Abstract:
In order to help companies to improve their competetiveness, it is important to develop new design methodologies. In this framework, a Functional And Robust Design (FARD) methodology dedicated to routine design of “highly productive” modular product ranges is proposed including principles of functional analysis, Design For Assembly (DFA), and techniques of modelling and simulation for ergonomics consideration. This paper focuses on the application of this original method applied to mechanical vibration and ergonomics problems of a scraper. Including biomechanical aspect in the design methodology, it is possible to identify the impact of a vibration tool on its users using numerical models of the tool coupled to a finite element model of the human hand. This method can proactively warn very early, in the design process, the risks of causing musculoskeletal disorders and facilitate an optimization of the mechanical tool. This study is a first step in a context of human-centered design.

We establish,
through solving semi-infinite programming problems, bounds on the probability
of safely reaching a desired level of wealth on a finite horizon,
when an investor starts with an optimal mean-variance financial investment
strategy under a non-negative wealth restriction.

The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in order to revisit the mathematical foundations of general relativity. Other engineering examples (control theory, elasticity theory, electromagnetism) will also be considered in order to illustrate the three fundamental results that we shall provide successively. 1) VESSIOT VERSUS CARTAN: The quadratic terms appearing in the “Riemann tensor” according to the “Vessiot structure equations” must not be identified with the quadratic terms appearing in the well known “Cartan structure equations” for Lie groups. In particular, “curvature+torsion” (Cartan) must not be considered as a generalization of “curvature alone” (Vessiot). 2) JANET VERSUS SPENCER: The “Ricci tensor” only depends on the nonlinear transformations (called “elations” by Cartan in 1922) that describe the “difference” existing between the Weyl group (10 parameters of the Poincaré subgroup + 1 dilatation) and the conformal group of space-time (15 parameters). It can be defined without using the indices leading to the standard contraction or trace of the Riemann tensor. Meanwhile, we shall obtain the number of components of the Riemann and Weyl tensors without any combinatoric argument on the exchange of indices. Accordingly and contrary to the “Janet sequence”, the “Spencer sequence” for the conformal Killing system and its formal adjoint fully describe the Cosserat equations, Maxwell equations and Weyl equations but General Relativity is not coherent with this result. 3) ALGEBRA VERSUS GEOMETRY: Using the powerful methods of “Algebraic Analysis”, that is a mixture of homological agebra and differential geometry, we shall prove that, contrary to other equations of physics (Cauchy

We start
recalling with critical eyes the mathematical methods used in gauge theory and
prove that they are not coherent with continuum mechanics, in particular the
analytical mechanics of rigid bodies (despite using the same group theoretical
methods) and the well known couplings existing between elasticity and
electromagnetism (piezzo electricity, photo elasticity, streaming
birefringence). The purpose of this paper is to avoid such contradictions by
using new mathematical methods coming from the formal theory of systems of
partial differential equations and Lie pseudo groups. These results finally
allow unifying the previous independent tentatives done by the brothers E. and
F. Cosserat in 1909 for elasticity or H. Weyl in 1918 for electromagnetism by
using respectively the group of rigid motions of space or the conformal group
of space-time. Meanwhile we explain why the Poincaré duality scheme existing between geometry and physics has to do with homological algebra
and algebraic analysis. We insist on the fact that these results could not have
been obtained before 1975 as the corresponding tools were not known before.

Abstract:
Different aspects of mathematical finance benefit from the use Hermite polynomials, and this is particularly the case where risk drivers have a Gaussian distribution. They support quick analytical methods which are computationally less cumbersome than a full-fledged Monte Carlo framework, both for pricing and risk management purposes. In this paper, we review key properties of Hermite polynomials before moving on to a multinomial expansion formula for Hermite polynomials, which is proved using basic methods and corrects a formulation that appeared before in the financial literature. We then use it to give a trivial proof of the Mehler formula. Finally, we apply it to no arbitrage pricing in a multi-factor model and determine the empirical futures price law of any linear combination of the underlying factors.

Abstract:
In this article, we start by a review of the circle group？ [1] and its topology induced [1] by the quotient metric, which we later use to define a topological structure on the unit circle . Using points on？ under the complex exponential map, we can construct orthogonal projection operators. We will show that under this construction, we arrive at a topological group, denoted？ of projection matrices. Together with the induced topology, it will be demonstrated that？ is Hausdorff and Second Countable forming a topological manifold. Moreover, I will use an example of a group action on？ to generate subgroups of？.

Abstract:
In this article, we wish to expand on some of the results obtained from the first article entitled Projection Theory. We have already established that one-parameter projection operators can be constructed from the unit circle . As discussed in the previous article these operators form a Lie group known as the Projection Group. In the first section, we will show that the concepts from my first article are consistent with existing theory [1] [2]. In the second section, it will be demonstrated that not only such operators are mutually congruent but also we can define a group action on ？by using the rotation group [3] [4]. It will be proved that the group acts on elements of ？in a non-faithful but ∞-transitive way consistent with both group operations. Finally, in the last section we define the group operation ？in terms of matrix operations using the operator and the Hadamard Product; this construction is consistent with the group operation defined in the first article.

Abstract:
The purpose of this paper is to give an explicit description of the irreducible decomposition of the multigraded S_n-module of coinvariants of S_n x S_n. Many of the results presented can be extended to analogous questions for other finite reflection group.

Abstract:
In this paper, we propose a new class of discrete time stochastic processes generated by a two-color generalized Pólya urn, that is reinforced every time. A single urn contains a white balls, b black balls and evolves as follows: at discrete times n=1,2,…, we sample M_{n }balls and note their colors, say R_{n }are white and M_{n}- R_{n} are black. We return the drawn balls in the urn. Moreover, N_{n}R_{n }new white balls and N_{n} (M_{n}- R_{n})new black balls are added in the urn. The numbers M_{n }and N_{n }are random variables. We show that the proportions of white balls forms a bounded martingale sequence which converges almost surely. Necessary and sufficient conditions for the limit to concentrate on the set {0,1} are given.

Abstract:
32 third year medical students were presented with a complex case of endocarditis. They were asked to synthesize the case and give the best formulation of the problem. They were then asked to provide a diagnosis. A subsequent group of 25 students were presented with the problem already formulated and were also asked for the diagnosis. We analyzed the student's problem formulations using the presence or absence of essential elements of the case, the use of higher-order concepts and the use of relations between concepts.12/32 students presented with the case made the correct diagnosis. Diagnostic accuracy was significantly associated with the use of higher-order concepts and relations between concepts. Establishing explicit relations was particularly important. Almost all students who missed the diagnosis could not elicit any relations between concepts but only reported factual observations. When presented with an already formulated problem, 19/25 students made the correct diagnosis. (p < 0.05)When faced with a complex new case, students may not have the structured knowledge to recognize the nature of the problem. They have to build new schema or problem representation. Our observations suggest that this process involves using higher-order concepts and establishing new relations between concepts. The fact that students could recognize the disease when presented with a formulated problem but had more difficulty when presented with the original complex case indicates that knowledge of the clinical features may be necessary but not sufficient for problem formulation. Our hypothesis is that problem formulation represents a distinct ability.Problem formulation is necessary because the world we experience is complex. The problems we face are often ill-structured. One or several elements of the problem may be unknown, the same elements may be different in different context, there is uncertainty about the concepts necessary for a solution, the relationship between concepts an