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1
Ma, Shiqi. "Determination of random schrödinger operators." HKBU Institutional Repository, 2019. https://repository.hkbu.edu.hk/etd_oa/671.
Full textAbstract:
Inverse problems arise in many fields such as radar imaging, medical imaging and geophysics. It draws much attention in both mathematical communities and industrial members. Mathematically speaking, many inverse problems can be formulated by one or several physical equations and mathematical models. For example, the signal used in radar imaging is governed by Maxwell's equation, and most of geophysical studies can be formulated using elastic equation. Therefore, rigorous mathematical theories can be applied to study the inverse problems coming from this complex world. Random inverse problem is a fascinating area studying how to extract useful statistical information from unknown object coming from real world. In this thesis, we focus on the study of inverse problem related to random Schrödinger operators. We are particularly interested in the case where both the source and the potential of the Schrödinger system are random. In our first topic, we are concerned with the direct and inverse scattering problems associated with a time-harmonic random Schrödinger equation with unknown random source and unknown potential. The well-posedness of the direct scattering problem is first established. Three uniqueness results are then obtained for the corresponding inverse problems in determining the variance of the source, the potential and the expectation of the source, respectively, by the associated far-field measurements. First, a single realization of the passive scattering measurement can uniquely recover the variance of the source without the a priori knowledge of the other unknowns. Second, if active scattering measurement can be further obtained, a single realization can uniquely recover the potential function without knowing the source. Finally, both the potential and the first two statistic moments of the random source can be uniquely recovered with full measurement data. Our second topic also focuses on the case where only the source is random. But in the second topic, the random model is different from our first topic. The second random model has received intensive study in recent years due to the reason that this random model has more flexibility fitting with different regularities. The recovering framework is similar to our first topic, but we shall develop different asymptotic estimates of the higher order terms, which is more difficult than the first one. Lastly, based on the previous two results, we study the case where both the source and the potential are random and unknown. The ergodicity is used to establish the single realization recovery. The asymptotic estimates of higher order terms are based on pseudodifferential operators and microlocal analysis. Three major novelties of our works in this thesis are that, first, we studied the case where both the source and the potential are unknown; second, both passive and active scattering measurements are used for the recovery in different scenarios; finally, only a single realization of the random sample is required to establish the recovery of useful information.
2
Chapman, Jacob W. "Spectral properties of random block operators." Thesis, The University of Alabama at Birmingham, 2013. http://pqdtopen.proquest.com/#viewpdf?dispub=3561259.
Full textAbstract:
Ever since the introduction of the Anderson model in 1958, physicists and mathematicians alike have been interested in the effects of disorder on quantum mechanical systems. For example, it is known that transport is suppressed for an electron moving about in a random environment, which follows from localization results proven for the Anderson model.
Quantum spin systems provide a relatively simple starting point when one is interested in studying many-body systems. Here we investigate a random block operator arising from the anisotropic xy-spin chain model. Allowing for arbitrary nontrivial single-site distributions, we prove a zero-velocity Lieb-Robinson bound under the assumption of dynamical localization at all energies.
After a preliminary study of basic properties and location of the almost-sure spectrum of this random block operator, we apply a transfer matrix formalism and prove contractivity and irreducibility properties of the Furstenberg group and, in particular, positivity of Lyapunov exponents at all nonzero energies. Then in the general setting of random block Jacobi matrices, we establish a Thouless formula, and under contractivity and irreducibility assumptions, we conclude dynamical localization via multiscale analysis by proving a Wegner estimate and an initial length scale estimate. Finally we apply our general results to prove localization for the special case of the Ising model, and we discuss a critical energy that arises.
3
Schmidt, Daniel F. "Eigenvalue Statistics for Random Block Operators." Diss., Virginia Tech, 2015. http://hdl.handle.net/10919/51851.
Full textAbstract:
The Schrodinger Hamiltonian for a single electron in a crystalline solid with independent, identically distributed (i.i.d.) single-site potentials has been well studied. It has the form of a diagonal potential energy operator, which contains the random variables, plus a kinetic energy operator, which is deterministic. In the less-understood cases of multiple interacting charge carriers, or of correlated random variables, the Hamiltonian can take the form of a random block-diagonal operator, plus the usual kinetic energy term. Thus, it is of interest to understand the eigenvalue statistics for such operators.
In this work, we establish a criterion under which certain random block operators will be guaranteed to satisfy Wegner, Minami, and higher-order estimates. This criterion is phrased in terms of properties of individual blocks of the Hamiltonian. We will then verify the input conditions of this criterion for a certain quasiparticle model with i.i.d. single-site potentials. Next, we will present a progress report on a project to verify the same input conditions for a class of one-dimensional, single-particle alloy-type models. These two results should be sufficient to demonstrate the utility of the criterion as a method of proving Wegner and Minami estimates for random block operators.Ph. D.
4
McCafferty, Andrew James. "Operators and special functions in random matrix theory." Thesis, Lancaster University, 2008. http://eprints.lancs.ac.uk/13101/.
Full textAbstract:
The Fredholm determinants of integral operators with kernel of the form (A(x)B(y) − A(y)B(x))/(x−y) arise in probabilistic calculations in Random Matrix Theory. These were extensively studied by Tracy and Widom, so we refer to them as Tracy–Widom operators. We prove that the integral operator with Jacobi kernel converges in trace norm to the integral operator with Bessel kernel under a hard edge scaling, using limits derived from convergence of differential equation coefficients. The eigenvectors of an operator with kernel of Tracy–Widom type can sometimes be deduced via a commuting differential operator. We show that no such operator exists for TW integral operators acting on L2(R). There are analogous operators for discrete random matrix ensembles, and we give sufficient conditions for these to be expressed as the square of a Hankel operator: writing an operator in this way aids calculation of Fredholm determinants. We also give a new example of discrete TW operator which can be expressed as the sum of a Hankel square and a Toeplitz operator.
5
Salim, Adil. "Random monotone operators and application to stochastic optimization." Thesis, Université Paris-Saclay (ComUE), 2018. http://www.theses.fr/2018SACLT021/document.
Full textAbstract:
Cette thèse porte essentiellement sur l'étude d'algorithmes d'optimisation. Les problèmes de programmation intervenant en apprentissage automatique ou en traitement du signal sont dans beaucoup de cas composites, c'est-à-dire qu'ils sont contraints ou régularisés par des termes non lisses. Les méthodes proximales sont une classe d'algorithmes très efficaces pour résoudre de tels problèmes. Cependant, dans les applications modernes de sciences des données, les fonctions à minimiser se représentent souvent comme une espérance mathématique, difficile ou impossible à évaluer. C'est le cas dans les problèmes d'apprentissage en ligne, dans les problèmes mettant en jeu un grand nombre de données ou dans les problèmes de calcul distribué. Pour résoudre ceux-ci, nous étudions dans cette thèse des méthodes proximales stochastiques, qui adaptent les algorithmes proximaux aux cas de fonctions écrites comme une espérance. Les méthodes proximales stochastiques sont d'abord étudiées à pas constant, en utilisant des techniques d'approximation stochastique. Plus précisément, la méthode de l'Equation Differentielle Ordinaire est adaptée au cas d'inclusions differentielles. Afin d'établir le comportement asymptotique des algorithmes, la stabilité des suites d'itérés (vues comme des chaines de Markov) est étudiée. Ensuite, des généralisations de l'algorithme du gradient proximal stochastique à pas décroissant sont mises au point pour resoudre des problèmes composites. Toutes les grandeurs qui permettent de décrire les problèmes à résoudre s'écrivent comme une espérance. Cela inclut un algorithme primal dual pour des problèmes régularisés et linéairement contraints ainsi qu'un algorithme d'optimisation sur les grands graphesThis thesis mainly studies optimization algorithms. Programming problems arising in signal processing and machine learning are composite in many cases, i.e they exhibit constraints and non smooth regularization terms. Proximal methods are known to be efficient to solve such problems. However, in modern applications of data sciences, functions to be minimized are often represented as statistical expectations, whose evaluation is intractable. This cover the case of online learning, big data problems and distributed computation problems. To solve this problems, we study in this thesis proximal stochastic methods, that generalize proximal algorithms to the case of cost functions written as expectations. Stochastic proximal methods are first studied with a constant step size, using stochastic approximation techniques. More precisely, the Ordinary Differential Equation method is adapted to the case of differential inclusions. In order to study the asymptotic behavior of the algorithms, the stability of the sequences of iterates (seen as Markov chains) is studied. Then, generalizations of the stochastic proximal gradient algorithm with decreasing step sizes are designed to solve composite problems. Every quantities used to define the optimization problem are written as expectations. This include a primal dual algorithm to solve regularized and linearly constrained problems and an optimization over large graphs algorithm
6
Montgomery-Smith, Stephen John. "The cotype of operators from C(K)." Thesis, University of Cambridge, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.305515.
