Les interfaces homme-machine (IHM) multimodales offrent à l’utilisateur la possibilité de combiner les modalités d’interaction afin d’augmenter la robustesse et l’utilisabilité de l’interface utilisateur d’un système. Plus particulièrement, en sortie, les IHM multimodales permettent au système de restituer à l’utilisateur, l’information produite par le noyau fonctionnel en combinant sémantiquement plusieurs modalités. Dans l’optique de concevoir de telles interfaces pour des systèmes critiques, nous avons proposé un modèle formel de conception des interfaces multimodales en sortie. Le modèle proposé se décompose en deux modèles : le modèle de fission sémantique qui décrit la décomposition de l’information à restituer en informations élémentaires, et le modèle d’allocation qui spécifie l’allocation des modalités et médias aux informations élémentaires. Nous avons également développé une formalisation B Événementiel détaillée des deux modèles : fission sémantique et allocation. Cette formalisation a été instanciée sur des études de cas puis généralisée dans un processus de développement B Événementiel cadre dans lequel s’inscrivent les modèles de fission sémantique et d’allocation. Cette formalisation a permis de procéder à la vérification de propriétés de sûreté, de vivacité et d’utilisabilité. / Multimodal Human-Computer Interfaces (HCI) offer to users the possibility to combine interaction modalities in order to increase user interface robustness and usability. Specifically, output multimodal HCI allow system to return to the user, the information generated by the functional core by combining semantically different modalities. In order to design such interfaces for critical systems, we proposed a formal model for the design of output multimodal interfaces. The proposed model consists of two models: the semantic fission model describes the decomposition of the information to return into elementary information and the allocation model specifies the allocation of the elementary information with modalities and media. We have also developed a detailed Event B formalization for the two models: semantic fission and allocation. This formalization has been instantiated on case studies and generalized in an Event B development process framework including semantic fission and allocation models. This formalization allows to carry out safety, liveness and usability properties verification.

Group Representation Theory is the mathematical language best suited to describing the symmetry properties of a structure, and a structural analysis can utilises Group Representation Theory to provide the most efficient and systematic method of exploiting the full symmetry properties of any symmetric structure. Group Representation Theory methods currently exist for the Stiffness Niethod of structural analysis, where the stiffness matrix of a structure is block-diagonalised into a number of independent submatrices, each of which relates applied loads and displacements with a particular type of symmetry. This dissertation extends the application of Group Representation Theory to the equilibrium and compatibility matrices which are commonly used in the Force Method of structural analysis. Group Representation Theory is used to find symmetry-adapted coordinate systems for both the external vector space which is suitable for representing the loads applied to a structure, and the internal vector space wh",t-k is-suitable for representing the internal forces. Using these symmetry-adapted coordinate systems the equilibrium matrix is block-diagonalised into a number of independent submatrix blocks, thus decomposing the analysis into a number of subproblems which require less computational effort. Each independent equilibrium submatrix block relates applied loads and internal forces with particular symmetry properties, and hence any states of self-stress or inextensional mechanisms in one of these equilibrium submatrix blocks will necessarily have ~rresponding symmetry properties. Thus, a symmetry analysis provides valuable insight into the behaviour of symmetric structures by helping to identify and classif:)'. any states of self-stress .or inextensional mechanisms present in a structure. In certain cases it is also possible for a symmetry analysis to identify when a structure contains a :ijnite rather than infinitesimal mechanism. To do this a symmetry analysis must b~ carried out using the symmetry properties of the inextensional mechanism of interest. If the analysis shows that any states of self-stress which exist in the structure have "lesser" symmetry properties, then the states of self-stress exist independently from the mechanism and cannot prevent its finite motion.

In Part I, we investigate the principal series representations of the n-fold covering groups of the special linear group over a p-adic field. Such representations are constructed via the Stone-von Neumann theorem. We have three interrelated results. We first compute the K-types of these representations. We then give a complete set of reducibility points for the unramified principal series representations. Among these are the unitary unramified principal series representations, for which we further investigate the distribution of the K-types among its irreducible components. In Part II, we demonstrate another application of the Stone-von Neumann theorem. Namely, we present a lower bound for the minimal degree of a faithful representation of an adjoint Chevalley group over a quotient ring of a non-Archimedean local field.

The main goal of this thesis is to present the theory of Locally Nilpotent Derivations and to show how it can be used to investigate the structure of the polynomial ring in two variables k[X;Y] over a field k of characteristic zero. The thesis gives a com- plete proof of Rentschler's Theorem, which describes all locally nilpotent derivations of k[X;Y]. Then we present Rentschler's proof of Jung's Theorem, which partially describes the group of automorphisms of k[X;Y]. Finally, we present the proof of the Structure Theorem for the group of automorphisms of k[X;Y].

