A geometric method in nonlinear programming

A differential geometric approach to the constrained function maximization problem is presented. The continuous analogue of the Newton-Raphson method due to Branin for solving a system of nonlinear equations is extended to the case where the system is under-determined. The method is combined with the continuous analogue of the gradient-projection method to obtain a constrained maximization method with enforced constraint restoration. Detailed analysis of the global behavior of both methods is provided. It is shown that the conjugate-gradient algorithm can take advantage of the sparse structure of the problem in the computation of a vector field, which constitutes the main computational task in the methods.This is part of a paper issued as Stanford University, Computer Science Department Report No. STAN-CS-77-643 (Ref. 45), which was presented at the Gatlinburg VII Conference, Asilomar, California, 1977. This work was supported in part by NSF Grant No. NAT BUR OF ECON RES/PO No. 4369 and by Department of Energy Contract No. EY-76-C-02-0016.The main part of this work was presented at the Japan-France Seminar on Functional Analysis and Numerical Analysis, Tokyo, Japan, 1976. The paper was prepared in part while the author was a visitor at the Department of Mathematics, North Carolina State University, Raleigh, North Carolina, 1976–77, and was completed while he was a visitor at the Computer Science Department, Stanford University, Stanford, California, 1977. He acknowledges the hospitality and stimulating environment provided by Professor G. H. Golub, Stanford University, and Professors N. J. Rose and C. D. Meyer, North Carolina State University.     

A transformed geometric distribution in stochastic modelling

The analytical concepts of infinite divisibility and (0) unimodality are fundamental to the study of probability distributions in general and to discrete distributions in particular. In this paper, a one-one correspondence is established between these two important properties which will permit any infinitely divisible discrete distribution (with finite mean value) to be transformed into a (0) unimodal discrete distribution. When this transformation is applied specifically to the geometric distribution, the result is a novel distribution, which can be fully and explicitly specified and whose factorial moments can be expressed in closed forms. This transformed geometric distribution is found to apply to underreported geometrically distributed decision processes, embedded renewal processes with logarithmically distributed components, and M/M/1 queues in which the service mechanism has been uniformly improved.     

Panel discussion on geometric modelling

Panel discussion on geometric modelling     

Broadcasting in geometric radio networks

We consider deterministic broadcasting in geometric radio networks (GRN) whose nodes know only a limited part of the network. Nodes of a GRN are situated in the plane and each of them is equipped with a transmitter of some range r. A signal from this node can reach all nodes at distance at most r from it but if a node u is situated within the range of two nodes transmitting simultaneously, then a collision occurs at u and u cannot get any message. Each node knows the part of the network within knowledge radius s from it, i.e., it knows the positions, labels and ranges of all nodes at distance at most s.The aim of this paper is to study the impact of knowledge radius s on the time of deterministic broadcasting in a GRN with n nodes and eccentricity D of the source. Our results show sharp contrasts between the efficiency of broadcasting in geometric radio networks as compared to broadcasting in arbitrary graphs. They also show quantitatively the impact of various types of knowledge available to nodes on broadcasting time in GRN. Efficiency of broadcasting is influenced by knowledge radius, knowledge of individual positions when knowledge radius is zero, and awareness of collisions.     

The voltage and current transfer ratios of RLC operator networks

The subject of this work is the voltage and current transfer ratios of three-terminal networks having no mutual coupling and whose impedances are analytic functions taking their values in an abelian self-adjoint algebra of bounded linear operators on a Hilbert space. Each such value is also assumed to be invertible. It is shown that these ratios have the form [I + A(ζ)]^?1, where, for each ζ in a sufficiently small open cone in the right-half complex plane with apex at the origin and the real axis as its bisector, the numerical range of A(ζ) is contained in a compact subset of the open right-half plane. This implies that the ratios are strictly contractive for each ζ in the cone. The angle of the cone is $\text{π}\text{(2k + 2)}$, where k is the number of internal nodes of a certain “surrogate” network; this result is best possible. For two-terminal-pair networks the ratios are shown to be strictly contractive for each ζ in a similar cone with angle $\text{π}\text{(2k + 4)}$.     

Judgemental modelling based on geometric least square

The Analytic Hierarchy Process developed by T.L. Saaty has received widespread attention in the literature. However this has not been without some criticisms, including the problems of meaning of consistency and large data requirements. This paper addresses the last two problems and presents a new method of calculating preference vectors. To the user the procedure is essentially the same, but the data requirements can be made less onerous and at the same time feedback is provided permitting a greater understanding of the data inputs. Preliminary results indicate the acceptability of the proposed methodology.     

