hermitian completions of band matrices and applications linear fractional descriptions are given for hermitian completions of partial banded matrices with prescribed cholesky signatures an inertia related notion the results are applied to obtain a linear fractional description for completions of partial triangular matrices with prescribed numbers of singular values 1

global completability with applications to self consistent quantum tomography let p 1 ldots p n in mathbb r d be unknown vectors and let omega subseteq 1 ldots n 2 assume that the inner products p i t p j are fixed for all i j in omega do these inner product constraints up to simultaneous rotation of all vectors determine p 1 ldots p n uniquely here we derive a necessary and sufficient condition for the uniqueness of p 1 ldots p n i e global completability which is applicable to a large class of practically relevant sets omega moreover given omega we show that the condition for global completability is universal in the sense that for almost all vectors p 1 ldots p n in mathbb r d the completability of p 1 ldots p n only depends on omega and not on the specific values of p i t p j for i j in omega this work was motivated by practical considerations namely matrix factorization techniques and self consistent quantum tomography

invertible completions of band matrices abstract starting with a hermitian band matrix r we fill it in as we try to complete it to an invertible hermitian matrix f often we can make f invertible even though all but one of the maximal hermitian submatrices in the band may be not invertible this paper is a continuation of the author s recent paper on bordered matrices

matrix sandwich problems abstract the matrix sandwich problem msp is introduced here as follows given a 0 1 valued matrix a where is interpreted as do not care does there exist a fill in of the asterisks with 0s and 1s such that the completed 0 1 valued matrix m satisfies property we study the computational complexity of this problem for several matrix properties including the ferrers property block decompositions and certain forbidden submatrices matrix sandwich problems are an important special case of matrix completion problems the latter being generally defined over the real numbers rather than simply 0 1

displacement structure and maximum entropy the study of matrices with a displacement structure is mainly concerned with recursions for the so called generator matrices the recursion usually involves free parameters which can be chosen in several ways so as to simplify the resulting algorithm in this correspondence we present a choice for the parameters that is motivated by a maximum entropy formulation this choice further motivates the introduction of the so called generalized reflection coefficients which are in general different from the better known schur coefficients

sine gordon theory in a semi strip abstract initial boundary value problems for sine gordon and complex sine gordon equations in a semi strip are treated the evolution of the weyl function and a uniqueness result are obtained for the complex sine gordon equation the evolution of the weyl function as well as an existence result and a procedure to recover solution are given for the sine gordon equation it is shown that for a wide class of examples the solutions of the sine gordon equation are unbounded in the quarter plane

finding positive matrices subject to linear restrictions abstract we characterize the existence of a positive definite l l matrix x the entries of which satisfy n nonhomogeneous linear conditions by the existence of a minimum for an associated function v smooth and strictly convex on r n if there exist solutions x 0 then lim x v x and the critical point x 0 of v can be approximated by the conjugate gradients method knowing x 0 provides by a simple analytic formula the unique solution x maximizing the entropy tr x ln x j 1 l j ln j where 1 l are the eigenvalues of x subject to the given restrictions related results are obtained in the semipositive definite case too

an efficient algorithm for maximum entropy extension of block circulant covariance matrices article history this paper deals with maximum entropy completion of partially specified block circulant matrices since positive definite symmet ric circulants happen to be covariance matrices of stationary peri odic processes in particular of stationary reciprocal processes this problem has applications in signal processing in particular to im age modeling in fact it is strictly related to maximum likelihood estimation of bilateral ar type representations of acausal signals subject to certain conditional independence constraints the maxi mum entropy completion problem for block circulant matrices has recently been solved by the authors although leaving open the problem of an efficient computation of the solution in this pa per we provide an efficient algorithm for computing its solution which compares very favorably with existing algorithms designed for positive definite matrix extension problems the proposed al gorithm benefits from the analysis of the relationship between our problem and the band extension problem for block toeplitz matri ces also developed in this paper

discrete skew selfadjoint canonical systems and the isotropic heisenberg magnet model abstract a discrete analog of a skew self adjoint canonical zakharov shabat or akns system with a pseudo exponential potential is introduced for the corresponding weyl function the direct and inverse problem are solved explicitly in terms of three parameter matrices as an application explicit solutions are obtained for the discrete integrable nonlinear equation corresponding to the isotropic heisenberg magnet model state space techniques from mathematical system theory play an important role in the proofs

positive definite hankel matrix completions and hamburger moment completions abstract in this paper we give solutions to hamburger moment problems with missing entries the problem of completing partial positive sequences is considered the main result is a characterization of positive definite completable patterns namely patterns that guarantee the existence of hamburger moment completion of a partial positive definite sequence moreover several patterns which are not positive definite completable are given

