The full infinite dimensional moment problem on semi-algebraic sets of generalized functions

We consider a generic basic semi-algebraic subset S of the space of generalized functions, that is a set given by (not necessarily countably many) polynomial constraints. We derive necessary and sufficient conditions for an infinite sequence of generalized functions to be realizable on S, namely to be the moment sequence of a finite measure concentrated on S. Our approach combines the classical results about the moment problem on nuclear spaces with the techniques recently developed to treat the moment problem on basic semi-algebraic sets of Rd. In this way, we determine realizability conditions that can be more easily verified than the well-known Haviland type conditions. Our result completely characterizes the support of the realizing measure in terms of its moments. As concrete examples of semi-algebraic sets of generalized functions, we consider the set of all Radon measures and the set of all the measures having bounded Radon–Nikodym density w.r.t. the Lebesgue measure.

A moment problem for random discrete measures

Let X be a locally compact Polish space. A random measure on X is a probability measure on the space of all (nonnegative) Radon measures on X. Denote by K(X) the cone of all Radon measures η on X which are of the form η =

Coherent states of a particle in a magnetic field and the Stieltjes moment problem

A solution to a version of the Stieltjes moment. problem is presented. Using this solution, we construct a family of coherent states of a charged particle in a uniform magnetic field. We prove that these states form an overcomplete set that is normalized and resolves the unity. By the help of these coherent states we construct the Fock-Bergmann representation related to the particle quantization. This quantization procedure takes into account a circle topology of the classical motion. (C) 2009 Elsevier B.V. All rights reserved.

Szego polynomials and the truncated trigonometric moment problem

We show how Szego polynomials can be used in the theory of truncated trigonometric moment problem.

On a moment problem associated with Chebyshev polynomials

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

A solution of the trigonometric moment problem via Tagamlitzki's "Theorem of the Cones"

In 1952 Y. Tagamlitzki gave an elegant proof of the classical Bochner’s theorem on the positively definite functions. Unfortunately, he never published his proof. In this paper we consider a related but simpler problem, the trigonometric moment problem, by using Tagamlitzki’s approach.

Parametrization of spectral surfaces of a class of periodic 5-diagonal matrices

The main result of this work is a parametric description of the spectral surfaces of a class of periodic 5-diagonal matrices, related to the strong moment problem. This class is a self-adjoint twin of the class of CMV matrices. Jointly they form the simplest possible classes of 5-diagonal matrices.

Necessary and sufficient conditions for realizability of point processes

We give necessary and sufficient conditions for a pair of (generali- zed) functions 1(r1) and 2(r1, r2), ri 2X, to be the density and pair correlations of some point process in a topological space X, for ex- ample, Rd, Zd or a subset of these. This is an infinite-dimensional version of the classical “truncated moment” problem. Standard tech- niques apply in the case in which there can be only a bounded num- ber of points in any compact subset of X. Without this restriction we obtain, for compact X, strengthened conditions which are necessary and sufficient for the existence of a process satisfying a further re- quirement—the existence of a finite third order moment. We general- ize the latter conditions in two distinct ways when X is not compact.

Polinômios de Szegö e análise de frequência

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Completeness for coherent states in a magnetic-solenoid field

This paper completes our study of coherent states in the so-called magnetic-solenoid field (a collinear combination of a constant uniform magnetic field and Aharonov-Bohm solenoid field) presented in Bagrov et al (2010 J. Phys. A: Math. Theor. 43 354016, 2011 J. Phys. A: Math. Theor. 44 055301). Here, we succeeded in proving nontrivial completeness relations for non-relativistic and relativistic coherent states in such a field. In addition, we solve here the relevant Stieltjes moment problem and present a comparative analysis of our coherent states and the well-known, in the case of pure uniform magnetic field, Malkin-Man'ko coherent states.

Sampling associated with resolvent-type kernels and lagrange-type interpolation series

In this paper a new class of Kramer kernels is introduced, motivated by the resolvent of a symmetric operator with compact resolvent. The article gives a necessary and sufficient condition to ensure that the associ- ated sampling formula can be expressed as a Lagrange-type interpolation series. Finally, an illustrative example, taken from the Hamburger moment problem theory, is included.

Density of Polynomials in the L^2 Space on the Real and the Imaginary Axes and in a Sobolev Space

2000 Mathematics Subject Classification: 41A10, 30E10, 41A65.

On Provable Algorithms for Determination of Continuous Protein Interdomain Motions from Residual Dipolar Couplings

Dynamics of biomolecules over various spatial and time scales are essential for biological functions such as molecular recognition, catalysis and signaling. However, reconstruction of biomolecular dynamics from experimental observables requires the determination of a conformational probability distribution. Unfortunately, these distributions cannot be fully constrained by the limited information from experiments, making the problem an ill-posed one in the terminology of Hadamard. The ill-posed nature of the problem comes from the fact that it has no unique solution. Multiple or even an infinite number of solutions may exist. To avoid the ill-posed nature, the problem needs to be regularized by making assumptions, which inevitably introduce biases into the result.

