Nondegenerate Limit Research Articles

Bivariate Limit Theorems for Record Values Based on Random Sample Sizes

In this paper, the class of limit distribution functions (df’s) of the joint upper record values with random sample size is fully characterized. Necessary and sufficient conditions, as well as the domains of attraction of the limit df’s are obtained. As an application of this result, the sufficient conditions for the weak convergence of the random of record quasi-ranges, record quasi-midranges, record extremal quasi-quotients and record extremal quasi-products are obtained. Moreover, the classes of the non-degenerate limit df’s of these statistics are derived.

Multivariate Extremes of Random Scores of Particles in Branching Processes with Max-Linear Heredity

The paper continues the author’s long-term study of the extrema of random scores of particles in branching processes. It is assumed that the particle scores are dependent via common heredity, the dependence being determined by the distance. The case in which the scores have distributions with heavy tails is considered. The max-linear score generation model is used. The asymptotic behavior of multivariate extremes of scores over generations is studied. Nondegenerate limit laws are obtained for the maxima under linear normalization, and examples are given for various types of branching processes.

Neutron star cooling via axion emission by nucleon–nucleon axion bremsstrahlung

Neutron stars generally cool off by the emission of $$\gamma $$ -rays and neutrinos. But axions can also be produced inside a neutron star by the process of nucleon–nucleon axion bremsstrahlung. The escape of these axions adds to the cooling process of the neutron star. We explore the nature of cooling of neutron stars including the axion emission and compare our result with the scenario when the neutron star is cooled by only the emission of $$\gamma $$ -rays and neutrinos. In our calculations we consider both the degenerate and non-degenerate limits for such axion energy loss rate and the resulting variation of luminosity with time and variation of surface temperature with time of the neutron star. In short, the thermal evolution of a neutron star is studied with three neutron star masses (1.0, 1.4 and 1.8 solar masses) and by including the effect of axion emission for different axion masses ( $$m_{a}=10^{-5}, 10^{-3}$$ and $$10^{-2}\,\mathrm{eV}$$ ) and compared with the same when the axion emission is not considered. We compared theoretical cooling curve with the observational data of three pulsars PSR B0656 $$+$$ 14, Geminga and PSR B1055-52 and finally gave an upper bound on axion mass limits $$m_{a}\le 10^{-3}\,\mathrm{eV}$$ which implies that the axion decay constant $$f_{a}\ge 0.6\times 10^{10}\,\mathrm{GeV}$$ .

Alfvén solitary waves with effect of arbitrary temperature degeneracy in spin quantum plasma

Nonlinear Alfvén waves are studied in a fluid model for nonrelativistic, magnetized spin-1/2 quantum plasmas with an arbitrary degeneracy effect. Following a local Fermi-Dirac distribution function, a modified equation of state is utilized which is applicable to both classical and degenerate limits. Using the fluid equations for Hall magnetohydrodynamics with quantum corrections due to statistical effects, Bohm potential, spin magnetization energy, and temperature degeneracy, a set of modified Zakharov equations are derived for circularly polarized nonlinear Alfvén waves. Ions are assumed to be cold, and the spin effects of electrons are incorporated through spin force along with spin magnetization current. A linear dispersion relation for finite amplitude Alfvén waves duly modified by spin magnetization and arbitrary temperature degeneracy effects is also obtained. Employing the Sagdeev potential approach, the properties of Alfvén solitary profiles in quantum plasmas with arbitrary degeneracy effects of electrons are analyzed. The amplitude of Sagdeev potential and of the associated soliton structure for both right and left-hand circularly polarized Alfvén waves is observed to decrease with the decrease in the value of the arbitrary temperature degeneracy factor G for the case of the nearly degenerate limit. Similarly, it is found that the amplitude of Sagdeev potential and of the related solitary profile increases for both kinds of circular polarized Alfvén waves with the increasing value of G in the case of the nearly non-degenerate limit.

Pointwise convergence of Birkhoff averages for global observables.

It is well-known that a strict analogue of the Birkhoff Ergodic Theorem in infinite ergodic theory is trivial; it states that for any infinite-measure-preserving ergodic system, the Birkhoff average of every integrable function is almost everywhere zero. Nor does a different rescaling of the Birkhoff sum that leads to a non-degenerate pointwise limit exist. In this paper, we give a version of Birkhoff's theorem for conservative, ergodic, infinite-measure-preserving dynamical systems where instead of integrable functions we use certain elements of , which we generically call global observables. Our main theorem applies to general systems but requires a hypothesis of "approximate partial averaging" on the observables. The idea behind the result, however, applies to more general situations, as we show with an example. Finally, by means of counterexamples and numerical simulations, we discuss the question of finding the optimal class of observables for which a Birkhoff theorem holds for infinite-measure-preserving systems.

