On Periodic Asymmetric Extrapolation

In this paper, we develop a new technique for the asymmetric approximation of discrete functions arising in seasonal customer demand extrapolation. We adapt the technique for two different settings, the so-called pull and push models. Our main goal here is to find effectively extrapolations minimizing the loss. For bothmodels, we discuss several features related to sampling, approximation, and extrapolation.     

Extrapolation Algorithms for Filtering Series of Functions,and Treating the Gibbs Phenomenon

In this paper, we study the application of some convergence acceleration methods to Fourier series, to orthogonal series, and, more generally, to series of functions. Sometimes, the convergence of these series is slow and, moreover, they exhibit a Gibbs phenomenon, in particular when the solution or its first derivative has discontinuities. It is possible to circumvent, at least partially, these drawbacks by applying a convergence acceleration method (in particular, the -algorithm) or by approximating the series by a rational function (in particular, a Padé approximant). These issues are discussed and some numerical results are presented. We will see that adding its conjugate series as an imaginary part to a Fourier series greatly improves the efficiency of the algorithms for accelerating the convergence of the series and reducing the Gibbs phenomenon. Conjugacy for series of functions will also be considered.     

On the Extrapolation of Perturbation Series

We discuss certain special cases of algebraic approximants that are given as zeroes of so-called effective characteristic polynomials and their generalization to a multiseries setting. These approximants are useful for the convergence acceleration or summation of quantum mechanical perturbation series. Examples are given and some properties discussed.     

On the Upper Bound for the Expectation of the Norm of a Vector Uniformly Distributed on the Sphere and the Phenomenon of Concentration of Uniform Measure on the Sphere

Mathematical Notes - We consider the problem of constructing upper bounds for the expectation of the norm of a vector uniformly distributed on the Euclidean unit sphere.     

和形式的等价无穷小代换

在本文中,我们针对微积分教学中碰到的关于和形式等价无穷小代换求极限的问题进行了探讨,并推广了已知的结果.     

On the Pinsky Phenomenon

When n>2 it is well known that the spherical partial sums of n-fold Fourier integrals of a characteristic function of the ball D={x:|x|²<1} do not converge at the origin. In the mathematical literature this result is called “the Pinsky phenomenon”. In 1993 Pinsky established necessary and sufficient conditions for a piecewise smooth function, supported on D, which guarantee the convergence at the origin its spherical partial sums. We prove this result for nonspherical partial sums, i.e. for Fourier integrals under summation over domains bounded by level surfaces of elliptic polynomials.     

On the Equivalence of Probability Spaces

For a general class of Gaussian processes W, indexed by a sigma-algebra \({\mathscr {F}}\) of a general measure space \((M,{\mathscr {F}}, \sigma )\), we give necessary and sufficient conditions for the validity of a quadratic variation representation for such Gaussian processes, thus recovering \(\sigma (A)\), for \(A\in {\mathscr {F}}\), as a quadratic variation of W over A. We further provide a harmonic analysis representation for this general class of processes. We apply these two results to: (i) a computation of generalized Ito integrals and (ii) a proof of an explicit and measure-theoretic equivalence formula, realizing an equivalence between the two approaches to Gaussian processes, one where the choice of sample space is the traditional path space, and the other where it is Schwartz’ space of tempered distributions.     

On Dark Computably Enumerable Equivalence Relations

We study computably enumerable (c.e.) relations on the set of naturals. A binary relation R on ω is computably reducible to a relation S (which is denoted by R ≤ _c S) if there exists a computable function f(x) such that the conditions (xRy) and (f(x)Sf(y)) are equivalent for all x and y. An equivalence relation E is called dark if it is incomparable with respect to ≤ _c with the identity equivalence relation. We prove that, for every dark c.e. equivalence relation E there exists a weakly precomplete dark c.e. relation F such that E ≤ _c F. As a consequence of this result, we construct an infinite increasing ≤ _c -chain of weakly precomplete dark c.e. equivalence relations. We also show the existence of a universal c.e. linear order with respect to ≤ _c .     

On Equivalence of Linear Functional-Differential Equations

For linear functional-differential equations of the first order there are derived canonical forms and an effective criterion of equivalence with respect to the pointwise transformations.     

