Polynomial-Time Algorithms for Multivariate Linear Problems with Finite-Order Weights: Worst Case Setting

We consider approximation of linear multivariate problems defined over weighted tensor product Hilbert spaces with finite-order weights. This means we consider functions of d variables that can be represented as sums of functions of at most q^* variables. Here, q^* is fixed (and presumably small) and d may be arbitrarily large. For the univariate problem, d = 1, we assume we know algorithms A1,ε that use O(ε−p) function or linear functional evaluations to achieve an error ε in the worst case setting. Based on these algorithms A_1,ε, we provide a construction of polynomial-time algorithms A_d,ε for the general d-variate problem with the number of evaluations bounded roughly by ε^−pd^q* to achieve an error ε in the worst case setting.     

Stability and bifurcation for periodic perturbations of the equilibrium of an oscillator with infinite or infinitesimal oscillation frequency

We consider small perturbations periodic in time of an oscillator whose restoring force has a leading term with exponent 3 or 1/3. The first case corresponds to oscillations with infinitesimal frequency and the second case to oscillations with infinite frequency. The smallness of the perturbation is determined both by the smallness of the considered neighborhood of the equilibrium point and by a small nonnegative parameter ε. For ε=0, the stability of the equilibrium point is studied. For ε>0, we find conditions for an invariant two-dimensional torus to branch off with “soft” or “rigid” loss of stability with loss index 1/2. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 323–335, March, 1999.     

Efficient piecewise-linear function approximation using the uniform metric

We given anO(n logn)-time method for finding a bestk-link piecewise-linear function approximating ann-point planar point set using the well-known uniform metric to measure the error, ε≥0, of the approximation. Our methods is based upon new characterizations of such functions, which we exploit to design an efficient algorithm using a plane sweep in “ε space” followed by several applications of the parametric-searching technique. The previous best running time for this problems wasO(n ²). This research was announced in preliminary form at the 10th ACM Symposium on Computational Geometry. The author was partially supported by the NSF and DARPA under Grant CCR-8908092, and by the NSF under Grants IRI-9116843 and CCR-9300079.     

Extremal Behavior of Diffusion Models in Finance

We investigate the extremal behavior of a diffusion X _t given by the SDE , where W is standard Brownian motion, μ is the drift term and σ is the diffusion coefficient. Under some appropriate conditions on X _t we prove that the point process of ε -upcrossings converges in distribution to a homogeneous Poisson process. As examples we study the extremal behavior of term structure models or asset price processes such as the Vasicek model, the Cox–Ingersoll–Ross model and the generalized hyperbolic diffusion. We also show how to construct a diffusion with pre-determined stationary density which captures any extremal behavior. As an example we introduce a new model, the generalized inverse Gaussian diffusion. This revised version was published online in July 2006 with corrections to the Cover Date.     

Cutting Plane Algorithms and Approximate Lower Subdifferentiability

A notion of boundedly ε-lower subdifferentiable functions is introduced and investigated. It is shown that a bounded from below, continuous, quasiconvex function is locally boundedly ε-lower subdifferentiable for every ε>0. Some algorithms of cutting plane type are constructed to solve minimization problems with approximately lower subdifferentiable objective and constraints. In those algorithms an approximate minimizer on a compact set is obtained in a finite number of iterations provided some boundedness assumption be satisfied.     

A poisson formula for solvable Lie groups

Given a probability measure μ on a locally compact second countable groupG the space of bounded μ-harmonic functions can be identified withL ^∞(η, α) where (η, α) is a BorelG-space with a σ-finite quasiinvariant measure α. Our goal is to show that when μ is an arbitrary spread out probability measure on a connected solvable Lie groupG then the μ-boundary (η, α) is a contractive homogeneous space ofG. Our approach is based on a study of a class of strongly approximately transitive (SAT) actions ofG. A BorelG-space η with a σ-finite quasiinvariant measure α is called SAT if it admits a probability measurev≪α, such that for every Borel set A with α(A)≠0 and every ε>0 there existsg∈G with ν(gA)>1−ε. Every μ-boundary is a standard SATG-space. We show that for a connected solvable Lie group every standard SATG-space is transitive, characterize subgroupsH⊆G such that the homogeneous spaceG/H is SAT, and establish that the following conditions are equivalent forG/H: (a)G/H is SAT; (b)G/H is contractive; (c)G/H is an equivariant image of a μ-boundary.     

Frequency-Domain Bounds for Nonnegative,Unsharply Band-Limited Functions

In this paper we present a technique for proving bounds of the Boas-Kac-Lukosz type for unsharply restricted functions with nonnegative Fourier transforms. Hence we consider functions F(x) ≥ 0, the Fourier transform f(u) of which satisfies |f(u)| ≤ ε for all u in a subset of (-∞,-1] ⋃ [1,∞), and are interested in bounds on |f(u)| for |u| ≤ 1. This technique gives rise to several "epsilonized" versions of the Boas-Kac-Lukosz bound (which deals with the case f(u) = 0, |u| ≥ 1). For instance, we find that |f(u)| ≤ L(u) + O(ε^2/3), where L(u) is the Boas-Kac-Lukosz bound, and show by means of an example that this version is the sharpest possible with respect to its behaviour as a function of ε as ε ↓ 0. The technique also turns out to be sufficiently powerful to yield the best bound as ε ↓ 0 in various other cases with less severe restrictions on f.     

Capturing the multivariate extremal index: bounds and interconnections

The multivariate extremal index function is a direction specific extension of the well-known univariate extremal index. Since statistical inference on this function is difficult it is desirable to have a broad characterization of its attributes. We extend the set of common properties of the multivariate extremal index function and derive sharp bounds for the entire function given only marginal dependence. Our results correspond to certain restrictions on the two dependence functions defining the multivariate extremal index, which are opposed to Smith and Weissman’s (1996) conjecture on arbitrary dependence functions. We show further how another popular dependence measure, the extremal coefficient, is closely related to the multivariate extremal index. Thus, given the value of the former it turns out that the above bounds may be improved substantially. Conversely, we specify improved bounds for the extremal coefficient itself that capitalize on marginal dependence, thereby approximating two views of dependence that have frequently been treated separately. Our results are completed with example processes.      

Length spectra and the Teichmüller metric for surfaces with boundary

We consider some metrics and weak metrics defined on the Teichmüller space of a surface of finite type with nonempty boundary, that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare these metrics and weak metrics with the Teichmüller metric. The comparison is on subsets of Teichmüller space which we call “ε ₀-relative e{\epsilon}-thick parts”, and whose definition depends on the choice of some positive constants ε ₀ and e{\epsilon}. Meanwhile, we give a formula for the Teichmüller metric of a surface with boundary in terms of extremal lengths of families of arcs.     

On the singularities of composition of entire functions

Let f be an entire function. A point zo is called a critical point of f if f'(zo) = 0, and f(zo) is called a critical value (or an algebraic singularity) of f. Next a ∈ (C^) is said to be an asymptotic value (or a transcendental singularity) of f if there exists a curve Γ:[0,1]→(C^) such that limt→1Γ(t)=∞ and limt→1(foΓ)(t)=a. In this paper we find relations between the asymptotic values of f, g and f o g, relations between critical points of f, g and f o g and also in the case when the two functions f and g are semi-conjugated with another entire function.     

Extremal configurations of certain problems on the capacity and harmonic measure

We study certain extremal problems concerning the capacity of a condenser and the harmonic measure of a compact set. In particular, we answer in the negative Tamrazov's question on the minimum of the capacity of a condenser. We find the solution to Dubinin's problem on the maximum of the harmonic measure of a boundary set in the family of domains containing no “long” segments of given inclination. It is also shown that the segment [1-L, 1] has the maximal harmonic measure at the point z=0 among all curves γ={z=z(t), 0≤t≤1}, z(0)=1, that lie in the unit disk and have given length L, 0<L<1. The proofs are based on Baernstein's method of *-functions, Dubinin's dissymmetrization method, and the method of extremal metrics. Bibliography: 21 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 170–195.     

The sum of coefficients of bounded univalent functions

We solve the maximal value problem for the functional in the class of functionsf(z)=z+a ₂z²+… that are holomorphic and univalent in the unit disk and satisfy the inequality |f(z)|<M. We prove that the Pick functions are extremal for this problem for sufficiently largeM whenever the set of indicesk ₁,…,k_m contains an even number. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 728–733, May, 1997. Translated by S. S. Anisov     

Dynamic Coresets

We give a dynamic data structure that can maintain an ε-coreset of n points, with respect to the extent measure, in O(log n) time per update for any constant ε>0 and any constant dimension. The previous method by Agarwal, Har-Peled, and Varadarajan requires polylogarithmic update time. For points with integer coordinates bounded by U, we alternatively get O(log log U) time. Numerous applications follow, for example, on dynamically approximating the width, smallest enclosing cylinder, minimum bounding box, or minimum-width annulus. We can also use the same approach to maintain approximate k-centers in time O(log n) (or O(log log U) if the spread is bounded by U) for any constant k and any constant dimension. For the smallest enclosing cylinder problem, we also show that a constant-factor approximation can be maintained in O(1) randomized amortized time on the word RAM. This work has been supported by NSERC. A preliminary version of this paper has appeared in Proc. 24th ACM Sympos Comput. Geom., 2008.     

