Sum Power Research Articles

DUAL RECURSIVE FORMULAS FOR THE SUMS OF POWERS OF INTEGERS

In this note, we introduce the concept of a dual pair of recursive formulas for the sums of powers of integers Central to this concept is the symmetry property exhibited by the power sum polynomial . We illustrate the concept by some examples taken from the literature, and derive our own pair of dual recursive formulas for .

Green Rate‐Splitting Multiple Access for Multicell and Rate Fairness Maximization

ABSTRACTThis paper investigates the green design of rate‐splitting multiple access (RSMA) for multicell, multiantenna downlink communication systems to maximize rate fairness under quality of service (QoS) constraints. An optimization framework is developed to minimize the weighted sum power consumption by jointly optimizing transmit beamformers, power allocation, and common/private rate split while ensuring per‐user minimum rate requirements. The algorithm design explicitly considers the rate fairness among users via a newly proposed green fairness index. The nonconvex transceiver design problems are transformed into tractable forms and efficiently solved by applying tools from alternating optimization and successive convex approximation. Extensive simulations demonstrate that the proposed green RSMA optimization framework can effectively balance the throughput among cell‐edge and cell‐center users, hence combating the near‐far effect. Simulation results show that at 20‐dB SNR, green RSMA achieves over 20% and 50% sum rate improvement compared with NOMA and OMA, respectively. Moreover, green RSMA maintains high fairness among users, even when the channel conditions are disparate. When the weakest user is located 120 m from the base station, green RSMA exhibits a 21.2% higher fairness index than NOMA, showcasing its ability to ensure equitable service quality across users. The framework also achieves substantial energy efficiency gains, consuming less power while providing higher data rates than baseline schemes. These results demonstrate the effectiveness of green RSMA in enhancing spectral efficiency, user fairness, and energy efficiency in multicell, multiantenna communication systems, making it a promising solution for sustainable and high‐performance wireless networks.

Recursive Formula for Sum of Powers of Natural Numbers and its Generalization to Arithmetic Progression

Recursive Formula for Sum of Powers of Natural Numbers and its Generalization to Arithmetic Progression

Effective results for polynomial values of (alternating) power sums of arithmetic progressions

Effective results for polynomial values of (alternating) power sums of arithmetic progressions

A binary tree of complete intersections with the strong Lefschetz property

In this paper we give a new family of complete intersections which have the strong Lefschetz property. The family consists of Artinian algebras defined by ideals generated by power sum symmetric polynomials of consecutive degrees and of certain ideals naturally derived from them. This family has a structure of a binary tree and this observation is a key to prove that all members in it have the strong Lefschetz property.

Giant graviton expansion for general Wilson line operator indices

We propose a giant graviton expansion for Wilson line operator indices in general representations. The inserted line operators are specified by power sum symmetric polynomials pλ labeled by partitions λ. We interpret the partitions as the structure of fundamental string worldsheets wrapping around the temporal circle. The strings may or may not end on giant gravitons, and by summing the contributions from all brane configurations consistent with the specified partitions, we obtain the finite N line operator index. The proposed formula is consistent with known results and passes highly non-trivial numerical tests.

Distribution of angular momenta ML and MS in non-relativistic configurations: statistical analysis using cumulants and Gram-Charlier series

Abstract The distributions P(ML,MS) of the total magnetic quantum numbers ML and MS for N electrons of angular momentum l, as well as the enumeration of LS spectroscopic terms and spectral lines, are crucial for the calculation of atomic structure and spectra, in particular for the modeling of emission or absorption properties of hot plasmas. However, no explicit formula for P(ML,MS) is known yet. In the present work, we show that the generating function for the cumulants, which characterize the distribution, obeys a recurrence relation, similar to the Newton-Girard identities relating elementary symmetric polynomials to power sums. This enables us to provide an explicit formula for the generating function. We also analyze the possibility of representing the P(ML,MS) distribution by a bi-variate Gram-Charlier series, which coefficients are obtained from the knowledge of the exact moments of P(ML,MS). It is shown that a simple approximation is obtained by truncating this series to the first few terms, though it is not convergent.

