Kac-Akhiezer formula for normal integral operators

The Kac-Akhiezer formula for finite section normal Wiener-Hopf integral operators is proved. This is an extension of the corresponding result for symmetric operator [2, 3, 4, 5, 6, 7].

Social-class inequalities in ill health - the contribution of physical workload

Tutkimuksen aiheita olivat yhteiskuntaluokkien väliset erot sairastavuudessa ja alentuneessa toimintakyvyssä, sekä fyysisen työkuormituksen ja joidenkin muiden työolojen vaikutus sairastavuuteen. Empiirisestä työstä on raportoitu myös neljässä kansainvälisissä tieteellisissä aikakauskirjoissa julkaistussa artikkelissa. Tässä julkaistu yhteenveto sisältää tulosten yhteenvedon lisäksi myös tutkimusta koskevien käsitteellisten ja teoreettisten kysymysten sekä tutkimustradition kriittisen katsauksen. Työn päätavoitteita olivat 1) tutkia fyysisesti kuormittavan työn, ja jossain määrin muiden työolojen osuutta yhteiskuntaluokkien välisiin eroihin sairaudessa ja toimintakyvyn alentuneisuudessa; 2) tutkia työn fyysisen kuormittavuuden, työhön liittyvien vaikutusmahdollisuuksien ja hallinnan (decision latitude), luokka-aseman, iän ja sukupuolen yhteisvaikutuksia heikentyneeseen terveydentilaan; sekä 3) tutkia missä määrin mekaanisten työaltisteiden ja tuki- ja liikuntaelinsairastavuuden välinen yhteys voi selittää yhteiskuntaluokkien välisiä eroja heikentyneessä yleisessä terveydentilassa. Tutkittavat olivat keski-ikäisiä Helsingin kaupungin työntekijöitä. Analyysit perustuivat poikittaisasetelmaan, ja käytetty aineisto oli Helsinki Health Studyn vuosien 2000 ja 2002 välillä kerättyä aineistoa. Analyyseihin käytetyssä aineistossa oli 3740:stä 8002:een tutkittavaa. Tulosten perusteella fyysisillä (sekä fysikaalisilla) työoloilla on merkittävä vaikutus yhteiskuntaluokkien välisiin eroihin yleisessä sairastavuudessa, toimintakyvyn heikentymisessä, tuki- ja liikuntaelinsairastavuudessa sekä itsearvioidussa terveydentilassa. Naisilla lähes puolet heikentyneen toimintakyvyn ja koetun terveydentilan luokkaeroista vaikutti olevan selitettävissä fyysisellä työkuormituksella. Hallintamahdollisuuksien ei havaittu merkittävästi muuttavan fyysisen kuormituksen vaikutusta toimintakykyyn. Fyysisen kuormittavuuden terveysvaikutus voimistui kasvavan iän mukaan enemmän naisilla kuin miehillä. Osa, mutta ei koko fyysisen kuormituksen vaikutus yhteiskuntaluokkien eroihin heikentyneessä terveydessä vaikutti välittyvän tuki- ja liikuntaelinsairastavuuden kautta. Terveys ja sairaus eivät ole yhtenäisiä tiloja, ja siksi monet eri sosiaalisesti ja rakenteellisesti määräytyvät olosuhteet todennäköisesti vaikuttavat yhteiskunnallisten terveyserojen syntymiseen. Fyysis-materiaalisten olojen vaikutusta terveyserojen syntyyn nyky-yhteiskunnassa on mahdollisesti aliarvioitu. Yhteiskuntaluokkien väliset erot fyysis-materiaalisissa olosuhteissa eivät ole kadonneet, ja nämä erot todennäköisesti vaikuttavat terveyserojen syntyyn.

On solitons with half integral charge

Can certain soliton states, with half integral expectation value of charge, be also eigenstates of charge X with half integral eigenvalue? It can be so only with a somewhat sophisticated definition of charge.

Some direct numerical approaches to the solution of the singular nonlinear transonic integral equation

The nonlinear singular integral equation of transonic flow is examined, noting that standard numerical techniques are not applicable in solving it. The difficulties in approximating the integral term in this expression were solved by special methods mitigating the inaccuracies caused by standard approximations. It was shown how the infinite domain of integration can be reduced to a finite one; numerical results were plotted demonstrating that the methods proposed here improve accuracy and computational economy.

Water-wave scattering by two submerged plane vertical barriers-Abel integral-equation approach

The classical problem of surface water-wave scattering by two identical thin vertical barriers submerged in deep water and extending infinitely downwards from the same depth below the mean free surface, is reinvestigated here by an approach leading to the problem of solving a system of Abel integral equations. The reflection and transmission coefficients are obtained in terms of computable integrals. Known results for a single barrier are recovered as a limiting case as the separation distance between the two barriers tends to zero. The coefficients are depicted graphically in a number of figures which are identical with the corresponding figures given by Jarvis (J Inst Math Appl 7:207-215, 1971) who employed a completely different approach involving a Schwarz-Christoffel transformation of complex-variable theory to solve the problem.

A Fredholm-type theory for third-kind linear integral equations

The third-kind linear integral equation Image where g(t) vanishes at a finite number of points in (a, b), is considered. In general, the Fredholm Alternative theory [[5.]] does not hold good for this type of integral equation. However, imposing certain conditions on g(t) and K(t, t′), the above integral equation was shown [[1.], 49–57] to obey a Fredholm-type theory, except for a certain class of kernels for which the question was left open. In this note a theory is presented for the equation under consideration with some additional assumptions on such kernels.

