General Polynomial Research Articles

Classifying Abelian Groups Through Acyclic Matchings

AbstractThe inquiry into identifying sets of monomials that can be eliminated from a generic homogeneous polynomial via a linear change of coordinates was initiated by E. K. Wakeford. This linear algebra problem prompted C. K. Fan and J. Losonczy to introduce the notion of acyclic matchings in the additive group $$\mathbb {Z}^n$$ Z n , subsequently extended to abelian groups by the latter author. Alon, Fan, Kleitman, and Losonczy established the acyclic matching property for $$\mathbb {Z}^n$$ Z n . This note aims to classify all abelian groups with respect to the acyclic matching property.

Predicting Housing Prices Using Supervised Machine Learning Models

Abstract: Variations in housing prices exert a profound impact on economic policies and personal financial decisions. This study aims to delve into the factors that affect housing prices and construct a predictive model using machine learning techniques. Machine learning enables computers to acquire knowledge from data and make predictions or decisions. This study forcasts house prices by analyzing the characteristics of data from Seattle, Washington using machine learning multiple linear regression, polynomial regression, and K-nearest neighbors regression (KNN). The findings of this investigation demonstrate that the polynomial regression model is the most accurate in predicting housing prices and can capture nonlinear relationships in the data more effectively than multiple linear regression and K-nearest neighbors regression. The main factors affecting housing prices are interior living space and the construction and design of buildings. These insights hold potential for enhancing government policies, facilitating effective land use decision-making by planners, and enabling investors to make more informed choices.

ON THE BINARY SEQUENCE $(1,1,0,1,0^3,1,0^7,1,0^{15},\ldots )$

Abstract Let $\mathbb {F}$ be a field and $(s_0,\ldots ,s_{n-1})$ be a finite sequence of elements of $\mathbb {F}$ . In an earlier paper [G. H. Norton, ‘On the annihilator ideal of an inverse form’, J. Appl. Algebra Engrg. Comm. Comput.28 (2017), 31–78], we used the $\mathbb {F}[x,z]$ submodule $\mathbb {F}[x^{-1},z^{-1}]$ of Macaulay’s inverse system $\mathbb {F}[[x^{-1},z^{-1}]]$ (where z is our homogenising variable) to construct generating forms for the (homogeneous) annihilator ideal of $(s_0,\ldots ,s_{n-1})$ . We also gave an $\mathcal {O}(n^2)$ algorithm to compute a special pair of generating forms of such an annihilator ideal. Here we apply this approach to the sequence r of the title. We obtain special forms generating the annihilator ideal for $(r_0,\ldots ,r_{n-1})$ without polynomial multiplication or division, so that the algorithm becomes linear. In particular, we obtain its linear complexities. We also give additional applications of this approach.

A dynamically consistent discretization method for the Goodwin model with nonlinear Phillips curve. Comparing qualitative and quantitative dynamics

AbstractThe Goodwin model is a widely used economic growth model able to explain endogenous fluctuations in employment rate and wage share; in its initial version, the standard Phillips curve is used. In the present work, we suggest a revised Phillips curve that takes into account how the wage share influences the rate of changes of the wage itself thus obtaining a continuous-time modified Goodwin model. Since applying models to real data often requires working in a discrete-time setup, we then move from the continuous-time to the discrete-time version of the proposed model, by using a general polynomial discretization method in backward and forward-looking (hybrid discretization). By comparing the continuous-time system to its discrete-time counterpart we prove that fixed points and local dynamics do not change, as long as the time step is not too high. Moreover, numerical simulations employing Dynamic Time Warping, cross-correlation, and semblance analysis consistently affirm that enhancing the similarity of quantitative dynamics is achieved by reducing the time step.