Weighted power summation and contrast normalization mechanisms account for short-latency eye movements to motion and disparity of sine-wave gratings and broadband visual stimuli in humans.

In this paper, we show that the model we proposed earlier to account for the disparity vergence eye movements (disparity vergence responses, or DVRs) in response to horizontal and vertical disparity steps of white noise visual stimuli also provides an excellent description of the short-latency ocular following responses (OFRs) to broadband stimuli in the visual motion domain. In addition, we reanalyzed the data and applied the model to several earlier studies that used sine-wave gratings (single or a combination of two or three gratings) and white noise stimuli. The model provides a very good account of all of these data. The model postulates that the short-latency eye movements-OFRs and DVRs-can be accounted for by the operation of two factors: an excitatory drive, determined by a weighted sum of contributions of stimulus Fourier components, scaled by a global contrast normalization mechanism. The output of the operation of these two factors is then nonlinearly scaled by the total contrast of the stimulus. Despite different roles of disparity (horizontal and vertical) and motion signals in visual scene analyses, the earliest processing stages of these different signals appear to be very similar.

Utilizing aggregation operators based on q-rung orthopair neutrosophic soft sets and their applications in multi-attributes decision making problems

Neutrosophic sets provide greater versatility in dealing with a variety of uncertainties, including independent, partially independent, and entirely dependent scenarios, which q-ROF soft sets cannot handle. Indeterminacy, on the other hand, is ignored completely or partially by q-ROF soft sets. To address this issue, this study offers a unique novel concept as known as q-RONSS, which combines neutrosophic set with q-ROF soft set. This technique addresses vagueness using a set of truth, indeterminacy, and false membership degrees associated with the parametrization tool, with the condition that the sum of the qth power of the truth, indeterminacy, and false membership degrees be less than or equal to one. In addition, this study outlines operational laws for the suggested structure. The main purpose this article is to define some averaging and geometric operators based on the q-rung orthopair neutrosophic soft set. Furthermore, this article provides a step-by-step method and a mathematical model for the suggested techniques. To solve a MADM issue, this research article proposes a numerical example of people selection for a specific position in a real estate business based on a variety of criteria. Finally, to demonstrate the proposed model's superiority and authenticity, this article performs several analyses, including sensitivity analysis, to address the reliability and influence of various parameter "q" values on the alternatives and the ultimate ranking outcomes using the averaging and geometric operators. A comparison of the proposed operators to current operators demonstrates the validity of the proposed structure. Furthermore, a comparison of the proposed structure to current theories demonstrates its superiority by overcoming their limits and offering a more flexible and adaptable framework. Finally, this study reviews the findings and consequences of our research.

Features of building millimeter-wave power combiners with an increased output power level

The problems of designing output power amplifiers of radio transmitting systems based on the summation of the power of individual monolithic integrated circuits are considered. An analysis of well-known solutions for the construction of multichannel power summation devices in the millimeter wave range is carried out. On its basis, proposals have been developed for the construction of multi-channel summation devices with an increased level of output power. The developed devices make it possible to increase the density of amplifier cells and the level of output power while maintaining the permissible thermal regime. Individual amplifier cells can also be replaced. The developed recommendations for the construction of multi-channel millimeter-wave power summation devices can be used in the creation and improvement of output power amplifiers of radio transmitting systems.

