve the problem for 3 d cases in the second step the normalized eigenvector matrix is utilized to obtain the normalized stiffness matrix in turn this matrix is decomposed into the stiffness and mass matrices of the system finally a numerical example is presented to demonstrate the efficiency of the proposed algorithm in determining the mass and stiffness matrices of a 3 d structural model for a given set of target vibrational frequencies

p adic coupling of mock modular forms and shadows a mock modular form is the holomorphic part of a harmonic maass form f the nonholomorphic part of f is a period integral of its shadow a cusp form g a direct method for relating the coefficients of shadows and mock modular forms is not known we solve these problems when the shadow is an integer weight newform our solution is p adic and it relies on our definition of an algebraic regularized mock modular form as an application we consider the modular solution to the cubic arithmetic geometric mean

on two geometric theta lifts the theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type in this paper we establish for the orthogonal group o p 2 an adjointness result between borcherds s singular theta lift and the kudla millson lift we extend this result to arbitrary signature by introducing a new singular theta lift for o p q on the geometric side this lift can be interpreted as a differential character in the sense of cheeger and simons for the cycles under consideration

hilbert class polynomials and traces of singular moduli where q e the values of j z at imaginary quadratic arguments in the upper half of the complex plane are known as singular moduli singular moduli are algebraic integers which play prominent roles in classical and modern number theory see c bcsh for example hilbert class fields of imaginary quadratic fields are generated by singular moduli furthermore isomorphism classes of elliptic curves with complex multiplication are distinguished by singular moduli throughout let d 0 3 mod 4 be a positive integer so that d is the discriminant of an order in an imaginary quadratic field and let h d be the hurwitz kronecker class number for the discriminant d let qd be the set of positive definite integral binary quadratic forms note including imprimitive forms if there are any

arithmetic properties of coefficients of half integral weight maass poincare series zagier 23 proved that the generating functions for the traces of level 1 singular moduli are weight 3 2 modular forms he also obtained generalizations for twisted traces and for traces of special non holomorphic modular functions using properties of kloosterman salie sums and a well known reformulation of salie sums in terms of orbits of cm points we systematically show that such results hold for arbitrary weakly holomorphic and cuspidal half integral weight poincare series in kohnen s 0 4 plus space these results imply the aforementioned results of zagier and they provide exact formulas for such traces

differential operators for harmonic weak maass forms and the vanishing of hecke eigenvalues for integers k 2 we study two differential operators on harmonic weak maass forms of weight 2 k the operator 2 k resp d k 1 defines a map to the space of weight k cusp forms resp weakly holomorphic modular forms we leverage these operators to study coefficients of harmonic weak maass forms although generic harmonic weak maass forms are expected to have transcendental coefficients we show that those forms which are dual under 2 k to newforms with vanishing hecke eigenvalues such as cm forms have algebraic coefficients using regularized inner products we also characterize the image of d k 1

on minimality and stable minimality of time varying linear systems with well posed boundary conditions for time varying linear systems with well posed boundary conditions the concept of stable minimality is introduced a complete characterization is given of stably minimal systems with analytic coefficients

sixth order mock theta functions two new mock theta functions of the sixth order are defined the main theorem in this paper theorem 1 1 provides four transformation formulas relating the new mock theta functions with ramanujan s mock theta functions of the sixth order two further representations of the new mock theta functions are established lastly a hitherto unproved entry from ramanujan s lost notebook related to sixth order mock theta functions is proved

realization of acausal weighting patterns with boundary value descriptor systems this paper examines the realization of acausal weighting patterns with two point boundary value descriptor systems tpbvdss attention is restricted to the subclass oftpbvdss that are stationary so that their input output weighting pattern is shift invariant and extendible i e their weighting pattern can be extended outwards indefinitely then given an infinite acausal shift invariant weighting pattern the realization problem consists of constructing a minimal tpbvds over a fixed interval whose extended weighting pattern matches the given pattern the realization method that is proposed relies on a new transform the s t transform which is better adapted to the analysis of descriptor dynamics than the standard z transform since it handles zero and infinite frequencies in a symmetric way this new transform is used to determine the dimension of a minimal realization and then to construct a minimal realization by obtaining state space representations for two homogeneous rational matrices in s and obtained from the causal and anticausal components of the weighting pattern

