Equisingular Deformations of Sandwiched Singularities

We relate the equisingular deformation theory of plane curve singularities and sandwiched surface singularities. We show the existence of a smooth map between the two corresponding deformation functors and study the kernel of this map. In particular we show that the map is an isomorphism when a certain invariant is large enough.     

Field Theory Approach to the Dynamics of a Continuous Medium: A Perturbation Theory

We propose a perturbation theory that allows solving the equations of motion for the displacement vector in the body of the Earth in the framework of the linear theory of elasticity. We show that tectonic processes are primarily determined by tidal actions. We analyze the tidal effects in the Earth–Moon–Sun system.     

On the cotangent cohomology of rational surface singularities with almost reduced fundamental cycle

We prove dimension formulas for the cotangent spaces T ¹ and T ² for a class of rational surface singularities by calculating a correction term in the general dimension formulas. We get that it is zero if the dual graph of the rational surface singularity X does not contain a particular type of configurations, and this generalizes a result of Theo de Jong stating that the correction term c (X ) is zero for rational determinantal surface singularities. In particular our result implies that c (X ) is zero for Riemenschneiders quasi‐determinantal rational surface singularities, and this also generalizes results for quotient singularities. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Remarks on the r and Δ convolutions

In this paper we study the properties of the r–deformation introduced in [B1]. We observe that the associated convolution coming from the conditionally free convolution is associative only for r = 1 and r = 0. We give the realization of some r–Gaussian random variables and obtain Haagerup–Pisier–Buchholz type inequalities. We also study another convolution defined with the use of the r–deformation through a moment–cumulant formula [KY1] and show that it is associative and in general not positive. Partially sponsored with KBN grant no 2P03A00723 and RTN HPRN-CT-2002-00279.     

Isomonodromic deformations and “antiquantization” for the simplest ordinary differential equations

We consider three different models of linear differential equations and their isomonodromic deformations. We show that each of the models has its own specificity, although all of them lead to the same final result. It turns out that isomonodromic deformations are closely related to the Hamiltonian structure of both classical mechanics and quantum mechanics. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 1, pp. 143–151, January, 2007.     

Bimodule Deformations, Picard Groups and Contravariant Connections

We study deformations of invertible bimodules and the behavior of Picard groups under deformation quantization. While K ₀-groups are known to be stable under formal deformations of algebras, Picard groups may change drastically. We identify the semiclassical limit of bimodule deformations as contravariant connections and study the associated deformation quantization problem. Our main focus is on formal deformation quantization of Poisson manifolds by star products.     

Instanton in the Field of a Pointlike Source of a Euclidean Non-Abelian Field

We consider the behavior of an (anti)instanton in the field of a pointlike source of a Euclidean non-Abelian field and investigate (anti)instanton deformations described by variations of its characteristic parameters. We formulate the variational problem of seeking the corresponding “crumpled” topological configurations and solve it algebraically using the Ritz method (of multipole decomposition of deformation fields). We investigate the domain of parameters specific for the instanton liquid model. We propose a simple method of taking contributions of such configurations in the functional integral into account approximately. In the framework of superposition analysis, we obtain an estimate for the mean energy of the source in the instanton liquid, and the estimate increases linearly with distance. In the case of a dipole in the color-singlet state, the energy is a linear function of the distance between the sources with the “strength” coefficient agreeing well with the known model and lattice estimates. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 267–298, February, 2006.     

The application of a Gauss-Eyring model to predict the behavior of thermoplastics in tensile experiments

The Gaussian expression for the isothermal tensile deformation of thermoplastics including the proposed strain hardening constant G_p, has been combined with the Eyring flow equation to provide a new relation describing the rate of strain of a thermoplastic in terms of the true stress and the extension ratio under isothermal conditions. In conventional mechanical tests this model can be used to quantify the tendency to strain localization, to predict the natural draw ratio and the inversion point where the true engineering stress passes through a minimum. The latter is expected to correlate with the value of the extension ratio where crazes do not propagate under tension. The equation is most easily demonstrated in constant load experiments where they agree well with published work. However, for a more precise evaluation of the theory the constant G_p should be measured separately and the calculated results compared with other tests on the same material. Where necking occurs it is possible to use a simplified plug flow model to calculate neck profiles. These show that no special assumptions are required to account for necking which results directly from the interaction of geometric thinning and strain hardening, even where true strain softening is absent. The procedure makes it possible to illustrate the way in which the form of the neck can be affected by the rate of extension or in a constant load experiment by the applied load. ©1995 John Wiley & Sons, Inc.     

Deformations of the canonical commutation relations

Deformations of the canonical commutation relations which have the effect of altering the spectrum of a standard Hamiltonian, bilinear in creation and annihilation operators are described. The problem of going over from an eigenvalue situation, as is the case in the vast majority of papers in the literature, to a theory with time evolution is discussed, and a special example with deformation parameter an Nth root of unity is constructed which possesses a consistent time evolution. This work is an account of some recent studies of associative deformations of the Heisenberg algebra of several creation and annihilation algebras, with Jean Nuyts of the University of Mons, Hainaut, together with some observations of my own concerning the difficulty of implementing time evolution in a quantum group context. It builds on earlier work with Cosmas Zachos (Argonne National Laboratory, USA), which in turn is re;ated to work of Manin, and Wess, Zumino and collaborators. The main idea is that, if quantum groups have any role in physics, then they must manifest themselves at the level of the basic rules of quantisation.     

Adaptive remeshing for ductile fracture prediction in metal formingRemaillage adaptatif pour la prévision de la rupture ductile en mise en forme

The analysis of mechanical structures using the Finite Element Method in the framework of large elastoplastic strain, needs frequent remeshing of the deformed domain during computation. Indeed, the remeshing is due to the large geometrical distortion of finite elements and the adaptation to the physical behavior of the solution. This paper gives the necessary steps to remesh a mechanical structure during large elastoplastic deformations with damage. An important part of this process is constituted by geometrical and physical error estimates. The proposed method is integrated in a computational environment using the ABAQUS/Explicit solver and the BL2D-V2 adaptive mesher. To cite this article: H. Borouchaki et al., C. R. Mecanique 330 (2002) 709–716.