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7
Kitagaki, Yoshihiko. "Generalized eigenvalue-counting estimates for some random acoustic operators." Kyoto University, 2011. http://hdl.handle.net/2433/142295.
Full textAbstract:
Kyoto University (京都大学)0048
新制・課程博士
博士(人間・環境学)
甲第16167号
人博第550号
新制||人||133(附属図書館)
22||人博||550(吉田南総合図書館)
28746
京都大学大学院人間・環境学研究科共生人間学専攻
(主査)准教授 上木 直昌, 教授 森本 芳則, 教授 髙﨑 金久
学位規則第4条第1項該当
8
Rambane, Daniel Thanyani. "Operators defined by conditional expectations and random measures / Daniel Thanyani Rambane." Thesis, North-West University, 2004. http://hdl.handle.net/10394/282.
Full textAbstract:
This study revolves around operators defined by conditional expectations
and operators generated by random measures.
Studies of operators in function spaces defined by conditional expectations
first appeared in the mid 1950's by S-T.C. Moy [22] and S. Sidak [26].
N. Kalton studied them in the setting of Lp-spaces 0 < p < 1 in [15, 131 and
in L1-spaces, [14], while W. Arveson [5] studied them in L2-spaces. Their
averaging properties were studied by P.G. Dodds and C.B. Huijsmans and
B. de Pagter in [7] and C.B. Huijsmans and B. de Pagter in [lo]. A. Lambert
[17] studied their relationship with multiplication operators in C*-modules.
It was shown by J.J. Grobler and B. de Pagter [8] that partial integral operators
that were studied A.S. Kalitvin et a1 in [2, 4, 3, 11, 121 and the special
cases of kernel operators that were, inter alia, studied by A.R. Schep in [25]
were special cases of conditional expectation operators.
On the other hand, operators generated by random measures or pseudo-integral
operators were studied by A. Sourour [28, 271 and L.W. Weis [29,30],
building on the studies of W. Arveson [5] and N. Kalton [14, 151, in the late
1970's and early 1980's.
In this thesis we extend the work of J.J. Grobler and B. de Pagter [8] on
Multiplication Conditional Expectation-representable (MCE-representable)
operators. We also generalize the result of A. Sourour [27] and show that
order continuous linear maps between ideals of almost everywhere finite
measurable functions on u-finite measure spaces are MCE-representable.
This fact enables us to easily deduce that sums and compositions of MCE-representable
operators are again MCE-representable operators. We also
show that operators generated by random measures are MCE-representable.
The first chapter gathers the definitions and introduces notions and concepts
that are used throughout. In particular, we introduce Riesz spaces and
operators therein, Riesz and Boolean homomorphisms, conditional expectation
operators, kernel and absolute T-kernel operators.
In Chapter 2 we look at MCE-operators where we give a definition different
from that given by J.J. Grobler and B. de Pagter in [8], but which we
show to be equivalent.
Chapter 3 involves random measures and operators generated by random
measures. We solve the problem (positively) that was posed by A. Sourour
in [28] about the relationship of the lattice properties of operators generated
by random measures and the lattice properties of their generating random
measures. We show that the total variation of a random signed measure
representing an order bounded operator T, it being the difference of two
random measures, is again a random measure and represents ITI.
We also show that the set of all operators generated by a random measure
is a band in the Riesz space of all order bounded operators.
In Chapter 4 we investigate the relationship between operators generated
by random measures and MCE-representable operators. It was shown by
A. Sourour in [28, 271 that every order bounded order continuous linear
operator acting between ideals of almost everywhere measurable functions is
generated by a random measure, provided that the measure spaces involved
are standard measure spaces. We prove an analogue of this theorem for
the general case where the underlying measure spaces are a-finite. We also,
in this general setting, prove that every order continuous linear operator is
MCE-representable. This rather surprising result enables us to easily show
that sums, products and compositions of MCE-representable operator are
again MCE-representable.
Key words: Riesz spaces, conditional expectations, multiplication conditional
expectation-representable operators, random measures.Thesis (Ph.D. (Mathematics))--North-West University, Potchefstroom Campus, 2004.
9
Poulin, Philippe. "Random SchrÜdinger operators of Anderson type with generalized Laplacians and sparse potentials." Thesis, McGill University, 2006. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=102719.
Full textAbstract:
The first part of the thesis concerns Green's functions of discrete Laplacians on lattices. In the continuous case, it is well known that the corresponding Green's functions decay polynomially. However, an identical proof of this fact fails in the discrete case, since the constant energy surfaces of the discrete Laplacian are not convex. Two approaches are presented to turn around this problem. One consists of adapting the stationary phase method in order to treat non convex surfaces admitting kappa > 0 non vanishing principal curvatures at each point; as suggested by Littman. The other consists of changing the discretization of the Laplacian, as suggested by Molchanov and Vainberg.The second part of the thesis concerns random Schrodinger operators of type Anderson on the d-dimensional lattice. Sufficient conditions are presented for such operators, H = Delta + V, to satisfy almost surely the following, remarkable spectral and scattering properties: (1) Outside spec(Delta), the spectrum of H is pure point with exponentially decaying eigenfunctions (so-called Anderson localization). Examples where the spectrum of H is equal to the whole real line are also exhibited, in which case the eigenvalues of H are in addition dense in R \spec(Delta); (2) Inside spec(Delta), the spectrum of H is purely absolutely continuous (so-called delocalization); (3) Inside spec(Delta), the wave operators between H and Delta exist and are complete. Such Anderson operators are exhibited for the first time in the literature. Using the estimate of the first, part of the thesis, the mentioned sufficient conditions appear to be sparseness conditions on the support of the potential.
10
Hagger, Raffael [Verfasser], and Marko [Akademischer Betreuer] Lindner. "Fredholm Theory with Applications to Random Operators / Raffael Hagger. Betreuer: Marko Lindner." Hamburg : Universitätsbibliothek der Technischen Universität Hamburg-Harburg, 2016. http://d-nb.info/1081423633/34.
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11
Dietlein, Adrian [Verfasser], and Peter [Akademischer Betreuer] Müller. "Spectral properties of localized continuum random Schrödinger operators / Adrian Dietlein ; Betreuer: Peter Müller." München : Universitätsbibliothek der Ludwig-Maximilians-Universität, 2018. http://d-nb.info/1166559777/34.
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12
Trinh, Tuan Phong. "Random and periodic operators in dimension 1 : Decorrelation estimates in spectal statistics and resonances." Thesis, Sorbonne Paris Cité, 2015. http://www.theses.fr/2015USPCD005/document.
