General Polynomial Research Articles

Численно-аналитический метод преобразований для анализа нелинейных математических моделей полиномиальной структуры

The article considers a scientific problem of investigating mathematical models with polynomial nonlinearity, which are represented by systems of nonlinear differential equations. There is presented a numericall-analytical transformation method for investigating nonlinear dynamical systems with many degrees of freedom of polynomial structure. As opposed to the analogues, this method allows solving a wide range of problems for nonlinear systems 
 of general polynomial structure while reducing the computational resource intensity. An algorithm of the method 
 of transformations for investigating the nonlinear dynamic systems with m degrees of freedom is given. The research of stability of solutions of nonlinear dynamic systems with m degrees of freedom is carried out. Algorithmic formulas of transformation method for the solution of systems with nonlinearity of the fourth degree are given. Computational experiments on solving systems of differential equations with a small nonlinear part in the form of a multinomial 
 of the sixth degree using the transformation method show fourth order accuracy in computation. A nonlinear mathematical model of the vibration protection system is investigated by the presented method of transformations. The theorem on determination of the stationary state for systems of differential equations of polynomial structure by transformation method is proved. Algorithmic formulas for computation are presented. A general matrix form for vector indices is presented. Formulas for economical computation of right-hand sides of polynomial structure are presented and it is proposed to apply Pan's scheme with coefficient preprocessing. This method allows studying the different modes 
 of nonlinear models dynamics, for example, to determine such extreme modes as resonance, sub-harmonic, and polyharmonic modes. As an example of the method application, the vibration protection problem of the tower from external periodic influences was solved. The transformed solution takes into account all nonlinear polynomial components. The method allows to investigate the dynamics of a wide range of nonlinear systems with the necessary accuracy

The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems

This paper studies the number of small limit cycles produced around an elementary center for Hamiltonian differential systems with the elliptic Hamiltonian function H=12y2+12x2−23x3+a4x4(a≠0) under two types of polynomial perturbations of degree m, respectively. It is proved that the Hamiltonian system perturbed in Liénard systems can have at least [3m−14] small limit cycles near the center, where m≤101, and that the related near-Hamiltonian system with general polynomial perturbations can have at least m+[m+12]−2 small-amplitude limit cycles, where m≤16. Furthermore, in any of the cases, the bounds for limit cycles can be reached by studying the isolated zeros of the corresponding first order Melnikov functions and with the help of Maple programs. Here, [·] represents the integer function.

Energetically Optimal Trajectory for a Redundant Planar Robot by Means of a Nested Loop Algorithm

This work presents an optimal point-to-point trajectory planning method based on fourth degree polynomials to reduce the energy consumption of a redundant planar robot in the XY plane. The contribution of this study focuses on the development of a two-level nested optimization algorithm, which manages to optimize three elements: the Optimal Weighting Vector that establishes the influence of each joint on the total energy consumption of the robotic equipment; the optimal start and finish configurations of the trajectory; and the vector corresponding to the fifth coefficient of the trajectory generator polynomial. The tests are carried out with different optimization techniques and objective functions related to energy consumption to determine their best combination and achieve the minimum possible computation time and energy savings, according to the test conditions used. Furthermore, the energetic performance of the redundant manipulator is compared with its non-redundant version, limiting its operation to only two degrees of freedom in all its variants.

A Scalable Digit-Parallel Polynomial Multiplier Architecture for NIST-Standardized Binary Elliptic Curves

This work presents a scalable digit-parallel finite field polynomial multiplier architecture with a digit size of 32 bits for NIST-standardized binary elliptic fields. First, a dedicated digit-parallel architecture is proposed for each binary field recommended by NIST, i.e., 163, 233, 283, 409 and 571. Then, a scalable architecture having support for all variants of binary fields of elliptic curves is proposed. For performance investigation, we have compared dedicated multiplier architectures with scalable design. After this, the dedicated and scalable architectures are compared with the most relevant state-of-the-art multipliers. All multiplier architectures are implemented in Verilog HDL using the Vivado IDE tool. The implementation results are reported on a 28 nm Virtex-7 FPGA technology. The dedicated multipliers utilize slices of 1182 (for m=163), 1451 (for m=233), 1589 (for m=283), 2093 (for m=409) and 3451 (for m=571). Moreover, our dedicated designs can operate at a maximum frequency of 500, 476, 465, 451 and 443 MHz. Similarly, for all supported binary fields, our scalable architecture (i) utilizes 3753 slices, (ii) achieves 305 MHz clock frequency, (iii) takes 0.013 μs for one finite field multiplication and (iv) consumes 3.905 W power. The proposed scalable digit-parallel architecture is more area-efficient than most recent state-of-the-art multipliers. Consequently, the reported results and comparison to the state of the art reveal that the proposed architectures are well suited for cryptographic applications.

