Polynomials Of Degree Research Articles

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.

Efficient entropy-stable discontinuous spectral-element methods using tensor-product summation-by-parts operators on triangles and tetrahedra

We present a new class of efficient and robust discontinuous spectral-element methods of arbitrary order for nonlinear hyperbolic systems of conservation laws on curved triangular and tetrahedral unstructured grids. Such discretizations employ a recently introduced family of sparse tensor-product summation-by-parts (SBP) operators in collapsed coordinates within an entropy-conservative modal formulation, which is rendered entropy stable when a dissipative numerical flux is used at element interfaces. The proposed algorithms exploit the structure of such SBP operators alongside that of the Proriol–Koornwinder–Dubiner polynomial basis used to represent the numerical solution on the reference triangle or tetrahedron, and a weight-adjusted approximation is employed in order to efficiently invert the local mass matrix for curvilinear elements. Using such techniques, we obtain an improvement in time complexity from O(p2d) to O(pd+1) relative to existing entropy-stable formulations using multidimensional SBP operators not possessing such a tensor-product structure, where p is the polynomial degree of the approximation and d is the number of spatial dimensions. The number of required entropy-conservative two-point flux evaluations between pairs of quadrature nodes is accordingly reduced by a factor ranging from 1.56 at p=2 to 4.57 at p=10 for triangles, and from 1.88 at p=2 to 10.99 at p=10 for tetrahedra. Through numerical experiments involving smooth solutions to the compressible Euler equations on isoparametric triangular and tetrahedral grids, the proposed methods using tensor-product SBP operators are shown to exhibit similar levels of accuracy for a given mesh and polynomial degree to those using multidimensional operators based on symmetric quadrature rules, with both approaches achieving order p+1 convergence with respect to the element size in the presence of interface dissipation as well as exponential convergence with respect to the polynomial degree. Furthermore, both operator families are shown to give rise to entropy-stable schemes which exhibit excellent robustness for test problems characteristic of under-resolved turbulence simulations. Such results suggest that the algorithmic advantages resulting from the use of tensor-product operators are obtained without compromising accuracy or robustness, enabling the efficient extension of the benefits of entropy stability to higher polynomial degrees than previously considered for triangular and tetrahedral elements.

A hp-adaptive finite element approach for 3D controlled source electromagnetic forward modeling

Accurate and efficient 3D controlled-source electromagnetic (CSEM) forward modeling is crucial for deciphering CSEM responses in complex geological contexts and for assessing the efficacy of survey designs. It also serves as the foundation for 3D inversion, a critical technique for interpreting CSEM data. We present a novel hp-adaptive finite element method for 3D CSEM forward modeling. This method enhances the traditional h-adaptive approach, which only refines the mesh while keeping the polynomial degree of the basis functions constant, by enabling simultaneous refining mesh and increasing polynomial degree of basis functions. High-order elements yield superior approximations when the solution exhibits sufficient smoothness, while mesh refinement enhances accuracy in regions with less smooth solutions. The adaptive refinement process is guided by a goal-oriented error estimator that relies on the residual of the governing equation and the continuity of electromagnetic fields. The hp-adaptive strategy, which determines whether to refine elements or increase their polynomial degree, is based on the refinement levels of the elements. Constraints are applied to hanging nodes, introduced during the refinement process, to ensure the continuity of the finite element space. We validate the accuracy, efficiency, and convergence of the hp-adaptive method through tests on two layered models. Further simulations on a topographic model and a realistic model corroborate its effectiveness. Our findings indicate that the hp-adaptive method outperforms the traditional h-adaptive method in terms of accuracy, convergence rate and computational cost, making it a promising technique for 3D CSEM forward modeling.