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

multidimensional maximum entropy covariance extension a universal characterization of maximum entropy covariances for multidimensional signals is presented it is shown that the maximum entropy extension of an arbitrary partial covariance of a nonstationary multidimensional signal always has a banded inverse i e the inverse is sparse and has the same support as the given partial covariance a dual formulation of the problem that makes it possible to approximate maximum entropy extensions with models selected from suitably constrained model sets is introduced it is proved that the best approximation in terms of multidimensional recursible autoregressive models can be determined by solving a set of linear equations a simple graph theoretic criterion is introduced to characterize those partial covariances whose maximum entropy extension coincides with its autoregressive approximation as in the conventional one dimensional stationary maximum entropy problem

one step completions of hermitian partial matrices with minimal negative signature abstract the negative signature of hermitian completions of the partial matrix a b b b e e d is computed

shift invariant subspaces factorization and interpolation for matrices i the canonical case abstract this article begins a program of classifying subspaces of block matrices that are invariant under multiplication on the right by the algebra of block upper or lower triangular matrices this gives a unifying point of view to several general kinds of factorization problems including lu qr and cholesky factorization a second application is to some matrix completion problems such as that of filling in the upper triangular entries of a partially defined matrix to make the result of norm less than or equal to 1 included is a linear fractional parametrization of the set of all solutions of this problem

rank one quadratic twists of an infinite family of elliptic curves a conjecture of goldfeld implies that a positive proportion of quadratic twists of an elliptic curve e q has analytic rank 1 this assertion has been confirmed by vatsal v1 and the first author by for only two elliptic curves here we confirm this assertion for infinitely many elliptic curves e q using the heegner divisors the 3 part of the class groups of quadratic fields and a variant of the binary goldbach problem for polynomials

rank one twists of a certain elliptic curve the purpose of this note is to give the first known example for a conjecture of goldfeld on the number of rank one curves appearing in a family of quadratic twists we show unconditionally that the curve x0 19 has the property that a positive proportion of its quadratic twists have analytic rank one this amounts to a strong nonvanishing statement for the derivatives of certain l functions when the sign of the functional equation is 1 we simultaneously obtain the fact that a positive proportion of twists have analytic rank zero when the sign of the functional equation is 1 let e be a modular elliptic curve over q with corresponding newform f z anq n s2 0 n let l s f l s e ann s be the usual l series then l s f satisfies a functional equation under s 2 s with sign 1 e let d be the fundamental discriminant of a quadratic field k k d and write d for the associated quadratic character in this setting we define the twisted l function l s fd d n ann if e is given by the equation y 2 p x then l s fd is the l function attached to the twisted elliptic curve ed given by dy 2 p x therefore l s fd also satisfies a functional equation and if d n 1 then the sign is given by d d n for r 0 1 and a positive real number x define m r f x d d x ords 1l s fd r the following conjecture of goldfeld gol79 is well known

positive semidefinite completions of partial hermitian matrices abstract we classify the ranks of positive semidefinite completions of hermitian band matrices and other partially specified hermitian matrices with chordal graphs and specified main diagonals completing a partially specified matrix means filling in the unspecified entries

spanning tree extensions of the hadamard fischer inequalities abstract all possible graph theoretic generalizations of a certain sort for the hadamard fischer determinantal inequalities are determined these involve ratios of products of principal minors which dominate the determinant furthermore the cases of equality in these inequalities are characterized and equality is possible for every set of values which can occur for the relevant minors this relates recent work of the authors on positive definite completions and determinantal identities when applied to the same collections of principal minors earlier generalizations give poorer more difficult to compute bounds than the present inequalities thus this work extends and in a certain sense completes a series of generalizations of hadamard fischer begun in the 1960s

