Factors of sums and alternating sums involving binomial coefficients and powers of integers

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer $r$, there holds {align*} \sum_{k=0}^{n_1}\epsilon^k (2k+1)^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}+1\choose n_i-k} \equiv 0 \mod (n_1+n_m+1){n_1+n_m\choose n_1}, {align*} and conjecture that for any nonnegative integer $r$ and positive integer $s$ such that $r+s$ is odd, $$\sum_{k=0}^{n}\epsilon ^k (2k+1)^{r}({2n\choose n-k}-{2n\choose n-k-1})^{s} \equiv 0 \mod{{2n\choose n}},$$ where $\epsilon=\pm 1$.Comment: 14 pages, to appear in Int. J. Number Theor

Multiple cyclical fractional structures in financial time series

This paper analyses multiple cyclical structures in financial time series. In particular, we focus on the monthly structure of the Nasdaq, the Dow Jones and the Standard&Poor stock market indices. The three series are modelled as long-memory processes with poles in the spectrum at multiple frequencies, including the long-run or zero frequency

Single Ion Mass Spectrometry at 100 ppt and Beyond

Abstract. Using a Penning trap single ion mass spectrometer, our group has measured the atomic masses of 14 isotopes with a fractional accuracy of about 10 −10 . The masses were extracted from 28 cyclotron frequency ratios of two ions altenately confined in our trap. The precision on these measurements was limited by the temporal fluctuations of our magnetic field during the 5-10 minutes required to switch from one ion to the other. By trapping two different ions in the same Penning trap at the same time, we can now simultaneously measure their two cyclotron frequencies and extract the ratio with a precision of about 10 −11 in only a few hours. We have developed novel techniques to measure and control the motion of the two ions in the trap and we are currently using these tools to carefully investigate the important question of systematic errors in those measurements. Overview Accuracy in mass spectrometry has been advanced over two orders of magnitude by the use of resonance techniques to compare the cyclotron frequencies of single trapped ions. This paper provides an overview of the MIT Penning trap apparatus, techniques and measurements. We begin by describing the various interesting applications of our mass measurements and the wide-ranging impact they have on both fundamental physics and metrology. In the same section, we also describe further scientific applications that an improved accuracy would open. This serves as a motivation for our most current work (described in Sect. 4) to increase our precision by about an order of magnitude. Before describing the latest results, we give in Sect. 3 an overview of our apparatus and methods, with special emphasis on the techniques which we have developed for making measurements with accuracy around 10 −10 . In those measurements, we alternately trapped two different ions (one at the time) and compared their cyclotron frequencies to obtain their mass ratio. The main limitation of this method was the fact that our stable magnetic field would typically fluctuate by several parts in 10 10 during the 5-10 minutes required to switch from one ion to the other. In order to eliminate this problem, we now confine both ions simultaneously in our Penning trap. In Sect. 4, we describe the various techniques that have allowed us to load a pair in the trap and demonstrate a significant gain in precision from simultaneously measuring both their cyclotron frequencies. New tools to measure and control the motion of the ions are also presented. Those tools are invaluable in our current investigation of the important questio

Scalar Casimir Effect on a D-dimensional Einstein Static Universe

We compute the renormalised energy momentum tensor of a free scalar field coupled to gravity on an (n+1)-dimensional Einstein Static Universe (ESU), RxS^n, with arbitrary low energy effective operators (up to mass dimension n+1). A generic class of regulators is used, together with the Abel-Plana formula, leading to a manifestly regulator independent result. The general structure of the divergences is analysed to show that all the gravitational couplings (not just the cosmological constant) are renormalised for an arbitrary regulator. Various commonly used methods (damping function, point-splitting, momentum cut-off and zeta function) are shown to, effectively, belong to the given class. The final results depend strongly on the parity of n. A detailed analytical and numerical analysis is performed for the behaviours of the renormalised energy density and a quantity `sigma' which determines if the strong energy condition holds for the `quantum fluid'. We briefly discuss the quantum fluid back-reaction problem, via the higher dimensional Friedmann and Raychaudhuri equations, observe that equilibrium radii exist and unveil the possibility of a `Casimir stabilisation of Einstein Static Universes'.Comment: 37 pages, 15 figures, v2: minor changes in sections 1, 2.5, 3 and 4; version published in CQ