Multiple Linear and Polynomial Models for Studying the Dynamics of the Soil Solution

The objective of the present work was to study the soil solution throughout time in pots under greenhouse conditions. The work consisted of monitoring the solution of calcareous soil and forest soil in the absence of plants, with different types of fertilization: treatment 1: absolute control (irrigation water); treatment 2: Steiner nutrient solution; treatment 3: solid fertilizers; and treatment 4: vermicompost tea (aqueous extract). The samples were collected weekly using lysimeters for 14 weeks. They were analyzed to determine the nitrate content, total nitrogen, calcium, potassium, magnesium, sodium, sulfur, zinc, boron, pH, electrical conductivity, and oxide-reduction potential. To understand the interactions between treatments, soil type, and time over ion behavior and availability, linear and polynomial models were used, selected by a cross-validation method, which resulted in robust models, where it was found that the pH behavior is associated with the type of fertilization and soil type, with the elapsed time being a nonsignificant factor. On the other hand, time influenced the dynamics of the remaining ions and their availability. It was found that the multiple polynomial model fit better for the variables: potassium, calcium, sodium (square degree), electrical conductivity, nitrates, sulfur (cubic degree), zinc, oxidation-reduction potential, nitrogen, magnesium, and boron (quartic degree).

Polynomial Multiplication Architecture with Integrated Modular Reduction for R-LWE Cryptosystems

Polynomial Multiplication Architecture with Integrated Modular Reduction for R-LWE Cryptosystems

An area-time efficient point-multiplication architecture for ECC over GF(2m) using polynomial basis

The modern age has seen an enormous progress in communications. Millions of devices communicate over the web transmitting confidential information which sometimes is of national importance. Securing these devices has become prime concern in this evolutionary digital world. Elliptic curve cryptography has grabbed quite attention for securing these devices primely because of its small-key size with relatively same-level of security when compared to other cryptosystems. The high performance of ECC relies on the finite-field arithmetic operations. Point-multiplication is the most resource consuming and time critical ECC operation. Many architectures and algorithms are presented in the literature to address the area complexity and time complexity of point-multiplication. In this paper, a point-multiplication architecture for generic irreducible polynomials is proposed based on the modified Montgomery-ladder algorithm. In addition, the FF-inversion operation of point multiplication is realized by employing modified Itoh–Tsujii algorithm to achieve reduction in the computation time. The hardware complexity and delay of the proposed point multiplication architecture are estimated, and a comparison with the corresponding point multiplication architectures available in the literature is presented. It is observed that the proposed architecture achieves area-time efficiency of around 17%–86% and 42%–98%, respectively, over GF(2163) and GF(2233) when compared to the architectures available in the literature.

An Efficient Crypto Processor Architecture for Side-Channel Resistant Binary Huff Curves on FPGA

This article presents an efficient crypto processor architecture for point multiplication acceleration of side-channel secured Binary Huff Curves (BHC) on FPGA (field-programmable gate array) over GF(2233). We have implemented six finite field polynomial multiplication architectures, i.e., (1) schoolbook, (2) hybrid Karatsuba, (3) 2-way-karatsuba, (4) 3-way-toom-cook, (5) 4-way-toom-cook and (6) digit-parallel-least-significant. For performance evaluation, each implemented polynomial multiplier is integrated with the proposed BHC architecture. Verilog HDL is used for the implementation of all the polynomial multipliers. Moreover, the Xilinx ISE design suite tool is employed as an underlying simulation platform. The implementation results are presented on Xilinx Virtex-6 FPGA devices. The achieved results show that the integration of a hybrid Karatsuba multiplier with the proposed BHC architecture results in lower hardware resources. Similarly, the use of a least-significant-digit-parallel multiplier in the proposed design results in high-speed (in terms of both clock frequency and latency). Consequently, the proposed BHC architecture, integrated with a least-significant-digit-parallel multiplier, is 1.42 times faster and utilizes 1.80 times lower FPGA slices when compared to the most recent BHC accelerator architectures.

