This dissertation is devoted to the study of self-similar groups and related topics. It consists of three parts. The first part is devoted to the study of examples of finitely generated amenable groups for which every finitely presented cover contains non-abelian free subgroups. The study of these examples was motivated by natural questions about finiteness properties of finitely generated groups. We show that many examples of amenable self-similar groups studied in the literature cannot be covered by finitely presented amenable groups. We investigate the class of contracting self-similar groups from this perspective and formulate a general result which is used to detect this property. As an application we show that almost all known examples of groups of intermediate growth cannot be covered by finitely presented amenable groups. The latter is related to the problem of the existence of finitely presented groups of intermediate growth. The second part focuses on the study of one important example of a self-similar group called the first Grigorchuk group G, from the viewpoint of pro finite groups. We investigate finite quotients of this group related to presentations and group (co)homology. As an outcome of this investigation we prove that the pro finite completion G_hat of this group is not finitely presented as a pro finite group. The last part focuses on a class of recursive group presentations known as L-presentations, which appear in the study of self-similar groups. We investigate the relation of such presentations with the normal subgroup structure of finitely presented groups and show that normal subgroups with finite cyclic quotient of finitely presented groups have such presentations. We apply this result to finitely presented indicable groups without free sub-semigroups.

The central purpose of this study is the integration of modem philosophical thinking with different chaos theory principles and definitions to form relational perspectives. Relations are used in different contexts to base the causes of deterministic chaos (chaost) in the laws of nature which constitutes order. The chaost-attractor is used as subjective conception to investigate the possibilities of hidden order in a seemingly chaotic state of the objective reality. Relevant definitions of the chaos theory were analysed methodically and transcendentally with the aid of concepts of order and relations. Attention is given to the broad associations and analogies from philosophy and other disciplines which relate to the connectivity of objects to form systems. Subjective model development was done which is used to consequentially analyse some statements from published research which applied principles of chaost. It is argued that: the intrinsic properties of objects determine the causality of forces which bind objects to compose systems; a web of interactive bonds functions subjective to laws of nature which determine whether a system is in a state of order, chaost or real chaos; a dynamical transfer of many intrinsic and asymmetric properties via internal bonds constitutes non-linear connectivity which causes a sensitivity for initial conditions. It is found that the chaost of the chaos theory is not the same as real, objective chaos. The random-like evolution of a dynamic system is determined by the occurrence of irregularities and uncertainties in its internal order. A web of interactive bonds distribute small changes self-similar and scale-relevant. The difficulty in describing and explaining the complex behaviour of composed entities is simplified by the proposed web-chaost model. / Thesis (M.A. (Philosophy))--North-West University, Potchefstroom Campus, 2006.

The thesis is made up of a number of studies involving long-range dependence (LRD), that is, a slow power-law decay in the temporal correlation of stochastic models. Such a phenomenon has been frequently observed in practice. The models with LRD often yield non-standard probabilistic and statistical results. The thesis includes in particular the following topics: Multivariate limit theorems. We consider a vector made of stationary sequences, some components of which have LRD, while the others do not. We show that the joint scaling limits of the vector exhibit an asymptotic independence property. Non-central limit theorems. We introduce new classes of stationary models with LRD through Volterra-type nonlinear filters of white noise. The scaling limits of the sum lead to a rich class of non-Gaussian stochastic processes defined by multiple stochastic integrals. Limit theorems for quadratic forms. We consider continuous-time quadratic forms involving continuous-time linear processes with LRD. We show that the scaling limit of such quadratic forms depends on both the strength of LRD and the decaying rate of the quadratic coefficient. Behavior of the generalized Rosenblatt process. The generalized Rosenblatt process arises from scaling limits under LRD. We study the behavior of this process as its two critical parameters approach the boundaries of the defining region. Inference using self-normalization and resampling. We introduce a procedure called "self-normalized block sampling" for the inference of the mean of stationary time series. It provides a unified approach to time series with or without LRD, as well as with or without heavy tails. The asymptotic validity of the procedure is established.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20154号 / 理博第4239号 / 新制||理||1610(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 坂上 貴之, 教授 上田 哲生, 教授 國府 寛司 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

