Bistability and Hysteresis of Intersubband Absorption in Strongly Interacting Electrons on Liquid Helium

We study nonlinear inter-subband microwave absorption of electrons bound to the liquid helium surface. Already for a comparatively low radiation intensity, resonant absorption due to transitions between the two lowest subbands is accompanied by electron overheating. The overheating results in a significant population of higher subbands. The Coulomb interaction between electrons causes a shift of the resonant frequency, which depends on the population of the excited states and thus on the electron temperature $T_e$. The latter is determined experimentally from the electron photoconductivity. The experimentally established relationship between the frequency shift and $T_e$ is in reasonable agreement with the theory. The dependence of the shift on the radiation intensity introduces nonlinearity into the rate of the inter-subband absorption resulting in bistability and hysteresis of the resonant response. The hysteresis of the response explains the behavior in the regime of frequency modulation, which we observe for electrons on liquid $^3$He and which was previously seen for electrons on liquid $^4$He

High order approximation for the coverage probability by a confident set centered at the positive-part James-Stein estimator

In this paper we continue our investigation connected with the new approach developed in Ahmed et al. [Ahmed, S.E., Saleh, A.K.Md.E., Volodin, A., Volodin, I., 2006. Asymptotic expansion of the coverage probability of James-Stein estimators. Theory Probab. Appl. 51 (4) 1-14] for asymptotic expansion construction of coverage probabilities, for confidence sets centered at James-Stein and positive-part James-Stein estimators. The coverage probabilities for these confidence sets depend on the noncentrality parameter τ2, the same as the risks of these estimators. In this paper we consider only the confidence set centered at the positive-part James-Stein estimator. As is shown in the above-mentioned reference, the new approach provides a method to obtain for the given confidence set, an asymptotic expansion of the coverage probability as one formula for both cases τ → 0 and τ → ∞. We obtain the third terms of the asymptotic expansion for both mentioned cases, that is, the coefficients at τ2 and τ- 2. Numerical illustrations show that the third term has only a small influence on the accuracy of the asymptotic estimation of coverage probability. Crown Copyright © 2009

Confidence intervals for a ratio of binomial proportions based on direct and inverse sampling schemes

© 2016, Pleiades Publishing, Ltd.A general problem of the interval estimation for a ratio of two proportions p1/p2 according to data from two independent samples is considered. Each sample may be obtained in the framework of direct or inverse binomial sampling. Asymptotic confidence intervals are constructed in accordance with different types of sampling schemes with an application, where it is possible, of unbiased estimations of success probabilities and also their logarithms. Since methods of constructing confidence intervals in the situations when values for the both samples are obtained for identical sample schemes (for only direct or only inverse binomial sampling) are already developed and well known, the main purpose of this paper is the investigation of constructing confidence intervals in two cases that correspond to different sampling schemes (one is direct, another is inverse). In this situation it is possible to plan the sample size for the second sample according to the number of successes in the first sample. This, as it is shown by the results of statistical modeling, provides the intervals with confidence level which closer to the nominal value. Our goal is to show that the normal approximations (which are relatively simple) for estimates of p1/p2 and their logarithms are reliable for a construction of confidence intervals. The main criterion of our judgment is the closeness of the confidence coefficient to the nominal confidence level. It is proved theoretically and shown by statistically modeled data that the scheme of inverse binomial sampling with planning of the size in the second sample is preferred. Main probability characteristics of intervals corresponding to all possible combinations of sampling schemes are investigated by the Monte-Carlo method. Estimations of coverage probability, expectation and standard deviation of interval widths are collected in tables and some recommendations for an application of each of the intervals obtained are presented. Finally, a sufficient and complete review of the literature for the problem is also presented

Confidence sets based on the positive part James–Stein estimator with the asymptotically constant coverage probability