Full textAbstract:
Cette thèse comporte deux parties qui correspondent à deux domaines distincts : les opérateurs aléatoires et les opérateurs périodiques en dimension 1. Dans la première partie, nous prouvons une estimée de décorrélation pour un opérateur aléatoire avec désordre hors diagonal en dimension 1. En se servant de cette estimée, nous déduisons l'indépendance asymptotique des statistiques locales des valeurs propres près d'énergies distinctes positives dans le régime localisé. Finalement, nous donnons une démonstration alternative de l'estimée de décorrélation pour le modèle d'Anderson discret unidimensionnel. La deuxième partie de cette thèse est liée à un problème de résonances pour l'opérateur de Schrödinger discret en dimension 1 avec potentiel périodique tronqué [...]This thesis consists of two parts : te random and periodic operators in dimension 1. In this part, we prove the decorrelation estimate for a 1D lattice Hamiltonian with off-diagonal disorder. Consequently, we deduce the asymptotic independance of the local level statistics near distinct positive energies in the localized regime. Finally, we revisit a known result on the decorrelation estimate for the 1D discret Anderson model. The second part on my thesis adresses questions on resonances for a 1D Schrödinger operators with truncated periodic potential [...]
13
Veselić, Ivan. "Existence and regularity properties of the integrated density of states of random Schrödinger operators /." Berlin [u.a.] : Springer, 2008. http://dx.doi.org/10.1007/978-3-540-72691-3.
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14
Glaffig, Clemens H. Simon Barry. "Smoothness of the integrated density of states for random Schrodinger operators on multidimensional strips /." Diss., Pasadena, Calif. : California Institute of Technology, 1988. http://resolver.caltech.edu/CaltechETD:etd-09012005-155238.
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15
Takahara, Jyunichi. "WEGNER ESTIMATES FOR GENERALIZED ALLOY TYPE POTENTIALS." Kyoto University, 2013. http://hdl.handle.net/2433/180367.
Full textAbstract:
Kyoto University (京都大学)0048
新制・課程博士
博士(人間・環境学)
甲第17837号
人博第658号
新制||人||158(附属図書館)
25||人博||658(吉田南総合図書館)
30652
京都大学大学院人間・環境学研究科共生人間学専攻
(主査)教授 上木 直昌, 教授 森本 芳則, 教授 髙﨑 金久
学位規則第4条第1項該当
16
Chonchaiya, Ratchanikorn. "Computing the Spectra aand Pseudospectra of Non-Self Adjoint Random Operators Arising in Mathematical Physics." Thesis, University of Reading, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.533744.
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17
Schwarzenberger, Fabian. "The Integrated Density of States for Operators on Groups." Universitätsbibliothek Chemnitz, 2014. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-138523.
Full textAbstract:
This book is devoted to the study of operators on discrete structures. The operators are supposed to be self-adjoint and obey a certain translation invariance property. The discrete structures are given as Cayley graphs via finitely generated groups. Here, sofic groups and amenable groups are in the center of our considerations. Note that every finitely generated amenable group is sofic. We investigate the spectrum of a discrete self-adjoint operator by studying a sequence of finite dimensional analogues of these operators. In the setting of amenable groups we obtain these approximating operators by restricting the operator in question to finite subsets Qn , n ∈ N. These finite dimensional operators are self-adjoint and therefore admit a well-defined normalized eigenvalue counting function. The limit of the normalized eigenvalue counting functions when |Qn | → ∞ (if it exists) is called the integrated density of states (IDS). It is a distribution function of a probability measure encoding the distribution of the spectrum of the operator in question on the real axis.
We prove the existence of the IDS in various geometric settings and for different types of operators. The models we consider include deterministic as well as random situations. Depending on the specific setting, we prove existence of the IDS as a weak limit of distribution functions or even as a uniform limit. Moreover, in certain situations we are able to express the IDS via a semi-explicit formula using the trace of the spectral projection of the original operator. This is sometimes referred to as the validity of the Pastur-Shubin trace formula.
In the most general geometric setting we study, the operators are defined on Cayley graphs of sofic groups. Here we prove weak convergence of the eigenvalue counting functions and verify the validity of the Pastur-Shubin trace formula for random and non-random operators . These results apply to operators which not necessarily bounded or of finite hopping range. The methods are based on resolvent techniques. This theory is established without having an ergodic theorem for sofic groups at hand. Note that ergodic theory is the usual tool used in the proof of convergence results of this type.
Specifying to operators on amenable groups we are able to prove stronger results. In the discrete case, we show that the IDS exists uniformly for a certain class of finite hopping range operators. This is obtained by using a Banach space-valued ergodic theorem. We show that this applies to eigenvalue counting functions, which implies their convergence with respect to the Banach space norm, in this case the supremum norm. Thus, the heart of this theory is the verification of the Banach space-valued ergodic theorem. Proceeding in two steps we first prove this result for so-called ST-amenable groups. Then, using results from the theory of ε-quasi tilings, we prove a version of the Banach space-valued ergodic theorem which is valid for all amenable groups.
Focusing on random operators on amenable groups, we prove uniform existence of the IDS without the assumption that the operator needs to be of finite hopping range or bounded. Moreover, we verify the Pastur-Shubin trace formula. Here we present different techniques. First we show uniform convergence of the normalized eigenvalue counting functions adapting the technique of the Banach space-valued ergodic theorem from the deterministic setting. In a second approach we use weak convergence of the eigenvalue counting functions and additionally obtain control over the convergence at the jumps of the IDS. These ingredients are applied to verify uniform existence of the IDS. In both situations we employ results from the theory of large deviations, in order to deal with long-range interactions.
18
Taraldsen, Gunnar. "Spectral theory of random operators : The energy spectrum of the quantum electron in a disordered solid." Doctoral thesis, Norwegian University of Science and Technology, Faculty of Information Technology, Mathematics and Electrical Engineering, 1992. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-670.
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19
Drabkin, Maxim [Verfasser], and Hermann [Akademischer Betreuer] Schulz-Baldes. "Analysis of certain random operators related to solid state physics / Maxim Drabkin. Gutachter: Hermann Schulz-Baldes." Erlangen : Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), 2015. http://d-nb.info/1076165257/34.
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20
Myers, Steven A. "On the development of block-ciphers and pseudo-random function generators using the composition and XOR operators." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape7/PQDD_0007/MQ45953.pdf.
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21
Soosten, Per von [Verfasser], Simone [Akademischer Betreuer] Warzel, László [Gutachter] Erdös, Herbert [Gutachter] Spohn, and Simone [Gutachter] Warzel. "Hierarchical Random Matrices and Operators / Per von Soosten ; Gutachter: László Erdös, Herbert Spohn, Simone Warzel ; Betreuer: Simone Warzel." München : Universitätsbibliothek der TU München, 2018. http://d-nb.info/1161846832/34.
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22
Parapayalage, Chandana Dinesh Kumara. "BUILDING EXTRACTION IN HAZARDOUS AREAS USING EXTENDED MORPHOLOGICAL OPERATORS WITH HIGH RESOLUTION OPTICAL IMAGERY." 京都大学 (Kyoto University), 2014. http://hdl.handle.net/2433/193579.
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23
Pettersson, Per, Alireza Doostan, and Jan Nordström. "On Stability and Monotonicity Requirements of Finite Difference Approximations of Stochastic Conservation Laws with Random Viscosity." Linköpings universitet, Beräkningsmatematik, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-90995.
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The stochastic Galerkin and collocation methods are used to solve an advection-diusion equation with uncertain and spatially varying viscosity. We investigate well-posedness, monotonicity and stability for the extended system resulting from the Galerkin projection of the advection-diusion equation onto the stochastic basis functions. High-order summationby- parts operators and weak imposition of boundary conditions are used to prove stability of the semi-discrete system. It is essential that the eigenvalues of the resulting viscosity matrix of the stochastic Galerkin system are positive and we investigate conditions for this to hold. When the viscosity matrix is diagonalizable, stochastic Galerkin and stochastic collocation are similar in terms of computational cost, and for some cases the accuracy is higher for stochastic Galerkin provided that monotonicity requirements are met. We also investigate the total spatial operator of the semi-discretized system and its impact on the convergence to steadystate
24
Schwarzenberger, Fabian. "The Integrated Density of States for Operators on Groups." Doctoral thesis, Universitätsbibliothek Chemnitz, 2013. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-123241.
