Equations of hyperelliptic type

This chapter is of an auxiliary nature, being mainly concerned with the relationship between bounds for linear forms in the logarithms of algebraic numbers in different (archimedean and non-archimedean) metrics. This material will later be used in the analysis of Thue and Thue-Mahler equations. Elliptic and hyperelliptic equations, and equations of hyperelliptic type, will be analysed using direct bounds for linear forms in the logarithms of algebraic numbers, and the necessary results are stated without proof at the end of the chapter. *** DIRECT SUPPORT *** A00I6B17 00002

In some practical situations, a searcher might have difficulties with implementing an optimal search plan of the form stipulated in the previous chapters. The plan might call for an instantaneous shift of search effort from one time period to the next. If the searcher requires a significant amount of time to carry out this shift, a relatively fast moving target would “get ahead” of the searcher. This situation is especially prevalent in robotic searches of buildings, where transit from room to room accounts for the majority of time expenditure, and searches using low-speed unmanned aerial systems, where the ratio of searcher speed to target speed is low. In this chapter, we describe methods for computing optimal search plans while accounting for real-world constraints on the agility of the searcher. In fact, we consider multiple searchers, each providing a discrete search effort, as well as multiple targets. The chapter starts, however, with the simpler situation of a single searcher looking for a single target. We formulate the optimal search problem as that of finding the optimal searcher path and describe a branch-and-bound algorithm for its solution. We proceed by generalizing the formulation to account for a searcher that operates at different “altitudes” with a more complex sensor. We also describe algorithmic enhancements that both handle the more general situation and provide computational speed-ups. The chapter then addresses the situation with multiple searchers, first of identical types and second of different types and also with multiple targets. These generalizations are most easily handled within a mathematical programming framework, which facilitates the consideration of a multitude of constraints including those related to airspace deconflication and also allows the leverage of well-developed optimization solvers for the determination of optimal searcher plans. The chapter ends with a description of some algorithms behind these solvers, with an emphasis on cutting-plane methods. Throughout the chapter we remain in the context of discrete time and space search.

The basic tools for describing and analyzing random processes have all been developed in the preceding chapters along with a variety of examples of random processes with and without memory. The goal of this chapter is to use these tools to describe a menagerie of useful random processes, usually by taking a simple random process and applying some form of signal processing such as linear filtering in order to produce a more complicated random process. In Chapter 5 the effect of linear filtering on second-order moments was considered. In this chapter we look in more detail at the resulting output process and consider other forms of signal processing as well. In the course of the development a few new tools and several variations on old tools for deriving distributions are introduced. Much of this chapter can be considered as practice of the methods developed in the previous chapters, with names often being given to the specific examples developed. In fact several processes with memory have been encountered previously: the binomial counting process and the discrete time Wiener process, in particular. The goal now is to extend the techniques used in these special cases to more general situations and to introduce a wider variety of processes. The development of examples begins with a continuation of the study of the output processes of linear systems with random process inputs.

In this paper we consider binary cubic diophantine equations of every form and shape, solely subjected to the requirement that they represent elliptic curves defined over Q. How to deal with standard Weierstras equations is well understood, but comparatively little is known about elliptic equations of different type. We give a detailed analysis of the general situation and subsequently apply the elliptic logarithm method to a variety of unusual elliptic equations, notably to some equations directly related to Krawtchouk polynomials. 1991 Mathematics subject classification: 11D25, 11G05, 11Y50, 12D10

The article proposes original solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem. And we use the matrix pseudo-inversion technique of Moore-Penrose. The posed problem in the article continues the study of conditions of solvability and schemes for finding solutions of the nonlinear Noetherian boundary-value problems given in the monographs by A. Poincare, A.M. Lyapunov, I.G. Malkin, J. Hale, Yu.A. Ryabov, A.M. Samoylenko, N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina and A.A. Boychuk. We studied a general case, when a linear bounded operator corresponding to the homogeneous part of the linear Noetherian differential-algebraic boundary value problem has no inverse. Sufficient conditions for reducibility of the differential algebraic equation to the system uniting a differential and algebraic equation are found. Thus, the differential-algebraic boundary value problem is reduced to the nonlinear Noetherian boundary value problem for the system of ordinary differential equations. We studied the case of the presence of simple roots of the equation for generating amplitudes. Constructive necessary and sufficient conditions of existence were obtained to find solutions to the problem in the critical case, and the converging iterative scheme was constructed. The proposed solvability conditions, and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem are illustrated in detail by the example from the nonlinear Noetherian differential-algebraic boundary value problem for Duffing type equations. For control of the rate of the iterative scheme convergence to the exact solution of the differential-algebraic boundary value problem for the Duffing type equation, we used the residuals of the obtained approximations in the Duffing type equation in the space of continuous functions.

