Harmonic Sums Research Articles

The [formula omitted] contributions to the gluonic massive operator matrix elements

The O(αs3nfTF2CA,F) terms to the massive gluonic operator matrix elements are calculated for general values of the Mellin variable N using a new summation technique. These twist-2 matrix elements occur as transition functions in the variable flavor number scheme at NNLO. The calculation uses sum-representations in generalized hypergeometric series turning into harmonic sums. The analytic continuation to complex values of N is provided.

New approach for approximating the continuum wave function by Gaussian basis set

A new approach for approximating the continuum wave functions for hydrogenic atoms with Gaussians basis sets is developed and tested. In this the plane wave is left unchanged and the distorting factor, represented by the Confluent Hypergeometric function, is expanded as a sum of Spherical Harmonics multiplied by a series of Gaussians. In this way the number of spherical waves and Gaussians will be significantly reduced and the plane wave will be responsible for the momentum conservation.As compared with previous methods that expand the full continuum, including the plane wave, our strategy does not require a great quantity of partial waves for convergence. Dense oscillations which are characteristic of the plane wave, are avoided. To test the performance of this approximation to describe a free-bound atomic form factor, the ionization cross section of hydrogen by impact of protons in first Born approximation is calculated. Compared with the exact results, a good agreement with just 4 spherical waves and ten Gaussians each is obtained. The method looks very interesting, especially to speed up atomic and molecular collision calculations involving the continuum.

Le produit harmonique des suites

By means of an involutary binomial transformation on complex sequences, we define a new product called “harmonic” because of its remarkable properties towards harmonic sums. The Euler series transformation allows one to deduce from these properties some new and remarkable identities.

Finite-frequency identification of plant with time delay

A method of finite-frequency identification for stable plants with time-delay in the presence of unknown-but-bounded disturbance and measurement noise is proposed. This method uses test signal which is a sum of harmonics. Quantity of harmonics does not exceed count of plant coefficients. Conditions of convergence of identification process are also given.

Massive 3-loop ladder diagrams for quarkonic local operator matrix elements

3-loop diagrams of the ladder-type, which emerge for local quarkonic twist-2 operator matrix elements, are computed directly for general values of the Mellin variable N using Appell-function representations and applying modern summation technologies provided by the package Sigma and the method of hyperlogarithms. In some of the diagrams generalized harmonic sums with ξ∈{1,1/2,2} emerge beyond the usual nested harmonic sums. As the asymptotic representation of the corresponding integrals shows, the generalized sums conspire giving well behaved expressions for large values of N. These diagrams contribute to the 3-loop heavy flavor Wilson coefficients of the structure functions in deep-inelastic scattering in the region Q2≫m2.

Large-N c quantum chromodynamics and harmonic sums

In the Large-Nc limit of QCD, two--point functions of local operators become Harmonic Sums. I review some properties which follow from this fact and which are relevant for phenomenological applications. This has led us to consider a class of Analytic Number Theory Functions as toy models of Large-Nc QCD which I also discuss.

Controlled electric field in multiply connected circular structure

This paper presents the analysis and calculation of homogeneity zone of controlled (rotating) electric field in the internal area with circular multiply connected border and equal lengths of boundary arcs depending on the order of border connectivity, potential distribution across the boundary arcs, and the ratio between the lengths of boundary arcs and gaps between the arcs. Mathematical expressions are presented both in the form of complex variable functions and as sums of spatial harmonics.

Harmonic Summation and Assessment Based on Probability Distribution

At the point of common coupling, there are many cases where the phase-angle difference between harmonic vectors is unknown. In practice, for example, in calculating the steady-state rating of the ac passive harmonic filter, the arithmetic summation method is often used in determining the maximal harmonic level. This summation law is very conservative as it may result in high cost of equipment ratings. Actually, a compromise should be made in taking into account the equipment safety and the risk of overrating as well as excessive costs. Based on uniform distribution of the difference in phase angles, this letter provides an algorithm to calculate the probability upon which the magnitude of summation vector of two harmonics may exceed a given value. This may also result in a recommendation of another harmonic summation method for the standard IEC/TR 61000-3-6.

