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 wake behind a rectangular forebody with variable inlet conditions is investigated. The perforated surface of the two-dimensional rectangular forebody, with a smooth leading edge and a blunt trailing edge, allows for boundary layer modification by means of wall suction. The test section, of which the rectangular forebody is the main part, is experimentally evaluated with a series of hot-wire and Prandtl tube measurements in the boundary layer and the wake. For a suction coefficient of Γ&gt;9, corresponding to 0.9% suction of the free stream velocity, the asymptotic suction boundary layer (ASBL) is obtained at the trailing edge of the forebody for laminar boundary layers (Rex=1.6×105−3.8×105). The key feature of the ASBL, a spatially invariant boundary thickness which can be modified independent of the Reynolds number, is used to perform a unique parametrical study. Turbulent boundary layers (Rex=4.5×105−3.0×106) subject to wall suction are also investigated. For a critical suction coefficient Γcrit, which depends on Rex, the boundary layer relaminarizes. Strong evidence is found to support the hypothesis that turbulent boundary layers will ultimately attain the ASBL as well, provided that the wall suction is strong enough. The effect of the modulated laminar and turbulent boundary layers on the wake characteristics is studied. The shape of the mean wake velocity profile, scaled with the velocity deficit U0and the wake half width ∆y1/2, is found tobe independent of x/h, for x/h&gt; 6 and Reh &gt;6.7×103. The wake width is shown to scale with the effective thickness of the body h+2δ1, where the ratio is expected to vary with the downstream location. A decrease of the displacement thickness leads to a decrease of the base pressure, with Cp,b = −0.36 in the ASBL limit. The Strouhal number based on the effective thickness becomes Sth+2δ1 ≈ 0.29 in the ASBL limit and independent of the plate thickness (h) Reynolds number, in the range Reh = 2.9×103 − 6.7×103. For the turbulent boundary Sth+2δ1 is found to be 25% lower, which shows that the wake characteristics depend on the state of the boundary layer at the trailing edge. The total drag is found to be reduced by as much as 30% for Reh = 2.7×104 when a wall normal velocity of only 3.5% of the free stream velocity is applied. Wall suction successively reduces the total drag with increasing wall suction, at least in the Reynolds number rangeReh = 8.0×103−5.5×104. / <p>QC 20140312</p>

Although many estimates of species numbers have been attempted using various techniques, many smaller phyla remain poorly known without such estimates. For most of these it is unclear if they are species-poor or just poorly studied. The phylum Tardigrada is one of these phyla. Specialists have created a regularly updated checklist for the known tardigrade species, which as of 15 July 2013 listed 1190 taxa (species and subspecies). Of these, 1008 are limnoterrestrial and 182 are marine. These were the most up-to-date data at the time of our analysis. As species accumulation curves show little sign of levelling out, they do not provide a useful tool for estimating global tardigrade diversity from existing species numbers. A new technique has recently been developed that uses the more complete knowledge of higher taxonomic levels to estimate the asymptotic number of species. We applied this technique to limnoterrestrial and marine tardigrades. We estimate that the global total for limnoterrestrial tardigrades is 1145 (upper 95% CI = 2101), and the global total for marine tardigrades is 936 (upper 95% CI = 1803). This yields 87% completeness for our knowledge of limnoterrestrial tardigrades, and only 19% completeness for our knowledge of marine tardigrades. Thus, although many more marine species remain to be discovered, it appears that tardigrades are both poorly studied and relatively species poor.

A series of composite beam models are constructed for efficient high-fidelity beam analysis based on the variational-asymptotic method (VAM). Without invoking any a priori kinematic assumptions, the original three-dimensional, geometrically nonlinear beam problem is rigorously split into a two-dimensional cross-sectional analysis and a one-dimensional global beam analysis, taking advantage of the geometric small parameter that is an inherent property of the structure. The thermal problem of composite beams is studied first. According to the quasisteady theory of thermoelasticity, two beam models are proposed: one for heat conduction analysis and the other for thermoelastic analysis. For heat conduction analysis, two different types of thermal loads are modeled: with and without prescribed temperatures over the crosssections. Then a thermoelastic beam model is constructed under the previously solved thermal field. This model is also extended for composite materials, which removed the restriction on temperature variations and added the dependence of material properties with respect to temperature based on Kovalenoko’s small-strain thermoelasticity theory. Next the VAM is applied to model the multiphysics behavior of beam structure. A multiphysics beam model is proposed to capture the piezoelectric, piezomagnetic, pyroelectric, pyromagnetic, and hygrothermal effects. For the zeroth-order approximation, the classical models are in the form of Euler-Bernoulli beam theory. In the refined theory, generalized Timoshenko models have been developed, including two transverse shear strain measures. In order to avoid ill-conditioned matrices, a scaling method for multiphysics modeling is also presented. Three-dimensional field quantities are recovered from the one-dimensional variables obtained from the global beam analysis. A number of numerical examples of different beams are given to demonstrate the application and accuracy of the present theory. Excellent agreements between the results obtained by the current models and those obtained by three-dimensional finite element analysis, analytical solutions, and those available in the literature can be observed for all the cross-sectional variables. The present beam theory has been implemented into the computer program VABS (Variational Asymptotic Beam Sectional Analysis).

