Academic literature on the topic 'Truncated moment problem'

Firstly, we recall the classical moment problem and some basic results related to it. By its formulation, this is an inverse problem: being given a sequence (yj)j∈ℕn of real numbers and a closed subset F⊆ℝn, n∈{1,2,…}, find a positive regular Borel measure μ on F such that ∫Ftjdμ=yj, j∈ℕn. This is the full moment problem. The existence, uniqueness, and construction of the unknown solution μ are the focus of attention. The numbers yj, j∈ℕn are called the moments of the measure μ. When a sandwich condition on the solution is required, we have a Markov moment problem. Secondly, we study the existence and uniqueness of the solutions to some full Markov moment problems. If the moments yj are self-adjoint operators, we have an operator-valued moment problem. Related results are the subject of attention. The truncated moment problem is also discussed, constituting the third aim of this work.

Abstract We study the truncated multidimensional moment problem with a general type of truncations. The operator approach to the moment problem is presented. The case where the associated operators form a commuting self-adjoint tuple is characterized in terms of the given moments. The case of the dimensional stability is characterized in terms of the prescribed moments as well. Some sufficient conditions for the solvability of the moment problem are presented. A construction of the corresponding solution is described by algorithms. Numerical examples of the construction are provided.

Let [Formula: see text], with [Formula: see text] and [Formula: see text], be a given complex-valued sequence. The complex moment problem (respectively, the general complex moment problem) associated with [Formula: see text] consists in determining necessary and sufficient conditions for the existence of a positive Borel measure (respectively, a charge) [Formula: see text] on [Formula: see text] such that [Formula: see text] In this paper, we investigate the notion of recursiveness in the two variable case. We obtain several useful results that we use to deduce new necessary and sufficient conditions for the truncated complex moment problem to admit a solution. In particular, we show that the general complex moment problem always has a solution. A concrete construction of the solution and an illustrating example are also given.

Inverse problems naturally occur in many branches of science and mathematics. An inverse problem entails finding the values of one or more parameters using the values obtained from observed data. A typical example of an inverse problem is the inversion of the Radon transform. Here a function (for example of two variables) is deduced from its integrals along all possible lines. This problem is intimately connected with image reconstruction for X-ray computerized tomography. Moment problems are a special class of inverse problems. While the classical theory of moments dates back to the beginning of the 20th century, the systematic study of truncated moment problems began only a few years ago. In this dissertation we will first survey the elementary theory of truncated moment problems, and then focus on those problems with cubic column relations. For a degree 2n real d-dimensional multisequence β ≡ β (2n) ={β i}i∈Zd+,|i|≤2n to have a representing measure μ, it is necessary for the associated moment matrix Μ(n) to be positive semidefinite, and for the algebraic variety associated to β, Vβ, to satisfy rank Μ(n)≤ card Vβ as well as the following consistency condition: if a polynomial p(x)≡ ∑|i|≤2naixi vanishes on Vβ, then Λ(p):=∑|i|≤2naiβi=0. In 2005, Professor Raúl Curto collaborated with L. Fialkow and M. Möller to prove that for the extremal case (Μ(n)= Vβ), positivity and consistency are sufficient for the existence of a (unique, rank Μ(n)-atomic) representing measure. In joint work with Professor Raúl Curto we have considered cubic column relations in M(3) of the form (in complex notation) Z3=itZ+ubar Z, where u and t are real numbers. For (u,t) in the interior of a real cone, we prove that the algebraic variety Vβ consists of exactly 7 points, and we then apply the above mentioned solution of the extremal moment problem to obtain a necessary and sufficient condition for the existence of a representing measure. This requires a new representation theorem for sextic polynomials in Z and bar Z which vanish in the 7-point set Vβ. Our proof of this representation theorem relies on two successive applications of the Fundamental Theorem of Linear Algebra. Finally, we use the Division Algorithm from algebraic geometry to extend this result to other situations involving cubic column relations.