Leading quantum correction to energy of ‘short’ spiky strings

We consider semiclassical quantization of spiky strings spinning in the AdS3 part of AdS5 × S5 using an integrability-based (algebraic curve) method. In the ‘short-string’ (small-spin) limit the expansion of string energy starts with its flat-space expression. We compute the leading quantum string correction to ‘short’ spiky string energy and find the explicit form of the corresponding one-loop coefficient a01. It turns out to be rational and expressed in terms of the harmonic sums as functions of the number n of spikes. In the special case of n = 2 when the spiky string reduces to the single-folded spinning string, the coefficient a01 takes the value (−1/4) found in Gromov et al (2011 J. High Energy Phys. JHEP08(2011)046). We also consider a similar computation for the m-folded string and more general spiky string with an extra ‘winding’ number, finding similar expressions for a01. These results may be useful for a description of energies of higher excited states in the quantum AdS5 × S5 string spectrum, generalizing earlier discussions of the string counterparts of the Konishi operator.

Distribution of alternative power sums and Euler polynomials modulo a prime

For a fixed integer s ≥ 2 , we estimate exponential sums with alternative power sums A s ( n ) = ∑ i = 0 n ( − 1 ) i i s individually and on average, where A s ( n ) is computed modulo p . Our estimates imply that, for any ϵ > 0 , the sets { A s ( n ) : n < p 1 / 2 + ϵ } and { ( − 1 ) n E s ( n ) : n < p 1 / 2 + ϵ } are uniformly distributed modulo a sufficient large p , where E s ( x ) are Euler polynomials. Comparing with the results in Garaev et al. (2006) [M. Z. Garaev, F. Luca and I. E. Shparlinski, Distribution of harmonic sums and Bernoulli polynomials modulo a prime, Math. Z., 253 (2006), 855–865], we see that the uniform distribution properties for the alternative power sums and Euler polynomials modulo a prime are better than those for the harmonic sums and Bernoulli polynomials.

Harmonic sums and polylogarithms generated by cyclotomic polynomials

The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable quantum field theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin transforms of Poincaré–iterated integrals, including denominators of higher cyclotomic polynomials. We derive the cyclotomic harmonic polylogarithms and harmonic sums and study their algebraic and structural relations. The analytic continuation of cyclotomic harmonic sums to complex values of N is performed using analytic representations. We also consider special values of the cyclotomic harmonic polylogarithms at argument x = 1, respectively, for the cyclotomic harmonic sums at N → ∞, which are related to colored multiple zeta values, deriving various of their relations, based on the stuffle and shuffle algebras and three multiple argument relations. We also consider infinite generalized nested harmonic sums at roots of unity which are related to the infinite cyclotomic harmonic sums. Basis representations are derived for weight \documentclass[12pt]{minimal}\begin{document}$\sf{w=1,2}$\end{document}w=1,2 sums up to cyclotomy \documentclass[12pt]{minimal}\begin{document}$\sf{l=20}$\end{document}l=20. This paper is dedicated to Martinus Veltman on the occasion of his 80th birthday.

Numerical simulation of core convection by a multi-layer semi-implicit spherical spectral method

A semi-implicit multi-layer spherical spectral method for simulating stellar core convection is described. The fully compressible three-dimensional hydrodynamic equations with rotation and energy generation are solved. Prognostic variables are expressed as finite sums of spherical harmonics in the horizontal directions and handled by the finite difference method in the radial direction. The stratified approximation is used to simplify the nonlinearity to quadratic. A multi-layer scheme is employed to overcome the time step problem arising from shrinking grid sizes in the physical space near the center of the star. Despite of the different spectral truncations in different layers, round-off conservation of the total mass and total angular momentum of the whole domain can be maintained, and were confirmed numerically. The code is parallelized; with 12 processors the speedup factor is about 9. The solutions of model core convection with and without rotation are discussed.

