scholarly journals Algorithms for computing sums of powers of arithmetic progressions by using Eulerian numbers

2021 ◽  
Vol 27 (4) ◽  
pp. 140-148
Author(s):  
Peter J. Shiue ◽  
◽  
Shen C. Huang ◽  
Jorge E. Reyes ◽  
◽  
...  
Keyword(s):  
Arithmetic Progressions ◽  
Experimental Results ◽  
Negative Integer ◽  
New Method ◽  
Complex Numbers ◽  
Eulerian Numbers ◽  
Sums Of Powers ◽  
Novel Algorithms

The sums of powers of arithmetic progressions is of the form a^p+(a+d)^p +(a+2d)^p+\cdots+(a+(n-1)d)^p, where n\geq 1, p is a non-negative integer, and a and d are complex numbers with d\neq 0. This sum can be computed using classical Eulerian numbers \cite{worpitzky1883studien} and general Eulerian numbers \cite{xiong2013general}. This paper gives a new method using classical Eulerian numbers to compute this sum. The existing formula that uses general Eulerian numbers are more algorithmically complex due to more numbers to compute. This paper presents and focuses on two novel algorithms involving both types of Eulerian numbers. This paper gives a comparison to Xiong \textit{et al.}’s result with general Eulerian numbers \cite{xiong2013general}. Moreover, an analysis of theoretical time complexities is presented to show our algorithm is less complex. Various values of p are analyzed in the proposed algorithms to add significance to the results. The experimental results show the proposed algorithm remains around 70\% faster as p increases.

scholarly journals On algorithms for computing the sums of powers of arithmetic progressions

2020 ◽  
Vol 26 (4) ◽  
pp. 113-121
Author(s):  
Peter J. Shiue ◽  
◽  
Shen C. Huang ◽  
Eric Jameson ◽  
◽  
...  
Keyword(s):  
Elementary Proof ◽  
Stirling Numbers ◽  
Arithmetic Progressions ◽  
Experimental Results ◽  
Negative Integer ◽  
Complex Numbers ◽  
Sums Of Powers

This paper is concerned with sums of powers of arithmetic progressions of the form a^p+(a+d)^p+(a+2d)^p+\cdots+(a+(n-1)d)^p, where n\geq 1, p is a non-negative integer, and a and d are complex numbers with d\neq 0. This paper gives an elementary proof to a theorem presented by Laissaoui and Rahmani [9] as well as an algorithm based on their formula. Additionally, this paper presents a simplification to Laissaoui and Rahmani’s formula that is better suited to computation, and a second algorithm based on this simplification. Both formulas use Stirling numbers of the second kind, which are the number of ways to partition p labelled objects into k nonempty unlabelled subsets [4]. An analysis of both algorithms is presented to show the theoretical time complexities. Finally, this paper conducts experiments with varying values of p. The experimental results show the proposed algorithm remains around 10% faster as p increases.

A new method for monitoring the redox potential of fuel salt based on the deposition of 95Nb on Hastelloy C276

2021 ◽  
Vol 109 (5) ◽  
pp. 357-365
Author(s):  
Zhiqiang Cheng ◽  
Zhongqi Zhao ◽  
Junxia Geng ◽  
Xiaohe Wang ◽  
Jifeng Hu ◽  
...  
Keyword(s):  
Redox Potential ◽  
Molten Salt ◽  
Specific Activity ◽  
Chemical Reduction ◽  
Quantitative Approach ◽  
Experimental Results ◽  
New Method ◽  
Molten Salt Reactor ◽  
Fuel Salt ◽  
Well Correlation

Abstract To develop the application of 95Nb as an indicator of redox potential for fuel salt in molten salt reactor (MSR), the specific activity of 95Nb in FLiBe salt and its deposition of 95Nb on Hastelloy C276 have been studied. Experimental results indicated that the amount of 95Nb deposited on Hastelloy C276 resulted from its chemical reduction exhibited a positive correlation with the decrease of 95Nb activity in FLiBe salt and the relative deposition coefficient of 95Nb to 103Ru appeared a well correlation with 95Nb activity in FLiBe salt. Both correlations implied that the measurement of 95Nb activity deposited on Hastelloy C276 specimen might provide a quantitative approach for monitoring the redox potential of fuel salt in MSR.

