Parametric Polynomial Research Articles

Positive dimensional parametric polynomial systems, connectivity queries and applications in robotics

In this paper we introduce methods and algorithms that will help us solve connectivity queries of parameterized semi-algebraic sets. Answering these connectivity queries is applied in the design of robotic structures having similar kinematic properties (e.g. topology of the kinematic-singularity-free space). From these algorithms one also obtain solutions to connectivity queries of a specific parameter which is in turn related to kinematic-singularity free path-planning of a specific manipulator belonging to the family of robots with these properties; i.e. we obtain paths joining two given singularity free configurations lying in the same connected component of the singularity-free space.We prove in the paper how one reduces the problems related to connectivity queries of parameterized semi-algebraic sets to closed and bounded semi-algebraic sets. We then design an algorithm using computer-algebra methods for “solving” positive dimensional polynomial system depending on parameters. The meaning of solving here means partitioning the parameter space into semi-algebraic components over which the number of connected components of the semi-algebraic set defined by the input system is invariant. The complexity of this algorithm is singly exponential in the dimension of the ambient space. The algorithm scales enough to analyze automatically the family of UR-series robots.Finally we provide manual analysis of the family of UR-series robots, proving that the number of connected components of the complementary of kinematic singularity set of a generic UR-robot is eight.

An extended GCRD algorithm for parametric univariate polynomial matrices and application to parametric Smith form

The first extended greatest common right divisor (GCRD) algorithm for parametric univariate polynomial matrices is presented. The starting point of this GCRD algorithm is the free property of submodules over univariate polynomial rings. We convert the computation of GCRDs to that of free basis for modules and prove that a free basis of the submodule generated by row vectors of input matrices forms just a GCRD of these matrices. The GCRD algorithm is obtained by computing a minimal Gröbner basis for the corresponding submodule since a minimal Gröbner basis of submodules is a free basis for univariate cases. While the key idea of extended algorithm is to construct a special module by adding the unit vectors which can record the representation coefficients. This method based on modules can be naturally generalized to the parametric case because of the comprehensive Gröbner systems for modules. As a consequence, we obtain an extended GCRD algorithm for parametric univariate polynomial matrices. More importantly, we apply the proposed extended GCD algorithm for univariate polynomials (as a special case of matrices) to the computation of Smith normal form, and give the first algorithm for reducing a univariate polynomial matrix with parameters to its Smith normal form.

Progressive approximation of bound states by finite series of square-integrable functions

We use the “tridiagonal representation approach” to solve the time-independent Schrödinger equation for bound states in a basis set of finite size. We obtain two classes of solutions written as a finite series of square integrable functions that support a tridiagonal matrix representation of the wave operator. The differential wave equation becomes an algebraic three-term recursion relation for the expansion coefficients of the series, which is solved in terms of finite polynomials in the energy and/or potential parameters. These orthogonal polynomials contain all physical information about the system. The basis elements in configuration space are written in terms of either the Romanovski–Bessel polynomial or the Romanovski–Jacobi polynomial. The maximum degree of both polynomials is limited by the polynomial parameter(s). This makes the size of the basis set finite but sufficient to give a very good approximation of the bound state wavefunctions that improves with an increase in the basis size.

Two-Dimensional Parametric Polynomial Chaotic System

When used in engineering applications, most existing chaotic systems may have many disadvantages, including discontinuous chaotic parameter ranges, lack of robust chaos, and easy occurrence of chaos degradation. In this article, we propose a two-dimensional (2-D) parametric polynomial chaotic system (2D-PPCS) as a general system that can yield many 2-D chaotic maps with different exponent coefficient settings. The 2D-PPCS initializes two parametric polynomials and then applies modular chaotification to the polynomials. Setting different control parameters allows the 2D-PPCS to customize its Lyapunov exponents in order to obtain robust chaos and behaviors with desired complexity. Our theoretical analysis demonstrates the robust chaotic behavior of the 2D-PPCS. Two illustrative examples are provided and tested based on numeral experiments to verify the effectiveness of the 2D-PPCS. A chaos-based pseudorandom number generator is also developed to illustrate the applications of the 2D-PPCS. The experimental results demonstrate that these examples of the 2D-PPCS can achieve robust and desired chaos, have better performance, and generate higher randomness pseudorandom numbers than some representative 2-D chaotic maps.

