Binomial Form Research Articles

On binomial Thue-Mahler equations

We give a rather sharp upper bound for the degree of a binomial Thue--Mahler equation in terms of the coefficients and the primes involved. Further, we establish explicit lower bounds for the greatest prime factor of a binomial binary form at integral points. Our estimates considerably generalize and improve the earlier results obtained in this direction.

Bayesian bootstrap for proportional hazards models

We propose two Bayesian bootstrap extensions, the binomial and Poisson forms, for proportional hazards models. The binomial form Bayesian bootstrap is the limit of the posterior distribution with a beta process prior as the amount of the prior information vanishes, and thus can be considered as a default nonparametric Bayesian analysis. It is also the same as Lo's Bayesian bootstrap for censored data when covariates are absent. The Poisson form Bayesian bootstrap is equivalent to the Bayesian analysis with Cox's profile likelihood. When the baseline distribution is discrete, thus when the data set has many ties, simulation study suggests that the binomial form Bayesian bootstrap performs better than standard frequentist procedures in the frequentist sense. An advantage of the proposed Bayesian bootstrap procedures over the standard Bayesian analysis is conceptual and computational simplicity. Finally, it is shown that both Bayesian bootstrap posteriors are asymptotically equivalent to the sampling distribution of the maximum likelihood estimator.

Rapidity spectra analysis in terms of non-extensive statistic approach

We provide the description of rapidity spectra of particles produced in pp̄ collisions using the anomalous diffusion approach to account for their non-equilibrium character. In particular, we exhibit connections between multiproduction processes and anomalous diffusion described through the nonlinear Focker-Planck equation with a nonlinearity given by the nonextensivity parameter q describing the underlying Tsallis q-statistics. We demonstrate how this leads to the Feynman scaling violation in these collisions. The q parameter obtained in this way turns out to be closely connected to the parameter 1 k converting the original poissonian multiplicity distribution to its observed Negative Binomial form. The inelasticity of the reaction has also been calculated and found to slightly decrease with increasing reaction energy.

A Frequency Distribution for the Number of Hematogenous Organ Metastases

The number of metastases associated with a primary tumor can be a major determinant for the chance of cure. A model is proposed here where the frequency distribution of the number of experimental organ metastases is affected by Poisson statistics, and by stochastic variations in regional blood flow. The model predicts that the mean E ( X) and variance var ( X) of the number of metastases per organ should relate according to a power function, var ( X)= E ( X) p + E ( X), where and p are the constants, and that the actual distribution has a Poisson-negative binomial form. This model was found consistent with the data derived from a meta-analysis of over 47 000 murine experimental metastases. This frequency distribution (together with knowledge of the size distribution for metastases and the fraction of clonogenic cells) could permit more accurate assessments for tumor control probabilities with adjuvant therapies.

Binomial level densities

It is shown that nuclear level densities in a finite space are described by a continuous binomial function, determined by the first three moments of the Hamiltonian, and the dimensionality of the underlying vector space. Experimental values for ${}^{55}\mathrm{Mn},$ ${}^{56}\mathrm{Fe},$ and ${}^{60}\mathrm{Ni}$ are very well reproduced by the binomial form, which turns out to be almost perfectly approximated by Bethe's formula with backshift. A proof is given for which binomial densities reproduce the low moments of Hamiltonians of any rank: A strong form of the famous central limit result of Mon and French. Conditions under which the proof may be extended to the full spectrum are examined.

The nucleon-airnucleus interaction probability law with rising cross section

The diffusion equation of cosmic-ray nucleons is exactly integrated usingthe successive approximation method for a general distribution of theprimary component, and taking into account the rising nucleon-aircross sections with energy. The interaction probability law for thenucleon in the atmosphere is obtained as a consequence of the respectivediffusion equation. If the nucleon-air cross sections rise logarithmically,this probability law assumes a binomial form, and for the constant crosssection, it is purely Poissonian. The well known approximate solution iscompared with our exact solution. It is found that the former always givesa nucleon number greater than ours by, for example, 15-25% in theenergy region 30-10 000 GeV at sea level in the case of the meaninelasticity ⟨κ⟩ = 0.60. It is also shown that a fairlyaccurate description of nucleon flux at sea level (1030 g cm-2) and hadron intensities at 840 g cm-2 and at1030 g cm-2 are obtained with ⟨κ⟩varying between 0.55 and 0.60.

On the forward–backward correlations in a two-stage scenario

It is demonstrated that in a two-stage scenario with elementary Poissonian emitters of particles (colour strings) arbitrarily distributed in their number and average multiplicities, the forward–backward correlations are completely determined by the final distribution of forward particles. The observed linear form of the correlations then necessarily requires this distribution to have a negative binomial form. For emitters with a negative binomial distribution in the produced particles distributed so as to give the final distribution also of the negative binomial form, the forward–backward correlations have an essentially non-linear form which disagrees with the experimental data.

