arXiv subjects
Publications and source records attributed to S. R. Mane.
We comment on various statements in a recent document (A.~Deur, S.~J.~Brodsky, C.~D.~Roberts and B.~Terzi{ć}, Regarding the Rotational Unruh Effect, \textit{arXiv:2607.25004v1 [hep.ph]}, (2026)). The examples cited treat electrons (or positrons) circulating and emitting photons in high-energy storage rings. The topic is also of interest in astrophysics, for electrons orbiting in magnetic fields around neutron stars.
We study the trinomial equation $x^n +px +q =0$. Here $p$ and $q$ are both real and nonzero. For $n\ge3$, expressions for the roots have been published as hypergeometric series in powers of the parameter $q^{n-1}/p^n$. For the special case of the cubic ($n=3$), we employ Kummer's identities to derive alternative series solutions in powers of the discriminant $D$, and also series in powers of $1/D$. We next derive new series, in powers of $D$ and also in powers of $1/D$, for all $n\ge 3$. The resulting series suggest identities analogous to Kummer's identities, for higher order hypergeometric series.
We present an exposition on the Fuss--Catalan numbers, which are a generalization of the well known Catalan numbers. The literature on the subject is scattered (especially for the case of multiple independent parameters, as will be explained in the text), with overlapping definitions by different authors and duplication of proofs. This paper collects the main theorems and identities, with a consistent notation. Contact is made with the works of numerous authors, including the early works of Lambert and Euler. We demonstrate the application of the formalism to solve algebraic equations by infinite series. Our main result in this context is a new necessary and sufficient formula for the domain of absolute convergence of the series solutions of algebraic equations, which corrects and extends previous work in the field. Some historical material is placed in an Appendix.
The recurrence for the $k$-Fibonacci polynomials is usually iterated upwards to positive values of $n$ only. When the recurrence is iterated downwards to $n<0$, there are indices where the polynomials vanish identically. This fact does not seem to have been noted in the literature. We derive the set of such indices. We establish the connection of our results to the solution of the Skolem problem for the $k$-Fibonacci numbers. For $k\ge3$ and $n<0$, we show that the degree of the polynomial does not increase monotonically with $|n|$. The so-called `left-justified $k$-nomial triangle' is extended to treat negative indices. We derive expressions for the individual polynomial coefficients (the elementary symmetric polynomials of the roots). We present results for the properties of the polynomials, for both $n>0$ and $n<0$, including factorization of the polynomials and properties of the roots. Results are also derived for real roots. We present new, tighter, bounds on the amplitudes of the nonzero roots. We derive new combinatorial sums for the polynomial coefficients, which are more concise and computationally efficient than previously published expressions.
The usual theoretical model for synchrotron radiation in circular accelerators (synchrotrons and storage rings) is to treat a single electron moving in a horizontal circle in a uniform vertical magnetic field, but the true situation in real storage rings is more complicated and exhibits much richer physics. The magnetic fields are inhomogeneous, and there is a bunch of many particles and they traverse a distribution of orbits (hence they encounter different magnetic fields). This results in so-called ``depolarizing spin resonances'' (which do not appear in a simple model of a uniform vertical magnetic field). The calculation of the equilibrium electron spin polarization requires a much more careful analysis. For example, a key insight is that, for motion in inhomogeneous magnetic fields, ``spin flip'' is in general \emph{not} a $180^\circ$ reversal of the spin orientation. The physics of radiative spin polarization involves a mix of many disciplines, and provides a good example of cross-disciplinary thinking. We shall also briefly note the connection to astrophysics. The astrophysics literature mainly treats electron motion in very strong magnetic fields, stronger than the Schwinger critical field (for example a neutron star). It is a problem of ongoing interest in astrophysics to study the radiation by electrons circulating in such strong magnetic fields. This article aims to provide the reader with a survey of the basic physics principles of radiative spin polarization, omitting low-level mathematical algebra as much as possible. Such details can be found in the literature, and are not relevant here.
