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An appropriate generalization ofCharnes-Cooper's [1962] variable transformation is introduced, by which a parameter-free convex program is associated to nonlinear fractional programs. The equivalent program also enables a direct approach toJagannathan's. [1973] duality theory simultaneously extending it. In particular for some special cases further duality theorems are derived.

The primary objective of this paper is to show that semicomplex structures may be employed to define a local fixed point index on the category whose objects are all finite unions of compact, convex subsets of locally convex topological vector spaces and whose morphisms are all continuous maps between its objects. This is established via the use of the lc∗ spaces introduced by Lefschetz to generalize absolute neighborhood retracts.

A theoretical study of the statistical mechanical description of systems composed of non-spherical convex molecules is made. Thermodynamic functions of one-component and multicomponent systems of particles interacting via the pair potential of the Kihara core type are expressed by integrals over the minimum distance between two interacting convex bodies and three angles characterizing the convex body geometry. The approach is applied to the hard convex body system where the averaged contact correlation function is introduced. Exploiting ideas of the scaled particle theory the approximate expressions for the averaged correlation functions are given in terms of the geometric functionals of hard convex bodies.

We examine linear controllable systems under the condition of incomplete information on the current phase state, when at each instant only certain functions of the phase coordinates are accessible to measurement. We assume that noise, for which there is no special description, enters into the controllable system and into the measurement (observation) channel. Only the exact deterministic constraints which this noise satisfies, are known. At each instant the optimal controls are synthesized from the information on the whole previous history of observation on the basis of a minimax criterion for a convex function of phase coordinate values at the termination instant of the process. Thus, in the paper we examine a specific information game problem of conflict control [1], The peculiarity of the problem manifests itself in the fact that here we are required to optimize as an aggregate both the synthesis process of the optimal control as well as the process of continuous estimation of the magnitudes of the current phase coordinates. The latter leads to the use of both the extremal constructions of the theory of differential-game problems of dynamics [1, 2] as well as the functional constructions for minimax problems of position observation and prediction [3, 4]. Fundamental attention is given to the constructive formation of the solution within the framework of convex analysis [5, 6]. The investigations in [7–9] were devoted to a different formulation of problems on the combination of control and observation.

The closed convex hull and extreme points are obtained for the functions which are convex, starlike, and close-to-convex and in addition are real on(−1,1)( - 1,1). We also obtain this result for the functions which are convex in the direction of the imaginary axis and real on(−1,1)( - 1,1). Integral representations are given for the hulls of these families in terms of probability measures on suitable sets. We also obtain such a representation for the functionsf(z)f(z)analytic in the unit disk, normalized and satisfyingRe⁡f′(z)>α\operatorname {Re} f’(z) > \alphaforα>1\alpha > 1. These results are used to solve extremal problems. For example, the upper bounds are determined for the coefficients of a function subordinate to some function satisfyingRe⁡f′(z)>α\operatorname {Re} f’(z) > \alpha.

A family {Ki¦ of convex domains in the Euclidean planeR2 is said to permit a plane covering if there exist rigid motions {τi} such that Ui∞=1τiKi =R2. Necessary and sufficient conditions that a given family of convex domains permits a plane covering are established.

