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

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

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

Monomiality principle, Sheffer-type polynomials and the normal ordering problem

We solve the boson normal ordering problem for $(q(a^\dag)a+v(a^\dag))^n$ with arbitrary functions $q(x)$ and $v(x)$ and integer $n$, where $a$ and $a^\dag$ are boson annihilation and creation operators, satisfying $[a,a^\dag]=1$. This consequently provides the solution for the exponential $e^{\lambda(q(a^\dag)a+v(a^\dag))}$ generalizing the shift operator. In the course of these considerations we define and explore the monomiality principle and find its representations. We exploit the properties of Sheffer-type polynomials which constitute the inherent structure of this problem. In the end we give some examples illustrating the utility of the method and point out the relation to combinatorial structures.Comment: Presented at the 8'th International School of Theoretical Physics "Symmetry and Structural Properties of Condensed Matter " (SSPCM 2005), Myczkowce, Poland. 13 pages, 31 reference

Effective superpotentials for compact D5-brane Calabi-Yau geometries

For compact Calabi-Yau geometries with D5-branes we study N=1 effective superpotentials depending on both open- and closed-string fields. We develop methods to derive the open/closed Picard-Fuchs differential equations, which control D5-brane deformations as well as complex structure deformations of the compact Calabi-Yau space. Their solutions encode the flat open/closed coordinates and the effective superpotential. For two explicit examples of compact D5-brane Calabi-Yau hypersurface geometries we apply our techniques and express the calculated superpotentials in terms of flat open/closed coordinates. By evaluating these superpotentials at their critical points we reproduce the domain wall tensions that have recently appeared in the literature. Finally we extract orbifold disk invariants from the superpotentials, which, up to overall numerical normalizations, correspond to orbifold disk Gromov-Witten invariants in the mirror geometry.Comment: 55 pages; v2: references added, typos correcte

A Methodology for Successful University Graduate CubeSat Programs

The University of Colorado Smead Department of Aerospace Engineering has over a decade of success in designing, building, and operating student led CubeSat missions. The experience and lessons learned from building and operating the CSSWE, MinXSS-1, MinXSS-2, and QB50-Challenger missions have helped grow a knowledge base on the most effective and efficient ways to manage some of the “tall poles” when it comes to student run CubeSat missions. Among these “tall poles” we have seen student turnover, software, and documentation become some of the hardest to knock-down and we present our strategies for doing so. We use the MAXWELL mission (expected to launch in 2021) as a road-map to detail the methodology we have built over the last decade to ensure the greatest chance of mission success

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