On Short Exponential Sums Involving Fourier Coefficients of Holomorphic Cusp Forms

By an exponential sum of the Fourier coefficients of a holomorphic cusp form we mean the sum which is formed by first taking the Fourier series of the said form,then cutting the beginning and the tail away and considering the remaining sum on the real axis. For simplicity’s sake, typically the coefficients are normalized. However, this isn’t so important as the normalization can be done and removed simply by using partial summation. We improve the approximate functional equation for the exponential sums of the Fourier coefficients of the holomorphic cusp forms by giving an explicit upper bound for the error term appearing in the equation. The approximate functional equation is originally due to Jutila [9] and a crucial tool for transforming sums into shorter sums. This transformation changes the point of the real axis on which the sum is to be considered. We also improve known upper bounds for the size estimates of the exponential sums. For very short sums we do not obtain any better estimates than the very easy estimate obtained by multiplying the upper bound estimate for a Fourier coefficient (they are bounded by the divisor function as Deligne [2] showed) by the number of coefficients. This estimate is extremely rough as no possible cancellation is taken into account. However, with small sums, it is unclear whether there happens any remarkable amounts of cancellation.

Demographic Modelling of Human Population Growth

This work presents models and methods that have been used in producing forecasts of population growth. The work is intended to emphasize the reliability bounds of the model forecasts. Leslie model and various versions of logistic population models are presented. References to literature and several studies are given. A lot of relevant methodology has been developed in biological sciences. The Leslie modelling approach involves the use of current trends in mortality,fertility, migration and emigration. The model treats population divided in age groups and the model is given as a recursive system. Other group of models is based on straightforward extrapolation of census data. Trajectories of simple exponential growth function and logistic models are used to produce the forecast. The work presents the basics of Leslie type modelling and the logistic models, including multi- parameter logistic functions. The latter model is also analysed from model reliability point of view. Bayesian approach and MCMC method are used to create error bounds of the model predictions.

Quasiconformal mappings and inequalities involving special functions

This PhD thesis in Mathematics belongs to the field of Geometric Function Theory. The thesis consists of four original papers. The topic studied deals with quasiconformal mappings and their distortion theory in Euclidean n-dimensional spaces. This theory has its roots in the pioneering papers of F. W. Gehring and J. Väisälä published in the early 1960’s and it has been studied by many mathematicians thereafter. In the first paper we refine the known bounds for the so-called Mori constant and also estimate the distortion in the hyperbolic metric. The second paper deals with radial functions which are simple examples of quasiconformal mappings. These radial functions lead us to the study of the so-called p-angular distance which has been studied recently e.g. by L. Maligranda and S. Dragomir. In the third paper we study a class of functions of a real variable studied by P. Lindqvist in an influential paper. This leads one to study parametrized analogues of classical trigonometric and hyperbolic functions which for the parameter value p = 2 coincide with the classical functions. Gaussian hypergeometric functions have an important role in the study of these special functions. Several new inequalities and identities involving p-analogues of these functions are also given. In the fourth paper we study the generalized complete elliptic integrals, modular functions and some related functions. We find the upper and lower bounds of these functions, and those bounds are given in a simple form. This theory has a long history which goes back two centuries and includes names such as A. M. Legendre, C. Jacobi, C. F. Gauss. Modular functions also occur in the study of quasiconformal mappings. Conformal invariants, such as the modulus of a curve family, are often applied in quasiconformal mapping theory. The invariants can be sometimes expressed in terms of special conformal mappings. This fact explains why special functions often occur in this theory.

Soodakattiloiden ilmakanavien hyödyntäminen jäykistävänä rakenteena

Työssä tutkittiin soodakattiloiden ilmakanavien hyödyntämistä jäykistävänä rakenteena. Työssä käsiteltiin yksittäisiä jäykistämättömiä ja jäykistettyjä levykenttiä ja niiden lommahduskestävyyttä Eurokoodi standardin mukaisesti ja elementtimenetelmän avulla. Lisäksi käsiteltiin lommahduksen teoriaa ja levykenttien käyttäytymistä yleisellä tasolla erilaisilla kuormituksilla ja reunaehdoilla. Työn tavoitteena oli selvittää kuinka lommahdus tutkitaan Eurokoodin mukaisesti ja elementtimenetelmää hyödyntäen, kun levykentän kuormituksena on poikittainen kuormitus tason suuntaisen kuormituksen lisäksi. Työssä tutkittiin kahden eri elementtimenetelmään pohjautuvan ratkaisuvaihtoehdon käyttöä lommahduslaskennassa. Työssä kehitettiin Eurokoodin sovellettu yhteisvaikutuskaavan käyttö lineaarisen ominaisarvotehtävän ratkaisun lisänä, jossa otetaan huomioon painekuorman vaikutus levykentän lommahduksessa. Kehitettyä menetelmää sovellettiin ilmakanavan esimerkkirakenteen mitoituksessa.

Stability Analysis in Multicriteria Discrete Portfolio Optimization.