Ruin Probabilities as Recurrence Sequences in a Discrete-Time Risk Process

The theory of linear recurrence sequences is applied to obtain an explicit formula for the ultimate ruin probability in a discrete-time risk process. It is assumed that the claims distribution is arbitrary but has finite support {0,1,…,m+1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\{0,1,\\ldots ,m+1\\}}$$\\end{document}, for some integer m≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{m\\ge 1}$$\\end{document}. The method requires finding the zeroes of an m degree polynomial and solving a system of m linear equations. An approximation is derived and some numerical results and plots are provided as examples.

Orthogonal polynomial bases in the mixed virtual element method

AbstractThe use of orthonormal polynomial bases has been found to be efficient in preventing ill‐conditioning of the system matrix in the primal formulation of Virtual Element Methods (VEM) for high values of polynomial degree and in presence of badly‐shaped polygons. However, we show that using the natural extension of a orthogonal polynomial basis built for the primal formulation is not sufficient to cure ill‐conditioning in the mixed case. Thus, in the present work, we introduce an orthogonal vector‐polynomial basis which is built ad hoc for being used in the mixed formulation of VEM and which leads to very high‐quality solution in each tested case. Furthermore, a numerical experiment related to simulations in Discrete Fracture Networks (DFN), which are often characterised by very badly‐shaped elements, is proposed to validate our procedures.

Comparative Analysis of Structure and Motion of Vision Logistics Robotic Arms

This paper combines visual recognition and robotic arm, optimizes the design of robotic arm structure, and researches the logistics robotic arm with six degrees of freedom and visual perception ability to enrich the logistics robotic use scene. Firstly, the visual recognition system is installed on the existing six-degree-of-freedom robotic arm, and the coordinate system is established by the light bar recognition method combined with the D-H parameter method. Secondly, a kinematic simulation model was established by using MATLAB, and the state of the joints was analyzed by using forward and inverse kinematic simulation and fifth degree polynomial interpolation. Finally, the actual logistics fixed-point transportation task is designed for trajectory planning. The results show that the robotic arm is able to be able to perform different grasping tasks in non-structural environments, around the visual analysis of the robotic arm as well as the optimization of the motion track, to enhance the efficiency of the comprehensive use of visual and tactile information, to improve the adaptability of the robotic hand grasping, and to be able to meet the precise control of the pick-up process and the use of the realization of the payload grasping and dexterous movement. The research in this paper provides a certain reference value for the propulsion robot arm to perform intelligent grasping tasks.

Investigation of the accuracy of integral models of the oculo-motor system

For mathematical modeling of the human oculomotor system (OMS), integral nonlinear models are used, which simultaneously take into account the nonlinear and inertial properties of the research object. Based on the data of experimental studies of the OMS "input-output", transient and diagonal intersections of transient functions of the second and third orders are determined. To obtain experimental data, an innovative eye tracking technology is used, which allows recording eye responses to test visual stimuli. Thus test signals are displayed on the computer monitor at different distances from the starting position in the horizontal direction. The aim of the research is to study the accuracy of OMS identification using eye-tracking data by evaluating the calculation errors of multidimensional transient functions when using methods of nonlinear dynamic identification based on models in the form of Volterra series and polynomials. The object of the study is the process of nonparametric identification of the OMS based on Volterra models in the time domain. The subject of the research is algorithmic and software tools for calculating the dynamic characteristics of OMS based on eye-tracking data, analyzing the accuracy of the obtained models using two identification methods: the approximation method and the least squares method (LSM). The means of nonlinear dynamic identification of the human OMS based on Volterra series and polynomials were developed in the Python programming environment. The accuracy estimates of various OMS (linear, quadratic, and cubic) models were obtained based on data from three responses to test signals of different amplitudes. For the same test signals, the same models in the form of the Volterra series and polynomials were obtained, as these models coincide in the region of convergence of the Volterra series. The analysis of errors in the assessment of the dynamic characteristics of the OMS demonstrated that the model in the form of an integral polynomial of the second degree, which was built using the LSM based on three responses, has an accuracy twice as high as the accuracy of similar models built using the data of two responses. Thus, in further studies of the psychophysiological state of a person based on nonlinear dynamic models of the OMS according to the data of three responses, it is advisable to use a model in the form of a quadratic Volterra polynomial. Figs.: 17. Tabl. 3. Refs.: 10 titles.