The present thesis intents on analysing the truncated matricial Hamburger power moment problem in the general (degenerate and non-degenerate) case. Initiated due to manifold lines of research, by this time, outnumbering results and thoughts have been established that are concerned with specific subproblems within this field. The resulting presence of such a diversity as well as an extensively considered topic si- multaneously involves advantageous as well as obstructive aspects: on the one hand, we adopt the favourable possibility to capitalise on essential available results that proved beneficial within subsequent research. Nevertheless, on the other hand, we are obliged to illustrate major preparatory work in order to illucidate the comprehension of the attaching examination. Moreover, treating the matricial cases of the respective problems requires meticulous technical demands, in particular, in view of the chosen explicit approach to solving the considered tasks. Consequently, the first part of this thesis is dedicated to furnishing the necessary basis arranging the prime results of this research paper. Compul- sary notation as well as objects are introduced and thoroughly explained. Furthermore, the required techniques in order to achieve the desired results are characterised and ex- haustively discussed. Concerning the respective findings, we are afforded the opportunity to seise presentations and results that are, by this time, elaborately studied. Being equipped with mandatory cognisance, the thematically bipartite second and pivo- tal part objectives to describe all the possible values of all the solution functions of the truncated matricial Hamburger power moment problem M P [R; (s j ) 2n j=0 , ≤]. Aming this, we realise a first paramount achievement epitomising one of the two parts of the main results: Capturing an established representation of the solution set R 0,q [Π + ; (s j ) 2n j=0 , ≤] of the assigned matricial Hamburger moment problem via operating a specific algorithm of Schur-type, we expand these findings. We formulate a parameterisation of the set R 0,q [Π + ; (s j ) 2n j=0 , ≤] which is compatible with establishing respective equivalence classes within a certain subset of Nevanlinna pairs and utilise specific systems of orthogonal polynomials in order to entrench novel representations. In conclusion, we receive a para- meterisation that is valid within the entire upper open complex half-plane Π + . The second of the two prime parts changes focus to analysing all possible values of the functions belonging to R 0,q [Π + ; (s j ) 2n j=0 , ≤] in an arbitrary point w ∈ Π + . We gain two decisive conclusions: We identify these respective values to exhaust particular matrix balls 2n K[(s j ) 2n j=0 , w] := {F (w) | F ∈ R 0,q [Π + ; (s j ) j=0 , ≤]} the parameters of which are feasable to being described by specific rational matrix-valued functions and, in this course, enhance formerly established analyses. Moreover, we compile an alternative representation of the semi-radii constructing the respective matrix balls which manifests supportive in further consideration. We seise the achieved parameterisation of the set K[(s j ) 2n j=0 , w] and examine the behaviour of the respective sequences of left and right semi-radii. We recognise that these sequences of semi-radii associated with the respective matrix balls in the general case admit a particular monotonic behaviour. Consequently, with increasing number of given data, the resulting matrix balls are identified as being nested. Moreover, a proper description of the limit case of an infinite number of prescribed moments is facilitated.:1. Brief Historic Embedding and Introduction 2. Part I: Initialising Compulsary Cognisance Arranging Principal Achievements 2.1. Notation and Preliminaries 2.2. Particular Classes of Holomorphic Matrix-Valued Functions 2.3. Nevanlinna Pairs 2.4. Block Hankel Matrices 2.5. A Schur-Type Algorithm for Sequences of Complex p × q Matrices 2.6. Specific Matrix Polynomials 3. Part II: Momentous Results and Exposition – Improved Parameterisations of the Set R 0,q [Π + ; (s j ) 2n j=0 , ≤] 3.1. An Essential Step to a Parameterisation of the Solution Set R 0,q [Π + ; (s j ) 2n j=0 , ≤] 3.2. Parameterisation of the Solution Set R 0,q [Π + ; (s j ) 2n j=0 3.3. Particular Matrix Polynomials 3.4. Description of the Solution Set of the Truncated Matricial Hamburger Moment Problem by a Certain System of Orthogonal Matrix Polynomials 4. Part III: Prime Results and Exposition – Novel Description Balls 4.1. Particular Rational Matrix-Valued Functions 4.2. Description of the Values of the Solutions 4.3. Monotony of the Semi-Radii and Limit Balls of the Weyl Matrix 5. Summary of Principal Achievements and Prospects A. Matrix Theory B. Integration Theory of Non-Negative Hermitian Measures

This work investigate two different approaches for the parametrization of a special moment problem of Stieltjes-type. On the one hand we deal with systems of Potapov's fundamental matrix inequalities. Thereby, we examine certain invariant subspaces, so-called Dubovoj subspaces, and special matrix polynomials as wells as their associated J- forms. On the other hand we consider a Schur-analytic approach and present a special one-step algorithm. Moreover, considerations on linear fractional transformations of matrices serve as an important tool for the development of the algorithm. Both representations aim at a description of the solution in the non-degenerate case as well as in the different degenerate cases.

