Tractability of Multivariate Approximation over a Weighted Unanchored Sobolev Space

We study d-variate L ₂-approximation for a weighted unanchored Sobolev space having smoothness m≥1. This space is equipped with an unusual norm which is, however, equivalent to the norm of the d-fold tensor product of the standard Sobolev space. One might hope that the problem should become easier as its smoothness increases. This is true for our problem if we are only concerned with asymptotic analysis: the nth minimal error is of order n ^?(m?δ) for any δ>0. However, it is unclear how long we need to wait before this asymptotic behavior kicks in. How does this waiting period depend on d and m? It is easy to prove that no matter how the weights are chosen, the waiting period is at least m ^d , even if the error demand ε is arbitrarily close to 1. Hence, for m≥2, this waiting period is exponential in d, so that the problem suffers from the curse of dimensionality and is intractable. In other words, the fact that the asymptotic behavior improves with m is irrelevant when d is large. So we will be unable to vanquish the curse of dimensionality unless m=1, i.e., unless the smoothness is minimal. In this paper, we prove the more difficult fact that our problem can be tractable if m=1. That is, we can find an ε-approximation using polynomially-many (in d and ε ^?1) information operations, even if only function values are permitted. When m=1, it is even possible for the problem to be strongly tractable, i.e., we can find an ε-approximation using polynomially-many (in ε ^?1) information operations, independently of d. These positive results hold when the weights of the Sobolev space decay sufficiently quickly or are bounded finite-order weights, i.e., the d-variate functions we wish to approximate can be decomposed as sums of functions depending on at most ω variables, where ω is independent of d.     

Polynomial Approximation of Functions on a Quasi-Smooth Arc with Hermitian Interpolation

We construct polynomial approximations for continuous functions f defined on a quasi-smooth (in the sense of Lavrentiev) arc L in the complex plane which simultaneously interpolate f and its derivatives at given points of L.     

Wavelet Approximation of Distributions with Bounded Variation Derivatives

Let BV ^r denote the space of distributions f such that the distributional derivatives D ^α f with |α|≤r exist as measures of bounded variation. This paper discusses estimates for wavelet coefficients of BV ^r distributions, direct (Jackson) and inverse (Bernstein) inequalities for n-term approximation of elements of BV ^r in the L _p spaces using compactly supported wavelets. In particular, optimal rates of approximation are established. Linear approximation in similar contexts is also considered for comparison. This research was supported by the 2003–2007 Academic Grant of Prof. P. Wojtaszczyk awarded by the Foundation for Polish Science. Part of this research was supported within the HASSIP framework.     

Infinite-Dimensional Quadrature and Approximation of Distributions

We study numerical integration of Lipschitz functionals on a Banach space by means of deterministic and randomized (Monte Carlo) algorithms. This quadrature problem is shown to be closely related to the problem of quantization and to the average Kolmogorov widths of the underlying probability measure. In addition to the general setting, we analyze, in particular, integration with respect to Gaussian measures and distributions of diffusion processes. We derive lower bounds for the worst case error of every algorithm in terms of its cost, and we present matching upper bounds, up to logarithms, and corresponding almost optimal algorithms. As auxiliary results, we determine the asymptotic behavior of quantization numbers and Kolmogorov widths for diffusion processes.      

Martingale Approximation and Optimality of Some Conditions for the Central Limit Theorem