Power Mean Inequalities and Sums of Squares

Power Mean Inequalities and Sums of Squares

Gaps Between Consecutive Primes and the Exponential Distribution

Based on the primes less than 4 × 10 18 , Oliveira e Silva et al. (Math. Comp., 83(288):2033–2060, 2014) conjectured an asymptotic formula for the sum of the kth power of the gaps between consecutive primes less than a large number x. We show that the conjecture of Oliveira e Silva holds if and only if the kth moment of the first n gaps is asymptotic to the kth moment of an exponential distribution with mean log n , though the distribution of gaps is not exponential. Asymptotically exponential moments imply that the gaps asymptotically obey Taylor’s law of fluctuation scaling: variance of the first n gaps ∼ (mean of the first n gaps)2. If the distribution of the first n gaps is asymptotically exponential with mean log n , then the expectation of the largest of the first n gaps is asymptotic to ( log n ) 2 . The largest of the first n gaps is asymptotic to ( log n ) 2 if and only if the Cramér-Shanks conjecture holds. Numerical counts of gaps and the maximal gap Gn among the first n gaps test these results. While most values of Gn are better approximated by ( log n ) 2 than by other models, seven exceptional values of n with G n > 2 e − γ ( log n ) 2 suggest that lim sup n → ∞ G n / [ 2 e − γ ( log n ) 2 ] may exceed 1.

Representations of Various Power Sums

Representations of Various Power Sums

Multi-row extremum seeking for wind farm power maximization

This paper presents results from wind tunnel experiments to evaluate power gains from wake steering via yaw control. An experimental scaled wind farm with 12 turbines in an aligned rectangular array is used. Wake steering is performed by yawing turbines using a closed-loop algorithm termed the Log-of-Power Proportional Integral Extremum Seeking Control (LP-PIESC). Two configurations are considered. In the first configuration, the turbines in the first two upstream rows are controlled. In the second case, yaw control is applied to the turbines in the first upstream row and the third row. For both cases, uncontrolled turbines have no yaw misalignment. The results show that by independent parallel maximization of the power sum of a reduced number of turbines, it is possible to obtain a close approximation of the true maximum power. The data shows that the LP-PIESC algorithm can converge relatively fast compared to traditional ESC algorithms.

Learning-based Property Estimation with Polynomials

The problem of estimating data properties using sampling frequency histograms has attracted extensive interest in the area of databases. The properties include the number of distinct values (NDV), entropy, and so on. In the field of databases, property estimation is fundamental to complex applications. For example, NDV estimation is the foundation of query optimization, and entropy estimation is the foundation of data compression. Among them, methods originating from statistics exhibit desirable theoretical guarantees but rely on specific assumptions about the distribution of data, resulting in poor performance in real-world applications. Learning-based methods, which use information from training data, are adaptable in the real world but often lack theoretical guarantees or explainability. In addition, a unified framework for estimating these frequency-based estimators with machine learning is lacking. Given the aforementioned challenges, it is natural to wonder if a unified framework with theoretical guarantees can be established for property estimation. The recent literature has presented theoretical studies that propose estimation frameworks based on polynomials. These studies also prove estimation errors with respect to the sample size. Motivated by the above polynomial estimation framework, we propose a learning-based estimation framework with polynomial approximation, which aims to learn the coefficients of the polynomial, providing theoretical guarantees to the learning framework. Through comprehensive experiments on both synthetic and real-world datasets for estimating various data properties like NDV, entropy, and power sum, our results show the superiority of our algorithms over previous estimators.

Power Summation in an Open Resonator with Aperture Coupling Elements

The paper considers the powers summation of two sources in a hemispherical open resonator (OR). A scheme with a E- tee waveguide was used. The study was carried out in the four-millimeter wavelength range. In the resonator, the first highest axially-asymmetric ТЕМ10q mode is excited with the help of aperture coupling elements. The coupling elements are pyramidal horns located in the center of a flat mirror, with apertures of 6.9×9.6 mm and length of 85 mm. There are one-dimensional E- polarized diffraction gratings with periods of 0.2 mm, 0.4 mm, 0.6 mm in the apertures of the coupling elements. It is shown that the maximum power summation coefficient of two sources in an OR using aperture coupling elements, in the apertures of which a one-dimensional diffraction grating with a period of 0.6 mm is located, is 90%. The high excitation efficiency in the resonator of the TEM10q mode with the help of aperture coupling elements ensures the angular selection of the oscillation spectrum of such a resonant system. It has been established that to sum the powers of two sources in the OR, it is advisable to use not the TEM10q mode but TEM20q.