The issue of accurately determining the effective properties of composite materials has received the attention of numerous researchers in the last few decades and continues to be in the forefront of material research. Micromechanics models have been proven to be very useful tools for design and analysis of composite materials. In the present work, a versatile micromechanics modeling framework, namely, the Variational Asymptotic Method for Unit Cell Homogenization (VAMUCH), has been invented and various micromechancis models have been constructed in light of this novel framework. Considering the periodicity as a small parameter, we can formulate the variational statements of the unit cell through an asymptotic expansion of the energy functional. It is shown that the governing differential equations and periodic boundary conditions of mathematical homogenization theories (MHT) can be reproduced from this variational statement. Finally, we employed the finite element method to solve the numerical solution of the constrained minimization problem. If the local fields within the unit cell are of interest, the proposed models can also accurately recover those fields based on the global behavior. In comparison to other existing models, the advantages of VAMUCH are: (1) it invokes only two essential assumptions within the concept of micromechanics for heterogeneous material with identifiable unit cells; (2) it has an inherent variational nature and its numerical implementation is shown to be straightforward; (3) it calculates the different material properties in different directions simultaneously, which is more efficient than those approaches requiring multiple runs under different loading conditions; and (4) it calculates the effective properties and the local fields directly with the same accuracy as the fluctuation functions. No postprocessing calculations such as stress averaging and strain averaging are needed. The present theory is implemented in the computer program VAMUCH, a versatile engineering code for the homogenization of heterogeneous materials. This new micromechanics modeling approach has been successfully applied to predict the effective properties of composite materials including elastic properties, coefficients of thermal expansion, and specific heat and the effective properties of piezoelectric and electro-magneto-elastic composites. This approach has also been extended to the prediction of the nonlinear response of multiphase composites. Numerous examples have been utilized to clearly demonstrate its application and accuracy as a general-purpose micromechanical analysis tool.

To seek better material behaviors, the research of material properties has been mas- sively carried out in both industrial and academic fields throughout the twentieth century. Composite materials are known for their abilities of combining constituent materials in or- der to fulfill the desirable overall material performance. One of the advantages of composite materials is the adjustment between stiffness and lightness of materials in order to meet the needs of various engineering designs. Even though the finite element analysis is mature, composites are heterogeneous in nature and can present difficulties at the structural level with the acceptable computational time. A way of simplifying such problems is to find a way to connect structural analysis with corresponding analysis of representative microstructure of the material, which is normally called micromechanics modeling or homogenization.Generally speaking, the goal of homogenization is to predict a precise material behavior by taking into account the information stored in both microscopic and macroscopic levels of the composites. Of special concern to researchers and engineers is the thermomechanical behavior of composite materials since thermal effect is almost everywhere in real practical cases of engineering. In aerospace engineering, the thermomechanical behaviors of compos- ites are even more important since flight under high speed usually produces a large amount of heat which will cause very high thermal-related deformation and stress.In this dissertation, the thermomechanical behavior of composites will be studied based on the variational asymptotic method for unit cell homogenization (VAMUCH) which was recently developed as an efficient and accurate micromechanics modeling tool. The theories and equations within the code are based on the variational asymptotic method invented by Prof. Berdichevsky. For problems involving small parameters, the traditional asymptotic method is often applied by solving a system of differential equations while the variational asymptotic method is using a variational statement that only solves one functional of such problems where the traditional asymptotic method may apply.First, we relax the assumption made by traditional linear thermoelasticity that not only a small overall strain is assumed to be small but also the temperature variation. Of course, in this case we need to add temperature dependent material properties to VAMUCH so that the secant material properties can be calculated. Then, we consider the temperature field to be point-wise different within the microstructure; a micromechanics model with nonuniformly distributed temperature field will be addressed. Finally, the internal and external loads induced energies are considered in order to handle real engineering structures under their working conditions.

Indiana University-Purdue University Indianapolis (IUPUI) / We consider the behavior of zeros of random polynomials of the from \begin{equation*} P_{n,m}(z) := \eta_0\varphi_m^{(m)}(z) + \eta_1 \varphi_{m+1}^{(m)}(z) + \cdots + \eta_n \varphi_{n+m}^{(m)}(z) \end{equation*} as \( n\to\infty \), where \( m \) is a non-negative integer (most of the work deal with the case \( m =0 \) ), \( \{\eta_n\}_{n=0}^\infty \) is a sequence of i.i.d. Gaussian random variables, and \( \{\varphi_n(z)\}_{n=0}^\infty \) is a sequence of orthonormal polynomials on the unit circle \( \mathbb T \) for some Borel measure \( \mu \) on \( \mathbb T \) with infinitely many points in its support. Most of the work is done by manipulating the density function for the expected number of zeros of a random polynomial, which we call the intensity function.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第19341号 / 情博第593号 / 新制||情||103(附属図書館) / 32343 / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 中村 佳正, 教授 矢ｹ崎 一幸, 教授 西村 直志 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第20739号 / 情博第653号 / 新制||情||113(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 中村 佳正, 教授 山下 信雄, 教授 西村 直志 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20886号 / 理博第4338号 / 新制||理||1623(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 長谷川 真人, 教授 向井 茂, 准教授 照井 一成 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM