Univariate Polynomials Research Articles

Noninteracting optimal and adaptive torque control using an online parameter estimation with help of polynomials in EKF for a PMSM.

This paper addresses a non-interacting torque control strategy to decouple the d- and q-axis dynamics of a permanent magnet synchronous machine (PMSM). The maximum torque per ampere (MTPA) method is used to determine the reference currents for the desired torque. To realize the noninteracting control, knowledge concerning the inductances Ld and Lq of the electrical machine is necessary. These two inductances are estimated by two extended Kalman filters (EKFs), which use a univariate polynomial as a model to describe the saturation effects of the PMSM. The Kalman filters (KF) are realized within a noninteracting control system to improve the observability of the inductance. Despite the non-perfect decoupling, thanks to the structural stochastic nature of the KFs, noninteracting cancellation errors are represented with its process noise and the inductances are estimated sufficiently well. In this sense, we can speak about KFs for and within noninteracting control. Estimating inductances is fundamental for optimal torque control, which is a viable approach to reducing mechanical vibration and disturbance. Moreover, the control strategy of model-based techniques must be adaptively tuned to work properly. Starting from the existing literature, a viable control structure is proposed in which the stability of the control loop using a Proportional Integral (PI) controller is shown for the resulting time-varying system. In fact, the model is represented as a time-varying system because of the presence of the variable inductances Ld and Lq and because of the presence of the velocity of the rotor which is not considered as a state. In this paper, a forward Euler discretization is used to realize the observer in the discrete experimental setup. Measures realized with hardware in the loop (HIL) show interesting results in the context of inductance estimation, due to the advantage of the reduction of the dimensions of the two decoupled EKFs resulting from the noninteraction control. Using the HIL simulator, the proposed torque control strategy is investigated, showing promising results in terms of increasing observability due to decoupling. This and the usage of univariate polynomials in EKF calculations lead to significant reduction of measurement points, reduction of oscillations and ripples, deviation between desired and achieved torque and reduction of disturbances. Moreover, the proposed control strategy using a very limited calculation load, at the same time, maintains the ripples inside the technical limits of the obtained torque. Both effects are due to the decoupled EKFs with simplified and reduced order of the models using univariate polynomials, which require significantly fewer measuring points in the run-up to the creation of the model of the inductances Ld and Lq.

Dimension-free discretizations of the uniform norm by small product sets

Let f be an analytic polynomial of degree at most K−1. A classical inequality of Bernstein compares the supremum norm of f over the unit circle to its supremum norm over the sampling set of the K-th roots of unity. Many extensions of this inequality exist, often understood under the umbrella of Marcinkiewicz–Zygmund-type inequalities for Lp,1≤p≤∞ norms. We study dimension-free extensions of these discretization inequalities in the high-dimension regime, where existing results construct sampling sets with cardinality growing with the total degree of the polynomial. In this work we show that dimension-free discretizations are possible with sampling sets whose cardinality is independent of deg(f) and is instead governed by the maximum individual degree of f; i.e., the largest degree of f when viewed as a univariate polynomial in any coordinate. For example, we find that for n-variate analytic polynomials f of degree at most d and individual degree at most K−1, ∥f∥L∞(Dn)≤C(X)d∥f∥L∞(Xn) for any fixed X in the unit disc D with |X|=K. The dependence on d in the constant is tight for such small sampling sets, which arise naturally for example when studying polynomials of bounded degree coming from functions on products of cyclic groups. As an application we obtain a proof of the cyclic group Bohnenblust–Hille inequality with an explicit constant O(logK)2d.

Exact H∞ optimization of dynamic vibration absorbers: Univariate-polynomial-based algorithm and operability analysis

H∞ optimization of dynamic vibration absorbers (DVAs) to minimize the maximum response amplitude of primary structures is a classic topic. The commonly used fixed-point method only provides approximate solutions requiring the primary structure is undamped. Instead, we perform exact optimization and investigate the less-reported parametric effects on optimization operability. To handle the known restrictions posed by grounded dampers, a typical DVA model mounted on a damped primary structure, with components connected to both the primary and the base, is considered. We explore three elaborated cases depending on the grounded dampers distributing in the primary structure and the DVA. Our findings reveal that the frequency responses of the primary structure with dual resonant peaks of equal height may not be the global optimum, and we establish a nontrivial necessary condition for operable exact optimization. Furthermore, we elucidate the effects of structural arrangements on the optimized results and provide design rules to maximize vibration suppression performance. The optimization follows the proposal of a so-called resultant-based algorithm that guarantees global optimum with high efficiency and a generalizable core by exclusively constructing univariate polynomial equations. This study contributes a systematic analysis framework alongside efficient calculation tools for exact optimization and DVA performance evaluation.

