Approximations of Lie algebras

It is proved that a finitely generated metabelian Lie algebra over an arbitrary field can be approximated by finite-dimensional algebras and a stronger result is also obtained over fields of nonzero characteristics.Translated from Matematicheskie Zametki, Vol. 12, No. 6, pp. 713–716, December, 1972.     

Compound Binomial Approximations

We consider the approximation of the convolution product of not necessarily identical probability distributions q _j I + p _j F, (j=1,...,n), where, for all j, p _j =1−q _j ∈[0, 1], I is the Dirac measure at point zero, and F is a probability distribution on the real line. As an approximation, we use a compound binomial distribution, which is defined in a one-parametric way: the number of trials remains the same but the p _j are replaced with their mean or, more generally, with an arbitrary success probability p. We also consider approximations by finite signed measures derived from an expansion based on Krawtchouk polynomials. Bounds for the approximation error in different metrics are presented. If F is a symmetric distribution about zero or a suitably shifted distribution, the bounds have a better order than in the case of a general F. Asymptotic sharp bounds are given in the case, when F is symmetric and concentrated on two points. An erratum to this article can be found at     

Approximations in hyperquasigroups

In this paper, we introduce the concept of fuzzy rough subhyperquasigroups of rough hyperquasigroups and obtain some interesting results. Moreover, we consider the relation β* defined on a hyperquasigroupG and interpret the lower and upper approximations as subsets of the quasigroupG/β*, and give some results in this connection.     

Adaptive Martingale Approximations

H-systems are those orthonormal systems which allow computation of conditional expectations via a Fourier expansion. These systems provide natural approximations to continuous stochastic processes and we indicate how they could be used to perform lossy compression on a set of given signals. More specifically, we show how to construct, for a given random variable, an adaptive martingale approximation by means of generalized Haar functions. We also indicate how the construction can be extended to a given random vector. A generalized multiresolution analysis algorithm is also described and numerical examples are provided.     

Approximations of Nash equilibria

Inspired by previous works on approximations of optimization problems and recent papers on the approximation of Walrasian and Nash equilibria and on stochastic variational inequalities, the present paper investigates the approximation of Nash equilibria and clarifies the conditions required for the convergence of the approximate equilibria via a direct approach, a variational approach, and an optimization approach. Besides directly addressing the issue of convergence of Nash equilibria via approximation, our investigation leads to a deeper understanding of various notions of functional convergence and their interconnections; more importantly, the investigation yields improved conditions for convergence of the approximate Nash equilibria via the variational approach. An illustrative application of our results to the approximation of a Nash equilibrium in a competitive capacity expansion model under uncertainty is presented.     

Approximations of inventory models

Summary A general procedure is presented for constructing approximations of discrete review single product dynamic inventory models. Bounds are derived for the approximations and compared with the ones ofHinderer [1978],Whitt [1978] for the approximation of a general dynamic program. Good order-policies are constructed. Finally, conditions are given under which the sequence of bounds associated with a sequence of approximating models converges to zero.

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Zusammenfassung Es wird ein allgemeines Verfahren zur Approximation periodisch inspizierter dynamischer Ein-Produkt-Lagerhaltungsmodelle betrachtet. Dazu werden Fehlerschranken gegeben. Diese werden verglichen mit denen vonHinderer [1978],Whitt [1978] für ein allgemeines dynamisches Entscheidungsmodell. Außerdem werden gute Bestellpolitiken konstruiert und hinreichende Bedingungen gegeben, unter denen die Fehlerschranken einer Folge von approximierenden Modellen gegen Null konvergiert.

     

Diophantine Approximations on Fractals

We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the natural measure) contains all finite patterns (hence is well approximable). Similarly, we show that for a variety of fractals in [0, 1]², possessing some symmetry, almost any point is not Dirichlet improvable (hence is well approximable) and has property C (after Cassels). We then settle by similar methods a conjecture of M. Boshernitzan saying that there are no irrational numbers x in the unit interval such that the continued fraction expansions of {nx mod 1}_{n ? \mathbb N}{\{nx\,{\rm mod}\,1\}_{n \in {\mathbb N}}} are uniformly eventually bounded.     

Two-Sided Pareto Front Approximations

A new approach to derive Pareto front approximations with evolutionary computations is proposed here. At present, evolutionary multiobjective optimization algorithms derive a discrete approximation of the Pareto front (the set of objective maps of efficient solutions) by selecting feasible solutions such that their objective maps are close to the Pareto front. However, accuracy of such approximations is known only if the Pareto front is known, which makes their usefulness questionable. Here we propose to exploit also elements outside feasible sets to derive pairs of such Pareto front approximations that for each approximation pair the corresponding Pareto front lies, in a certain sense, in-between. Accuracies of Pareto front approximations by such pairs can be measured and controlled with respect to distance between elements of a pair. A rudimentary algorithm to derive pairs of Pareto front approximations is presented and the viability of the idea is verified on a limited number of test problems.     