schur products and matrix completions we prove that a necessary and sufficient condition for a given partially positive matrix to have a positive completion is that a certain schur product map defined on a certain subspace of matrices is a positive map by analyzing the positive elements of this subspace we obtain new proofs of the results of h dym and i gohberg and grone johnson sa and wolkowitz linear algebra appl 58 1984 109 124 linear algebra appl 36 1981 1 24 we also obtain a new proof of a result of u haagerup decomposition of completely bounded maps on operation algebras preprint characterizing the norm of schur product maps and a new hahn banach type extension theorem for these maps finally we obtain generalizations of many of these results to matrices of operators which we apply to the study of representations of certain subalgebras of the n n matrices

model validation of multirate systems from time domain experimental data the model validation problem using time domain experimental data is studied for multirate linear fractional uncertain models in this note as a technical tool the caratheodory fejer cf interpolation problem with a nest operator constraint is first investigated this problem is itself of interest mathematically and has potential applications in addressing other problems in control signal processing and circuit theory a necessary and sufficient solvability condition for this interpolation problem is given the validation tests are then presented based on this condition and the lifting technique tractable convex optimization methods can be used to solve the validation problems

the circulant rational covariance extension problem the complete solution the rational covariance extension problem to determine a rational spectral density given a finite number of covariance lags can be seen as a matrix completion problem to construct an infinite dimensional positive definite toeplitz matrix the northwest corner of which is given the circulant rational covariance extension problem considered in this paper is a modification of this problem to partial stochastic realization of periodic stationary process which are better represented on the discrete unit circle rather than on the discrete real line the corresponding matrix completion problem then amounts to completing a finite dimensional toeplitz matrix that is circulant another important motivation for this problem is that it provides a natural approximation involving only computations based on the fast fourier transform for the ordinary rational covariance extension problem potentially leading to an efficient numerical procedure for the latter the circulant rational covariance extension problem is an inverse problem with infinitely many solutions in general each corresponding to a bilateral arma representation of the underlying periodic process in this paper we present a complete smooth parameterization of all solutions and convex optimization procedures for determining them a procedure to determine which solution that best matches additional data in the form of logarithmic moments is also presented

positive semidefinite matrix completions on chordal graphs and constraint nondegeneracy in semidefinite programming let g v e be a graph in matrix completion theory it is known that the following two conditions are equivalent i g is a chordal graph ii every g partial positive semidefinite matrix has a positive semidefinite matrix completion in this paper we relate these two conditions to constraint nondegeneracy condition in semidefinite programming and prove that they are each equivalent to iii for any g partial positive definite matrix that has a positive semidefinite completion constraint nondegeneracy is satisfied at each of its positive semidefinite matrix completions

inheritance principles for chordal graphs abstract we prove two results concerning positive completions of partial positive matrices first we give a parametrization of all positive invertible completions of a partial positive matrix having as support a chordal graph along a fixed increasing chordal sequence then a formula for the determinant of each completion is obtained in terms of the associated parameters as an application we obtained a proof of a conjecture of w w barrett c r johnson and l rodman regarding an inheritance principle for chordal graphs

direct approach to the band completion problem we study the problem of completion of a matrix with a specified band in such a way that the inverse be a band matrix with the same width for the general block matrix case a new approach to solve this and related problems is proposed the relations with the class of quasiseparable matrices play an important role

completion of partial matrices to contractions an n by m partially specified complex matrix is called a partial contraction if every rectangular submatrix consisting entirely of specified entries is itself a contraction necessary and sufficient condition are given for the pattern of specified entries such that any n by m partial contraction with this pattern may be completed to a full n by m contraction

blind separation of gaussian sources with general covariance structures bounds and optimal estimation we consider the separation of gaussian sources exhibiting general arbitrary not necessarily stationary covariance structures first assuming a semi blind scenario in which the sources covariance structures are known we derive the maximum likelihood estimate of the separation matrix as well as the induced cramer rao lower bound icrlb on the attainable interference to source ratio isr we then extend our results to the fully blind scenario in which the covariance structures are unknown we show that under a scaling convention the fisher information matrix in this case is block diagonal implying that the same icrlb as in the semi blind scenario applies in this case as well subsequently we demonstrate that the same semi blind optimal performance can be approached asymptotically in the fully blind scenario if the sources are sufficiently ergodic or if multiple snapshots are available