Polynomial Moving Regression Band Stocks Trading System

In this research, we attempted to fit a trading system based on polynomial moving regression bands (MRB) to Nasdaq100 stocks from 2017 till the end of March 2024. Since stocks movement does not follow a linear behavior, we used multiple degree polynomial regression models to identify the stocks’ trends and two standard deviations from the regression model to generate the trading signals. This way, the MRB was transformed into a momentum indicator designed to identify strong uptrends that can be used by a fully automated trading system. Our results indicate that the behavior of Nasdaq100 stocks can be tracked using all three examined polynomial models and can be traded profitably using fully automated systems based on those models. The best performing model was the model that used a four-degree polynomial MRB achieving the highest average net profit (USD 162.73). Regarding the risks involved, the third model has the lowest loss in dollar value (USD −95.52), and the highest minimum percent of profitable trades (41.51%) and profit factor (0.55) that indicates that this strategy is relatively less risky than the other two strategies.

Study of Fifth-Order Nonlinear Caudrey–Dodd–Gibbon Model for Multiwave, M-Shaped Solitons, Lumps, Generalized Breathers, Manifold Periodic, Rogue Wave and Kink Wave Interactions

Numerous novel wave structures, such as lump, lump with one-kink, lump with two-kink soliton, generalized breather, Ma breather, Kuznetsov-Ma breather, Akhmediev breather, manifold periodic, and rogue wave solutions to [Formula: see text]-dimensional fifth-order nonlinear Caudrey–Dodd–Gibbon (FNCDG) model via symbolic computation are studied based on the ansatz function scheme. Studying the lump-type solution involves selecting the function as the generic quadratic polynomial function. The lump with one- and two-kink solutions is created by mixing a quadratic function with one or two exponential functions. By adding hyperbolic function with a quadratic function, the rogue wave solutions are manipulated. In addition, the multiwave, [Formula: see text]-shaped rational solitons and their interactions with kink waves are analytically evaluated. Furthermore, different 3D, 2D, and contour profiles are created in different dimensions by changing the parameter values.

A Generalized Number-Theoretic Transform for Efficient Multiplication in Lattice Cryptography

The Number Theoretic Transform (NTT) has emerged as a powerful tool for efficiently computing convolutions of digital signals, due to its inherent advantages such as numerical stability, reliance on simple integer operations, and proven efficiency. Its applications have extended to accelerating polynomial multiplication in lattice-based cryptography. However, existing NTT multiplication algorithms impose restrictions on the underlying moduli, potentially affecting key and ciphertext sizes as well as computational overhead. Therefore, enabling NTT with small moduli holds significant potential for enhancing the overall system performance. This study introduces a novel reduction framework for NTT computation in cyclotomic rings employing field extensions. Our approach replaces the underlying polynomial ring with a two-dimensional isomorphic ring, effectively relaxing the restrictions imposed on the NTT moduli. The proposed framework is evaluated through two case studies relevant to the LAC and NTTRU lattice-based cryptographic schemes. Comprehensive theoretical analysis is provided, demonstrating the effectiveness of our approach in enabling NTT with small moduli and its potential to improve the efficiency of lattice-based cryptography.

Abelian- and Tauberian-type results for the generalized Stockwell and wavelet transforms

Tauberian-type results characterizing the quasiasymptotic behavior of polynomial multiplication of Lizorkin distributions in terms of their Stockwell transforms are obtained. Some Abelian-type results relating the quasiasymptotic behavior and quasiasymptotic boundedness of Lizorkin distributions to the asymptotic behavior of their Stockwell transforms are given. Several Abelian-type results for the generalized wavelet transform are also presented.

Unpacking Needs Protection

Most of the previous attacks on Dilithium exploit side-channel information which is leaked during the computation of the polynomial multiplication cs1, where s1 is a small-norm secret and c is a verifier's challenge. In this paper, we present a new attack utilizing leakage during secret key unpacking in the signing algorithm. The unpacking is also used in other post-quantum cryptographic algorithms, including Kyber, because inputs and outputs of their API functions are byte arrays. Exploiting leakage during unpacking is more challenging than exploiting leakage during the computation of cs1 since c varies for each signing, while the unpacked secret key remains constant. Therefore, post-processing is required in the latter case to recover a full secret key. We present two variants of post-processing. In the first one, a half of the coefficients of the secret s1 and the error s2 is recovered by profiled deep learning-assisted power analysis and the rest is derived by solving linear equations based on t = As1 + s2, where A and t are parts of the public key. This case assumes knowledge of the least significant bits of t, t0. The second variant uses lattice reduction to derive s1 without the knowledge of t0. However, it needs a larger portion of s1 to be recovered by power analysis. We evaluate both variants on an ARM Cortex-M4 implementation of Dilithium-2. The experiments show that the attack assuming the knowledge of t0 can recover s1 from a single trace captured from a different from profiling device with a non-negligible probability.