We treat success runs of independent identically distributed Bernoulli trials (with success parameter $p$) distributed according to the Type II binomial distribution of order $k$. However, the success runs are separated by a gap $g\ge1$ (a failure followed by $g-1$ arbitrary outcomes). Most of the literature treats the case $g=1$ only. Our main results are expressions for the probability mass function (we present two derivations) and the distribution of the longest success run. We also present more concise expressions for previously published results for the factorial moments. We present results for the mean, variance, probability mass function and factorial moments for $\textrm{NB}_{\rm II}(k,g,r)$, the Type II negative binomial distribution of order $k$, where the number of success runs $r$ is fixed and the number of trials $n$ is variable. Let $L$ denote the length of the longest success run. We present a recurrence and generating function for the distribution of $L$ and derive expressions for the mean, variance and factorial moments of $L$.
We present new expressions for the $k$-generalized Fibonacci numbers, say $F_k(n)$. They satisfy the recurrence $F_k(n) = F_k(n-1) +\dots+F_k(n-k)$. Explicit expressions for the roots of the auxiliary (or characteristic) polynomial are presented, using Fuss-Catalan numbers. Properties of the roots are enumerated. We quantify the accuracy of asymptotic approximations for $F_k(n)$ for $n\gg1$. Our results subsume and extend some results published by previous authors. We also present a basis (or `fundamental solutions') to solve the above recurrence for arbitrary initial conditions. We comment on the use of generating functions and multinomial sums for the $k$-generalized Fibonacci numbers and related sequences. We note that the resulting multinomial sums are Dickson polynomials of the second kind in several variables. We also present what may be a new identity for companion matrices.
Compound Poisson distributions have been employed by many authors to fit experimental data, typically via the method of moments or maximum likelihood estimation. We propose a new technique and apply it to several sets of published data. It yields better fits than those obtained by the original authors for a set of widely employed compound Poisson distributions (in some cases, significantly better). The technique employs the power spectrum (the absolute square of the characteristic function). The new idea is suggested as a useful addition to the tools for parameter estimation of compound Poisson distributions.
We present a general formula for the particular solution of an inhomogeneous linear difference equation with variable coefficients. The answer is expressed as a weighted sum of fundamental solutions of the associated linear difference equation. This corresponds to an initial value problem in the case of linear differential equations. We remark that Green's functions are naturally suited for solving such problems. This note presents a Green's function formalism to solve an inhomogeneous linear difference equation with variable coefficients. Both the retarded and advanced Green's functions are required, to obtain a complete solution. We independently confirm previous work for the case of linear difference equations with constant coefficients.
When polarized electron bunches are injected and circulated in a high-energy storage ring, the polarization of the bunches relaxes to the asymptotic value of the radiative polarization, caused by the synchrotron radiation. Hence the bunches must be refreshed periodically, to maintain a predetermined time-averaged value of the bunch polarization. In general, the refresh rates of the "up" and "down" polarization bunches are different. We suggest an alternative policy. We point out that the total bunch refresh rate is almost independent of the asymptotic level of the radiative polarization. We also note that the true goal of so-called "spin matching" is to maximize the buildup time constant (not the asymptotic level) of the radiative polarization. We suggest a scheme to equalize the refresh rates of the "up" and "down" polarization bunches, which may be (i) helpful for accelerator operations, and also (ii) reduce systematic errors in HEP experiments.
The negative binomial distribution NB$(k,r)$ of Type I is the probability distribution for a sequence of independent Bernoulli trials (with success parameter $p\in(0,1)$) with $r$ nonoverlapping success runs of length $\ge k$. We present a new, more concise, expression for its probability mass function. We show it can also be succinctly written using hypergeometric functions. We also present new expressions (combinatorial sums) for its moments and factorial moments, as opposed to only the mean and variance (which are already known). Next, we present an alternative non-combinatorial viewpoint, which yields expressions for the factorial moments not only for nonoverlapping success runs, but also for runs with an overlap of $\ell$, where $\ell\in[0,k-1]$. The case $\ell=k-1$ is the negative binomial distribution NB$(k,r)$ of Type III. The results also yield the solution for the negative binomial distribution NB$(k,r)$ with a minimum gap between the success runs (explained in the text). Addendum 1/23/2024: The probability mass function and factorial moments are derived from the probability generating function. Addendum 1/26/2024: Alternative expressions are presented for the negative binomial distribution NB$(k,r)$ of Type II.