The Number of Limit Cycles for a Class of Cubic Systems with Multiple Parameters

In this paper, we study the number of limit cycles for a class of cubic systems with general quadratic polynomial perturbations. By using the Melnikov function theory we obtain that five limit cycles can be bifurcated from a period annulus. We also study the Hopf bifurcation at the center surrounded by the annulus.

Construction D’ Lattices for Power-Constrained Communications

Designs and methods for nested lattice codes using Construction D&#x2019; lattices for coding and convolutional code lattices for shaping are described. Two encoding methods and a decoding algorithm for Construction D&#x2019; coding lattices that can be used with shaping lattices for power-constrained channels are given. We construct nested lattice codes with good coding properties, high shaping gain, and low-complexity encoding and decoding. Convolutional code generator polynomials for Construction A lattices with the greatest shaping gain are given, as a result of an extensive search. It is shown that rate 1/3 convolutional codes provide a more favorable performance-complexity trade-off than rate 1/2 convolutional codes. Tail-biting convolutional codes have higher shaping gain than that of zero-tailed convolutional codes. A design for quasi-cyclic low-density parity-check (QC-LDPC) codes to form Construction D&#x2019; lattices which have efficient encoding and indexing is presented. The resulting QC-LDPC Construction D&#x2019; lattices are evaluated using four shaping lattices: the <inline-formula> <tex-math notation="LaTeX">$E_{8}$ </tex-math></inline-formula> lattice, the <inline-formula> <tex-math notation="LaTeX">$BW_{16}$ </tex-math></inline-formula> lattice, the Leech lattice and our best-found convolutional code lattice, showing a shaping gain of approximately 0.65 dB, 0.86 dB, 1.03 dB and 1.25 dB at dimension 2304.

DIRECT: A Differential Dynamic Programming Based Framework for Trajectory Generation

This letter introduces a differential dynamic programming (DDP) based framework for polynomial trajectory generation for differentially flat systems. In particular, instead of using a linear equation with increasing size to represent multiple polynomial segments as in literature, we take a new perspective from state-space representation such that the linear equation reduces to a finite horizon control system with a fixed state dimension and the required continuity conditions for consecutive polynomials are automatically satisfied. Consequently, the constrained trajectory generation problem (both with and without time optimization) can be converted to a discrete-time finite-horizon optimal control problem with inequality constraints, which can be approached by a recently developed interior-point DDP (IPDDP) algorithm. Furthermore, for unconstrained trajectory generation with preallocated time, we show that this problem is indeed a linear-quadratic tracking (LQT) problem (DDP algorithm with exact one iteration). All these algorithms enjoy linear complexity with respect to the number of segments. Both numerical comparisons with state-of-the-art methods and physical experiments are presented to verify and validate the effectiveness of our theoretical findings. The implementation code will be open-sourced. <xref ref-type="fn" rid="fn1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¹</xref> <fn id="fn1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><label>¹</label> [Online] Available: <uri>https://github.com/ntu-caokun/DIRECT</uri> </fn>

Different Stochastic Resonances Induced by Multiplicative Polynomial Trichotomous Noise in a Fractional Order Oscillator with Time Delay and Fractional Gaussian Noise