This thesis focuses on matrix generalizations of Hurwitz polynomials. A real polynomial with all its roots in the open left half plane of the complex plane is called a Hurwitz polynomial. The study of these Hurwitz polynomials has a long and abundant history, which is associated with the names of Hermite, Routh, Hurwitz, Liénard, Chipart, Wall, Gantmacher et al. The direct matricial generalization of Hurwitz polynomials is naturally defined as follows: A p by p matrix polynomial F is called a Hurwitz matrix polynomial if the determinant of F is a Hurwitz polynomial. Recently, Choque Rivero followed another line of matricial extensions of the classical Hurwitz polynomial, called matrix Hurwitz type polynomials. However, the notion “matrix Hurwitz type polynomial” is still irrelative to “Hurwitz matrix polynomial” due to the totally unclear zero location of the former notion. So the main goal of this thesis is to discover the relation between the two notions “matrix Hurwitz-type polynomials” and “Hurwitz matrix polynomials' and provide some criteria to identify Hurwitz matrix polynomials. The central idea is to determine the inertia triple of matrix polynomials in terms of some related matrix sequences. Suppose that F is a p by p matrix-valued polynomial of degree n. We split F into the odd part and the even part, which allow us to introduce an essential rational matrix function of right type G. From the matrix coefficients of the Laurent series of G we construct the (n-1)-th extended sequence of right Markov parameters (SRMP) of F. Then we show that the inertia triple of F can be characterized by a combination of the inertia triples of two block Hankel matrices generated by the (n-1)-th SRMP of F and the number of zeros (counting for multiplicities) of greatest right common divisors of the even part and the odd part of F lying on the left half of the real axis. By an analogous approach we also obtain the dual results for the inertia triple of F in terms of the SLMP of F. Then we demonstrate that F is a Hurwitz matrix polynomial of degree n if and only if the (n − 1)-th SRMP (resp. SLMP) of F is a Stieltjes positive definite sequence. On this account, the two notions “Hurwitz matrix polynomials” and “matrix Hurwitz type polynomials” are equivalent. In addition, we investigate quasi-stable matrix polynomials appearing in the theory of stability, which contain Hurwitz matrix polynomials as a special case. We seek a correspondence between quasi-stable matrix polynomials, Stieltjes moment problems and multiple Nevanlinna-Pick interpolation in the Stieltjes class. Accordingly, we prove that F is a quasi-stable matrix polynomial if and only if the (n − 1)-th SRMP (resp. SLMP) of F is a Stieltjes non-negative definite extendable sequence and the zeros of right (resp. left) greatest common divisors of the even part and the odd part of F are located on the left half of the real axis.:1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Matrix polynomials and greatest common divisors. . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Greatest common divisors of matrix polynomials . . . . . . . . . . . . . . . . . . . . . 8 3 Matrix sequences and their connection to truncated matricial moment problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4 Matrix fraction description and some related topics . . . . . . . . . . . . . . . . . . 19 4.1 Realization of Matrix fraction description from Markov parameters . . . . . . . 19 4.2 The interrelation between Hermitian transfer function matrices and monic orthogonal system of matrix polynomials . . . . . . . . . . . . . . . . . . . . . . . .27 5 The Bezoutian of matrix polynomials and the inertia problem of matrix polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33 5.2 The Anderson-Jury Bezoutian matrices in connection to special transfer function matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38 6 Para-Hermitian strictly proper transfer function matrices and their related monic Hurwitz matrix polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 7 Solution of matricial Routh-Hurwitz problems in terms of the Markov pa- rameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 8 Matrix Hurwitz type polynomials and some related topics . . . . . . . . . . . . . . 67 9 Hurwitz matrix polynomials and some related topics . . . . . . . . . . . . . . . . . . 77 9.1 Hurwitz matrix polynomials, Stieltjes positive definite sequences and matrix Hurwitz type polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 9.2 S -system of Hurwitz matrix polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 82 10 Quasi-stable matrix polynomials and some related topics . . . . . . . . . . . . 95 10.1 Particular monic quasi-stable matrix polynomials and Stieltjes moment problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95 10.2 Particular monic quasi-stable matrix polynomials and multiple Nevanlinna- Pick interpolation in the Stieltjes class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101 10.3 General description of monic quasi-stable matrix polynomials . . . . . . . . .104 List of terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 List of notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Selbständigkeitserklärung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