Finiteness of conical algorithms with ω-subdivisions

In this paper the problem of finding the global optimum of a concave function over a polytope is considered. A well-known class of algorithms for this problem is the class of conical algorithms. In particular, the conical algorithm based on the so called ω-subdivision strategy is considered. It is proved that, for any given accuracy ε>0, this algorithm stops in a finite time by returning an ε-optimal solution for the problem, while it is convergent for ε=0. Received January 24, 1996 / Revised version received December 9, 1998 Published online June 11, 1999     

Extremal functions as divisors for kernels of Toeplitz operators

In this paper, an extremal function of a Banach space of analytic functions in the unit disk (not all functions vanishing at 0) is a function solving the extremal problem for functions f of norm 1. We study extremal functions of kernels of Toeplitz operators on Hardy spaces H^p, 1<p<∞. Such kernels are special cases of so-called nearly invariant subspaces with respect to the backward shift, for which Hitt proved that when p=2, extremal functions act as isometric divisors. We show that the extremal function is still a contractive divisor when p<2 and an expansive divisor when p>2 (modulo p-dependent multiplicative constants). We give examples showing that the extremal function may fail to be a contractive divisor when p>2 and also fail to be an expansive divisor when p<2. We discuss to what extent these results characterize the Toeplitz operators via invariant subspaces for the backward shift.     

Beyond all orders: Singular perturbations in a mapping

Summary We consider a family ofq-dimensional (q>1), volume-preserving maps depending on a small parameterε. Asε → 0⁺ these maps asymptote to flows which attain a heteroclinic connection. We show that for smallε the heteroclinic connection breaks up and that the splitting between its components scales withε likeε ^γexp[-β/ε]. We estimateβ using the singularities of theε → 0⁺ heteroclinic orbit in the complex plane. We then estimateγ using linearization about orbits in the complex plane. These estimates, as well as the assertions regarding the behavior of the functions in the complex plane, are supported by our numerical calculations. Deceased.     

Polynomial-Time Algorithms for Multivariate Linear Problems with Finite-Order Weights: Average Case Setting

We study multivariate linear problems in the average case setting with respect to a zero-mean Gaussian measure whose covariance kernel has a finite-order weights structure. This means that the measure is concentrated on a Banach space of d-variate functions that are sums of functions of at most q ^* variables and the influence of each such term depends on a given weight. Here q ^* is fixed whereas d varies and can be arbitrarily large. For arbitrary finite-order weights, based on Smolyak’s algorithm, we construct polynomial-time algorithms that use standard information. That is, algorithms that solve the d-variate problem to within ε using of order function values modulo a power of ln ε ⁻¹. Here p is the exponent which measures the difficulty of the univariate (d=1) problem, and the power of ln ε ⁻¹ is independent of d. We also present a necessary and sufficient condition on finite-order weights for which we obtain strongly polynomial-time algorithms, i.e., when the number of function values is independent of d and polynomial in ε ⁻¹. The exponent of ε ⁻¹ may be, however, larger than p. We illustrate the results by two multivariate problems: integration and function approximation. For the univariate case we assume the r-folded Wiener measure. Then p=1/(r+1) for integration and for approximation.      

Zero-Sum Ergodic Semi-Markov Games with Weakly Continuous Transition Probabilities

Zero-sum ergodic semi-Markov games with weakly continuous transition probabilities and lower semicontinuous, possibly unbounded, payoff functions are studied. Two payoff criteria are considered: the ratio average and the time average. The main result concerns the existence of a lower semicontinuous solution to the optimality equation and its proof is based on a fixed-point argument. Moreover, it is shown that the ratio average as well as the time average payoff stochastic games have the same value. In addition, one player possesses an ε-optimal stationary strategy (ε>0), whereas the other has an optimal stationary strategy. A. Jaśkiewicz is on leave from Institute of Mathematics and Computer Science, Wrocław University of Technology. This work is supported by MNiSW Grant 1 P03A 01030.     

Existence of limits of maximal means

For the class II(ℝ ^m ) of continuous almost periodic functionsf: ℝ ^m → ℝ, we consider the problem of the existence of the limit

Wannier Functions for Quasiperiodic Finite-Gap Potentials

We consider Wannier functions of quasiperiodic g-gap (g ≥ 1) potentials and investigate their main properties. In particular, we discuss the problem of averaging that underlies the definition of the Wannier functions for both periodic and quasiperiodic potentials and express Bloch functions and quasimomenta in terms of hyperelliptic σ-functions. Using this approach, we derive a power series for the Wannier function for quasiperiodic potentials valid for |x| ≃ 0 and an asymptotic expansion valid at large distances. These functions are important in a number of applied problems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 234–256, August, 2005.