Multi-aspect assessment and optimization of a poly-generation layout composed of a geothermal power plant and a wind turbine

The current research is an effort to propose a novel poly-generation layout by hybridization of wind power and geothermal energy as two renewable heat sources. The geothermal unit comprises a steam flash cycle as the topping system and a Kalina flash cycle as the bottoming system. Hot water is produced through the utilization of the energy of the geothermal water before reinjection to the ground. An E126/7580 wind turbine is accountable for generating the favorable output electricity. Moreover, the waste heat from the generator of the wind turbine is recovered in a trilateral flash cycle. The sum of power produced by the steam flash cycle and the trilateral flash cycle is employed to generate hydrogen in an electrolysis unit. Moreover, the output electricity of the Kalina flash cycle is employed to produce cooling and freshwater in a transcritical CO2 refrigeration cycle and a reverse osmosis desalination system. The final evaluation of the system utilizing energy and exergy-economic analyses and three-objective optimization of the layout revealed an exergetic efficiency of 37.11 % in the optimum case. The configuration produces 771.7 kW, 454.3 kW, 3585 kW, 1.29 kgh-1, and 3 kgs-1 of electricity, cooling, heating, hydrogen, and freshwater, correspondingly. The economic analysis of the system indicates a total cost rate of 140.2 $h-1, a specific cost of products equal to 19.5 $GJ-1, and a payback period of 1.03 years, suggesting an excellent economic performance for such a complex system.

Non-vanishing of multiple zeta values for higher genus curves over finite fields

In this paper, we show that ∞-adic multiple zeta values associated to the function field of an algebraic curve of higher genus over a finite field are not zero, under certain assumption on the gap sequence associated to the rational point ∞ on the given curve. Using arguments and results of Sheats and Thakur for the case of the projective line, we calculate the absolute values of power sums in the series defining multiple zeta values, and show that the calculation implies the non-vanishing result.

Joint subchannel power allocation for downlink NOMA systems based on quantum carnivorous plant algorithm

Currently, most existing research on non-orthogonal multiple access (NOMA) systems converts power allocation into two independent procedures, i.e., inter-subchannel power allocation and intra-subchannel power allocation. However, improper inter-subchannel power allocation will adversely affect the power allocation process of intra-subchannel. If the power allocations of inter-subchannel and intra-subchannel are optimized simultaneously, it usually needs high computational complexity, and is hard to obtain an optimal solution. To tackle this issue, this paper studies the joint subchannel power allocation for NOMA systems and proposes a new intelligent algorithm called quantum carnivorous plant algorithm (QCPA) for simultaneously optimizing inter-subchannel and intra-subchannel power allocations. Analysis of the convergence of QCPA verified its superior performance on the test functions. By utilizing the optimal solution obtained by QCPA as the power allocation scheme, energy efficiency and sum transmission rate in NOMA systems are significantly increased compared to existing traditional power allocation methods. The results above demonstrate that QCPA effectively addresses test functions and the power allocation problem for NOMA. QCPA possesses favorable convergence in comparison to other comparison algorithms and methods.

Dense clusters of zeros near the zero-free region of ζ(s)

The methods of Korobov and Vinogradov produce a zero-free region for the Riemann zeta function [Formula: see text] of the form [Formula: see text] For many decades, the general shape of the zero-free region has not changed (although explicit known values for [Formula: see text] have improved over the years). In this paper, we show that if the zero-free region cannot be widened substantially, then there exist infinitely many distinct dense clusters of zeros of [Formula: see text] lying close to the edge of the zero-free region. Our proof provides specific information about the location of these clusters and the number of zeros contained in them. To prove the result, we introduce and apply a variant of the original method of de la Vallée Poussin combined with ideas of Turán to control the real parts of power sums. We also prove similar results for [Formula: see text]-functions associated to nonquadratic Dirichlet characters [Formula: see text] modulo [Formula: see text].