© 2014 Taylor & Francis. The asymptotic expansions for the coverage probability of a confidence set centred at the James–Stein estimator presented in our previous publications show that this probability depends on the non-centrality parameter τ2 (the sum of the squares of the means of normal distributions). In this paper we establish how these expansions can be used for a construction of confidence region with constant confidence level, which is asymptotically (the same formula for both case τ→0 and τ→∞) equal to some fixed value 1−α. We establish the shrinkage rate for the confidence region according to the growth of the dimension p and also the value of τ for which we observe quick decreasing of the coverage probability to the nominal level 1−α. When p→∞ this value of τ increases as O(p1/4). The accuracy of the results obtained is shown by the Monte-Carlo statistical simulations

James-Stein confidence set: Equal area approach to the global approximation of coverage probability

In [S. A. Ahmed, A. K. MD. E. Saleh, A. I. Volodin, and I. N. Volodin, "Asymptotic Expansion of the Coverage Probability of James-Stein Estimators," Theory Probab. Appl. 51, 683-695 (2007)], an asymptotic expansion of coverage probabilities for the James-Stein confidence sets was constructed, which was asymptotically exact for both large and small values of the noncentrality parameter τ 2, that is, the sum of squares of the means of p ≥ 3 normal distributions subject to confidence estimation. As numerical examples show, this expansion can be used on the almost entire domain of values τ 2 for computing the coverage probability with error of order 10 -2. In this paper, a similar asymptotic expansion is suggested, which computes the coverage probability with much smaller global error in the domain of small and moderate values of p. The accuracy of approximations is illustrated by statistical modeling data. © 2011 Pleiades Publishing, Ltd

Simulation of Atomic Structure Near Nanovoids in BCC Iron

Generally, displacement fields near voids are determined by the equations of elasticity theory. Such a description has its disadvantages as it does not take into account the discrete atomic structure of materials. In this work, we use a new variant of Molecular Static method for investigation of the atomic structure near nanovoids. In our model an iterative procedure is employed, in which the atomic structure in the void vicinity and the parameter determining the displacement of atoms embedded into an elastic continuum are obtained in a self-consistent manner. Results show that the displacements are significantly different for varies crystallographic directions. Keywords: voids; iron; simulation; atomic structure; vacancie

Sonoluminescence and collapse dynamics of multielectron bubbles in helium

Multielectron bubbles (MEBs) differ from gas-filled bubbles in that it is the Coulomb repulsion of a nanometer thin layer of electrons that forces the bubble open rather than the pressure of an enclosed gas. We analyze the implosion of MEBs subjected to a pressure step, and find that despite the difference in the underlying processes the collapse dynamics is similar to that of gas-filled bubbles. When the MEB collapses, the electrons inside it undergo strong accelerations, leading to the emission of radiation. This type of sonoluminescence does not involve heating and ionisation of any gas inside the bubble. We investigate the conditions necessary to obtain sonoluminescence from multielectron bubbles and calculate the power spectrum of the emitted radiation.Comment: 6 figure

Local asymptotic efficiency of a sequential probability ratio test for d-guarantee discrimination of composite hypotheses

A sequential Wald test for discrimination of two simple hypotheses θ = θ1 and θ = θ2 with boundaries A and B is applied to distinguish composite hypotheses θ θ0, the parameters θ1, θ2, A, and B being chosen in such a way that d-posteriori probabilities of errors do not exceed the given restrictions β0 and β1. An asymptotic behavior of boundaries A, B and the average observation time are studied when β= max{β0, β1} → 0. An asymptotic (β → 0) comparison is made of Eθv with the least given number of observations necessary for discrimination of composite hypotheses with the same restrictions β0, β1 on d-posteriori probabilities of errors. It is shown that the minimum (in a neighborhood of the point θ = θ0) gain of the average observation time makes up 25%. Therefore, there are sequential tests within the bounds of a d-posteriori approach that give a gain in the size of observations for every value of a parameter tested

Asymptotics of the necessary sample size under small error probabilities

The asymptotics of the necessary sample size is considered in testing close hypotheses when the error probabilities vanish. © 1997 Plenum Publishing Corporation