Let (X _i ) be a stationary and ergodic Markov chain with kernel Q and f an L ² function on its state space. If Q is a normal operator and f=(I?Q)^1/2 g (which is equivalent to the convergence of \(\sum_{n=1}^{\infty}\frac{\sum_{k=0}^{n-1}Q^{k}f}{n^{3/2}}\) in L ²), we have the central limit theorem [cf. (Derriennic and Lin in C.R. Acad. Sci. Paris, Sér. I 323:1053–1057, 1996; Gordin and Lif?ic in Third Vilnius conference on probability and statistics, vol. 1, pp. 147–148, 1981)]. Without assuming normality of Q, the CLT is implied by the convergence of \(\sum_{n=1}^{\infty}\frac{\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}}{n^{3/2}}\), in particular by \(\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}=o(\sqrt{n}/\log^{q}n)\), q>1 by Maxwell and Woodroofe (Ann. Probab. 28:713–724, 2000) and Wu and Woodroofe (Ann. Probab. 32:1674–1690, 2004), respectively. We show that if Q is not normal and f∈(I?Q)^1/2 L ², or if the conditions of Maxwell and Woodroofe or of Wu and Woodroofe are weakened to \(\sum_{n=1}^{\infty}c_{n}\frac{\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}}{n^{3/2}}<\infty\) for some sequence c _n ↘0, or by \(\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}=O(\sqrt{n}/\log n)\), the CLT need not hold.     

Approximation and contamination bounds for probabilistic programs

In many applications of manufacturing and service industries, the quality of a process is characterized by the functional relationship between a response variable and one or more explanatory variables. Profile monitoring is for checking the stability of this relationship over time. In some situations, multiple profiles are required in order to model the quality of a product or process effectively. General multivariate linear profile monitoring is particularly useful in practice due to its simplicity and flexibility. However, in such situations, the existing parametric profile monitoring methods suffer from a drawback in that when the profile parameter dimensionality is large, the detection ability of the procedures commonly used T ²-type charting statistics is likely to decline substantially. Moreover, it is also challenging to isolate the type of profile parameter change in such high-dimensional circumstances. These issues actually inherit from those of the conventional multivariate control charts. To resolve these issues, this paper develops a new methodology for monitoring general multivariate linear profiles, including the regression coefficients and profile variation. After examining the connection between the parametric profile monitoring and multivariate statistical process control, we propose to apply a variable-selection-based multivariate control scheme to the transformations of estimated profile parameters. Our proposed control chart is capable of determining the shift direction automatically based on observed profile data. Thus, it offers a balanced protection against various profile shifts. Moreover, the proposed control chart provides an easy but quite effective diagnostic aid. A real-data example from the logistics service shows that it performs quite well in the application.     

Fourier Series and Approximation on Hexagonal and Triangular Domains

Several problems on Fourier series and trigonometric approximation on regular hexagonal and triangular domains are studied. The results include Abel and Cesàro summability of Fourier series, degree of approximation, and best approximation by trigonometric functions with both direct and inverse theorems. One of the objectives of this study is to demonstrate that Fourier series on spectral sets enjoy a rich structure that permits an extensive theory for Fourier series and approximation.     

Recursive Algorithms for Trailing Stop: Stochastic Approximation Approach

Trailing stops are often used in stock trading to limit the maximum of a possible loss and to lock in a profit. This work develops stochastic approximation algorithms to estimate the optimal trailing stop percentage. A stochastic optimization approach is proposed to recursively estimate the desired trailing stop percentage. A modification using projection is developed to ensure that the approximation sequence constructed stays in a reasonable range. Convergence of the algorithm is obtained. Moreover, interval estimates are constructed. Simulation examples are presented to compare our algorithm with Monte Carlo methods. Finally, we use real market data to demonstrate the algorithms.     

On Tightness and Weak Convergence in the Approximation of the Occupation Measure of Fractional Brownian Motion

In this paper we consider approximations of the occupation measure of the Fractional Brownian motion by means of some functionals defined on regularizations of the paths. In a previous article Berzin and León proved a cylindrical convergence to a Wiener process of conveniently rescaled functionals. Here we show the tightness of the approximation in the space of continuous functions endowed with the topology of uniform convergence on compact sets. This allows us to simplify the identification of the limit.     