two dimensional multivariate parametric models for radar applications part ii maximum entropy extensions for hermitian block matrices in a series of two papers a new class of parametric models for two dimensional multivariate matrix valued space time adaptive processing is introduced this class is based on the maximum entropy extension and or completion of partially specified matrix valued hermitian covariance matrices in both the space and time dimensions this first paper considers the more restricted class of toeplitz hermitian covariance matrices that model stationary clutter if the clutter is stationary only in time then we deal with a toeplitz block matrix whereas clutter that is stationary in time and space is described by a toeplitz block toeplitz matrix we first derive exact expressions for this new class of 2 d models that act as approximations for the unknown true covariance matrix second we propose suboptimal but computationally simpler relaxed 2 d time varying autoregressive models ldquorelaxationsrdquo that directly use the non toeplitz hermitian sample covariance matrix the high efficiency of these parametric models is illustrated by simulation results using true ground clutter covariance matrices provided by the darpa kassper dataset 1 which is a trusted phenomenological airborne radar model and a complementary afrl dataset

nonlinear schrodinger equation in a semi strip evolution of the weyl titchmarsh function and recovery of the initial condition and rectangular matrix solutions from the boundary conditions abstract rectangular matrix solutions of the defocusing nonlinear schrodinger equation dnls are studied in quarter plane and semi strip evolution of the corresponding weyl titchmarsh weyl function is described in terms of the initial weyl function and boundary conditions in the next step the initial weyl function is recovered for the quarter plane case from the long time asymptotics of the wave function considered at the boundary thus it is shown that the evolution of the weyl function is uniquely defined by the boundary conditions moreover a procedure to recover solutions of dnls uniquely defined by the boundary conditions is given in a somewhat different way the same boundary value problem is also dealt with in a semi strip for the case of a quasi analytic initial condition

inner outer factorization and the inversion of locally finite systems of equations abstract we consider the problem of computing the inverse of a large class of infinite systems of linear equations which are described by a finite set of data the class consists of equations in which the linear operator is represented by a discrete time varying dynamical system whose local state space is of finite dimension at each time point k and which reduces to time invariant systems for time points k in this generalization of classical matrix inversion theory inner outer factorizations of operators play the role that qr factorization plays in classical linear algebra numerically they lead to so called square root implementations for which attractive algorithms can be derived which do not require the determination of spurious multiple eigenvalues as would be the case if the problem was converted to a discrete time riccati equation by squaring we give an overview of the theory and the derivation of the main algorithms the theory contains both the standard lti case and the case of a finite set of linear equations as special instances a particularly instance of which is called matrices of low hankel rank recently sometimes called quasi separable matrices however in the general case considered here new phenomena occur which are not observed in these classical cases namely the occurrence of defect spaces we describe these and give an algorithm to compute them as well in all cases the algorithms given are linear in the amount of data

the positive definite completion problem revisited abstract in r grone c r johnson e sa h wolkowicz positive definite completions of partial hermitian matrices linear algebra appl 58 1984 109 124 the positive definite semi completion problem in which the underlying graph is chordal was solved for the positive definite case the process was constructive and the completion was obtained by completing the partial matrix an entry at a time for the positive semidefinite case they obtained completions of a particular sequence of partial positive definite matrices with the same underlying graph and noted that there is a convergent subsequence of these completions that converges to the desired completion here in the chordal case we provide a constructive solution based entirely on matrix graph theoretic methods to the positive semi definite completion problem our solution associates a specific tree called the clique tree c r johnson m lundquist matrices with chordal inverse zero patterns linear and multilinear algebra 36 1993 1 17 with the chordal graph of the given partial positive semi definite matrix this tree structure allows us to complete the matrix a block at a time as opposed to an entry at a time as in grone et al 1984 for the positive definite case in grone et al 1984 using complex analytic techniques the completion for the positive definite case was shown to be the unique determinant maximizing completion and was shown to be the unique completion that has zeros in its inverse in the positions corresponding to the unspecified entries of the partial matrix here we show the same using only matrix graph theoretic tools

