Extraneous fixed points,basin boundaries and chaotic dynamics for Schröder and König rational iteration functions

Summary The Schröder and König iteration schemes to find the zeros of a (polynomial) functiong(z) represent generalizations of Newton's method. In both schemes, iteration functionsf _m (z) are constructed so that sequencesz _n+1 =f _m (z _n ) converge locally to a rootz ^* ofg(z) asO(|z _n –z ^*|^m). It is well known that attractive cycles, other than the zerosz ^*, may exist for Newton's method (m=2). Asm increases, the iteration functions add extraneous fixed points and cycles. Whether attractive or repulsive, they affect the Julia set basin boundaries. The König functionsK _m (z) appear to minimize such perturbations. In the case of two roots, e.g.g(z)=z ²–1, Cayley's classical result for the basins of attraction of Newton's method is extended for allK _m (z). The existence of chaotic {z _n } sequences is also demonstrated for these iteration methods.     

Diffusion approximation forGI/G/1 controlled queues

A queueing model is considered in which a controller can increase the service rate. There is a holding cost represented by functionh and the service cost proportional to the increased rate with coefficientl. The objective is to minimize the total expected discounted cost.Whenh andl are small and the system operates in heavy traffic, the control problem can be approximated by a singular stochastic control problem for the Brownian motion, namely, the so-called reflected follower problem. The optimal policy in this problem is characterized by a single numberz ^* so that the optimal process is a reflected diffusion in [0,z ^*]. To obtainz ^* one needs to solve a free boundary problem for the second order ordinary differential equation. For the original problem the policy which increases to the maximum the service rate when the normalized queue-length exceedsz ^* is approximately optimal.     

Boundary values of a convergent sequence of J-contractive matrix-functions

In this note it is proved that if W_n(z) are J-contractive matrix-functions which are meromorphic in the disk ¦z¦<1 (j–w=">_n ^* (z)JW_n(z) W^* (z)JW(z) \leqslant W_n^* (z)JW_n (z)W^* (z)JW(z) \leqslant W_n^* (z)JW_n (z)     

An extension of Karmarkar's algorithm for solving a system of linear homogeneous equations on the simplex

We present an extension of Karmarkar's algorithm for solving a system of linear homogeneous equations on the simplex. It is shown that in at most O(nL) steps, the algorithm produces a feasible point or proves that the problem has no solution. The complexity is O(n ² m ² L) arithmetic operations. The algorithm is endowed with two new powerful stopping criteria.     

Speeding up Karmarkar's algorithm for multicommodity flows

We show how to speed up Karmarkar's linear programming algorithm for the case of multicommodity flows. The special structure of the constraint matrix is exploited to obtain an algorithm for the multicommodity flow problem which requires O(s ^3.5 v ^2.5 eL) arithmetic operations, each operation being performed to a precision of O (L) bits. Herev is the number of vertices ande is the number of edges in the given network,s is the number of commodities, andL is bounded by the number of bits in the input. We obtain a speed up of the order of (e ^0.5/v ^0.5)+(e ^2.5/v ^2.5s²) over Karmarkar's modified algorithm which is substantial for dense networks. The techniques in the paper can also be used to speed up any interior point algorithm for any linear programming problem whose constraint matrix is structurally similar to the one in the multicommodity flow problem. Research supported by a fellowship from the Shell Foundation. Research supported by NSF under grant NSF DCR-8404239.     

The Operator Factorization Method in Inverse Obstacle Scattering

The standard factorization method from inverse scattering theory allows to reconstruct an obstacle pointwise from the normal far field operator F. The kernel of this method is the study of the first kind Fredholm integral equation (F^* F)^1/4 f = Φ_z with the right-hand part In this paper we extend the factorization method to cover some kinds of boundary conditions which leads to non-normal far field operators. We visualize the scatterer explicitly in terms of the singular system of the selfadjoint positive operator F_# = [(ReF)^* (ReF)]^1/2 + ImF. The following characterization criterium holds: a given point z is inside the obstacle if and only if the function Φ_z belongs to the range of F_#^1/2. Our operator approach provides the tool for treatment of a wide class of inverse elliptic problems.     

