Searcharxiv⌕ Search

arXiv subjects

S. R. Mane

Publications and source records attributed to S. R. Mane.

32 records · Page 2Linked to original sources
arXiv2023

Asymptotic results for the Poisson distribution of order k

The Poisson distribution of order $k$ is a special case of a compound Poisson distribution. Its mean and variance are known, but results for its median and mode are difficult to obtain, although a few cases have been solved and upper/lower bounds for the mode have been established. This note points out that asymptotic results, for both the median and mode, exhibit simple patterns. The calculations are numerical, hence the results are presented as conjectures. The purpose of this note is to discern patterns for the median and mode, including expressions for exact limits and rates of convergence and (possibly sharp) upper/lower bounds, in a sense to be made precise in the text. The derivation of proofs of the results is left for future work.

math.PR↗
arXiv2023

Structure of the probability mass function of the Poisson distribution of order $k$

The Poisson distribution of order $k$ is a special case of a compound Poisson distribution. For $k=1$ it is the standard Poisson distribution. Although its probability mass function (pmf) is known, what is lacking is a $visual$ interpretation, which a sum over terms with factorial denominators does not supply. Unlike the standard Poisson distribution, the Poisson distribution of order $k$ can display a maximum of $four$ peaks simultaneously, as a function of two parameters: the order $k$ and the rate parameter $λ$. This note characterizes the shape of the pmf of the Poisson distribution of order $k$. The pmf can be partitioned into a single point at $n=0$, an increasing sequence for $n \in [1,k]$ and a mountain range for $n>k$ (explained in the text). The ``parameter space'' of the pmf is mapped out and the significance of each domain is explained, in particular the change in behavior of the pmf as a domain boundary is crossed. A simple analogy (admittedly unrelated) is that of the discriminant of a quadratic with real coefficients: its domains characterize the nature of the roots (real or complex), and the domain boundary signifies the presence of a repeated root. Something similar happens with the pmf of the Poisson distribution of order $k$. As an application, this note explains the mode structure of the Poisson distribution of order $k$. Improvements to various inequalities are also derived (sharper bounds, etc.). New conjectured upper and lower bounds for the median and the mode are also proposed.

math.PR↗
arXiv2023

First double mode of the Poisson distribution of order $k$

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 focus in this note is for $k\ge2$. For sufficiently small values of the rate parameter $λ$, both the median and mode equal zero. The median is zero if and only if $λ\le (\ln2)/k$. The supremum value of $λ$ for the mode to be zero is known only for small values of $k$. This note presents results for the "first double mode" by which is meant the first occasion (smallest value of $λ$) the distribution is bimodal, with modes at $0$ and $m>0$. Next, an almost complete answer is supplied to the question "which positive integers cannot be modes of the Poisson distribution of order $k$?" The term "almost complete" signifies that some parts of the answer are conjectures based on numerical searches. However, if the conjectures are proved to be correct, the solution presented in this note is complete: all parameter values are covered.

math.PR↗
arXiv2016

Relativistic Stern-Gerlach Deflection: Hamiltonian Formulation

A Hamiltonian formalism is employed to elucidate the effects of the Stern-Gerlach force on beams of relativistic spin-polarized particles, for passage through a localized region with a static magnetic or electric field gradient. The problem of the spin-orbit coupling for nonrelativistic bounded motion in a central potential (hydrogen-like atoms, in particular) is also briefly studied.

physics.acc-ph↗
arXiv2015

Decay Lifetimes of the Vector and Tensor Polarization of a Stored Deuteron Beam

An rf solenoid was operated in the IUCF Cooler to induce a depolarizing spin resonance for a stored polarized deuteron beam. The decay lifetimes of the vector and tensor polarizations were found to be in the ratio $1.9:1$, as opposed to the expected ratio of $3:1$ from standard angular momentum theory. This report describes my investigations and attempts to formulate a theoretical explanation of the above phenomenon.

physics.acc-ph↗
arXiv2015

Spin Wheel

A distillation of Koop's idea of the Spin Wheel.

