Harmonic Sums Research Articles

Real Valued Functions for the BFKL Eigenvalue

We consider known expressions for the eigenvalue of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation in N=4 super Yang-Mills theory as a real valued function of two variables. We define new real valued functions of two complex conjugate variables that have a definite complexity analogous to the weight of the nested harmonic sums. We argue that those functions span a general space of functions for the BFKL eigenvalue at any order of the perturbation theory.

Non-vanishing of certain cyclotomic multiple harmonic sums and application to the non-vanishing of certain p-adic cyclotomic multiple zeta values

We define and apply a method to study the non-vanishing of p-adic cyclotomic multiple zeta values. We prove the non-vanishing of certain cyclotomic multiple harmonic sums, and, via a formula proved in another paper, which expresses certain cyclotomic multiple harmonic sums as infinite sums of products of p-adic cyclotomic multiple zeta values, this implies the non-vanishing of certain p-adic cyclotomic multiple zeta values.

An Alternative Approach to Obtain a New Gain in Step-Size of LMS Filters Dealing with Periodic Signals

Partial updates (PU) of adaptive filters have been successfully applied in different contexts to lower the computational costs of many control systems. In a PU adaptive algorithm, only a fraction of the coefficients is updated per iteration. Particularly, this idea has been proved as a valid strategy in the active control of periodic noise consisting of a sum of harmonics. The convergence analysis carried out here is based on the periodic nature of the input signal, which makes it possible to formulate the adaptive process with a matrix-based approach, the periodic least-mean-square (P-LMS) algorithm In this paper, we obtain the upper bound that limits the step-size parameter of the sequential PU P-LMS algorithm and compare it to the bound of the full-update P-LMS algorithm. Thus, the limiting value for the step-size parameter is expressed in terms of the step-size gain of the PU algorithm. This gain in step-size is the quotient between the upper bounds ensuring convergence in the following two scenarios: first, when PU are carried out and, second, when every coefficient is updated during every cycle. This step-size gain gives the factor by which the step-size can be multiplied so as to compensate for the convergence speed reduction of the sequential PU algorithm, which is an inherently slower strategy. Results are compared with previous results based on the standard sequential PU LMS formulation. Frequency-dependent notches in the step-size gain are not present with the matrix-based formulation of the P-LMS. Simulated results confirm the expected behavior.

Explicit relations between multiple zeta values and related variants

In this paper we present some new identities for multiple polylogarithms (abbr. MPLs) and multiple harmonic star sums (abbr. MHSSs) by using the methods of iterated integral computations of logarithm functions. Then, by applying these formulas obtained, we establish some explicit relations between Kaneko-Yamamoto type multiple zeta values (abbr. K-Y MZVs), multiple zeta values (abbr. MZVs) and MPLs. Further, we find some explicit relations between MZVs and multiple zeta star values (abbr. MZSVs). Furthermore, we define an Apéry-type variant of MZSVs ζB⋆(k) (called multiple zeta B-star values, abbr. MZBSVs) which involve MHSSs and central binomial coefficients, and establish some explicit connections among MZVs, alternating MZVs and MZBSVs by using the method of iterated integrals. Finally, some interesting consequences and illustrative examples are presented.

Инфрадианные ритмы динамики нарастания побегов в клонах ивы трехтычинковой (Salix triandra)

Abstract. Purpose of research – harmonic analysis of the structure of seasonal dynamics of daily increment of shoots in almond willow (Salix triandra) clones. Research methodology and methods. Object is a model population created by cuttings of sibs-seedlings from the same family obtained by regular inbreeding over three generations. Material – growing long shoots. Experimental group: seven clones of one-year saplings from cuttings; 6-fold repetition. Control group – seedlings from the same family on their own roots of the fourth year of life. Methods: comparative morphological, chronobiological, numerical analysis of time series. Observations were made during the growing season of 2020. Results. The beginning of the growth of shoots – the end of the first decade of May. The maximum daily increase is in early summer (06.06…14.06). Further, the daily growth decreased unevenly until the end of August. Seasonal dynamics of daily increment is determined by the interaction of linear and nonlinear components. Linear components determine the seasonal trend of daily increment dynamics. They are approximated by the corresponding regression equations with different reliability. Nonlinear components determine the cyclical nature of the seasonal dynamics of daily increment. They are approximated by sums of harmonics with an oscillation period of 9–144 days with very high reliability. Scientific novelty. The cyclical nature of the seasonal dynamics of daily increment is determined by the interaction of biorhythms with different periods. Subannual biorhythms with a period of more than 48 days correct seasonal trends of daily increment. Infradian biorhythms with a period of 9...36 days determine the alternation of peaks and dips in the seasonal dynamics of daily increment. Biorhythms with a period of 29...36 days were synchronized in the experimental and control groups, but shifted in phase when comparing the experiment and control. Biorhythms with a period of 21...24 days are synchronized on all the researched shoots. Biorhythms with a period of 9…18 days are not synchronized, but their resulting fluctuations affect the dynamics of daily increment at the beginning and end of the growing season.