Full textAbstract:
This thesis is devoted to the study of operators on discrete structures. The operators are supposed to be self-adjoint and obey a certain translation invariance property. The discrete structures are given as Cayley graphs via finitely generated groups. Here, sofic groups and amenable groups are in the center of our considerations. Note that every finitely generated amenable group is sofic. We investigate the spectrum of a discrete self-adjoint operator by studying a sequence of finite dimensional analogues of these operators. In the setting of amenable groups we obtain these approximating operators by restricting the operator in question to finite subsets Qn , n ∈ N. These finite dimensional operators are self-adjoint and therefore admit a well-defined normalized eigenvalue counting function. The limit of the normalized eigenvalue counting functions when |Qn | → ∞ (if it exists) is called the integrated density of states (IDS). It is a distribution function of a probability measure encoding the distribution of the spectrum of the operator in question on the real axis.
In this thesis, we prove the existence of the IDS in various geometric settings and for different types of operators. The models we consider include deterministic as well as random situations. Depending on the specific setting, we prove existence of the IDS as a weak limit of distribution functions or even as a uniform limit. Moreover, in certain situations we are able to express the IDS via a semi-explicit formula using the trace of the spectral projection of the original operator. This is sometimes referred to as the validity of the Pastur-Shubin trace formula.
In the most general geometric setting we study, the operators are defined on Cayley graphs of sofic groups. Here we prove weak convergence of the eigenvalue counting functions and verify the validity of the Pastur-Shubin trace formula for random and non-random operators . These results apply to operators which not necessarily bounded or of finite hopping range. The methods are based on resolvent techniques.
This theory is established without having an ergodic theorem for sofic groups at hand. Note that ergodic theory is the usual tool used in the proof of convergence results of this type.
Specifying to operators on amenable groups we are able to prove stronger results. In the discrete case, we show that the IDS exists uniformly for a certain class of finite hopping range operators. This is obtained by using a Banach space-valued ergodic theorem. We show that this applies to eigenvalue counting functions, which implies their convergence with respect to the Banach space norm, in this case the supremum norm. Thus, the heart of this theory is the verification of the Banach space-valued ergodic theorem. Proceeding in two steps we first prove this result for so-called ST-amenable groups. Then, using results from the theory of ε-quasi tilings, we prove a version of the Banach space-valued ergodic theorem which is valid for all amenable groups.
Focusing on random operators on amenable groups, we prove uniform existence of the IDS without the assumption that the operator needs to be of finite hopping range or bounded. Moreover, we verify the Pastur-Shubin trace formula. Here we present different techniques. First we show uniform convergence of the normalized eigenvalue counting functions adapting the technique of the Banach space-valued ergodic theorem from the deterministic setting.
In a second approach we use weak convergence of the eigenvalue counting functions and additionally obtain control over the convergence at the jumps of the IDS. These ingredients are applied to verify uniform existence of the IDS.
In both situations we employ results from the theory of large deviations, in order to deal with long-range interactions.
25
Kim, Jinho D. "Centralized random backoff for collision free wireless local area networks." Thesis, University of Edinburgh, 2018. http://hdl.handle.net/1842/31055.
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Over the past few decades, wireless local area networks (WLANs) have been widely deployed for data communication in indoor environments such as offices, houses, and airports. In order to fairly and efficiently use the unlicensed frequency band that Wi-Fi devices share, the devices follow a set of channel access rules, which is called a wireless medium access control (MAC) protocol. It is known that wireless devices following the 802.11 standard MAC protocol, i.e. the distributed coordination function (DCF), suffer from packet collisions when multiple nodes simultaneously transmit. This significantly degrades the throughput performance. Recently, several studies have reported access techniques to reduce the number of packet collisions and to achieve a collision free WLAN. Although these studies have shown that the number of collisions can be reduced to zero in a simple way, there have been a couple of remaining issues to solve, such as dynamic parameter adjustment and fairness to legacy DCF nodes in terms of channel access opportunity. Recently, In-Band Full Duplex (IBFD) communication has received much attention, because it has significant potential to improve the communication capacity of a radio band. IBFD means that a node can simultaneously transmit one signal and receive another signal in the same band at the same time. In order to maximize the performance of IBFD communication capability and to fairly share access to the wireless medium among distributed devices in WLANs, a number of IBFD MAC protocols have been proposed. However, little attention has been paid to fairness issues between half duplex nodes (i.e. nodes that can either transmit or receive but not both simultaneously in one time-frequency resource block) and IBFD capable nodes in the presence of the hidden node problem.
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Gioev, Dimitri. "Generalizations of Szego Limit Theorem : Higher Order Terms and Discontinuous Symbols." Doctoral thesis, KTH, Mathematics, 2001. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3123.
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Baker, Steven Jeffrey. "Spectral properties of displacement models." Birmingham, Ala. : University of Alabama at Birmingham, 2007. https://www.mhsl.uab.edu/dt/2007p/baker.pdf.
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Thesis (Ph. D.)--University of Alabama at Birmingham, 2007.Additional advisors: Richard Brown, Ioulia Karpechina, Ryoichi Kawai, Boris Kunin. Description based on contents viewed Feb. 5, 2008; title from title screen. Includes bibliographical references (p. 73-75).
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Giunti, Arianna. "Green\'s function estimates for elliptic and parabolic operators: Applications to quantitative stochastic homogenization and invariance principles for degenerate random environments and interacting particle systems." Doctoral thesis, Universitätsbibliothek Leipzig, 2017. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-225533.
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This thesis is divided into two parts: In the first one (Chapters 1 and 2), we deal with problems arising from quantitative homogenization of the random elliptic operator in divergence form $-\\nabla \\cdot a \\nabla$. In Chapter 1 we study existence and stochastic bounds for the Green function $G$ associated to $-\\nabla \\cdot a \\nabla$ in the case of systems. Without assuming any regularity on the coefficient field $a= a(x)$, we prove that for every (measurable) uniformly elliptic tensor field $a$ and for almost every point $y \\in \\mathbb^d$, there exists a unique Green\'s function centred in $y$ associated to the vectorial operator $-\\nabla \\cdot a\\nabla $ in $\\mathbb{R}^d$, $d> 2$. In addition, we prove that if we introduce a shift-invariant ensemble $\\langle\\cdot \\rangle$ over the set of uniformly elliptic tensor fields, then $\\nabla G$ and its mixed derivatives $\\nabla \\nabla G$ satisfy optimal pointwise $L^1$-bounds in probability.
Chapter 2 deals with the homogenization of $-\\nabla \\cdot a \\nabla$ to $-\\nabla \\ah \\nabla$ in the sense that we study the large-scale behaviour of $a$-harmonic functions in exterior domains $\\{ |x| > r \\}$ by comparing them with functions which are $\\ah$-harmonic. More precisely, we make use of the first and second-order correctors to compare an $a$-harmonic function $u$ to the two-scale expansion of suitable $\\ah$-harmonic function $u_h$. We show that there is a direct correspondence between the rate of the sublinear growth of the correctors and the smallness of the relative homogenization error $u- u_h$.
The theory of stochastic homogenization of elliptic operators admits an equivalent probabilistic counterpart, which follows from the link between parabolic equations with elliptic operators in divergence form and random walks. This allows to reformulate the problem of homogenization in terms of invariance principle for random walks. The second part of thesis (Chapters 3 and 4) focusses on this interplay between probabilistic and analytic approaches and aims at exploiting it to study invariance principles in the case of degenerate random conductance models and systems of interacting particles.
In Chapter 3 we study a random conductance model where we assume that the conductances are independent, stationary and bounded from above but not uniformly away from $0$. We give a simple necessary and sufficient condition for the relaxation of the environment seen by the particle to be diffusive in the sense of every polynomial moment.
