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INTEGRAL REPRESENTATION OF EVEN positive-definite functions OF TWO VARIABLES

UKRAINIAN MATHEMATICAL JOURNAL 2011年第6期63卷 981-992页

作者： Lopotko, O. V. Ukrainian Natl Forestry Univ Lvov Ukraine

We obtain an integral representation of even functions of two variables for which the kernel [k(1)(x + y) + k(2)(x - y)], x, y is an element of R(2), is positive definite.

关键词： INTEGRAL representations positive-definite functions VARIABLES (Mathematics) KERNEL functions ALGEBRAIC number theory functions (Mathematics) MATHEMATICAL analysis

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Subequivalence relations and positive-definite functions

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GROUPS GEOMETRY AND DYNAMICS 2009年第4期3卷 579-625页

作者： Ioana, Adrian Kechris, Alexander S. Tsankov, Todor CALTECH Dept Math Pasadena CA 91125 USA Univ Paris 06 F-75252 Paris 05 France

We study a positive-definite function associated with a countable, measure-preserving equivalence relation, which can be used to measure quantitatively the proximity of subequivalence relations. Combined with a co-inducing construction introduced by Epstein and earlier work of Ioana, this can be used to construct many mixing actions of countable groups and establish the non-classifiability, in a strong sense, of orbit equivalence of actions of nonamenable groups. We also discuss connections with percolation on Cayley graphs and the theory of costs.

关键词： Subequivalence relations positive-definite functions co-induced action orbit equivalence non-amenable groups classification percolation property (T) cost

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INEQUALITIES AND positive-definite functions ARISING FROM A PROBLEM IN MULTIDIMENSIONAL-SCALING

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ANNALS OF STATISTICS 1994年第1期22卷 406-438页

作者： BUJA, A LOGAN, BF REEDS, JA SHEPP, LA AT&T BELL LABS MURRAY HILLNJ 07974

We solve the following variational problem: Find the maximum of E E\\X-Y\\ subject to E\\X\\2 less-than-or-equal-to 1, where X and Y are i.i.d. random n-vectors, and \\ . \\ is the usual Euclidean norm on R(n). This problem arose from an investigation into multidimensional scaling, a data analytic method for visualizing proximity data. We show that the optimal X is unique and is (1) uniform on the surface of the unit sphere, for dimensions n greater-than-or-equal-to 3, (2) circularly symmetric with a scaled version of the radial density rho/(1 - rho2)1/2, 0 less-than-or-equal-to rho less-than-or-equal-to 1, for n = 2, and (3) uniform on an interval centered at the origin, for n = 1 (Plackett's theorem). By proving spherical symmetry of the solution, a reduction to a radial problem is achieved. The solution is then found using the Wiener-Hopf technique for (real) n < 3. The results are reminiscent of classical potential theory, but they cannot be reduced to it. Along the way, we obtain results of independent interest: for any i.i.d. random n-vectors X and Y, E \\ X - Y \\ less-than-or-equal-to E \\ X + Y \\. Further, the kernel K(p,beta)(x,y) = \\ x + y \\ p(beta) - \\ x - y \\ p(beta),x,y is-an-element-of R(n) and \\ x \\ p = (SIGMA\x(i)\p)1/p, is positive-definite, that is, it is the covariance of a random field, K(p,beta)(x,y) = E[Z(x)Z(y)] for some real-valued random process Z(x), for 1 less-than-or-equal-to p less-than-or-equal-to 2 and 0 < beta less-than-or-equal-to p less-than-or-qual-to 2 (but not for beta > p or p > 2 in general). Although this is an easy consequence of known results, it appears to be new in a strict sense. In the radial problem, the average distance D(r1, r2) between two spheres of radii r1 and r2 is used as a kernel. We derive properties of D(r1, r2), including nonnegative definiteness on signed measures of zero integral.

关键词： MULTIDIMENSIONAL SCALING MAXIMAL EXPECTED DISTANCE POTENTIAL THEORY INEQUALITIES positive-definite functions WIENER-HOPF TECHNIQUE

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Harmonic analysis on semigroups : theory of positive definite and related functions = 半群上的调...

