A SIMPLICIAL HOMOTOPY ALGORITHM FOR COMPUTING ZERO POINTS ON POLYTOPES

In this paper a triangulation of continuous and arbitrary refinement of grid sizes is proposed for simplicial homotopy algorithms to compute zero points on a polytope P. The proposed algorithm generates a piecewise linear path in P × [1,∞) from any chosen interior point x0 of P on level {1} to a solution of the underlying problem. The path is followed by making linear programming pivot steps in a linear system and replacement steps in the triangnlation.The starting point x0 is left in a direction to one vertex of P. The direction in which x0 leaves depends on the function value at x0 and the polytope P. Moreover, we also give a new equivalent form of the Brouwer fixed point theorem on polytopes. This form has many important applications in mathematical programming and the theory of differential equations.

ON THE COMPLEXITY OF A PL HOMOTOPY ALGORITHM FOR ZEROS OF POLYNOMIALS

A PL homotopy algorithm is modified to yield a polynomial-time result on its computational complexity.We prove that the cost of locating all zeros of a polynomial of degree n to an accuracy of ε(measured by the number of evaluations of the polynomial)grows no faster than O(max{n^4,n^3log_2(n/ε)}).This work is in response to a question raised in a paper by S.Smale as to the efficiency of piecewise linear methods in solving equations.In comparison with a few results reported,the algorithm under discussion is the only one providing correct multiplicities and the only one employing vector labelling.

HOMOTOPY CONTINUATION ALGORITHM FOR STRUCTURAL DAMAGE IDENTIFICATION

A new method is put forward for structural damage identification based on the homotopy continuation algorithm. A numerical example is presented to verify the method. The beams with different damage locations and different damage extents are identified by this method. The numerical examples have proved that this new method is capable of easy convergence, which is not sensitive to the initial iterative values. It is effective for accurately identifying multiple damages. By incorporating the finite element method into the homotopy continuation algorithm, the damage identifying ability of the new method can be greatly enhanced.

Volterra Feedforward Neural Networks:Theory and Algorithms

The Volterra feedforward neural network with nonlinear interconnections and related homotopy learning algorithm are proposed in the paper. It is shown that Volterra neural network and the homolopy learning algorithms are significant potentials in nonlinear approximation ability,convergent speeds and global optimization than the classical neural networks and the standard BP algorithm, and related computer simulations and theoretical analysis are given too.

An Iteration Method Based on Homotopy Function for Solving Polynomial Systems and Application to Mechanisms Problems

A new iterating method based on homotopy function is developed in this paper. All solutions can be found easily without the need of choosing proper initial values. Compared to the homotopy continuation method, the solution process of the present method is simplified, and the computation efficiency as well as the reliability for obtaining all solutions is also improved. By application of the method to the mechanisms problems, the results are satisfactory.

Thermophysical properties of pure gases and mixtures at temperatures of 300-30000K and atmospheric pressure:thermodynamic properties and solution of equilibrium compositions

The equilibrium compositions and thermodynamic properties(density,enthalpy,etc at constant pressure)of plasma of pure gases and mixtures under local thermodynamic nonequilibrium have been calculated in this paper.The homotopy Levenberg-Marquardt algorithm was proposed to accurately solve nonlinear equations with singular Jacobian matrices,and is constructed by the Saha equation and Guldberg-Waage equation combined with mass conservation,the electric neutrality principle and Dalton’s partial pressure law,to solve the problem of dependence on the initial value in the process of iteration calculation.In this research,the equations at a higher temperature were solved and used as the auxiliary equations,and the homotopy control parameters’sequence of the homotopy equations was selected by equal ratios.For auxiliary equations,the iterative initial value was obtained by assuming that there were only the highestvalence atomic cations and electrons at this temperature,and the plasma equilibrium composition distribution with the required accuracy was ultimately solved under the current conditions employing the Levenberg-Marquardt algorithm.The control parameter sequence was arranged according to the geometric sequence and the homotopy step was gradually shortened to ensure continuity of the homotopy process.Finally,the equilibrium composition and thermodynamic properties of pure N_(2),Mg(30%)-CO_(2)(70%)and Mg(40%)-CO(50%)-N_(2)(10%)mixture plasma at atmospheric pressure were calculated and the calculation process of some specified temperatures was shown and analyzed.The calculation accuracy of equilibrium composition is higher than other findings in the literature.The results for the thermodynamic properties are in good agreement with data reported by the literature.

HOMOTOPY SIMPLEX PIVOT ALGORITHM

We prove the following theorems. Theorem 1. Let Im= [a1, b1]×...×[am, bm], {Tδi} be a proper simplicial subdivision sequence of Im×[0,1],H: Im×[0,1]→Rm be a homotopy,

Darcy-Forchheimer mangetized flow based on differential type nanoliquid capturing Ohmic dissipation effects

Hydromagnetic nanoliquid establish an extraordinary category of nanoliquids that unveil both liquid and magnetic attributes.The interest in the utilization of hydromagnetic nanoliquids as a heat transporting medium stem from a likelihood of regulating its flow along with heat transportation process subjected to an externally imposed magnetic field.This analysis reports the hydromagnetic nanoliquid impact on differential type(second-grade)liquid from a convectively heated extending surface.The well-known Darcy-Forchheimer aspect capturing porosity characteristics is introduced for nonlinear analysis.Robin conditions elaborating heat-mass transportation effect are considered.In addition,Ohmic dissipation and suction/injection aspects are also a part of this research.Mathematical analysis is done by implementing the basic relations of fluid mechanics.The modeled physical problem is simplified through order analysis.The resulting systems(partial differential expressions)are rendered to the ordinary ones by utilizing the apposite variables.Convergent solutions are constructed employing homotopy algorithm.Pictorial and numeric result are addressed comprehensively to elaborate the nature of sundry parameters against physical quantities.The velocity profile is suppressed with increasing Hartmann number(magnetic parameter)whereas it is enhanced with increment in material parameter(second-grade).With the elevation in thermophoresis parameter,temperature and concentration of nanoparticles are accelerated.