The spectral distribution function of random matrices is an information-carrying object widely studied within Random matrix theory. In this thesis we combine the results of the theory together with the idea of free independence introduced by Voiculescu (1985). Important theoretical part of the thesis consists of the introduction to Free probability theory, which justifies use of asymptotic freeness with respect to particular matrices as well as the use of Stieltjes and R-transform. Both transforms are presented together with their properties. The aim of thesis is to point out characterizations of those classes of the matrices, which have closed form expressions for the asymptotic spectral distribution function. We consider all matrices which can be decomposed to the sum of asymptotically free independent summands. In particular, explicit calculations are performed in order to illustrate the use of asymptotic free independence to obtain the asymptotic spectral distribution for a matrix Q and generalize Marcenko and Pastur (1967) theorem. The matrix Q is defined as <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?Q%20=%20%5Cfrac%7B1%7Dn%20X_1X%5E%5Cprime_1%20+%20%5Ccdot%5Ccdot%5Ccdot%20+%20%5Cfrac%7B1%7Dn%20X_kX%5E%5Cprime_k," />  where Xi is p × n matrix following a matrix normal distribution, Xi ~ Np,n(0, \sigma^2I, I). Finally, theorems pointing out classes of matrices Q which lead to closed formula for the asymptotic spectral distribution will be presented. Particularly, results for matrices with inverse Stieltjes transform, with respect to the composition, given by a ratio of polynomials of 1st and 2nd degree, are given.

The spectral distribution function of random matrices is an information-carrying object widely studied within Random matrix theory. In this thesis we combine the results of the theory together with the idea of free independence introduced by Voiculescu (1985). Important theoretical part of the thesis consists of the introduction to Free probability theory, which justifies use of asymptotic freeness with respect to particular matrices as well as the use of Stieltjes and R-transform. Both transforms are presented together with their properties. The aim of thesis is to point out characterizations of those classes of the matrices, which have closed form expressions for the asymptotic spectral distribution function. We consider all matrices which can be decomposed to the sum of asymptotically free independent summands. In particular, explicit calculations are performed in order to illustrate the use of asymptotic free independence to obtain the asymptotic spectral distribution for a matrix Q and generalize Marcenko and Pastur (1967) theorem. The matrix Q is defined as <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?Q%20=%20%5Cfrac%7B1%7Dn%20X_1X%5E%5Cprime_1%20+%20%5Ccdot%5Ccdot%5Ccdot%20+%20%5Cfrac%7B1%7Dn%20X_kX%5E%5Cprime_k," />  where Xi is p × n matrix following a matrix normal distribution, Xi ~ Np,n(0, \sigma^2I, I). Finally, theorems pointing out classes of matrices Q which lead to closed formula for the asymptotic spectral distribution will be presented. Particularly, results for matrices with inverse Stieltjes transform, with respect to the composition, given by a ratio of polynomials of 1st and 2nd degree, are given.

We introduce real second-order freeness in second-order noncommutative probability spaces. We demonstrate that under this definition, independent ensembles of the three real models of random matrices which we consider, namely real Ginibre matrices, Gaussian orthogonal matrices, and real Wishart matrices, are asymptotically second-order free. These ensembles do not satisfy the complex definition of second-order freeness satisfied by their complex analogues. This definition may be used to calculate the asymptotic fluctuations of products of matrices in terms of the fluctuations of each ensemble. We use a combinatorial approach to the matrix calculations similar to genus expansion, but in which nonorientable surfaces appear, demonstrating the commonality between the real ensembles and the distinction from their complex analogues, motivating this distinct definition. We generalize the description of graphs on surfaces in terms of the symmetric group to the nonorientable case. In the real case we find, in addition to the terms appearing in the complex case corresponding to annular spoke diagrams, an extra set of terms corresponding to annular spoke diagrams in which the two circles of the annulus are oppositely oriented, and in which the matrix transpose appears. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2011-09-09 11:07:37.414

