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Register a new userLa theorie floue et la derivation fractionnaire : Application aux equations differentielles hybrides
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Aziz Elghazouani
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On this thesis we present the fuzzy sets, fuzzy numbers, the fractional derivative and also we discuss the solution of the first order of fuzzy hybrid equation.
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The goal of this Habilitation `a diriger des recherches is to present two different applications, namely computations of certain partition functions in probability and applications to integrable systems, of the topological recursion developed by B. Eynard and N. Orantin in 2007. Since its creation, the range of applications of the topological recursion has been growing and many results in different fields have been obtained. The first aspect that I will develop deals with the historical domain of the topological recursion: random matrix integrals. I will review the formalism of the topological recursion as well as how it can be used to obtain asymptotic $frac{1}{N}$ series expansion of various matrix integrals. In particular, a key feature of the topological recursion is that it can recover from the leading order of the asymptotic all sub-leading orders with elementary computations. This method is particularly well known and fruitful in the case of hermitian matrix integrals, but I will also show that the general method can be used to cover integrals with hard edges, integrals over unitary matrices and much more. In the end, I will also briefly mention the generalization to $beta$-ensembles. In a second chapter, I will review the connection between the topological recursion and the study of integrable systems having a Lax pair representation. Most of the results presented there will be illustrated by the case of the famous six Painleve equations. Though the formalism used in this chapter may look completely disconnected from the previous one, it is well known that the local statistics of eigenvalues in random matrix theory exhibit a universality phenomenon and that the encountered universal systems are precisely driven by some solutions of the Painlev{e} equations. As I will show, the connection can be made very explicit with the topological recursion formalism.
This article is a brief presentation of results surrounding the fundamental gap. We begin by recalling Bakry-Emery geometry and demonstrate connections between eigenvalues of the Laplacian with the Dirichlet and Neumann boundary conditions. We then show a connection between the fundamental gap and Bakry-Emery geometry, concluding with a presentation of the key ideas in Andrewss and Clutterbucks proof of the fundamental gap conjecture. We conclude with a presentation of results for the fundamental gap of triangles and simplices.
Observation of the workings of productive organizations shows that the characteristics of a trade, backed by nature given to a technological environment, determine the productive combination implemented by the decision maker, and the structure of the operating cycle which is related. The choice of the production function and the choice of the ring structure strain the operating conditions under which the firms cash flow will evolve. New tools for financial control - leverage cash and operating cash surplus - provide the entrepreneur the information relevant to the efficiency of the strategic choices of the firm.
During the whole of 1874, Camille Jordan and Leopold Kronecker quar- relled vigorously over the organisation of the theory of bilinear forms. That theory promised a general and homogeneous treatment of numerous questions arising in various 19th-century theoretical contexts, and it hinged on two theorems, stated independently by Jordan and Weierstrass, that would today be considered equivalent. It was, however, the perceived difference between those two theorems that sparked the 1874 controversy. Focusing on this quarrel allows us to explore the algebraic identity of the polynomial practices of the manipulations of forms in use before the advent of structural approaches to linear algebra. The latter approaches identified these practices with methods for the classification of similar matrices. We show that the prac- tices -- Jordans canonical reduction and Kroneckers invariant computation -- reflect identities inseparable from the social context of the time. Moreover, these practices reveal not only tacit knowledge, local ways of thinking, but also -- in light of a long history tracing back to the work of Lagrange, Laplace, Cau- chy, and Hermite -- two internal philosophies regarding the significance of generality which are inseparable from two disciplinary ideals opposing algebra and arithmetic. By interrogating the cultural identities of such practices, this study aims at a deeper understanding of the history of linear algebra without focusing on issues related to the origins of theories or structures.
Let $X$ be a real-analytic manifold and $gcolon Xto{mathbf R}^n$ a proper triangulable subanalytic map. Given a subanalytic $r$-form $omega$ on $X$ whose pull-back to every non singular fiber of $g$ is exact, we show tha $omega$ has a relative primitive: there is a subanalytic $(r-1)$-form $Omega$ such that $dgLambda (omega-dOmega)=0$. The proof uses a subanalytic triangulation to translate the problem in terms of relative Whitney forms associated to prisms. Using the combinatorics of Whitney forms, we show that the result ultimately follows from the subanaliticity of solutions of a special linear partial differential equation. The work was inspired by a question of Franc{c}ois Treves.
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