On a Primal-Dual Analytic Center Cutting Plane Method for Variational Inequalities

We present an algorithm for variational inequalities VI( , Y) that uses a primal-dual version of the Analytic Center Cutting Plane Method. The point-to-set mapping is assumed to be monotone, or pseudomonotone. Each computation of a new analytic center requires at most four Newton iterations, in theory, and in practice one or sometimes two. Linear equalities that may be included in the definition of the set Y are taken explicitly into account.We report numerical experiments on several well—known variational inequality problems as well as on one where the functional results from the solution of large subproblems. The method is robust and competitive with algorithms which use the same information as this one.     

A weakened asymptotic monogeny condition

Functions of one complex variable that satisfy a weakened asymptotic monogeny condition are proved to be holomorphic. Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 902–911, December, 1996. The author thanks Proffessor Dolzhenko for discussing the results of the paper. The research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00236.     

A Derivative-Coderivative Inclusion in Second-Order Nonsmooth Analysis

For twice smooth functions, the symmetry of the matrix of second partial derivatives is automatic and can be seen as the symmetry of the Jacobian matrix of the gradient mapping. For nonsmooth functions, possibly even extended-real-valued, the gradient mapping can be replaced by a subgradient mapping, and generalized second derivative objects can then be introduced through graphical differentiation of this mapping, but the question of what analog of symmetry might persist has remained open. An answer is provided here in terms of a derivative-coderivative inclusion.     

Null-quadrics of codimension 4 in ℂ7

In this paper we study the automorphisms of nondegenerate quadrics of type (4, 3). In particular, we classify the so-called null-quadrics (we prove that there are exactly two null-quadrics up to equivalence), and find their automorphism groups. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 657–665, November, 1997. Translated by S. S. Anisov     

Fréchet Spaces of Holomorphic Functions without Copies of l1

Let X be a Banach space. Let H_w*(X*) the Fréchet space whose elements are the holomorphic functions defined on X* whose restrictions to each multiple mB(X*), m = 1,2, …, of the closed unit ball B(X*) of X* are continuous for the weak-star topology. A fundamental system of norms for this space is the supremum of the absolute value of each element of H_w*(X*) in mB(X*), m = 1,2,…. In this paper we construct the bidual of l₁ when this space contains no copy of l¹. We also show that if X is an Asplund space, then H_w*(X*) can be represented as the projective limit of a sequence of Banach spaces that are Asplund.     

Convergence Groups, Hausdorff Dimension, and a Theorem of Sullivan and Tukia

We show that a discrete, quasiconformal group preserving ⁿ has the property that its exponent of convergence and the Hausdorff dimension of its limit set detect the existence of a non-empty regular set on the sphere at infinity to ⁿ . This generalizes a result due separately to Sullivan and Tukia, in which it is further assumed that the group act isometrically on ⁿ , i.e. is a Kleinian group. From this generalization we are able to extract geometric information about infinite-index subgroups within certain of these groups.     

On the Existence of Solutions of Quasivariational Inclusion Problems

Quasivariational inclusion problems are formulated and sufficient conditions on the existence of solutions are shown. As special cases, we obtain several results on the existence of solutions of general vector ideal, proper, Pareto, weak quasioptimization problems, quasivariational inequalities, and vector quasiequilibrium problems. Further, we prove theorems on the existence for solutions of systems of these inclusions. As a corollary, we obtain an ideal minimax theorem concerning vector functions.     

New classes of domains with explicit Bergman kernel

We introduce two classes of egg type domains, built on general bounded sym-metric domains, for which we obtain the Bergman kernel in explicit formulas. As an aux-iliary tool, we compute the integral of complex powers of the generic norm on a boundedsymmetric domains using the well-known integral of Selberg. This generalizes matrix in-tegrals of Hua and leads to a special polynomial with integer or half-integer coefficientsattached to each irreducible bounded symmetric domain.     

The identity is isolated among composition operators

Let be the Banach algebra of bounded holomorphic functions on the open unit ball of a Banach space. We show that the identity operator is an isolated point in the space of composition operators on . This answers a conjecture of Aron, Galindo and Lindström.