Differential subordinations concerning starlike functions

Denote byS ^* (⌕), (0≤⌕<1), the family consisting of functionsf(z)=z+a ₂z²+...+a_nzⁿ+... that are analytic and starlike of order ⌕, in the unit disc ⋎z⋎<1. In the present article among other things, with very simple conditions on μ, ⌕ andh(z) we prove the f’(z) (f(z)/z)^μ−1<h(z) implies f∈S^*(⌕). Our results in this direction then admit new applications in the study of univalent functions. In many cases these results considerably extend the earlier works of Miller and Mocanu [6] and others.     

The Generalised Dyson Circular Unitary Ensemble: Asymptotic Distribution of the Eigenvalues at the Origin of the Spectrum

The first part of this paper is devoted to the study of F_N{\Phi_N} the orthogonal polynomials on the circle, with respect to a weight of type f = (1 − cos θ) ^α c where c is a sufficiently smooth function and ${\alpha > -\frac{1}{2}}${\alpha > -\frac{1}{2}}. We obtain an asymptotic expansion of the coefficients F^*(p)_N(1){\Phi^{*(p)}_{N}(1)} for all integer p where F^*_N{\Phi^*_N} is defined by F^*_N (z) = z^N [`(F)]_N(\frac1z) (z 1 0){\Phi^*_N (z) = z^N \bar \Phi_N(\frac{1}{z})\ (z \not=0)}. These results allow us to obtain an asymptotic expansion of the associated Christofel–Darboux kernel, and to compute the distribution of the eigenvalues of a family of random unitary matrices. The proof of the results related to the orthogonal polynomials are essentially based on the inversion of the Toeplitz matrix associated to the symbol f.     

A new full-Newton step <Emphasis Type="Italic">O</Emphasis>(<Emphasis Type="Italic">n</Emphasis>) infeasible interior-point algorithm for semidefinite optimization

Interior-point methods for semidefinite optimization have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, the second author designed a primal-dual infeasible interior-point algorithm with the currently best iteration bound for linear optimization problems. Since the algorithm uses only full Newton steps, it has the advantage that no line-searches are needed. In this paper we extend the algorithm to semidefinite optimization. The algorithm constructs strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem, close to their central paths. Two types of full-Newton steps are used, feasibility steps and (ordinary) centering steps, respectively. The algorithm starts from strictly feasible iterates of a perturbed pair, on its central path, and feasibility steps find strictly feasible iterates for the next perturbed pair. By using centering steps for the new perturbed pair, we obtain strictly feasible iterates close enough to the central path of the new perturbed pair. The starting point depends on a positive number ζ. The algorithm terminates either by finding an ε-solution or by detecting that the primal-dual problem pair has no optimal solution (X ^*,y ^*,S ^*) with vanishing duality gap such that the eigenvalues of X ^* and S ^* do not exceed ζ. The iteration bound coincides with the currently best iteration bound for semidefinite optimization problems.     

The generating function for traces of singular moduli and an application to Borcherds products

Let p be a prime and f(z)=∑ n a(n)q n be a weakly holomorphic modular function for \varGamma 0*(p2)\varGamma _{0}^{*}(p^{2}) with a(0)=0. We use Bruinier and Funke’s work to find the generating series of modular traces of f(z) as Jacobi forms. And as an application we construct Borcherds products related to the Hauptmoduln jp2*j_{p^{2}}^{*} for genus zero groups \varGamma 0*(p2)\varGamma _{0}^{*}(p^{2}).     

Near-circularity of the error curve in complex Chebyshev approximation

Let f(z) be analytic on the unit disk, and let p^*(z) be the best (Chebyshev) polynomial approximation to f(z) on the disk of degree at most n. It is observed that in typical problems the “error curve,” the image of the unit circle under (f − p^*)(z), often approximates to a startling degree a perfect circle with winding number n + 1. This phenomenon is approached by consideration of related problems whose error curves are exactly circular, making use of a classical theorem of Carathéodory and Fejér. This leads to a technique for calculating approximations in one step that are roughly as close to best as the best approximation error curve is close to circular, and hence to strong theorems on near-circularity as the radius of the domain shrinks to 0 or as n increases to ∞. As a computational example, very tight bounds are given for approximation of e^z on the unit disk. The generality of the near-circularity phenomenon (more general domains, rational approximation) is discussed.     