Geodesic metrics for RBF approximation of some physical quantities measured on sphere

The radial basis function (RBF) approximation is a rapidly developing field of mathematics. In the paper, we are concerned with the measurement of scalar physical quantities at nodes on sphere in the 3D Euclidean space and the spherical RBF interpolation of the data acquired. We employ a multiquadric as the radial basis function and the corresponding trend is a polynomial of degree 2 considered in Cartesian coordinates. Attention is paid to geodesic metrics that define the distance of two points on a sphere. The choice of a particular geodesic metric function is an important part of the construction of interpolation formula.We show the existence of an interpolation formula of the type considered. The approximation formulas of this type can be useful in the interpretation of measurements of various physical quantities. We present an example concerned with the sampling of anisotropy of magnetic susceptibility having extensive applications in geosciences and demonstrate the advantages and drawbacks of the formulas chosen, in particular the strong dependence of interpolation results on condition number of the matrix of the system considered and on round-off errors in general.

[formula omitted]-adaptive discontinuous Galerkin solution of transonic viscous flows with variable time step-size

The discontinuous Galerkin methods demonstrated to be well suited for scale resolving simulations of complex configurations, characterized by different fluid dynamics phenomena, such as transition, separation, shock/boundary layer interaction. Unfortunately, solvers based on these methods cannot yet reach the computational efficiency of well-established standard solvers, e.g., based on finite volume methods. To reduce the computational cost and the memory footprint, while not spoiling the accuracy, different approaches were investigated, which act on the spatial or the temporal discretization, and on the linear algebra. The above approaches have been extensively investigated, but rarely considering their mutual interactions. In this work, a p-adaptation algorithm is coupled with a time step-size adaptation algorithm to mitigate robustness issues arising after each adaptation cycle, probably related to the projection of the old solution on the new polynomial orders distribution. Moreover, this strategy is able to control the transient phase before each adaptation cycle, and can handle also the presence of initial steady solution, when the mesh is too coarse and the polynomial degree has not yet compensated for the lack of spatial accuracy. The effectiveness and accuracy of the proposed algorithm are demonstrated on transonic viscous flows with shock-wave/boundary-layer interaction.

On 2-superirreducible polynomials over finite fields

We investigate k-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most k. Let F be a finite field of characteristic p. We show that no 2-superirreducible polynomials exist in F[t] when p=2 and that no such polynomials of odd degree exist when p is odd. We address the remaining case in which p is odd and the polynomials have even degree by giving an explicit formula for the number of monic 2-superirreducible polynomials having even degree d. This formula is analogous to that given by Gauss for the number of monic irreducible polynomials of given degree over a finite field. We discuss the associated asymptotic behaviour when either the degree of the polynomial or the size of the finite field tends to infinity.

GOLD PRICE PREDICTION IN INDONESIA BASED ON INTEREST RATE USING DISTRIBUTED LAG ALMON TRANSFORMATION

Gold is valued for its safety and profitability, driven by steady price changes and influenced by interest rates. Accurately predicting gold prices is very important to make the right investment decisions. This study aims to build a gold price model in Indonesia using the Almon transformation lag distribution and see gold price predictions based on the model that has been built. We used the data on gold prices and interest rates from January 2016 to December 2023. Based on the results of the analysis, the best Almon transformation model used in this study is the Almon model with a maximum lag length of 16 and the second polynomial degree. The prediction results have a MAPE of 16.49%, which shows that the Almon model can predict gold prices well for one year. This study contributes to the understanding of gold price dynamics amid economic variations. However, limitations in the model assumptions should be considered.

