duality in generalized nonlinear fractional programming abstract for generalized nonlinear fractional programming two duals are introduced under four sets of assumptions thus four theorems of strong duality are established all the theorems are proved with the help of alternative theorems in the convex case though the numerators and denominators of the ratios in the objective functions need not necessarily be either convex or concave

optimization theory for n set functions abstract an optimization theory is developed for functions of n sets optimality conditions are established and a lagrangian duality is obtained

saddle points and lagrangian type duality for discrete minmax fractional subset programming problems with generalized convex functions abstract in this paper we present four sets of saddle point type optimality conditions construct two lagrangian type dual problems and prove weak and strong duality theorems for a discrete minmax fractional subset programming problem we establish these optimality and duality results under appropriate b convexity hypotheses

the rings r x and r x abstract the rings r x and r x are investigated of particular importance is the question of when r x or r x is an arithmetical ring a prufer ring a hilbert ring or a euclidean ring answers are sought in terms of the base ring r

optimality and duality for a multi objective programming problem involving generalized d type i and related n set functions in this paper we introduce several generalized convexity for a real valued set function and establish optimality and duality results for a multi objective programming problem involving generalized d type i and related n set functions

assignable polynomials to linear systems over von neumann regular rings abstract given a system a b over a commutative von neumann regular ring r it is proved that there exist a matrix f and a vector u such that the single input system a b f b u and the original system a b have the same module of reachable states and the same set of polynomials assignable by state feedback moreover there is a bijection between reachable states and assignable polynomials in the form of a certain isomorphism of r modules and the existence of this isomorphism for all systems actually characterizes von neumann regular rings finally the set of assignable polynomials to a single input system is completely described for arbitrary commutative rings which in the case of von neumann regular rings completes the study of assignable polynomials to multi input systems

on the optimality conditions of vector valued n set functions abstract in a finite atomless measure space x the optimization problem of vector valued n set functions defined on a convex subfamily s of n is investigated the necessary and sufficient conditions of pareto optimal solution or proper r p solution of optimization problem with differentiable vector valued n set functions are given

on multiple objective programming problems with set functions abstract the convexity of a subset of a algebra and the convexity of a set function on a convex subset are defined related properties are also examined a farkas minkowski theorem for set functions is then proved these results are used to characterize properly efficient solutions for multiple objective programming problems with set functions by associated scalar problems

trace properties in rings with zero divisors an integral domain d is said to have the radical trace property if i d i is a radical ideal for each noninvertible nonzero ideal i of d for a commutative ring r with nonzero identity the dual of a dense ideal i is the set r i t q r ti r where q r is the complete ring of quotients of r in this article the radical trace property is extended to rings with nonzero zero divisors specifically r has the radical trace property for regular ideals if i r i is a radical ideal for each noninvertible regular ideal i and r has the radical trace property for semiregular ideals if the same conclusion holds for noninvertible semiregular ideals so to those ideals that contain finitely generated dense ideals alternately r is said to be a rtp ring in the regular case and a q0 rtp in the semiregular case if r is an rtp ring and s is an overring of r so between r and the total quotient ring t r that is r flat then s is an rtp ring a similar statement holds for q0 rtp rings in the case s is r flat and between r and the ring of finite fractions q0 r

the complete integral closure of r x for a reduced ring r that is completely integrally closed it is not always the case that the corresponding polynomial ring r x is completely integrally closed in this paper the question of when r x is completely integrally closed is shown to be related to the question of when r is completely integrally closed in t r x the total quotient ring of r x a characterization of the complete integral closure of r x is given in the main theorem and this result is used to characterize the complete integral closure of the semigroup ring r s when s is a torsion free cancellative monoid

optimality and duality for multiobjective fractional programming involvingn set functions abstract we consider a multiobjective fractional programming problem mfp involving vector valued objective n set functions in which their numerators are different from each other but their denominators are the same by using the concept of proper efficiency we establish optimality conditions and duality relations for our problem mfp under convexity assumptions on objective and constrained functions