Accident Models for Two-Lane Rural Segments and Intersections

Data collected from the states of Minnesota and Washington on rural two-lane highways are used to build accident models for segments and three-legged and four-legged intersections stop-controlled on the minor legs. The quantity, quality, and variety of data collected, together with the advanced techniques applied in the analysis, make this study of special interest. Variables include traffic, horizontal and vertical alignments, lane and shoulder widths, roadside hazard rating, channelization, and number of driveways. Models are of negative binomial and extended negative binomial form and yield R2 values from 0.42 to 0.73 and overdispersion parameters from 0.20 to 0.51. A segment model combining both states and including state as a variable, and intersection models derived from Minnesota data, are featured, along with summary statistics, goodness-of-fit measures, and cross-validation between the states. Segment accidents depend significantly on most of the roadway variables collected, while intersection accidents depend primarily on traffic. The study recommends development of adjustment factors for different regions and times and further development of extended negative binomial models.

Deflections of Timoshenko beam with varying cross-section

Closed form solutions are presented for bending beams with linearly and (in the binomial form) parabolically varying depth and for bending beams with linearly varying width along the beam's length. The solutions are developed taking into account the shear deformation of the beam. The solutions are achieved, in an original way, by transforming the fourth-order differential equations with variable coefficients into fourth-order differential equations with constant coefficients. Though the solutions presented refer to three recurrent variations in the beam cross-section shape, the procedure outlined can be applied to beams with binomial variation (with any exponent) in the depth or width of the cross-section. Moreover, the solutions can be achieved for polynomial, exponential and sinusoidal load conditions. The solutions can be utilized to obtain the stiffness factors and the flexibility coefficients of beams in the analysis of frames. Closed form solutions for longitudinal displacements are also presented. The analytical solutions are applied to four recurrent beams commonly used in civil engineering practice and a comparison with a numerical procedure is made.

Negative binomial distributions and clan model in proton-nucleus interactions

Experimental results on multiplicity distributions in different pseudorapidity intervals of charged shower particles produced in proton-AgBr and proton-CNO collisions at 800 GeV are presented. The different distributions are described well by the negative binomial form. We successfully interpret our results in terms of the clan model. The values of the rapidity gap probability in terms of average number of clans in different pseudorapidity intervals are also determined.

New highly charged fullerene ions: Production and fragmentation by slow ion impact.

Fullerene ions, ${\mathrm{C}}_{60}^{\mathit{q}+}$, with q up to 9, have been observed in a study of their production by slow (v0.5 au) impact of the projectiles $^{40}\mathrm{Ar}^{4,5,8,12,16,17+}$, $^{136}\mathrm{Xe}^{27+}$, $^{86}\mathrm{Kr}^{28+}$, $^{209}\mathrm{Bi}^{20,38,44,46+}$, and $^{238}\mathrm{U}^{46+}$ on a neutral fullerene beam. The distribution of ion yields for each projectile is representable by a binomial form; variation of the biniomial fit parameters with projectile charge suggests the maximum positive charge for the fullerene ion. Correlations between the time of flight of first and second ions are shown to provide details of the fragmentation of fullerenes in close collisions. \textcopyright{} 1996 The American Physical Society.

On quantum shot noise

We discuss the distribution function for the current noise in quantum point contacts. Special interest is paid to a contact of a superconductor with a normal metal. A new derivation of the Lesovik-Levitov formulae is suggested. It is shown, for the SN contacts, that the distribution of the noise describes independent processes when charge ± e 0 or ±2 e 0 passes through the contact. At low temperature and voltage only processes with double charge transfer are relevant. At zero temperature and low voltage the distribution has a binomial form.

Quantum shot noise in a normal-metal-superconductor point contact.

The distribution function for the current noise in a normal-metal--superconductor quantum point contact is calculated. It is shown that this distribution describes independent processes when a charge \ifmmode\pm\else\textpm\fi{}${\mathit{e}}_{0}$ or \ifmmode\pm\else\textpm\fi{}2${\mathit{e}}_{0}$ passes through the contact. At low temperature and voltage only processes with double-charge transfer are relevant. At zero temperature and low voltage the distribution has a binomial form.

On the Distribution of Births in Human Populations

Mathematical models are invaluable tools in the study of the population dynamics of places with inadequate or defective data. Brass (1958) was probably the first to conclude that in suitable conditions the number of women with no children as well as the birth distribution for mothers follows approximately the negative binomial form. In 1978 Onyemaekeogum contradicted Brasss finding that the negative binomial distribution (NBD) reasonably well describes the distribution of births in human populations. Both used data from Africa countries but collected at different points in time. This paper investigates the adequacy of NBD to fit data for India. The study finds that the model both in its complete and truncated forms provides a reasonably good fit to most of the observed distributions of births. The maximum likelihood estimators of the parameters of the model were used for the fitting.

Multiplicity distributions in small phase-space domains in central nucleus-nucleus collisions

Multiplicity distributions of negatively charged particles have been studied in restricted phase space intervals for central S+S, O+Au and S+Au collisions at 200 GeV/nucleon. It is shown that multiplicity distributions are well described by a negative binomial form irrespectively of the size and dimensionality of phase space domain. A clan structure analysis reveals interesting similarities between complex nuclear collisions and a simple partonic shower. The lognormal distribution agrees reasonably well with the multiplicity data in large domains, but fails in the case of small intervals. No universal scaling function was found to describe the shape of multiplicity distributions in phase space intervals of varying size.

