General Polynomial Research Articles

Day-ahead dynamic operating envelopes using stochastic unbalanced optimal power flow

Driven by the energy transition, distribution networks are dealing with increasing uptake of distributed energy resources, including solar photovoltaic generation. Residential rooftop solar allows customers to minimize exposure to price increases in the market, which has led to high penetration of PV in regions with high amounts of solar hours such as Australia. Eventually, this leads to congestion in the network, either due to voltage rise, or due to overcurrent in lines or transformers. Distribution utilities are now moving on from static export limits for customers in congested networks, to dynamic limits that are based on the state of the network. It is considered thoughtful to give customers advance warning, e.g. day-ahead, of the moments and degrees of export limitation, despite uncertainty surrounding the future state of the network. Therefore, in this paper, we consider the application of general polynomial chaos expansion based stochastic unbalanced optimal power flow to the day-ahead determination of dynamic export limits. We perform a numerical study on a European style low voltage feeder and illustrate the impact of fairness principles on chance-constrained stochastic nonlinear optimization without the need of sampling, linearizing the power flow equations, or applying relaxations. Case studies show the necessity of considering unbalanced study of distribution system. It was observed that equality measures reduce the overall output of the system in an attempt to achieve equal relative injection, while alpha fairness with a higher value of alpha is a compromise between the efficiency and fairness in DOEs.

Analog In-memory Circuit Design of Polynomial Multiplication for Lattice Cipher Acceleration Application

As the core operation of lattice cipher, large-scale polynomial multiplication is the biggest computational bottleneck in its realization process. How to quickly calculate polynomial multiplication under resource constraints has become an urgent problem to be solved in the hardware implementation of lattice ciphers. Therefore, an analog in-memory circuit for fast polynomial multiplication calculation is proposed. First, an in-memory computing circuit for Discrete Fourier Transform and Inverse Discrete Fourier Transform based on memristor array is designed. On this basis, a fully analog circuit that can realize polynomial multiplication in one step is designed. Compared with traditional hardware implementation, the in-memory calculation method used in this article decreases the calculation time of polynomial multiplication to the microsecond level, which greatly improves the speed of lattice cipher encryption and decryption. For the specific examples in this article, PSPICE simulation shows that the average accuracy of the calculation result is above 99.90%.

Comparison and development of the descriptive model with polynomial structures to fit multi-component dynamic breakthrough curves with roll-up, stepwise, and saddle-shaped structure

Dynamic breakthrough curves with roll-up, stepwise, and saddle-shaped structures have been commonly observed in the adsorption/separation process, but could be rarely described by the descriptive model, which limits further analysis and explanation of the breakthrough data. In this work, two new descriptive models, including a general polynomial logistic equation (GPLE) and a general polynomial Gompertz equation (GPGE) are developed to correlate these roll-up, stepwise, and saddle-shaped breakthrough curves. Their correlation coefficients (R2) between the descriptive model and breakthrough data are higher than 0.95, indicating the good applicability of the two descriptive models. Taking some difficult-to-fit literature data as an example, by adjusting these characteristic parameters, six types of breakthrough curves can be fitted by the GPLE or GPGE model. Representatively, satisfying N=2.0, a1≠a2 or b1≠b2, CN<0 or CN<1.0 (∑1NCN=1.0), various roll-up or stepwise breakthrough curves could be fitted accurately. If N=3.0, the GPGE model could explain the multiple roll-up, the multiple stepwise, and the saddle-shaped breakthrough curves. As a result, two new descriptive models can be easily used by workers in various fields to analyze and calculate breakthrough data. Although some reasonable physical explanations need to be explored in the future, they will provide a modeling basis for studying multi-component dynamic adsorption behavior.