On Equivalence of Moduli of Smoothness

It is known that iffW^k_p, thenω_m(f, t)_ptω_m−1(f′, t)_p…. Its inverse with any constants independent offis not true in general. Hu and Yu proved that the inverse holds true for splinesSwith equally spaced knots, thusω_m(S, t)_ptω_m−1(S′, t)_pt²ω_m−2(S″, t)_p…. In this paper, we extend their results to splines with any given knot sequence, and further to principal shift-invariant spaces and wavelets under certain conditions. Applications are given at the end of the paper.     

On Abelian Branched Coverings of the Sphere

 We obtain an enumeration formula for the number of weak equivalence classes of the branched (?×ℬ)-covering of the sphere with m-branch points, when ? and ℬ are finite abelian groups with (|?|,|ℬ|)=1. From this, we can deduce an explicit formula for enumerating the weak equivalence classes of pseudofree spherical (ℤ _p ×ℤ _q )-actions on a given surface, when p and q are distinct primes. Received: August 10, 1999 Final version received: June 19, 2000     

On the Equivalence of Borel Sets

Mathematical Notes - A theorem on the existence of a definable selector for the relation $$E_{aleph_0}$$ on the $$boldsymbol{Delta}^0_2$$ -sets of the Baire space $$omega^omega$$ is proved.     

On Equivalence Relations in Ordered Sets

A relationship between equivalence relations in finite ordered sets and linear decompositions is presented.     

关于Finsler流形中的距离函数与测地球

利用 Finsler流形中的切曲率和旗曲率 ,研究了距离函数与测地球的凸性 ;指出了在单连通完备 Minkowski空间中测地球正好是平面的一部分     

On Filtered Polynomial Approximation on the Sphere

This paper considers filtered polynomial approximations on the unit sphere \(\mathbb {S}^d\subset \mathbb {R}^{d+1}\), obtained by truncating smoothly the Fourier series of an integrable function f with the help of a “filter” h, which is a real-valued continuous function on \([0,\infty )\) such that \(h(t)=1\) for \(t\in [0,1]\) and \(h(t)=0\) for \(t\ge 2\). The resulting “filtered polynomial approximation” (a spherical polynomial of degree \(2L-1\)) is then made fully discrete by approximating the inner product integrals by an N-point cubature rule of suitably high polynomial degree of precision, giving an approximation called “filtered hyperinterpolation”. In this paper we require that the filter h and all its derivatives up to \(\lfloor \tfrac{d-1}{2}\rfloor \) are absolutely continuous, while its right and left derivatives of order \(\lfloor \tfrac{d+1}{2}\rfloor \) exist everywhere and are of bounded variation. Under this assumption we show that for a function f in the Sobolev space \(W^s_p(\mathbb {S}^d),\ 1\le p\le \infty \), both approximations are of the optimal order \( L^{-s}\), in the first case for \(s>0\) and in the second fully discrete case for \(s>d/p\), conditions which in both cases cannot be weakened.     

On a Problem Related to Sphere and Circle Packing

The paper proves that a set which contains spheres centeredat all points of a set of Hausdorff dimension greater than 1must have positive Lebesgue measure. It also proves the correspondingresult for circles, provided that the set of centers has Hausdorffdimension greater than 3/2.     

On the Equivalence of Multiparameter Gaussian Processes

Our purpose is to characterize the multiparameter Gaussian processes, that is Gaussian sheets, that are equivalent in law to the Brownian sheet and to the fractional Brownian sheet. We survey multiparameter analogues of the Hitsuda, Girsanov and Shepp representations. As an application, we study a special type of stochastic equation with linear noise.      

On a Property of Functions on the Sphere

According to the Knaster conjecture, for any continuous function $f:S^{n - 1} \to \mathbb{R}$ and any $n$ -point subset of the sphere $S^{n - 1}$ , there exists a rotation mapping all the points of this subset to a level surface of the function $f$ . In the present paper, this conjecture is proved for the case in which ${n = p^1 }$ for an odd prime $p$ and the points lie on a circle and divide it into equal parts.     

On Unitary Equivalence of Analytic Toeplitz Operators

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On Morita Equivalence of Partially Ordered Monoids

We show that there is one-to-one correspondence between certain algebraically and categorically defined subobjects, congruences and admissible preorders of S-posets. Using preservation properties of Pos-equivalence functors between Pos-categories we deduce that if S and T are Morita equivalent partially ordered monoids and F:Pos _S →Pos _T is a Pos-equivalence functor then an S-poset A _S and the T-poset F(A _S ) have isomorphic lattices of (regular, downwards closed) subobjects, congruences and admissible preorders. We also prove that if A _S has some flatness property then F(A _S ) has the same property.