scholarly journals A Novel Concept Acquisition Approach Based on Formal Contexts

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Ting Qian ◽  
Ling Wei
Keyword(s):  
Data Analysis ◽  
Formal Concept Analysis ◽  
Real Life ◽  
Experimental Results ◽  
New Method ◽  
Concept Acquisition ◽  
Formal Concept ◽  
Knowledge Processing ◽  
Formal Concepts ◽  
Novel Concept

As an important tool for data analysis and knowledge processing, formal concept analysis (FCA) has been applied to many fields. In this paper, we introduce a new method to find all formal concepts based on formal contexts. The amount of intents calculation is reduced by the method. And the corresponding algorithm of our approach is proposed. The main theorems and the corresponding algorithm are examined by examples, respectively. At last, several real-life databases are analyzed to demonstrate the application of the proposed approach. Experimental results show that the proposed approach is simple and effective.

A modified Runge-Kutta method

SIMULATION ◽  
1968 ◽  
Vol 10 (5) ◽  
pp. 221-223 ◽  
Author(s):  
A.S. Chai
Keyword(s):  
Error Estimate ◽  
Linear Combination ◽  
Experimental Results ◽  
The Other ◽  
New Method ◽  
Kutta Method ◽  
Starting Values ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
A Minor

It is possible to replace k2 in a 4th-order Runge-Kutta for mula (also Nth-order 3 ≤ N ≤ 5) by a linear combination of k1 and the ki's in the last step, using the same procedure for computing the other ki's and y as in the standard R-K method. The advantages of the new method are: It re quires one less derivative evaluation, provides an error estimate at each step, gives more accurate results, and needs a minor change to switch to the RK to obtain the starting values. Experimental results are shown in verification of the for mula.

scholarly journals Obtaining Easily Sums of Powers on Arithmetic Progressions and Properties of Bernoulli Polynomials by Operator Calculus

2017 ◽  
Vol 9 (5) ◽  
pp. 73
Author(s):  
Do Tan Si
Keyword(s):  
Differential Operator ◽  
Generating Function ◽  
Arithmetic Progression ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Arithmetic Progressions ◽  
Operator Calculus ◽  
Sums Of Powers ◽  
The Way

We show that a sum of powers on an arithmetic progression is the transform of a monomial by a differential operator and that its generating function is simply related to that of the Bernoulli polynomials from which consequently it may be calculated. Besides, we show that it is obtainable also from the sums of powers of integers, i.e. from the Bernoulli numbers which in turn may be calculated by a simple algorithm.By the way, for didactic purpose, operator calculus is utilized for proving in a concise manner the main properties of the Bernoulli polynomials. 

Cutting Speed of Electric Discharge Machining for SiC Ingot

2012 ◽  
Vol 717-720 ◽  
pp. 861-864 ◽  
Author(s):  
Hideki Yamada ◽  
Satarou Yamaguchi ◽  
Norimasa Yamamoto ◽  
Tomohisa Kato
Keyword(s):  
Silicon Carbide ◽  
Electric Discharge ◽  
Cutting Speed ◽  
Experimental Results ◽  
New Method ◽  
Electric Discharge Machining ◽  
Hard Materials ◽  
Diameter Ingot ◽  
Diamond Saw

A new method based on electric discharge machining (EDM) was developed for cutting a silicon carbide (SiC) ingot. The EDM method is a very useful technique to cut hard materials like SiC. By cutting with the EDM method, kerf loss and roughness of sample are generally smaller than those obtained by cutting with a diamond saw. Moreover, the warpage is smaller than that by the diamond saw cutting, and the cutting speed can be 10 times faster than that of the diamond saw at the present time. We used wires of 50 mm and 100 mm diameters in the experiments, and the experimental results of the cutting speed and the kerf losses are presented. The kerf loss of the 50 mm wire is less than 100 mm, and the cutting speed is about 0.8 mm/min for the thickness of a 6 mm SiC ingot. If we can maintain the cutting speed, the slicing time of a 2 inches diameter ingot would be about seven hours.

