one two sided residue interpolation for matrix valuedh2 functions with symmetries abstract the two sided lagrange sylvester interpolation problem introduced by ball gohberg and rodman is solved in the framework ofh2functions when some symmetry constraints are added

laurent interpolation for rational matrix functions and a local factorization principle abstract a regular rational matrix function is constructed when a finite part of the laurent series of the function and of its inverse are given at a finite number of points provided an obvious necessary condition is satisfied the construction is such that the function will have at most one extra pole or zero possibly of high multiplicity this construction is further applied to develop local principles for factorizations of rational matrix functions

on the solution of the minimal rational interpolation problem it is shown that the reachability and the observability indices of appropriately defined pairs of matrices provide the key for solving the minimal rational bitangen tial interpolation problem

cuspidal class number of the tower of modular curves x1 npn we consider a cuspidal class number which is the order of a subgroup of the full cuspidal divisor class group of x1 npn with p nmid n and n 1 by studying the second generalized bernoulli numbers we obtain results similar to ones ferrero and washington in ann math 2 109 2 377 395 1979 washington in invent math 49 87 97 1978 about the relative class numbers of cyclotomic mathbb z p extension of an abelian number field

matrix polynomial solutions of tangential lagrange sylvester interpolation conditions of low mcmillan degree abstract a matrix polynomial of mcmillan degree n 1 which satisfies n tangential lagrange interpolation conditions is constructed explicitly in realization form analogous results are given for more general bitangential lagrange sylvester interpolation conditions a corollary is a type of euclidean algorithm for matrix polynomials and mcmillan degree the main tool is the recent parametrization of all matrix polynomial solutions of a set of lagrange sylvester interpolation conditions due to ball gohberg and rodman

identification and estimation with contaminated data when do covariate data sharpen inference abstract contaminated or corrupted data typically require strong assumptions to identify parameters of interest however weaker assumptions often identify bounds on these parameters this paper addresses when covariate data variables in addition to the one of interest tighten these bounds first we construct the identification region for the distribution of the variable of interest this region demonstrates that covariate data are useless without knowledge about the distribution of erroneous data conditional on the covariates then we develop bounds both on probabilities and on parameters of this distribution that respect stochastic dominance

structured matrices and unconstrained rational interpolation problems we describe a fast recursive algorithm for the solution of an unconstrained rational interpolation problem by exploiting the displacement structure concept we use the interpolation data to implicitly define a convenient non hermitian structured matrix and then apply a computationally efficient procedure for its triangular factorization this leads to a transmission line interpretation that makes evident the interpolation properties we further discuss connections with the lagrange interpolating polynomial as well as questions regarding the minimality and the admissible degrees of complexity of the solutions

bitangential interpolation problems for symmetric rational matrix functions abstract we develop the zero pole structure of rational matrix functions s that satisfy an identity of the form s t v s v for some square matrix v for which v vt where 1 as an application we describe the set of symmetric or antisymmetric solutions of a set of bitangential interpolation conditions solutions of bitangential interpolation conditions satisfying other types of symmetries are also discussed

the iwasawa conjecture for totally real fields let f be a totally real number field let p be a prime number and for any integer n let fun denote the group of nth roots of unity let 41 be a p adic valued artin character for f and let f be the extension of f attached to 4 i e so that 4 is the character of a faithful representation of gal f f we will assume that f is also totally real for a number field k let k denote the cyclotomic zp extension of k following greenberg we say that 4 is of type s if f n fc f and of type w if 4 is one dimensional with f p c fcc deligne and ribet in dr following kubota and leopoldt for the case f q have proved the existence of a p adic l function associated to a one dimensional artin character 4 with f totally real this function lp s 4 is continuous for s e zp 1 and even at s 1 if 4 is not trivial and satisfies the following interpolation property