a maximum entropy solution of the covariance extension problem for reciprocal processes stationary reciprocal processes defined on a finite interval of the integer line can be seen as a special class of markov random fields restricted to one dimension nonstationary reciprocal processes have been extensively studied in the past especially by jamison et al the specialization of the nonstationary theory to the stationary case however does not seem to have been pursued in sufficient depth in the literature stationary reciprocal processes and reciprocal stochastic models are potentially useful for describing signals which naturally live in a finite region of the time or space line estimation or identification of these models starting from observed data seems still to be an open problem which can lead to many interesting applications in signal and image processing in this paper we discuss a class of reciprocal processes which is the acausal analog of auto regressive ar processes familiar in control and signal processing we show that maximum likelihood identification of these processes leads to a covariance extension problem for block circulant covariance matrices this generalizes the famous covariance band extension problem for stationary processes on the integer line as in the usual stationary setting on the integer line the covariance extension problem turns out to be a basic conceptual and practical step in solving the identification problem we show that the maximum entropy principle leads to a complete solution of the problem

factorization of block matrices abstract an lu type factorization theorem due to elsner and to gohberg and goldberg is generalized to block matrices one form of the general factorization takes the form lmu where l is block lower triangular u is block upper triangular and m is a subpermutation matrix each of whose blocks is diagonal a factorization is also given where the middle term is a block diagonal subpermutation matrix and the factorization is applied to wiener hopf equations the nonuniqueness of the middle term in the factorization is analyzed a special factorization for self adjoint block matrices is also obtained

mihaly matrix completions moments and sums of hermitian squares by mihaly bakonyi and hugo j woerdeman princeton series in applied mathematics princeton university press 2011 xii 518 pp hardback isbn 978 0 691 12889 4

the maximum distance problem and band sequences abstract we solve the following problem for 1 j k n and j k m let a jk be a given complex number with a kj jk we wish to find linearly independent vectors x 1 x n such that x k x j a jk for j k m and such that the distance from x k to the linear span of x 1 x k 1 is maximal for 2 k n we construct and characterize all such sequences of vectors our solution leads naturally to the class of m band sequences of vectors in an inner product space we study these sequences and characterize their equivalence classes under unitary transformations

invertible completions of partial operator matrices the nonsymmetric case abstract we consider nonsymmetric partial operator matrices r whose directed graphs belong to a certain class and for which certain key fully specified principal submatrices are invertible under these circumstances we prove the existence of a unique invertible completion f of r such that f 1 ij is block zero whenever rij is unspecified in this way the existing results of johnson and lundquist are generalized to the nonsymmetric case

a maximum entropy principle in the general framework of the band method abstract a maximum entropy principle is developed in the general framework of the band method known maximum entropy principles appear as corollaries and new ones are derived

linearly constrained positive definite completions abstract a partial matrix is a rectangular array consisting of specified entries which are fixed elements of the complex numbers c and unspecified entries which are free variables over c given a partial hermitian matrix a z in which z is the vector of unspecified entries of a a fixed matrix b and a fixed vector c the optimal solution of the problem maximize det a z subject to a z positive definite bz c is unique provided the problem is feasible and has a simple characterization this class of problems includes the case in which a z is constrained to be toeplitz

on the compatibility condition for linear systems and a factorization formula for wave functions the well known compatibility condition for linear systems wx gw and wt fw is considered and new results are obtained in this way a factorization formula for wave functions which is basic in the inverse spectral transform approach to initial boundary value problems is proved in greater generality than before some applications follow

dualistic differential geometry of positive definite matrices and its applications to related problems abstract analysis on the set of positive definite matrices is involved in many engineering problems we develop a differential geometric theory of the set of positive definite matrices by means of a specific class of connections introduced into it consequently various dualistic aspects of the set are elucidated next using the theory we derive new results and interesting geometrical interpretations for matrix approximation positive definite matrix completion and linear matrix inequality problems

a note on extensions of band matrices with maximal and submaximal invertible blocks abstract completions or extensions of band matrices with invertible maximal and submaximal blocks are considered linear fractional descriptions for invertible minimal rank and lu ul decomposable extensions are derived self adjoint and toeplitz cases are included

the negative signature of some hermitian matrices abstract it is proved that the range of the negative signature of the hermitian completions of the partial matrix a b b c d d e is an interval and explicit formulae for the bounds of this interval are obtained as an application the solvability of the problem of lifting with prescribed negative signature of defect is settled

