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 / Doctor of Human and Environmental Studies / Kyoto University / DFAM

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).

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.

A stochastic operator is a positive linear contraction, P : L1 --+ L1, such that llPfII2 = llfll1 for f > 0. It is called asymptotically stable if the iterates pn f of each density converge in the norm to a fixed density. Pf(x) = f K(x,y)f(y)dy, where K( ·, y) is a density, defines a stochastic kernel operator. A general probabilistic/ deterministic model for biological systems is considered. This leads to the LMT operator P f(x) = Jo - Bx H(Q(>.(x)) - Q(y)) dy, where -H'(x) = h(x) is a density. Several particular examples of cell cycle models are examined. An operator overlaps supports iffor all densities f,g, pn f APng of 0 for some n. If the operator is partially kernel, has a positive invariant density and overlaps supports, it is asymptotically stable. It is found that if h( x) > 0 for x ~ xo ~ 0 and ["'" x"h(x) dx < liminf(Q(A(x))" - Q(x)") for a E (0, 1] lo x-oo then P is asymptotically stable, and an opposite condition implies P is sweeping. Many known results for cell cycle models follow from this. / Mathematical Science / M. Sc. (Mathematics)

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.

A stochastic operator is a positive linear contraction, P : L1 --+ L1, such that llPfII2 = llfll1 for f > 0. It is called asymptotically stable if the iterates pn f of each density converge in the norm to a fixed density. Pf(x) = f K(x,y)f(y)dy, where K( ·, y) is a density, defines a stochastic kernel operator. A general probabilistic/ deterministic model for biological systems is considered. This leads to the LMT operator P f(x) = Jo - Bx H(Q(>.(x)) - Q(y)) dy, where -H'(x) = h(x) is a density. Several particular examples of cell cycle models are examined. An operator overlaps supports iffor all densities f,g, pn f APng of 0 for some n. If the operator is partially kernel, has a positive invariant density and overlaps supports, it is asymptotically stable. It is found that if h( x) > 0 for x ~ xo ~ 0 and ["'" x"h(x) dx < liminf(Q(A(x))" - Q(x)") for a E (0, 1] lo x-oo then P is asymptotically stable, and an opposite condition implies P is sweeping. Many known results for cell cycle models follow from this. / Mathematical Science / M. Sc. (Mathematics)

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.

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.