nonparametric maximum likelihood estimation from samples with irrelevant data and verification bias abstract suppose that some measurements come from a distribution f that is of interest and others come from another irrelevant distribution g some measurements from f are verified and known to be from f the other unverified measurements may be from f or from g not all measurements from f are equally likely to be verified and no measurement from g is ever verified this model applies to measurements of low concentrations obtained using gas chromatography mass spectroscopy for example as is shown in this article how well a feature t f of f can be estimated when there are unverified data depends on what can be assumed about f g and the conditional probability v x of verifying a measurement of x from f if f g and v are unrestricted then more than one choice of f g v gives the same distribution p of the observable x and thus t f cannot be uniquely estimated from data but if the set of values of t f that correspond to a distribution p of x is small enough then it is reasonable to try t

quantitative non vanishing of dirichlet l values modulo p let p be an odd prime and k a non negative integer let n be a positive integer such that p not mid n and lambda a dirichlet character modulo n we obtain quantitative lower bounds for the number of dirichlet character chi modulo f with the critical dirichlet l value l k lambda chi being p indivisible here f rightarrow infty with n f 1 and p not mid f phi f we explore the indivisibility via an algebraic and a homological approach the latter leads to a bound of the form f 1 2 the p indivisibility yields results on the distribution of the associated p selmer ranks we also consider an iwasawa variant it leads to an explicit upper bound on the lowest conductor of the characters factoring through the iwasawa mathbb z ell extension of mathbb q for an odd prime ell ne p with the corresponding critical l value twists being p indivisible

column reduced rational matrix functions with given null pole data in the complex plane abstract explicit formulas are given for rational matrix functions which have a prescribed null pole structure in the complex plane and are column reduced at infinity a full parametrization of such functions is obtained the results are specified and developed further for matrix polynomials

the nevanlinna pick interpolation problems and power moment problems for matrix valued functions abstract this paper is concerned with the study of several types of matrix interpolation problems including the matrix nevanlinna pick interpolation problem in the nevanlinna class n p the matrix nevanlinna pick interpolation problem in the stieltjes class l p the truncated matrix power moment problem on the real axis the stieltjes matrix power moment problem we use the so called block hankel vector approach for the treatment of these problems and make explicit the correspondence between solutions to the nevanlinna pick interpolation problem and solutions to an associated power moment problem with the block hankel vector under consideration this enables one not only to find existence criteria for the solutions of the nevanlinna pick interpolation problem but also to describe the solutions based on the results of power moment problem and vice versa

a nonsplit sum of coefficients of modular forms we introduce and study certain truncated sums of hecke eigenvalues of gl2 automorphic forms along quadratic polynomials a power saving estimate is established and new applications to moments of critical l values associated to quadratic fields are derived an application to the asymptotic behavior of the height of heegner points and singular moduli is discussed in detail

generalized hermite formula for the two sided lagrange sylvester interpolation the classical hermite formula of the scalar polynomial interpolation is generalized to the setting of two sided lagrange sylvester interpolation for matrix polynomials the formula then is applied to a description of the set of all matrix polynomial solutions to that two sided interpolation

multiple nevanlinna pick interpolation with both interior and boundary data and its connection with the power moment problem abstract the so called hankel vector approach is used to handle three boundary versions of the multiple nevanlinna pick interpolation problem in the nevanlinna class n involving both interior and boundary data it turns out that each of these boundary interpolation problems can be reduced to what amounts to a certain truncated standard or nonstandard hamburger moment problem associated with the hankel vector of the former with some possible constraint on distribution functions that assign no mass to any of the real nodes in particular this leads to solvability criteria for each of these interpolation problems and the description of solutions by using results from theory of moments

an easy to compute factorization of rational generalized positive functions we give a simple constructive proof of a factorization theorem for scalar rational functions with a non negative real part on the imaginary axis a mathematica program performing this factorization is provided

polynomial interpolation and gaussian quadrature for matrix valued functions abstract the techniques for polynomial interpolation and gaussian quadrature are generalized to matrix valued functions it is shown how the zeros and rootvectors of matrix orthonormal polynomials can be used to get a quadrature formula with the highest degree of precision

