Calculation Of Sum Research Articles

Computation of Sum of Squares Polynomials from Data Points

We propose an iterative algorithm for the numerical computation of sums of squares of polynomials approximating given data at prescribed interpolation points. The method is based on the definition ...

Computation of Sums of Natural Powers of the Inverses of Roots of an Equation Connected with a Spectral Problem

We consider equations arising in the oblique derivative spectral problem of the form$$ az{J}_v^{\prime }(z)+b{J}_v(z)=0,\kern1em z\in \mathbb{C}, $$ where ν, a, b ∈ ℂ are parameters such that |a|+|b| > 0 and Jν(z) is the Bessel function. For roots of the equation we prove summation relations. The results obtained agree with the theory of Rayleigh sums which are calculated in terms of zeros of the Bessel functions.

Improved Particle Filter Resampling Architectures

The most challenging aspect of particle filtering hardware implementation is the resampling step. This is because of high latency as it can be only partially executed in parallel with the other steps of particle filtering and has no inherent parallelism inside it. To reduce the latency, an improved resampling architecture is proposed which involves pre-fetching from the weight memory in parallel to the fetching of a value from a random function generator along with architectures for realizing the pre-fetch technique. This enables a particle filter using M particles with otherwise streaming operation to get new inputs more often than 2M cycles as the previously best approach gives. Results show that a pre-fetch buffer of five values achieves the best area-latency reduction trade-off while on average achieving an 85% reduction in latency for the resampling step leading to a sample time reduction of more than 40%. We also propose a generic division-free architecture for the resampling steps. It also removes the need of explicitly ordering the random values for efficient multinomial resampling implementation. In addition, on-the-fly computation of the cumulative sum of weights is proposed which helps reduce the word length of the particle weight memory. FPGA implementation results show that the memory size is reduced by up to 50%.

Some New Identities of Second Order Linear Recurrence Sequences

The main purpose of this paper is using the combinatorial method, the properties of the power series and characteristic roots to study the computational problem of the symmetric sums of a certain second-order linear recurrence sequences, and obtain some new and interesting identities. These results not only improve on some of the existing results, but are also simpler and more beautiful. Of course, these identities profoundly reveal the regularity of the second-order linear recursive sequence, which can greatly facilitate the calculation of the symmetric sums of the sequences in practice.

Joint Communication and Computation Optimization for Wireless Powered Mobile Edge Computing with D2D Offloading

This paper studies a wireless powered mobile edge computing (MEC) system with device-to-device (D2D)-enabled task offloading. In this system, a set of distributed multi-antenna energy transmitters (ETs) use collaborative energy beamforming to wirelessly charge multiple users. By using the harvested energy, the actively computing user nodes can offload their computation tasks to nearby idle users (as helper nodes) via D2D communication links for self-sustainable remote computing. We consider the frequency division multiple access (FDMA) protocol, such that the D2D communications of different user-helper pairs are implemented over orthogonal frequency bands. Furthermore, we focus on a particular time block for task execution, which is divided into three slots for computation task offloading, remote computing, and result downloading, respectively, at different user-helper pairs. Under this setup, we jointly optimize the collaborative energy beamforming at ETs, the communication and computation resource allocation at users and helpers, and the user-helper pairing, so as to maximize the sum computation rate (i.e., the number of task input-bits executed over this block) of the users, subject to individual energy neutrality constraints at both users and helpers. First, we consider the computation rate maximization problem under any given user-helper pairs, for which an efficient solution is proposed by using the techniques of alternating optimization and convex optimization. Next, we develop the optimal user-helper pairing scheme based on exhaustive search and a low-complexity scheme based on greedy selection. Numerical results show that the proposed design significantly improves the sum computation rate at users, as compared to benchmark schemes without such joint optimization.

Computation Efficiency Maximization in OFDMA-Based Mobile Edge Computing Networks

Computation-efficient resource allocation strategies are of crucial importance in mobile edge computing networks. However, few works have focused on this issue. In this letter, weighted sum computation efficiency (CE) maximization problems are formulated in a mobile edge computing (MEC) network with orthogonal frequency division multiple access (OFDMA). Both partial offloading mode and binary offloading mode are considered. The closed-form expressions for the optimal subchannel and power allocation schemes are derived. In order to address the intractable non-convex weighted sum-of ratio problems, an efficiently iterative algorithm is proposed. Simulation results demonstrate that the CE achieved by our proposed resource allocation scheme is better than that obtained by the benchmark schemes.