As a consequence, we derive polynomial moment estimates on the corrector which imply that the discrete elliptic operator homogenises or, equivalently, that the random conductance model satisfies a quenched invariance principle.
In Chapter 4 we turn to a more complicated model, namely the symmetric exclusion process. We show a diffusive upper bound on the transition probability of a tagged particle in this process. The proof relies on optimal spectral gap estimates for the dynamics in finite volume, which are of independent interest. We also show off-diagonal estimates of Carne-Varopoulos type.
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Vogel, Martin. "Propriétés spectrales des opérateurs non-auto-adjoints aléatoires." Thesis, Dijon, 2015. http://www.theses.fr/2015DIJOS018/document.
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Dans cette thèse, nous nous intéressons aux propriétés spectrales des opérateurs non-auto-adjoints aléatoires. Nous allons considérer principalement les cas des petites perturbations aléatoires de deux types des opérateurs non-auto-adjoints suivants :1. une classe d’opérateurs non-auto-adjoints h-différentiels Ph, introduite par M. Hager [32],dans la limite semiclassique (h→0); 2. des grandes matrices de Jordan quand la dimension devient grande (N→∞). Dans le premier cas nous considérons l’opérateur Ph soumis à de petites perturbations aléatoires. De plus, nous imposons que la constante de couplage δ vérifie e (-1/Ch) ≤ δ ⩽ h(k), pour certaines constantes C, k > 0 choisies assez grandes. Soit ∑ l’adhérence de l’image du symbole principal de Ph. De précédents résultats par M. Hager [32], W. Bordeaux-Montrieux [4] et J. Sjöstrand [67] montrent que, pour le même opérateur, si l’on choisit δ ⪢ e(-1/Ch), alors la distribution des valeurs propres est donnée par une loi de Weyl jusqu’à une distance ⪢ (-h ln δ h) 2/3 du bord de ∑. Nous étudions la mesure d’intensité à un et à deux points de la mesure de comptage aléatoire des valeurs propres de l’opérateur perturbé. En outre, nous démontrons des formules h-asymptotiques pour les densités par rapport à la mesure de Lebesgue de ces mesures qui décrivent le comportement d’un seul et de deux points du spectre dans ∑. En étudiant la densité de la mesure d’intensité à un point, nous prouvons qu’il y a une loi de Weyl à l’intérieur du pseudospectre,une zone d’accumulation des valeurs propres dûe à un effet tunnel près du bord du pseudospectre suivi par une zone où la densité décroît rapidement. En étudiant la densité de la mesure d’intensité à deux points, nous prouvons que deux valeurs propres sont répulsives à distance courte et indépendantes à grande distance à l’intérieur de ∑. Dans le deuxième cas, nous considérons des grands blocs de Jordan soumis à des petites perturbations aléatoires gaussiennes. Un résultat de E.B. Davies et M. Hager [16] montre que lorsque la dimension de la matrice devient grande, alors avec probabilité proche de 1, la plupart des valeurs propres sont proches d’un cercle. De plus, ils donnent une majoration logarithmique du nombre de valeurs propres à l’intérieur de ce cercle. Nous étudions la répartition moyenne des valeurs propres à l’intérieur de ce cercle et nous en donnons une description asymptotique précise. En outre, nous démontrons que le terme principal de la densité est donné par la densité par rapport à la mesure de Lebesgue de la forme volume induite par la métrique de Poincaré sur la disque D(0, 1)In this thesis we are interested in the spectral properties of random non-self-adjoint operators. Weare going to consider primarily the case of small random perturbations of the following two types of operators: 1. a class of non-self-adjoint h-differential operators Ph, introduced by M. Hager [32], in the semiclassical limit (h→0); 2. large Jordan block matrices as the dimension of the matrix gets large (N→∞). In case 1 we are going to consider the operator Ph subject to small Gaussian random perturbations. We let the perturbation coupling constant δ be e (-1/Ch) ≤ δ ⩽ h(k), for constants C, k > 0 suitably large. Let ∑ be the closure of the range of the principal symbol. Previous results on the same model by M. Hager [32], W. Bordeaux-Montrieux [4] and J. Sjöstrand [67] show that if δ ⪢ e(-1/Ch) there is, with a probability close to 1, a Weyl law for the eigenvalues in the interior of the pseudospectrumup to a distance ⪢ (-h ln δ h) 2/3 to the boundary of ∑. We will study the one- and two-point intensity measure of the random point process of eigenvalues of the randomly perturbed operator and prove h-asymptotic formulae for the respective Lebesgue densities describing the one- and two-point behavior of the eigenvalues in ∑. Using the density of the one-point intensity measure, we will give a complete description of the average eigenvalue density in ∑ describing as well the behavior of the eigenvalues at the pseudospectral boundary. We will show that there are three distinct regions of different spectral behavior in ∑. The interior of the of the pseudospectrum is solely governed by a Weyl law, close to its boundary there is a strong spectral accumulation given by a tunneling effect followed by a region where the density decays rapidly. Using the h-asymptotic formula for density of the two-point intensity measure we will show that two eigenvalues of randomly perturbed operator in the interior of ∑ exhibit close range repulsion and long range decoupling. In case 2 we will consider large Jordan block matrices subject to small Gaussian random perturbations. A result by E.B. Davies and M. Hager [16] shows that as the dimension of the matrix gets large, with probability close to 1, most of the eigenvalues are close to a circle. They, however, only state a logarithmic upper bound on the number of eigenvalues in the interior of that circle. We study the expected eigenvalue density of the perturbed Jordan block in the interior of thatcircle and give a precise asymptotic description. Furthermore, we show that the leading contribution of the density is given by the Lebesgue density of the volume form induced by the Poincarémetric on the disc D(0, 1)
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Zalczer, Sylvain. "Propriétés spectrales de modèles de graphène périodique et désordonné." Thesis, Toulon, 2020. http://www.theses.fr/2020TOUL0003.
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Cette thèse traite de différents aspects de la théorie spectrale d’opérateurs utilisés pour modéliser le graphène. Elle est constituée de deux parties. La première traite du cas périodique. Je commence par présenter la théorie générale des systèmes périodiques. J’introduis ensuite les différents modèles de graphène en les comparant. Enfin, je m’intéresse à différentes façons de rendre le graphène semi-conducteur. Je fais en particulier une étude de nanorubans de divers types et présente un résultat d’ouverture d’une lacune spectrale pour un opérateur pseudo-différentiel. La deuxième partie traite du cas désordonné. Je commence par présenter la théorie générale des opérateurs aléatoires. J’explique ensuite succinctement l’analyse multi-échelles qui est la méthode permettant de montrer le résultat essentiel de cette théorie, appelé localisation d’Anderson. Enfin, je donne la preuve de cette localisation pour un modèle de graphène ainsi qu’un résultat sur la densité d’états intégréeThis thesis deals with various aspects of spectral theory of operators used to model graphene. It is made of two parts.The first parts deals with the periodic case. I begin by presenting a general theory of periodic systems. I introduce then different models of graphene and compare them. Finally, I look at various ways to make graphene a semiconductor. In particular, I study different types of nanoribbons and I give a result of gap opening for a pseudodifferential operator. The second part deals with the disordered case. I begin by presenting a general theory of random operators. Then, I briefly explain multiscale analysis, which is the method used to prove the main result of this theory, which is called Anderson localization. Finally, I give a proof of this localization for a model of graphene and a result on the integrated density of states
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Shirley, Christopher. "Statistiques spectrales d'opérateurs de Schrödinger aléatoires unidimensionnels." Thesis, Paris 6, 2014. http://www.theses.fr/2014PA066434/document.
