Extension of CR-functions into weighted wedges through families of nonsmooth analytic discs

The goal of this paper is to develop a theory of nonsmooth analytic discs attached to domains with Lipschitz boundary in real submanifolds of . We then apply this technique to establish a propagation principle for wedge extendibility of CR-functions on these domains along CR-curves and along boundaries of attached analytic discs. The technique from this paper has been also extensively used by the authors recently to obtain sharp results on wedge extension of CR-functions on wedges in prescribed directions extending results of BOGGESS-POLKING and EASTWOOD-GRAHAM.

     

Uniqueness theorem for CR-functions on generating CR-manifolds

Translated from Matematicheskie Zametki, Vol. 48, No. 6, pp. 47–50, December, 1990.     

Comultiplications on free groups and wedges of circles

By means of the fundamental group functor, a co-H-space structure or a co-H-group structure on a wedge of circles is seen to be equivalent to a comultiplication or a cogroup structure on a free group . We consider individual comultiplications on and their properties such as associativity, coloop structure, existence of inverses, etc. as well as the set of all comultiplications of . For a comultiplication of we define a subset of quasi-diagonal elements which is basic to our investigation of associativity. The subset can be determined algorithmically and contains the set of diagonal elements . We show that is a basis for the largest subgroup of on which is associative and that is a free factor of . We also give necessary and sufficient conditions for a comultiplication on to be a coloop in terms of the Fox derivatives of with respect to a basis of . In addition, we consider inverses of a comultiplication, the collection of cohomomorphisms between two free groups with comultiplication and the action of the group on the set of comultiplications of . We give many examples to illustrate these notions. We conclude by translating these results from comultiplications on free groups to co-H-space structures on wedges of circles.

     

Holomorphic extension of integrable CR-functions from part of the boundary of the domain

Translated from Matematicheskie Zametki, Vol. 48, No. 2, pp. 64–71, August, 1990.     

Nonuniform supersonic flow past wedges

Summary The problem of steady, inviscid, nonuniform, supersonic flow past pointed wedges with attached shock waves is considered. The solution is obtained by the one-strip approximation of the method of integral relations. The results agree well with experimental data.

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Résumé Le problème de l'écoulement supersonique non-uniforme non-visqueux stationnaire autour d'un dièdre est considéré dans le cas des ondes de choc attachées. La solution est obtenue par la première approximation de la méthode des relations intégrales. Le rapport entre les résultats théoriques et expérimentaux est bon.

     

Factor normal wedges of semi-ordered spaces

A concept of a factor normal cone in a linear topological space is introduced and basic properties of semiordered spaces possessing a positive factor normal cone are studied. The main aim of the paper is to investigate topologies of semiordered spaces whose dual positive cone in the conjugate space is factor normal.     

Diffraction of normal shock wave by yawed wedges

The diffraction of normal shock with yawed wedges of small angles have been considered in this paper. Vorticity distribution of the fluid particle over the diffracted shock has been determined for several Mach numbers of the shock wave. The Mach reflection effects have also been investigated when the bend is concave to the flow.     

Diffraction of normal shock wave by yawed wedges

The diffraction of normal shock with yawed wedges of small angles have been considered in this paper. Vorticity distribution of the fluid particle over the diffracted shock has been determined for several Mach numbers of the shock wave. The Mach reflection effects have also been investigated when the bend is concave to the flow.     

On the antiplane shear deformation of finite wedges

Antiplane shear deformation of finite wedges is considered under different boundary conditions. First, the assertions and results of a recent paper, namely Chue and Liu [C.H. Chue, W.J. Liu, Comments on “Analysis of an isotropic finite wedge under antiplane deformation”, Int. J. Solids Struct. 41 (2004) 5023–5034] are invalidated. Then, closed form solutions are extracted for the stress distribution in the wedge. These closed forms have the advantages of showing the possible geometric stress singularity as well as the load singularity explicitly, in addition to the continuity or discontinuity as well as the convergence of the results in the entire region. Finally, the stress intensity factors are extracted in the special case of a circular shaft containing an edge crack under different boundary conditions.     

Image of the Nash map in terms of wedges

M. Lejeune-Jalabert (Lecture Notes in Math., vol. 777, Springer-Verlag, 1980, pp. 303–336) proposed the following ‘problem of wedges’: let X be a surface over an algebraically closed field k of characteristic zero. Given a wedge $\text{φ}\text{:}\text{Spec}\text{k[[ξ,t]]→X}$, whose special arc lifts to the minimal resolution Y of X in an arc transversal to an irreducible component of the exceptional locus in a general point, does φ lift to Y? The main result in this Note is to extend this problem to a problem of wedges in a variety X of any dimension and to prove that, if the wedge problem is true for X, then the Nash problem is true for X. From this, necessary and sufficient conditions are given for an essential divisor to belong to the image of the Nash map, and we conclude that the Nash problem is true for sandwiched surface singularities. To cite this article: A.J. Reguera, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Reflected Brownian motion in generic triangles and wedges

Consider a generic triangle in the upper half of the complex plane with one side on the real line. This paper presents a tailored construction of a discrete random walk whose continuum limit is a Brownian motion in the triangle, reflected instantaneously on the left and right sides with constant reflection angles. Starting from the top of the triangle, it is evident from the construction that the reflected Brownian motion lands with the uniform distribution on the base. This raises some questions on the possible distributions of hulls generated by local processes.     

Limiting Amplitude principle in the scattering by wedges

We consider a nonstationary scattering of plane waves by a wedge. We prove that the Sommerfeld‐type integral, constructed in (Math. Meth. Appl. Sci. 2005; 28 :147–183; Proc. Int. Seminar ‘Day on Diffraction‐2003’, University of St. Petersburg, 2003; 151–162), is a classical smooth solution from a functional space, and prove the Limiting Amplitude principle. Copyright © 2006 John Wiley & Sons, Ltd.     

Inner coefficient of quasiconformality of a pair of dihedral wedges

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 6, pp. 12–16, November–December, 1988.