Architecting Effectual Computation for Machine Learning Accelerators

Inference efficiency is the predominant design consideration for modern machine learning accelerators. The ability of executing multiply-and-accumulate (MAC) significantly impacts the throughput and energy consumption during inference. However, MAC operation suffers from significant ineffectual computations that severely undermines the inference efficiency and must be appropriately handled by the accelerator. The ineffectual computations are manifested in two ways: first, zero values as the input operands of the multiplier, waste time and energy but contribute nothing to the model inference; second, zero bits in nonzero values occupy a large portion of multiplication time but are useless to the final result. In this article, we propose an ineffectual-free yet cost-effective computing architecture, called split-and-accumulate (SAC) with two essential bit detection mechanisms to address these intractable problems in tandem. It replaces the conventional MAC operation in the accelerator by only manipulating the essential bits in the parameters (weights) to accomplish the partial sum computation. Besides, it also eliminates multiplications without any accuracy loss, and supports a wide range of precision configurations. Based on SAC, we propose an accelerator family called Tetris and demonstrate its application in accelerating state-of-the-art deep learning models. Tetris includes two implementations designed for either high performance (i.e., cloud applications) or low power consumption (i.e., edge devices), respectively, contingent to its built-in essential bit detection mechanism. We evaluate our design with Vivado HLS platform and achieve up to $6.96\times $ performance enhancement, and up to $55.1\times $ energy efficiency improvement over conventional accelerator designs.

Analytic continuation for multiple zeta values using symbolic representations

We introduce a symbolic representation of [Formula: see text]-fold harmonic sums at negative indices. This representation allows us to recover and extend some recent results by Duchamp et al., such as recurrence relations and generating functions for these sums. This approach is also applied to the study of the family of extended Bernoulli polynomials, which appear in the computation of harmonic sums at negative indices. It also allows us to reinterpret the Raabe analytic continuation of the multiple zeta function as both a constant term extension of Faulhaber’s formula, and as the result of a natural renormalization procedure for Faulhaber’s formula.

Offloading and system resource allocation optimization in TDMA based wireless powered mobile edge computing

• A WP-MEC system with N WDs and one HAP is considered, where WDs and HAP are single antenna devices. • The goal is to maximize the weighted sum computation rate by joint optimization of system resources management and task computing time allocation. • An alternating direction multiplier method (ADMM) based distributed optimization method is proposed. • Experimental results show that the proposed method greatly increases the weighted sum computation rate while keeping the energy consumption at a low level. In this paper, a wireless powered mobile edge computing (WP-MEC) system is considered, in which a hybrid access point integrated with MEC servers can charge N wireless devices (WDs) by broadcasting radio-frequency signals, and the time division multiple access (TDMA) protocol is used for task offloading of WDs. The goal of this paper is to maximize the weighted sum computation rate by joint optimization of system resources management and task computing time allocation. To solve this optimization problem, an alternating direction multiplier method (ADMM) based distributed optimization method is proposed. The proposed method can decompose the optimization problem into N sub-problems, which are solved by N WDs. Experimental results show that the proposed method outperforms the benchmarks and greatly increases the weighted sum computation rate while keeping the energy consumption at a low level under the premise of time complexity O ( N ).

Structure and Properties of the α-Cs2Mo2−xWxO7 Solid Solution

The tungsten substitution for molybdenum in cesium dimolybdate Cs2Mo2O7 with the formation of an α-Cs2Mo2−xWxO7 solid solution is studied, and its homogeneity region at 540 °C is found to reach 0 ≤ x ≤ 0.6. It is established that the volume of the monoclinic (pseudo-orthorhombic) unit cell and the solid solution melting point monotonically increase with increasing tungsten content, whereas the phase transition temperature to the orthorhombic phase remains practically constant and is ∼390 °C. The crystals of solid solutions with several compositions are obtained, their structure is determined, and the tungsten atoms are found to be mainly located in octahedral positions of the α-Cs2Mo2O7 structure. As a result of the phase transition at 390 °C the electrical conductivity of Cs2Mo2O7 increases approximately by two orders of magnitude reaching 1.2·10−4S/cm at 460 °C. Due to a large size of Cs+ ions and a high charge of molybdenum cations it is supposed that the charge carriers are oxygen anions. The calculation of bond valence sum (BVS) maps for possible positions of oxygen atoms in the α-Cs2Mo2O7 cell shows the formation of a 3D system of oxygen conductivity channels at boundary BVS isosurface values ≥ 2.13.