Some Orthogonal Polynomials Arising from Coherent States

We explore in this paper some orthogonal polynomials which are naturally associated to certain families of coherent states, often referred to as nonlinear coherent states in the quantum optics literature. Some examples turn out to be known orthogonal polynomials but in many cases we encounter a general class of new orthogonal polynomials for which we establish several qualitative results.Comment: 21 page

A new class of coherent states with Meixner-Pollaczek polynomials for the Gol'dman-Krivchenkov Hamiltonian

A class of generalized coherent states with a new type of the identity resolution are constructed by replacing the labeling parameter zn/n! of the canonical coherent states by Meixner-Pollaczek polynomials with specific parameters. The constructed coherent states belong to the state Hilbert space of the Gol'dman-Krivchenkov Hamiltonian.Comment: 10 pages, Submitte

Optimized Planar Penning Traps for Quantum Information Studies

A one-electron qubit would offer a new option for quantum information science, including the possibility of extremely long coherence times. One-quantum cyclotron transitions and spin flips have been observed for a single electron in a cylindrical Penning trap. However, an electron suspended in a planar Penning trap is a more promising building block for the array of coupled qubits needed for quantum information studies. The optimized design configurations identified here promise to make it possible to realize the elusive goal of one trapped electron in a planar Penning trap for the first time - a substantial step toward a one-electron qubit

A Feynman integral in Lifshitz-point and Lorentz-violating theories in R^D ⨁ R<i>^m</i>

We evaluate a 1-loop, 2-point, massless Feynman integral ID,m(p,q) relevant for perturbative field theoretic calculations in strongly anisotropic d=D+m dimensional spaces given by the direct sum RD ⨁ Rm . Our results are valid in the whole convergence region of the integral for generic (noninteger) codimensions D and m. We obtain series expansions of ID,m(p,q) in terms of powers of the variable X:=4p2/q4, where p=|p|, q=|q|, p Є RD, q Є Rm, and in terms of generalised hypergeometric functions 3F2(−X), when X&lt;1. These are subsequently analytically continued to the complementary region X≥1. The asymptotic expansion in inverse powers of X1/2 is derived. The correctness of the results is supported by agreement with previously known special cases and extensive numerical calculations

Geometry of Schroedinger Space-Times II: Particle and Field Probes of the Causal Structure

We continue our study of the global properties of the z=2 Schroedinger space-time. In particular, we provide a codimension 2 isometric embedding which naturally gives rise to the previously introduced global coordinates. Furthermore, we study the causal structure by probing the space-time with point particles as well as with scalar fields. We show that, even though there is no global time function in the technical sense (Schroedinger space-time being non-distinguishing), the time coordinate of the global Schroedinger coordinate system is, in a precise way, the closest one can get to having such a time function. In spite of this and the corresponding strongly Galilean and almost pathological causal structure of this space-time, it is nevertheless possible to define a Hilbert space of normalisable scalar modes with a well-defined time-evolution. We also discuss how the Galilean causal structure is reflected and encoded in the scalar Wightman functions and the bulk-to-bulk propagator.Comment: 32 page

Orbitofrontal cortex mediates pain inhibition by monetary reward

Pleasurable stimuli, including reward, inhibit pain, but the level of the neuraxis at which they do so and the cerebral processes involved are unknown. Here, we characterized a brain circuitry mediating pain inhibition by reward. Twenty-four healthy participants underwent functional magnetic resonance imaging while playing a wheel of fortune game with simultaneous thermal pain stimuli and monetary wins or losses. As expected, winning decreased pain perception compared to losing. Inter-individual differences in pain modulation by monetary wins relative to losses correlated with activation in the medial orbitofrontal cortex (mOFC). When pain and reward occured simultaneously, mOFCs functional connectivity changed: the signal time course in the mOFC condition-dependent correlated negatively with the signal time courses in the rostral anterior insula, anterior-dorsal cingulate cortex and primary somatosensory cortex, which might signify momentto-moment down-regulation of these regions by the mOFC. Monetary wins and losses did not change the magnitude of pain-related activation, including in regions that code perceived pain intensity when nociceptive input varies and/or receive direct nociceptive input. Pain inhibition by reward appears to involve brain regions not typically involved in nociceptive intensity coding but likely mediate changes in the significance and/or value of pain