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Dans cette thèse, nous allons prouver des estimations de décorrelation des valeurs propres pour plusieurs modèles d'opérateurs de Schrödinger aléatoires en dimension un, dans le régime localisé, tant que nous avons des estimations de Wegner. Ceci permet l'étude des statistiques spectrales.Nous commencerons donc par présenter les hypothèses sur lesquelles nous nous appuyons et les différents modèles considérés.Nous étudierons ensuite les estimations de Minami, qui peuvent être vues comme des estimations de décorrélation des valeurs propres proches. Nous montrerons qu'en dimension un, elles sont conséquences des estimations de Wegner et de l'hypothèse de localisation. Les estimations prouvées ici ont un domaine de validité plus restreint que les estimations de Minami classiques, mais sont suffisantes pour notre étude.Nous étudierons ensuite les estimations de décorrélation des valeurs propres éloignées pour les différents modèles présentés. Nous montrerons qu'elles sont conséquences des estimations de Minami, des estimations de Wegner et de l'hypothèse de localisation. Les preuves données seront différentes selon les modèles étudiés.Enfin, nous montrerons que ces résultats permettent d'étudier les statistiques spectrales, dans le régime localisé. Par exemple, les estimations de décorrélation permettent de montrer que les statistiques locales des niveaux d'énergies, prises à deux énergies différentes, convergent faiblement vers deux processus de Poisson indépendants sur $\R$ d'intensité la mesure de LebesgueIn this thesis, we will prove decorrelation estimates of eigenvalues for several models of random Schrödinger operators in dimension one, in the localized regime, provided we have Wegner estimates. This will allow us to study spectral statistics.We will begin with the presentation of the hypotheses needed in our proofs and the models under consideration.We will continue with the study of the Minami estimates, which can be seen as decorrelation estimates of close eigenvalues. We will show that, in dimension one and in the localized regime, they are the consequences of the Wegner estimates. The results proven here have a area of validity smaller than the usual Minami estimates, but it will suffice for our study.Next, we will study the decorrelation estimates of distant eigenvalues for the models under consideration. We will show that they are consequences of the Minami estimates and the Wegner estimates, in the localized regime. The proofs will be different from one model to another.Eventually, we will show that these results allow us to study spectral statistics in the localized regime. For instance, the decorrelation estimates will be used to prove that the local energy level statistics, taken at two distincts energy levels, converge weakly to two independent Poisson processes on $\R$ with intensity the Lebesgue measure
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Nakano, Yushi. "Stochastic Stability of Partially Expanding Maps via Spectral Approaches." Kyoto University, 2015. http://hdl.handle.net/2433/200463.
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Kyoto University (京都大学)0048
新制・課程博士
博士(人間・環境学)
甲第19200号
人博第741号
新制||人||178(附属図書館)
27||人博||741(吉田南総合図書館)
32192
京都大学大学院人間・環境学研究科共生人間学専攻
(主査)教授 宇敷 重廣, 教授 森本 芳則, 准教授 木坂 正史
学位規則第4条第1項該当
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Giunti, Arianna [Verfasser], Felix [Gutachter] Otto, and Antoine [Gutachter] Gloria. "Green\'s function estimates for elliptic and parabolic operators: Applications to quantitative stochastic homogenization and invariance principles for degenerate random environments and interacting particle systems : Green\''s function estimates for elliptic and parabolic operators:Applications to quantitative stochastic homogenization andinvariance principles for degenerate random environments andinteracting particle systems / Arianna Giunti ; Gutachter: Felix Otto, Antoine Gloria." Leipzig : Universitätsbibliothek Leipzig, 2017. http://d-nb.info/1241064598/34.
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Rojas, Molina Constanza. "Etude mathématique des propriétés de transport des opérateurs de Schrödigner aléatoires avec structure quasi-cristalline." Thesis, Cergy-Pontoise, 2012. http://www.theses.fr/2012CERG0565/document.
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Cette thèse est consacrée à l'étude du transport électronique dans des modèles désordonnés non ergodiques, dans le cadre de la théorie des opérateurs de Schrödinger aléatoires.Pour commencer, nous reformulons l'outil principal pour notre étude, l'analyse multi-échelles, dans le cadre non ergodique. Nous établissons les conditions d'homogénéité que l'opérateur doit vérifier pour appliquer cette méthode. Ensuite, nous étudions les propriétés spectrales des opérateurs de Delone-Anderson non ergodiques. Ces systèmes modélisent l'énergie d'une particule en interaction avec un milieu dont la structure atomique est quasi-cristalline et la nature des impuretés est désordonnée. Dans le cas où les mesures de probabilité associées au potentiel de simple site sont régulières, en dimension 2 et sous l'effet d'un champ magnétique, nous établissons une transition métal-isolant et l'existence d'une énergie de mobilité qui sépare les régions de localisation et de délocalisation dynamiques. Pour des mesures de simple site régulières et celle de Bernoulli, nous démontrons la localisation dynamique en bas du spectre. De plus, nous obtenons une description quantitative de la région de localisation dynamique en termes de paramètres géométriques de l'ensemble de Delone de base.Nous concluons ce travail avec l'étude de la densité d'états intégrée pour des modèles de Delone-Anderson, en combinaison avec des outils de la théorie des systèmes dynamiques associés aux quasi-cristaux. Sous certaines conditions sur la géométrie de l'ensemble de Delone sous-jacent, nous montrons l'existence de la densité d'états intégrée. De plus, dans le cas d'une perturbation de Delone-Anderson du Laplacien libre, nous démontrons qu'elle a un comportement asymptotique de Lifshitz en bas du spectreHis thesis is devoted to the study of electronic transport in non ergodic disordered models, in the framework of random Schrödinger operators.We start by reformulating the main tool in our study, the multiscale analysis, in the non ergodic setting. We establish suitable homogeneity conditions on the operator, in order to apply this method.Next, we study the spectral properties of non ergodic Delone-Anderson operators. These models represent a particle interacting with a medium whose atomic structure is quasi-crystalline and the nature of its impurities is disordered. In the case where the probability measures associated to the single-site potential are regular, in dimension 2 and under the effect of a magnetic field, we establish a metal-insulator transition and the existence of a mobility edge that separates the localization and delocalization regions. In arbitrary dimension, for regular and for Bernoulli single-site measures, we show dynamical localization at the bottom of the spectrum. Moreover, we obtain a quantitative lower bound on the size of the localization region in terms of the geometric parameters of the underlying Delone structure.We conclude this essay by studying the integrated density of states for Delone-Anderson models, using tools from the theory of dynamical systems associated to quasicrystals. Under certain conditions on the geometry of the underlying Delone set, we show the existence of the integrated density of states. Furthermore, in the case of a Delone-Anderson perturbation of the free Laplacian, we show it exhibits Lifshitz tails at the bottom of the spectrum
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Sutton, Brian D. (Brian David). "The stochastic operator approach to random matrix theory." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/33094.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 147-150) and index.
Classical random matrix models are formed from dense matrices with Gaussian entries. Their eigenvalues have features that have been observed in combinatorics, statistical mechanics, quantum mechanics, and even the zeros of the Riemann zeta function. However, their eigenvectors are Haar-distributed-completely random. Therefore, these classical random matrices are rarely considered as operators. The stochastic operator approach to random matrix theory, introduced here, shows that it is actually quite natural and quite useful to view random matrices as random operators. The first step is to perform a change of basis, replacing the traditional Gaussian random matrix models by carefully chosen distributions on structured, e.g., tridiagonal, matrices. These structured random matrix models were introduced by Dumitriu and Edelman, and of course have the same eigenvalue distributions as the classical models, since they are equivalent up to similarity transformation. This dissertation shows that these structured random matrix models, appropriately rescaled, are finite difference approximations to stochastic differential operators. Specifically, as the size of one of these matrices approaches infinity, it looks more and more like an operator constructed from either the Airy operator, ..., or one of the Bessel operators, ..., plus noise. One of the major advantages to the stochastic operator approach is a new method for working in "general [beta] " random matrix theory. In the stochastic operator approach, there is always a parameter [beta] which is inversely proportional to the variance of the noise.