Immersion and Invariance Adaptive Control through Polynomial Adaptation

Abstract In conventional immersion and invariance (I&amp;I) adaptive control design, control parameter adaptation is typically linear with respect to the parameter-error-induced perturbation, resulting in quadratic-rate dissipation of energy associated with the off-the-manifold variable. Departing from such a convention, this article contributes a novel strategy - polynomial adaptation. As the name suggests, control parameter adaptation in this approach takes the form of a general polynomial in relation to the perturbation. Accordingly, this new design induces polynomial-rate energy dissipation, which is faster than the quadratic one in the conventional scheme, thereby enhancing closed-loop control performance. The theoretical underpinnings of the new approach are demonstrated through the design of an I&amp;I adaptive tracking control law for a general nth-order, single-input-single-output, parametrically uncertain, nonlinear system in the controllable canonical form. Additionally, a numerical study of the proposed method on the second-order forced Duffing oscillator showcases its improved transient performance in comparison to a baseline controller developed with the standard I&amp;I adaptive control technique.

Developments of the I-Function Several Complex Variables

The purpose of this study is to present the I-function of r-variables, a natural extension of thepopular, intriguing, and practical Fox H-function. The results are quite generalized when r=1, at which pointwe get the I-function. Second, we showed that an integral is formed by multiplying the I-function of a singlevariable by the Bessel function and the General polynomial class. A two-variable I-function may be obtainedby evaluating the Sine transform of the product of the Hermite polynomial and Fox's H-function, and itsspecial case, the Cosine transform of the product of the Legendre polynomial and Fox's H-function. Wehave also provided exceptional examples, basic features, and circumstances under which convergenceoccurs. Results for H-functions in r variables are generalized in this study.

An ODE Approach to Multiple Choice Polynomial Programming

An ODE Approach to Multiple Choice Polynomial Programming

An Area-Efficient and Configurable Number Theoretic Transform Accelerator for Homomorphic Encryption

Homomorphic Encryption (HE) allows for arbitrary computation of encrypted data, offering a method for preserving privacy in cloud computations. However, efficiency remains a significant obstacle, particularly with the polynomial multiplication of large parameter sets, which occupies substantial computing and memory overhead. Prior studies proposed the use of Number Theoretic Transform (NTT) to accelerate polynomial multiplication, which proved efficient, owing to its low computational complexity. However, these efforts primarily focused on NTT designs for small parameter sets, and configurability and memory efficiency were not considered carefully. This paper focuses on designing a unified NTT/Inverse NTT (INTT) architecture with high area efficiency and configurability, which is more suitable for HE schemes. We adopt the Constant-Geometry (CG) NTT algorithm and propose a conflict-free access pattern, demonstrating a 16.7% reduction in coefficients of storage capacity compared to the state-of-the-art CG NTT design. Additionally, we propose a twiddle factor generation strategy to minimize storage for Twiddle Factors (TFs). The proposed architecture offers configurability of both compile time and runtime, allowing for the deployment of varying parallelism and parameter sets during compilation while accommodating a wide range of polynomial degrees and moduli after compilation. Experimental results on the Xilinx FPGA show that our design can achieve higher area efficiency and configurability compared with previous works. Furthermore, we explore the performance difference between precomputed TFs and online-generated TFs for the NTT architecture, aiming to show the importance of online generation-based NTT architecture in HE applications.

Computation Zagreb, Randic, and general sum-connectivity polynomials on symmetrical chemical dendrimer

Computation Zagreb, Randic, and general sum-connectivity polynomials on symmetrical chemical dendrimer

Even grade generic skew-symmetric matrix polynomials with bounded rank

We show that the set of m×m complex skew-symmetric matrix polynomials of even grade d, i.e., of degree at most d, and (normal) rank at most 2r is the closure of the single set of matrix polynomials with certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic m×m complex skew-symmetric matrix polynomials of even grade d and rank at most 2r. The analogous problem for the case of skew-symmetric matrix polynomials of odd grade is solved in [24].

General Structure and Properties of Cyclic Codes Over \(GF_2\)

This article provides a comprehensive analysis of cyclic codes in GF(2), focusing on their general structure and key properties. Cyclic codes are characterized by their generator polynomials, which define their structure and play a crucial role in encoding and decoding processes. The cyclical shift property, inherent in cyclic codes, facilitates efficient implementation using shift register circuits, making them practical for real-world applications. The discussion highlights the algebraic properties that distinguish cyclic codes from other linear block codes, emphasizing their ability to detect and correct errors.