Almost every problem of design, planning and management in the technical and organizational systems has several conflicting goals or interests. Nowadays, multicriteria decision models represent a rapidly developing area of operation research. While solving practical optimization problems, it is necessary to take into account various kinds of uncertainty due to lack of data, inadequacy of mathematical models to real-time processes, calculation errors, etc. In practice, this uncertainty usually leads to undesirable outcomes where the solutions are very sensitive to any changes in the input parameters. An example is the investment managing. Stability analysis of multicriteria discrete optimization problems investigates how the found solutions behave in response to changes in the initial data (input parameters). This thesis is devoted to the stability analysis in the problem of selecting investment project portfolios, which are optimized by considering different types of risk and efficiency of the investment projects. The stability analysis is carried out in two approaches: qualitative and quantitative. The qualitative approach describes the behavior of solutions in conditions with small perturbations in the initial data. The stability of solutions is defined in terms of existence a neighborhood in the initial data space. Any perturbed problem from this neighborhood has stability with respect to the set of efficient solutions of the initial problem. The other approach in the stability analysis studies quantitative measures such as stability radius. This approach gives information about the limits of perturbations in the input parameters, which do not lead to changes in the set of efficient solutions. In present thesis several results were obtained including attainable bounds for the stability radii of Pareto optimal and lexicographically optimal portfolios of the investment problem with Savage's, Wald's criteria and criteria of extreme optimism. In addition, special classes of the problem when the stability radii are expressed by the formulae were indicated. Investigations were completed using different combinations of Chebyshev's, Manhattan and Hölder's metrics, which allowed monitoring input parameters perturbations differently.

Lauseen aspektin ilmaiseminen suomen kielessä. Tutkimus olla-, tehdä- ja tulla-verbillisistä tekstilauseista käyttökonteksteissaan

Tutkimuksessa tarkastellaan lauseen aspektin ilmaisemista suomen kielessä. Aspektia käsitellään merkityskategoriana, joka osoittaa lauseen kuvaaman asiaintilan ajallisen keston, ja perustavanlaatuisena aspektuaalisena erontekona pidetään rajattuuden ja rajaamattomuuden vastakohtaisuutta. Tutkimuksessa selvitetään, millä perusteella lauseet saavat joko rajatun tai rajaamattoman aspektitulkinnan ja miten konteksti vaikuttaa tähän tulkintaan. Lauseen kontekstina käsitellään kielellistä kontekstia eli tekstiä. Työ on aineistopohjainen tutkimus kirjoitetusta nykysuomesta, ja tarkastelun kohteena on sanomalehtiteksteistä koottu lauseaineisto. Lauseiden pääverbit ovat olla, tehdä ja tulla. Aineistosta on mahdollista esittää sekä kvalitatiivisia että kvantitatiivisia huomioita. Tutkimuksen teoreettisen ja metodologisen taustan muodostavat eräiden kognitiivisen kielitieteen suuntausten kuvauskäsitteet ja -metodit sekä fennistinen aspektin kuvaamisen perinne. Tutkimuksessa tarkastellaan kahta fennistiikassa esitettyä tapaa määritellä lauseen aspektimerkitys ja osoitetaan, että ne ovat toisiaan täydentäviä. Molemmat lähestymistavat huomioon ottamalla on siis mahdollista kuvata lauseen aspektimerkityksen määräytyminen täsmällisemmin kuin vain yhteen kuvaustapaan keskittymällä. Lisäksi osoitetaan, että keskeisinä aspektin ilmaisemisen keinoina pidetyt keston ja toistuvuuden adverbiaalit jäävät aineistossa marginaalisiksi. Ajankohdan adverbiaaleja puolestaan käsitellään aiemmasta tutkimuksesta poiketen rajattuina tarkastelunäkökulmina kuvattuun asiaintilaan, ja ne toimivat tässä tehtävässä liittyessään aspektiltaan rajaamattomiin lauseisiin. Lisäksi tutkimus osoittaa, että aspektin ilmaisemisen kerroksellisuutta voidaan aspektin ilmaisemiseen osallistuvien lauseenjäsenten kerrostumisen ohella tarkastella lausekokonaisuuden eri semanttisten tasojen kerrostumisena. Lausetta laajemman kontekstin vaikutusta aspektitulkintaan ei ole aiemmin tutkittu suomen kielessä. Tutkimus osoittaa, että aspektiltaan monitulkintaisten lauseiden konteksti voi selventää tulkinnan tai mahdollistaa samanaikaisesti vaihtoehtoiset tulkinnat. Lisäksi erilaisten lauseenulkoisten rajan ilmausten avulla on mahdollista osoittaa lauseen aspektin rajattuutta siinä tapauksessa, että lause muutoin ymmärrettäisiin aspektiltaan rajaamattomaksi.

Spectral Analysis of Symmetric and Anti-Symmetric Pairwise Kernels

This work investigates theoretical properties of symmetric and anti-symmetric kernels. First chapters give an overview of the theory of kernels used in supervised machine learning. Central focus is on the regularized least squares algorithm, which is motivated as a problem of function reconstruction through an abstract inverse problem. Brief review of reproducing kernel Hilbert spaces shows how kernels define an implicit hypothesis space with multiple equivalent characterizations and how this space may be modified by incorporating prior knowledge. Mathematical results of the abstract inverse problem, in particular spectral properties, pseudoinverse and regularization are recollected and then specialized to kernels. Symmetric and anti-symmetric kernels are applied in relation learning problems which incorporate prior knowledge that the relation is symmetric or anti-symmetric, respectively. Theoretical properties of these kernels are proved in a draft this thesis is based on and comprehensively referenced here. These proofs show that these kernels can be guaranteed to learn only symmetric or anti-symmetric relations, and they can learn any relations relative to the original kernel modified to learn only symmetric or anti-symmetric parts. Further results prove spectral properties of these kernels, central result being a simple inequality for the the trace of the estimator, also called the effective dimension. This quantity is used in learning bounds to guarantee smaller variance.