physics.acc-ph↗
arXiv2015

Notes on orbit and spin tracking in an electrostatic storage ring

Two documents have recently been posted on the arXiv describing a numerical integration algorithm: "symplectic orbit/spin tracking code for all-electric storage rings" [1] and some computational results therefrom [2]. This note comments critically on some of the claims in [1] and [2]. In particular, it is not clear that the orbit tracking algorithm described in [1] is really symplectic. Specifically, for electrostatic beamline elements, the so-called "zero length elements," which are treated as position dependent kicks in the formalism in [1], are in fact {\em not} symplectic.

physics.acc-ph↗
arXiv2015

Some notes on beam dynamics due to vertical oscillations in an all-electric storage ring

A document has recently been posted on the arXiv [1], describing analytical formulas and results of particle tracking simulations, for precision tests of numerical integration algorithms for an EDM (electric dipole moment) storage ring. In the context of an all-electric storage ring, the authors cite theoretical formulas by Orlov [2]. However, the reference to Orlov is to a talk at a workshop in 2012, and is unpublished and difficult for independent researchers to access and validate. This note rederives and generalizes some of Orlov's principal results, using a Hamiltonian formalism, and also corrects some details in both Orlov's note [2] and the arXiv post [1].

physics.acc-ph↗
arXiv2015

Generalizations of the McMillan map to $N$-body systems

The McMillan map is a well-known example of a rational integrable system for one particle in a two-dimensional phase space. An elegant recent paper presented a generalization of the McMillan map to an $N$-body system, for particles moving in $d$ space dimensions. This paper presents some alternative generalizations (also completely integrable) of the McMillan map to $N$-body systems. In all cases, the phase space is foliated by a biquadratic curve in the dynamical variables (and a set of suitably chosen angular momentum variables). It is also demonstrated that the constraints to generalize the McMillan map to $N$-body systems are not trivial.

physics.acc-ph↗
arXiv2014

Spin decoherence rate in a weak focusing all-electric ring

I previously derived the expression for the spin decoherence rate (for orbital and spin motion in the horizontal plane) in a weak focusing all-electric storage ring with a radial field (logarithmic potential, field index $n=0$). Here I generalize the calculation to an arbitrary field index $n\ge0$. I also solve the model for the relativistic Kepler problem (field index $n=1$), where the solution for the orbit is known analytically, and I verify that it confirms the solution from perturbation theory.

physics.acc-ph↗
arXiv2014

Polarization at TLEP/FCC-ee: ideas and estimates

I examine various ideas for radiative polarization at TLEP/FCC-ee and formulate some estimates for the polarization buildup time and the asymptotic polarization. Using wigglers, a useful degree of polarization (for energy calibration), with a time constant of about 1 h, may be possible up to the threshold of $W$ pair production. At higher energies such as the threshold of Higgs production, attaining a useful level of polarization may be difficult in a planar ring. With Siberian Snakes, wigglers and some imagination, polarization of reasonable magnitude, with a reasonable time constant (of not more than about 1 h), may be achievable at very high energies.

physics.acc-ph↗
arXiv2014

Equation for spin decoherence rate in an all-electric ring

There is a quantitative error in the derivation for the spin decoherence rate by Talman in IPAC2012. The crucial point is a subtle confusion between the concept of `longitudinal' as in `along the reference orbit' and `parallel to the particle velocity.' They are $\it not$ the same direction, and the distinction is significant for high-precision experiments to search for a possible nonzero electric dipole moment (EDM) of a charged particle in a storage ring.

physics.acc-ph↗
arXiv2013

Fits to data for a stored uncooled polarized deuteron beam

I perform tracking simulations to fit various measurements of the polarization for a stored uncooled polarized deuteron beam, published in the recent paper by Benati et. al (P. Benati {\em et al.}, {\it Phys. Rev. ST Accel. Beams} {\bf 15}, 124202 (2012)). The collaboration kindly sent me datafiles of the polarization measurements, and also pertinent details of the experimental data acquisition procedure. The latter are essential to obtain quantitative fits to the data. I describe my findings and inferences from the data. In some cases I offer alternative interpretations of the data from that given by Benati et. al. I also correct some mistakes in my recent paper (S. R. Mane, {\em Nucl. Inst. Meth.} {\bf 726} 104--112 (2013)).

physics.acc-ph↗