More Accurate Skin-Depth Approximations

The penetration of a high-frequency alternating or rotatingmagnetic field into a conductor is considered. The magneticReynolds number is assumed to be small. The standard high-frequency(or skin-depth) approximation is shown to be incorrect in theinterior of the conductor, leading to large relative errors.Two terms of the correct expansion are calculated. It is foundthat a ‘focusing’ effect occurs, in a manner similarto the formation of caustics in geometrical optics. As a result,at some interior points the field is several times larger thanexpected. The behaviour near such singular points is found.The implications for electromagnetically driven fluid flow arebriefly discussed.     

Approximations for chromatic polynomials

A sequence of finite graphs may be constructed from a given graph by a process of repeated amalgamation. Associated with such a sequence is a transfer matrix whose minimum polynomial gives a recursion for the chromatic polynomials of the graphs in the sequence. Taking the limit, a generalised “chromatic polynomial” for infinite graphs is obtained.     

Two-Parametric Compound Binomial Approximations

We consider two-parametric compound binomial approximation of the generalized Poisson binomial distribution. We show that the accuracy of approximation essentially depends on the symmetry or shifting of distributions and construct asymptotic expansions. For the proofs, we combine the properties of norms with the results for convolutions of symmetric and shifted distributions. In the lattice case, we use the characteristic function method. In the case of almost binomial approximation, we apply Steins method.__________Published in Lietuvos Matematikos Rinkinys, Vol. 44, No. 4, pp. 443–466, October–December, 2004.     

Approximations for stop-loss premiums

We derive an approximation for stop-loss premiums for a number of specific cases. Both exponential and subexponential estimates are derived while special emphasis is given to the compound Poisson case. The full set of examples should provide a wide variety of situations covering most cases occurring in practice.     

功率和的强逼近

设$\{\xi_n, n\geq 1\}$是正的随机变量序列, $\ep \xi_1=\theta>0$, 设$S_n = \sum\limits_{i=1}^n \xi_i, Y_n=n\theta\log (S_n/(n\theta))$. 在该文中, 当$\{\xi_n\}$是独立同分布或强平稳$\varphi$ -混合的正随机变量序列时,作者给出功率和$\{Y_n\}$用Wiener过程的强逼近结果.     

Approximations of the Riley slice

Adapting the ideas of L. Keen and C. Series used in their study of the Riley slice of Schottky groups generated by two parabolics, we explicitly identify ‘half-space’ neighbourhoods of pleating rays which lie completely in the Riley slice. This gives a provable method to determine if a point is in the Riley slice or not. We also discuss the family of Farey polynomials which determine the rational pleating rays and their root set which determines the Riley slice; this leads to a dynamical systems interpretation of the slice. Adapting these methods to the case of Schottky groups generated by two elliptic elements in subsequent work facilitates the programme to identify all the finitely many arithmetic generalised triangle groups and their kin.     

线性过程的强逼近

该文对由独立同分布随机变量序列所生成的线性过程建立了泛函重对数律和用Wiener过程对线性过程的强逼近结果.     

Piecewise-Linear Approximations of Multidimensional Functions

We develop explicit, piecewise-linear formulations of functions f(x):ℝ ⁿ ↦ℝ, n≤3, that are defined on an orthogonal grid of vertex points. If mixed-integer linear optimization problems (MILPs) involving multidimensional piecewise-linear functions can be easily and efficiently solved to global optimality, then non-analytic functions can be used as an objective or constraint function for large optimization problems. Linear interpolation between fixed gridpoints can also be used to approximate generic, nonlinear functions, allowing us to approximately solve problems using mixed-integer linear optimization methods. Toward this end, we develop two different explicit formulations of piecewise-linear functions and discuss the consequences of integrating the formulations into an optimization problem.     

Approximations of Relaxed Optimal Control Problems

In the present paper, we investigate an approximation technique for relaxed optimal control problems. We study control processes governed by ordinary differential equations in the presence of state, target, and integral constraints. A variety of approximation schemes have been recognized as powerful tools for the theoretical studying and practical solving of Infinite-dimensional optimization problems. On the other hand, theoretical approaches to the relaxed optimal control problem with constraints are not sufficiently advanced to yield numerically tractable schemes. The explicit approximation of the compact control set makes it possible to reduce the sophisticated relaxed problem to an auxiliary optimization problem. A given trajectory of the relaxed problem can be approximated by trajectories of the auxiliary problem. An optimal solution of the introduced optimization problem provides a basis for the construction of minimizing sequences for the original optimal control problem. We describe how to carry out the numerical calculations in the context of nonlinear programming and establish the convergence properties of the obtained approximations.The authors thank the referees for helpful comments and suggestions.