The monopole equations and -holomorphic curves on weakly convex almost Kähler 4-manifolds

We prove that a weakly convex almost Kähler 4-manifold contains a compact, non-constant -holomorphic curve if the corresponding monopole invariant is not zero and if the corresponding line bundle is non-trivial.

     

A theta function identity and its implications

In this paper we prove a general theta function identity with four parameters by employing the complex variable theory of elliptic functions. This identity plays a central role for the cubic theta function identities. We use this identity to re-derive some important identities of Hirschhorn, Garvan and Borwein about cubic theta functions. We also prove some other cubic theta function identities. A new representation for is given. The proofs are self-contained and elementary.

     

Parametric Borwein-Preiss variational principle and applications

A parametric version of the Borwein-Preiss smooth variational principle is presented, which states that under suitable assumptions on a given convex function depending on a parameter, the minimum point of a smooth convex perturbation of it depends continuously on the parameter. Some applications are given: existence of a Nash equilibrium and a solution of a variational inequality for a system of partially convex functions, perturbed by arbitrarily small smooth convex perturbations when one of the functions has a non-compact domain; a parametric version of the Kuhn-Tucker theorem which contains a parametric smooth variational principle with constraints; existence of a continuous selection of a subdifferential mapping depending on a parameter.

The tool for proving this parametric smooth variational principle is a useful lemma about continuous -minimizers of quasi-convex functions depending on a parameter, which has independent interest since it allows direct proofs of Ky Fan's minimax inequality, minimax equalities for quasi-convex functions, Sion's minimax theorem, etc.

     

Maximal monotonicity, conjugation and the duality product

Recently, the authors studied the connection between each maximal monotone operator and a family of convex functions. Each member of this family characterizes the operator and satisfies two particular inequalities.

The aim of this paper is to establish the converse of the latter fact. Namely, that every convex function satisfying those two particular inequalities is associated to a unique maximal monotone operator.

     

On the continuity of biconjugate convex functions

We show that a Banach space is a Grothendieck space if and only if every continuous convex function on has a continuous biconjugate function on , thus also answering a question raised by S. Simons. Related characterizations and examples are given.

     

Holomorphic approximation on compact, holomorphically convex, real-analytic varieties

Every continuous function on a compact, holomorphically convex, real-analytic subset of can be approximated uniformly by functions holomorphic on the set.

     

Universal spaces for almost -dimensionality

We find universal functions for the class of lower semi-continuous (LSC) functions with at most -dimensional domain. In an earlier paper we proved that a space is almost -dimensional if and only if it is homeomorphic to the graph of an LSC function with an at most -dimensional domain. We conclude that the class of almost -dimensional spaces contains universal elements (that are topologically complete). These universal spaces can be thought of as higher-dimensional analogues of complete Erdos space.

     

Tame combing and almost convexity conditions

We give the first examples of groups which admit a tame combing with linear radial tameness function with respect to any choice of finite presentation, but which are not minimally almost convex on a standard generating set. Namely, we explicitly construct such combings for Thompson??s group F and the Baumslag?CSolitar groups BS(1, p) with p ??? 3. In order to make this construction for Thompson??s group F, we significantly expand the understanding of the Cayley complex of this group with respect to the standard finite presentation. In particular we describe a quasigeodesic set of normal forms and combinatorially classify the arrangements of 2-cells adjacent to edges that do not lie on normal form paths.     

Weak compactness is equivalent to the fixed point property in

A nonempty, closed, bounded, convex subset of has the fixed point property if and only if it is weakly compact.

     

A Minkowski-style theorem for focal functions of compact convex reflectors

This paper continues the study of a class of compact convex hypersurfaces in which are boundaries of compact convex bodies obtained by taking the intersection of (solid) confocal paraboloids of revolution. Such hypersurfaces are called reflectors. In reflectors arise naturally in geometrical optics and are used in design of light reflectors and reflector antennas. They are also important in rendering problems in computer graphics.