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2012年

作者： Berg Christian

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Concentration of L ^p -bandlimited functions on discrete sets

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LITHUANIAN MATHEMATICAL JOURNAL 2014年第4期54卷 471-481页

作者： Norvidas, Saulius Vilnius State Univ Inst Math & Informat LT-08663 Vilnius Lithuania

For sigma > 0 and 1 a parts per thousand currency sign p a parts per thousand currency sign a, let B (sigma) (p) be the Bernstein space of all f a L (p) (a"e) such that the Fourier transform (in a distributional sense) is supported on [-sigma, sigma]. In this paper, we study a problem of maximal concentration in the l (p) -norm of irregular discrete fraction for f a B (sigma) (p) . More precisely, for a given sequence of real numbers {lambda (k) }kaZ that is sparse in the sense that inf {|lambda m -aEuro parts per thousand lambda k| : m not equal aEuro parts per thousand k} > 0, we investigate the inequality a (k) |f(lambda (k) )| (p) a parts per thousand currency signaEuro parts per thousand C (p) aEuro-faEuro- (p) (p) . An estimate of the concentration bound C (p) is given.

关键词： bandlimited functions entire functions of exponential type Plancherel-Polya inequalities positive-definite functions hermitian operators in Banach spaces

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Correlation functions for atmospheric data analysis

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QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY 1999年第559期125卷 2449-2464页

作者： Gneiting, T Univ Washington Dept Stat Seattle WA 98195 USA

Atmospheric data assimilation techniques rely on parametric models for spatial correlation functions. This article proposes and discusses various families of homogeneous and isotropic correlation models on Euclidean spaces and on the sphere. In particular, three simply parametrized classes of compactly supported, smooth, and analytically simple correlation functions are proposed. The first two classes approximate standard second- and third-order autoregressive functions, and a member of the third family approximates the Gaussian function within a maximal error of 0.0056. Furthermore, correlation models suggested previously for meteorological applications are checked for permissibility, with both positive and negative results.

关键词： compact support data assimilation isotropic correlation functions parametrized model positive-definite functions

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Non-linear Mittag-Leffler stabilisation of commensurate fractional-order non-linear systems

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IET CONTROL THEORY AND APPLICATIONS 2015年第5期9卷 681-690页

作者： Ding, Dongsheng Qi, Donglian Wang, Qiao Zhejiang Univ Coll Elect Engn Hangzhou 310027 Peoples R China

Mittag-Leffler stability is a property of fractional-order dynamical systems, also called fractional Lyapunov stability, requiring the evolution of the positive-definite functions to be Mittag-Leffler, rather than the exponential meaning in Lyapunov stability theory. Similarly, fractional Lyapunov function plays an important role in the study of Mittag-Leffler stability. The aim of this study is to create closed-loop systems for commensurate fractional-order non-linear systems (FONSs) with Mittag-Leffler stability. We extend the classical backstepping to fractional-order backstepping for stabilising (uncertain) FONSs. For this purpose, several conditions of control fractional Lyapunov functions for FONSs are investigated in terms of Mittag-Leffler stability. Within this framework, (uncertain) FONSs Mittag-Leffler stabilisation is solved via fractional-order backstepping and the global convergence of closed-loop systems is guaranteed. Finally, the efficiency and applicability of the proposed fractional-order backstepping are demonstrated in several examples.

关键词： nonlinear control systems uncertain systems stability Lyapunov methods closed loop systems convergence nonlinear Mittag–Leffler stabilisation commensurate fractional-order nonlinear systems Mittag–Leffler stability global convergence uncertain FONS fractional-order backstepping closed-loop systems fractional Lyapunov function Lyapunov stability theory positive-definite functions fractional Lyapunov stability fractional-order dynamical systems

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The Impact of positive-definite Moisture Transport on NWP Precipitation Forecasts

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MONTHLY WEATHER REVIEW 2009年第1期137卷 488-494页

作者： Skamarock, William C. Weisman, Morris L. Natl Ctr Atmospher Res Boulder CO 80307 USA

A positive-definite transport scheme for moisture is tested in a nonhydrostatic forecast model using convection-permitting resolutions. Use of the positive-definite scheme is found to significantly reduce the large positive bias in surface precipitation forecasts found in the non-positive-definite model forecasts, in particular at high precipitation thresholds. The positive-definite scheme eliminates spurious sources of water arising from the clipping of negative moisture values in the non-positive-definite model formulation, leading to the bias reduction.