O. Introduction Let (X, Y) be a locally convex linear topological space with topology Y and denote by C its cone of continuous seminorms. Then the two main results of this paper are a bipolar theorem for subsets of C and a Grothendieck-type representation of the completion of C with respect to some topologies of uniform convergence on subsets of X. The completeness theorem is based on an approximation lemma for seminorms corresponding to the one for linear functionals; we prove: If M is a convex circled subset of X and h a seminorm on X whose restriction to M is continuous then for every e > 0 there exists a continuous seminorm h~ on X such that [(h-h~) (m)l < e for all m e M. These results will be established in Sections 1-3 by sublinear techniques in the somewhat more general framework of sublinear functionals. The completeness theorem will provide some new insight in properties of locally convex spaces from the point of view of seminorms and will, consequently, turn out to be a very useful tool in considering special types of those spaces, both in characterizing and handling them. Moreover it will yield a unified approach to several problems in locally convex theory. It is the primary object of Sections 4-6 to carry out some work in this direction. In Section 4 some types of locally convex spaces are characterized by completeness properties of their cones of continuous seminorms. Section 5 is devoted to the problem of characterizing the finest locally convex topology q on X which agrees with the initial topology ~of X on every member Ak of an increasing sequence (Ak)k~ of convex circled subsets of X whose union is absorbing. This problem has been investigated in detail in the papers of Garling [5] and Roelcke [14]. As for this problem the above mentioned Grothendieck-type representation theorem will show that the cone C, of t/-continuous seminorms on X is nothing else but the completion of C with respect to the topology of uniform convergence on the sets Ak, k e N. This result is entirely analogous to the corresponding one concerning the dual space of X with respect to t/(see [14]).

The object of this paper is to extend some recent generalizations of Petrović’s inequality by Vasić and others. We shall also use our technique to obtain some results which have interesting applications in the theory of Fourier series as well as the theory of approximations.

In this paper FENCHEL'S duality theorem is proved for convex optimization problems in a real vector space and its algebraical dual space, respectively; it is not necessary for this vector space to be finite dimensional. Moreover, we give some results on polar cones and projection cones of convex sets.

The closed convex hull and extreme points of families of analytic functions, which are defined in terms of subordination, are determined. Integral representations are given for the hulls of these families in terms of probability measures on suitable sets. These results are used to solve extremal problems. The functions we consider are defined by subordination to certain classes of starlike and convex mappings.

A mathematical programming theory is developed for dual minimax problems of Fenchel type and a related Lagrangian saddle point problem. Questions of existence, characterization and duality are treated. The results extend the known, purely convex case. The proofs use the theory of conjugate saddle functions. (Author)

We prove a Marx conjecture for the closed convex hull of the family of functions which are starlike of order α \alpha and k-fold symmetric. We obtain precise results for the functions which are starlike, starlike of order 1 2 \tfrac {1}{2} , and starlike with 2-fold symmetry in their power series expansions.

For certain types of mathematical programming problems, a related dual problem can be constructed in which the objective value of the dual problem is equal to the objective function of the given problem. If these two problems do not have equal values, a duality gap is said to exist. No such gap exists for pairs of ordinary dual linear programming problems, but this is not the case for linear programming problems in which the nonnegativity conditionx ⩾ 0 is replaced by the condition thatx lies in a certain convex setK. Duffin (Ref. 1) has shown that, whenK is a cone and a certain interiority condition is fulfilled, there will be no duality gap. In this note, we show that no duality gap exists when the interiority condition is satisfied andK is an arbitrary closed convex set inRn.

Convex prestructures, a mathematical framework that extends the usual concept of convex subset of a real linear space, are employed to generalize methods used in the study of axiomatic quantum mechanics. A brief summary of the mathematical framework of convex prestructures is given. Convex prestructures are classified, and those which are isomorphic to a convex subset of a real linear space are characterized. The operational quantum mechanics of Davies and Lewis is generalized within the framework of convex prestructures and the existence of a physically motivated orthomodular poset is given. Mielnik's beams and filters are also discussed within the framework of convex prestructures. An error in Mielnik's formulation of beam mixtures is pointed out and it is shown that his beam mixtures and one classification of convex prestructures are equivalent. Also his concept of a filter is generalized in the framework of convex prestructures and geometric requirements needed on the set of normalized states so that they may correspond to a physical system are investigated. Finally, Mackey's axioms are discussed and reformulated in the language of P‐convex structures.

The complementarity problem is defined and studied for cases where the constraints involve convex cones, thus extending the real and complex complementarity problems. Special cases of the problem are equivalent to dual, linear or quadratic, programs over polyhedral cones.

For an exponential functiona(z) we consider the convolutiona* x in the function space H(G), consisting of functions analytic in convex domains G. We obtain conditions (close to necessary and sufficient) on G and G1 subject to which the equationa* (H(G1)) = H(G) is satisfied.