parameter reconstruction of vibration systems from partial eigeninformation quadratic matrix polynomials are fundamental to vibration analysis because of the predetermined interconnectivity among the constituent elements and the mandatory nonnegativity of the physical parameters most given vibration systems will impose some inherent structure on the coefficients of the corresponding quadratic matrix polynomials in the inverse problem of reconstructing a vibration system from its observed or desirable dynamical behavior respecting the intrinsic structure becomes important and challenging both theoretically and practically the issue of whether a structured inverse eigenvalue problem is solvable is problem dependent and has to be addressed structure by structure in an earlier work physical systems that can be modeled under the paradigm of a serially linked mass spring system have been considered via specifically formulated inequality systems in this paper the framework is generalized to arbitrary generally linked systems in particular given any configuration of interconnectivity in a mass spring system this paper presents a mechanism that systematically and automatically generates a corresponding inequality system a numerical approach is proposed to determine whether the inverse problem is solvable and if it is so computes the coefficient matrices while providing an estimate of the residual error the most important feature of this approach is that it is problem independent that is the approach is general and robust for any kind of physical configuration the ideas discussed in this paper have been implemented into a software package by which some numerical experiments are reported

on the reconstruction of a damped vibrating system from two complex spectra part 1 theory abstract the paper concerns an n degree of freedom damped vibrating system consisting ofn 1 masses connected in parallel by springs and dampers to an n th mass the paper analyzes the construction of such a system from the given complex eigenvalue data the analysis has two parts the establishment of the conditions on the eigenvalues which ensure that they correspond to an actual system the derivation of the system parameters from the eigenvalues

characterization of integral operators with semi separable kernels with symmetries we characterize integral operators with semi separable kernels in a certain class that have different symmetries we treat the selfadjoint case the positive case the j unitary case the positive real case the dissipative case and the contractive case

construction of band matrices from spectral data abstract a stable numerical algorithm is presented to generate a symmetric p band matrix from the given eigenvalues of the p greatest leading submatrices the algorithm consists of two parts first a matrix with the given spectral data is constructed then this matrix is transformed into a p band matrix leaving invariant the eigenvalues of the p greatest leading submatrices

on an inverse spectral problem for a quadratic jacobi matrix pencil abstract given two monic polynomials p 2 n and p 2 n 2 of degree 2n and 2 n 2 n 2 with complex coefficients and with disjoint zero sets we give necessary and sufficient conditions on these polynomials such that there exist two n n jacobi matrices b and c for which p 2 n det 2 i n b c p 2 n 2 det 2 i n 1 b 1 c 1 where b 1 and c 1 are the n 1 n 1 jacobi matrices obtained from b and c by deleting the last row and the last column the zeros of p 2 n and p 2 n 2 are the eigenvalues of the quadratic jacobi matrix pencils on the right hand side of the equalities whence the title of the paper the problem is formulated and solved in a slightly more general form

an inverse eigenvalue problem for the symmetric tridiagonal quadratic pencil with application to damped oscillatory systems a method is presented which constructs an n by n tridiagonal symmetric quadratic pencil which has its 2n eigenvalues and the 2n 2 of its n 1 dimensional leading principal subpencil prescribed it is shown that if the given eigenvalues are distinct there are at most 2 n 2n 3 n 2 different solutions in the degenerate case where some of the given eigenvalues are common there are an infinite number of solutions apart from finding the roots of certain polynomials the problem is solved in a finite number of steps where the problem has only a finite number of solutions they can all be found in a systematic manner the method is demonstrated with a simple example and its use is illustrated with a practical engineering application in vibrations

reachability observability minimality and extendibility for two point boundary value descriptor systems a deterministic system theory is developed for two point boundary value descriptor systems tpbvdss in particular detailed characterizations of the properties of reachability observability and minimality are obtained in addition extendibility i e the concept of considering tpbvds as being defined on a sequence of intervals of increasing length is defined and studied this system theory generalizes our earlier results for the class of stationary systems for which the input output map weighting pattern is shift invariant

constructing the physical parameters of a damped vibrating system from eigendata abstract in this paper we consider the inverse problem for a discrete damped mass spring system where the mass damping and stiffness matrices are all symmetric tridiagonal we first show that the model can be constructed from two real eigenvalues and three real eigenvectors or two complex conjugate eigenpairs and a real eigenvector then we study the general under determined and over determined problems in particular we provide the sufficient and necessary conditions on the given two real or complex conjugate eigenpairs so that the under determined problem has a physical solution however for large model order the construction from these data may be sensitive to perturbations to reduce the sensitivity we propose the minimum norm solution over the under determined noisy data and the least squares solution to the over determined measured data we also discuss the physical realizability of the required model by the positivity constrained regularization method for the ill posed under determined problem and the least squares optimization problems with positivity constraints for the ill posed over determined problem finally we give simple numerical examples to illustrate the effectiveness of our methods