On the Generalized Covering Radii of Reed-Muller Codes

We study generalized covering radii, a fundamental property of linear codes that characterizes the trade-off between storage, latency, and access in linear data-query protocols such as PIR. We prove lower and upper bounds on the generalized covering radii of Reed-Muller codes, as well as finding their exact value in certain extreme cases. With the application to linear data-query protocols in mind, we also construct a covering algorithm that gets as input a set of points in space, and find a corresponding set of codewords from the Reed-Muller code that are jointly not farther away from the input than the upper bound on the generalized covering radius of the code. We prove that the algorithm runs in time that is polynomial in the code parameters.

Reducing Energy Consumption of Pressure Sensor Calibration Using Polynomial HyperNetworks with Fourier Features

Our research aims to reduce the cost of pressure sensor calibration through machine learning. Pressure sensor calibration is a standard process whereby freshly manufactured pressure sensors are subjected to various controlled temperature and pressure setpoints to compute a mapping between the sensor's output and true pressure. Traditionally this mapping is calculated by fitting a polynomial with calibration data. Obtaining this data is costly since a large spectrum of temperature and pressure setpoints are required to model the sensor's behavior. We present a machine learning approach to predict a pre-defined calibration polynomial's parameters while requiring only one-third of the calibration data. Our method learns a pattern from past calibration sessions to predict the calibration polynomial's parameters from partial calibration setpoints for any newly manufactured sensor. We design a novel polynomial hypernetwork coupled with Fourier features and a weighted loss to solve this problem. We perform extensive evaluations and show that the current industry-standard method fails under similar conditions. In contrast, our approach saves two-thirds of the calibration time and cost. Furthermore, we conduct comprehensive ablations to study the effect of Fourier mapping and weighted loss. Code and a novel calibration dataset validated by calibration engineers are also made public.

Quantum Circuit Transformation: A Monte Carlo Tree Search Framework

In the noisy intermediate-scale quantum era, quantum processing units suffer from, among others, highly limited connectivity between physical qubits. To make a quantum circuit effectively executable, a circuit transformation process is necessary to transform it, with overhead cost the smaller the better, into a functionally equivalent one so that the connectivity constraints imposed by the quantum processing unit are satisfied. Although several algorithms have been proposed for this goal, the overhead costs are often very high, which degenerates the fidelity of the obtained circuits sharply. One major reason for this lies in that, due to the high branching factor and vast search space, almost all of these algorithms only search very shallowly, and thus, very often, only (at most) locally optimal solutions can be reached. In this article, we propose a Monte Carlo Tree Search (MCTS) framework to tackle the circuit transformation problem, which enables the search process to go much deeper. The general framework supports implementations aiming to reduce either the size or depth of the output circuit through introducing SWAP or remote CNOT gates. The algorithms, called MCTS-Size and MCTS-Depth , are polynomial in all relevant parameters. Empirical results on extensive realistic circuits and IBM Q Tokyo show that the MCTS-based algorithms can reduce the size (respectively, depth) overhead by, on average, 66% (respectively, 84%) when compared with t \( \left| {\mathrm{ket}} \right\rangle \) , an industrial-level compiler.

АНАЛИЗ СПУТНИКОВЫХ ИЗМЕРЕНИЙ ВЕГЕТАЦИОННОГО ИНДЕКСА В ПРЕДГОРНЫХ И РАВНИННЫХ РАЙОНАХ РСО-АЛАНИЯ

To analyze the seasonal and long-term variability of the normalized relative vegetation index NDVI, the multichannel MODIS spectroradiometer measurements with a 10x10 km resolution over a 20-year period of time are used. On the territory of the RNO-Alania in the foothill areas and on the plains near the border of the Stavropol Territory five geographical locations are selected, located at different distances from the Northern spurs of the Caucasian Ridge. For each geographical location the sets of time series are obtained for: 16-day NDVI composites, daily temperature at an altitude of 2 m and precipitation for the 20-years period from 01.01.2000 to 31.12.2019. By calculating the integral characteristic for an annual period of time for every location, the series of 20 annual values are obtained. According to the obtained series, three 20-year trends are built and the trend parameters of a first-order polynomial are calculated using the least squares method.
 The NDVI trend slope switch is observed in the Mozdok-Russkoe location for transition from the time period 2000–2009 to the period 2010–2019, when the slope changes from positive to negative. Similar NDVI trend slope switches are observed for other locations. This may be due changing the level of soil moisture over time, taking into account mainly the downward trends in precipitation. The trend slope of the NDVI integral characteristic 5–15 % increases with the increasing of the distance from the mountain ridge. The temperature trends for all locations are ascending, that indicates an average increase of the number of warm days from year to year. At the same time, the maximum temperature during the summer period does not significantly increase. The trend of the integral characteristic for the amount of precipitation is mainly downward. However, its specific values have a fairly large spread, which means that the amount of precipitation varies significantly from year to year. In addition, the amount of precipitation decreases with distance from mountain ridges.
 These factors should be taken into account when choosing agricultural products for cultivation and planning additional land-reclamations in the flat and foothill areas on the south of the Russian Federation.

