On a Generalized Class of Minimax Inequalities

Some results on a generalized class of minimax inequalities based on the r–I–G–H-KKM mapping theorems in a G–H-space setting are presented. The r–I–G–H-KKM mappings represent a new class of KKM mappings in G–H-spaces as well as in the interval spaces.     

Radial Index and Euler Obstruction of a 1-Form on a Singular Variety

A notion of the radial index of an isolated singular point of a 1-form on a singular (real or complex) variety is discussed. For the differential of a function it is related to the Euler characteristic of the Milnor fibre of the function. A connection between the radial index and the local Euler obstruction of a 1-form is described. This gives an expression for the local Euler obstruction of the differential of a function in terms of Euler characteristics of some Milnor fibres.Mathematics Subject Classifications (2000). 14B05, 32S60, 58A10*Partially supported by the DFG-programme “Global methods in complex geometry” (Eb 102/4–2), grants RFBR–04–01–00762, NSh–1972.2003.1     

Solution of a problem of Ulam

In this paper we solve the following Ulam problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist” and establish results involving a product of powers of norms [[5.]; [5.]; [5.]]. There has been much activity on a similar “ -isometry” problem of Ulam [ [1.], 633–636; [2.], 263–277; [4.]]. This work represents an improvement and generalization of the work of [3.], 222–224].     

On a certain algebra on a set of graphs

Translated fromSibirskii Matematicheskii Zhurnal, Vol. 35, No. 4, pp. 793–800, July–August, 1994.     

Mass-spectrometric investigation of solid lubricating coatings subjected to friction in a deep vacuum

The life of solid lubricating coatings of the VNII NP type, based on molybdenum disulfides and various binders, has been experimentally investigated under deep vacuum conditions (10^–8–5 · 10^–9 torr) together with the composition of the gas released in the friction process. It is shown that both under atmospheric conditions and in a deep vacuum the life of the coatings depends on the chemical nature of the film-former. The depth of the vacuum also has an important influence on the life of the coatings, both the mechanism and the end result of this effect depending to a large extent on the physicochemical properties of the bind. On the interval 10^–1–10^–2 torr there is a sudden change in the life of the coating.Physicotechnical Institute of Low Temperatures, Academy of Sciences of the Ukrainian SSR, Khar'kov. Translated from Mekhanika Polimerov, No. 6, pp. 1070–1075, November–December, 1970.     

Subrings of invariants of a finite group of automorphisms of a Jordan ring

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 1, pp. 207–211, January–February, 1991.     

Mathematical description of a charged-particle system near a porous membrane

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 40, No. 1, pp. 114–117, January–February, 1988.     

Singular decomposition of a differential operator on a semiaxis

Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 34, No. 6, pp. 34–48, November–December, 1993.     

Existence of a normal complement of a group in the multiplicative group of its group ring

Suppose G is the adjoint group of a locally nilpotent algebra over a ring K of characteristic 0 without zero-divisors. If the orders of the elements of G are not invertible in K, then the normalized multiplicative group of the group ring KG is a semidirect product of the torsion-free congruence subgroup and G.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7–8, pp. 884–889, July–August, 1991.     

Question of well-posedness of a certain inverse problem for a hyperbolic integrodifferential equation

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 33, No. 3, pp. 69–77, May–June, 1992.     

Interaction of a granular flow with a rectangular obstacle

In this work we studied the cluster formation of particles flowing between two parallel plates hitting a rectangular obstacle. After a short time a cluster is formed on top of the obstacle. Quantification of the cluster formation, shows that the number of clusters decrease with time, whereas the mean and maximum cluster size grow during the simulation. Our results indicate that for the development of clusters the value of the restitution coefficient for particle–particle collisions is most important parameter for this formation process. We found that changes in the value of the restitution parameters for particle–obstacle and particle–wall did not show significant changes in the cluster development.     

Uniqueness of a solution to the problem of evolution of a point vortex

Translated from Sibirskii Matematicheskii, Vol. 35, No. 3, pp. 696–701, May–June, 1994.     

Mechanical properties of a composite with a carbonized matrix

The effect of reinforcement schemes that differ in the magnitude of the disorientation angle of neighboring filler layers on the mechanical properties of a "carbon-carbon" composite under extension, compression, or bending is shown.Translated from Mekhanika Polimerov, No. 2, pp. 235–240, March–April, 1976.     

On a class of elliptic operators with nonpolynomial degeneracy on a boundary

Rostov-on-Don. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 5, pp. 91–99, September–October, 1992.     

On a Commutative Class of Search Directions for Linear Programming over Symmetric Cones

The commutative class of search directions for semidefinite programming was first proposed by Monteiro and Zhang (Ref. 1). In this paper, we investigate the corresponding class of search directions for linear programming over symmetric cones, which is a class of convex optimization problems including linear programming, second-order cone programming, and semidefinite programming as special cases. Complexity results are established for short-step, semilong-step, and long-step algorithms. Then, we propose a subclass of the commutative class for which we can prove polynomial complexities of the interior-point method using semilong steps and long steps. This subclass still contains the Nesterov–Todd direction and the Helmberg–Rendl–Vanderbei–Wolkowicz/Kojima–Shindoh–Hara/Monteiro direction. An explicit formula to calculate any member of the class is also given.     

On a certain boundary value problem of N. P. Vekua with a piecewise-smooth shift on a piecewise Lyapunov contour

Odessa. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 33, No. 2, pp. 108–115, March–April, 1992.     

Asymptotic approximation of a logarithmic potential

Ufa. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 5, pp. 142–154, September–October, 1991.     

Forced vibrations of a shell immersed in a viscous compressible fluid

Institute for Mechanics Problems, Academy of Sciences of the USSR. Moscow Physico-Technological Institute. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 4, pp. 93–95, October–December, 1991.     

Hunting for a Smaller Convex Subdifferential

Certain useful basic results of the gradient (in the smooth case), the Clarkesubdifferential, the Michel–Penot subdifferential, which is also known asthe "small" subdifferential, and the directional derivative(in the nonsmooth case) are stated and discussed. One of the advantages ofthe Michel–Penot subdifferential is the fact that it is in general "smaller"than the Clarke subdifferential. In this paper it is shown that there existsubdifferentials which may be smaller than the Michel–Penot subdifferentialandwhich have certain useful calculus. It isfurther shown that in the case of quasidifferentiability, the Michel–Penotsubdifferential enjoys calculus whichhold for the Clarke subdifferential only in the regular case.     

Bounds for the Betti numbers of a level surface of a harmonic polynomial

Institute of Problems for Information Transmission, RAN. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 26, No. 4, pp. 86–88, October–December, 1992.