On a Problem of Approximation by Linear Means

is valid, where ω(f,δ) is the modulus of continuity of f. Furthermore they asked that: (1.3) If Theorem A can be extended to matrix summability? Later, P. D. Kathal, A. S. B. Holland and B. N. Sahney partially answered this problem. They proved the following Theorem B If {Λ_(n,k)} is monotonic non-increasing for all 0≤k≤n and Λ_(n,k)≥0 (0≤k≤n),Λ_(n,k)=0 (k>n), then we have     

Polynomial Birth–Death Distribution Approximation in the Wasserstein Distance

The polynomial birth–death distribution (abbreviated, PBD) on ℐ={0,1,2,…} or ℐ={0,1,2,…,m} for some finite m introduced in Brown and Xia (Ann. Probab. 29:1373–1403, 2001) is the equilibrium distribution of the birth–death process with birth rates {α _i } and death rates {β _i }, where α _i ≥0 and β _i ≥0 are polynomial functions of i∈ℐ. The family includes Poisson, negative binomial, binomial, and hypergeometric distributions. In this paper, we give probabilistic proofs of various Stein’s factors for the PBD approximation with α _i =a and β _i =i+bi(i−1) in terms of the Wasserstein distance. The paper complements the work of Brown and Xia (Ann. Probab. 29:1373–1403, 2001) and generalizes the work of Barbour and Xia (Bernoulli 12:943–954, 2006) where Poisson approximation (b=0) in the Wasserstein distance is investigated. As an application, we establish an upper bound for the Wasserstein distance between the PBD and Poisson binomial distribution and show that the PBD approximation to the Poisson binomial distribution is much more precise than the approximation by the Poisson or shifted Poisson distributions.      

Approximation algorithms for MAX RES CUT with limited unbalanced constraints

Two kinds of MAX RES CUT problems, the MAX s?t CUT and the MAX s?t?v CUT, with limited unbalanced constraints are considered. Approximation algorithms used in Frieze and Jerrum (Integer Programming and Combinatorial Optimization, vol. 920, pp. 1–13, Springer, Berlin, 1995), Galbiati and Maffioli (Theor. Comput. Sci. 385:78–87, 2007), Han et al. (Math. Program. Ser. B 92:509–535, 2002) and Ye (Math. Programm. 90:101–111, 2001) are extended to the two MAX RES CUT problems. A special matrix P is constructed by which it can ensure that the given nodes s,t are feasible to equality constraints with probability one for the MAX s?t CUT and s,t,v are feasible to equality constraints with at least probability 0.912 for the MAX s?t?v CUT. A fussy greedy sizing-adjusted procedure is then proposed to confirm that the round solution is feasible for all constraints. We find these extensions are nontrivial and some interesting results about performance ratio are obtained for the MAX RES CUT problem with limited unbalanced constraints.     

On the order of a non-abelian representation group of a slim dense near hexagon

In this paper we study the possible orders of a non-abelian representation group of a slim dense near hexagon. We prove that if the representation group R of a slim dense near hexagon S is non-abelian, then R is a 2-group of exponent 4 and |R|=2 ^β , 1+NPdim(S)≤β≤1+dimV(S), where NPdim(S) is the near polygon embedding dimension of S and dimV(S) is the dimension of the universal representation module V(S) of S. Further, if β=1+NPdim(S), then R is necessarily an extraspecial 2-group. In that case, we determine the type of the extraspecial 2-group in each case. We also deduce that the universal representation group of S is a central product of an extraspecial 2-group and an abelian 2-group of exponent at most 4. This work was partially done when B.K. Sahoo was a Research Fellow at the Indian Statistical Institute, Bangalore Center with NBHM fellowship, DAE Grant 39/3/2000-R&D-II, Govt. of India.     

The Accuracy of Merging Approximation in Generalized St. Petersburg Games

Merging asymptotic expansions of arbitrary length are established for the distribution functions and for the probabilities of suitably centered and normalized cumulative winnings in a full sequence of generalized St. Petersburg games, extending the short expansions due to Csörg? (Acta Sci. Math. (Szeged) 73:297–331, 2007). These expansions are given in terms of suitably chosen members from the classes of subsequential semistable infinitely divisible asymptotic distribution functions and certain derivatives of these functions. The length of the expansion depends upon the tail parameter. Both uniform and nonuniform bounds are presented.     