Cam displacement curve optimization for minimal jerk using search and rescue optimization algorithm

Polynomials of a degree higher than 8th used for the rise phase of displacement curve of a cam profile were optimized for gaining minimal jerk using the search and rescue optimization algorithm. The algorithm was used to obtain the polynomial coefficients for the polynomials of the higher degree that cannot be calculated by using the analytical approach without adding additional boundary conditions other than displacement, velocity and acceleration in characteristic positions. The degree of the polynomial from which the optimal value of jerk using the optimization algorithm can be obtained was determined, as well as the degree after which the results of the optimization give worse values of the jerk.

Polynomial Representation and Degree Sequence of Graphs Resulting from Some Graph Operations

Let $G = (V(G),E(G))$ be a graph with degree sequence $\langle d_1,d_2, \cdots, d_n \rangle$, where $d_1 \ge d_2 \ge \cdots \ge d_n$. The polynomial representation of $G$ is given by $f_G(x) = \displaystyle \sum_{i=1}^n x^{d_i} = \sum_{k=1}^{\Delta(G)}a_kx^{k}$, where $a_k$ is the number of vertices of $G$ having degree $k$ for each $i = 1,2,\cdots n = \Delta(G)$. In this paper, we give the polynomial representation of the complement and line graph of a graph, the shadow graph, complementary prism, edge corona, strong product, symmetric product, and disjunction of two graphs.

Rheological effects in peristaltic flow of Prandtl fluid through elliptical duct: A comprehensive analysis

AbstractThis research venture comprehends a theoretical examination of non‐Newtonian fluid flowing peristaltic via an elliptical channel. Furthermore, the Prandtl fluid method for this elliptic duct problem is thoroughly considered. This mathematical inquiry adopts a non‐Newtonian Prandtl fluid model. A polynomial methodology is used to analyze partial differential equations that appear in nondimensional form and deliver an exact analytical solution for the temperature and velocity profile. This study is the first to utilize a novel order polynomial of degree eight having eleven constants to precisely solve the temperature equation for Prandtl fluid flow via an elliptic domain. A comprehensive graphical analysis is also provided to understand the mathematical conclusions fully. The graphs of the velocity profiles clearly show that the non‐Newtonian effects are more potent along the minor axis of the elliptical duct. The streamlined graphs accentuating the trapping phenomenon show specific closed contours close to the boundary wall of the peristaltic duct.

Density of power-free values of polynomials II

In this paper we prove that polynomials F(x1,⋯,xn)∈Z[x1,⋯,xn] of degree d≥3, satisfying certain hypotheses, take on the expected density of (d−1)-free values. This extends the authors' earlier result in [14] where a different method implied the similar statement for polynomials of degree d≥5.

On a class of constacyclic codes of length 4ps over 𝔽pm[u] 〈u3〉

Let [Formula: see text] be a prime such that [Formula: see text], where [Formula: see text] is a positive integer. For any nonzero element [Formula: see text] of [Formula: see text], we determine the algebraic structure of all [Formula: see text]-constacyclic codes of length [Formula: see text] over the finite commutative chain ring [Formula: see text], where [Formula: see text] and [Formula: see text] is a positive integer. If the unit [Formula: see text] is a square, [Formula: see text], each [Formula: see text]-constacyclic code of length [Formula: see text] is expressed as a direct sum of an [Formula: see text]-constacyclic code and an [Formula: see text]- constacyclic code of length [Formula: see text]. In the main case that the unit [Formula: see text] is not a square, it is shown that any nonzero polynomial of degree at most [Formula: see text] over [Formula: see text] is invertible in the ambient ring [Formula: see text]. It is also proven that the ambient ring [Formula: see text] is a local ring with the unique maximal ideal [Formula: see text], where [Formula: see text]. Such [Formula: see text]-constacyclic codes are then classified into eight distinct types of ideals, and the detailed structures of ideals in each type are provided. Among other results, the number of codewords, and the dual of each [Formula: see text]-constacyclic code are obtained. The non-existence of self-dual and isodual [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text], when the unit [Formula: see text] is not a square, is likewise proved.