order estimation and discrimination between stationary and time varying tvar autoregressive models for a set of t independent observations of the same n variate correlated gaussian process we derive a method of estimating the order of an autoregressive ar model of this process regardless of its stationary or time varying nature we also derive a test to discriminate between stationary ar models of order m ar m and time varying autoregressive models of order m tvar m we demonstrate that within this technique the number t of independent identically distributed data samples required for order estimation and discrimination just exceeds the maximum possible order mmax which in many cases is significantly fewer than the dimension of the problem n

extensions of band matrices with band inverses abstract let r ij be a given set of i j matrices for i j 1 n and i j m where 0 m n 1 necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix f ij i j 1 n such that f ij r ij for i j m f admits either a left or right block triangular factorization and f 1 ij 0 for i j m the well known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m n 1 the special case in which the given r ij are positive definite in an appropriate sense is explored in detail and an inequality which corresponds to burg s maximal entropy inequality in the theory of covariance extension is deduced the block toeplitz case is also studied

on the covariance completion problem under a circulant structure covariance matrices with a circulant structure arise in the context of discrete time periodic processes and their significance stems also partly from the fact that they can be diagonalized via a fourier transformation this note deals with the problem of completion of partially specified circulant covariance matrices the particular completion that has maximal determinant i e the so called maximum entropy completion was considered in carli where it was shown that if a single band is unspecified and to be completed the algebraic restriction that enforces the circulant structure is automatically satisfied and that the inverse of the maximizer has a band of zero values that corresponds to the unspecified band in the data i e it has the dempster property the purpose of the present note is to develop an independent proof of this result which in fact extends naturally to any number of missing bands as well as arbitrary missing elements more specifically we show that this general fact is a direct consequence of the invariance of the determinant under the group of transformations that leave circulant matrices invariant a description of the complete set of all positive extensions of partially specified circulant matrices is also given and certain connections between such sets and the factorization of certain polynomials in many variables facilitated by the circulant structure is highlighted

positive semidefinite matrices with a given sparsity pattern abstract this paper concerns the cone of positive semidefinite matrices which have zeros in prescribed entries one of the main purposes is to obtain information and if possible describe the ranks of external matrices in such cones in terms of the pattern of prescribed zeros in particular we study those patterns for which the ranks of extremal matrices are less than a given number

unitary rank structured matrices in this paper we describe how one can represent a unitary rank structured matrix in an efficient way as a product of elementary unitary or givens transformations we also provide some basic operations for manipulating the representation such as the transition to zero creating form the transition to a unitary givens weight representation as well as an internal pull through process of the two branches of the representation finally we characterize how to determine the shift correction term to the rank structure and we provide some applications to this result

l series with nonzero central critical value suppose that f zn 1 an f qtn is a cusp form of weight 2k k e n we denote by l f s the l function of f for re s sufficiently large the value of l f s is given by l f s en 1 an and one can show that l f s has analytic continuation to the entire complex plane the value of l f s at s k will be of particular interest to us and we will refer to this value as the central critical value of l f s let xd denote the dirichlet character associated to the extension q d q that is xd n va where ad denotes the discriminant of q d q define the dth quadratic twist of f to be fxd en 1 an f xd n qn for any integer d the l function of fx is the dth quadratic twist of l f s that is l fxd s zn aan f xd n we will be interested in determining how often l fxd s has nonzero central critical value as d varies over all integers since xd 2 xd we will restrict our attention to the square free integers d we expect that as we let d vary over all of the square free integers a positive proportion of the l functions l fxd s will have nonzero central critical value indeed goldfeld 7 conjectures that for newforms f of weight 2 l fxd 1 78 0 for 1 of the square free integers given an elliptic curve e y2 x3 ax2 bx c a b c e z with conductor ne and an integer d we define the dth quadratic twist of e to be the curve ed y2 x3 adx2 bd2x cd3 let l ed s denote the l function associated to ed for square free d coprime to 2ne l ed s is simply the dth quadratic twist of l ei s if f e s2 n is a newform with integer coefecients we know via the theory of eichler and shimura that there is an elliptic curve e over q having conductor n so that l e s l f s thus if d is coprime to 2n then l ed s l fxd s also one knows from the work of kolyvagin 13 as supplemented by the work of murty and murty 17 or that of bump friedberg and hoffstein 3 see also 10 for a shorter proof that if e is a modular elliptic curve and if l e 1 7 0 then the rank of e is 0 thus if f has the property that a positive proportion of the twists of l f s have nonzero central critical value then this implies that a positive density of the quadratic twists ed have rank 0 there have been many papers which have proved results in this direction for example in 2 3 6 10 16 17 19 28 one can find general theorems on