Linear and nonlinear analysis of kinetic Alfven waves in quantum magneto-plasmas with arbitrary temperature degeneracy

Linear and nonlinear kinetic Alfven waves (KAWs) are studied in collisionless, non-relativistic two fluid quantum magneto-plasmas by considering arbitrary temperature degeneracy. A general coupling parameter is applied to discuss the range of validity of the proposed model in nearly degenerate and nearly non-degenerate plasma limits. Linear analysis of KAWs shows an increase (decrease) in frequency with the increase in parameter ζ(δ) for the nearly non-degenerate (nearly degenerate) plasma limit. The energy integral equation in the form of Sagdeev potential is obtained by using the approach of the Lorentz transformation. The analysis reveals that the amplitude of the Sagdeev potential curves and soliton structures remains the same, but the potential depth and width of soliton structure change for both the limiting cases. It is further observed that only density hump structures are formed in the sub-alfvenic region for value Kz2>1. The effects of parameters ζ, δ on the nonlinear properties of KAWs are shown in graphical plots. New results for comparison with earlier work have also been highlighted. The significance of this work to astrophysical plasmas is also emphasized.

On the calculation of Lorenz numbers for complex thermoelectric materials

A first-principles informed approach to the calculation of Lorenz numbers for complex thermoelectric materials is presented and discussed. Example calculations illustrate the importance of using accurate band structures and energy-dependent scattering times. Results obtained by assuming that the scattering rate follows the density-of-states show that in the non-degenerate limit, Lorenz numbers below the commonly assumed lower limit of 2(kB/q)2 can occur. The physical cause of low Lorenz numbers is explained by the shape of the transport distribution. The numerical and physical issues that need to be addressed in order to produce accurate calculations of the Lorenz number are identified. The results of this study provide a general method that should contribute to the interpretation of measurements of total thermal conductivity and to the search for materials with low Lorenz numbers, which may provide improved thermoelectric figures of merit, zT.

Biggins’ martingale convergence for branching Lévy processes

A branching L\'evy process can be seen as the continuous-time version of a branching random walk. It describes a particle system on the real line in which particles move and reproduce independently in a Poissonian manner. Just as for L\'evy processes, the law of a branching L\'evy process is determined by its characteristic triplet $(\sigma^2,a,\Lambda)$, where the branching L\'evy measure $\Lambda$ describes the intensity of the Poisson point process of births and jumps. We establish a version of Biggins' theorem in this framework, that is we provide necessary and sufficient conditions in terms of the characteristic triplet $(\sigma^2,a,\Lambda)$ for additive martingales to have a non-degenerate limit.

Growth of normalizing sequences in limit theorems for conservative maps

We consider normalizing sequences that can give rise to nondegenerate limittheorems for Birkhoff sums under the iteration of a conservative map. Mostclassical limit theorems involve normalizing sequences that are polynomial,possibly with an additional slowly varying factor. We show that, ingeneral, there can be no nondegenerate limit theorem with a normalizingsequence that grows exponentially, but that there are examples where itgrows like a stretched exponential, with an exponent arbitrarily close to 1.

Limit Theorem for the Process of Additive Replacements

A transient discrete-time Markov chain is considered that describes the evolution of the content of an urn with balls having $n$ different colors. At each step the number of balls of a randomly selected color is increased by the number of balls of another randomly selected color. For the case when colors are chosen independently and uniformly, formulas for the first two moments of the numbers of balls are obtained. Under weaker assumptions on the distribution of colors chosen, it is shown that the vector formed by the fractions of balls of $n$ colors has a nondegenerate limit distribution.

Nondegenerate centers and limit cycles of cubic Kolmogorov systems

We characterize the center conditions for a cubic Kolmogorov differential system. We also study the number of small limit cycles that can bifurcate from the singular point of focus type for such systems.

Carrier relaxation time modelling of monolayer black phosphorene

Phosphorene as an innovative structure that can be exfoliated similarly to the graphene with a direct, inherent and suitable bandgap presents exceptional prospects for future generations of electronic devices. Phosphorene possess high carrier mobility, therefore, in this work its carrier statistics in the form of monolayer phosphorene in the non-degenerate limit is analytically modelled and the mobility relation with carrier relaxation time is investigated. Energy dispersion relation is used to develop and calculate the required parameters for carrier relaxation time model which is an important parameter in conduction theory. On the other hand, the dependency of carrier velocity and mobility to voltage, normalised Fermi energy and temperature are modelled. Finally, the carrier relaxation time as a function of carrier mobility is modelled and its dependency towards temperature and normalised Fermi energy is discussed. It is shown that the relaxation time is strongly dependent on the carrier mobility which increases by increasing the mobility.

Electronic transport coefficients in plasmas using an effective energy-dependent electron-ion collision-frequency

We present a model to calculate the electrical and thermal electronic conductivities in plasmas using the Chester-Thellung-Kubo-Greenwood approach coupled with the Kramers approximation. The divergence in photon energy at low values is eliminated using a regularization scheme with an effective energy-dependent electron-ion collision-frequency. Doing so, we interpolate smoothly between the Drude-like and the Spitzer-like regularizations. The model still satisfies the well-known sum rule over the electrical conductivity. Such kind of approximation is also naturally extended to the average-atom model. A particular attention is paid to the Lorenz number. Its nondegenerate and degenerate limits are given and the transition towards the Drude-like limit is proved in the Kramers approximation.

