Construction of multivariate compactly supported orthonormal wavelets

We propose a constructive method to find compactly supported orthonormal wavelets for any given compactly supported scaling function φ in the multivariate setting. For simplicity, we start with a standard dilation matrix 2I_2×2 in the bivariate setting and show how to construct compactly supported functions ψ₁,. . .,ψ_n with n>3 such that {2^kψ_j(2^kx−ℓ,2^ky−m), k,ℓ,m∈Z, j=1,. . .,n} is an orthonormal basis for L₂(ℝ²). Here, n is dependent on the size of the support of φ. With parallel processes in modern computer, it is possible to use these orthonormal wavelets for applications. Furthermore, the constructive method can be extended to construct compactly supported multi-wavelets for any given compactly supported orthonormal multi-scaling vector. Finally, we mention that the constructions can be generalized to the multivariate setting. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C15, 42C30.     

用边际法求解－类设备的最优分配问题

本文针对一类设备的最优分配问题，将利润表转化为边际利润表，并建立了相应的寻优方程，通过实例说明，在一定情况下，用改进后的算法比常规算法节省一定的计算工作量．     

二阶完全对称张量空间中可分元素的坐标间的齐次关系

本文给出了二阶完全对称张量间Ｖ^（２）中可分元素的坐标间齐次关系，特别地，这种齐次关系可以由行列式表示出来。并由此得到了Ｖ^（２）中可分元素的一个等价条件。     

关于Witt向量之序列

对出现于公式Π_ｎ≥１１/(１＋ｄ_ｎｔ^ｎ／ｎ！)＝（１－ｔ）ｅ^ｔ中的整数ｄ_ｎ给出了精确估计     

Semi-parametric multivariate modelling when the marginals are the same

A model is developed for multivariate distributions which have nearly the same marginals, up to shift and scale. This model, based on “interpolation” of characteristic functions, gives a new notion of “correlation”. It allows straightforward nonparametric estimation of the common marginal distribution, which avoids the “curse of dimensionality” present when nonparametically estimating the full multivariate distribution. The method is illustrated with environmental monitoring network data, where multivariate modelling with common marginals is often appropriate.     

Infinite-variate wide-sense Markov processes and functional analysis for bounded operator-forming vectors

Let p, q be arbitrary parameter sets, and let $\text{H}$ be a Hilbert space. We say that x = (xⁱ)_i?q, xⁱ ? $\text{H}$, is a bounded operator-forming vector (?$\text{H}$_F^q) if the Gram matrix 〈x, x〉 = [(xⁱ, x^j)]_i?q,j?q is the matrix of a bounded (necessarily ≥ 0) operator on $\text{l}{}_{\mathrm{q}}{}^2$, the Hilbert space of square-summable complex-valued functions on q. Let A be p × q, i.e., let A be a linear operator from $\text{l}{}_{\mathrm{q}}{}^2$ to $\text{l}{}_{\mathrm{p}}{}^2$. Then exists a linear operator ǎ from (the Banach space) $\text{H}$_F^q to $\text{H}$_F^p on $\text{D}$(A) = {x:x ? $\text{H}$_F^q, $\text{A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is p × q bounded on $\text{l}{}_{\mathrm{q}}{}^2$} such that y = ǎx satisfies y^j?σ(x) = {space spanned by the xⁱ}, 〈y, x〉 = A〈x, x〉 and $\text{〈y, y〉 = A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}^1$. This is a generalization of our earlier [J. Multivariate Anal.4 (1974), 166–209; 6 (1976), 538–571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes.     

Some properties of bivariate Gumbel Type A distributions with proportional hazard rates

We call a set of univariate distributions with the same mathematical form but different parameter values a family $\text{J}$. Consider a bivariate Gumbel Type A survival distribution, S₁₂(x₁, x₂), defined in (2.1), for which both marginal distributions, S₁(x₁), S₂(x₂), belong to the same family, $\text{J}$ of distributions. It is proved in this paper that subject to weak conditions, the crude hazard rates, h₁(t) and h₂(t), are proportional if and only if the marginal hazard rates, λ₁(t) and λ₂(t), are proportional (Theorem 1). It is also shown that the survival functions of W = min(X₁, X₂), and of the identified minimum, W_i = X_i, for X_i < X_j, j ≠ i, belong to the same family $\text{J}$ as do S₁(x₁), S₂(x₂) (Corollary 1). Counter-examples of distributions other than Gumbel Type A, for which these properties do not hold, are given. Some applications to the analysis of competing risks, using a family of Gompertz distributions, are discussed.     

The time path of the marginal cost of oil: The turning point and the subsequent upward drift

The gradual exhaustion of existing deposits of a depletable non-renewable resource such as oil tends to shift the supply price curve of the resource upwards, increasing its marginal cost. Advances in technologies for exploration and production act as a brake on such upward shifts. Thus, there is a tug-of-war between the gradual exhaustion of existing deposits and technological progress. Using a recently developed constrained least-squares regression technique, we demonstrate that technological progress was the dominant force of the two during the first part of this century, causing a secular drop in marginal costs, but that this situation eventually was reversed, and that the gradual exhaustion of deposits gained the upper hand, causing marginal costs to increase. The turning point occurred around 1971–72. We also discuss the forecasting of the possible current upward drift of marginal costs.     

Conditioning convex and nonconvex problems

Two ways of defining a well-conditioned minimization problem are introduced and related, with emphasis on the quantitative aspects. These concepts are used to study the behavior of the solution sets of minimization problems for functions with connected sublevel sets, generalizing results of Attouch-Wets in the convex case. Applications to continuity properties of subdifferentials and to projection mappings are pointed out.We are grateful to M. Valadier for pointing out, during a lecture by the author in Montpellier in October 1990 presenting the main results of the present paper, that existence results in Section 2 of the present paper can be dissociated from estimates.     

The modes of an ultra-cold strongly magnetized charged Bose gas

The modes of a strongly magnetized charged Bose gas are presented for ultra-low temperatures. For longitudinal oscillations propagating parallel to the magnetic field the dispersion relation is found to be dominated by the one-dimensional field-free plasmon dispersion relation as found by Alexandrov, Beere and Kabanov recently in reference [1], while for propagation perpendicular to the magnetic field they are found to be influenced by the cyclotron motion of the particles. Dispersion relations for these modes known as Bernstein modes are given near the cyclotron frequency and its first two harmonics. The dispersion relations for transverse modes in the system are then presented for the cases of photon propagation perpendicular and parallel to the direction of the magnetic field. Received: 3 July 1997 / Revised: 12 August 1997 / Accepted: 4 November 1997