biholomorphic maps between teichmuller spaces in this paper we study biholomorphic maps between teichmuller spaces and the induced linear isometries between the corresponding tangent spaces the first main result in this paper is the following classification theorem if m and n are two riemann surfaces that are not of exceptional type and if there exists a biholomorphic map between the corresponding teichmuller spaces teich m and teich n then m and n are quasiconformally related also every such biholomorphic map is geometric in particular we have that every automorphism of the teichmuller space teich m must be geometric this result generalizes the previously known results see 2 5 7 and enables us to prove the well known conjecture that states that the group of automorphisms of teich m is isomorphic to the mapping class group of m whenever the surface m is not of exceptional type in order to prove the above results we develop a method for studying linear isometries between l 1 type spaces our focus is on studying linear isometries between banach spaces of integrable holomorphic quadratic differentials which are supported on riemann surfaces our main result in this direction theorem 1 1 states that if m and n are riemann surfaces of nonexceptional type then every linear isometry between a 1 m and a 1 n is geometric that is every such isometry is induced by a conformal map between m and n

characterizing when the rings r and r t are integrally closed via linear equations over r let r be a commutative ring with an identity element this paper is w x devoted to study when r and the polynomial ring r t are integrally closed in terms of the systems of linear equations over r denote by t r the total quotient ring of r i e the ring of fractions of r with respect to the multiplicative set of all nonzero divisors of r the ring r is integrally closed if every x g t r which is integral over r belongs to r if r is an integrally closed domain then so is the polynomial w x ring an one indeterminate r t on the other hand if r is an integrally w x closed ring with nonzero nilradical then r t is never integrally closed w x the element art is integral over but does not belong to r t for every nonzero nilpotent a in r the question is when is the polynomial ring w x r t integrally closed w x necessary conditions for r t to be integrally closed are that r is reduced and integrally closed however these conditions are not sufficient w x w x see 6 p 187 t akiba 1 proves that if r is a reduced ring such that for each maximal ideal m of r the local ring r is an integrally closed m w x domain then r t is integrally closed in the same paper it is proven that w x if r is a reduced integrally closed with min r compact then r t is integrally closed if and only if t r is a von neumann regular ring t lucas gave necessary and sufficient conditions in order that the w x w x w x polynomial ring r t be integrally closed in 7 it is shown that r t is w x w x integrally closed if and only if r is integrally closed in t r t in 8 it is

sufficiency criteria and duality for nonlinear programs involving n set functions abstract several sufficiency criteria and duality results are established under generalized convexity conditions for a class of nonlinear programming problems involving differentiable n set functions

amenability poincare series and quasiconformal maps any coveringy x of a hyperbolic riemann surfacex of finite area determines an inclusion of teichmuller spaces teich x teich y we show this map is an isometry for the teichmuller metric if the covering isamenable and contracting otherwise in particular we establish 1 for classical poincare series kra s theta conjecture

duality for minmaxb vex programming involvingn set functions abstract a minmax programming problem involving several b vex n set functions is considered necessary and sufficient optimality theorems and wolfe type duality results are established duality for the n set generalized minmax fractional programming problem is derived as a special case of the main problem

duality in mathematical programming of set functions on fenchel duality theorem abstract the optimization problem of a set function defined on a family a of measurable subsets in an atomless finite measure space x a m is investigated the generalized fenchel theorem is formulated and proved in this note

duality for multiobjective fractional programming involving n set functions abstract in the present paper we consider a class of multiobjective fractional programming problems involving differentiable n set functions and establish duality results in terms of properly efficient solutions further we relate the problem to a certain saddle point of a lagrangian

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

efficiency conditions and duality models for multiobjective fractional subset programming problems with generalized ϝ α ϱ θ v convex functions abstract in this paper we first introduce a new class of generalized convex n set functions called v convex functions and then present numerous sets of parametric and semiparametric sufficient efficiency conditions under generalized v convexity assumptions for a multiobjective fractional subset programming problem moreover we construct three parametric and three semiparametric duality models and prove appropriate duality theorems

parameter free sufficient optimality conditions and duality models for minmax fractional subset programming problems with generalized f ϱ θ convex functions abstract in this paper we present numerous sets of nonparametric sufficient optimality conditions and duality results under various generalized f convexity assumptions for a minmax fractional subset programming problem

on minimax fractional programming of generalized convex set functions abstract we derive necessary and sufficient optimality conditions for the discrete minimax programming problem with subdifferentiable i convex set functions then we apply these optimality criteria to construct two models of parameter free dual problems and a third model of another dual problem we also establish weak strong and strictly converse duality theorems