the asymptotic distribution of traces of maass poincare series abstract we establish an asymptotic formula with a power savings in the error term for traces of cm values of a family of maass poincare series which contains the modular j function as a special case by work of borcherds 1998 2 zagier 2002 31 and bringmann and ono 2007 4 these traces are fourier coefficients of half integral weight weakly holomorphic modular forms and maass forms

mock period functions sesquiharmonic maass forms and non critical values of l functions we introduce a new technique of completion for 1 cohomology which parallels the corresponding technique in the theory of mock modular forms this technique is applied in the context of non critical values of l functions of gl 2 cusp forms we prove that a generating series of non critical values can be interpreted as a mock period function we define in analogy with period polynomials further we prove that non critical values can be encoded into a sesquiharmonic maass form finally we formulate and prove an eichler shimura type isomorphism for the space of mock period functions

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

a solution of the affine quadratic inverse eigenvalue problem abstract the quadratic inverse eigenvalue problem qiep is to find the three matrices m c and k given a set of numbers closed under complex conjugations such that these numbers become the eigenvalues of the quadratic pencil p 2 m c k the affine inverse quadratic eigenvalue problem aqiep is the qiep with an additional constraint that the coefficient matrices belong to an affine family that is these matrices are linear combinations of substructured matrices an affine family of matrices very often arise in vibration engineering modeling and analysis research on qiep and aqiep are still at developing stage in this paper we propose three methods and the associated mathematical theories for solving aqiep a newton method an alternating projections method and a hybrid method combining the two validity of these methods are illustrated with results on numerical experiments on a spring mass problem and comparisons are made with these three methods amongst themselves and with another newton method developed by elhay and ram 2002 12 the results of our experiments show that the hybrid method takes much smaller number of iterations and converges faster than any of these methods

maximal rank hermitian completions of partially specified hermitian matrices abstract in this note it is shown that for a given partially specified hermitian matrix p the maximal rank for arbitrary possibly nonhermitian complex completions can be attained by hermitian completions a simple formula for the maximal rank for nonhermitian completions was computed previously by cohen et al we also discuss the same situation for symmetric matrices over an arbitrary field and show that the field size may be critical in establishing the same formulas finally we discuss the same questions under toeplitz structure and show that for the matrix h 1 1 1 1 1 the maximal completion rank is 3 for complex hermitian toeplitz completions 3 for real symmetric completions 3 for real toeplitz completions but only 2 for real symmetric toeplitz completions

completing triangular block matrices with maximal and minimal ranks abstract this paper considers two completion problems on block matrices with maximal and minimal ranks let a a ij be an n n block matrix where a ij n i j 1 is given and a ij 1 i is a variant block entry then determine all these variant block entries such that a a ij has maximal and minimal possible ranks respectively by making use of the theory of generalized inverses of matrices we present complete solutions to these two problems as applications we also determine maximal and minimal ranks of the matrix expression a bxc when x is a variant triangular block matrix and then present a necessary and sufficient condition for the matrix equation bxc a to have a triangular block solution

positive extensions fejer riesz factorization and autoregressive filters in two variables in this paper we treat the two variable positive extension problem for trigonometric polynomials where the extension is required to be the reciprocal of the absolute value squared of a stable polynomial this problem may also be interpreted as an autoregressive filter design problem for bivariate stochastic processes we show that the existence of a solution is equivalent to solving a finite positive definite matrix completion problem where the completion is required to satisfy an additional low rank condition as a corollary of the main result a necessary and sufficient condition for the existence of a spectral fejerriesz factorization of a strictly positive two variable trigonometric polynomial is given in terms of the fourier coefficients of its reciprocal tools in the proofs include a specific two variable kronecker theorem based on certain elements from algebraic geometry as well as a two variable christoffel darboux like formula the key ingredient is a matrix valued polynomial that appears in a parametrized version of the schur cohn test for stability the results also have consequences in the theory of two variable orthogonal polynomials where a spectral matching result is obtained as well as in the study of inverse formulas for doubly indexed toeplitz matrices finally numerical results are presented for both the autoregressive filter problem and the factorization problem

lifting cusp forms to maass forms with an application to partitions for 2 k abstract see text we define lifts of cuspidal poincare series in s k gamma 0 n to weight 2 k harmonic weak maass forms this construction answers a question of dyson by providing the general framework explaining ramanujan s mock theta functions as an application we show that the number of partitions of a positive integer n is the trace of singular moduli of a maass form arising from the lift of a weight 4 cusp form corresponding to a calabi yau threefold