Representation of an Integer by a Quadratic Form through the Cornacchia Algorithm

Cornachia’s algorithm can be adapted to the case of the equation x2+dy2=nand even to the case of ax2+bxy+cy2=n. For the sake of completeness, we have given modalities without proofs (the proof in the case of the equation x2+y2=n). Starting from a quadratic form with two variables f(x,y)=ax2+bxy+cy2and n an integer. We have shown that a primitive positive solution (u,v)of the equation f(x,y)=nis admissible if it is obtained in the following way: we take α modulo n such that f(α,1)≡0modn, u is the first of the remainders of Euclid’s algorithm associated with n and α that is less than 4cn/| D |) (possibly α itself) and the equation f(x,y)=n. has an integer solution u in y. At the end of our work, it also appears that the Cornacchia algorithm is good for the form n=ax2+bxy+cy2if all the primitive positive integer solutions of the equation f(x,y)=nare admissible, i.e. computable by the algorithmic process.

Identities on q-Harmonic Numbers

With the help of the classical Abel’s lemma on summation by parts and algorithm of q-hypergeometric summations, we deal with the summation, which can be written as multiplication of a q-hypergeometric term and q-harmonic numbers. This enables us to construct and prove identities on q-harmonic numbers. Several examples are also given.

Z-numbers的否定及其在多属性决策中的应用

否定命题是数理逻辑中的重要表示方式,也是智能信息处理中刻画信息的关键工具.Znumber是一种新的不确定性语言信息表示方式.回顾了否定的相关研究,并介绍了可能性分布的否定在Z-number理论中的应用.通过最大香农熵和粒子群算法,提出了一种优化模型,用于发现Z-number中隐藏的可能性分布.研究表明,Z-number的否定为语言决策分析和解决复杂问题提供了有效工具.

A New Image Watermarking Scheme Using Genetic Algorithm and Residual Numbers with Discrete Wavelet Transform

Transmission of data over the internet has become a critical issue as a result of the advancement in technology, since it is possible for pirates to steal the intellectual property of content owners. This paper presents a new digital watermarking scheme that combines some operators of the Genetic Algorithm (GA) and the Residue Number (RN) System (RNS) to perform encryption on an image, which is embedded into a cover image for the purposes of watermarking. Thus, an image watermarking scheme uses an encrypted image. The secret image is embedded in decomposed frames of the cover image achieved by applying a three-level Discrete Wavelet Transform (DWT). This is to ensure that the secret information is not exposed even when there is a successful attack on the cover information. Content creators can prove ownership of the multimedia content by unveiling the secret information in a court of law. The proposed scheme was tested with sample data using MATLAB2022 and the results of the simulation show a great deal of imperceptibility and robustness as compared to similar existing schemes.

Collatz Conjecture Redefinition on Prime Numbers

The definition of Collatz Operator, the mathematical avatar of the Collatz Algorithm, permits the transformation of the Collatz conjecture, which is delineated over the whole natural number set, into an equivalent inference restricted to the odd prime number set only. Based on this redefinition, one can describe an empirical-heuristic proof of the Collatz conjecture.

Ensuring Quality of Random Numbers from TRNG: Design and Evaluation of Post-Processing Using Genetic Algorithm

Random numbers generated by pseudo-random and true random number generators (TRNG) are used in a wide variety of important applications. A TRNG relies on a non-deterministic source to sample random numbers. In this paper, we improve the post-processing stage of TRNGs using a heuristic evolutionary algorithm. Our post-processing algorithm decomposes the problem of improving the quality of random numbers into two phases: (i) Exact Histogram Equalization: it modifies the random numbers distribution with a specified output distribution;(ii) Stationarity Enforcement: using genetic algorithms, the output of (ii) is permuted until the random numbers meet wide-sense stationarity. We ensure that the quality of the numbers generated from the genetic algorithm is within a specified level of error defined by the user. We parallelize the genetic algorithm for improved performance. The post-processing is based on the power spectral density of the generated numbers used as a metric. We propose guideline parameters for the evolutionary algorithm to ensure fast convergence, within the first 100 generations, with a standard deviation over the specified quality level of less than 0.45. We also include a TestU01 evaluation over the random numbers generated.

RESEARCH ON THE PACKING ALGORITHM FOR ANTI-SEU OF FPGA BASED ON TRIPLE MODULAR REDUNDANCY AND THE NUMBERS OF FAN-OUTS OF THE NET

Static Random Access Memory(SRAM) based Field Programmable Gate Array(FPGA) is widely applied in the field of aerospace, whose anti-SEU(Single Event Upset) capability becomes more and more important. To improve anti-FPGA SEU capability, the registers of the circuit netlist are tripled and divided into three categories in this study. By the packing algorithm, the registers of triple modular redundancy are loaded into different configurable logic block. At the same time, the packing algorithm considers the effect of large fan-out nets. The experimental results show that the algorithm successfully realize the packing of the register of Triple Modular Redundancy(TMR). Comparing with Timing Versatile PACKing(TVPACK), the algorithm in this study is able to obtain a 11% reduction of the number of the nets in critical path, and a 12% reduction of the time delay in critical path on average when TMR is not considered. Especially, some critical path delay of circuit can be improved about 33%.

