Decompositions of Substochastic Transition Functions

This article addresses the stability of a nonlinear electron plasma wave (EPW) against the growth of longitudinal sidebands. The electron distribution function consistent with the EPW is assumed to only depend on the dynamical action. Consequently, the EPW is either stationary (a so-called Berstein-Greene-Kruskal mode) or varies very slowly in space and time (a so-called adiabatic wave). The sideband growth rates and the unstable spectrum are calculated theoretically by accounting for the exact nonlinear electron orbits in the EPW. Our theoretical results are compared against those from previous theories, and also against those from Vlasov simulations when the distribution function is differentiable. The latter comparisons show that our theory may also apply when the electron distribution function depends on, both the action and the angle. Moreover, our theory allows for discontinuous distributions, which are consistent with important classes of EPWs, e.g., those resulting from stimulated Raman scattering. Addressing such distributions using kinetic codes remains a challenge. Nevertheless, using our theoretical results, in this article we discussthe impact of discontinuities on the unstable spectrum in the linear regime, and on the range of validity of the linear approximation.

A method for simulation of vlasov systems with discontinuous distribution functions

Let $\rho(x, y)$ be a positive definite symmetric kernel defined over the unit square such that $\rho(x, y) = K(x, y) - \sum^k_{i=1} \psi_i(x)\psi_i (y), 0 \leqq x, y \leqq 1$, where $K(x, y)$ is a bounded symmetric positive definite kernel defined over the unit square, and $\psi_i(x) \in L_2(0, 1)$. Methods of finding Fredholm determinant $D(\lambda)$ of $\rho(x, y)$ in terms of the eigenvalues and the eigenfunctions of $K(x, y)$ are given. A kernel of the type of $\rho(x, y)$ arises as the covariance function of a Gaussian process in the limiting distribution of the modified Cramer-Smirnov test statistic in the $k$-parameter case which may be described as follows: Let $X_1, \cdots, X_n$ be $n$ independent observations (random variables) from a population with a continuous distribution function $G(x)$. Suppose for every $\theta = (\theta_1, \cdots, \theta_k) \in \mathbf{I,I}$ being an open interval in the $k$-dimensional Euclidean space $R^k, F(x, \theta)$ is a continuous distribution function. Let $\hat{\theta}_n$ be an estimate of $\theta$ obtained from the sample. A test of the hypothesis $H: G(x) = F(x, \theta)$ for some unspecified $\theta \in \mathbf{I}$ based on the statistic $C_n^2 = n \int^{+\infty}_{-\infty} \lbrack F_n(x) - F(x, \hat{\theta}_n) \rbrack^2 dF(x, \hat{\theta}_n),$ is considered and the characteristic function of the asymptotic distribution of $C_n^2$ is shown to be the Fredholm determinant of a kernel of the type of $\rho(x, y)$ with $K(x, y) = \min (x, y) - xy$ whose eigenvalues and eigenfunctions are known. Results are also used to obtain the limiting distribution of $l$-sample analogue of $C_n^2$.

This paper studies the structure of downlink sum-rate maximizing selective decentralized feedback policies for opportunistic beamforming under finite feedback constraints on the average number of mobile users feeding back. First, it is shown that any sum-rate maximizing selective decentralized feedback policy must be a threshold feedback policy. This result holds for all fading channel models with continuous distribution functions. Second, the resulting optimum threshold selection problem is analyzed in detail. This is a nonconvex optimization problem over finite-dimensional Euclidean spaces. By utilizing the theory of majorization, an underlying Schur-concave structure in the sum-rate function is identified, and the sufficient conditions for the optimality of homogenous threshold feedback policies are obtained. Applications of these results are illustrated for well-known fading channel models such as Rayleigh, Nakagami, and Rician fading channels. Rather surprisingly, it is shown that using the same threshold value at all mobile users is not always a rate-wise optimal feedback strategy, even for a network in which mobile users experience statistically the same channel conditions. For the Rayleigh fading channel model, on the other hand, homogenous threshold feedback policies are proven to be rate-wise optimal if multiple orthonormal data carrying beams are used to communicate with multiple mobile users simultaneously.

Rank based estimators of the change-point

Suppose we have a random sample from a continuous distribution function with support on the positive reals. This paper will investigate a semi-parametric estimator of the distribution function that is based on mixtures of lognormal distributions. This estimator will be constructed so that a specified number of moments of the empirical distribution function are reproduced. We will give several asymptotic results for this moment-type estimator including a central limit theorem.

[2], [3], G. Ishii [5] . In a previous paper, we had treated the test of fit in life test in the case that we stop when the s-th death has occurred, G. Ishii [4] . In the present paper we treat the case when we stop the life test at a certain preassigned time T. Let X be a random variable having the continuous distribution function. In order to test the hypothesis Ho that the distribution function is a known function F(x), F. N. David [1] and M. Okamoto [6] have proposed the following non-parametric test : Let xi (i =1, 2, , M) be M independent observation of a random variable X. There are real numbers i =1, 2, ••• , m— 1 such that F(a,) —F(at _i) —1/m, i = 1, 2, --, m where ao= — co, am= + (ai_i, ai] will be called part. Let v be the number of parts which contain no x's. If v is too large, we reject Ho. Now we shall apply the above non-parametric test to life test.