A general investigation on the mechanism of stochastic resonance is reported in a time-delay fractional Langevin system, which endues a nonlinear form multiplicative colored noise and fractional Gaussian noise. In terms of theoretical analysis, both the expressions of output steady-state amplitude and that of the first moment of system response are obtained by utilizing stochastic averaging method, fractional Shapiro and Laplace methods. Due to the presence of trichotomous colored noise, the excitation frequency can induce fresh multimodal Bona fide stochastic resonance, exhibiting much more novel dynamical behaviors than the non-disturbance case. It is verified that multimodal pattern only appears with small noise switching rate and memory damping order. The explicit expressions of critical noise intensity corresponding to the generalized stochastic resonance are given for the first time, by which it is determined that nonlinear form colored noise induces much more of a comprehensive resonant phenomena than the linear form. In the case of slow transfer rate noise, a newfangled phenomenon of double hypersensitive response induced by a variation in noise intensity is discovered and verified for the first time, with the necessary range of parameters for this phenomenon given. In terms of numerical scheme, an efficient and feasible algorithm for generating trichotomous noise is proposed, by which an algorithm based on the Caputo fractional derivative are applied. The numerical results match well with the analytical ones.

ℤ4R-additive cyclic and constacyclic codes and MDSS codes

In this paper, we will study the structure of [Formula: see text]-additive codes where [Formula: see text] is the well-known ring of 4 elements and [Formula: see text] is the ring given by [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text]. We will classify all maximum distance separable codes with respect to the Singleton bound (MDSS) over [Formula: see text] Then we will focus on [Formula: see text]-additive cyclic and constacyclic codes. We will find a unique set of generator polynomials for these codes and determine minimum spanning sets for them. We will also find the generator polynomials for the dual of any [Formula: see text]-additive cyclic or constacyclic code.

Gröbner bases and critical values: The asymptotic combinatorics of determinantal systems

Determinantal polynomial systems are those involving maximal minors of some given matrix. An important situation where these arise is the computation of the critical values of a polynomial map restricted to an algebraic set. This leads directly to a strategy for, among other problems, polynomial optimisation.Computing Gröbner bases is a classical method for solving polynomial systems in general. For practical computations, this consists of two main stages. First, a Gröbner basis is computed with respect to a DRL (degree reverse lexicographic) ordering. Then, a change of ordering algorithm, such as Sparse-FGLM, designed by Faugère and Mou, is used to find a Gröbner basis of the same system but with respect to a lexicographic ordering. The complexity of this latter step, in terms of the number of arithmetic operations in the ground field, is O(mD2), where D is the degree of the ideal generated by the input and m is the number of non-trivial columns of a certain D×D matrix.While asymptotic estimates are known for m in the case of generic polynomial systems, thus far, the complexity of Sparse-FGLM was unknown for the class of determinantal systems.By assuming Fröberg's conjecture, thus ensuring that the Hilbert series of generic determinantal ideals have the necessary structure, we expand the work of Moreno-Socías by detailing the structure of the DRL staircase in the determinantal setting. Then we study the asymptotics of the quantity m by relating it to the coefficients of these Hilbert series. Consequently, we arrive at a new bound on the complexity of the Sparse-FGLM algorithm for generic determinantal systems and, in particular, for generic critical point systems.We consider the ideal inside the polynomial ring K[x1,…,xn], where K is some infinite field, generated by p generic polynomials of degree d and the maximal minors of a p×(n−1) polynomial matrix with generic entries of degree d−1. Then, in this setting, for the case d=2 and for n≫p we establish an exact formula for m in terms of n and p. Moreover, for d≥3, we give a tight asymptotic formula, as n→∞, for m in terms of n,p and d.

Fused-Filament Fabrication of Short Carbon Fiber-Reinforced Polyamide: Parameter Optimization for Improved Performance under Uniaxial Tensile Loading.

This study intends to contribute to the state of the art of Fused-Filament Fabrication (FFF) of short-fiber-reinforced polyamides by optimizing process parameters to improve the performance of printed parts under uniaxial tensile loading. This was performed using two different approaches: a more traditional 2k full factorial design of experiments (DoE) and multiple polynomial regression using an algorithm implementing machine learning (ML) principles such as train-test split and cross-validation. Evaluated parameters included extrusion and printing bed temperatures, layer height and printing speed. It was concluded that when exposed to new observations, the ML-based model predicted the response with higher accuracy. However, the DoE fared slightly better at predicting observations where higher response values were expected, including the optimal solution, which reached an UTS of 117.1 ± 5.7 MPa. Moreover, there was an important correlation between process parameters and the response. Layer height and printing bed temperatures were considered the most influential parameters, while extrusion temperature and printing speed had a lower influence on the outcome. The general influence of parameters on the response was correlated with the degree of interlayer cohesion, which in turn affected the mechanical performance of the 3D-printed specimens.