strictly contractive and positive completions for block matrices abstract this paper deals with the following two problems of completion of block matrices 1 let a ij i j q be given matrices find additional matrices a ij such that the block matrix a a ij n i 1 m j 1 has norm less than one 2 let b ij j i q be given matrices find additional b ij such that the completion b b ij n i j 1 is positive definite the analysis is based on a study of certain linear fractional maps this approach leads to an explicit description of all solutions by linear fractional maps of which the coefficients are given directly in terms of the given data the maximum entropy principle appears as a corollary of a general formula for the determinant of a completion special attention is paid to the toeplitz case

the sine gordon equation on time scales abstract we formulate and discuss integrable analogue of the sine gordon equation on arbitrary time scales this unification contains the sine gordon equation discrete sine gordon equation and the hirota equation doubly discrete sine gordon equation as special cases we present the lax pair check compatibility conditions and construct the darboux backlund transformation finally we obtain a soliton solution on arbitrary time scale the solution is expressed by the so called cayley exponential function

canonical periods and congruence formulae the purpose of this article is to show how congruences between the fourier coefficients of hecke eigenforms give rise to corresponding congruences between the algebraic parts of the critical values of the associated l functions this study was initiated by b mazur in his fundamental work on the eisenstein ideal see maz77 and maz79 where it was made clear that congruences for analytic l values were closely related to the integral structure of certain hecke rings and cohomology groups the results of maz79 also showed that congruences were useful in the study of nonvanishing of l functions this idea was then further developed by stevens ste82 and rubin wiles rw82 the work of rubin and wiles in particular used congruences to study the behavior of elliptic curves in towers of cyclotomic fields a key ingredient here was a theorem of washington which states roughly that almost l values in certain families are nonzero modulo p this theme has recently been taken up again in the work of ono skinner osa osb james jam and kohnen koh97 while the earlier history was primarily concerned with cyclotomic twists the current emphasis is on families of twists by quadratic characters here one wants quantitative estimates for the number of quadratic twists of a given modular form which have nonvanishing l function at s 1 we continue this trend in the present work by using our general results to obtain a strong nonvanishing theorem for the quadratic twists of modular elliptic curves with rational points of order three this generalizes a beautiful example due to kevin james and provides new evidence for a conjecture of goldfeld gol79 it should however be pointed out that even the study of quadratic twists may be traced back to mazur the reader is urged to look at pages 212 213 of maz79 and especially at the footnote at the bottom of page 213 the theorems of davenport heilbronn dh71 and washington was78 which are crucial in this paper are both mentioned in mazur s article we want to begin by discussing the congruences that lie at the heart of this article thus let f anq n be an elliptic modular cuspform of level m and weight k 2 assume that f is a simultaneous eigenform for all the hecke operators and that a1 f 1 the l function associated to f is defined by the dirichlet series l s f ann s which converges for the real part of s sufficiently large and has analytic continuation to s c a fundamental theorem of shimura shi76 states that l s f enjoys the following algebraicity property

maximum entropy kernels for system identification bayesian nonparametric approaches have been recently introduced in system identification scenario where the impulse response is modeled as the realization of a zero mean gaussian process whose covariance kernel has to be estimated from data in this scheme quality of the estimates crucially depends on the parametrization of the covariance of the gaussian process a family of kernels that have been shown to be particularly effective in the system identification framework is the family of diagonal correlated dc kernels maximum entropy properties of a related family of kernels the tuned correlated tc kernels have been recently pointed out in the literature in this technical note we show that maximum entropy properties indeed extend to the whole family of dc kernels the maximum entropy interpretation can be exploited in conjunction with results on matrix completion problems in the graphical models literature to shed light on the structure of the dc kernel in particular we prove that the dc kernel admits a closed form factorization inverse and determinant these results can be exploited both to improve the numerical stability and to reduce the computational complexity associated with the computation of the dc estimator

schur analysis of some completion problems abstract the complete schur analysis for contractive block matrices is given together with its connections with positiveness these are used for reobtaining some results on completion problems sometimes in a more general setting these completion problems include positive completions contractive and isometric completions and the realizability of upper triangular contractions