The notion of a focal function for reflectors plays a central role similar to that of the Minkowski support function for convex bodies. In this paper the basic question of when a given function is a focal function of a convex reflector is answered by establishing necessary and sufficient conditions. In addition, some smoothness properties of reflectors and of the associated directrix hypersurfaces are also etablished.

     

On the shellability of the order complex of the subgroup lattice of a finite group

We show that the order complex of the subgroup lattice of a finite group is nonpure shellable if and only if is solvable. A by-product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups.

     

Characterization of the Hilbert ball by its automorphism group

Let be a bounded, convex domain in a separable Hilbert space. The authors prove a version of the theorem of Bun Wong, which asserts that if such a domain admits an automorphism orbit accumulating at a strongly pseudoconvex boundary point, then it is biholomorphic to the ball. Key ingredients in the proof are a new localization argument using holomorphic peaking functions and the use of new ``normal families' arguments in the construction of the limit biholomorphism.

     

Weakly abelian lattice-ordered groups

Every nilpotent lattice-ordered group is weakly Abelian; i.e., satisfies the identity . In 1984, V. M. Kopytov asked if every weakly Abelian lattice-ordered group belongs to the variety generated by all nilpotent lattice-ordered groups [The Black Swamp Problem Book, Question 40]. In the past 15 years, all attempts have centred on finding counterexamples. We show that two constructions of weakly Abelian lattice-ordered groups fail to be counterexamples. They include all preiously considered potential counterexamples and also many weakly Abelian ordered free groups on finitely many generators. If every weakly Abelian ordered free group on finitely many generators belongs to the variety generated by all nilpotent lattice-ordered groups, then every weakly Abelian lattice-ordered group belongs to this variety. This paper therefore redresses the balance and suggests that Kopytov's problem is even more intriguing.

     

Jack polynomials and some identities for partitions

We prove an identity about partitions involving new combinatorial coefficients. The proof given is using a generating function. As an application we obtain the explicit expression of two shifted symmetric functions, related with Jack polynomials. These quantities are the moments of the ``-content' random variable with respect to some transition probability distributions.

     

The complex Hessian equation with infinite Dirichlet boundary value

The existence and nonexistence of the -subharmonic solutions for the complex Hessian equations with infinite Dirichlet boundary value are proved in the certain bounded domain in . We calculate the k-Hessian of the radially symmetric function and use radial functions to construct various barrier functions in this paper. Moreover, it is shown that the growth rate conditions are nearly optimal.

     

Generalized subdifferential of the distance function

We derive the proximal normal formula for almost proximinal sets in a smooth and locally uniformly convex Banach space. Our technique leads us to show the generic Fréchet smoothness of the distance function in the case the norm is Fréchet smooth, and we derive a necessary and sufficient condition for the convexity of a Chebyshev set in a Banach space with norms on and locally uniformly convex.

     

Group algebras with units satisfying a group identity II

We classify group algebras of torsion groups over a field of characteristic with units satisfying a group identity.

     

Semidirect sum of groups in which endomorphisms are generated by inner automorphisms

An I-E group is a group in which every endomorphism is finitely generated by its inner automorphisms. In this paper a characterization for a semidirect sum of I-E groups to be an I-E group is obtained and some well-known results are generalized. We then use this characterization to prove that a semidirect sum of finite I-E groups will again be an I-E group if the normal semidirect summand is unique and fully invariant. Conditions for a group to be an I-E group are also given.

     

Commutators for the maximal and sharp functions

The class of functions for which the commutator with the Hardy-Littlewood maximal function or the maximal sharp function are bounded on are characterized and proved to be the same.

     

Continuous selections and reflexive Banach spaces

Every l.s.c. mapping from a space into the non-empty closed convex subsets of a reflexive Banach space admits a continuous selection provided it satisfies a ``weak' u.s.c. condition. This result partially generalizes some known selection theorems. Also, it is successful in solving a problem concerning the set of proper lower semi-continuous convex functions on a reflexive Banach space.