Karmarkar's linear programming algorithm and Newton's method

This paper describes a full-dimensional version of Karmarkar's linear programming algorithm, theprojective scaling algorithm, which is defined for any linear program in ⁿ having a bounded, full-dimensional polytope of feasible solutions. If such a linear program hasm inequality constraints, then it is equivalent under an injective affine mappingJ: ^{n m} to Karmarkar's original algorithm for a linear program in ^m havingm—n equality constraints andm inequality constraints. Karmarkar's original algorithm minimizes a potential functiong(x), and the projective scaling algorithm is equivalent to that version of Karmarkar's algorithm whose step size minimizes the potential function in the step direction.The projective scaling algorithm is shown to be a global Newton method for minimizing a logarithmic barrier function in a suitable coordinate system. The new coordinate system is obtained from the original coordinate system by a fixed projective transformationy = (x) which maps the hyperplaneH _opt ={x:c, x =c ₀} specified by the optimal value of the objective function to the hyperplane at infinity. The feasible solution set is mapped under to anunbounded polytope. Letf _LB(y) denote the logarithmic barrier function associated to them inequality constraints in the new coordinate system. It coincides up to an additive constant with Karmarkar's potential function in the new coordinate system. Theglobal Newton method iterate y ^* for a strictly convex functionf(y) defined on a suitable convex domain is that pointy ^* that minimizesf(y) on the search ray {y+ v _N(y): 0} wherev _N(y) =–(² f(y))^–1(f(y)) is the Newton's method vector. If {x ^(k)} are a set of projective scaling algorithm iterates in the original coordinate system andy ^(k) =(x ^(k)) then {y ^(k)} are a set of global Newton method iterates forf _LB(y) and conversely.Karmarkar's algorithm with step size chosen to minimize the potential function is known to converge at least at a linear rate. It is shown (by example) that this algorithm does not have a superlinear convergence rate.     

多叶解析函数族子类的一些结果(英文)

本文研究了在单位开圆盘U={z:|z|1}内多叶解析的函数族G_(p,c)~*(a,b,σ)的性质.利用函数论的方法,获得了G_(p,c)~*(a,b,σ)族相关的准哈达玛乘积的一般化结果及G_(p,c)~*(a,b,σ)的极值点与支撑点.推广了先前相应的一些研究工作。     

Convex Optimal Designs for Compound Polynomial Extrapolation

The extrapolation design problem for polynomial regression model on the design space [–1,1] is considered when the degree of the underlying polynomial model is with uncertainty. We investigate compound optimal extrapolation designs with two specific polynomial models, that is those with degrees |m, 2m}. We prove that to extrapolate at a point z, |z| > 1, the optimal convex combination of the two optimal extrapolation designs | _m ^* (z), _2m ^* (z)} for each model separately is a compound optimal extrapolation design to extrapolate at z. The results are applied to find the compound optimal discriminating designs for the two polynomial models with degree |m, 2m}, i.e., discriminating models by estimating the highest coefficient in each model. Finally, the relations between the compound optimal extrapolation design problem and certain nonlinear extremal problems for polynomials are worked out. It is shown that the solution of the compound optimal extrapolation design problem can be obtained by maximizing a (weighted) sum of two squared polynomials with degree m and 2m evaluated at the point z, |z| > 1, subject to the restriction that the sup-norm of the sum of squared polynomials is bounded.     

A randomized scheme for speeding up algorithms for linear and convex programming problems with high constraints-to-variables ratio

We extend Clarkson's randomized algorithm for linear programming to a general scheme for solving convex optimization problems. The scheme can be used to speed up existing algorithms on problems which have many more constraints than variables. In particular, we give a randomized algorithm for solving convex quadratic and linear programs, which uses that scheme together with a variant of Karmarkar's interior point method. For problems withn constraints,d variables, and input lengthL, ifn = (d ²), the expected total number of major Karmarkar's iterations is O(d ²(logn)L), compared to the best known deterministic bound of O( L). We also present several other results which follow from the general scheme.     

Solving a class of variational inequalities with inexact oracle operators

Consider a class of variational inequality problems of finding ${x^*\in S}Consider a class of variational inequality problems of finding x^* ? S{x^*\in S}, such that