Fine-grained polynomial functional encryption

In this work, we present fine-grained secure polynomial functional encryption (PFE) for degree d≥1 over a field F: a ciphertext encrypts x∈Fn, a key is associated with a degree-d polynomial P and decryption recovers P(x)∈F. Fine-grained cryptographic primitives are secure against a resources bounded class of adversaries and computed by honest users with less resources than adversaries. In this paper, we construct the fine-grained PFE in these two fine-grained settings:(1) NC1 PFE: Based on the worst-case assumption NC1⊊⊕L/poly, we construct public-key polynomial functional encryption and achieve (i) selective simulation-based security and (ii) static function-hiding against adversary in NC1 where all honest algorithms are computable in AC0[2] and ciphertext sizes are O(n).(2) AC0 PFE: We construct a private-key polynomial functional encryption achieve unconditionally selective simulation-based security against adversary in AC0 where all honest algorithms are computable in AC0 and ciphertext sizes are O(n).

Strictly positive polynomials in the boundary of the SOS cone

We study the boundary of the cone of real polynomials that can be decomposed as a sum of squares (SOS) of real polynomials. This cone is included in the cone of nonnegative polynomials and both cones share a part of their boundary, which corresponds to polynomials that vanish at at least one point. We focus on the part of the boundary which is not shared, corresponding to strictly positive polynomials.For the cases of polynomials of degree 6 in 3 variables and degree 4 in 4 variables, this boundary has been completely characterized by G. Blekherman. For the cases of a polynomial f in more variables or of higher degree, results by G. Blekherman, R. Sinn and M. Velasco and other authors based on a conjecture by Ottaviani and Paoletti give bounds for the maximum number of linearly independent polynomials that can appear in an SOS decomposition of f, or equivalently the maximum rank of the matrices in the Gram spectrahedron of f. We show that the same bounds can be obtained from the Eisenbud-Green-Harris conjecture. Combining theoretical results and computational techniques, we compute examples that allow us to prove the optimality of the bounds for all degrees and number of variables. Additionally, we give examples for the following problems: examples in the boundary of the cone that are the sum of less than n squares and have common complex roots, and examples of polynomials in the boundary with SOS length larger than the expected one from the dimension.

Practically secure linear-map vector commitment and its applications

The primitive of vector commitment scheme allows a user to commit to an ordered sequence of messages (i.e., a vector) and later open the commitment at any position subset of the vector. The most important and desirable feature of vector commitment schemes is that the size of the opening proof is sublinear in the length of the committed vector. The original vector commitment scheme has now been extended to support several new functionalities like aggregation, updatability and homomorphism, and has applications ranging from verifiable data streaming to stateless cryptocurrency. Among these extensions, the linear-map vector commitment (LVC) scheme enables a user to open a general linear map evaluated on the committed vector, rather than those messages of the committed vector as in the original vector commitment scheme. However, the existing LVC schemes are only proved to be secure under the idealized assumptions, i.e., using the algebraic group model, which might be unpractical in the real world. To this end, we eliminate the use of algebraic group model, and propose a practically secure LVC construction. Our construction achieves practical security by additionally generating degree proofs for polynomials that enable a verifier to check the degree of polynomials publicly. We prove the security of the proposed LVC construction in the standard model under a q-type complexity assumption over bilinear groups. Moreover, we demonstrate how to use the proposed LVC scheme to construct maintainable vector commitments and verifiable data streaming protocols. The theoretical comparison and experimental results indicate that our proposal provides stronger security guarantee, while being competitive in terms of efficiency.

Lyapunov’s stability analysis for first degree polynomial systems, subject to risk-sensitive control

This work presents a novel methodology to verify the stability of first-degree stochastic polynomial systems under a Risk-Sensitive (RS) optimal control using Lyapunov’s stability analysis theory. The so-called Lyapunov’s indirect method is applied to prove its stability when it is combined with the dynamics of Riccati’s gain equation. The proposed methodology guarantees both the exponential stability of deterministic systems and the robustness of stochastic systems. Simulation results demonstrate the robustness, the effectiveness and the feasibility of this proposal.