Application of Mathematics for Robust Stability and for Robustly Strictly Positive Real on an Uncertain Interval Plant

ABSTRACTIn this article, we present a robust control problem wherein represents a family of interval plants. For this particular problem, we introduce four Kharitonov polynomials uniquely by minimizing and maximizing the concept of multilinear functions with uncertain parameters and for the plant and where denotes the conjugate of and , with representing frequency within a specified domain. This technique yields a Kharitonov rectangle or box whose every four vertices are represented by four unique Kharitonov polynomials , each is denoted by and and this rectangle characterizes both robust stability and robustly strictly positive real (SPR) for the family of interval plants (). Thus, the introduction of this Kharitonov rectangle stands as a novel innovation in this article. Our contribution encompasses two key aspects. Initially, we demonstrate the stability of the four unique Kharitonov polynomials employing Hurwitz stability criteria, followed by the utilization of Kharitonov's theorem to ensure robust stability of . Subsequently, to establish robustly SPR of , we analyze the SPR of each interval plant concerning the stability of and , adhering to the condition . Additionally, we provide a detailed illustrative example. Furthermore, we demonstrate the motion of the Kharitonov rectangle and the robust stability test through simulation.

A polynomial proxy model approach to verifiable decentralized federated learning

Decentralized Federated Learning improves data privacy and eliminates single points of failure by removing reliance on centralized storage and model aggregation in distributed computing systems. Ensuring the integrity of computations during local model training is a significant challenge, especially before sharing gradient updates from each local client. Current methods for ensuring computation integrity often involve patching local models to implement cryptographic techniques, such as Zero-Knowledge Proofs. However, this approach becomes highly complex and sometimes impractical for large-scale models that use techniques such as random dropouts to improve training convergence. These random dropouts create non-deterministic behavior, making it challenging to verify model updates under deterministic protocols. We propose ProxyZKP, a novel framework combining Zero-Knowledge Proofs with polynomial proxy models to provide computation integrity in local training to address this issue. Each local node combines a private model for online deep learning applications and a proxy model that mediates decentralized model training by exchanging gradient updates. The multivariate polynomial nature of proxy models facilitates the application of Zero-Knowledge Proofs. These proofs verify the computation integrity of updates from each node without disclosing private data. Experimental results indicate that ProxyZKP significantly reduces computational load. Specifically, ProxyZKP achieves proof generation times that are 30–50% faster compared to established methods like zk-SNARKs and Bulletproofs. This improvement is largely due to the high parallelization potential of the univariate polynomial decomposition approach. Additionally, integrating Differential Privacy into the ProxyZKP framework reduces the risk of Gradient Inversion attacks by adding calibrated noise to the gradients, while maintaining competitive model accuracy. The results demonstrate that ProxyZKP is a scalable and efficient solution for ensuring training integrity in decentralized federated learning environments, particularly in scenarios with frequent model updates and the need for strong model scalability.

Power boundedness and related properties for weighted composition operators on [formula omitted]

We characterize those pairs (ψ,φ) of smooth mappings ψ:Rd→C,φ:Rd→Rd for which the corresponding weighted composition operator Cψ,φf=ψ⋅(f∘φ) acts continuously on S(Rd). Additionally, we give several easy-to-check necessary and sufficient conditions of this property for interesting special cases. Moreover, we characterize power boundedness and topologizablity of Cψ,φ on S(Rd) in terms of ψ,φ. Among other things, as an application of our results we show that for a univariate polynomial φ with deg(φ)≥2, power boundedness of Cψ,φ on S(R) for every ψ∈OM(R) only depends on φ and that in this case power boundedness of Cψ,φ is equivalent to (Cψ,φn)n∈N converging to 0 in Lb(S(R)) as well as to the uniform mean ergodicity of Cψ,φ. Additionally, we give an example of a power bounded and uniformly mean ergodic weighted composition operator Cψ,φ on S(R) for which neither the multiplication operator f↦ψf nor the composition operator f↦f∘φ acts on S(R). Our results complement and considerably extend various results of Fernández, Galbis, and the second named author.