Increased performance of Matsubara space calculations: A case study within Eliashberg theory

We present a method to considerably improve the numerical performance for solving Eliashberg-type coupled equations on the imaginary axis. Instead of the standard practice of introducing a hard numerical cutoff for treating the infinite summations involved, our scheme allows for the efficient calculation of such sums extended formally up to infinity. The method is first benchmarked with isotropic Migdal-Eliashberg theory calculations and subsequently applied to the solution of the full-bandwidth, multiband and anisotropic equations focusing on the FeSe/SrTiO$_3$ interface as a case study. Compared to the standard procedure, we reach similarly well converged results with less than one fifth of the number of frequencies for the anisotropic case, while for the isotropic set of equations we spare approximately ninety percent of the complexity. Since our proposed approximations are very general, our numerical scheme opens the possibility of studying the superconducting properties of a wide range of materials at ultra-low temperatures.

Minkowski sum computation for planar freeform geometric models using $$G^1$$-biarc approximation and interior disk culling

We present an efficient algorithm for computing the Minkowski sum of two planar geometric models bounded by B-spline curves. The boundary curves are first approximated by \(G^1\)-biarc splines within a given error bound \(\epsilon > 0\). A superset of Minkowski sum boundary is then generated using the biarc approximations. For non-convex models, the superset contains redundant arcs. An efficient and robust elimination of the redundancies is the main challenge of Minkowski sum computation. For this purpose, we use the Minkowski sum of interior disks of the two input models, which are again disks in the Minkowski sum interior. The majority of redundant arcs are eliminated by testing each against a small number of interior disks selected for efficiency. From the planar arrangement of remaining arcs, we construct the Minkowski sum boundary in a correct topology. We demonstrate a real-time performance and the stability of circle-based Minkowski sum computation using a large set of test data.

Unicyclic graphs with second largest and second smallest permanental sums

Let A(G) be an adjacency matrix of a graph G. Then the polynomial π(G,x)=per(xI−A(G)) is called the permanental polynomial of G, and the permanental sum of G is the sum of the absolute values of the coefficients of π(G, x). In this paper, the second largest and second smallest permanental sums among connected unicyclic graphs and the corresponding extremal graphs are determined. In addition, we show that the computation of permanental sum is #P-complete.

Dynamic multi-user computation offloading for wireless powered mobile edge computing

The combination of mobile edge computing (MEC) and wireless energy transfer (WET) has been recognized as a prominent solution to liberate massive wireless devices from limited battery capacities and compute-intensive workloads in the Internet of Things (IoT) environment. This new paradigm is called wireless powered mobile edge computing (WP-MEC). In this paper, an online computation rate maximization (OCRM) algorithm for multi-user WP-MEC systems is proposed by jointly managing the radio, computational resources, allocating time for energy transfer and data transmission. In each time block, the optimal bandwidth allocation and data transmit powers for offloading are achieved by Bisection method with linear time complexity. In addition, the closed-form solutions of optimal time allocation and CPU-cycle frequencies of wireless devices are obtained. The theoretical performance analysis shows that the time average sum computation rate and task queue backlog obey an O1/V,OV tradeoff. Numerical results verify that, our proposed OCRM algorithm outperforms the baseline schemes without joint optimization, which dramatically improves the computation rate while making the system stable.