A Jacobi-Ritz Approach for Aeroelastic Analysis of Swept Distributed Propulsion Aircraft Wing

Abstract This paper presents a Jacobi-Ritz approach for conducting flutter and divergence analysis of a complex distributed propulsion aircraft wing like NASA X-57. The general orthogonal Jacobi polynomials are used to approximate the bending displacement and torsional rotation angle in the Ritz method based structural and aeroelastic analysis. The Jacobi polynomials satisfy the orthogonality condition using weight functions which are easily modified to satisfy different essential and natural boundary conditions. Compared to simple polynomials, Jacobi polynomials can eliminate the well-known ill-conditioning numerical issues when considering higher-order polynomials. The Jacobi-Ritz method is also found to alleviate mode switching often encountered in tracking the changes of modes with the varying airspeed. The Jacobi-Ritz method is later used to investigate the flutter and divergence speeds under distributed propulsor mass and their locations, nonuniform aerodynamic model in the presence of multiple propulsors, and the sweep angle. Results show that placing the distributed propulsors on the wing's leading edge increases the flutter speed even though the bending and torsion modal frequencies are decreased compared to those of the wing without propulsors. The presence of pods for the middle high-lift motors causes an extra aerodynamic moment which reduces the flutter speed. Parametric studies also show that the divergence speed is lower than the flutter speed for a uniform and straight distributed propulsor wing. Using swept-back wing configuration and placing the tip propulsor near the wing's leading edge can help to increase both flutter and divergence speeds.

Modeling and Prediction of The Mechanical Properties of Feedstock by Cooling-Slope Casting Process using MOJaya Algorithm

Cooling-slope (CS) casting parameters optimization is very challenging due to the uncertainties of parameter values that are influenced by many factors. Therefore, a combination of estimation and optimization models that can solve multi-objective problems using limited data is proposed to reduce the real casting operational cost and time. In casting optimization, modeling and optimization of CS parameters have been considered to identify optimal CS parameters that would lead to better feedstock performance. Computational techniques have been suggested by previous researchers to solve the optimization in the manufacturing process. The study assessed the CS process to determine the feedstock's quality by evaluating its mechanical qualities, namely Tensile strength and Impact strength. Hence, computational methods namely the MOJaya algorithm are utilized to model and optimize the parameters of CS to address the CS problem and forecast the performance of the feedstock. This work conducted a thorough analysis of the impact of CS process parameters on the tensile strength and impact strength using an experimental design. The experiment was conducted using a three-level full factorial design (FFD) of the experiment (DOE), for the parameters of pouring temperature, slanting angle, and pouring distance. The modeling technique involved the development of multiple polynomial regression (MPR) as an objective function in the MOJaya Algorithm. The outcome of the optimization procedure revealed that the optimal solutions obtained from MOJaya were projected to surpass both the initial experimental data and the MOJaya algorithm in terms of the highest values of tensile strength and impact strength. The difference between the two sets of values is 10.92% and 15.76%, respectively. Subsequently, this technique may be employed by a caster to forecast the performance of the feedstock without using repeated experiments that are costly and time-consuming.

The distribution of polynomials in monotone-independent elements

Building on the work of Arizmendi and Celestino (2021), we derive the ∗-distributions of polynomials in monotone independent and infinitesimally monotone independent elements. For non-zero complex numbers [Formula: see text] and [Formula: see text], we derive explicitly the ∗-distribution of [Formula: see text] whenever [Formula: see text] and [Formula: see text] are monotone or infinitesimally monotone independent elements. This encompasses both cases of the commutator and anti-commutator. This approach can be pushed to study more general polynomials. As applications, we derive the limiting distribution with respect to the partial trace of polynomials in a certain class of random matrices.