f(x*)T (z-x*) 3 0,    "z ? S,f(x^*)^\top (z-x^*)\geq 0,\quad \forall z\in S,      17. Area of Julia sets of holomorphic self-maps onC *      Fang Liping 《数学学报(英文版)》1993,9(2):160-165 It is a general problem to study the measure of Julia sets. There are a lot of results for rational and entire functions. In this note, we describe the measure of Julia set for some holomorphic self-maps onC *. We'll prove thatJ(f) has positive area, wheref:C *→C *,f(z)=z m c P(z)+Q(1/z) ,P(z) andQ(z) are monic polynomials of degreed, andm is an integer.      18. Utility maximization under g*-expectation      Yongxu Jiang  Peng Luo  Lihe Wang 《随机分析与应用》2016,34(4):644-661 In this article, we introduce a nonlinear expectation, called g*-expectation, based on g-expectation and consider the optimal utility under g*-expectation in the market with a risk-free bond and d risky stocks in finite trading interval [0, T]. We construct a stochastic family by taking advantage of the comparison theorem of backward stochastic differential equations and the g*-martingale. We generalize the results of Hu et al. (Annals of Applied Probability 28(2):1691–1712, 2005), and obtain the explicit forms of the optimal trading strategies both for exp?-utility and the power utility, when g(t, z) = βt|z|2 + γtz.      19. The Compact Law of the Iterated Logarithm for Multivariate Stochastic Approximation Algorithms      《随机分析与应用》2013,31(1):181-203 Abstract We consider a sequence (Z n ) n≥1 defined by a general multivariate stochastic approximation algorithm and assume that (Z n ) converges to a solution z* almost surely. We establish the compact law of the iterated logarithm for Z n by proving that, with probability one, the limit set of the sequence (Z n  ? z*) suitably normalized is an ellipsoid. We also give the law of the iterated logarithm for the l p norms, p ∈ [1, ∞], of (Z n  ? z*).      20. Trust region algorithms for the nonlinear least distance problem      H. -P. Helfrich  D. Zwick 《Numerical Algorithms》1995,9(1):171-179 The nonlinear least distance problem is a special case of equality constrained optimization. Let a curve or surface be given in implicit form via the equationf(x)=0,x R d , and letz R d be a fixed data point. We discuss two algorithms for solving the following problem:Find a point x * such that f(x * )=0and z–x *2 is minimal among all such x. The algorithms presented use thetrust region approach in which, at each iteration, an, approximation to the objective function or merit function is minimized in a given neighborhood (the trust region) of the current iterate. Among other things, this allows one to prove global convergence of the algorithm.Both authors were supported in part by the Deutsche Forschungsgemeinschaft, SFB 350.

Area of Julia sets of holomorphic self-maps onC *

It is a general problem to study the measure of Julia sets. There are a lot of results for rational and entire functions. In this note, we describe the measure of Julia set for some holomorphic self-maps onC ^*. We'll prove thatJ(f) has positive area, wheref:C ^*→C ^*,f(z)=z ^m c ^P(z)+Q(1/z) ,P(z) andQ(z) are monic polynomials of degreed, andm is an integer.     

Utility maximization under g*-expectation

In this article, we introduce a nonlinear expectation, called g*-expectation, based on g-expectation and consider the optimal utility under g*-expectation in the market with a risk-free bond and d risky stocks in finite trading interval [0, T]. We construct a stochastic family by taking advantage of the comparison theorem of backward stochastic differential equations and the g*-martingale. We generalize the results of Hu et al. (Annals of Applied Probability 28(2):1691–1712, 2005), and obtain the explicit forms of the optimal trading strategies both for exp?-utility and the power utility, when g(t, z) = β_t|z|² + γ_tz.     

The Compact Law of the Iterated Logarithm for Multivariate Stochastic Approximation Algorithms

Abstract

We consider a sequence (Z _n ) _n≥1 defined by a general multivariate stochastic approximation algorithm and assume that (Z _n ) converges to a solution z* almost surely. We establish the compact law of the iterated logarithm for Z _n by proving that, with probability one, the limit set of the sequence (Z _n  ? z*) suitably normalized is an ellipsoid. We also give the law of the iterated logarithm for the l ^p norms, p ∈ [1, ∞], of (Z _n  ? z*).     

Trust region algorithms for the nonlinear least distance problem

The nonlinear least distance problem is a special case of equality constrained optimization. Let a curve or surface be given in implicit form via the equationf(x)=0,x R ^d , and letz R ^d be a fixed data point. We discuss two algorithms for solving the following problem:Find a point x ^* such that f(x ^* )=0and z–x ^*₂ is minimal among all such x. The algorithms presented use thetrust region approach in which, at each iteration, an, approximation to the objective function or merit function is minimized in a given neighborhood (the trust region) of the current iterate. Among other things, this allows one to prove global convergence of the algorithm.Both authors were supported in part by the Deutsche Forschungsgemeinschaft, SFB 350.