Showing papers on "Monotone polygon published in 1972"

Monotone solutions of the parametric linear complementarity problem

Abstract: The parametric linear complementarity problem under study here is given by the conditionsq + αp + Mz ≥ 0,α ≥ 0,z ≥ 0,z T (q + αp + Mz) = 0 whereq is nonnegative. This paper answers three questions including the following one raised by G. Maier: What are the necessary and sufficient conditions onM guaranteeing that for every nonnegative starting pointq and every directionp the components of the solution to the parametric linear complementarity problem are nondecreasing functions of the parameterα?

Gaussian Sample Functions: Uniform Dimension and Hölder Conditions Nowhere

Abstract: Let X(t), t≥0, be a real Gaussian process with mean 0, stationary increments, and a2(t) = E|X(t) - X(0)|2. Here dH(λ), for some bounded monotone H. We summarize the main results.

Minimal Negative Gate Networks

Abstract: A negative gate is a gate that can realize an arbitrary negative function (or monotone decreasing function) and a positive gate is one that can realize an arbitrary positive function (or monotone increasing function). This paper discusses methods of realizing a given logical function or a given set of logical functions using a minimum number of negative gates alone or using a minimum number of negative and positive gates.

Open mappings of the universal curve onto continuous curves

Abstract: A criterion for the existence of an open mapping from one compact metric space onto another is established in this paper. This criterion is then used to establish the existence of a monotone open mapping of the universal curve onto any continuous curve and the existence of a light open mapping of the universal curve onto any nondegenerate continuous curve. These examples show that iff is a monotone open or a light open mapping of one compact space X onto another Y, then it will not necessarily be the case that dim Y? dim X+ k, where k is some positive integer.

Two-body problem with slowly decreasing mass

Abstract: Evolution of the orbital elements of a two-body system with slowly decreasing mass according to Jeans' mode is described by a non-linear, non-autonomous system of differential equations. In general the system contains one stationary solution (e=1,f=π), for which an instability criterion is derived. For example the stationary solution is unstable for all Jeans-Eddington functionsm n (t) with 1≦n≦3 which characterize the loss of mass. Furthermore, it is possible to describe the quantitative behaviour ofE+ω,e anda for arbitrarym(t) in a large number of cases. In the case of the Jeans-Eddington functions we find that the amplitude of the oscillations ine is monotone decreasing with time ifn>3 and it is monotone increasing with time ifn<3. By comparing these analytical results with the numerical calculations of Hadjidemetriou we explain the rapid rotation of the line of apsides which occurs if the initial value ofe is nearly-circular.

Representation theorems for equivalent optimization problems

Abstract: In conjunction with the problem of transforming a given optimization problem into a form from which the functional equations of dynamic programming are obtainable, Karp and Held (1967) made clear the relation between a certain class of decision processes and dynamic programming from the view point of automata theory. This paper also follows the line of Karp and Held, and presents a number of new concepts. First we assume that a given optimization problem is discrete and deterministic: it is given in the form of discrete decision process (ddp). Then we define six classes of decision processes: sdp (sequential decision process), msdp (monotone sdp), smsdp (strictly monotone sdp), pmsdp (positively monotone sdp), ap (additive process), and lmsdp (loop-free msdp). The sdp is considered as a general model of a decision process with finite states. The msdp is a subclass of sdp's from which the functional equations of dynamic programming are obtainable. The smsdp, pmsdp, ap, and lmsdp are subclasses of msdp's, which have simpler structures than that of msdp. In fact, simpler solution methods for solving the resulting functional equations are available for these subclasses. Two types of representation theorems are first proved for each class of decision processes: one is the w (weak)-representation theorem which is a necessary and sufficient condition for a given ddp to be realized by a decision process of the specific class in the sense that both have the same set of optimal policies, and the other is the s (strong)-representation theorem, which assumes the coincidence of cost value for each feasible policy in addition to the above condition. Based on the w -representation theorems, various properties of sets of optimal policies are investigated for each class. In particular, it is shown that although sets of optimal policies of sdp and msdp are not closed under most of operations, they are closed for smsdp, pmsdp, ap, and lmsdp. In fact, a set of policies can be a set of optimal policies of an smsdp, pmsdp, or ap if and only if it is regular (i.e., accepted by a finite automaton). For an lmsdp, a set can be a set of optimal policies if and only if it is finite.

Some transformations of monotone sequences

Abstract: In this paper, we discuss a class of methods for summing sequences which are generalizations of a method due to Salzer. The methods are not regular, and in contrast to the classical regular methods, seem to work best on sequences which are monotone. In our main theorem, we determine a class of convergent sequences for which the methods yield sequences which converge to the same sum.

Über die näherungsweise Lösung nichtlinearer Integralgleichungen

Abstract: In this paper we study the iterative solvability of nonlinear systems of equations which arise from the discretization of Hammerstein integral equations. It is shown that, for a large class of equations satisfying monotonicity assumptions, it is possible to solve these systems by means of a linearly convergent iteration method. Moreover, for general monotone operators on a Hilbert space a globally convergent variant of Newton's method is given. Finally it is shown that this method effectively can be applied in a natural way to the systems of equations under consideration.

