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

A probabilistic approach to some results by Nieto and Truax

In this paper, we reconsider some results by Nieto and Truax about generating functions for arbitrary order coherent and squeezed states. These results were obtained using the exponential of the Laplacian operator; more elaborated operational identities were used by Dattoli et al. \cite{Dattoli} to extend these results. In this note, we show that the operational approach can be replaced by a purely probabilistic approach, in the sense that the exponential of derivatives operators can be identified with equivalent expectation operators. This approach brings new insight about the kinks between operational and probabilistic calculus.Comment: 2nd versio

A New Class of Non-Linear Stability Preserving Operators

We extend Br\"and\'en's recent proof of a conjecture of Stanley and describe a new class of non-linear operators that preserve weak Hurwitz stability and the Laguerre-P\'olya class.Comment: Fixed typos, spelling, and updated links in reference

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

Multicenter assessment of quantitative sensory testing (QST) for the detection of neuropathic-like pain responses using the topical capsaicin model

Background: The use of quantitative sensory testing (QST) in multicenter studies has been quite limited, due in part to lack of standardized procedures among centers. Aim: The aim of this study was to assess the application of the capsaicin pain model as a surrogate experimental human model of neuropathic pain in different centers and verify the variation in reports of QST measures across centers. Methods: A multicenter study conducted by the Quebec Pain Research Network in six laboratories allowed the evaluation of nine QST parameters in 60 healthy subjects treated with topical capsaicin to model unilateral pain and allodynia. The same measurements (without capsaicin) were taken in 20 patients with chronic neuropathic pain recruited from an independent pain clinic. Results: Results revealed that six parameters detected a significant difference between the capsaicin-treated and the control skin areas: (1) cold detection threshold (CDT) and (2) cold pain threshold (CPT) are lower on the capsaicin-treated side, indicating a decreased in cold sensitivity; (3) heat pain threshold (HPT) was lower on the capsaicin-treated side in healthy subjects, suggesting an increased heat pain sensitivity; (4) dynamic mechanical allodynia (DMA); (5) mechanical pain after two stimulations (MPS2); and (6) mechanical pain summation after ten stimulations (MPS10), are increased on the capsaicin-treated side, suggesting an increased in mechanical pain (P < 0.002). CDT, CPT and HPT showed comparable effects across all six centers, with CPT and HPT demonstrating the best sensitivity. Data from the patients showed significant difference between affected and unaffected body side but only with CDT. Conclusion: These results provide further support for the application of QST in multicenter studies examining normal and pathological pain responses

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

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

Solvable Systems of Linear Differential Equations

The asymptotic iteration method (AIM) is an iterative technique used to find exact and approximate solutions to second-order linear differential equations. In this work, we employed AIM to solve systems of two first-order linear differential equations. The termination criteria of AIM will be re-examined and the whole theory is re-worked in order to fit this new application. As a result of our investigation, an interesting connection between the solution of linear systems and the solution of Riccati equations is established. Further, new classes of exactly solvable systems of linear differential equations with variable coefficients are obtained. The method discussed allow to construct many solvable classes through a simple procedure.Comment: 13 page