Optimization-aided construction of multivariate Chebyshev polynomials

This article is concerned with an extension of univariate Chebyshev polynomials of the first kind to the multivariate setting, where one chases best approximants to specific monomials by polynomials of lower degree relative to the uniform norm. Exploiting the Moment-SOS hierarchy, we devise a versatile semidefinite-programming-based procedure to compute such best approximants, as well as associated signatures. Applying this procedure in three variables leads to the values of best approximation errors for all monomials up to degree six on the euclidean ball, the simplex, and the cross-polytope. Furthermore, inspired by numerical experiments, we obtain explicit expressions for Chebyshev polynomials in two cases unresolved before, namely for the monomial x12x22x3 on the euclidean ball and for the monomial x12x2x3 on the simplex.

Small Public Exponent Brings More: Improved Partial Key Exposure Attacks against RSA

Let (N,e) be a public key of the RSA cryptosystem, and d be the corresponding private key. In practice, we usually choose a small e for quick encryption. In this paper, we improve partial private key exposure attacks against RSA with a small public exponent e. The key idea is that under such a setting we can usually obtain more information about the prime factor of N and then by solving a univariate modular polynomial with Coppersmith's method, N can be factored in polynomial time. Compared to previous results, we reduce the number of d's leaked bits needed to mount the attack by log_2 (e) bits. Furthermore, our experiments show that for 1024-bit N, our attack can achieve the theoretical bound on a personal computer, which verified our attack.

7796 The Profile And Associated Risk Factors Of Diabetic Foot Disease Among Patients With Type 2 Diabetes Mellitus Seen At Outpatient Clinics In A Tertiary Private Hospital In Quezon City, Philippines

Abstract Disclosure: D.C. Lopez: None. K.T. Lim: None. B.C. S.: None. O.C. Dampil: None. The Profile and Associated Risk Factors of Diabetic Foot Disease Among Patients with Type 2 Diabetes Mellitus Seen at Outpatient Clinics in a Tertiary Private Hospital in Manila, Philippines. Background: Diabetic foot disease is a debilitating complication of diabetes mellitus and is a major source of morbidity, mortality, and reduced quality of life. Determining the risk factors of diabetic foot ulcer is important to prevent the devastating consequences among patients with diabetes. This study aims to determine the profile and associated risk factors of diabetic foot disease among patients diagnosed with type 2 diabetes mellitus being seen in private specialty clinics. Methodology: A single center, analytical cross-sectional study was conducted among patients with type 2 diabetes mellitus at outpatient clinics in a tertiary private hospital in the Philippines. Clinical information and laboratory results were obtained during the time of consult; and patients were classified as either low risk, moderate risk, high risk for diabetic foot disease or in active foot disease. Interaction between various risk factors was explored by multivariate analysis. Results: A total of 129 adult patients with diabetes mellitus were evaluated using the Diabetes Foot Screening and Risk Stratification Tool from January to May 2021. The demographic and clinical profile of the study showed that proportion of patients with neuropathy (χ2=60.66, p=0.001), long duration of diabetes (F=5.92, p=0.004), and low estimated glomerular filtration rate (eGFR) (F=3.70, p=0.023) was significantly higher in patients with moderate to high risk for diabetic foot disease. Univariate polynomial logistic regression analyses showed that neuropathy greatly increased the odds of having moderate (OR=115.63, p=0.001) and high risk stratification (OR=166.50, p=0.001). Other factors were also noted to affect the risk stratification for diabetic foot disease such as duration of diabetes (OR=1.09, p=0.004) and hypertension (OR=2.45, p=0.027). On the other hand, normal eGFR (OR=–1.02, p=0.011), left normal ankle brachial pressure index (OR=–4.32, p=0.042), and good glycemic control (OR=–5.41, p=0.050) significantly decrease the likelihood of having high risk stratification for diabetic patients. Conclusion: Those most susceptible to developing high risk for diabetic foot disease were patients with neuropathy, hypertension, and long duration of diabetes. High-risk patients with the given profile should be followed closely in order to prevent complications brought by diabetes. Keywords:Diabetes mellitus, Diabetic foot disease, Foot ulcer Presentation: 6/3/2024