(cont.) In contrast, the traditional Gaussian random matrix models identify the parameter [beta] with the real dimension of the division algebra of elements, limiting much study to the cases [beta] = 1 (real entries), [beta] = 2 (complex entries), and [beta] = 4 (quaternion entries). An application to general [beta] random matrix theory is presented, specifically regarding the universal largest eigenvalue distributions. In the cases [beta] = 1, 2, 4, Tracy and Widom derived exact formulas for these distributions. However, little is known about the general [beta] case. In this dissertation, the stochastic operator approach is used to derive a new asymptotic expansion for the mean, valid near [beta] = [infinity]. The expression is built from the eigendecomposition of the Airy operator, suggesting the intrinsic role of differential operators. This dissertation also introduces a new matrix model for the Jacobi ensemble, solving a problem posed by Dumitriu and Edelman, and enabling the extension of the stochastic operator approach to the Jacobi case.
by Brian D. Sutton.
Ph.D.
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Kegel, Tobias [Verfasser]. "Simulation and estimation of operator scaling stable random fields / Tobias Kegel." Siegen : Universitätsbibliothek der Universität Siegen, 2011. http://d-nb.info/101902786X/34.
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Stahl, Andreas [Verfasser], and Peter [Gutachter] Scheffler. "Tempered operator scaling stable random fields / Andreas Stahl ; Gutachter: Peter Scheffler." Siegen : Universitätsbibliothek der Universität Siegen, 2019. http://d-nb.info/121026823X/34.
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Zhu, Xueyun. "Vlist and Ering: compact data structures for simplicial 2-complexes." Thesis, Georgia Institute of Technology, 2013. http://hdl.handle.net/1853/50389.
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Various data structures have been proposed for representing the connectivity of manifold triangle meshes. For example, the Extended Corner Table (ECT) stores V+6T references, where V and T respectively denote the vertex and triangle counts. ECT supports Random Access and Traversal (RAT) operators at Constant Amortized Time (CAT) cost. We propose two novel variations of ECT that also support RAT operations at CAT cost, but can be used to represent and process Simplicial 2-Complexes (S2Cs), which may represent star-connecting, non-orientable, and non-manifold triangulations along with dangling edges, which we call sticks. Vlist stores V+3T+3S+3(C+S-N) references, where S denotes the stick count, C denotes the number of edge-connected components and N denotes the number of star-connecting vertices. Ering stores 6T+3S+3(C+S-N) references, but has two advantages over Vlist: the Ering implementation of the operators is faster and is purely topological (i.e., it does not perform geometric queries). Vlist and Ering representations have two principal advantages over previously proposed representations for simplicial complexes: (1) Lower storage cost, at least for meshes with significantly more triangles than sticks, and (2) explicit support of side-respecting traversal operators which each walks from a corner on the face of a triangle t across an edge or a vertex of t, to a corner on a faces of a triangle or to an end of a stick that share a vertex with t, and this without ever piercing through the surface of a triangle.
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Hoffmann, Alexander [Verfasser]. "Operator Scaling Stable Random Sheets with application to binary mixtures / Alexander Hoffmann." Siegen : Universitätsbibliothek der Universität Siegen, 2011. http://d-nb.info/1017706352/34.
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Sönmez, Ercan [Verfasser]. "Hausdorff dimension results for operator-self-similar stable random fields / Ercan Sönmez." Düsseldorf : Universitäts- und Landesbibliothek der Heinrich-Heine-Universität Düsseldorf, 2017. http://d-nb.info/1128293935/34.
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Tran, Tat Dat. "Information Geometry and the Wright-Fisher model of Mathematical Population Genetics." Doctoral thesis, Universitätsbibliothek Leipzig, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-90508.
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My thesis addresses a systematic approach to stochastic models in population genetics; in particular, the Wright-Fisher models affected only by the random genetic drift. I used various mathematical methods such as Probability, PDE, and Geometry to answer an important question: \"How do genetic change factors (random genetic drift, selection, mutation, migration, random environment, etc.) affect the behavior of gene frequencies or genotype frequencies in generations?”.
In a Hardy-Weinberg model, the Mendelian population model of a very large number of individuals without genetic change factors, the answer is simple by the Hardy-Weinberg principle: gene frequencies remain unchanged from generation to generation, and genotype frequencies from the second generation onward remain also unchanged from generation to generation.
With directional genetic change factors (selection, mutation, migration), we will have a deterministic dynamics of gene frequencies, which has been studied rather in detail. With non-directional genetic change factors (random genetic drift, random environment), we will have a stochastic dynamics of gene frequencies, which has been studied with much more interests. A combination of these factors has also been considered.
We consider a monoecious diploid population of fixed size N with n + 1 possible alleles at a given locus A, and assume that the evolution of population was only affected by the random genetic drift. The question is that what the behavior of the distribution of relative frequencies of alleles in time and its stochastic quantities are.
When N is large enough, we can approximate this discrete Markov chain to a continuous Markov with the same characteristics. In 1931, Kolmogorov first introduced a nice relation between a continuous Markov process and diffusion equations. These equations called the (backward/forward) Kolmogorov equations which have been first applied in population genetics in 1945 by Wright.
Note that these equations are singular parabolic equations (diffusion coefficients vanish on boundary). To solve them, we use generalized hypergeometric functions. To know more about what will happen after the first exit time, or more general, the behavior of whole process, in joint work with J. Hofrichter, we define the global solution by moment conditions; calculate the component solutions by boundary flux method and combinatorics method.
One interesting property is that some statistical quantities of interest are solutions of a singular elliptic second order linear equation with discontinuous (or incomplete) boundary values. A lot of papers, textbooks have used this property to find those quantities. However, the uniqueness of these problems has not been proved. Littler, in his PhD thesis in 1975, took up the uniqueness problem but his proof, in my view, is not rigorous. In joint work with J. Hofrichter, we showed two different ways to prove the uniqueness rigorously. The first way is the approximation method. The second way is the blow-up method which is conducted by J. Hofrichter.
By applying the Information Geometry, which was first introduced by Amari in 1985, we see that the local state space is an Einstein space, and also a dually flat manifold with the Fisher metric; the differential operator of the Kolmogorov equation is the affine Laplacian which can be represented in various coordinates and on various spaces. Dynamics on the whole state space explains some biological phenomena.
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Cao, Zhenwei. "Quantum evolution: The case of weak localization for a 3D alloy-type Anderson model and application to Hamiltonian based quantum computation." Diss., Virginia Tech, 2012. http://hdl.handle.net/10919/19205.
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Over the years, people have found Quantum Mechanics to be extremely useful in explaining various physical phenomena from a microscopic point of view. Anderson localization, named after physicist P. W. Anderson, states that disorder in a crystal can cause non-spreading of wave packets, which is one possible mechanism (at single electron level) to explain metalinsulator transitions. The theory of quantum computation promises to bring greater computational power over classical computers by making use of some special features of Quantum Mechanics. The first part of this dissertation considers a 3D alloy-type model, where the Hamiltonian is the sum of the finite difference Laplacian corresponding to free motion of an electron and a random potential generated by a sign-indefinite single-site potential. The result shows that localization occurs in the weak disorder regime, i.e., when the coupling parameter λ is very small, for energies E ≤ −Cλ² . The second part of this dissertation considers adiabatic quantum computing (AQC) algorithms for the unstructured search problem to the case when the number of marked items is unknown. In an ideal situation, an explicit quantum algorithm together with a counting subroutine are given that achieve the optimal Grover speedup over classical algorithms, i.e., roughly speaking, reduce O(2n ) to O(2n/2 ), where n is the size of the problem. However, if one considers more realistic settings, the result shows this quantum speedup is achievable only under a very rigid control precision requirement (e.g., exponentially small control error).Ph. D.
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Kwon, Dennis Oshuk 1979. "Intrusion detection by random dispersion and voting on redundant Web server operations." Thesis, Massachusetts Institute of Technology, 2002. http://hdl.handle.net/1721.1/8112.
