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.

Hyperbolic type metrics and distortion of quasiconformal map pings

This Ph.D. thesis consists of four original papers. The papers cover several topics from geometric function theory, more specifically, hyperbolic type metrics, conformal invariants, and the distortion properties of quasiconformal mappings. The first paper deals mostly with the quasihyperbolic metric. The main result gives the optimal bilipschitz constant with respect to the quasihyperbolic metric for the M¨obius self-mappings of the unit ball. A quasiinvariance property, sharp in a local sense, of the quasihyperbolic metric under quasiconformal mappings is also proved. The second paper studies some distortion estimates for the class of quasiconformal self-mappings fixing the boundary values of the unit ball or convex domains. The distortion is measured by the hyperbolic metric or hyperbolic type metrics. The results provide explicit, asymptotically sharp inequalities when the maximal dilatation of quasiconformal mappings tends to 1. These explicit estimates involve special functions which have a crucial role in this study. In the third paper, we investigate the notion of the quasihyperbolic volume and find the growth estimates for the quasihyperbolic volume of balls in a domain in terms of the radius. It turns out that in the case of domains with Ahlfors regular boundaries, the rate of growth depends not merely on the radius but also on the metric structure of the boundary. The topic of the fourth paper is complete elliptic integrals and inequalities. We derive some functional inequalities and elementary estimates for these special functions. As applications, some functional inequalities and the growth of the exterior modulus of a rectangle are studied.

The solutions and tunneling properties for a linear potential and an inverted harmonic oscillator in an infinite well

In this work we look at two different 1-dimensional quantum systems. The potentials for these systems are a linear potential in an infinite well and an inverted harmonic oscillator in an infinite well. We will solve the Schrödinger equation for both of these systems and get the energy eigenvalues and eigenfunctions. The solutions are obtained by using the boundary conditions and numerical methods. The motivation for our study comes from experimental background. For the linear potential we have two different boundary conditions. The first one is the so called normal boundary condition in which the wave function goes to zero on the edge of the well. The second condition is called derivative boundary condition in which the derivative of the wave function goes to zero on the edge of the well. The actual solutions are Airy functions. In the case of the inverted oscillator the solutions are parabolic cylinder functions and they are solved only using the normal boundary condition. Both of the potentials are compared with the particle in a box solutions. We will also present figures and tables from which we can see how the solutions look like. The similarities and differences with the particle in a box solution are also shown visually. The figures and calculations are done using mathematical software. We will also compare the linear potential to a case where the infinite wall is only on the left side. For this case we will also show graphical information of the different properties. With the inverted harmonic oscillator we will take a closer look at the quantum mechanical tunneling. We present some of the history of the quantum tunneling theory, its developers and finally we show the Feynman path integral theory. This theory enables us to get the instanton solutions. The instanton solutions are a way to look at the tunneling properties of the quantum system. The results are compared with the solutions of the double-well potential which is very similar to our case as a quantum system. The solutions are obtained using the same methods which makes the comparison relatively easy. All in all we consider and go through some of the stages of the quantum theory. We also look at the different ways to interpret the theory. We also present the special functions that are needed in our solutions, and look at the properties and different relations to other special functions. It is essential to notice that it is possible to use different mathematical formalisms to get the desired result. The quantum theory has been built for over one hundred years and it has different approaches. Different aspects make it possible to look at different things.

The solutions and tunneling properties for a linear potential and an inverted harmonic oscillator in an infinite well

In this work we look at two different 1-dimensional quantum systems. The potentials for these systems are a linear potential in an infinite well and an inverted harmonic oscillator in an infinite well. We will solve the Schrödinger equation for both of these systems and get the energy eigenvalues and eigenfunctions. The solutions are obtained by using the boundary conditions and numerical methods. The motivation for our study comes from experimental background. For the linear potential we have two different boundary conditions. The first one is the so called normal boundary condition in which the wave function goes to zero on the edge of the well. The second condition is called derivative boundary condition in which the derivative of the wave function goes to zero on the edge of the well. The actual solutions are Airy functions. In the case of the inverted oscillator the solutions are parabolic cylinder functions and they are solved only using the normal boundary condition. Both of the potentials are compared with the particle in a box solutions. We will also present figures and tables from which we can see how the solutions look like. The similarities and differences with the particle in a box solution are also shown visually. The figures and calculations are done using mathematical software. We will also compare the linear potential to a case where the infinite wall is only on the left side. For this case we will also show graphical information of the different properties. With the inverted harmonic oscillator we will take a closer look at the quantum mechanical tunneling. We present some of the history of the quantum tunneling theory, its developers and finally we show the Feynman path integral theory. This theory enables us to get the instanton solutions. The instanton solutions are a way to look at the tunneling properties of the quantum system. The results are compared with the solutions of the double-well potential which is very similar to our case as a quantum system. The solutions are obtained using the same methods which makes the comparison relatively easy. All in all we consider and go through some of the stages of the quantum theory. We also look at the different ways to interpret the theory. We also present the special functions that are needed in our solutions, and look at the properties and different relations to other special functions. It is essential to notice that it is possible to use different mathematical formalisms to get the desired result. The quantum theory has been built for over one hundred years and it has different approaches. Different aspects make it possible to look at different things.