It is shown that if E[7] is a locally convex space ordered by a closed, generating positive cone K satisfying certain mild hypotheses relating K and 7, then E′ is lattice-ordered by the dual cone K′ whenever the identity linear transformation on E is the pointwise limit of sums of transformations x→〈x,x′〉z where z∈K and x′∈K′. The converse is true for certain classes of spaces, e.g., Frechet spaces.

The notion of an asymptotic center is used to prove a number of results concerning the existence of fixed points under certain selfmappings of a closed and bounded convex subset of a uniformly convex Banach space.

New distortion theorems are obtained for the class of functions p ( z ) = 1 + c n z n + ⋯ ( n ≥ 1 ) p(z) = 1 + {c_n}{z^n} + \cdots (n \geq 1) which are analytic and Re p ( z ) > α ( 0 ≤ α > 1 ) \text {Re} p(z) > \alpha (0 \leq \alpha > 1) in the unit disk | z | > 1 |z| > 1 . These are used to obtain new results regarding the partial sums of univalent convex functions.

Starting from works in Rockafellar (1967), a duality theorem, existence theorem, reciprocity principle and a max–min principle are obtained in a class of dynamic programming which has a possibility of wide economic applications. These theorems serve to summarize the theory of the efficient price vectors developed in activity analysis and to provide a linkage between that theory and the maximum principle in the optimal control theory.

We prove the solvability of convolution equations in some spaces of holomorphic functions of n variables. We clarify the structure of solutions of the homogeneous equation.

Kuhn-Tucker necessary and sufficient conditions for the nonlinear programming problem are applied to the project cost-duration analysis problem for project networks with convex costs. These conditions give an optimality curve for the problem. A solution is optimal if and only if when the values for activities are plotted on their optimality diagram, the values lie on the optimality curve. An algorithm is given here when the cost is convex and quadratic. The algorithm is also generalized to the case when the cost is convex and piecewise quadratic. The algorithm can be used to solve problems with convex cost functions by approximating them by piecewise quadratic functions.

Supported in part by the U.S. Army Research Office (Durham) under Contract No. DAHC04-73-C-0032.

This paper presents an algorithm for the global maximization of a convex function subject to linear inequality constraints. It is computationally finite and is designed to converge rapidly on problems in which there are few local optima or the global optimum is significantly better than most of the other local optima.

A stationary Poisson process of hyperplanes in Rn is characterized (up to an equivalence) by the function θ such that θ(s) is the density of the Poisson point process induced on the straight lines with direction s. The set of these functions θ is a convex cone ℛ1, a basis of which is a simplex Θ, and a given function θ belongs to ℛ1 if and only if it is the supporting function of a symmetrical compact convex set which is a finite Minkowski sum of line segments or the limit of such finite sums. Another application is given concerning the tangential cone at h = 0 of a coveriance function.

It is shown that a convex body in n-dimensional Euclidean space can be approximated by a sequence of smooth convex bodies in such a way that the principal radii of curvature converge in a certain sense. This fact is used to characterize those first surface measures of convex bodies which belong to polytopes. Furthermore it is proved that the support function of a convex body whose first surface measure has bounded density must have continuous first partial derivatives.

In [2] those locally convex spaces E, called GN-spaces, were investigated, for which every closed linear mapping from E to any normed space F is continuous. Here we study the smaller class of spaces E, called GM-spaces, which arise by admitting now for F all metrizable locally convex spaces. The GM-spaces have characterizations and permanence properties similar to those for GN-spaces. Main results are the barrelledness of every dense subspace of a GM-space, the finite dimension of the bounded subsets of separated GM-spaces, an embedding theorem., and the existence of separated GM-spaces which do not have the finest locally convex topology.

Let Z n denote the set of all ordered n -tuples of integers. Let us call any finite subset of Z n a body in Z n , and any finite set of bodies in Z n a family in Z n . Consider the following problem: Give a decision procedure which for any family ℱ in Z n decides the following.