Natural Discrete One Parameter Polynomial Exponential Family of Distributions and the Application

Natural Discrete One Parameter Polynomial Exponential Family of Distributions and the Application

Classification of 3-Degree-of-Freedom 3-UPU Translational Parallel Mechanisms Based on Constraint Singularity Loci Using Gröbner Cover

Abstract A 3-degree-of-freedom (DOF) 3-UPU translational parallel mechanism (TPM) is one of the typical TPMs. Despite comprehensive studies on 3-UPU TPMs in which the joint axes on the base and the moving platform are coplanar, only a few 3-UPU TPMs with skewed base and moving platform have been proposed, and the impact of link parameters on constraint singularity loci of such TPMs has not been systematically investigated. The advances in computing comprehensive Gröbner system (CGS) or Gröbner cover of parametric polynomial systems provide an efficient tool for solving this problem. This paper presents a systematic classification of 3-UPU TPMs with skewed base and moving platform based on constraint singularity loci. First, the constraint singularity equation of a 3-UPU TPM is derived. Using Gröbner cover, the 3-UPU TPMs are classified into 12 types. Finally, a novel 3-UPU TPM is proposed. Reconfiguration analysis shows that unlike most existing 3-UPU TPMs which can transit from a 3-DOF translational mode to two or more 3-DOF operation modes, the proposed 3-UPU TPM can only transit from a 3-DOF translational mode to one general 3-DOF operation mode. The singularity locus divides the workspace of this 3-UPU TPM into two constraint singularity-free regions. As a by-product, a 3-UPU parallel mechanism that the moving platform can undergo 3-DOF translation and 1-DOF infinitesimal rotation is revealed. This work provides a solid foundation for the design of 3-UPU TPMs and a starting point for the classification of 3-UPU parallel mechanisms.

Mathematical analysis of the van der Waals equation

The parametric cubic van der Waals polynomial pV3−(RT+bp)V2+aV−ab is analysed mathematically and some new generic features (theoretically, for any substance) are revealed: the temperature range for applicability of the van der Waals equation, T>a/(4Rb), and the isolation intervals, at any given temperature between a/(4Rb) and the critical temperature 8a/(27Rb), of the three volumes on the isobar–isotherm: 3b/2<VA≤3b, 2b<VB<(3+5)b, and 3b<VC<b+RT/p. The unstable states of the van der Waals model have also been generically localized: they lie in an interval within the isolation interval of VB. In the case of a unique intersection point of an isotherm with an isobar, the isolation interval of this unique volume is also determined. A discussion on finding the volumes VA,B,C, on the premise of Maxwell’s hypothesis, is also presented.

SPPGKM: A secure polynomial function powered group key management scheme for dynamic user environments in cloud

AbstractIn this modern era of smart cities supported by digital locker facilities, e‐governance, e‐healthcare, e‐banking, e‐learning systems, and other digital systems, People use mobile phones to upload and share sensitive documents with others. Previous group key management schemes suffer from problems such as expensive double encryptions, the need for re‐encryption during every user leave operation and becoming vulnerable to attacks. Hence, to address these three issues, this manuscript introduces a pairing based cryptography based computationally efficient protocol powered by polynomial equations. First, to reduce computational overhead during group communication, the document is encrypted only once with minimal complexity before uploaded to cloud. Second, during a user leave or join operation, the re‐encryption of documents in cloud is avoided in the group manager side by tactically relieving the group key parameters from the polynomial parameters. Finally, the security analysis shows that the careful handling of secure parameters of users averts the collusion attack, man‐in‐the‐middle attack, message modification attacks, and all other possible attacks. The implementation results portray the capacity of this work to handle secure group communications in cloud storage.

Fitting the seasonal distribution of wind direction in central region of Kerala using Bernstein Polynomials

Non parametric Bernstein polynomial estimator for circular data was applied for fitting daily wind directions. Since Thrissur District of Kerala faces a special type of wind called Vruschikakkaatu in November to February season and there is a decreases of such wind speed year after year, an analysis was conducted to check whether there is variation in wind direction. The seasonwise distribution of wind directions during morning (7.25 am) and evening (2.25 pm) for the period from 1983 to 1997 and 2003 to 2017 were separately fitted. Study revealed there is no change in the seasonal distributions.&#x0D;

Robust 2D and 3D images zero - watermarking using dual Hahn moment invariants and Sine Cosine Algorithm.