Sulfaattimassan D/Q(EOP)D-valkaisimon optimointi

Työssä tutkittiin sulfaattisellutehtaan ECF-valkaisimon toimintaan vaikuttavia tekijöitä. Tavoitteena oli selvittää tekijöiden vaikutus massan laatuominaisuuksiin, valkaisimon kemikaalikustannuksiin ja klooridioksidin kulutukseen. Työn kirjallisuusosassa on tarkasteltu nykyaikaisen sulfaattisellutehtaan eri prosesseja. Erityistä huomiota on kiinnitetty sellun ECF-valkaisuun ja kyseisessä valkaisussa käytettäviin kemikaaleihin. Lisäksi on tarkasteltu TCF-valkaisua ja mahdollisuuksia sulkea sellutehtaan vesikiertoja. Kokeellisessa osassa selvitettiin Oy Metsä-Botnia Ab Joutsenon tehtaan kolmivaiheisen ECF-valkaisimon massan laatuominaisuuksiin, kustannuksiin ja klooridioksidin kulutukseen vaikuttavia tekijöitä. Tavoitteena oli vaikuttaa positiivisesti massan laatuominaisuuksiin valkaisun keinoin. Samalla pyrittiin minimoimaan valkaisusta aiheutuvia kemikaalikustannuksia ja vähentämään klooridioksidin kulutusta. Työssä käytettiin Taguchi-menetelmää. Tehdyn tutkimuksen myötä saatiin runsaasti tietoa mihin eri ominaisuuksiin tutkitut tekijät vaikuttivat. Metallien poistolla eli kelatoinnilla havaittiin olevan suuri merkitys hapettavan alkaliuuttovaiheen ja samalla koko valkaisimon toimintaan. Suurella kelatointiaineannoksella valkaisimon selektiivisyys ja tehokkuus paranivat. Muista tekijöistä happiannoksella havaittiin olevan vaikutusta vain massan paperiteknisiin ominaisuuksiin ja hiilihydraattien koostumukseen. Happiannoksen kasvattaminen paransi repäisyindeksiä ja vähensi massan jauhatustarvetta yli valkaisimon. Kyseisiin ominaisuuksiin vaikuttivat myös klooridioksidi ja sen toimintaolosuhteet. Kemikaalikustannuksiin vaikuttavista tekijöistä valkaisimon tulokapalla ja ensimmäisen klooridioksidivaiheen kemikaaliannoksella havaittiin olevan suuri merkitys. Samat tekijät vaikuttivat myös valkaisimon klooridioksidin kokonaiskulukseen.

Creating possibilities for collective creativity : Brokerage Functions in Practice-Based Innovation