Continuous thermodynamics provides a simple way for the thermodynamic treatment of polydisperse systems. Such systems consist of a very large number of similar species whose composition is described not by the mole fractions of the individual components but by continuous distribution functions. For copolymers, multivariate distribution functions have to be used for describing the dependence of thermodynamic behavior on molar mass, chemical composition, sequence length, branching, etc. In this paper, we review continuous thermodynamics as applied to copolymer systems. Special attention is focused on liquid-liquid equilibria and thermodynamic stability. Equilibria in solutions of random copolymers, blends of random copolymers with homo- or copolymers, and also the high pressure phase equilibrium in the mixture of copoly(ethene vinylacetate) with its monomers are also discussed. A special examination of polydispersity effects in solutions with block copolymers is made. Thus, the paper reviews in a comprehensive way how to build up continuous thermodynamics with multivariate distribution functions and how to derive relations necessary for solving special problems. Some short remarks on possible future prospects will round up the paper.

13 Polydisperse fluids

1.1. Kuiper (1960) has proposed VN, an adaptation of the Kolmogorov statistic, to test the null hypothesis that a random sample of size N comes from a population with given continuous distribution function F(x). If the sample distribution function is FN(x), VN is defined by VN= sup (FN(x)-F(x))- inf (FN(x) -F(x)). (1) -co<x<Co -co<X<C* Kuiper showed that:

This paper introduces first-order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally non-linear class of generalised weakly differentiable functions and share key functional analytic properties with their Euclidean counterparts. Assuming the varifold to satisfy a uniform lower density bound and a dimensionally critical summability condition on its mean curvature, the following statements hold. Firstly, continuous and compact embeddings of Sobolev spaces into Lebesgue spaces and spaces of continuous functions are available. Secondly, the geodesic distance associated to the varifold is a continuous, not necessarily Ho&amp;x0308;lder continuous Sobolev function with bounded derivative. Thirdly, if the varifold additionally has bounded mean curvature and finite measure, then the present Sobolev spaces are isomorphic to those previously available for finite Radon measures yielding many new results for those classes as well. Suitable versions of the embedding results obtained for Sobolev functions hold in the larger class of generalised weakly differentiable functions.

Three body break up

A velocity-space adaptive unified gas kinetic scheme for continuum and rarefied flows

This chapter takes up several independent topics in Euclidean Fourier analysis, all having some bearing on the subject of partial differential equations. Section 1 elaborates on the relationship between the Fourier transform and the Schwartz space, the subspace of $L^{1}(\mathbb{R}^{N})$ consisting of smooth functions with the property that the product of any iterated partial derivative of the function with any polynomial is bounded. It is possible to make the Schwartz space into a metric space, and then one can consider the space of continuous linear functionals; these continuous linear functionals are called “tempered distributions.” The Fourier transform carries the space of tempered distributions in one-one fashion onto itself. Section 2 concerns weak derivatives, and the main result is Sobolev’s Theorem, which tells how to recover information about ordinary derivatives from information about weak derivatives. Weak derivatives are easy to manipulate, and Sobolev’s Theorem is therefore a helpful tool for handling derivatives without continually having to check the validity of interchanges of limits. Sections 3–4 concern harmonic functions, those functions on open sets in Euclidean space that are annihilated by the Laplacian. The main results of Section 3 are a characterization of harmonic functions in terms of a mean-value property, a reflection principle that allows the extension to all of Euclidean space of any harmonic function in a half space that vanishes at the boundary, and a result of Liouville that the only bounded harmonic functions in all of Euclidean space are the constants. The main result of Section 4 is a converse to properties of Poisson integrals for half spaces, showing that harmonic functions in a half space are given as Poisson integrals of functions or of finite complex measures if their $L^{p}$ norms over translates of the bounding Euclidean space are bounded. Sections 5–6 concern the Calderon–Zygmund Theorem, a far-reaching generalization of the theorem concerning the boundedness of the Hilbert transform. Section 5 gives the statement and proof, and two applications are the subject of Section 6. One of the applications is to Riesz transforms, and the other is to the Beltrami equation, whose solutions are “quasiconformal mappings.” Sections 7–8 concern multiple Fourier series for smooth periodic functions. The theory is established in Section 7, and an application to traces of integral operators is given in Section 8.

Anderson and Darling proposed the use of the statistic for testing the hypothesis that a sample of size n has been drawn from a population with a specified continuous cumulative distribution function Gn(x) is the empirical distribution function defined on the sample on the size n Author consider here the problem of determining and tabulating the distribution function of this statistics.