All Binomial‐Type Painlevé Equations of the Second Order and Degree Three or Higher

In this paper we construct all rational Painlevé‐type differential equations which take the binomial form, (d2y/dx2)n = F(x,y,dy/dx), where n ≥ 3, the case n = 2 having previously been treated in Cosgrove and Scoufis [1]. While F is assumed to be rational in the complex variables y and y′ and locally analytic in x, it is shown that the Painlevé property together with the absence of intermediate powers of y″ forces F to be a polynomial in y and y′. In addition to the six classes of second‐degree equations found in the aforementioned paper, we find nine classes of higher‐degree binomial Painlevé equations, denoted BP‐VII,..., BP‐XV, of which the first seven are new. Two of these equations are of the third degree, two of the fourth degree, three of the sixth degree, and two of arbitrary degree n. All equations are solved in terms of the first, second or fourth Painlevé transcendents, elliptic functions, or quadratures. In the appendices, we discuss certain closely related classes of second‐order nth equations (not necessarily of Painlevé type) which can also be solved in terms of Painlevé transcendents or elliptic functions.

The Probability of Tumor Cure: Response to Comments by Dr. Yakovlev

In response to our paper concerning the distribution of tumor clonogens at the end of treatment, Dr. Yakovlev points out that the probability-generating function, and hence the distribution, can be calculated exactly, and thus we could have reached our conclusions without using computer simulation. We thank Dr. Yakovlev for making this useful observation. The possibility of doing this was also pointed out to us by Elliot Landaw, a colleague at UCLA, soon after our paper appeared. In our paper, we derived a formula for the probability of tumor cure after fractionated treatment [our Eq. (8)] that approximated the simulation results quite well and represented a clear improvement over the usual Poisson formula [our Eqs. (1) and (2)] for cure probability among tumors in which proliferation occurs during treatment. A slightly different formula for tumor cure probability can be obtained from the probability-generating function derived by Dr. Yakovlev by substituting s = 0 into his Eqs. (1), (2), (5), and (6). The only difference between our formula, which could also be obtained using generating functions (1), and Dr. Yakovlev's formula is that we used a stochastic term for the cell-killing component but a deterministic term for the growth component, whereas Dr. Yakovlev used stochastic elements for both of these components. The good agreement between our formula and the simulation results is a reflection of the stronger relative influence of cell killing than of cell proliferation for the treatment schedules we considered. Our model and Dr. Yakovlev's model are identical when there is no cell division during treatment (i.e., a = 0), in which case both formulae reduce to a binomial form. We noted in our paper [(2), p. 277 and Appendix B], as Dr. Yakovlev points out again, that the Poisson model provides an excellent approximation in this case. Thus we too were not at all surprised by the good agreement between the predictions of Poisson statistics and the numerical results in the assumed absence of tumor cell proliferation during treatment. To assist readers who might be interested in investigating the differences in tumor cure between various fractionation schemes (3) and are not familiar with generating functions, we gi e below the explicit formula for calculating the probability of tumor cure for fractionation regimens in which e ch dose fraction might be of a different size and in which the time between dose fractions could vary. The formula g ven below applies only to the situation where there is a rel tively short interval between fractions, such that the probability of a cell undergoing two successive divisions in that interval is negligibly small. We are currently investigating the more general situation where the interval between fractions could be much larger, for example, in a splitcourse radiotherapy regimen. The formula given below is derived from the generating functions given by Dr. Yakovlev. Assume there are n fractions and let p, be the probability of cell survival associated with the ith dose and ai be the probability of cell division between the ith and (i + 1)st dose. Let

Description of large multiplicity fluctuations in small phase-space domains

We discuss the regularities of multiplicity distributions and the correlation pattern in different phase space domains. The distributions which describe reasonably well the multiplicity data in large domains (as the lognormal and negative binomial forms) fail in the case of small domains with large multiplicity fluctuations. Two alternative simple cascade models which lead to the logbinomial and Levy-stable distributions appear to have the minimal freedom (3 parameters) necessary to represent the data.

Deflections of Beams with Varying Rectangular Cross Section

Closed-for solutions are presented for bending beams with linearly and (in the binomial form) parabolically varying depth and for bending beams with linearly varying width. The solutions are achieved by transforming the fourth-order differential equations with function coefficients into fourth-order differential equations with constant coefficients. For each law of variability of the cross section, four load conditions are considered: Concentrated, uniformly distributed, linearly, and parabolically varying distributed. The solutions depend on four integration constants, and they can be applied to any mechanical and kinematical end conditions. Dimensionless graphs for maximum deflection of three special beams under typical load conditions are given as a function of the ratio between the minimum and the maximum heights. For the statically indeterminate beam, dimensionless stress (of the flexural stress component) is also given. Though the solution expressions require a little computational effort (a pocket calculator is sufficient) the dimensionless graphs are useful for analyzing similar problems.

Linear differential equations of binomial form

Linear differential equations of binomial form