Goal-driven method for decoding the configuration of coherent wave groups required for the generation of arbitrary-order vortex lattices.

Together, the number of waves, wave vectors, amplitudes, and additional phases constitute the coherent wave group configuration and determine the pattern of the interference field. Identifying an appropriate wave group configuration is key to generating vortex lattices via interferometry. Previous studies have approached this task by first assigning the four elements, then calibrating the vortex state of the interference field. However, this method has failed to progress beyond generating third-order vortex lattices, which are insufficient for some practical applications. Therefore, this study proposes a method for determining the proper wave group configurations corresponding to arbitrary-order vortex lattices. We adopt a goal-driven approach: First, we set a vortex lattice as the target field and model it, before decomposing the target field into a sum of multiple harmonics using Fourier transforms. These harmonics constitute the wave group required to generate the target vortex lattice. As vortex lattices of any order can be set as the target field, the proposed method is compatible with any mode order. Simulations and experiments were conducted for fourth- and fifth-order vortex lattices, thus demonstrating the effectiveness of the proposed method.

Harmonic sums from the Kummer theorem

The Kummer summation theorem for F12(−1)-series is extended with three integer parameters and then examined by means of power series expansions. Quite a variety of new alternating series involving harmonic-like numbers and squared central binomial coefficients are evaluated in closed form, by making use of coefficient-extraction methods that have recently been applied in the determination of explicit identities for infinite families of rational series for 1π involving harmonic numbers. The same approach is also applied to obtain new identities for integrals involving complete elliptic integrals with complex arguments.

Reflection identities of harmonic sums and pole decomposition of BFKL eigenvalue

We analyze the recent results of next-to-next-to-leading (NNLO) singlet BFKL eigenvalue in [Formula: see text] SYM written in terms of harmonic sums. The nested harmonic sums building known NNLO BFKL eigenvalue for specific values of the conformal spin have poles at negative integers. We sort the harmonic sums according to the complexity with respect to their weight and depth and use their pole decomposition in terms of the reflection identities to find the most complicated terms of NNLO BFKL eigenvalue for an arbitrary value of the conformal spin. The obtained result is compatible with the Bethe–Salpeter approach to the BFKL evolution.

Recovering the instantaneous images of underwater objects distorted by surface waves

ABSTRACT Method for recovering the single instantaneous images of underwater objects distorted by surface waves is presented. This method consists of a combination of two independent parts. In the first part, the algorithm for the determination of an instantaneous relief of waved sea surface from sunglints characteristics has been developed. The instantaneous relief is presented as a sum of harmonics with unknown amplitudes, wavenumbers and phases. To find those unknowns, a system of nonlinear equations that includes the parameters of sunglints, such as the coordinates of glints and their sizes, was derived. The number of unknowns depends on a number of harmonics in a sum, approximating the instantaneous surface relief. In the second part, the algorithm for recovering of the instantaneous image of an underwater object distorted by surface waves is developed. The correctness of the method is tested by a natural experiment.

Some results on q-multiple harmonic sums

In this paper the author present some new identities for q-analogues of multiple harmonic (star) sums whose indices are the sequences ({p}a) and ({p}a, p+1, {p}b). Then we use these formulas to establish some relations between multiple harmonic (star) sums and classical harmonic numbers and binomial coefficients. As an application we give some explicit formulas involving multiple zeta star values. Some interesting consequences and illustrative examples are also given.