Statistical physics has been the foundation for much of our understanding about plasma physics. Often, plasma physics phenomena are explained using statistical physics principles and theories. Here, I reverse this paradigm to instead use plasma experiments to test statistical physics principles. In this thesis, I test statistical physics principles with an experimental dusty plasma, which is a four-component mixture of micron-sized ``dust'' particles, electrons, ions, and neutral gas molecules. When immersed in the plasma, the dust particles acquire large negative charges, since they accumulate more electrons than ions. Due to their large electric charges, the dust particles have interparticle potential energies that greatly exceed their kinetic energies, so that the collection of dust particles is considered to be a strongly coupled plasma. Like other strongly coupled plasma, the collection of dust particles can exhibit solid-like or liquid-like behavior. A key advantage offered by dusty plasma experiments is the ability to track the motion of individual dust particles. Dust particles are sufficiently large to allow for direct imaging using a video camera, so that time series data can be obtained for particle positions and velocities. These particle-level data provide a richer description of the dynamics and structure than can be obtained for most other strongly coupled plasmas, simple liquids, or solid materials. In particular, the particle-level data of positions and velocities are often required inputs for testing statistical physics theories or principles. The dusty plasma data I analyze are from the experiment of Haralson~\textit{et al.} [1,2], where dust particles were electrically levitated in a single horizontal layer within a vacuum chamber. The collection of dust particles initially settled into a crystalline lattice with solid-like behavior. To reach a liquid-like state, or to drive a shear flow, dust particles were manipulated using the radiation pressure force of lasers. In this thesis, I test three different statistical physics principles using an experimental dusty plasma. First, I test the fluctuation theorem, which was first was presented in 1993 by Evans, Cohen, and Morriss [3]. The fluctuation theorem, which is one of the most important recent developments in statistical physics, quantifies the probability that the entropy production rate will temporarily fluctuate to negative values in ``violations'' of the second law of thermodynamics. The original formulation of the fluctuation theorem described the entropy production due to viscous heating in a shear flow; this version of the fluctuation theorem had never been experimentally demonstrated in a liquid of any kind. In Chapter 2, I provide the first such demonstration by showing that the entropy production rate in a liquid-like dusty plasma shear flow satisfies the fluctuation theorem. This result also serves as the first demonstration that a strongly coupled plasma obeys the fluctuation theorem. Second, I measure the Einstein frequency $\Omega_E$, which describes the stochastic process of collisions in a strongly coupled plasma, and I compare my measurement to predictions made in the literature that used simulation data. Often, for weakly coupled plasma, a collision frequency is obtained to provide a measure of the strength of particle-particle interactions. However, for strongly coupled plasma (and likewise for liquids and solids), a collision frequency is not well defined since collisions are multibody and occur continuously. Another quantity is needed to describe the strength of particle-particle interactions. I propose that the Einstein frequency $\Omega_E$, a concept more commonly used in solid physics, is better suited for describing particle-particle interactions in a strongly coupled plasma. In Chapter 3, I present and use a new method to obtain the Einstein frequency of a 2D dusty plasma in both a liquid-like state and a crystalline state. My measurement of the Einstein frequency, which serves as a proxy for a collision frequency, is consistent with simulation predictions in the literature. Third, I present particle-coordination survival functions, which provide a richer description of microscopic dynamics in a liquid than the commonly used relaxation time. Relaxation times have been used, for example the Maxwell relaxation time, to describe the characteristic time scale for the crossover between elastic and viscous behavior in viscoelastic liquids. However, relaxation times are single-value measures that cannot fully describe the complexity of a liquid. In Chapter 4, using a survival function that retains temporal information about the local structural in a liquid, I discover that the microscopic arrangements in a liquid-like 2D dusty plasma have multiple time scales. Unexpectedly, non-defects have two time scales, while defects have one. My survival functions are time-series graphs of the probability that a particle's number of nearest neighbors, i.e., its coordination, remains the same. The two time scales for non-defects are revealed by an elbow in their survival-function curve. As a spinoff with a considerable amount of importance, I performed the simplest fluctuation theorem experiment to date, using an aerosol. An aerosol is simply a particle that is immersed in air. In Chapter 5, I show that the fluctuation theorem is applicable for an aerosol particle undergoing free-fall in air due to gravity. While the particle typically fell downwards, it is observed to occasionally fall upwards, against the force of gravity. For such upward displacements, the work done on the particle is negative, which is a temporary violation of the second law. I find that the probability of these temporarily violations obeys the work fluctuation theorem. This result also allowed an application: a novel diagnostic method to measure the mass of aerosol particles.