The ultimate goal of our study is to obtain suggestions for the development of devices capable of the automatic modeling of natural phenomena. It is therefore of fundamental importance to develop a theoretical basis for the description of their optimal performance. In the previous chapters it was stated that an empirical modeling of natural phenomena includes three main tasks: Estimation, storage, and application of a probability distribution. Each of these tasks can be optimized by using the methods described in the previous chapter. The aim of this section is to present the problems and the solutions related with an optimal storage of empirical information about a continuous probability distribution in a system comprised of a finite number of discrete memory units. Such a system can be considered as a basic building block of an automatic modeler of natural phenomena. For example, we can imagine the brain of a biological organism or the digital memory of a computer that continually obtains signals from its surroundings and it optimally stores the corresponding empirical information. The first problem is the estimation of the probability density function of a continuous variable from the empirical data. We have already seen that this can be solved using Parzen’s window. [2] The second problem is the storage of the continuous probability density in a discrete system. This task is generally related with a loss of information and thus the question arises, how one can minimize this loss.KeywordsWindow FunctionExcited NeuronReference VectorDiscrete Random VariableMemory UnitThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Among commutative rings, the polynomial rings in a finite number of indeterminates enjoy important special properties and are frequently used in applications. As they are also of paramount importance in Algebraic Geometry, polynomial rings have been intensively studied. On the other hand, rings of formal power series have been extensively used in “algebroid geometry” and have many properties which are parallel to those of polynomial rings. In the first section of this chapter we shall define formal power series rings and we shall show that the main properties of polynomial rings which have been derived in previous chapters (see, in particular, Vol. I, Ch. I, §§ 16–18) hold also for formal power series rings. In the later sections of this chapter we shall give deeper properties of polynomial rings and, whenever possible, the parallel properties of power series rings.

The transmission of constants a x (j) by asystem of orthogonal functions {f(j,θ)} has been discussed in the previous chapter. The transmission of time functions F(θ) or F x (θ) by means of a system of time functions {Φ(k, θ)} will be discussed now. These functions Φ(k,θ) will be called carriers. F(θ) denotes any time function, e.g. the output voltage of a microphone. The notation F x (θ)isused to emphasize time functions that contain a finite number of coefficients a x (j), such as teletype signals: $${F_x}\left( \theta \right) = \sum\limits_{j = 0}^{m - 1} {{a_x}} \left( j \right)f\left( {j 9} \right)$% MathType!End!2!1!$$

In this book we study the CAPM as a general equilibrium model in which the natural questions of existence and uniqueness of an equilibrium play an important role. Similar to Nielsen [1990b] and Allingham [1991] in the previous chapter we have given general proofs of existence within this framework where consumption sets are not bounded below and satiation is possible. As shown by an example with two equilibria constructed by Nielsen [1988], there is, however, no reason to expect uniqueness in general. We will continue along this line of research and will analyze the structure of market demand in the CAPM. We will show that given any choice of a finite number of normalized price systems and the respective demands satisfying Walras’ Law and the Tobin Separation Property, there exist two variance-averse agents whose market demand coincides with the preassigned values. This result parallels the result proved in Chapter 3 which itself was a generalization of results known in the general equilibrium literature as the Sonnenschein-Mantel-Debreu result on the structure of market excess demand functions. The result proved in this chapter for the CAPM considers the market excess demand function on a finite set of prices which is then similar to the result of Andreu [1982].KeywordsDemand FunctionMarket DemandAsset AllocationIndifference CurveMarket PortfolioThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The system of partial differential equations which forms the basic semiconductor equations together with appropriate boundary conditions has been investigated and characterized analytically in the previous chapter. This system cannot be solved explicitly in general. Therefore, the solution must be calculated by means of numerical approaches. We shall consider in this chapter such Solution procedures for the sealed equations which read: $$\lambda ^2 \cdot \text{div}\,\text{grad}\,\psi - (n - p - C) = 0$$ (6-1) $$\text{div}(D_n \cdot \text{grad}\,n - \mu _\text{n} \cdot n \cdot \text{grad}\,\psi ) - R(\psi ,n,p) = \frac{{\partial n}} {{\partial t}}$$ (6-2) $$\text{div}(D_p \cdot \text{grad}\,p - \mu _\text{p} \cdot p \cdot \text{grad}\,\psi ) - R(\psi ,n,p) = \frac{{\partial p}} {{\partial t}}$$ (6-3) Any numerical approach for the Solution of such a system consists essentially of three tasks. First, the domain, i.e. the Simulation geometry of the device, has to be partitioned into a finite number of subdomains, in which the Solution can be approximated easily with a desired accuracy. Secondly, the differential equations have to be approximated in each of the subdomains by algebraic equations which involve only values of the continuous dependent variables at discrete points in the domain and knowledge of the structure of the chosen funetions which approximate the dependent variables within each of the subdomains. In that way one obtains a fairly large system of, in general nonlinear, algebraic equations with unknowns comprised of approximations of the continuous dependent variables at discrete points. The Solution of this system is the final third task to be carried out. As this problem can be viewed rather independently it will be treated separately in Chapter 7.