Compact and Low-Latency FPGA-Based Number Theoretic Transform Architecture for CRYSTALS Kyber Postquantum Cryptography Scheme

In the modern era of the Internet of Things (IoT), especially with the rapid development of quantum computers, the implementation of postquantum cryptography algorithms in numerous terminals allows them to defend against potential future quantum attack threats. Lattice-based cryptography can withstand quantum computing attacks, making it a viable substitute for the currently prevalent classical public-key cryptography technique. However, the algorithm’s significant time complexity places a substantial computational burden on the already resource-limited chip in the IoT terminal. In lattice-based cryptography algorithms, the polynomial multiplication on the finite field is well known as the most time-consuming process. Therefore, investigations into efficient methods for calculating polynomial multiplication are essential for adopting these quantum-resistant lattice-based algorithms on a low-profile IoT terminal. Number theoretic transform (NTT), a variant of fast Fourier transform (FFT), is a technique widely employed to accelerate polynomial multiplication on the finite field to achieve a subquadratic time complexity. This study presents an efficient FPGA-based implementation of number theoretic transform for the CRYSTAL Kyber, a lattice-based public-key cryptography algorithm. Our hybrid design, which supports both forward and inverse NTT, is able run at high frequencies up to 417 MHz on a low-profile Artix7-XC7A100T and achieve a low latency of 1.10μs while achieving state-of-the-art hardware efficiency, consuming only 541-LUTs, 680 FFs, and four 18 Kb BRAMs. This is made possible thanks to the newly proposed multilevel pipeline butterfly unit architecture in combination with employing an effective coefficient accessing pattern.

Analysing All-Optical Random Bit Sequences Using Gap-Based Approaches.

Quantum mechanical phenomena are revolutionizing classical engineering fields such as signal processing or cryptography. When randomness plays an important role, like in cryptography where random bit sequences guarantee certain levels of security, quantum mechanical phenomena allow new ways of generating random bit sequences. Such sequences have a lot of applications in the communication sector, e.g., regarding data transmission, simulation, sensors or radars, and beyond. They can be generated deterministically (e.g., by using polynomials, resulting in pseudo-random sequences) or in a non-deterministic way (e.g., by using physical noise sources like external devices or sensors, resulting in random sequences). Important characteristics of such binary sequences can be modelled by gap processes in conjunction with the probability theory. Recently, all-optical approaches have attracted a lot of research interest. In this work, an adaptation of the quantum key distribution setup is utilized for generating randomised bit sequences. The simulation results show that all-optically generated sequences very well resemble the theoretically ideal probability density characteristic. Additionally, an experimental optical setup is developed that confirms the simulation results. Furthermore, m-sequences show very promising results as well as Gold sequences. Additionally, the level of burstiness, i.e., the distribution of ones and zeros throughout the sequence, is studied for the different sequences. The results enable the finding that generator polynomials with concentrated non-zero coefficients lead to more bursty bit sequences.

Robust adaptive control of delayed systems with multiple uncertainties and polynomial nonlinearities

This work discusses the issue of robust adaptive control for delayed systems with multiple uncertain factors. The system uncertainties cover uncertain dynamics, unknown system parameters, and external disturbances. Besides, the growing conditions have both low-order and high-order polynomial nonlinearities, which bring in numerous challenges for control design. The suitable transformations are first introduced to obtain a new system with a dynamic gain. Then, the control design of an auxiliary system is considered. By introducing the Lyapunov–Krasovskii (L–K) functional method, and providing an important result for an auxiliary system, the adaptive homogeneous control strategy is proposed to achieve global asymptotic stability (GAS) of the entire system. Besides, by considering the external disturbances, the method is also extended to design robust adaptive tracking controller. Numerical and practical examples are provided to verify the theory.

A Survey of Polynomial Multiplications for Lattice-Based Cryptosystems

We survey various mathematical tools used in software works multiplying polynomials in ℤ q [ x ] ⟨ x n − α x − β ⟩ . In particular, we survey implementation works targeting polynomial multiplications in lattice-based cryptosystems Dilithium, Kyber, NTRU, NTRU Prime, and Saber with instruction set architectures/extensions Armv7-M, Armv7E-M, Armv8-A, and AVX2. There are three emphases in this paper: (i) modular arithmetic, (ii) homomorphisms, and (iii) vectorization. For modular arithmetic, we survey Montgomery, Barrett, and Plantard multiplications. For homomorphisms, we survey (a) various homomorphisms such as Cooley–Tukey FFT, Good–Thomas FFT, Bruun's FFT, Rader's FFT, Karatsuba, and Toom–Cook; (b) various algebraic techniques for adjoining nice properties to the coefficient rings, including localization, Schönhage's FFT, Nussbaumer's FFT, and coefficient ring switching; and (c) various algebraic techniques related to the polynomial moduli, including twisting, composed multiplication, evaluation at ∞ , truncation, incomplete transformation, striding, and Toeplitz matrix-vector product. For vectorization, we survey the relations between homomorphisms and vector arithmetic. We then go through several case studies: We compare the implementations of modular multiplications used in Dilithium and Kyber, explain how the matrix-to-vector structure was exploited in Saber, and review the design choices of transformations for NTRU and NTRU Prime with vectorization. Finally, we outline several interesting implementation projects.