A philosophy for the modelling of realistic nonlinear systems

A nonlinear dynamical system is modelled as a nonlinear mapping from a set of input signals into a corresponding set of output signals. Each signal is specified by a set of real number parameters, but such sets may be uncountably infinite. For numerical simulation of the system each signal must be represented by a finite parameter set and the mapping must be defined by a finite arithmetical process. Nevertheless the numerical simulation should be a good approximation to the mathematical model. We discuss the representation of realistic dynamical systems and establish a stable approximation theorem for numerical simulation of such systems.

     

A geometric characterization of the nonlinear Schrodinger equation and its applications

We prove that the nonlinear Schrodinger equation of attractive type (NLS+ describes just spher-ical surfaces (SS) and the nonlinear Schrodinger equation of repulsive type (NLS-) determines only pseudo-spherical surfaces (PSS). This implies that, though we show that given two differential PSS (resp. SS) equationsthere exists a local gauge transformation (despite of changing the independent variables or not) which trans-forms a solution of one into any solution of the other, it is impossible to have such a gauge transformationbetween the NLS+ and the NLS-.     

The modulation equations of nonlinear geometric optics

The purpose of this paper is to present an introduction to the central aspects of the modulation equations of nonlinear geometric optics. That means regularity properties and existence in the large.     

A new approach to some nonlinear geometric equations in dimension two

We give new arguments for several Liouville type results related to the equation −Δ u = Ke^2u. The new approach is based on the holomorphic function associated with any solution, which plays a similar role as the Hopf differential for harmonic maps from a surface.     

拟似然非线性模型的某些渐近推断:几何方法

Abstract. A modified Bates and Watts geometric framework is proposed for quasi-likelihoodnonlinear models in Euclidean inner product space. Based on the modified geometric framework,some asymptotic inference in terms of curvatures for quasi-likelihood nonlinear models is stud-ied. Several previous results for nonlinear regression models and exponential family nonlinearmodels etc. are extended to quasi-likelihood nonlinear models.     

Computational modelling of nonlinear calcium waves

The calcium transport in biological systems is modelled as a reaction–diffusion process. Nonlinear calcium waves are then simulated using a stochastic cellular automaton whose rules are derived from the corresponding coupled partial differential equations. Numerical simulations show self-organized criticality in the complex calcium waves and patterns. Both the stochastic cellular automaton approach and the equation-based simulations can predict the characteristics of calcium waves and complex pattern formation. The implication of locality of calcium distribution with positional information in biological systems is also discussed.     

A three-dimensional autonomous nonlinear dynamical system modelling equatorial ocean flows

We investigate a nonlinear three-dimensional model for equatorial flows, finding exact solutions that capture the most relevant geophysical features: depth-dependent currents, poleward or equatorial surface drift and a vertical mixture of upward and downward motions.     

SOME ASYMPTOTIC INFERENCE IN MULTINOMIAL NONLINEAR MODELS （A GEOMETRIC APPROACH）

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A geometric computing method for nonlinear optimal regulator problems with singular arcs

A computing algorithm, based on the geometry of certain reachable sets, is presented for fixed terminal time optimal regular problems having differential equations \(\dot x = f(x ,u , t)\) . Admissible controls must be measurable and have values in a setU, which must be compact, but need not be convex. Functionsf(x, u, t) andf _x (x, u, t) must be continuous and Lipschitz inx andu, but existence off _u (x, u, t) or second derivatives is not required. The algorithm is based on taking a sequence of nonlinear steps, each of which linearizes \(\dot x = f(x ,u , t)\) in state only, about a current nominal control and trajectory. Small perturbations are assured by keeping the perturbed controlclose to the nominal control. In each nonlinear step, a regulator problem,linear in state, is solved by a convexity method of Barr and Gilbert (Refs. 1–2), which is undeterred by the possibility of singular arcs. The resulting control function is substituted into the original nonlinear differential equations, producing an improved trajectory. Convergence of the algorithm is not proved, but demonstrated by a computing example, known to be singular. In addition, procedures are described for choosing parameters in the algorithm and for testing for theplausibility of convergence.     

A geometric characterization of the nonlinear Schr?dinger equation and its applications

We prove that the nonlinear Schr?dinger equation of attractive type (NLS⁺) describes just spherical surfaces (SS) and the nonlinear Schr?dinger equation of repulsive type (NLS^-) determines only pseudospherical surfaces (PSS). This implies that, though we show that given two differential PSS (resp. SS) equations there exists a local gauge transformation (despite of changing the independent variables or not) which transforms a solution of one into any solution of the other, it is impossible to have such a gauge transformation between the NLS⁺ and the NLS^-.