Patterned sparse random matrices: A moment approach

We consider four specific [Formula: see text] sparse patterned random matrices, namely the Symmetric Circulant, Reverse Circulant, Toeplitz and the Hankel matrices. The entries are assumed to be Bernoulli with success probability [Formula: see text] such that [Formula: see text] with [Formula: see text]. We use the moment approach to show that the expected empirical spectral distribution (EESD) converges weakly for all these sparse matrices. Unlike the Sparse Wigner matrices, here the random empirical spectral distribution (ESD) converges weakly to a random distribution. This weak convergence is only in the distribution sense. We give explicit description of the random limits of the ESD for Reverse Circulant and Circulant matrices. As in the non-sparse case, explicit description of the limits appears to be difficult to obtain in the Toeplitz and Hankel cases. We provide some properties of these limits. We then study the behavior of the largest eigenvalue of these matrices. We prove that for the Reverse Circulant and Symmetric Circulant matrices the limit distribution of the largest eigenvalue is a multiple of the Poisson. For Toeplitz and Hankel matrices we show that the non-degenerate limit distribution exists, but again it does not seem to be easy to obtain any explicit description.

Thermosize potentials in semiconductors

A thermosize junction consists of two different sized structures made using the same material. Classical and quantum thermosize effects (CTSEs and QTSEs), which are opposite to each other, induce a thermosize potential in a thermosize junction. A semi-analytical method is proposed to calculate thermosize potentials in wide ranges of degeneracy and confinement by considering both CTSEs and QTSEs in thermosize junctions made using semiconductors. Dependencies of thermosize potential on temperature, size, and degeneracy are examined. It is shown that a potential difference in millivolt scale can be induced as a combined effect of CTS and QTS. The highest potential is obtained in nondegenerate limit where the full analytical solution is obtained. The model can be used to design semiconductor thermosize devices for a possible experimental verification of CTSEs and QTSEs, which may lead to new nano energy conversion devices.

A note on the asymptotics of the maxima for the St. Petersburg game

In this note, we consider the maxima of payoffs for the generalized St. Petersburg game. The maxima for the original St. Petersburg game cannot be normalized to converge to a nondegenerate limit distribution. However, tuning the parameters appearing in the generalization, we show the normalized maxima converge to the Fréchet distribution.

Density-functional calculations of transport properties in the nondegenerate limit and the role of electron-electron scattering.

We compute electrical and thermal conductivities of hydrogen plasmas in the nondegenerate regime using Kohn-Sham density functional theory (DFT) and an application of the Kubo-Greenwood response formula, and demonstrate that for thermal conductivity, the mean-field treatment of the electron-electron (e-e) interaction therein is insufficient to reproduce the weak-coupling limit obtained by plasma kinetic theories. An explicit e-e scattering correction to the DFT is posited by appealing to Matthiessen's Rule and the results of our computations of conductivities with the quantum Lenard-Balescu (QLB) equation. Further motivation of our correction is provided by an argument arising from the Zubarev quantum kinetic theory approach. Significant emphasis is placed on our efforts to produce properly converged results for plasma transport using Kohn-Sham DFT, so that an accurate assessment of the importance and efficacy of our e-e scattering corrections to the thermal conductivity can be made.

Asymptotic behavior of the joint record values, with applications

The class of limit distribution functions of the joint upper record values, as well as the joint of lower record values, is fully characterized. Sufficient conditions for the weak convergence are obtained. As an application of this result, the sufficient conditions for the weak convergence of the record quasi-range, record quasi-midrange, record extremal quasi-quotient and record extremal quasi-product are obtained. Moreover, the classes of the non-degenerate limit distribution functions of these statistics are derived.

Convergence of complex martingales in the branching random walk: the boundary

Biggins [Uniform convergence of martingales in the branching random walk. Ann. Probab., 20(1):137–151, 1992] proved local uniform convergence of additive martingales in $d$-dimensional supercritical branching random walks at complex parameters $\lambda $ from an open set $\Lambda \subseteq \mathbb{C} ^d$. We investigate the martingales corresponding to parameters from the boundary $\partial \Lambda $ of $\Lambda $. The boundary can be decomposed into several parts. We demonstrate by means of an example that there may be a part of the boundary, on which the martingales do not exist. Where the martingales exist, they may diverge, vanish in the limit or converge to a non-degenerate limit. We provide mild sufficient conditions for each of these three types of limiting behaviors to occur. The arguments that give convergence to a non-degenerate limit also apply in $\Lambda $ and require weaker moment assumptions than the ones used by Biggins.

The slow regime of randomly biased walks on trees

We are interested in the randomly biased random walk on the supercritical Galton–Watson tree. Our attention is focused on a slow regime when the biased random walk $(X_{n})$ is null recurrent, making a maximal displacement of order of magnitude $(\log n)^{3}$ in the first $n$ steps. We study the localization problem of $X_{n}$ and prove that the quenched law of $X_{n}$ can be approximated by a certain invariant probability depending on $n$ and the random environment. As a consequence, we establish that upon the survival of the system, $\frac{|X_{n}|}{(\log n)^{2}}$ converges in law to some non-degenerate limit on $(0,\infty)$ whose law is explicitly computed.