This paper deals with the estimation of second order wave excitation forces on a free floating triple-hull barge using software developed based on finite element method. The wave-hull interaction nonlinear problem is presented here using the perturbation method, where potential flow theory is used. In the finite element model, the absorbing or nonreflecting far boundary condition is applied at a truncated surface in the form of boundary damper. The software developed for the solution of this nonlinear problem is validated for two and three-dimensional cases for which analytical and other numerical solutions are known. A convergence study on the three-dimensional cylinder problem is carried out to derive a guideline in selecting finite element mesh density and its grading. A triple-hull barge problem is selected here as a practical problem to study the nonlinear wave effects on the forces and motions. A grid independent study on this problem is carried out by using four finite element meshes of different density and grading. The optimum mesh selected from this study is used for further analysis of the problem. Bandwidth optimization is carried out on the generated meshes in order to reduce the computational effort, as the finite element algorithm used here is based on the banded solver technique. The second order wave excitation forces and moments on the barge estimated for different wave steepness (H/Lw - 0.08 to 0.012) in oblique sea condition shows that the second order surge and heave forces amounts up to 49% and 17.5% respectively and the second order yaw moment up to 39% when compared with the first order (linear) wave forces. Similar trend is observed for the forces and moments in beam sea and head sea conditions.

An innovative time integration method that incorporates spurious high-frequency dissipation capability into the so called “high precision direct integration algorithm” is presented, and its numerical stability and accuracy is discussed. The integration algorithm is named “high precision” to emphasize its numerical capability in reaching computer hardware precision. The proposed procedure employs the well-known state space approach to solve the simultaneous ordinary differential equations, the exact solution of which contains an exponential matrix to be efficiently computed using the truncated Taylor series expansion together with the power-of-two algorithm. The proposed method, belonging to the category of explicit methods, is found to provide better accuracy than many other existing time integration methods, and the integration scheme remains numerically stable over a wide range of frequencies of engineering interest. This paper is also devoted to study numerical accuracy of the Precise Integration Method in solving forced vibration problems, particularly near resonance conditions. The numerically obtained transfer functions are then compared with the analytical exact solution to detect spurious resonance. Finally, numerical examples are used to illustrate its high performance in numerical stability and accuracy. The proposed method carries the merit that can be directly applied to solve momentum equations of motion with exactly the same procedure.

Low-frequency pitch motions of a moored semisubmersible in irregular sea states are analyzed. Physical mechanisms and significance to air-gap problems are addressed. Excitation from wave drift and from moorings/risers is primarily considered, Effects from current and wind are also addressed. Related challenges in deepwater model testing of semis with truncated moorings are discussed. Motion and air-gap data from two previously performed model tests are analysed. Catenary moorings in 335m water depth and in 1100m water depth, respectively, are considered. Model scales are 1:55 and 1:150, respectively. Observed slow-drift pitch components are of the same magnitude level as the wave-frequency components. Comparisons to coupled numerical analysis models are made. Wave drift moment coefficients calibrated empirically according to experiments were used, since the original coefficients gave too low results. The final comparisons show good agreement for the 1:55 case. For the 1:150 case, fairly good agreement is found, but some deviations are observed and believed to be due to poorer wave repeatablity. Tests with truncated moorings at half of the two actual depths were also included, for a check of methods for deepwater model tests performed at reduced depths and combine with numerical analysis (hybrid verification). The importance of proper experimental reproduction at reduced depths, of full-depth pitch and air-gap, is addressed. The results show that with the actual truncation designs, reasonable agreements are obtained, but use of the scale 1:150 seems to give too large uncertainties due to the poorer wave repeatability.