Signals Featuring Harmonics With Random Frequencies – Spectral, Distributional and Ergodic Properties

An interesting class of non-Gaussian stationary processes is obtained when in the harmonics of a signal with random amplitudes and phases, one allows also for frequencies to vary randomly. In the resulting models, the statistical distribution of frequencies determines the process spectrum while the distribution of amplitudes governs the process distributional properties. Since decoupling the distribution from the spectrum can be advantageous in applications, we thoroughly investigate a variety of properties exhibited by these models. A process in the considered class of models is uniquely defined by a triple consisting of a positive scale, a normalized spectrum (which is also the distribution of the frequencies), and a normalized Levy measure determining the process distribution. We extend previous work that represented processes as a finite sum of harmonics, by conveniently embedding them into the class of harmonizable processes. Harmonics are integrated with respect to independently scattered second-order non-Gaussian random measures. We present a proper mathematical framework that allows for studying spectral, distributional, and ergodic properties. The mathematical elegance of these representations avoids serious conceptual and technical difficulties with the limiting behavior for discretized models while, at the same time, facilitates the derivation of their fundamental properties. In particular, the multivariate distributions are obtained and the asymptotic behavior of time averages is formally derived through the strong ergodic theorem. Several deficiencies following from the previous approaches are resolved and some of the results appearing in the literature are corrected and extended. It is shown that due to the lack of ergodicity, processes exhibit an interesting property of non-trivial randomness remaining in the limit of time averages.

Hermitian separability of BFKL eigenvalue in Bethe\u2013Salpeter approach

We consider the Bethe–Salpeter approach to the BFKL evolution in order to naturally incorporate the property of the Hermitian separability in the BFKL approach. We combine the resulting all order ansatz for the BFKL eigenvalue together with reflection identities for harmonic sums and derive the most complicated term of the next-to-next-to-leading order BFKL eigenvalue in SUSY N=4. We also suggest a numerical technique for reconstructing the unknown functions in our ansatz from the known results for specific values of confomal spin.

Analytic continuation of harmonic sums near the integer values

We present a simple method for analytic continuation of harmonic sums near negative and positive integer numbers. We provide a precomputed database for the exact expansion of harmonic sums over a small parameter near these integer numbers, along with MATHEMATICA code, which shows the application of the database for actual problems. We also provide the FORM code that was used to obtain the database mentioned above. The applications of the obtained database for the study of evolution equations in the quantum field theory are discussed.

A novel algorithm for nested summation and hypergeometric expansions

We consider a class of sums over products of Z-sums whose arguments differ by a symbolic integer. Such sums appear, for instance, in the expansion of Gauss hypergeometric functions around integer indices that depend on a symbolic parameter. We present a telescopic algorithm for efficiently converting these sums into generalized polylogarithms, Z-sums, and cyclotomic harmonic sums for generic values of this parameter. This algorithm is illustrated by computing the double pentaladder integrals through ten loops, and a family of massive self-energy diagrams through mathcal{O}left({epsilon}^6right) in dimensional regularization. We also outline the general telescopic strategy of this algorithm, which we anticipate can be applied to other classes of sums.

Коррекция режимов резания по данным численного моделирования динамики системы инструмент-заготовка с учетом резонансов

The purpose of the work is to adjust the cutting modes for the manufacturing process of the Cover-Bracket part, which are set in accordance with the recommendations of the electronic catalog CoroPlus ToolGuide by the tool manufacturer Sandvik. This correction is required in order to improve the dynamic stability of machining. The problem of cutting mode adjustment is solved through the use of the methods of numerical simulation of the tool-workpiece system dynamics with regard to the resonances in the Femap engineering analysis program with Nastran. Recommendations are given for cutting modes taking into account the technical capabilities of the machine-tool and tooling, as well as the volume of the material removed with no reference for the dynamic properties of the tool and machine-tool. It is shown that at the 7th and 8th technological transitions the resonant vibrations are observed in the milling tool-workpiece system and the ratio of forces on the cutting edges of milling tools changes up to 245%, which leads to their uneven wear and decreases machining quality. It is found out that the oscillations of the processing forces can be represented as a sum of several harmonics of the rotational and tooth mesh frequencies. The results obtained show that the problem of deviation from the resonant frequencies can be solved not only by lowering the spindle speed, but also by changing the manufacturing process. Recommended change in the sequence of the technological transitions 7 and 8 allows to avoid resonance frequencies without reducing the performance level and machine at the maximum permissible cutter speed of 18,000 rpm. Under this approach, the first harmonics of the tooth mesh frequencies will be outside the resonant zone. In the future, it is planned to supplement the model with the dynamic characteristics of the machine-tool, tool and equipment.