Расчет статистической суммы и аппроксимация многочастичных функций распределения для магнетиков в модели Гейзенберга

Расчет статистической суммы и аппроксимация многочастичных функций распределения для магнетиков в модели Гейзенберга

Короткое вычисление кратной суммы Кривоколеско – Лейнартаса с линейными ограничениями на индексы суммирования

Короткое вычисление кратной суммы Кривоколеско – Лейнартаса с линейными ограничениями на индексы суммирования

Computation of jacobi sums and cyclotomic numbers with reduced complexity

Jacobi sums and cyclotomic numbers are the important objects in number theory. The determination of all the Jacobi sums and cyclotomic numbers of order $e$ are merely intricate to compute. This paper presents the lesser numbers of Jacobi sums and cyclotomic numbers which are enough for the determination of all Jacobi sums and the cyclotomic numbers of a particular order.

Fast algorithm for Minkowski sum computation of polyhedron and vertical cylinder

Fast algorithm for Minkowski sum computation of polyhedron and vertical cylinder

A fusion approach to classify hyperspectral oil spill data

Oil spills in the ocean are one of the major environmental concerns as they pose significant threat to the ecosystem. In the recent year’s hyperspectral sensors have been used to detect oil spills due to their ability to capture reflected solar signal with very narrow bandwidth, that enables to differentiate materials even with subtle signature differences. However high spectral dimensionality affects classification accuracy due to insufficient training samples. Conventional methods have dealt with this problem through band (or feature) selection or feature extraction (or transformation) approaches. In the case of an oil emulsion signal, as evident from the literature the oil characteristics are present only in certain bands of a spectral signature. Hence feature selection without focusing on right bands may lead to selecting features of less significance for target identification. Moreover, in feature extraction approaches during transformation from high dimensional space to low dimension space some of the important features could be lost. Hence, the proposed research takes advantage of different but complimentary benefits of both feature selection and feature extraction methods to obtain final features effectively. In the proposed research, features are selected by modelling the oil emulsion signal using derivative spectrum and calculation of partial sum of energies. Derivative spectrum represents variation in signal energies and calculation of partial sum of signal energies for each band facilitates the application of filters bank to capture oil sensitive signal characteristics. Feature extraction is done through wavelet transform by gradual multi-scale zooming of signals through partial analysis of time (space) frequencies. The obtained features are fused together and fed to Gaussian Mixture Model (GMM) classifier to classify oil emulsions. The proposed approach gives 5% to–10% higher classification accuracy as compared to some of the conventional techniques.

Resource minimization and power reduction of ESPFFIR filter using unified adder/subtractor

Digital filters are the important processing elements in most of the digital signal processing system designs, ranging from medical signal processing to wireless communication. Low power and area efficient Finite Impulse Response filter is designed with the aid of parallel processing which increases the throughput. Symmetric property reduces the multipliers into half in Even Symmetric Parallel Fast Finite Impulse Response (ESPFFIR) filters. The adder and the subtractor present in the pre/post-processing blocks increase in accordance with the block size L of ESPFFIR filter which in turn increases the numbers of unified operators. The canonical signed digit representation of the coefficient in the design Multiple Constant Multiplications (MCM) based multipliers has the advantages of utilizing unified operators in sub-filter block of ESPFFIR. To reduce the power consumption and resource utilization of the ESPFFIR filter, the unified operator used in the pre/post processing and MCM blocks are replaced by unified adder subtractor (UAS). The UAS utilizes fewer resources for simultaneous calculation of sum and difference of two operands with less area and low power consumption. This paper shows the design of 16-bit Square Root (SQRT) UAS CSLA adder using UAS logical element to reduce the area utilization, power consumption and delay. Then the Heuristic of cumulative benefit (Hcub) based MCM multiplier is designed using UAS. The existing SQRT BEC CSLA adder and subtractor with same input present in the pre/post processing block of the ESPFFIR filters are replaced by a single SQRT UAS CSLA in suitable places. The Hcub MCM used in the sub-filter blocks of the ESPFFIR are replaced using SQRT UAS CSLA based MCM multiplier. The SQRT UAS CSLA consumes 78% less area, 29.51% less power and 24% less delay when compared to the BEC CSLA 2 (one BEC based CSLA Adder plus one BEC based CSLA Subtractor). Finally, the SQRT UAS CSLA adder and Hcub based UAS MCM multipliers are used to design the filter, the overall area and power consumption of UAS based ESPFFIR is reduced compared with the existing BEC based ESPFFIR filter. Cadence RC tool is used for analysis.