𝒫𝒯 and anti-𝒫𝒯 symmetries for astrophysical waves

Context. Discrete symmetries have found numerous applications in photonics and quantum mechanics, but remain little studied in fluid mechanics, particularly in astrophysics. Aims. We aim to show how 𝒫𝒯 and anti-𝒫𝒯 symmetries determine the behaviour of linear perturbations in a wide class of astrophysical problems. They set the location of ‘exceptional points’ in the parameter space and the associated transitions to instability, and are associated with the conservation of quadratic quantities that can be determined explicitly. Methods. We study several classical local problems: the gravitational instability of isothermal spheres and thin discs, the Schwarzschild instability, the Rayleigh-Bénard instability and acoustic waves in dust–gas mixtures. We calculate the locations and the order of the exceptional points using the resultant of two univariate polynomials, as well as the conserved quantities in the different regions of the parameter space using Krein theory. Results. All problems studied here exhibit discrete symmetries, even though Hermiticity is broken by different physical processes (self-gravity, buoyancy, diffusion, and drag). This analysis provides genuine explanations for certain instabilities, and for the existence of regions in the parameter space where waves do not propagate. Those two aspects correspond to regions where 𝒫𝒯 and anti-𝒫𝒯 symmetries are broken respectively. Not all instabilities are associated to symmetry breaking (e.g. the Rayleigh-Benard instability).

Subresultants of several univariate polynomials in Newton basis

In this paper, we consider the problem of formulating the subresultant polynomials for several univariate polynomials in Newton basis. It is required that the resulting subresultant polynomials be expressed in the same Newton basis as that used in the input polynomials. To solve the problem, we devise a particular matrix with the help of the companion matrix of a polynomial in Newton basis. Meanwhile, the concept of determinant polynomial in power basis for formulating subresultant polynomials is extended to that in Newton basis. It is proved that the generalized determinant polynomial of the specially designed matrix provides a new formula for the subresultant polynomial in Newton basis, which is equivalent to the subresultant polynomial in power basis. Furthermore, we show an application of the new formula in devising a basis-preserving method for computing the gcd of several Newton polynomials.

Impact of preoperative anxiety on postoperative outcomes in patients undergoing minimally invasive thoracoscopic surgery: A prospective cohort study

BackgroundPreoperative anxiety is a common preoperative psychological state in patients with cancer and associated with worsening perioperative outcomes. However, high-quality prospective studies on preoperative anxiety in patients undergoing lung surgery are scarce. MethodsWe conducted a prospective cohort study, enrolling a total of 540 patients. Preoperative anxiety in patients undergoing thoracic surgery was measured using the Hospitalization Anxiety Scale. Patients were grouped according to the Hospitalization Anxiety Scale scores as follows: no anxiety (score <8) and anxiety (score ≥8). The association of preoperative anxiety with postoperative complications and non-complicated adverse events was determined by univariate regression and polynomial regression analyses. ResultsA total of 121 patients (22.4 %) experienced preoperative anxiety. The anxiety group had a longer average hospital stay (4.33 vs. 3.85 days). Postoperative complications were similar between groups, but the anxiety group reported worse sleep quality (measured by the Athens Insomnia Scale). Regarding postoperative pain, both groups had comparable rates of mild and severe pain on postoperative day 1. However, the anxiety group experienced significantly higher rates of severe pain on postoperative day 2 and mild pain on postoperative day 3. Additionally, the incidence of postoperative nausea and vomiting was significantly higher in the anxiety group on postoperative day 1. ConclusionsPreoperative anxiety may not increase the rates of postoperative complications in patients undergoing lung surgery. However, it may be associated with postoperative sleep disturbances, pain, nausea, and vomiting, as well as prolong the length of postoperative hospitalization.

Griffiths polynomials of Racah type

Bivariate Griffiths polynomials of Racah type are constructed from univariate Racah polynomials. The bispectral properties of the former are deduced from simple properties of the latter. A duality relation and the orthogonality of these polynomials are provided. The domain of validity for the indices and variables of these polynomials is also determined. Particular limits on the parameters entering the polynomials allow to define several Griffiths polynomials of other types. One special limit connects them to the original Griffiths polynomials (of Krawtchouk type). Finally, a connection with the 9j symbols is made.