Total positivity of some polynomial matrices that enumerate labeled trees and forests I: forests of rooted labeled trees

We consider the lower-triangular matrix of generating polynomials that enumerate k-component forests of rooted trees on the vertex set [n] according to the number of improper edges (generalizations of the Ramanujan polynomials). We show that this matrix is coefficientwise totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. More generally, we define the generic rooted-forest polynomials by introducing also a weight m! , phi _m for each vertex with m proper children. We show that if the weight sequence varvec{phi } is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays.

Efficient high-precision integer multiplication on the GPU

The multiplication of large integers, which has many applications in computer science, is an operation that can be expressed as a polynomial multiplication followed by a carry normalization. This work develops two approaches for efficient polynomial multiplication: one approach is based on tiling the classical convolution algorithm, but taking advantage of new CUDA architectures, a novelty approach to compute the multiplication using integers without accuracy lossless; the other one is based on the Strassen algorithm, an algorithm that multiplies large polynomials using the FFT operation, but adapting the fastest FFT libraries for current GPUs and working on the complex field. Previous studies reported that the Strassen algorithm is an effective implementation for “large enough” integers on GPUs. Additionally, most previous studies do not examine the implementation of the carry normalization, but this work describes a parallel implementation for this operation. Our results show the efficiency of our approaches for short, medium, and large sizes.

Polynomial Multiplication over Finite Fields in Time \( O(n \log n \)

Assuming a widely believed hypothesis concerning the least prime in an arithmetic progression, we show that polynomials of degree less than \( n \) over a finite field \( \mathbb {F}_q \) with \( q \) elements can be multiplied in time \( O (n \log q \log (n \log q)) \) , uniformly in \( q \) . Under the same hypothesis, we show how to multiply two \( n \) -bit integers in time \( O (n \log n) \) ; this algorithm is somewhat simpler than the unconditional algorithm from the companion paper [ 22 ]. Our results hold in the Turing machine model with a finite number of tapes.

Multi-Unit Serial Polynomial Multiplier to Accelerate NTRU-Based Cryptographic Schemes in IoT Embedded Systems.

Concern for the security of embedded systems that implement IoT devices has become a crucial issue, as these devices today support an increasing number of applications and services that store and exchange information whose integrity, privacy, and authenticity must be adequately guaranteed. Modern lattice-based cryptographic schemes have proven to be a good alternative, both to face the security threats that arise as a consequence of the development of quantum computing and to allow efficient implementations of cryptographic primitives in resource-limited embedded systems, such as those used in consumer and industrial applications of the IoT. This article describes the hardware implementation of parameterized multi-unit serial polynomial multipliers to speed up time-consuming operations in NTRU-based cryptographic schemes. The flexibility in selecting the design parameters and the interconnection protocol with a general-purpose processor allow them to be applied both to the standardized variants of NTRU and to the new proposals that are being considered in the post-quantum contest currently held by the National Institute of Standards and Technology, as well as to obtain an adequate cost/performance/security-level trade-off for a target application. The designs are provided as AXI4 bus-compliant intellectual property modules that can be easily incorporated into embedded systems developed with the Vivado design tools. The work provides an extensive set of implementation and characterization results in devices of the Xilinx Zynq-7000 and Zynq UltraScale+ families for the different sets of parameters defined in the NTRUEncrypt standard. It also includes details of their plug and play inclusion as hardware accelerators in the C implementation of this public-key encryption scheme codified in the LibNTRU library, showing that acceleration factors of up to 3.1 are achieved when compared to pure software implementations running on the processing systems included in the programmable devices.

Some Spectral Properties of Multiplicative Hermite Equation

We reconstruct the Multiplicative Hermite Equation from multiplicative Sturm-Liouville equation. A new representation of eigenfunctions for the constructed problem are obtained by the power series solution technique. While making these solutions, multiplicative Hermite polynomials were used strongly. We get a generator for multiplicative Hermite polynomials and construct integration representations for these polynomials. Finally, some spectral properties of the multiplicative Hermite problem are examined in detail.