Multiparameter Fuss--Catalan numbers with application to algebraic equations

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.Comment: 50 pages, 4 figure

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.Comment: 57 pages, 4 figures. v2: I solved the model also for off-energy dispersion orbits. The dispersion orbits are circles and I calculated the spin decoherence rate exactly. v3: I solved the model also for vertical motion, for $n=0$ (logarithmic potential). I have also solved for the spin precession the Kepler problem (inverse square law) exactly; the paper is in press. Extra appendices have been added; in particular, Ivan Koop kindly gave permission to reproduce his elegant analysis of motion in a vertical spiral in a logarithmic potential. v4: Added synchrotron oscillations. arXiv admin note: substantial text overlap with arXiv:1403.502

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.Comment: 38 pages, 3 figure

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.Comment: 8 pages, no figure

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].Comment: 8 pages, no figures, fix typos in version

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)).Comment: 24 figure

Comment on "rf Wien filter in an electric dipole moment storage ring: The "partially frozen spin'' effect''

The Jacobi-Anger identity is employed to quantify the findings of Morse, Orlov and Semertzidis [1]. The secular term in the longitudinal spin component is proportional to a Bessel function. A suggestion for values of machine parameters to optimize the secular term is presented.Comment: 5 pages, no figure

Orbit and spin tracking in a radial electric field

I derive an expression for the spin decoherence rate for a smooth focusing electrostatic ring with a radial electric field. The answer contains some salutary lessons about spin decoherence calculations in electrostatic storage rings: the numerical tracking results disagreed with naive analytical calculations, but it was the tracking results which were correct, and more careful analytical calculations (canonical transformations) were required.Comment: 13 pages, 1 figur

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.Comment: 8 pages, no figure

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.Comment: 6 pages. no figure