In the network era, creative achievements like innovations are more and more often created in interaction among different actors. The complexity of today‘s problems transcends the individual human mind, requiring not only individual but also collective creativity. In collective creativity, it is impossible to trace the source of new ideas to an individual. Instead, creative activity emerges from the collaboration and contribution of many individuals, thereby blurring the contribution of specific individuals in creating ideas. Collective creativity is often associated with diversity of knowledge, skills, experiences and perspectives. Collaboration between diverse actors thus triggers creativity and gives possibilities for collective creativity. This dissertation investigates collective creativity in the context of practice-based innovation. Practice-based innovation processes are triggered by problem setting in a practical context and conducted in non-linear processes utilising scientific and practical knowledge production and creation in cross-disciplinary innovation networks. In these networks diversity or distances between innovation actors are essential. Innovation potential may be found in exploiting different kinds of distances. This dissertation presents different kinds of distances, such as cognitive, functional and organisational which could be considered as sources of creativity and thus innovation. However, formation and functioning of these kinds of innovation networks can be problematic. Distances between innovating actors may be so great that a special interpretation function is needed – that is, brokerage. This dissertation defines factors that enhance collective creativity in practice-based innovation and especially in the fuzzy front end phase of innovation processes. The first objective of this dissertation is to study individual and collective creativity at the employee level and identify those factors that support individual and collective creativity in the organisation. The second objective is to study how organisations use external knowledge to support collective creativity in their innovation processes in open multi-actor innovation. The third objective is to define how brokerage functions create possibilities for collective creativity especially in the context of practice-based innovation. The research objectives have been studied through five substudies using a case-study strategy. Each substudy highlights various aspects of creativity and collective creativity. The empirical data consist of materials from innovation projects arranged in the Lahti region, Finland, or materials from the development of innovation methods in the Lahti region. The Lahti region has been chosen as the research context because the innovation policy of the region emphasises especially the promotion of practice-based innovations. The results of this dissertation indicate that all possibilities of collective creativity are not utilised in internal operations of organisations. The dissertation introduces several factors that could support collective creativity in organisations. However, creativity as a social construct is understood and experienced differently in different organisations, and these differences should be taken into account when supporting creativity in organisations. The increasing complexity of most potential innovations requires collaborative creative efforts that often exceed the boundaries of the organisation and call for the involvement of external expertise. In practice-based innovation different distances are considered as sources of creativity. This dissertation gives practical implications on how it is possible to exploit different kinds of distances knowingly. It underlines especially the importance of brokerage functions in open, practice-based innovation in order to create possibilities for collective creativity. As a contribution of this dissertation, a model of brokerage functions in practice-based innovation is formulated. According to the model, the results and success of brokerage functions are based on the context of brokerage as well as the roles, tasks, skills and capabilities of brokers. The brokerage functions in practice-based innovation are also possible to divide into social and cognitive brokerage.

On the Properties and Cosmology of Q-balls

The properties and cosmological importance of a class of non-topological solitons, Q-balls, are studied. Aspects of Q-ball solutions and Q-ball cosmology discussed in the literature are reviewed. Q-balls are particularly considered in the Minimal Supersymmetric Standard Model with supersymmetry broken by a hidden sector mechanism mediated by either gravity or gauge interactions. Q-ball profiles, charge-energy relations and evaporation rates for realistic Q-ball profiles are calculated for general polynomial potentials and for the gravity mediated scenario. In all of the cases, the evaporation rates are found to increase with decreasing charge. Q-ball collisions are studied by numerical means in the two supersymmetry breaking scenarios. It is noted that the collision processes can be divided into three types: fusion, charge transfer and elastic scattering. Cross-sections are calculated for the different types of processes in the different scenarios. The formation of Q-balls from the fragmentation of the Aflieck-Dine -condensate is studied by numerical and analytical means. The charge distribution is found to depend strongly on the initial energy-charge ratio of the condensate. The final state is typically noted to consist of Q- and anti-Q-balls in a state of maximum entropy. By studying the relaxation of excited Q-balls the rate at which excess energy can be emitted is calculated in the gravity mediated scenario. The Q-ball is also found to withstand excess energy well without significant charge loss. The possible cosmological consequences of these Q-ball properties are discussed.

The discourse on political pluralism in early eighteenth-century England : a conceptual study with special reference to terminology of religious origin

Doctoral dissertation, University of Jyväskylä

Genetic parameters for growth traits in pigs estimated using third degree polynomial functions

Selostus: Sian kasvuominaisuuksien perinnölliset tunnusluvut arvioituna kolmannen asteen polynomifunktion avulla

Properties of slow-release phosphorus fertilizers with special references to their use on drained peatland forests : A review

Tiivistelmä: Hidasliukoisten fosforilannoitteiden ominaisuudet ja käyttökelpoisuus suometsien lannoituksessa. Kirjallisuuteen perustuva tarkastelu

Responses of yield and N use of spring sown crops to N fertilization, with special reference to the use of plant growth regulators

Selostus: Typpilannoituksen ja kasvunsääteiden vaikutukset kevätviljojen ja rypsin satoon sekä typen käyttöön