Numerical Investigation of Superharmonic Imaging Using Chirp Excitation

At present, along with the fundamental method of ultrasound imaging, tissue harmonic imaging based on imprinting of the second harmonic component is established as the standard method of ultrasound imaging. Recently introduced superharmonic imaging (SHI) has an edge over this standard method and has improved spatial resolution. To explore further advancement in ultrasound imaging for enhanced resolution and penetration depth. SHI is the summation of higher harmonics generated due to nonlinear tissue propagation of ultrasound. Here, the addition of third, fourth and fifth harmonics has been done for the superharmonic field. The modeling of nonlinear wave propagation is done in frequency domain on the basis of the second order operator splitting approach. The higher harmonic components are picked up using the inverse filters. The standard evaluation parameters were significantly improved with linear frequency modulation (FM) chirp. The axial level was 2.8 times higher with linear FM chirp as compared to conventional short pulse. Minor benefits such as a 5% increase in beam width and suppression of side lobes were also recorded. This numerical investigation confirms that linear FM chirp in SHI provides better resolution and penetration depth. State of the art fundamental and second harmonic ultrasound imaging techniques were also improved with linear FM chirp as compared with conventional excitation. SHI is superior to the existing diagnostic ultrasound techniques.

Tracing the quarter-diurnal signatures of nutrients and dissolved organic matter to evaluate their nonconservative behaviors in coastal seawaters

[1] In June 2009, six consecutive observations with duration times of 13 h or 26 h were performed at the sea surface of outer Hangzhou Bay in the East China Sea. Physical (current direction and velocity), hydrological (salinity, suspended particulate matter (SPM) turbidity, total volume and mean size of suspended flocs), and chemical tracer (nutrients, chromophoric dissolved organic matter, and dissolved organic nitrogen) data were collected. Harmonic analysis that included diurnal, semidiurnal, and quarter‐diurnal constituents was conducted on time series data for all of the parameters measured. Spectral analysis was performed to determine whether the three harmonics could characterize the variations observed in the data. The amplitude ratio of quarter‐diurnal harmonic to the sum of diurnal and semidiurnal harmonics was expected to quantify the nonconservative extent of the chemical tracers. This ratio generally was higher in turbid seawater, whereas the lowest ratio occurred at a station where the seawater displayed zero turbidity throughout the experiment. The quarter‐diurnal phase lags between SPM and chemical tracers were believed to provide information about how SPM and chemicals interact. How field processes such as resuspension and flocculation/ disaggregation affected the kinetics of the chemical tracers also was explored by plotting the quarter‐diurnal phase spectra along the 32 size classes obtained from Laser In Situ Scattering and Transmissometry measurements. This study provided an interesting and potentially useful technique to estimate the nonconservative behaviors of biogenic elements and to study the relationships between the fields of physical oceanography and biogeochemistry in estuarine and coastal seawaters. Citation: Gao, L., D. Fan, Y. Zhang, D. Li, and J. Cai (2011), Tracing the quarter‐diurnal signatures of nutrients and dissolved organic matter to evaluate their nonconservative behaviors in coastal seawaters, J. Geophys. Res., 116, G03015,

Simulation of terahertz generation in corrugated plasma waveguides

We simulate the response of a corrugated plasma channel to an ultrashort laser pulse in two dimensions with the goal of demonstrating the production of terahertz frequency electromagnetic modes. Corrugated channels support electromagnetic modes that have a Floquet-type dispersion relation and thus consist of a sum of spatial harmonics with subluminal phase velocities. This allows the possibility of phase matching between the ponderomotive potential associated with the laser pulse and the electromagnetic modes of the channel. Since the bandwidth of an ultrashort pulse includes terahertz frequencies, significant excitation of terahertz radiation is possible. Here we consider realistic density profiles to obtain predictions of the terahertz power output and mode structure for a channel with periodic boundary conditions. We then estimate pulse depletion effects from our simulation results. The fraction of laser energy converted to terahertz is independent of laser pulse energy in the linear regime, and we find it to be around 1%. Extrapolating to a pulse energy of 0.5 J gives a terahertz power output of 6 mJ with a pulse depletion length of less than 20 cm.