Analysis of two degrees of freedom systems through weighted mean square linearization approach

In this paper a study of the free, forced and self-excited vibrations of non-linear, two degrees of freedom systems is reported. The responses are obtained by linearizing the nonlinear equations using the weighted mean square linearization approach. The scope of this approach, in terms of the type of non-linearities the method can tackle, is also discussed.

On the Solution of Certain Simultaneous Pairs of Dual Integral Equations

Mit einer direkten Methode, bei der der Erdelyi-Kober- und der modifizierte Hankel-Operator Anwendung finden, werden gewisse Systeme aus zwei bzw. drei Paaren dualer Integralgleichungen mit Bessel-Kernen in geschlossener Form gelöst. Für bestimmte Funktionenklassen und Ordnungen der Bessel-Funktionen ist die Vorgehensweise angebrachter und geeigneter als die bereits existierenden Methoden.

Analysis of two degrees of freedom systems through weighted mean square linearization approach

In this paper a study of the free, forced and self-excited vibrations of non-linear, two degrees of freedom systems is reported. The responses are obtained by linearizing the nonlinear equations using the weighted mean square linearization approach. The scope of this approach, in terms of the type of non-linearities the method can tackle, is also discussed.

Determination of integral fringe order in photoelasticity

The paper describes a method to determine the integral fringe order associated with a fractional fringe value that is measured using Tardy or other similar compensators. The method makes use of two different wavelengths of light to determine the fractional fringe values. Further, it does not assume the independence of the material fringe constant on the wavelength of light used. From these measured fractional fringe values, the associated integral fringe order is determined. A method to construct a ready-reckoner table is also described which helps to identify the integral fringe order from any two measured fractional fringe values.

Boundedness of weakly singular integral operators on domains

The monograph dissertation deals with kernel integral operators and their mapping properties on Euclidean domains. The associated kernels are weakly singular and examples of such are given by Green functions of certain elliptic partial differential equations. It is well known that mapping properties of the corresponding Green operators can be used to deduce a priori estimates for the solutions of these equations. In the dissertation, natural size- and cancellation conditions are quantified for kernels defined in domains. These kernels induce integral operators which are then composed with any partial differential operator of prescribed order, depending on the size of the kernel. The main object of study in this dissertation being the boundedness properties of such compositions, the main result is the characterization of their Lp-boundedness on suitably regular domains. In case the aforementioned kernels are defined in the whole Euclidean space, their partial derivatives of prescribed order turn out to be so called standard kernels that arise in connection with singular integral operators. The Lp-boundedness of singular integrals is characterized by the T1 theorem, which is originally due to David and Journé and was published in 1984 (Ann. of Math. 120). The main result in the dissertation can be interpreted as a T1 theorem for weakly singular integral operators. The dissertation deals also with special convolution type weakly singular integral operators that are defined on Euclidean spaces.

Aspects of atomic decompositions and Bergman projections

The concept of an atomic decomposition was introduced by Coifman and Rochberg (1980) for weighted Bergman spaces on the unit disk. By the Riemann mapping theorem, functions in every simply connected domain in the complex plane have an atomic decomposition. However, a decomposition resulting from a conformal mapping of the unit disk tends to be very implicit and often lacks a clear connection to the geometry of the domain that it has been mapped into. The lattice of points, where the atoms of the decomposition are evaluated, usually follows the geometry of the original domain, but after mapping the domain into another this connection is easily lost and the layout of points becomes seemingly random. In the first article we construct an atomic decomposition directly on a weighted Bergman space on a class of regulated, simply connected domains. The construction uses the geometric properties of the regulated domain, but does not explicitly involve any conformal Riemann map from the unit disk. It is known that the Bergman projection is not bounded on the space L-infinity of bounded measurable functions. Taskinen (2004) introduced the locally convex spaces LV-infinity consisting of measurable and HV-infinity of analytic functions on the unit disk with the latter being a closed subspace of the former. They have the property that the Bergman projection is continuous from LV-infinity onto HV-infinity and, in some sense, the space HV-infinity is the smallest possible substitute to the space H-infinity of analytic functions. In the second article we extend the above result to a smoothly bounded strictly pseudoconvex domain. Here the related reproducing kernels are usually not known explicitly, and thus the proof of continuity of the Bergman projection is based on generalised Forelli-Rudin estimates instead of integral representations. The minimality of the space LV-infinity is shown by using peaking functions first constructed by Bell (1981). Taskinen (2003) showed that on the unit disk the space HV-infinity admits an atomic decomposition. This result is generalised in the third article by constructing an atomic decomposition for the space HV-infinity on a smoothly bounded strictly pseudoconvex domain. In this case every function can be presented as a linear combination of atoms such that the coefficient sequence belongs to a suitable Köthe co-echelon space.

Integral Transformations of Volterra Gaussian Processes

We study integral representations of Gaussian processes with a pre-specified law in terms of other Gaussian processes. The dissertation consists of an introduction and of four research articles. In the introduction, we provide an overview about Volterra Gaussian processes in general, and fractional Brownian motion in particular. In the first article, we derive a finite interval integral transformation, which changes fractional Brownian motion with a given Hurst index into fractional Brownian motion with an other Hurst index. Based on this transformation, we construct a prelimit which formally converges to an analogous, infinite interval integral transformation. In the second article, we prove this convergence rigorously and show that the infinite interval transformation is a direct consequence of the finite interval transformation. In the third article, we consider general Volterra Gaussian processes. We derive measure-preserving transformations of these processes and their inherently related bridges. Also, as a related result, we obtain a Fourier-Laguerre series expansion for the first Wiener chaos of a Gaussian martingale. In the fourth article, we derive a class of ergodic transformations of self-similar Volterra Gaussian processes.