We derive a simple expression for the $r^{th}$ factorial moment $μ_{(r)}$ of the geometric distribution of order $k$ with success parameter $p\in(0,1)$ (and $q=1-p$) in terms of its probability mass function $f_k(n)$. Specifically, $μ_{(r)} = r!f_k((r+1)k+r)/((qp^k)^{r+1})$.
The factorial moments of the standard Poisson distribution are well known and are simple, but the raw moments are considered to be more complicated (Touchard polynomials). The present note presents a recurrence relation and an explicit combinatorial sum for the raw moments of the Poisson distribution of order $k$. Unlike the standard Poisson distribution (the case $k=1$), for $k>1$ the structure of the raw and factorial moments have many similarities and the raw moments are not more complicated (formally, at least) than the factorial moments. We remark briefly on the central moments (i.e.~moments centered on the mean) of the Poisson distribution of order $k$.
The factorial moments of the standard Poisson distribution are well known. The present note presents an explicit combinatorial sum for the factorial moments of the Poisson distribution of order $k$. Unlike the standard Poisson distribution (the case $k=1$), for $k>1$ the structure of the factorial moments is much more complicated. Some properties of the factorial moments of the Poisson distribution of order $k$ are elucidated in this note.
This note analyzes properties of the median $ν$ of the Poisson distribution of order $k$. Given a value for the median in the interval $ν\in[1,k]$, an equation to calculate the corresponding value of the rate parameter $λ$ is derived. Numerical evidence is presented that the value of the median exhibits many scaling properties, which permit one to formulate parameterizations of the value of the median in various domains of the parameter space $(k,λ)$. In all cases, the relevant quantities to calculate are $ν/k$ and $μ/k$, where $μ$ is the mean.
Kostadinova and Minkova published an expression for the probability mass function (pmf) of the Poisson distribution of order $k$, as a combinatorial sum ($\mathit{Pliska~Stud.~Math.~Bulgar.}\ {\bf 22},\ 117-128\ (2013)$). Inspired by their elegant solution, this note presents an alternative combinatorial sum for the pmf of the Poisson distribution of order $k$. The terms are partitioned into blocks of length $k$ (as opposed to $k+1$ by Kostadinova and Minkova). The new sum offers an advantage in the following sense. For $n\in[rk+1,(r+1)k]$, the lowest power of $λ$ in the pmf is $λ^{r+1}$. Hence the lower limit of summation can be increased, to avoid needlessly calculating terms which cancel to identically zero.
This note focuses on the properties of two blocks of elements of the probability mass function (pmf) of the Poisson distribution of order $k\ge2$. The first block is the elements for $n\in[1,k]$ and the second block is the elements for $n\in[k+1,2k]$. It is proved that elements in the first block form an ``absolutely monotonic sequence'' by which is meant that all the finite differences of the sequence are positive. Next, the properties of the elements in the second block are analyzed. It is shown that for sufficiently small $λ>0$, the sequence of elements for $n\in[k+1,2k]$ is strictly decreasing and also concave. The purpose of the analysis is to help determine a supremum value for $λ$, such that the pmf of the Poisson distribution of order $k\ge2$ decreases strictly for all $n \ge k$. A conjectured criterion for the supremum is given. Numerical calculations indicate it is the optimal bound, i.e.~the supremum. In addition, a simple expression is proposed, based on numerical calculation, which is sufficient (but not necessary) and is a good approximation for the supremum.
The Poisson distribution of order $k$ is a special case of a compound Poisson distribution. For $k=1$ it is the standard Poisson distribution. Our main result is a proof that for sufficiently small values of the rate parameter $λ$, the probability mass function (pmf) decreases monotonically for all $n\ge k$ (it is known that the pmf increases strictly for $1\le n \le k$, for fixed $k\ge2$ and all $λ>0$). The second main result is a partial proof that the difference (mean $-$ mode) does not exceed $k$. The term `partial proof' signifies that the derivation is conditional on an assumption which, although plausible and supported by numerical evidence, is as yet not proved. This note also presents new inequalities, and sharper bounds for some published inequalities, for the Poisson distribution of order $k$.