This thesis deals with reachability and freeness problems for systems with linear dynamics, including hybrid systems and matrix semigroups. Hybrid systems are a type of dynamical system that exhibit both continuous and discrete dynamic behaviour. Thus they are particularly useful in modelling practical real world systems which can both flow (continuous behaviour) and jump (discrete behaviour). Decision questions for matrix semigroups have attracted a great deal of attention in both the Mathematics and Theoretical Computer Science communities. They can also be used to model applications with only discrete components. For a computational model, the reachability problem asks whether we can reach a target point starting from an initial point, which is a natural question both in theoretical study and for real-world applications. By studying this problem and its variations, we shall prove in a formal mathematical sense that many problems are intractable or even unsolvable. Thus we know when such a problem appears in other areas like Biology, Physics or Chemistry, either the problem itself needs to be simplified, or it should by studied by approximation. In this thesis we concentrate on a specific hybrid system model, called an HPCD, and its variations. The objective of studying this model is twofold: to obtain the most expressive system for which reachability is algorithmically solvable and to explore the simplest system for which it is impossible to solve. For the solvable sub-cases, we shall also study whether reachability is in some sense easy or hard by determining which complexity classes the problem belongs to, such as P, NP(-hard) and PSPACE(-hard). Some undecidable results for matrix semigroups are also shown, which both strengthen our knowledge of the structure of matrix semigroups, and lead to some undecidability results for other models.

Abstract The filtration of pulp suspension is one of the key papermaking processes. It is measured using many kinds of analyzers and modeled using both physical and empirical mathematical expressions. The main target of this thesis was to develop an empirical mathematical model to describe the filtration of a drainage analyzer. In this work the diameter of the screen (10 cm), as well as the amount (1000 cm3), consistency (0.3%) and temperature (20 ˚C) of the sample were (about) the same as those of the Canadian Standard Freeness analyzer. The analyzer was equipped with the measurements of filtrated volume, driving pressure and filtrate consistency. The pressures from the underpressure of 80 kPa to the overpressure of 80 kPa could be used. Pulp types of SGW, PGW, TMP and chemical pulp with freeness levels from 30 to 600 ml were analyzed. A conditioner of the sample volume, temperature and consistency measurements was constructed. The error of freeness value caused by the errors of the conditioner was evaluated to be less than 1%. The equations for the calculation of the possible initial values of the controlled quantities were derived. The optically measured consistencies of the filtrates were different for different pulp types and decreased close to zero before the filtrated volume of 150 cm3. The filtration seemed to change the portion of optically active fines. In the model of this work the filtration time (t) is proportional to an experimental power of the filtrated volume (V): Ve = kt. An auxiliary constant (Ve = kt + V0e) was used during the regression analyses to overcome the disturbances and non-validity of the model in the beginning of the filtration. The correlation coefficients of the fit were higher than 0.999. The exponent 2 suggested by Darcy’s law is a special case while the exponent has also been close to 1 or 3. The curves of specific filtration resistances, based on the model and pressure measurements, were shown to change during the course of drainage and as a function of pressure additionally dependent on the pulp type. The model was successfully used to calculate freeness values and to filter noise from the measurements. / Tiivistelmä Paperimassasulpun suotauttaminen on paperinvalmistuksen avainprosesseja. Sitä on mitattu monenlaisilla analysaattoreilla ja kuvattu sekä fysikaalisilla että kokeellisilla matemaattisilla malleilla. Tämän tutkimuksen päätavoite on kehittää suotautuvuusanalysaattorin kokeellista matemaattista mallia. Tutkimuksessa viiran halkaisija (10 cm) sekä näytteen määrä (1000 cm3), sakeus (0.3 %) ja lämpötila (20&#160;˚C) olivat suunnilleen samat kuin Canadian standard freeness –analysaattorissa. Järjestelmä mittasi suotautettua tilavuutta, suotauttavaa painetta ja suodoksen sakeutta. Suotautuspaineet olivat 80 kPa:n ali- ja ylipaineen väliltä. Testattavana oli hioketta, painehioketta ja termohierrettä sekä kemiallista massaa, joiden freeness oli 30 ml:sta 600 ml:aan. Analysaattoriin rakennettu vakiointijärjestelmä sääti näytteen sakeuden, tilavuuden ja lämpötilan niin tarkasti halutuiksi, että näiden vaihtelu ei olisi muuttanut freeness-arvoa edes prosenttia. Suureiden kehittymiselle johdettiin kaavat. Niiden avulla voidaan laskea ne näytteen arvoalueet, joilta halutut tavoitearvot voidaan saavuttaa. Optisesti mitattu suodoksen sakeus riippui massatyypistä ja hiipui lähes nollaksi ennen kuin 150 ml oli suotautettu. Suotautus muutti optisesti aktiivisen hienoaineen osuutta. Tämän työn suodoskäyrän mallissa aika (t) on verrannollinen suotautetun tilavuuden (V) kokeellisen potenssiin: Ve = kt. Mallinnuksen ajaksi lisätään apuparametri (Ve = kt+V0e), jotta suotautuksen alku ei huononna mallia. Sovituksen korrelaatiokerroin oli yli 0.999. Eksponentin arvo vaihteli vähän yli yhdestä melkein kolmeen, joten Darcyn lain mukainen eksponentin arvo 2 osoittautui erikoistapaukseksi. Mallin ja painemittauksen avulla lasketut ominaisresistanssit muuttuivat suotautuksen kuluessa ja riippuivat myös massatyypistä. Mallin avulla voitiin laskea näytteen freeness sekä suodattaa mittauskohinaa.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第24394号 / 理博第4893号 / 新制||理||1699(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 小林 佑輔, 教授 牧野 和久, 教授 長谷川 真人 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