Uni/multi variate polynomial embeddings for zkSNARKs

A zero-knowledge proof is a cryptographic primitive that enables a prover to convince a verifier the validity of a mathematical statement (an NP statement) without revealing any secret inputs to the verifier. A special case, called zero-knowledge Succinct Non-interactive ARgument of Knowledge (zkSNARK) is particularly designed for arithmetic circuit proof systems which have important applications in blockchain privacy. The major computations in this type of zkSNARK proofs with post-quantum security are polynomial evaluations and Lagrange interpolations over finite fields. Given a sequence over a finite field, in the field of coding and sequences research, we understand that there are two representations of the sequence, one is a univariate polynomial and the other, a multivariate polynomial. This is exactly what is done in those zero-knowledge proof systems to transform the proof of a R1CS relation to evaluate uni/multi variate polynomials at some random points in the finite field. In this paper, we present a comparative analysis on how to convert a rank 1 constrained satisfiability (R1CS) system (more general than a circuit system) into a polynomial equality and provide analysis on the concrete complexities of provers, proof sizes and verifiers. We use two concrete zkSNARK schemes, i.e., Polaris, univariate polynomial encodings and Spartan, multivariate polynomial encodings, as examples to show our analysis. Secondly, we propose to select interpolating sets as subfields instead of affine spaces of a large field for Lagrange interpolation. This new method has improved the performance of R1CS encodings largely. We comment that post-quantum secure zkSNARKs yield post-quantum digital signatures with security only depending on symmetric-key schemes. Some open problems are proposed at the end of the paper.

An Investigation of Molecular Property Prediction and Classification Based on Machine Learning Algorithms

With the rapid advancement of science and technology, chemical and physical research has entered an era of complexity and high dimensionality, where traditional research paradigms struggle to optimize vast chemical parameter searches. The Machine Chemist platform was developed in this context, leveraging big data and intelligent models to automate chemical synthesis, characterization, and testing processes. This study aims to predict the y1, y2 and y3 properties and classes of 2,580 molecules based on their physicochemical properties and improve model accuracy through data analysis and modeling. Data preprocessing involved removing missing values, duplicates, and outliers using the quartile method, resulting in an analyzable dataset. A scatter plot of y2 and id suggested a univariate polynomial function relationship, leading to the construction of a univariate nonlinear regression model. The model achieved a high prediction accuracy, with a low root mean square error and a high coefficient of determination. Additionally, the relationship between class and y1~y3 , x1~x100 indicators was examined, revealing mostly nonlinear relationships. A random forest model was established to classify 2,580 molecules into 1 to 4 classes based on the properties of 200,000 chemical molecules. The model's performance was evaluated using decision trees, confusion matrices, precision, and recall metrics. Finally, the SHAP method assessed the impact of feature indicators on classification outcomes, contributing to the development of a reliable molecular category prediction model.

Orbit determination from one position vector and a very short arc of optical observations

In this paper, we address the problem of computing a preliminary orbit of a celestial body from one topocentric position vector P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{P}_1$$\\end{document} and a very short arc (VSA) of optical observations A2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{A}_2$$\\end{document}. Using the conservation laws of the two-body dynamics, we write the problem as a system of 8 polynomial equations in 6 unknowns. We prove that this system is generically consistent, namely, for a generic choice of the data P1,A2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{P}_1, \\mathcal{A}_2$$\\end{document}, it always admits solutions in the complex field, even when P1,A2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{P}_1, \\mathcal{A}_2$$\\end{document} do not correspond to the same celestial body. The consistency of the system is shown by deriving a univariate polynomial v\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {v}$$\\end{document} of degree 8 in the unknown topocentric distance at the mean epoch of the observations of the VSA. Through Gröbner bases theory, we also show that the degree of v\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {v}$$\\end{document} is minimum among the degrees of all the univariate polynomials solving this problem. Even though we can find solutions to our problem for a generic choice of P1,A2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{P}_1, \\mathcal{A}_2$$\\end{document}, most of these solutions are meaningless. In fact, acceptable solutions must be real and have to fulfill other constraints, including compatibility with Keplerian dynamics. We also propose a way to select or discard solutions taking into account the uncertainty in the data, if present. The proposed orbit determination method is relevant for different purposes, e.g., the computation of a preliminary orbit of an Earth satellite with radar and optical observations, the detection of maneuvres of an Earth satellite, and the recovery of asteroids which are lost due to a planetary close encounter. We conclude by showing some numerical tests in the case of asteroids undergoing a close encounter with the Earth.

