The fundamental theorem of algebra (FTA) is an important theorem in algebra. This theorem asserts that the complex field is algebracially closed. This thesis will include historical research of proofs of the fundamental theorem of algebra and provide information about the first proof given by Gauss of the theorem and the time when it was proved.

Comment fonctionne le cerveau ? Peut-on créer un cerveau artificiel ? Une étape essentielle en vue d'obtenir une réponse à ces questions est la modélisation mathématique des phénomènes à l'œuvre dans le cerveau. Ce manuscrit se focalise sur l'étude de modèles de réseaux de neurones inspirés de la réalité.Cette thèse se place à la rencontre entre trois grands domaines des mathématiques - l'étude des équations aux dérivées partielles (EDP), les probabilités et la statistique - et s'intéresse à leur application en neurobiologie. Dans un premier temps, nous établissons les liens qui existent entre deux échelles de modélisation neurobiologique. À un niveau microscopique, l'activité électrique de chaque neurone est représentée par un processus ponctuel. À une plus grande échelle, un système d'EDP structuré en âge décrit la dynamique moyenne de ces activités. Il est alors montré que le modèle macroscopique peut se retrouver de deux manières distinctes : en étudiant la dynamique moyenne d'un neurone typique ou bien en étudiant la dynamique d'un réseau de $n$ neurones en champ-moyen quand $n$ tend vers l’infini. Dans le second cas, la convergence vers une dynamique limite est démontrée et les fluctuations de la dynamique microscopique autour de cette limite sont examinées. Dans un second temps, nous construisons une procédure de test d'indépendance entre processus ponctuels, ces derniers étant destinés à modéliser l'activité de certains neurones. Ses performances sont contrôlées théoriquement et vérifiées d'un point de vue pratique par une étude par simulations. Pour finir, notre procédure est appliquée sur de vraies données / How does the brain compute complex tasks? Is it possible to create en artificial brain? In order to answer these questions, a key step is to build mathematical models for information processing in the brain. Hence this manuscript focuses on biological neural networks and their modelling. This thesis lies in between three domains of mathematics - the study of partial differential equations (PDE), probabilities and statistics - and deals with their application to neuroscience. On the one hand, the bridges between two neural network models, involving two different scales, are highlighted. At a microscopic scale, the electrical activity of each neuron is described by a temporal point process. At a larger scale, an age structured system of PDE gives the global activity. There are two ways to derive the macroscopic model (PDE system) starting from the microscopic one: by studying the mean dynamics of one typical neuron or by investigating the dynamics of a mean-field network of $n$ neurons when $n$ goes to infinity. In the second case, we furthermore prove the convergence towards an explicit limit dynamics and inspect the fluctuations of the microscopic dynamics around its limit. On the other hand, a method to detect synchronisations between two or more neurons is proposed. To do so, tests of independence between temporal point processes are constructed. The level of the tests are theoretically controlled and the practical validity of the method is illustrated by a simulation study. Finally, the method is applied on real data