This chapter advances the analytical understanding of cyber-physical systems to cover control loops. While the syntax and semantics of hybrid programs from previous chapters already discussed loops, their logical characterization was so far limited to unfolding by the iteration axiom. That suffices for systems with a fixed finite number of control actions in a fixed finite number of repetitions of the control loop, but is not enough to understand and analyze the most interesting CPSs with unbounded time-horizons reaching an unbounded number of control decisions over time. This chapter uses the fundamental concept of invariants to handle loops and develops their operational intuition. CPS invariants are developed systematically based on inductive formulations of dynamic axioms for repetitions.

In this paper we give an elementary proof of the famous theorem of J. Malmquist from 1913 and its refinement given by N. Steinmetz 1978. This result says, that if a differential equation of type (1.1) has a transcendental meromorphic solution, then it can be transformed into one of the six normalforms given in theorem 3.2 or a power of it. In this note we make no use of Nevanlinna's theory or the theory of central index used in former proofs. We only need the theorem of Liouville and the wellknown inequalities for meromorphic functions g with a finite number of poles, the Euclidean algorithm and a lemma of E Borel about monotone continuous functions.

In this paper, Euler-Lagrange type equations are used to describe the jet flow of a mixture of pulverized-coal and gas, which is an unsteady axisymmetric two-phase flow. By means of the finite-difference method, the coal particle's distribution, velocity and trajectory in the flow field are obtained. The coal particles are represented by a finite number of computational particles. Each particle's diameter is randomly assigned according to a given distribution. The states of the computational particles are different from each other. Turbulence is accounted for in a stochastic model. Explicit time-splitting scheme is used to calculate the strongly coupling interphase term. The numerical results are reasonable. The comparison between the numerical results and the experiment data for the case of the oil droplet injection shows good agreement. This numerical technique can be extended to the calculation of other two-phase flows of dilute particles or a droplet system.

The characteristics of the 1D geometric random walk of a finite number of agents are investigated by assuming constant variance. Firstly, the characteristics of the steady state solution of the distribution function, which is obtained using the extended geometric Brownian motion (EGBM), are investigated in the framework of the 1D Fokker–Planck type equation. The uniqueness and existence of the steady state solution of the distribution function requires the number of particles to be finite. To avoid the divergence of the steady state solution of the distribution function at the mean value in the 1D Fokker–Planck type equation, the hybrid model, which is a combination of EGBM and normal BM, is proposed. Next, the steady state solution of the distribution function, which is obtained using the geometric Lévy flight, is investigated under constant variance in the framework of the space fractional 1D Fokker–Planck type equation. Additionally, we confirm that the solution of the distribution function obtained using the super-elastic and inelastic (SI-) Boltzmann equation under constant variance approaches the Cauchy distribution, when the power law number of the relative velocity increases. Finally, dissipation processes of the pressure deviator and heat flux are numerically investigated using the 2D space fractional Fokker–Planck type equations for Lévy flight and SI-Boltzmann equation by assuming their linear response relations.

Given an equivariant oriented cohomology theory $h$, a split reductive group $G$, a maximal torus $T$ in $G$, and a parabolic subgroup $P$ containing $T$, we explain how the $T$-equivariant oriented cohomology ring $h_T(G/P)$ can be identified with the dual of a coalgebra defined using exclusively the root datum of $(G,T)$, a set of simple roots defining $P$ and the formal group law of $h$. In two papers [Push-pull operators on the formal affine Demazure algebra and its dual, arXiv:1312.0019] and [A coproduct structure on the formal affine Demazure algebra, arXiv:1209.1676], we studied the properties of this dual and of some related operators by algebraic and combinatorial methods, without any reference to geometry. The present paper can be viewed as a companion paper, that justifies all the definitions of the algebraic objects and operators by explaining how to match them to equivariant oriented cohomology rings endowed with operators constructed using push-forwards and pull-backs along geometric morphisms. Our main tool is the pull-back to the $T$-fixed points of $G/P$ which injects the cohomology ring in question into a direct product of a finite number of copies of the $T$-equivariant oriented cohomology of a point.