Pro-unipotent harmonic actions and dynamical properties of p-adic cyclotomic multiple zeta values

$p$-adic cyclotomic multiple zeta values depend on the choice of a number of iterations of the crystalline Frobenius of the pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - \{0,\mu_{N},\infty\}$. In this paper we study how the iterated Frobenius depends on the number of iterations, in relation with the computation of $p$-adic cyclotomic multiple zeta values in terms of cyclotomic multiple harmonic sums. This provides new results on that computation and the definition of a new pro-unipotent harmonic action.

Spatial-temporal variability of ice cover of the Bering sea

In this paper, we study the spatial and temporal variability of the Bering sea ice cover. The Bering sea is a major transportation importance as a link in the Northern sea route in turn every activity at sea, meeting the challenges of hydro-meteorological research and predictions in this region largely depend on the knowledge about ice conditions. Observing changes in the ice environment can also serve to assess climate change. For the study, the methods of mathematical statistics applied to data on the amount of ice as a percentage for the Bering sea area in the 1x1 degree grid for the period November 1981goda-April 2014. In the course of the work, the trend component of the time variability of sea ice was identified, spectral and harmonic analysis was performed, and autocorrelation analysis was performed. As a result, for long-term variability, the existence of a trend in the development of ice conditions was revealed, and the presence of a linear trend is unlikely. Analysis of local trends showed a decrease in ice cover in the period 1992-2003, after 2003 there was a large positive trend. Spectral analysis 4 peaks of the spectrum for each were calculated characteristics of harmonics. The sum of harmonics gives a good result when combined with the original data. Long-term variability is low-inertia. when performing an auto forecast for average annual values with a 10-year lead time, it was noted that the prognostic model does not give good results.

Formation and Evolutionary Changes of Coronal Holes in the Growth Phase of Cycle 23

Regularities of formation of coronal holes (CH) at the rising phase of cycle 23 are investigated. The period from 01.01.1997 to 01.03.2000 (Carrington rotations (CRs) 1918-2059) is considered in detail. The evolution of the global magnetic field (GMF) of the Sun from the zonal to the sectorial structure is analyzed. It is shown that the zonal structure is quasi-stable. The sum of zonal harmonics dominated up to CR 1941, although a stable four-sector structure of GMF was formed in 1932. In CRs 1941-1950 the contribution of the zonal and sectorial components becomes approximately equal and from CR 1950 the sectorial structure of GMF was dominated. Sectorial structure of GMF undergoing sharp changes from four-sector, at the beginning of growth of the sectorial harmonics (CR 1926), to the two-sector structure, then back to four-sector and then again to the two-sector structure. CHs clearly trace all evolutionary changes in the GMF. The structure of the polarity of the GMF uniquely determines the zones of photospheric magnetic fields where the CHs are formed.

Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs

We develop the technology for Polyakov-Mellin (PM) bootstrap in one- dimensional conformal field theories (CFT1). By adding appropriate contact terms, we bootstrap various effective field theories in AdS2 and analytically compute the CFT data to one loop. The computation can be extended to higher orders in perturbation theory, if we ignore mixing, for any external dimension. We develop PM bootstrap for O(N ) theories and derive the necessary contact terms for such theories (which also involves a new higher gradient contact term absent for N = 1). We perform cross-checks which include considering the diagonal limit of the 2d Ising model in terms of the 1d PM blocks. As an independent check of the validity of the results obtained with PM bootstrap, we propose a suitable basis of transcendental functions, which allows to fix the four-point correlators of identical scalar primaries completely, up to a finite number of ambiguities related to the number of contact terms in the PM basis. We perform this analysis both at tree level (with and without exchanges) and at one loop. We also derive expressions for the corresponding CFT data in terms of harmonic sums. Finally, we consider the Regge limit of one-dimensional correlators and derive a precise connection between the latter and the large-twist limit of CFT data. Exploiting this result, we study the crossing equation in the three OPE limits and derive some universal constraints for the large-twist limit of CFT data in Regge-bounded theories with a finite number of exchanges.

New congruences on multiple harmonic sums and Bernoulli numbers

Let Pn denote the set of positive integers which are prime to n. Let Bn be the n-th Bernoulli number. For any prime p � 11 and integer r � 2, we prove that X