Identities on harmonic and q-harmonic number sums

By partial fraction approach, we derive q-analog for several well known results on harmonic number sums.

Addition theorems for spin spherical harmonics: II. Results

Based on the results of part I (2011 J. Phys. A: Math. Theor. 44 165301), we obtain the general form of the addition theorem for spin spherical harmonics and give explicit results in the cases involving one spin-s′ and one spin-s spherical harmonics with s′, s = 1/2, 1, 3/2, and |s′ − s| = 0, 1. We also obtain a fully general addition theorem for one scalar and one tensor spherical harmonic of arbitrary rank. A variety of bilocal sums of ordinary and spin spherical harmonics are given in explicit form, including a general explicit expression for bilocal spherical harmonics.

Mod p structure of alternating and non-alternating multiple harmonic sums

Le theoreme bien connu de Wolstenholme affirme que, pour tout premier p > 3, la (p-1)-ieme somme partielle de la serie harmonique est congrue a 0 modulo p 2 . Si on remplace la serie harmonique par ∑ k≥1 1/n k pour k pair alors la congruence est vraie seulement modulo p au lieu de modulo p 2 . On peut considerer des generalisations aux sommes harmoniques multiples (SHM) et aux aux sommes harmoniques multiples alternees (SHMA) qui sont des sommes partielles de series zeta multiples et, respectivement, de sommes d'Euler alternees. Beaucoup de resultats dans cette direction ont ete obtenus dans les articles recents [6, 7, 8, 10, 11, 12], que nous recapitulerons dans ce papier. Il apparait que, pour un premier p, la (p-1)-ieme somme partielle d'une SHM ou SHMA generale n'est plus congrue a 0 modulo p; cependant elle peut souvent etre exprimee en terme de nombres de Bernoulli. Donc c'est un probleme assez interessant de trouver qui ils sont. Dans cet article, nous fournirons un cadre theorique dans lequel ce genre de resulats peut s'exprimer et etre etudie plus longuement. Nous calculerons aussi quelques SHM supplementaires modulo un premier p quand le poids est inferieur a 13.

First normal stress difference of entangled polymer solutions in large amplitude oscillatory shear flow

The first normal stress difference (N1) in large amplitude oscillatory shear (LAOS) consists of two contributions: one from non-oscillating nonzero mean value and the other from oscillating even harmonics, while the nonlinear shear stress can be expressed as a sum of odd harmonics. In this study, the two nonlinear contributions of N1 in entangled polymer solutions have been analyzed by systematical comparison with their shear stress counterparts (storage modulus and odd harmonics, respectively) as a function of strain amplitude and of Deborah number (De). The contributions of N1 were greater than those of the shear stress in the LAOS region. The nonzero mean value of N1 decreased with an increase in Deborah number, but conversely, the second harmonic of N1 increased. The relative intensity (the second harmonic over the fourth harmonic of N1) as an indicator of the nonlinearity of N1 decreased with the Deborah number. We also compared experimental results to model predictions from the eight-mode Giesekus c...

The [formula omitted] massive operator matrix elements of [formula omitted] for the structure function [formula omitted] and transversity

The contributions ∝ n f to the O ( α s 3 ) massive operator matrix elements describing the heavy flavor Wilson coefficients in the limit Q 2 ≫ m 2 are computed for the structure function F 2 ( x , Q 2 ) and transversity for general values of the Mellin variable N. Here, for two matrix elements, A q q , Q PS ( N ) and A q g , Q ( N ) , the complete result is obtained. A first independent computation of the contributions to the 3-loop anomalous dimensions γ q g ( N ) , γ q q PS ( N ) , and γ q q NS , ( TR ) ( N ) is given. In the computation advanced summation technologies for nested sums over products of hypergeometric terms with harmonic sums have been used. For intermediary results generalized harmonic sums occur, while the final results can be expressed by nested harmonic sums only.