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京都大学 / 新制・課程博士 / 博士(理学) / 甲第24386号 / 理博第4885号 / 新制||理||1699(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 平岡 裕章, 教授 COLLINS Benoit Vincent Pierre, 教授 坂上 貴之 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Work is one of the cornerstones of classical thermodynamics. However, a direct transfer of this concept to quantum systems has proved problematic, especially for non-equilibrium processes. Unlike in the classical case, quantum work cannot be defined unambiguously. Depending on the specific setting and the imposed assumptions, different definitions are well motivated. In particular, in quantum thermodynamics, a clear distinction must be made between the measurement, storage, and use of work, since these three facets of the concept are not necessarily compatible with each other. The present thesis is mainly concerned with the measurement aspect. With the help of illustrative scenarios several approaches to quantum work measurements, their advantages and drawbacks are discussed. The focus will be on the question to what extent quantumness plays a decisive role in such scenarios, both in a qualitative and quantitative sense. Based on the gedankenexperiment of a Szilárd machine a criterion is proposed which can be used to verify genuine quantum correlations between the work medium in a heat engine and its thermal environment. In a Szilárd scenario a Maxwell's demon determines the state of the work medium and uses this information to extract work. We split this model into a bipartite setting. The demon only has access to the environment and, thus, can only indirectly measure the state of the work medium. By sharing the acquired information with another agent, the latter can extract work. The question of the quantumness of the experiment can then be reduced to the question of the maximum attainable work in the context of a suitable quantum steering scenario. For the constructed setting a bound for the work output achievable for classical correlations between the engine and the environment is derived. Work extraction beyond this classical limit thus proves the quantum nature of the machine. The verification of non-classical correlations by means of quantum steering is motivated by the fact that such a scenario reflects the typical asymmetry of a thermodynamic setup. While the machine itself is considered to be controllable and characterized in detail, no requirements are imposed on the correlated environment and the measurements performed on it. Consequently, this verification of a truly quantum heat engine is semi-device-independent. In a second scenario, the compatibility of average work and work fluctuations in a driven system is discussed. Fluctuation theorems play an important role in classical non-equilibrium thermodynamics. The best-known example is the Jarzynski equality. This equation establishes a connection between the free energy difference of two equilibrium states and the fluctuating work measured in a non-equilibrium process. A transfer of the Jarzynski equality to quantum systems succeeds most simply if the work definition is based on a so-called two-point measurement scheme. This approach determines the work as the difference of two projective energy measurements. The disadvantage of this definition is the unavoidable disturbance of the quantum state by the measurement, which makes a determination of the correct average work impossible. By means of a generalized two-point measurement scheme, it is shown how this contradiction between fluctuating and average work can be overcome. The approach is based on the concept of joint measurability. Unsharp measurements with a smaller disturbance of the quantum state can be measured jointly and allow for the determination of the correct average work. Nevertheless, the connection between measured fluctuations and the change of free energy can be preserved by means of a modified Jarzynski equality, as elucidated in this thesis. Even though the two-point measurement scheme - both in its projective form and in the generalized variant presented in this thesis - satisfies a Jarzynski equality, the operationality and the associated experimental significance are to be assessed differently than in the classical case. In classical thermodynamics, the Jarzynski relation can be used practically to determine, for example, the change of free energy in RNA molecules. However, it is crucial for such an experiment that the non-equilibrium work can actually be measured without requiring detailed knowledge of the system under consideration. In contrast, the two-point measurement scheme defines work as the energy difference of the system between the beginning and the end of the process. Crucially, for the measurement of these energies the Hamiltonians have to be known and the free energy difference could therefore be calculated directly from this knowledge without reference to the Jarzynski equality. Thus, the operationality of the quantum Jarzynski relation differs fundamentally from its classical counterpart. In this thesis we develop a measurement scheme which, in principle, allows us to employ a quantum version of the Jarzynski equation without knowledge of the Hamiltonians. The crucial point is to include the apparatus that drives the system out of equilibrium in the quantum picture and to define the work measurement on that very apparatus. Such a work measurement can only be meaningfully defined as a quantum expectation value and work fluctuations cannot directly be measured, in contrast to the classical case. The work along a classical microstate trajectory can be determined in a single run. The trajectory itself does not need to be known for this purpose; its existence is sufficient. Quantum trajectories do not exist unless they are objectified by a measurement. It is shown how measurements on the environment of the system can provide information about the trajectories. A conditioning of the measured work on these trajectories then allows for the determination of work fluctuations in the quantum system. For these fluctuations an inequality is conjectured whose limit is given by the classical Jarzynski equation. Numerical results support the conjecture. A proof is still missing. By means of the presented framework, the free energy difference of a quantum system can, in principle, be determined without knowledge of the underlying Hamiltonian. However, as is shown, this requires an optimization over several external parameters, since the inequality in general provides only an upper bound. Thus, the operationality of the model enforces a quantum disadvantage. The methods presented in this thesis can be applied to various scenarios in quantum thermodynamics. Especially the framework for work measurements on an external apparatus offers an alternative to common approaches when the system under investigation and especially its Hamiltonian is not known in advance. The focus on operationality will help to better understand to what extend the work quantities defined and measured in quantum thermodynamic systems differ from the classical concept of work.

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京都大学 / 新制・課程博士 / 博士(理学) / 甲第23682号 / 理博第4772号 / 新制||理||1684(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)准教授 矢野 孝次, 教授 泉 正己, 教授 日野 正訓 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Thesis (Ph.D. (Mathematics)) -- University of Limpopo, 2021 / A finite group G is called (l, m, n)-generated, if it is a quotient group of the triangle group T(l, m, n) = x, y, z|xl = ym = zn = xyz = 1-. In [43], Moori posed the question of finding all the (p, q, r) triples, where p, q and r are prime numbers, such that a non-abelian finite simple group G is a (p, q, r)-generated. In this thesis, we will establish all the (p, q, r)-generations of the following groups, the Mathieu sporadic simple group M23, the alternating group A11 and the symplectic group Sp(6, 2). Let X be a conjugacy class of a finite group G. The rank of X in G, denoted by rank(G : X), is defined to be the minimum number of elements of X generating G. We investigate the ranks of the non-identity conjugacy classes of the above three mentioned finite simple groups. The Groups, Algorithms and Programming (GAP) [26] and the Atlas of finite group representatives [55] are used in our computation / University of Limpopo

In this thesis we shall examine the role of measurerability in the theory of Lebesgue Integration. This shall be done in the context of the real line where we define the notion of an integral of a bounded real-valued function over a set of bounded outer measure without a prior assumption of measurability concerning the function and the domain of integration. / Mathematical Sciences / M. Sc. (Mathematics)