scholarly journals Experimental Results on a New Method for Analysis of Heart Rate Variability

2014 ◽  
Vol 04 (08) ◽  
pp. 385-389 ◽  
Author(s):  
Elio Conte ◽  
William Giroldini ◽  
Vincenza Laterza ◽  
Sergio Conte ◽  
Maria Pieralice ◽  
...  
Keyword(s):  
Heart Rate ◽  
Heart Rate Variability ◽  
Experimental Results ◽  
New Method

Ethanol Synthesis from Dimethyl Oxalate Hydrogenation on Cu/SiO2 Catalyst

2012 ◽  
Vol 550-553 ◽  
pp. 175-178 ◽  
Author(s):  
Wen Wen Guo ◽  
Qian Qian Yin ◽  
Ling Jun Zhu ◽  
Shu Rong Wang
Keyword(s):  
High Activity ◽  
Experimental Results ◽  
New Method ◽  
Dimethyl Oxalate ◽  
Mild Conditions ◽  
Ethanol Synthesis

A new method of sustainable ethanol synthesis by hydrogenating dimethyl oxalate (DMO), which is easily obtained from syngas, over a Cu/SiO2catalyst is proposed based on previous works. The experimental results indicate that the Cu/SiO2catalyst exhibited a high activity under the relative mild conditions of 270°C and 2MPa with ethanol selectivity as high as 88% and extremely high DMO conversion.

scholarly journals Artificial Bee Colony Optimizer with Bee-to-Bee Communication and Multipopulation Coevolution for Multilevel Threshold Image Segmentation

2015 ◽  
Vol 2015 ◽  
pp. 1-23 ◽  
Author(s):  
Jun-yi Li ◽  
Yi-ding Zhao ◽  
Jian-hua Li ◽  
Xiao-jun Liu
Keyword(s):  
Image Segmentation ◽  
Information Exchange ◽  
Artificial Bee Colony ◽  
Experimental Results ◽  
New Method ◽  
Communication Pattern ◽  
Communication Model ◽  
Von Neumann ◽  
Bee Colony ◽  
Segmentation Problem

This paper proposes a modified artificial bee colony optimizer (MABC) by combining bee-to-bee communication pattern and multipopulation cooperative mechanism. In the bee-to-bee communication model, with the enhanced information exchange strategy, individuals can share more information from the elites through the Von Neumann topology. With the multipopulation cooperative mechanism, the hierarchical colony with different topologies can be structured, which can maintain diversity of the whole community. The experimental results on comparing the MABC to several successful EA and SI algorithms on a set of benchmarks demonstrated the advantage of the MABC algorithm. Furthermore, we employed the MABC algorithm to resolve the multilevel image segmentation problem. Experimental results of the new method on a variety of images demonstrated the performance superiority of the proposed algorithm.

Incremental Hyperplane Partitioning for Classification

2013 ◽  
Vol 4 (2) ◽  
pp. 67-79 ◽  
Author(s):  
Tao Yang ◽  
Sheng-Uei Guan ◽  
Jinghao Song ◽  
Binge Zheng ◽  
Mengying Cao ◽  
...  
Keyword(s):  
Genetic Algorithm ◽  
Linear Equations ◽  
Curse Of Dimensionality ◽  
Experimental Results ◽  
New Method ◽  
Complex Pattern ◽  
Classification Problems ◽  
Encoding Scheme ◽  
Incremental Approach ◽  
Higher Dimensional

The authors propose an incremental hyperplane partitioning approach to classification. Hyperplanes that are close to the classification boundaries of a given problem are searched using an incremental approach based upon Genetic Algorithm (GA). A new method - Incremental Linear Encoding based Genetic Algorithm (ILEGA) is proposed to tackle the difficulty of classification problems caused by the complex pattern relationship and curse of dimensionality. The authors solve classification problems through a simple and flexible chromosome encoding scheme, where the partitioning rules are encoded by linear equations rather than If-Then rules. Moreover, an incremental approach combined with output portioning and pattern reduction is applied to cope with the curse of dimensionality. The algorithm is tested with six datasets. The experimental results show that ILEGA outperform in both lower- and higher-dimensional problems compared with the original GA.

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