Investigation on the Blade Number of Pico-scale Crossflow Turbine for Low Head by Numerical Method

Pico-scale crossflow turbines (CFT) can be an alternative solution to meet electrical energy needs, especially in remote rural areas. CFT is recommended because of its suitability in low head (&lt; 5 m) conditions and fluctuating discharge conditions. One of the parameters that influences the performance of a CFT is the number of blades of the runner. CFT was discovered in 1903 and is still developing; however, the study of the physical phenomena of flow due to the blade number on the energy conversion process has yet to be comprehensively depicted. Therefore, this study aims to analyze the effect of the blade's number of runners on CFT performance using the computational fluid dynamics (CFD) method. The CFD method can visualize the flow field more detail than analytical and experimental. The CFD method is run with a moving mesh feature (transient) and pressure-based solver with a head condition of 3 m. The blades number studied were 16, 18, 22, 24, 26, and 30. Based on the results, the relationship of the CFT efficiency to blade number is described using a second-order multiple regression polynomial, and runner rotation is parabolic. Based on the performance curve, the CFT with 26 blades has the highest performance for low-head conditions.

Biosorption of pb(II) from aqueous solution by citrus reticulate: adsorption studies, and modeling

Removing toxic Pb(II) ions from aqueous solution by the peels of citrus reticulate (mandarin orange), a fruit industry waste, presents suitable scale-up possibilities. The Scanning Electron Microscope (SEM) and Brunauer-Emmett-Teller (BET) studies reflected that the mandarin orange peel powder had a porous surface area (32.46 m2g−1), average pore size and pore volume was 38.6 Å and 0.402 cm3g−1, respectively, favorable for binding Pb(II) ions. Fourier-transform infrared spectroscopy (FTIR) showed C-Br stretching, primary alcohol (C-O), phenolic O-H, and carbodimide N = C = N bands primarily helped to bind Pb(II) ions. The study evaluated and optimized the parametric influences of pH, adsorbate and biosorbent concentration, contact time and temperature on the removal efficiency of Pb(II) ions. A maximum of 97.08% Pb(II) was removed from 20 mg L−1 solution when 2.5 g L−1 adsorbent was present. The reaction obeyed the pseudo-second-order kinetic model. The intra-particle diffusion was involved in lead sorption. The Langmuir isotherm model resulted in an adsorption capacity of 23.04 mg g−1. 35.28% Pb(II) was removed in the 3rd adsorption-desorption cycle with 0.4 M HCl. The adsorption process was natural, impulsive and endothermic. The statistical investigation used Multiple Polynomial Regression (MPR) and Genetic Algorithm (GA). The analysis effectively forecasted the percentage removal at the optimized condition.

Novel Methods for Synthesizing Self-Checking Combinational Circuits by Means of Boolean Signal Correction and Polynomial Codes

This paper proposes the use of a polynomial code for synthesizing self-checking digital devices. The code is chosen for its error detection characteristics in data symbols and is used for Boolean signals correction in embedded control circuits. In practice, it is possible to equip the device with the ability to detect faults. In contrast to the approaches found in the world literature to solve this problem, this proposal suggests identifying groups of structurally independent outputs to distinguish between convertible and non-convertible outputs of the diagnosed block in the embedded control circuit. The only outputs that can be converted are those that are used as checking symbols for the polynomial code in the embedded control circuit. The other functions remain unchanged. The polynomial codes are used to select them. The authors present algorithms for synthesizing fault detection devices using the proposed approach.