Here, I present two continuous probability density function approaches to solve an important inverse problem called the RDC trigonometric moment problem. By focusing on interdomain orientations we reduced the problem to determination of a distribution on the 3D rotational space from residual dipolar couplings (RDCs). We derived an analytical equation that relates alignment tensors of adjacent domains, which serves as the foundation of the two methods. In the first approach, the ill-posed nature of the problem was avoided by introducing a continuous distribution model, which enjoys a smoothness assumption. To find the optimal solution for the distribution, we also designed an efficient branch-and-bound algorithm that exploits the mathematical structure of the analytical solutions. The algorithm is guaranteed to find the distribution that best satisfies the analytical relationship. We observed good performance of the method when tested under various levels of experimental noise and when applied to two protein systems. The second approach avoids the use of any model by employing maximum entropy principles. This 'model-free' approach delivers the least biased result which presents our state of knowledge. In this approach, the solution is an exponential function of Lagrange multipliers. To determine the multipliers, a convex objective function is constructed. Consequently, the maximum entropy solution can be found easily by gradient descent methods. Both algorithms can be applied to biomolecular RDC data in general, including data from RNA and DNA molecules.

Muscle moment-arms: a key element in muscle-force estimation.

A clear and rigorous definition of muscle moment-arms in the context of musculoskeletal systems modelling is presented, using classical mechanics and screw theory. The definition provides an alternative to the tendon excursion method, which can lead to incorrect moment-arms if used inappropriately due to its dependency on the choice of joint coordinates. The definition of moment-arms, and the presented construction method, apply to musculoskeletal models in which the bones are modelled as rigid bodies, the joints are modelled as ideal mechanical joints and the muscles are modelled as massless, frictionless cables wrapping over the bony protrusions, approximated using geometric surfaces. In this context, the definition is independent of any coordinate choice. It is then used to solve a muscle-force estimation problem for a simple 2D conceptual model and compared with an incorrect application of the tendon excursion method. The relative errors between the two solutions vary between 0% and 100%.

Measurement Problems and Problem-Solving in Kraft Pulping

Tämän työn tarkoituksena on koota yhteen selluprosessin mittausongelmat ja mahdolliset mittaustekniikat ongelmien ratkaisemiseksi. Pääpaino on online-mittaustekniikoissa. Työ koostuu kolmesta osasta. Ensimmäinen osa on kirjallisuustyö, jossa esitellään nykyaikaisen selluprosessin perusmittaukset ja säätötarpeet. Mukana on koko kuitulinja puunkäsittelystä valkaisuun ja kemikaalikierto: haihduttamo, soodakattila, kaustistamo ja meesauuni. Toisessa osassa mittausongelmat ja mahdolliset mittaustekniikat on koottu yhteen ”tiekartaksi”. Tiedot on koottu vierailemalla kolmella suomalaisella sellutehtaalla ja haastattelemalla laitetekniikka- ja mittaustekniikka-asiantuntijoita. Prosessikemian paremmalle ymmärtämiselle näyttää haastattelun perusteella olevan tarvetta, minkä vuoksi konsentraatiomittaukset on valittu jatkotutkimuskohteeksi. Viimeisessä osassa esitellään mahdollisia mittaustekniikoita konsentraatiomittausten ratkaisemiseksi. Valitut tekniikat ovat lähi-infrapunatekniikka (NIR), fourier-muunnosinfrapunatekniikka (FTIR), online-kapillaarielektroforeesi (CE) ja laserindusoitu plasmaemissiospektroskopia (LIPS). Kaikkia tekniikoita voi käyttää online-kytkettyinä prosessikehitystyökaluina. Kehityskustannukset on arvioitu säätöön kytketylle online-laitteelle. Kehityskustannukset vaihtelevat nollasta miestyövuodesta FTIR-tekniikalle viiteen miestyövuoteen CE-laitteelle; kehityskustannukset riippuvat tekniikan kehitysasteesta ja valmiusasteesta tietyn ongelman ratkaisuun. Työn viimeisessä osassa arvioidaan myös yhden mittausongelman – pesuhäviömittauksen – ratkaisemisen teknis-taloudellista kannattavuutta. Ligniinipitoisuus kuvaisi nykyisiä mittauksia paremmin todellista pesuhäviötä. Nykyään mitataan joko natrium- tai COD-pesuhäviötä. Ligniinipitoisuutta voidaan mitata UV-absorptiotekniikalla. Myös CE-laitetta voitaisiin käyttää pesuhäviön mittauksessa ainakin prosessikehitysvaiheessa. Taloudellinen tarkastelu pohjautuu moniin yksinkertaistuksiin ja se ei sovellu suoraan investointipäätösten tueksi. Parempi mittaus- ja säätöjärjestelmä voisi vakauttaa pesemön ajoa. Investointi ajoa vakauttavaan järjestelmään on kannattavaa, jos todellinen ajotilanne on tarpeeksi kaukana kustannusminimistä tai jos pesurin ajo heilahtelee eli pesuhäviön keskihajonta on suuri. 50 000 € maksavalle mittaus- ja säätöjärjestelmälle saadaan alle 0,5 vuoden takaisinmaksuaika epävakaassa ajossa, jos COD-pesuhäviön vaihteluväli on 5,2 – 11,6 kg/odt asetusarvon ollessa 8,4 kg/odt. Laimennuskerroin vaihtelee tällöin välillä 1,7 – 3,6 m3/odt asetusarvon ollessa 2,5 m3/odt.