The thesis consists of four main chapters. The first chapter includes an introduction to inhomogeneous self-similar sets and measures. In particular, we show that these sets and measures are natural generalizations of the well known self-similar sets and measures. We then investigate the structure of these sets and measures. In the second chapter we study various fractal dimensions (Hausdorff, packing and box dimensions) of inhomogeneous self-similar sets and compare our results with the well-known results for (ordinary) self-similar sets. In the third chapter we investigate the L {q} spectra and the Renyi dimensions of inhomogeneous self-similar measures and prove that new multifractal phenomena, not exhibited by (ordinary) self-similar measures, appear in the inhomogeneous case. Namely, we show that inhomogeneous self-similar measures may have phase transitions which is in sharp contrast to the behaviour of the L {q} spectra of (ordinary) self-similar measures satisfying the Open Set Condition. Then we study the significantly more difficult problem of computing the multifractal spectra of inhomogeneous self-similar measures. We show that the multifractal spectra of inhomogeneous self-similar measures may be non-concave which is again in sharp contrast to the behaviour of the multifractal spectra of (ordinary) self-similar measures satisfying the Open Set Condition. Then we present a number of applications of our results. Many of them are related to the notoriously difficult problem of computing (or simply obtaining non-trivial bounds) for the multifractal spectra of self-similar measures not satisfying the Open Set Condition. More precisely, we will show that our results provide a systematic approach to obtain non-trivial bounds (and in some cases even exact values) for the multifractal spectra of several large and interesting classes of self-similar measures not satisfying the Open Set Condition. In the fourth chapter we investigate the asymptotic behaviour of the Fourier transforms of inhomogeneous self-similar measures and again we present a number of applications of our results, in particular to non-linear self-similar measures.

We introduce methods to cope with self-similar sets when we do not assume any separation condition. For a self-similar set K ⊆ ℝᵈ we establish a similarity dimension-like formula for Hausdorff dimension regardless of any separation condition. By the application of this result we deduce that the Hausdorff measure and Hausdorff content of K are equal, which implies that K is Ahlfors regular if and only if Hᵗ (K) > 0 where t = dim[sub]H K. We further show that if t = dim[sub]H K < 1 then Hᵗ (K) > 0 is also equivalent to the weak separation property. Regarding Hausdorff dimension, we give a dimension approximation method that provides a tool to generalise results on non-overlapping self-similar sets to overlapping self-similar sets. We investigate how the Hausdorff dimension and measure of a self-similar set K ⊆ ℝᵈ behave under linear mappings. This depends on the nature of the group T generated by the orthogonal parts of the defining maps of K. We show that if T is finite then every linear image of K is a graph directed attractor and there exists at least one projection of K such that the dimension drops under projection. In general, with no restrictions on T we establish that Hᵗ (L ∘ O(K)) = Hᵗ (L(K)) for every element O of the closure of T , where L is a linear map and t = dim[sub]H K. We also prove that for disjoint subsets A and B of K we have that Hᵗ (L(A) ∩ L(B)) = 0. Hochman and Shmerkin showed that if T is dense in SO(d; ℝ) and the strong separation condition is satisfied then dim[sub]H (g(K)) = min {dim[sub]H K; l} for every continuously differentiable map g of rank l. We deduce the same result without any separation condition and we generalize a result of Eroğlu by obtaining that Hᵗ (g(K)) = 0. We show that for the attractor (K1, … ,Kq) of a graph directed iterated function system, for each 1 ≤ j ≤ q and ε > 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dim[sub]H Kj - ε < dim[sub]H K. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of K. Using this property we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets. We study the situations where the Hausdorff measure and Hausdorff content of a set are equal in the critical dimension. Our main result here shows that this equality holds for any subset of a set corresponding to a nontrivial cylinder of an irreducible subshift of finite type, and thus also for any self-similar or graph directed self-similar set, regardless of separation conditions. The main tool in the proof is an exhaustion lemma for Hausdorff measure based on the Vitali's Covering Theorem. We also give several examples showing that one cannot hope for the equality to hold in general if one moves in a number of the natural directions away from `self-similar'. Finally we consider an analogous version of the problem for packing measure. In this case we need the strong separation condition and can only prove that the packing measure and δ-approximate packing pre-measure coincide for sufficiently small δ > 0.