How Blockchain-Enabled Drivers Stimulate Consumers’ Organic Food Purchase Intention: An Integrated Framework of Information Systems Success Model Within Stimulus-Organism-Response Theory in the Context of Vietnam

Integrating the Information System Success (ISS) model with the Stimulus-Organism-Response (SOR) paradigm, this study aims to explore how blockchain-enabled stimuli (blockchain-enabled traceability and blockchain-enabled transparency) affect ISS-related organisms (perceived information quality and perceived system quality), which individually, congruently, and incongruently trigger consumers’ behavioral responses (organic food purchase intentions). Using a purposive sampling of 5326 Vietnamese consumers, multiple linear regression and polynomial regression with response surface analysis are employed to examine the hypothesized model. The research indicates that both blockchain-enabled traceability and blockchain-enabled transparency significantly and positively affect perceived information quality and perceived system quality, which both significantly mediate the impacts of blockchain-enabled traceability and blockchain-enabled transparency on organic food purchase intention. The research also finds that higher congruence between perceived information quality and perceived system quality leads to increased organic food purchase intention, while greater disparity between perceived information quality and perceived system quality results in lower organic food purchase intention. This study contributes to the extant literature by integrating the ISS model with the SOR model to provide a comprehensive framework for understanding how blockchain technology affects consumer behavior in the organic food market. It also highlights the crucial role of both information and system quality in leveraging blockchain technology to enhance consumer trust and purchase intentions.

A geometric algorithm for the factorization of rational motions in conformal three space

Rational motions in conformal three space can be parametrized by polynomials with coefficients in a suitable Clifford algebra. We call them “spinor polynomials.” In this text we present a new algorithm to decompose generic spinor polynomials into linear factors. The factorization algorithm is based on the “kinematics at infinity”. Factorizations exist generically but not generally and are typically not unique. We prove that generic multiples of non-factorizable spinor polynomials admit factorizations and we demonstrate at hand of an example how our ideas can be used to tackle the hitherto unsolved problem of “factorizing” algebraic motions.

On the trace-zero doubly stochastic matrices of order 5

We propose a graph theoretic approach to determine the trace of the product of two permutation matrices through a weighted digraph representation for a pair of permutation matrices. Consequently, we derive trace-zero doubly stochastic (DS) matrices of order 5 whose k-th power is also a trace-zero DS matrix for k∈{2,3,4,5}. Then, we determine necessary conditions for the coefficients of a generic polynomial of degree 5 to be realizable as the characteristic polynomial of a trace-zero DS matrix of order 5. Finally, we approximate the eigenvalue region of trace-zero DS matrices of order 5.

Using a Multivariate Virtual Experiment for Uncertainty Evaluation with Unknown Variance

Virtual experiments are a digital representation of a real measurement and play a crucial role in modern measurement sciences and metrology. Beyond their common usage as a modeling and validation tool, a virtual experiment may also be employed to perform a parameter sensitivity analysis or to carry out a measurement uncertainty evaluation. For the latter to be compliant with statistical principles and metrological guidelines, the procedure to obtain an estimate and a corresponding measurement uncertainty requires careful consideration. We employ a Monte Carlo sampling procedure using a virtual experiment that allows one to perform a measurement uncertainty evaluation according to the Monte Carlo approach of JCGM-101 and JCGM-102, two widely applied guidelines for uncertainty evaluation in metrology. We extend and formalize a previously published approach for simple additive models to account for a large class of non-linear virtual experiments and measurement models for multidimensionality of the data and output quantities, and for the case of unknown variance of repeated measurements. With the algorithm developed here, a simple procedure for the evaluation of measurement uncertainty is provided that may be applied in various applications that admit a certain structure for their virtual experiment. Moreover, the measurement model commonly employed for uncertainty evaluation according to JCGM-101 and JCGM-102 is not required for this algorithm, and only evaluations of the virtual experiment are performed to obtain an estimate and an associated uncertainty of the measurand. We demonstrate the efficacy of the developed approach and the effect of the underlying assumptions for a generic polynomial regression example and an example of a simplified coordinate measuring machine and its virtual representation. The results of this work highlight that considerable effort, diligence, and statistical considerations need to be invested to make use of a virtual experiment for uncertainty evaluation in a way that ensures equivalence with the accepted guidelines.

