Admissible slopes for monotone and convex interpolation

Summary In many applications, interpolation of experimental data exhibiting some geometric property such as nonnegativity, monotonicity or convexity is unacceptable unless the interpolant reflects these characteristics. This paper identifies admissible slopes at data points of variousC ¹ interpolants which ensure a desirable shape. We discuss this question, in turn for the following function classes commonly used for shape preserving interpolations: monotone polynomials,C ¹ monotone piecewise polynomials, convex polynomials, parametric cubic curves and rational functions.     

Eigenvalues of Hermite and Laguerre ensembles: large beta asymptotics

In this paper we examine the zero and first order eigenvalue fluctuations for the β-Hermite and β-Laguerre ensembles, using tridiagonal matrix models, in the limit as β→∞. We prove that the fluctuations are described by multivariate Gaussians of covariance O(1/β), centered at the roots of a corresponding Hermite (Laguerre) polynomial. The covariance matrix itself is expressed as combinations of Hermite or Laguerre polynomials respectively.We show that the approximations are of real value even for small β; we can use them to approximate the true functions even for the traditional β=1,2,4 values.     

On characterizing collections arising from N-gons in the plane

Consider a labeled convex n-gon in the plane and a point x in general position in the interior. We associate with this a collection (x) of 3-element sets, namely those 3-element subsets of the vertices which contain x in their interior. We investigate the problem of characterizing those collections of subsets which can be represented in this matter. We give a complete characterization of this and a related problem.School of Mathematics, University of Minnesota, Minneapolis, MN 55455. This work was supported in part by NSF grant DMS-8700995.Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, England. Work done while Hill Visiting Professor at the University of Minnesota.     

Free arrangements and rhombic tilings

Let Z be a centrally symmetric polygon with integer side lengths. We answer the following two questions:

1. When is the associated discriminantal hyperplane arrangementfree in the sense of Saito and Terao?
2. When areall of the tilings of Z by unit rhombicoherent in the sense of Billera and Sturmfels?

Surprisingly, the answers to these two questions are very similar. Furthermore, by means of an old result of MacMahon on plane partitions and some new results of Elnitsky on rhombic tilings, the answer to the first question helps to answer the second. These results then also give rise to some interesting geometric corollaries. Consideration of the discriminantal arrangements for some particular octagons leads to a previously announced counterexample to the conjecture by Saito [ER2] that the complexified complement of a real free arrangement is aK (π, 1) space.     

Zeeman effect and stark splitting of the electronic states of the rare-earth ion in the paramagnetic terbium garnets Tb<Subscript>3</Subscript>Ga<Subscript>5</Subscript>O<Subscript>12</Subscript> and Tb<Subscript>3</Subscript>Al<Subscript>5</Subscript>O<Subscript>12</Subscript>

The Zeeman effect in the ⁷ F ₆ → ⁵ D ₄ absorption band of the Tb³⁺ ion in the paramagnetic garnets Tb₃Ga₅O₁₂ and Tb₃Al₅O₁₂ was studied. The field dependences of the Zeeman splitting of some absorption lines are found to exhibit unusual behavior: as the magnetic field increases, the band splitting decreases rather than increases. Symmetry analysis relates these lines to 4f → 4f electron transitions of the doublet-quasi-doublet or quasi-doublet-doublet type, for which the field dependences of the splitting differ radically from the well-known field dependences of the Zeeman splitting for quasi-doublet-quasi-doublet or quasi-doublet-singlet transitions in a longitudinal magnetic field.     

Dynamics of the chain of forced oscillators with long-range interaction: from synchronization to chaos

We consider a chain of nonlinear oscillators with long-range interaction of the type 1l(1+alpha), where l is a distance between oscillators and 0相似文献   

Structural and magnetic ordering in nanoparticles of complex 3<Emphasis Type="Italic">d</Emphasis>- and 4<Emphasis Type="Italic">f</Emphasis>-element oxides in germanate glass

Germanate glasses doped simultaneously with Fe and rare earth elements Gd, Tb, Dy, and Ho in concentrations of several percent have been investigated. It is shown that the phase state of paramagnetic elements changes from isolated ions and small clusters in the initial glass to crystalline magnetic nanoparticles in the samples subjected to heat treatments. The field dependences of the magnetization of such samples are characterized by magnetic saturation in fields of about 1 kOe and large Faraday rotation in the near-IR region (up to 10 deg/cm in the range from 700 to 800 nm).     

Behavioral and computational aspects of language and its acquisition

One of the greatest challenges facing the cognitive sciences is to explain what it means to know a language, and how the knowledge of language is acquired. The dominant approach to this challenge within linguistics has been to seek an efficient characterization of the wealth of documented structural properties of language in terms of a compact generative grammar—ideally, the minimal necessary set of innate, universal, exception-less, highly abstract rules that jointly generate all and only the observed phenomena and are common to all human languages. We review developmental, behavioral, and computational evidence that seems to favor an alternative view of language, according to which linguistic structures are generated by a large, open set of constructions of varying degrees of abstraction and complexity, which embody both form and meaning and are acquired through socially situated experience in a given language community, by probabilistic learning algorithms that resemble those at work in other cognitive modalities.     

Adapting support vector machine methods for horserace odds prediction

The methodology of Support Vector Machine Methods is adapted in a straightforward manner to enable the analysis of stratified outcomes such as horseracing results. As the strength of the Support Vector Machine approach lies in its apparent ability to produce generalisable models when the dimensionality of the inputs is large relative to the the number of observations, such a methodology would appear to be particularly appropriate in the horseracing context, where often the number of input variables deemed as being potentially relevant can be difficult to reconcile with the scarcity of relevant race results. The methods are applied to a relatively small (200 races in-sample) sample of Australian racing data and tested on 100 races out-of-sample with promising results, especially considering the relatively large number (12) of input variables used.     

Optical absorption spectrum and local symmetry of Tm ion in TmAl3(BO3)4 crystal

With the help of group theory analysis of absorption spectra of the transition ³H₆ → ³F₃ of Tm³⁺ ion in TmAl₃(BO₃)₄ crystal, measured at several temperatures from 1.8 till 293 K, it has been shown, that the local symmetry of the Tm³⁺ environment is C₃ and it decreases to C₁ at a low temperature. Effective selection rules and polarizations of lines at high enough temperatures (when the line-width is larger than the splitting in C₃ symmetry) have been obtained.