关键词： positive-definite functions PRECIPITATION forecasting CONVECTIVE clouds WEATHER forecasting METEOROLOGY

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A Polya criterion for (strict) positive-definiteness on the sphere

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IMA JOURNAL OF NUMERICAL ANALYSIS 2014年第2期34卷 550-568页

作者： Beatson, R. K. Castell, Wolfgang Zu Xu, Yuan Univ Canterbury Dept Math & Stat Christchurch 1 New Zealand German Res Ctr Environm Hlth Helmholtz Zentrum Munchen Sci Comp Res Unit D-85764 Neuherberg Germany Univ Oregon Dept Math Eugene OR 97403 USA

positive-definite functions are very important in both the theory and applications of approximation theory, probability and statistics. In particular, identifying strictly positive-definite kernels is of great interest as interpolation problems based upon such kernels are guaranteed to have a unique solution whenever the nodes {x(j)} are distinct. A Bochner-type result of Schoenberg characterizes continuous positive-definite zonal functions, f(cos center dot), on the sphere Sd-1, as those with non-negative Gegenbauer coefficients. More recent results characterize strictly positive-definite functions on Sd-1 by stronger conditions on the signs of the Gegenbauer coefficients. Unfortunately, given a function f, checking the signs of all the Gegenbauer coefficients can be an onerous, or impossible, task. Therefore, it is natural to seek simpler sufficient conditions which guarantee (strict) positive-definiteness. We state a conjecture which leads to a Polya-type criterion for functions to be (strictly) positive definite on the sphere Sd-1. In analogy to the case of Euclidean space, the conjecture claims positivity of a certain integral involving Gegenbauer polynomials. We provide a proof of the conjecture for d from 3 to 8.

关键词： positive-definite functions strictly positive-definite Polya criterion sphere compactly supported kernel

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Evaluation of Scalar Advection Schemes in the Advanced Research WRF Model Using Large-Eddy Simulations of Aerosol-Cloud Interactions

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MONTHLY WEATHER REVIEW 2009年第8期137卷 2547-2558页

作者： Wang, Hailong Skamarock, William C. Feingold, Graham Univ Colorado Cooperat Inst Res Environm Sci Boulder CO 80309 USA NOAA Earth Syst Res Lab Boulder CO USA Natl Ctr Atmospher Res Boulder CO 80307 USA

In the Advanced Research Weather Research and Forecasting Model (ARW), versions 3.0 and earlier, advection of scalars was performed using the Runge-Kutta time-integration scheme with an option of using a positive-definite (PD) flux limiter. Large-eddy simulations of aerosol-cloud interactions using the ARW model are performed to evaluate the advection schemes. The basic Runge-Kutta scheme alone produces spurious oscillations and negative values in scalar mixing ratios because of numerical dispersion errors. The PD flux limiter assures positive definiteness but retains the oscillations with an amplification of local maxima by up to 20% in the tests. These numerical dispersion errors contaminate active scalars directly through the advection process and indirectly through physical and dynamical feedbacks, leading to a misrepresentation of cloud physical and dynamical processes. A monotonic flux limiter is introduced to correct the generally accurate but dispersive solutions given by high-order Runge-Kutta scheme. The monotonic limiter effectively minimizes the dispersion errors with little significant enhancement of numerical diffusion errors. The improvement in scalar advection using the monotonic limiter is discussed in the context of how the different advection schemes impact the quantification of aerosol-cloud interactions. The PD limiter results in 20% (10%) fewer cloud droplets and 22% (5%) smaller cloud albedo than the monotonic limiter under clean (polluted) conditions. Underprediction of cloud droplet number concentration by the PD limiter tends to trigger the early formation of precipitation in the clean case, leading to a potentially large impact on cloud albedo change.

关键词： METEOROLOGY -- Research WEATHER forecasting RUNGE-Kutta formulas REYNOLDS stress positive-definite functions LIMITER circuits

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