Optimizing crystals-dilithium implementation in 16-bit MSP430 environment utilizing hardware multiplier

Dilithium was selected as one of NIST standard Post Quantum Digital Signature algorithms and is undergoing standardization as a Module Lattice Digital Signature Algorithm (ML-DSA). However, until now research on optimization in embedded environments has primarily been conducted on ARM architectures, which are the basic benchmark targets. To prepare for future quantum secure Internet of Things environments, performance optimization on resource-constrained must be considered. Thus, in this paper, for the first time, we propose an optimized implementation of Dilithium in the 16-bit MSP430 environment, a low-resource device. We redesign the state-of-the-art optimization strategies for Dilithium to suit the MSP430 environment. By taking full advantage of MSP430’s hardware multiplier in the NTT-based polynomial multiplication, we achieve 73.0% and 80.1% of performance improvement for NTT and NTT−1 compared to those in the reference implementation, which contributes about 5.5%–7.0%, 15.3%–17.5%, and 7.5%–10.0% of performance improvement compared to Dilithium’s public reference implementation for keypair generation, signing, and verification, respectively.

Stability Analysis and Stabilization of General Conformable Polynomial Fuzzy Models with Time Delay

This paper introduces a sum-of-squares (S-O-S) approach to Stability Analysis and Stabilization (SAS) of nonlinear dynamical systems described by General Conformable Polynomial Fuzzy (GCPF) models with a time delay. First, we present GCPF models, which are more general compared to the widely recognized Takagi–Sugeno Fuzzy (T-SF) models. Then, we establish SAS conditions for these models using a Lyapunov–Krasovskii functional and the S-O-S approach, making the SAS conditions in this work less conservative than the Linear Matrix Inequalities (LMI)-based approach to the T-SF models. In addition, the SAS conditions are found by satisfying S-O-S conditions dependent on membership functions that are reliant on the polynomial fitting approximation algorithm. These S-O-S conditions can be solved numerically using the recently developed SOSTOOLS. To demonstrate the effectiveness and practicality of our approach, two numerical examples are provided to demonstrate the effectiveness and practicality of our approach.

A hybrid memory polynomial digital predistortion model for RF transmitters

The power amplifier (PA), a key component of the transmitter system, operates near the saturation region, resulting in nonlinear distortion of the output signal, which affects the quality of the transmitter system. For this, a series of linearization techniques are used to compensate for distortion, one of the most effective and widely applied is the digital predistortion (DPD) technique. The traditional DPD models can be categorized into a single model or multiple models cascade or parallel. In this letter, a hybrid memory polynomial (HMP) model is proposed to further enhance the accuracy of the model, which is composed of multiple memory polynomial (MP) models by cascading and parallelizing. The experimental results show that the HMP model has better accuracy than the traditional MP model at the same complexity.

Some Results for Double Cyclic Codes over Fq+vFq+v2Fq.

Let Fq be a finite field with an odd characteristic. In this paper, we present a new result about double cyclic codes over a finite non-chain ring. Specifically, we study the double cyclic code over Fq+vFq+v2Fq with v3=v, which is isomorphic to Fq×Fq×Fq. This study mainly involves generator polynomials and generator matrices. The generating polynomial of the dual code is also obtained. We show the relationship between the generating polynomials of the double cyclic codes and those of their dual codes. Finally, as an application of these results, we construct some optimal codes over F3.

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.