Magnetically-Induced Buckling of a Whirling Conducting Rod with Applications to Electrodynamic Space Tethers

We study the effect of a magnetic field on the behaviour of a slender conducting elastic structure, motivated by stability problems of electrodynamic space tethers. Both static (buckling) and dynamic (whirling) instability are considered and we also compute post-buckling configurations. The equations used are the geometrically exact Kirchhoff equations. Magnetic buckling of a welded rod is found to be described by a surprisingly degenerate bifurcation, which is unfolded when both transverse anisotropy of the rod and angular velocity are considered. By solving the linearised equations about the (quasi-) stationary solutions, we find various secondary instabilities. Our results are relevant for current designs of electrodynamic space tethers and potentially for future applications in nano- and molecular wires.     

Approximation of integrable functions by general linear operators of their Fourier series at the Lebesgue points

Pointwise estimates of the deviation T _n,A,B f(⋅)−f(⋅) in terms of moduli of continuity [`(w)]_·f\bar{w}_{\cdot}f and w _⋅ f are proved. Analog results on norm approximation with remarks and corollaries are also given. In the results essentially weaker conditions than those in [2, Theorem 1, p. 437] are used.     

Remarks on a surplus-sharing rule from the Mishneh Torah

Rabbi Moshe ben Maimon (1135–1204), known as Moses Maimonides, ranks among the most distinguished philosophers of the Middle Ages. He is the renowned author of the Mishneh Torah, a comprehensive code of Jewish law. Book 12 (“Book of Acquisition”), Treatise 4 (“Agents and Partners”), of the Code of Maimonides is devoted in Chapter 4 to the allocation of the surplus from funds which a partnership invests in an indivisible input. The Rabbi’s case translates into a cooperative game where all intermediate coalitions are inessential. Standard axioms for cooperative-game solutions then suggest that the surplus be shared equally by the players, which is precisely the Maimonidean ruling. We show that this outcome can be preserved in spirit under much weaker assumptions on the worth of a game’s intermediate coalitions. We present results both for the nucleolus and the Shapley value of the underlying class of games.     

Stability of a Turnpike Phenomenon for a Discrete-Time Optimal Control System

We study the structure of solutions of a discrete-time control system with a compact metric space of states X which arises in economic dynamics. This control system is described by a nonempty closed set Ω⊂X×X which determines a class of admissible trajectories (programs) and by a bounded upper semicontinuous objective function v:Ω→R ¹ which determines an optimality criterion. We are interested in turnpike properties of the approximate solutions which are independent of the length of the interval, for all sufficiently large intervals. In the present paper, we show that these turnpike properties are stable under perturbations of the objective function v.     

Approximation by the Bernstein–Durrmeyer Operator on a Simplex

For the Jacobi-type Bernstein–Durrmeyer operator M _n,κ on the simplex T ^d of ℝ ^d , we proved that for f∈L ^p (W _κ ;T ^d ) with 1<p<∞,

Approximation Properties of a Special Class of Mathematical Expectation Operators

61-IntroductionLet(X.}"cNbeasequenceofindependentrandomvariableseachhavingthesamelatticepointdistributionwithprobabilityfunctionP(X,=a kh)=P*,expectationEX"=),l(x)andvarianceE(X"-ax")'=a2(x).WhereKeZ:={o,tl,l2,..'},a,harethepositivecon-stants,xeI=Y'=Ri8arealcontinuousparameter.LettheprobabilitydistributionofS,=ZX,beForacontinuousfunctionfontheinterval7'(possiblyinfinite)definethemathematicalex-pectationoperatorRemarks(l)UsingChebychev,sinequalitywegeteasily.(See[1,P,,,f,,.j)-(2)Manyo…