An NC Algorithm for Sorting Real Numbers in <em>O</em>(nlogn/√<span style="font-size: 14px;font-weight: bold;margin-left:-2px;margin-right:2px;border-top:2px solid black;">loglogn</span>) Operations

We apply the recent important result of serial sorting of n real numbers in time to the design of a parallel algorithm for sorting real numbers in time and operations. This is the first NC algorithm known to take operations for sorting real numbers.

Diversity of Pareto front: A multiobjective genetic algorithm based on dominating information

In this paper, the diversity information included by dominating number is analyzed, and the probabilistic relationship between dominating number and diversity in the space of objective function is proved. A ranking method based on dominating number is proposed to build the Pareto front. Without increasing basic Pareto method’s computation complexity and introducing new parameters, a new multiobjective genetic algorithm based on proposed ranking method (MOGA-DN) is presented. Simulation results on function optimization and parameters optimization of control system verify the efficiency of MOGA-DN.

A two-grid algorithm based on Newton iteration for the stream function form of the Navier-Stokes equations

In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navies-Stokes equations. The algorithm is constructed by reducing the original system to one small, nonlinear system on the coarse mesh space and two similar linear systems （with same stiffness matrix but different right-hand side） on the fine mesh space. The convergence analysis and error estimation of the algorithm are given for the case of conforming elements. Furthermore, the Mgorithm produces a numerical solution with the optimal asymptotic H^2-error. Finally, we give a numerical illustration to demonstrate the effectiveness of the two-grid algorithm for solving the Navier-Stokes equations.

Towards Post-Quantum Cryptography Using Thermal Noise Theory and True Random Numbers Generation

The advent of quantum computers and algorithms challenges the semantic security of symmetric and asymmetric cryptosystems. Thus, the implementation of new cryptographic primitives is essential. They must follow the breakthroughs and properties of quantum calculators which make vulnerable existing cryptosystems. In this paper, we propose a random number generation model based on evaluation of the thermal noise power of the volume elements of an electronic system with a volume of 58.83 cm3. We prove through the sampling of the temperature of each volume element that it is difficult for an attacker to carry out an exploit. In 12 seconds, we generate for 7 volume elements, a stream of randomly generated keys of 187 digits that will be transmitted from source to destination through the properties of quantum cryptography.

A New Evolutionary Algorithm Based on the Decimal Coding

Traditional Evolutionary Algorithm (EAs) is based on the binary code, real number code, structure code and so on. But these coding strategies have their own advantages and disadvantages for the optimization of functions. In this paper a new Decimal Coding Strategy (DCS), which is convenient for space division and alterable precision, was proposed, and the theory analysis of its implicit parallelism and convergence was also discussed. We also redesign several genetic operators for the decimal code. In order to utilize the historial information of the existing individuals in the process of evolution and avoid repeated exploring, the strategies of space shrinking and precision alterable, are adopted. Finally, the evolutionary algorithm based on decimal coding (DCEAs) was applied to the optimization of functions, the optimization of parameter, mixed-integer nonlinear programming. Comparison with traditional GAs was made and the experimental results show that the performances of DCEAS are better than the tradition GAs.

A Novel Power Optimization Method by Minimum Comparator Number Algorithm for Pipeline ADCs

The effects of stage numbers on power dissipation of pipeline analog-to-digital converter （ADC） are studied and a novel design method aiming for power optimization is presented. In this method, a minimum comparator number algorithm （MCNA） is first introduced, and then the optimum distribution of resolutions through pipeline ADC stages is deduced by MCNA. Based on the optimum stage-resolution distribution, an optimization method is established, which examines the precise function between ADC power and stage resolutions with a parameter of power ratio （Rp）. For 10-bit pipeline ADC with scaling down technology, the simulation results by using MATLAB CAD tools show that an eight-stage topology with 1-bit RSD correction achieves the power optimization indicated by the power reduction ratio.