multirate periodic systems and constrained analytic function interpolation problems multirate periodic systems and some related constrained analytic function interpolation problems are studied in this paper after showing how to convert a general multirate periodic system to an equivalent linear time invariant lti system with a structural constraint we formulate some analytic function interpolation problems with such a constraint that can find various applications in the study of multirate and periodic systems both the solvability conditions and characterization of all solutions are presented to these constrained interpolation problems

the intertwining lifting theorem for ordered groups abstract sz nagy s dilation theorem for a contraction on a hilbert space has been extended by mlak to the case of a contraction semigroup whose indices are the positive elements of a totally ordered discrete group we generalize to this case the intertwining lifting theorem of sz nagy and foias some previous interpolation results on ordered groups are obtained as consequences

time varying autoregressive tvar models for multiple radar observations we consider the adaptive radar problem where the properties of the nonstationary clutter signals can be estimated using multiple observations of radar returns from a number of sufficiently homogeneous range azimuth resolution cells we derive a method for approximating an arbitrary hermitian covariance matrix by a time varying autoregressive model of order m tvar m that is based on the dym gohberg band matrix extension technique which gives the unique tvar m model for any nondegenerate covariance matrix we demonstrate that the dym gohberg transformation of the sample covariance matrix gives the maximum likelihood ml estimate of the tvar m covariance matrix we introduce an example of tvar m clutter modeling for high frequency over the horizon radar that demonstrates its practical importance

the zakharov shabat spectral problem on the semi line hilbert formulation and applications the inverse spectral transform for the zakharov shabat equation on the semi line x 0 is reconsidered as a hilbert problem the boundary data induce an essential singularity as k to one of the basic solutions then solving the inverse problem means solving a hilbert problem with particular prescribed behaviour it is demonstrated that the direct and inverse problems are solved in a consistent way as soon as the spectral transform vanishes as 1 k at infinity in the whole upper half plane where it may possess single poles and is continuous and bounded on the real k axis the method is applied to stimulated raman scattering and sine gordon light cone for which it is demonstrated that time evolution conserves the properties of the spectral transform

spn completable graphs abstract an spn matrix is a matrix which is the sum of a real positive semidefinite matrix and a symmetric nonnegative one we solve the spn completion problem we show that the spn completable graphs are the graphs in which every odd cycle induces a complete subgraph

on the davenport heilbronn theorems and second order terms we give simple proofs of the davenport heilbronn theorems which provide the main terms in the asymptotics for the number of cubic fields having bounded discriminant and for the number of 3 torsion elements in the class groups of quadratic fields having bounded discriminant we also establish second main terms for these theorems thus proving a conjecture of roberts our arguments provide natural interpretations for the various constants appearing in these theorems in terms of local masses of cubic rings

sparse matrix decompositions and graph characterizations abstract zeros in positive definite correlation matrices arise frequently in probability and statistics and are intimately related to the notion of stochastic independence the question of when zeros i e sparsity in a positive definite matrix a are preserved in its cholesky decomposition and vice versa was addressed by paulsen et al v i paulsen s c power r r smith schur products and matrix completions j funct anal 85 1989 151 178 in particular they prove that for the pattern of zeros in a to be retained in the cholesky decomposition of a the pattern of zeros in a has to necessarily correspond to a chordal or decomposable graph associated with a specific type of vertex ordering this result therefore also yields a characterization of chordal graphs in terms of sparse positive definite matrices and has proved to be extremely useful in probabilistic and statistical analysis of markov random fields now consider a positive definite matrix a and its cholesky decomposition given by a ldl t where l is lower triangular with unit diagonal entries and d a diagonal matrix with positive entries in this paper we prove that a necessary and sufficient condition for zeros i e sparsity in a positive definite matrix a to be preserved in its associated cholesky matrix l and in addition also preserved in the inverse of the cholesky matrix l 1 is that the pattern of zeros corresponds to a co chordal or homogeneous graph associated with a specific type of vertex ordering we proceed to provide a second characterization of this class of graphs in terms of determinants of submatrices that correspond to cliques in the graph these results add to the growing body of literature in the field of sparse matrix decompositions and also prove to be critical ingredients in the probabilistic analysis of an important class of markov random fields