on l functions of elliptic curves and cyclotomic towers in this paper we shall strengthen a recent theorem of ralph greenberg 1 concerning l functions of elliptic curves and anticyclotomic towers let k be an imaginary quadratic field and m an abelian extension of k possibly of infinite degree we say that the extension m k is anticyclotomic if m is galois over q and if the nontrivial element of gal k q acts on gal m k by inversion now fix a finite set p of rational primes and consider the compositum in an algebraic closure of k of all anticyclotomic extensions of k which are unramified outside p we denote this compositum by l and refer to it as the maximal anticyclotomic extension of k unramified outside p the field l may also be described as the union of all ring class fields of k with conductor divisible only by primes in p class field theory shows that gal l k is isomorphic to the product of a finite group and the group

uniform distribution of heegner points let e be a modular elliptic curve over q of conductor n let k denote an imaginary quadratic field of discriminant d with n d 1 if p is a prime then there exists a unique zp extension k k such that gal k q acts nontrivially on gal k k the field k is called the anticyclotomic zp extension of k let e k denote the mordell weil group of e over k then a fundamental conjecture of mazur maz predicts that the size of e k is controlled by the prime factorization of n in k equivalently mazur s conjecture relates the size of the mordell weil group to the sign in the functional equation of certain l series the conjecture was verified by greenberg rohrlich and rubin in what mazur calls the exceptional case when e has complex multiplication by k more generally they settled the conjecture for certain abelian varieties with complex multiplication for a discussion of this cm case we refer the reader to gre roh and rub our goal in this paper is to treat the generic case which occurs either when e has no cm or when the field of complexmultiplications is distinct from k under certain conditions on e and k mazur s conjecture predicts that the group e k is finitely generated our main result asserts that this is in fact the case at least when p is an ordinary prime for e or when the class number of k is prime to p the main new ingredient we introduce is that of equidistribution following ideas used by ferrero andwashington to study the cyclotomic invariant more precisely we show that theheegner points associated to definite quaternion algebras are uniformly distributed on the components of a certain curve x and that the elements of a certain galois group act independently in a suitable sense this combined with a special value formula due to gross allows us to conclude that the special values of anticyclotomic l functions are almost always nonzero so that the statement about the mordell weil groups follows from the machinery of euler systems as developed by bertolini and darmon bd

a recursive method for solving unconstrained tangential interpolation problems an efficient recursive solution is presented for the one sided unconstrained tangential interpolation problem the method relies on the triangular factorization of a certain structured matrix that is implicitly defined by the interpolation data the recursive procedure admits a physical interpretation in terms of discretized transmission lines in this framework the generating system is constructed as a cascade of first order sections singular steps occur only when the input data is contradictory i e only when the interpolation problem does not have a solution various pivoting schemes can be used to improve numerical accuracy or to impose additional constraints on the interpolants the algorithm also provides coprime factorizations for all rational interpolants and can be used to solve polynomial interpolation problems such as the general hermite matrix interpolation problem a recursive method is proposed to compute a column reduced generating system that can be used to solve the minimal tangential interpolation problem

the nevanlinna pick interpolation problems and power moment problems for matrix valued functions iii the infinitely many data case abstract the objective of this paper is to establish an intrinsic and simple correspondence between a multiple nevanlinna pick matrix interpolation np problem in the nevanlinna class n p with a denumerable set of nodes and a certain full hamburger matrix moment problem with the help of the indicated correspondence the solution of an infinite np problem in n p can be reduced to what amounts to the study of the associated full hamburger matrix moment problem with the so called block hankel vector of the former and vice versa these investigations can be regarded as a natural extension of our previous work on both the np problems with a finite set of interpolation data and the truncated power matrix moment problems

nondetects detection limits and the probability of detection abstract when chemists cannot quantify the concentration in a field sample they report nondetect instead of a numerical measurement a data analyst faced with environmental data containing nondetects might assume that all nondetects are zeros all nondetects are smaller than the smallest detect numerical measurement or if a detection limit is reported that all nondetects are below the detection limit this article shows that these assumptions can be incorrect and suggests better alternatives the measurements that are likely to be reported as nondetect can be described by a new concept the probability of acceptance where acceptance means that a measurement passes the requirements for being reported as a detect the 90th percentile of the probability of acceptance curve is a reasonable upper bound or censoring limit for a measurement reported as nondetect the probability of acceptance also suggests the complexity of the data analysis task for example assuming that all nondetects are smaller than