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Ce travail porte sur la dynamique de la matière granulaire dans l'état fluide, et sa réponse à une excitation localisée dans deux limites : une faible perturbation suite à laquelle le système relaxe rapidement vers un état homogène, ou une agitation intense donnant lieu à une onde de choc telle qu'un souffle d'explosion. Cette réponse est affectée par deux caractéristiques des fluides granulaires : les particules macroscopiques qui les composent sont d'une part inélastiques, de sorte que leur dynamique est dissipative et ne possède pas d'état d'équilibre, et d'autre part polydisperses, c'est-à-dire hétérogènes en taille et en masse. Nous isolons d'abord un effet dynamique de la polydispersité en montrant qu'il existe un mélange optimal qui minimise le temps de relaxation du fluide vers son état asymptotique. Nous nous intéressons ensuite au cas où une seule des espèces est perturbée par l'application d'un champ extérieur, et étudions l'état stationnaire hors d'équilibre ainsi établi, dans la limite du traceur où les autres espèces constituent un bain stationnaire. Enfin, nous modélisons la croissance de formes autosimilaires dans ce bain suite à une intense libération ponctuelle d'énergie, que nous comparons au souffle d'une explosion dans un gaz moléculaire. / This work focuses on the dynamics of the fluid state of granular matter, and its response to a localized perturbation in two limiting cases : relaxation toward a homogeneous state or growth of a blast wave. This response is shaped by two distinctive features of granular media: their macroscopic constituent particles are both inelastic, entailing intrinsically non-equilibrium dynamics, and polydisperse or heterogeneous in their material properties. First, we isolate the effects of polydispersity in the return of a gas to its homogeneous asymptotic state, and evidence the existence of an optimal mixture for which the relaxation time is minimal. We then restrict the perturbation to accelerating a single species with an external field in order to study the induced non-equilibrium stationary state in the tracer limit, where other species are undisturbed by this process. Finally, we model the self-similar shock wave generated in such a dissipative bath by an intense yet localized release of energy, and contrast it with blast waves in molecular gases.

The effect of chamber length and Reynolds number on the stability and behavior of the flow field generated by a precessing jet nozzle was studied using stereoscopic particle image velocimetry (StereoPIV). An algorithm was developed to determine the mode of the flow based on the distribution of axial velocity. The optimal chamber length for precession to occur was found to be between 2 and 2.75 chamber-diameters. There is no precession at a chamber length of one diameter, and the occurrence of precession was found to be strongly related to Reynolds number. Conditionally averaged velocity distributions for the flow in precessing mode were calculated. The effect of initial condition on downstream behavior of axisymmetric jets was examined. Variations in spread and decay rates were found for jets issuing from different nozzles. Self-similar solutions for axisymmetric jets are therefore not universal, and are instead dependent upon initial conditions at the source.

The purpose of this thesis is to gain understanding of the load on flat target surfaces produced by projectile impact. Models are proposed from which upper and lower bounds can be derived for the transition be-tween interface defeat and normal penetration. It is shown that the dominating contribution to the normal load is generally provided by the hydrodynamic pressure due to the effect of inertia. In addition it is shown that the contributions from yield strength and compressibility are also significant. For a cylindrical tungsten alloy projectile at an impact velocity representative of to-day’s ordnance velocities, the contributions to the load intensity on the axis of symmetry from yield strength and compressibility are shown to be 15% and 3.4%, respectively, of that of inertia. Impact tests have shown that for conical projectiles transition from interface defeat to penetration occurs at a significantly lower impact velocity than for cylindrical projectiles. In order to better understand the influence of projectile shape, a conical projectile in axi-symmetric impact is studied by use of an analytical model for self-similar flow, and the results obtained are compared to results of numerical simula-tions. It is shown how the maximum load intensity, and the position of the maximum, depends on the apex angle. For an apex angle of 90º, the maximum load intensity is found to be almost three times that pro-duced by a cylindrical projectile with the same impact velocity. This maximum occurs well off the axis of symmetry and is 20% larger than the load intensity at this axis. Both the self-similar model and the nu-merical simulations show that the contribution to the load intensity from compressibility is positive below and negative above an apex angle of around 80º. The contribution of yield strength to the load in-tensity at centre of impact depends only weakly on the apex angle and is therefore similar to that of a cylindrical projectile.