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Thesis (M.Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2002.Includes bibliographical references (p. 126-128).
Until now, conventional approaches to the problem of computer security and intrusion tolerance have either tried to block intrusions altogether, or have tried to detect an intrusion in progress and stop it before the execution of malicious code could damage the system or cause it to send corrupted data back to the client. The goal of this thesis is to explore the question of whether voting, in conjunction with several key concepts from the study of fault-tolerant computing - namely masking, redundancy, and dispersion - can be effectively implemented and used to confront the issues of detecting and handling such abnormalities within the system. Such a mechanism would effectively provide a powerful tool for any high-security system where it could be used to catch and eliminate the majority of all intrusions before they were able to cause substantial damage to the system. There are a number of subgoals that pertain to the issue of voting. The most significant are those of syntactic equivalence and tagging. Respectively, these deal with the issues of determining the true equivalence of two objects to be voted on, and "marking" multiple redundant copies of a single transaction such that they can be associated at a later time. Both of these subgoals must be thoroughly examined in order to design the optimal voting system. The results of this research were tested in a simulation environment. A series of intrusions were then run on the voting system to measure its performance. The outcome of these tests and any gains in intrusion tolerance were documented accordingly.
by Dennis Oshuk Kwon.
M.Eng.
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He, Yuanjie. "Tradeoffs and Random Yield in Supply Chain Management." online version, 2005. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=case1121438339.
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Balanesković, Nenad [Verfasser], Gernot [Akademischer Betreuer] Alber, and Reinhold [Akademischer Betreuer] Walser. "Random Unitary Operations and Quantum Darwinism / Nenad Balanesković. Betreuer: Gernot Alber ; Reinhold Walser." Darmstadt : Universitäts- und Landesbibliothek Darmstadt, 2016. http://d-nb.info/1112141200/34.
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Lagro, Matthew Patrick. "A Perron-Frobenius Type of Theorem for Quantum Operations." Diss., Temple University Libraries, 2015. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/339694.
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MathematicsPh.D.
Quantum random walks are a generalization of classical Markovian random walks to a quantum mechanical or quantum computing setting. Quantum walks have promising applications but are complicated by quantum decoherence. We prove that the long-time limiting behavior of the class of quantum operations which are the convex combination of norm one operators is governed by the eigenvectors with norm one eigenvalues which are shared by the operators. This class includes all operations formed by a coherent operation with positive probability of orthogonal measurement at each step. We also prove that any operation that has range contained in a low enough dimension subspace of the space of density operators has limiting behavior isomorphic to an associated Markov chain. A particular class of such operations are coherent operations followed by an orthogonal measurement. Applications of the convergence theorems to quantum walks are given.
Temple University--Theses
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Samavat, Reza. "Mean Eigenvalue Counting Function Bound for Laplacians on Random Networks." Doctoral thesis, Universitätsbibliothek Chemnitz, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-159578.
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Spectral graph theory widely increases the interests in not only discovering new properties of well known graphs but also proving the well known properties for the new type of graphs. In fact all spectral properties of proverbial graphs are not acknowledged to us and in other hand due to the structure of nature, new classes of graphs are required to explain the phenomena around us and the spectral properties of these graphs can tell us more about the structure of them. These both themes are the body of our work here. We introduce here three models of random graphs and show that the eigenvalue counting function of Laplacians on these graphs has exponential decay bound. Since our methods heavily depend on the first nonzero eigenvalue of Laplacian, we study also this eigenvalue for the graph in both random and nonrandom cases.
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Rachakonda, Ravi Kanth. "Crew Rostering Problem: A Random Key Genetic Algorithm With Local Search." The Ohio State University, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=osu1230931714.
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Le, Masson Etienne. "Ergodicité et fonctions propres du laplacien sur les grands graphes réguliers." Phd thesis, Université Paris Sud - Paris XI, 2013. http://tel.archives-ouvertes.fr/tel-00866843.
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Dans cette thèse, nous étudions les propriétés de concentration des fonctions propres du laplacien discret sur des graphes réguliers de degré fixé dont le nombre de sommets tend vers l'infini. Cette étude s'inspire de la théorie de l'ergodicité quantique sur les variétés. Par analogie avec cette dernière, nous développons un calcul pseudo-différentiel sur les arbres réguliers : nous définissons des classes de symboles et des opérateurs associés, et nous prouvons un certain nombre de propriétés de ces classes de symboles et opérateurs. Nous montrons notamment que les opérateurs sont bornés dans L², et nous donnons des formules de l'adjoint et du produit. Nous nous servons ensuite de cette théorie pour montrer un théorème d'ergodicité quantique pour des suites de graphes réguliers dont le nombre de sommets tend vers l'infini. Il s'agit d'un résultat de délocalisation de la plupart des fonctions propres dans la limite des grands graphes réguliers. Les graphes vérifient une hypothèse d'expansion et ne comportent pas trop de cycles courts, deux hypothèses vérifiées presque sûrement par des suites de graphes réguliers aléatoires.
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Giulini, Ilaria. "Generalization bounds for random samples in Hilbert spaces." Thesis, Paris, Ecole normale supérieure, 2015. http://www.theses.fr/2015ENSU0026/document.
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Ce travail de thèse porte sur l'obtention de bornes de généralisation pour des échantillons statistiques à valeur dans des espaces de Hilbert définis par des noyaux reproduisants. L'approche consiste à obtenir des bornes non asymptotiques indépendantes de la dimension dans des espaces de dimension finie, en utilisant des inégalités PAC-Bayesiennes liées à une perturbation Gaussienne du paramètre et à les étendre ensuite aux espaces de Hilbert séparables. On se pose dans un premier temps la question de l'estimation de l'opérateur de Gram à partir d'un échantillon i. i. d. par un estimateur robuste et on propose des bornes uniformes, sous des hypothèses faibles de moments. Ces résultats permettent de caractériser l'analyse en composantes principales indépendamment de la dimension et d'en proposer des variantes robustes. On propose ensuite un nouvel algorithme de clustering spectral. Au lieu de ne garder que la projection sur les premiers vecteurs propres, on calcule une itérée du Laplacian normalisé. Cette itération, justifiée par l'analyse du clustering en termes de chaînes de Markov, opère comme une version régularisée de la projection sur les premiers vecteurs propres et permet d'obtenir un algorithme dans lequel le nombre de clusters est déterminé automatiquement. On présente des bornes non asymptotiques concernant la convergence de cet algorithme, lorsque les points à classer forment un échantillon i. i. d. d'une loi à support compact dans un espace de Hilbert. Ces bornes sont déduites des bornes obtenues pour l'estimation d'un opérateur de Gram dans un espace de Hilbert. On termine par un aperçu de l'intérêt du clustering spectral dans le cadre de l'analyse d'imagesThis thesis focuses on obtaining generalization bounds for random samples in reproducing kernel Hilbert spaces. The approach consists in first obtaining non-asymptotic dimension-free bounds in finite-dimensional spaces using some PAC-Bayesian inequalities related to Gaussian perturbations and then in generalizing the results in a separable Hilbert space. We first investigate the question of estimating the Gram operator by a robust estimator from an i. i. d. sample and we present uniform bounds that hold under weak moment assumptions. These results allow us to qualify principal component analysis independently of the dimension of the ambient space and to propose stable versions of it. In the last part of the thesis we present a new algorithm for spectral clustering. It consists in replacing the projection on the eigenvectors associated with the largest eigenvalues of the Laplacian matrix by a power of the normalized Laplacian. This iteration, justified by the analysis of clustering in terms of Markov chains, performs a smooth truncation. We prove nonasymptotic bounds for the convergence of our spectral clustering algorithm applied to a random sample of points in a Hilbert space that are deduced from the bounds for the Gram operator in a Hilbert space. Experiments are done in the context of image analysis