Study on Prediction Model of Number of Rainstorm Days in Summer Based on C5.0 Decision Tree Algorithm

Based on the data of daily precipitation in Lianyungang area from 1951 to 2012 and various climate signal data from the National Climate Center website and the NOAA website,a model for predicting whether the number of rainstorm days in summer in Lianyungang area is large was established by the classical C5. 0 decision tree algorithm. The data samples in 48 years( accounting for about 80% of total number of samples)was as the training set of a model,and the training accuracy rate of the model was 95. 83%. The data samples in the remaining 14 years( accounting for about 20% of total number of samples) were used as the test set of the model to test the model,and the test accuracy of the model was 85. 71%. The results showed that the prediction model of number of rainstorm days in summer constructed by C5. 0 algorithm had high accuracy and was easy to explain. Moreover,it is convenient for meteorological staff to use directly. At the same time,this study provides a new idea for short-term climate prediction of number of rainstorm days in summer.

Convergence analysis of a nonlinear Lagrange algorithm for general nonlinear constrained optimization problems

The convergence analysis of a nonlinear Lagrange algorithm for solving nonlinear constrained optimization problems with both inequality and equality constraints is explored in detail. The estimates for the derivatives of the multiplier mapping and the solution mapping of the proposed algorithm are discussed via the technique of the singular value decomposition of matrix. Based on the estimates, the local convergence results and the rate of convergence of the algorithm are presented when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions. Furthermore, the condition number of the Hessian of the nonlinear Lagrange function with respect to the decision variables is analyzed, which is closely related to efficiency of the algorithm. Finally, the preliminary numericM results for several typical test problems are reported.

Modeling and Validation of Base Pressure for Aerodynamic Vehicles Based on Machine Learning Models

The application of abruptly enlarged flows to adjust the drag of aerodynamic vehicles using machine learning models has not been investigated previously.The process variables(Mach number(M),nozzle pressure ratio(η),area ratio(α),and length to diameter ratio(γ))were numerically explored to address several aspects of this process,namely base pressure(β)and base pressure with cavity(βcav).In this work,the optimal base pressure is determined using the PCA-BAS-ENN based algorithm to modify the base pressure presetting accuracy,thereby regulating the base drag required for smooth flow of aerodynamic vehicles.Based on the identical dataset,the GA-BP and PSO-BP algorithms are also compared to thePCA-BAS-ENNalgorithm.The data for training and testing the algorithmswas derived using the regression equation developed using the Box-Behnken Design(BBD).The results show that the PCA-BAS-ENN model delivered highly accurate predictions when compared to the other two models.As a result,the advantages of these results are two-fold,providing:(i)a detailed examination of the efficiency of different neural network algorithms in dealing with a genuine aerodynamic problem,and(ii)helpful insights for regulating process variables to improve technological,operational,and financial factors,simultaneously.

Development of New Method for Generating Prime Numbers

The article is devoted to actual problems of prime numbers. A theorem that allows generating a sequence of prime numbers is proposed. An algorithm for generating prime numbers has been developed. A comparison of the proposed theorem, with Wilson’s theorem is also provided.

Performance Evaluation and Analysis of Switching Algorithms in MIMO-OFDM System with Ideal and Non-Ideal CSI

In this paper we analyzed the bit error rate performance of a switching algorithm between spatial multiplex-ing and diversity for an OFDM MIMO system with ideal channel state information. The effect of channel estimation error was studied and we verified by simulations that the spatial multiplexing outperforms the switching algorithm. Given that the switching algorithm is based on the comparison of the channel matrix Demmel condition number to a threshold, its accuracy is compromised when channel estimation error in-creases. As a first intuitive solution, we proceeded to the adaptation of the threshold, but this didn’t lead to a pertinent improvement for the main reason that channel estimation errors did affect the MIMO techniques which use different constellation. Based on that, we proposed a new estimation technique that improved the bit error rate performance significantly.

There Are Infinitely Many Mersnne Composite Numbers with Prime Exponents

By extending both arithmetical operations into finite sets of natural numbers, from the entire set of natural numbers successively deleting some residue classes modulo a prime, we invented a recursive sieve method or algorithm on natural numbers and their sets. The algorithm mechanically yields a sequence of sets, which converges to the set of all primes p such that 2p + 1 divides the Mersenne number Mp. The cardinal sequence corresponding to the sequence of sets is strictly increasing. So that we have captured enough usable structures, without any estimation, the existing theories of those structures allow us to prove an exact result: there are infinitely many Mersenne composite numbers with prime exponents Mp.

On Iterative Algorithm and Perturbation Analysis for the Nonlinear Matrix Equation

In this study,an iterative algorithm is proposed to solve the nonlinear matrix equation X+A∗eXA=In.Explicit expressions for mixed and componentwise condition numbers with their upper bounds are derived to measure the sensitivity of the considered nonlinear matrix equation.Comparative analysis for the derived condition numbers and the proposed algorithm are presented.The proposed iterative algorithm reduces the number of iterations significantly when incorporated with exact line searches.Componentwise condition number seems more reliable to detect the sensitivity of the considered equation than mixed condition number as validated by numerical examples.