In this paper, we initially provide significant improvements on the computational aspects of dual Hahn moment invariants (DHMIs) in both 2D and 3D domains. These improvements ensure the numerical stability of DHMIs for large-size images. Then, we propose an efficient method for optimizing the local parameters of dual Hahn polynomials (DHPs) when computing DHMIs using the Sine-Cosine Algorithm (SCA). DHMIs optimized via SCA are used to propose new and robust zero-watermarking scheme applied to both 2D and 3D images. On one hand, the simulation results confirm the efficiency of the proposed computation of 2D and 3D DHMIs regarding the numerical stability and invariability. Indeed, the proposed computation method of 2D DHMIs allows to reach a relative error (RE) of the order ≈10−10 for images of size 1024 × 1024 with an execution time improvement ratio exceeds 70% (ETIR ≥ 70%), which validates the efficiently of the proposed computation method. On the other hand, the simulation and comparison outcomes clearly demonstrate the robustness of the proposed zero-watermarking scheme against various geometric attacks (translation, rotation, scaling and combined transformations), as well as against other common 2D and 3D image processing attacks (compression, filtering, noise addition, etc.).Graphical abstract

Coordinated trajectory planning of a dual-arm space robot with multiple avoidance constraints

This paper presents a coordinated trajectory planning method for a free-floating dual-arm space robot system. The research purpose of this paper is to enable a space robot to complete the specified goals while avoiding collision with an obstacle by planning the motion of the joints. The coordinated trajectory planning problem for a space robot is transformed into an optimization problem to find the optimal trajectory of the base and manipulators by using particle swarm optimization (PSO). Since the optimization method is used to find an optimal trajectory, it is unnecessary to calculate the position of the base and the inverse kinematics of the joints. Therefore, the kinematic singular problem in seeking the inverse solution can be avoided. The joint trajectories are parametrized by a sine function with seventh-order polynomial parameters. Multiple constraints are added to the objective function as penalty items, and a new increasing strategy for a penalty factor is proposed to balance the penalty and search ability under multiple constraints. Finally, numerical simulations are carried out to verify the effectiveness of the proposed method.

Solid−liquid phase equilibrium for the ternary system (ammonium dihydrogen phosphate + agricultural grade ammonium polyphosphate (degree of polymerization ranged from 1 to 2) + water) at (278.2 and 313.2) K

Water-soluble ammonium polyphosphate (APP) as a promising high phosphorus fertilizer source in agricultural application can be used with ammonium dihydrogen phosphate (MAP) during fertilizer preparation and irrigation. In this work, the pH, density and solubility properties of the phase equilibrium of ternary APP-MAP-water system were systematically determined at the temperature of 278.2–313.2K. A new series of parameters of Simplified Apelblat model, empirical polynomials and Rational 2D function were used to express the solubility data of MAP, solution density and pH value, respectively. The maximum relative deviations of these three models were successively within 1.21, 0.49, and 0.54% and the corresponding model correlation coefficient R2 was close to or greater than 0.9990, which indicating the accuracy and reliability. The experimental results were further verified by observing the effect of APP on MAP supersaturated solution in OptiMax 1001 reactor and through the online detection techniques of FBRM and PVM. With the addition of water-soluble APP can significantly improve the content of P2O5 when it was combined well with MAP into fertilizer application. Therefore, APP can be better used with MAP to improve the utilization rate of phosphorus during irrigation following the basic thermodynamic data and models established in this research.

Efficient optimisation framework for convolutional neural networks with secure multiparty computation

In recent years, deep learning has become an increasingly popular approach to modelling data due to its ability to detect abstract underlying patterns in data. Its practical applications have been limited, however, by data privacy concerns, restricting its use in major sectors such as healthcare and banking. Secure multiparty computation (MPC) is a scheme which allows multiple parties to perform joint computations over private data, while keeping the content of their data secret. MPC can enable privacy-preserving machine learning, however current implementations are rarely applied in practice due to the prohibitively high cost of performing thousands of computations and transmitting data between parties. In this paper we propose a framework incorporating various optimisation approaches from the wider field of privacy-preserving deep learning, including privacy-preserving batch normalisation and polynomial approximation of activation functions, and evaluate their performance when applied to a privacy-preserving convolutional neural network (CNN), discussing the trade-off each offers in terms of their accuracy and efficiency. We experiment with parametric polynomial (PPoly) activations by deriving polynomial approximations to activation functions and allowing the network to tune the coefficients as learning weights. We will show that, in shallow CNNs, the application of batch normalisation in combination with a PPoly activation layer can result in faster convergence, with testing accuracy exceeding that achieved with an unencrypted network, at the cost of longer running times.