In a multi-receiver encryption environment, a sender can randomly choose a set of authorized receivers while distributing messages to them efficiently and securely. Recently, more and more researchers concern the privacy of receivers. They mentioned that an authorized receiver does not want other entities, except the service provider, to be able to derive her/his identity in many applications such as pay-TV. However, most of these protocols either provide no formal security proofs or are inefficient owing to high computation cost. In this thesis, we construct two provably secure and efficient anonymous multi-receiver certificated-based encryption schemes, PMCE and SCMCE, which avoid the key escrow problem while preserving the implicit certification of identity-based setting. The proposed PMCE and SCMCE get rid of pairing computation to encrypt a message and only need one and two pairing computations to decrypt the ciphertext, respectively. Finally, we define the security models and offer formal proofs to all properties including receiver anonymity.

This thesis examines verifiability and receipt freeness in cryptographic voting protocols in detail and points out the contradiction between these requirements. Firstly, an extensive electronic voting requirement set is clearly defined, and then the voting dilemma is described. This is followed by a suggestion of an applicable solution to overcome the voting dilemma by introducing Predefined Fake Vote (PreFote) scheme. Based on a comprehensive literature review, a classification of the existing privacy preserving approaches and a taxonomy of the existing cryptographic voting protocols extending the previous studies are provided. Thereby, a complete and secure cryptographic voting protocol satisfying all electronic voting security requirements at the same time seems non-existent. Hence, an alternative privacy preserving approach is highly needed. Pseudo-Voter Identity (PVID) scheme, proposed in the present study, is a practical and low cost one. The PVID scheme is based on RSA blind signature, and it allows recasting without sacrificing uniqueness. Furthermore, this study proposes a dynamic ballot mechanism including an extension with PreFotes. This study, wherein the PVID scheme and extended dynamic ballots with PreFotes are employed, proposes a practical, complete and secure cryptographic voting protocol over a network for large scale elections, which fulfils all of the electronic voting security requirements: privacy, eligibility, uniqueness, fairness, uncoercibility, receipt-freeness, individual verifiability and accuracy. Lastly, a method to analyse voting systems based on security requirements is suggested, and a detailed analysis of the proposed protocol, which uses this method, concludes this study.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21328号 / 理博第4424号 / 新制||理||1635(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 並河 良典, 教授 上 正明, 教授 森脇 淳 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

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