Thesis advisor: Kevin S. Bedell / One of the many achievements of renowned physicist L.D. Landau was the formulation of Fermi Liquid Theory (FLT). Originally debuted in the 1950s, FLT has seen abundant success in understanding degenerate Fermi systems and is still used today when trying to understand the physics of a new interacting Fermi system. Of its many advantages, FLT excels in explaining why interacting Fermi systems behave like their non-interacting counterparts, and understanding transport phenomena without cumbersome and confusing mathematics. In this work, FLT is applied to systems whose low energy excitations obey the massless Dirac equation; i.e. the energy dispersion is linear in momentum, ε α ρ, as opposed to the normal quadratic, ε α ρ². Such behavior is seen in numerous, seemingly unrelated, materials including graphene, high T[subscript]c superconductors, Weyl semimetals, etc. While each of these materials possesses its own unique properties, it is their low energy behavior that provides the justification for their grouping into one family of materials called Dirac materials (DM). As will be shown, the linear spectrum and massless behavior leads to profound differences from the normal Fermi liquid behavior in both equilibrium and transport phenomena. For example, with mass having no meaning, we see the usual effective mass relation from FLT being replaced by an effective velocity ratio. Additionally, as FLT in d=2 has been poorly studied in the past, and since the most famous DM in graphene is a d=2 system, a thorough analysis of FLT in d=2 is presented. This reduced dimensionality leads to substantial differences including undamped collective modes and altered quasiparticle lifetime. In chapter 3, we apply the Virial theorem to DM and obtain an expression for the total average ground state energy $E=\frac{B}{r_s}$ where $B$ is a constant independent of density and $r_s$ is a dimensionless parameter related to the density of the system: the interparticle spacing $r$ is related to $r_s$ through $r=ar_s$ where $a$ is a characterstic length of the system (for example, in graphene, $a=1.42$ \AA). The expression derived for $E$ is unusual in that it's typically impossible to obtain a closed form for the energy with all interactions included. Additionally, the result allows for easy calculation of various thermodynamic quantities such as the compressibility and chemical potential. From there, we use the Fermi liquid results from the previous chapter and obtain an expression for $B$ in terms of constants and Fermi liquid parameters $F_0^s$ and $F_1^s$. When combined with experimental results for the compressibility, we find that the Fermi liquid parameters are density independent implying a unitary like behavior for DM. In chapter 4, we discuss the alleged universal KSS lower bound in DM. The bound, $\frac{\eta}{s}\geq\frac{\hbar}{4\pi k_B}$, was derived from high energy/string theory considerations and was conjectured to be obeyed by all quantum liquids regardless of density. The bound provides information on the interactions in the quantum liquid being studied and equality indicates a nearly perfect quantum fluid. Since its birth, the bound has been highly studied in various systems, mathematically broken, and poorly experimented on due to the difficult nature of measuring viscosity. First, we provide the first physical example of violation by showing $\frac{\eta}{s}\rightarrow 0$ as $T\rightarrow T_c$ in a unitary Fermi gas. Next, we determine the bound in DM in d=2,3 and show unusual behavior that isn't seen when the bound is calculated for normal Fermi systems. Finally we conclude in chapter 5 and discuss the outlook and other avenues to explore in DM. Specifically, it must be pointed out that the physics of what happens near charge neutrality in DM is still poorly understood. Our work in understanding the Fermi liquid state in DM is necessary in understanding DM as a whole. Such a task is crucial when we consider the potential in DM, experimentally, technologically, and purely for our understanding. / Thesis (PhD) — Boston College, 2020. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

The interplay between generalizations of Riezs' famous representation theorem and Radon-Nikodým type theorems has a long history. This paper will explore certain aspects of the theory of bounded linear operators on continuous function spaces, Radon-Nikodým type properties, and their connections.

Let 1, 2, 3, 5, 8, … denote the Fibonacci sequence beginning with 1 and 2, and then setting each subsequent number to the sum of the two previous ones. Every positive integer n can be expressed as a sum of distinct Fibonacci numbers in one or more ways. Setting R(n) to be the number of ways n can be written as a sum of distinct Fibonacci numbers, we exhibit certain regularity properties of R(n), one of which is connected to the Euler φ-function. In addition, using a theorem of Fine and Wilf, we give a formula for R(n) in terms of binomial coefficients modulo two.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22982号 / 理博第4659号 / 新制||理||1669(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 熊谷 隆, 教授 長谷川 真人, 小澤 登高 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

This thesis studies complex harmonic polynomials of the form $f(z) = az^n + b\bar{z}^k+z$ where $n, k \in \mathbb{Z}$ with $n > k$ and $a, b > 0$. We show that the sum of the orders of the zeros of such functions is $n$ and investigate the locations of the zeros, including whether the zeros are in the sense-preserving or sense-reversing region and a set of conditions under which zeros have the same modulus. We also show that the number of zeros ranges from $n$ to $n+2k+2$ as long as certain criteria are met.