symmetric matrix polynomial equation interpolation results new numerical procedures are proposed to solve the symmetric matrix polynomial equation a t s x s x t s a s 2b s that is frequently encountered in control and signal processing an interpolation approach is presented that takes full advantage of symmetry properties and leads to an equivalent reduced size linear system of equations it results in a simple and general characterization of all solutions of expected column degrees several new theoretical results concerning stability theory and reduced sylvester resultant matrices are also developed and used to conclude a priori on the existence of a solution by means of numerical experiments it is shown that our algorithms are more efficient than older methods and namely appear to be numerically reliable

the iwasawa mu invariant of p adic hecke l functions for an odd prime p we compute the invariant of the anticyclotomic katz p adic l function of a p ordinary cm field if the conductor of the branch character is a product of primes split over the maximal real subfield except for rare cases where the root number of the p adic functional equation is congruent to 1 modulo p the invariant vanishes

sylvester and lyapunov equations and some interpolation problems for rational matrix functions abstract the main theme of this paper is the interrelations between interpolation problems for rational matrix functions and the linear matrix equations of sylvester and lyapunov type with suitable controllability and observability conditions using these interrelations we are able to solve certain interpolation problems for rational matrix functions which include for example the problem of constructing a rational matrix function of minimal size with prescribed zeros poles and their multiplicities as to matrix equations we prove for example a result which is in some sense the inverse to the well known chen wimmer inertia theorem an application to system theory is also included

interpolation by polynomials with symmetries abstract we here specialize the standard matrix valued polynomial interpolation to the case where the interpolating polynomials admit various symmetries the introduced approach allows for non convex families of interpolating polynomials if this family happens to be convex all minimal degree interpolating polynomials are parameterized

regular rational matrix functions with prescribed null and pole data except at infinity regular rational matrix functions are constructed in realized form with prescribed null and pole data in the complex and with no restrictions at infinity the construction is based on extension theorems for null and pole pairs an application to an interpolation for regular matrix polynominals is included

polynomial interpolation over quaternions abstract interpolation theory for complex polynomials is well understood in the non commutative quaternionic setting the polynomials can be evaluated on the left and on the right if the interpolation problem involves interpolation conditions of the same left or right type the results are very much similar to the complex case a consistent problem has a unique solution of a low degree less than the number of interpolation conditions imposed and the solution set of the homogeneous problem is an ideal in the ring h z the problem containing both left and right interpolation conditions is quite different there may exist infinitely many low degree solutions and the solution set of the homogeneous problem is a quasi ideal in h z

minimal mcmillan degree rational matrix functions with prescribed local zero pole structure abstract given a rational m n matrix function w z and a subset of the complex plane c we give an explicit algorithm using column operations to produce a rational matrix function h z of minimal possible mcmillan degree which has the same left zero pole structure on as w z i e for which h wq where q and q 1 are rational n n matrix functions with no poles in

horizontal non vanishing of heegner points and toric periods abstract let f be a totally real number field and a a modular gl 2 type abelian variety over f let k f be a cm quadratic extension let be a class group character over k such that the rankin selberg convolution l s a is self dual with root number 1 we show that the number of class group characters with bounded ramification such that l 1 a 0 increases with the absolute value of the discriminant of k we also consider a rather general rank zero situation let be a cuspidal cohomological automorphic representation over gl 2 a f let be a hecke character over k such that the rankin selberg convolution l s is self dual with root number 1 we show that the number of hecke characters with fixed type and bounded ramification such that l 1 2 0 increases with the absolute value of the discriminant of k the gross zagier formula and the waldspurger formula relate the question to horizontal non vanishing of heegner points and toric periods respectively for both situations the strategy is geometric relying on the zariski density of cm points on self products of a quaternionic shimura variety the recent result 26 31 1 on the andre oort conjecture is accordingly fundamental to the approach