A Saddle-Point Optimality Criterion for Nonconvex Programming in Normed Spaces

Abstract: By confining Lagrange multipliers to a class of two-parameter continuous monotone nondecreasing functionals, a new necessary and sufficient saddle-point optimality criterion may be obtained for nonconvex programming problems. This device also allows the solution of a nonconvex dual problem under general conditions. New necessary and sufficient conditions are given, as part of this development, for an extended form of stability in nonlinear programming. The theory is applied to the solution of finite- and infinite-dimensional programming problems.

A rank two algorithm for unconstrained minimization

Abstract: A stable second-order unconstrained minimization algorithm with quadratic termination is given. The algorithm does not require any one-dimensional minimizations. Computational results presented indicate that the performance of this algorithm compares favorably with other well-known unconstrained minimization algorithms. Introduction. Algorithms for unconstrained minimization have enjoyed a great deal of attention in recent years. The fundamental philosophy behind most of these algorithms is the exploitation of the locally quadratic nature of a well-behaved func- tion at an unconstrained local minimum. The Davidon-Fletcher-Powell method (1) was highly successful in this regard and guarantees finite convergence when the function is quadratic, monotone decrease of function value when it is not. However, a potentially time consuming one-dimensional minimization is required in each iteration. Recently developed rank one (2) and rank two (3) methods eliminate the need for this minimization. The rank one method, however, sacrifices finite con- vergence for stability, and the rank two method eliminates the one-dimensional minimization by trading finite convergence for monotone convergence. The amount of computation in each iteration is reduced, but the number of iterations required may be increased. In this paper, we present a rank two method which has the combined virtues of finite convergence for quadratic functions and stability for any function, and does not require a one-dimensional search at each iteration. This combination of de- sirable properties is achieved by making full use of the flexibility of a rank two algorithm. The algorithm given here is cyclic, i.e., it repeats itself every N iterations (when minimizing a function of N variables) unlike the Davidon-Fletcher-Powell algorithm, which is the same in every iteration. Kelley and Myers (8) presented a cyclic method which is a special case of the algorithm given here.

Computation of Best Monotone Approximations

Abstract: A numerical procedure to compute the best uniform approximation to a given continuous function by algebraic polynomials with nonnegative rth derivative is presented and analyzed. The method is based on discretization and linear programming. Several numerical experiments are discussed.

On time-free functions

Abstract: By regarding as equivalent any two real-valued functions of a real variable that can be obtained from each other by a monotone continuous transformation of the independent variable, time-free finctions are defined. A convenient maximal invariant is presented, and applied to some time-free functional equations.

A Remark on a Class of Linear Monotone Operators

Abstract: For applications of angle-bounded mappings and operators satisfying condition (,) to nonlinear equations of Hammerstein type in Banach spaces we refer to [1-3]. It is clear by definition that the adjoint of an angle-bounded mapping is again angle-bounded. The purpose of this paper is to show that the class of linear operators satisfying condition (,) is invariant under passage to the adjoint operator.

On the monotone nature of boundary value functions for $n$th-order differential equations

Abstract: We are concerned with the nth (n≥3) order linear differential equation 1 where the coefficients are continuous on (∞, ∞). Our main result is to give conditions under which the two-point boundary value function rij(t) (see Definition 2) are strictly increasing continuously differentiable functions of t. Levin [1] states without proof a similar theorem concerning just the monotone nature of the rij(t) but assumes that the coefficients in (1) satisfy the standard differentiability conditions when one works with the formal adjoint of (1). Bogar [2] looks at the same problem for an nth-order quasi-differential equation where he makes no assumption concerning the differentiability of the coefficients in the quasi differential equation that he considers. Bogar gives conditions under which the rij(t) are strictly increasing and continuous. The different approach of the author to this problem also enables the author to establish the continuous differentiability of the rij(t) and to express the derivatives in terms of the principal solutions Uj(x, t), j=0, 1,…,n-1 (see Definition 4).

Positive and Monotone Approximation

Abstract: Let T be a linear bounded operator that maps C[a, b] into C(k)[a, b] (with the natural topology of C(k)).

Retractions and quasi-monotone mappings of unicoherent spaces

Abstract: It is shown that retractions of connected, locally connected, unicoherent spaces are unieoherent, and that quasimonotone maps preserve the unicoherence of any connected uni-

Vertical Asymptotes and Bounds for Certain Solutions of a Class of Second Order Differential Equations

Abstract: In papers of Wong and Hille results appear concerning the existence of vertical asymptotes of certain solutions of second order nonlinear differential equations. Specifically, Hille considers the Thomas–Fermi equation. Results here, which in part involve methods of these authors, provide upper and lower bounds for solutions of the initial value problem \[ y'' = p(x)y^\gamma ,\quad y(a) > 0,\quad y'(a) = 0,\] as well as bounds for the first vertical asymptote $b > a$ of such solutions. The coefficient function p is positive and continuous on an appropriate interval under consideration and $\gamma \geqq 1$. The bounds are given in terms of integral functionals involving the coefficient function p and solutions, computable in terms of the incomplete beta-function, of a similar initial value problem where the coefficient function is constant. The results are applicable in their most complete form when the coefficient function is either monotone increasing or monotone decreasing. Specific results are obtained ...