Interpolation by decomposable univariate polynomials

The usual univariate interpolation problem of finding a monic polynomial f of degree n that interpolates n given values is well understood. This paper studies a variant where f is required to be composite, say, a composition of two polynomials of degrees d and e, respectively, with de=n, and with d+e−1 given values. Some special cases are easy to solve, and for the general case, we construct a homotopy between it and a special case. We compute a geometric solution of the algebraic curve presenting this homotopy, and this also provides an answer to the interpolation task. The computing time is polynomial in the geometric data, like the degree, of this curve. A consequence is that for almost all inputs, a decomposable interpolation polynomial exists.

The centre of the Dunkl total angular momentum algebra

For a finite dimensional representation V of a finite reflection group W, we consider the rational Cherednik algebra Ht,c(V,W) associated with (V,W) at the parameters t≠0 and c. The Dunkl total angular momentum algebra Ot,c(V,W) arises as the centraliser algebra of the Lie superalgebra osp(1|2) containing a Dunkl deformation of the Dirac operator, inside the tensor product of Ht,c(V,W) and the Clifford algebra generated by V.We show that when dim⁡V≥3 and for every value of the parameter c, the centre of Ot,c(V,W) is isomorphic to a univariate polynomial ring. Notably, the generator of the centre changes depending on whether or not (−1)V is an element of the group W. Using this description of the centre, and using the projection of the pseudo scalar from the Clifford algebra into Ot,c(V,W), we establish results analogous to “Vogan's conjecture” for a family of operators depending on suitable elements of the double cover W˜.

A Novel Spatiotemporal Periodic Polynomial Model for Predicting Road Traffic Speed

Accurate and fast traffic prediction is the data-based foundation for achieving traffic control and management, and the accuracy of prediction results will directly affect the effectiveness of traffic control and management. This paper proposes a new spatiotemporal periodic polynomial model for road traffic, which integrates the temporal, spatial, and periodic features of speed time series and can effectively handle the nonlinear mapping relationship from input to output. In terms of the model, we establish a road traffic speed prediction model based on polynomial regression. In terms of spatial feature extraction methods, we introduce a maximum mutual information coefficient spatial feature extraction method. In terms of periodic feature extraction methods, we introduce a periodic trend modeling method into the prediction of speed time series, and effective fusion is carried out. Four strategies are evaluated based on the Guangzhou road speed dataset: a univariate polynomial model, a spatiotemporal polynomial model, a periodic polynomial model, and a spatiotemporal periodic polynomial model. The test results show that the three methods proposed in this article can effectively improve prediction accuracy. Comparing the spatiotemporal periodic polynomial model with multiple machine learning models and deep learning models, the prediction accuracy is improved by 5.94% compared to the best feedforward neural network. The research in this article can effectively deal with the temporal, spatial, periodic, and nonlinear characteristics of speed prediction, and to a certain extent, improve the accuracy of speed prediction.

Algebraic properties of the maps .

The Boolean map defined by (where ) is used in various permutations that are part of cryptographic schemes, e.g., Keccak-f (the SHA-3-permutation), ASCON (the winner of the NIST Lightweight competition), Xoodoo, Rasta and Subterranean (2.0). In this paper, we study various algebraic properties of this map. We consider (through vectorial isomorphism) as a univariate polynomial. We show that it is a power function if and only if . We furthermore compute bounds on the sparsity and degree of these univariate polynomials, and the number of different univariate representations. Secondly, we compute the number of monomials of given degree in the inverse of (if it exists). This number coincides with binomial coefficients. Lastly, we consider as a polynomial map, to study whether the same rule ( ) gives a bijection on field extensions of . We show that this is not the case for extensions whose degree is divisible by two or three. Based on these results, we conjecture that this rule does not give a bijection on any extension field of .