dynamics on teichmuller spaces and self covering of riemann surfaces a non injective holomorphic self cover of a riemann surface induces a non surjective holomorphic self embedding of its teichmuller space we investigate the dynamics of such self embeddings by applying our structure theorem of self covering of riemann surfaces and examine the distribution of its isometric vectors on the tangent bundle over the teichmuller space we also extend our observation to quasiregular self covers of riemann surfaces and give an answer to a certain problem on quasiconformal equivalence to a holomorphic self cover

geometric isomorphisms between infinite dimensional teichmuller spaces let x and y be the interiors of bordered riemann surfaces with finitely generated fundamental groups and nonempty borders we prove that every holomorphic isomorphism of the teichmiiller space of x onto the teichmiiller space of y is induced by a quasiconformal homeomorphism of x onto y these teichmiiller spaces are not finite dimensional and their groups of holomorphic automorphisms do not act properly discontinuously so the proof presents difficulties not present in the classical case to overcome them we study weak continuity properties of isometries of the tangent spaces to teichmiiller space and special properties of teichmiiller disks

duality for multiobjectiveb vex programming involvingn set functions abstract in the present paper we consider a class of multiobjectiveb vex programming problems involving differentiableb vexn set functions and establish duality results in terms of properly efficient solutions further we relate the problem to a certain saddle point of a lagrangian and show multiobjective fractional program as a special case of the main problem

linear equations over commutative rings and determinantal ideals in this paper we give a characterization of the class of rings which have the property that the existence of a solution of any system of linear equations is equivalent to the equality of the determinantal ideals of the coefficient matrix and the augmented matrix of the system when one only considers domains this characterization is equivalent to that given in s however the techniques used by the authors of this quoted paper are different from those we use below our main technique is the fact that the existence of a solution for a system of equations becomes a local property

a criterion for exactness of a complex of length two throughout this paper r denotes a commutative ring with an identity element a aij a m n matrix with entries in r and a r r the homomorphism of r modules defined by a x a x the rank of a rankr a is the largest nonnegative integer s such that is a 0 where is a is the ideal of r generated by all the s s minors of a we put i a is a where s rankr a this paper is motivated by the question what makes the complex

efficiency and duality for nonlinear multiobjective programs involving n set functions abstract the concept of efficiency is used to formulate duality for multiobjective nonlinear programs involving n set functions these results are obtained for convex functions convex functions and pseudoconvex functions

a proof of thurston s topological characterization of rational functions the criterion proved in this paper was stated by thurston in november 1982 thurston lectured on its proof on several occasions notably at the nsf summer conference in duluth 1983 where one of the authors jhh was present using the notes of various attendants at these lectures we have reconstructed a proof that we have made as precise as we could since this required a certain amount of work on our part we thought it might be of some use to present this proof to the reader we thank dennis sullivan for useful conversations and fritz yon haesseler and especially ben wittner for help with the writing and valuable suggestions after the first version was written clifford earle pointed out that better estimates than what we had were to be found in b

on the optimality of differentiable nonconvex n set functions abstract our main contribution is the extension of the concepts of quasiconvexity and pseudoconvexity to n set functions some properties of differentiable nonconvex n set functions are established necessary and sufficient conditions for the existence of an optimal solution of the nonconvex program with n set functions are characterized by derivatives of the n set functions involved a duality theorem for the nonconvex program with n set functions is also developed in this paper

amenable coverings of complex manifolds and holomorphic probability measures let y x be a covering map between complex manifolds x and y for many holomorphic objects such as functions or forms one can define a pushforward operator carrying objects on y to objects on x by summing over the fiber the pushforward is efficient if there is little cancellation in the sum to achieve efficiency there must be coherence in phase on different sheets of the covering in this paper we will demonstrate a direct connection between efficiency of the push forward and amenability of the covering the latter is a purely combinatorial property of the pair of groups 1 y 1 x the covering is amenable if there exists a 1 x invariant finitely additive probability measure on the coset space 1 x 1 y amenability is discussed in more detail in 3 the connection will be established by studying complex valued measures on the fibers of the covering varying holomorphically with respect to the base