Symmetric dual quadratic program in complex space

Symmetric dual quadratic program in complex space is presented and some duality theorems are proved. Self-dual linear and quadratic programs in complex space are formed and self-duality theorem is extended to these cases.     

The dual of a fractional linear program

A linear fractional program is shown, under certain restrictions, to have a fractional linear program as its dual.     

A non-linear interpolation formula

In 1915, E. T. Whittaker showed that an L² function, whose Fourier tranform has compact support, can be interpolated on an arithmetic progression: the well-known Cardinal series, or Sampling Representation. In the present paper, a class of non-linear interpolation formulae like the sampling representation are derived, in which the points of interpolation consist, in part, of the eigenvalues of certain differential operators acting on a Riemann surface of genus, g > 1.     

On solving a primal geometric program by partial dual optimization

The several published methods for mapping a dual solution estimate to a primal solution estimate in posynomial geometric programming provide no criteria for deciding how much deviation from primal feasibility, or discrepancy between the primal and dual objective function values, should be permitted before the primal solution estimate is accepted by the designer. This paper presents a new and simple dual-to-primal conversion method that uses the cost coefficients to provide a sound economic criterion for determining when to accept a primal solution estimate. The primal solution estimate generated is the exact solution to a modified primal obtained from the given primal by modifying the cost coefficients, with the exponent matrix left unchanged. The method is shown to have desirable properties when coupled with a convergent dual algorithm.     

A coupled non-linear hyperbolic-sobolev system

Summary A boundary-initial value problem for a quasilinear hyperbolic system in one space variable is coupled to a boundary-initial value problem for a quasilinear equation of Sobolev type in two space variables of the form Mu_t(x, t)+L(t) u (x, t)=f(x, t, u(x, t)) where M and L(t) are second order elliptic spacial operators. The coupling occurs through one of the boundary conditions for the hyperbolic system and the source term in the equation of Sobolev type. Such a coupling can arise in the consideration of oil flowing in a fissured medium and out of that medium via a pipe. Barenblatt, Zheltov, and Kochina[2] have modeled flow in a fissured medium via a special case of the above equation. A local existence and uniqueness theorem is demonstrated. The proof involves the method of characteristics, some applications of results of R. Showalter and the contraction mapping theorem. Entrata in Redazione il 28 luglio 1976.     

A non-linear circle-preserving subdivision scheme

We describe a new method for constructing a sequence of refined polygons, which starts with a sequence of points and associated normals. The newly generated points are sampled from circles which approximate adjacent points and the corresponding normals. By iterating the refinement procedure, we get a limit curve interpolating the data. We show that the limit curve is , and that it reproduces circles. The method is invariant with respect to group of Euclidean similarities (including rigid transformations and scaling). We also discuss an experimental setup for a construction and various possible extensions of the method.      

A comparison of non-linear equation solvers

A comparative study of 10 FORTRAN and ALGOL programs for solving non-linear equations with one unknown, without using derivatives, was made. This paper gives the results and conclusions of the study.     

A dual decomposition algorithm

Suppose that a large-scale block-diagonal linear programming problem has been solved by the Dantzig—Wolfe decomposition algorithm and that an optimal solution has been attained. Suppose further that it is desired to perform a post-optimality analysis or a complete parametric analysis on the cost-coefficients or the RHS of the linking constraints. Efficient techniques for performing these analyses for the ordinary simplex case have not been easily applied to this case as one operation involves doing a minimizing ratio between all columns of two rows of the tableau. As the columns are not readily known in Dantzig—Wolfe decomposition, other techniques must be used. To date, suggested methods involve solving small linear programs to find these minimizing ratios. In this paper a method is presented which requires solving no linear programs (except possibly in the case of degeneracy of a subproblem) using and utilizing only the information typically stored for Dantzig—Wolfe decomposition.     

A computational comparison of some non-linear programs

This article deals mainly with a comparison of certain computational techniques used for the solution of non-linear constrained mathematical programming problems.     

A non-linear version of the amir-lindenstrauss method

A non-linear version of the Amir-Lindenstrauss method of projections for weakly compactly generated Banach spaces is proved, that implies immediately the Benyamini-Rudin-Wage result on continuous images of Eberlein compact spaces.     

A non-linear population model of diabetes mellitus

A mathematical study of the size of a population ofdiabetes mellitus patients is carried out in this paper. The study also monitors the number of patients with complications. By appropriate definition of a parameter, the mathematical model may be classified as linear or non-linear. The non-linear case is discussed and the critical values of the population are analysed for stability. Numerical methods are developed for solving the model equations and the results of numerical simulations are reported.     

A dual open coloring axiom

We discuss a dual of the Open Coloring Axiom introduced by Abraham et al. [U. Abraham, M. Rubin, S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of ₁-dense real order types, Ann. Pure Appl. Logic 29 (2) (1985) 123–206] and show that it follows from a statement about continuous colorings on Polish spaces that is known to be consistent. We mention some consequences of the new axiom and show that implies that all cardinal invariants in Cichoń’s diagram are at least ₂.     

A polynomial dual of partitions

Let P(x) = Σ_i=0ⁿa_ixⁱ be a nonnegative integral polynomial. The polynomial P(x) is m-graphical, and a multi-graph G a realization of P(x), provided there exists a multi-graph G containing exactly P(1) points where a_i of these points have degree i for 0≤i≤n. For multigraphs G, H having polynomials P(x), Q(x) and number-theoretic partitions (degree sequences) π, ?, the usual product P(x)Q(x) is shown to be the polynomial of the Cartesian product G × H, thus inducing a natural product π? which extends that of juxtaposing integral multiple copies of ?. Skeletal results are given on synthesizing a multi-graph G via a natural Cartesian product G₁ × … × G_k having the same polynomial (partition) as G. Other results include an elementary sufficient condition for arbitrary nonnegative integral polynomials to be graphical.     

A characterization of dual spaces

A Banach space is a dual space if and only if it is isometric to the space of uniform functions on a set with graded constraints.     

A renorming of dual spaces

If Banach spacesX,X ^* are both weakly compactly generated, thenX has an equivalent norm whose dual onX ^* is locally uniformly rotund.     

A construction scheme for linear and non-linear codes

A scheme for construcing linear and non-linear codes is presented. It constructs a code of block length 2n from two constituent codes of block length n. Codes so constructed can be either linear or non-linear even when the constituent codes are linear. The construction of many known linear and non-linear codes using this scheme will be shown.