N-Dimensional Polynomial Chaotic System With Applications

Designing high-dimensional chaotic maps with expected dynamic properties is an attractive but challenging task. The dynamic properties of a chaotic system can be reflected by the Lyapunov exponents (LEs). Using the inherent relationship between the parameters of a chaotic map and its LEs, this paper proposes an <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-dimensional polynomial chaotic system (<inline-formula> <tex-math notation="LaTeX">$n\text{D}$ </tex-math></inline-formula>-PCS) that can generate <inline-formula> <tex-math notation="LaTeX">$n\text{D}$ </tex-math></inline-formula> chaotic maps with any desired LEs. The <inline-formula> <tex-math notation="LaTeX">$n\text{D}$ </tex-math></inline-formula>-PCS is constructed from <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> parametric polynomials with arbitrary orders, and its parameter matrix is configured using the preliminaries in linear algebra. Theoretical analysis proves that the <inline-formula> <tex-math notation="LaTeX">$n\text{D}$ </tex-math></inline-formula>-PCS can produce high-dimensional chaotic maps with any desired LEs. To show the effects of the <inline-formula> <tex-math notation="LaTeX">$n\text{D}$ </tex-math></inline-formula>-PCS, two high-dimensional chaotic maps with hyperchaotic behaviors were generated. A microcontroller-based hardware platform was developed to implement the two chaotic maps, and the test results demonstrated the randomness properties of their chaotic signals. Performance evaluations indicate that the high-dimensional chaotic maps generated from <inline-formula> <tex-math notation="LaTeX">$n\text{D}$ </tex-math></inline-formula>-PCS have the desired LEs and more complicated dynamic behaviors compared with other high-dimensional chaotic maps. In addition, to demonstrate the applications of <inline-formula> <tex-math notation="LaTeX">$n\text{D}$ </tex-math></inline-formula>-PCS, we developed a chaos-based secure communication scheme. Simulation results show that <inline-formula> <tex-math notation="LaTeX">$n\text{D}$ </tex-math></inline-formula>-PCS has a stronger ability to resist channel noise than other high-dimensional chaotic maps.

Trajectory Planning For Car-like Robots Through Curve Parametrization And Genetic Algorithm Optimization With Applications To Autonomous Parking

Parallel parking a car is a difficult task and may be frustrating and stressful for the driver, while commonly causes traffic jam. One way to mitigate such negative effects is to provide vehicles with self-driving capabilities. As a cornerstone of a mobile robot's ability to move autonomously stands trajectory planning, which despite many works in the literature, is still considered an open problem especially with regards to nonholonomic vehicles such as car-like robots. Based on this scenario, this work presents a trajectory planning algorithm to parallel park car-like mobile robots based on polynomial parametrization and genetic algorithm optimization. The aim is to define a law of motion to lead the vehicle from an initial pose near a parking space to a final pose within the latter in a smooth way, with no interruption and avoiding any obstacles in the way. Simulation results validate the feasibility of the proposed algorithm which lays the foundation to broader studies.

Heuristics to sift extraneous factors in Dixon resultants

Dixon resultant is an efficient and practical method for simultaneously eliminating many variables from a parametric polynomial system P with coefficients in a field. It has been widely used in a variety of scientific fields including automated theorem proving, biological systems, computer vision, robot kinematics, and so on. However, Dixon resultant method is subjected to extraneous factors likewise other resultant methods. These extraneous factors are undesirable and they create the troublesome problems in certain applications. In other words, we compute the Dixon resultant possibly multiplied with some extraneous factors. Therefore, there is a need to develop or construct techniques that can eliminate, or at least reduce the number of extraneous factors from the Dixon resultant. Given a set of factors in Dixon resultant, we present a heuristic method to sift extraneous factors. It relies on the specialized system P by using the specialization of the parameters in the polynomial systems. Meanwhile, it needs to obtain the regular chain from the specialized system P. From this approach, the parallelization of the method arises naturally. By using the specialization formula this principle can be generalized to other resultant methods. We report on some benchmark examples of our algorithm sifting extraneous factors in Dixon resultants.