Single Polynomial Research Articles

On possible limit functions on a Fatou component in non-autonomous iteration

Abstract The possibilities for limit functions on a Fatou component for the iteration of a single polynomial or rational function are well understood and quite restricted. In non-autonomous iteration, where one considers compositions of arbitrary polynomials with suitably bounded degrees and coefficients, one should observe a far greater range of behavior. We show this is indeed the case and we exhibit a bounded sequence of quadratic polynomials which has a bounded Fatou component on which one obtains as limit functions every member of the classical Schlicht family of normalized univalent functions on the unit disc. The proof is based on quasiconformal surgery and the use of high iterates of a quadratic polynomial with a Siegel disc which closely approximate the identity on compact subsets. Careful bookkeeping using the hyperbolic metric is required to control the errors in approximating the desired limit functions and ensure that these errors ultimately tend to zero.

Products and powers of principal symmetric ideals

Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case.

A comparison between physics-based Li-ion battery models

Physics-based electrochemical battery models, such as the Doyle-Fuller-Newman (DFN) model, are valuable tools for simulating Li-ion battery behavior and understanding internal battery processes. However, the complexity and computational demands of such models limit their applicability for battery management systems and long-term aging simulations. Reduced-order models (ROMs), such as the Extended Single Particle Model (ESPM), Single Particle Model (SPM) and Polynomial and Padé approximations, here all referred to as simplifications, lead to faster computational speeds. Choosing the appropriate simplification method for a specific cell type and operating condition is a challenge. This study investigates the simulation accuracy and calculation speed of various simplifications for high-energy (HE) and high-power (HP) batteries at different current loading conditions and compares those to the full-order DFN model. The results indicate that among the ROMs, the ESPM consistently offers the best combination of high computational speed and relatively good accuracy in most conditions in comparison to the full-order DFN model. Among the approximations, higher-order polynomial approximation, third and fourth-order Padé approximation perform the best in terms of accuracy. The higher-order polynomial approximation shows an advantage in terms of computing speed, while the fourth-order Padé approximation achieves the highest overall accuracy among the different approximations.

Analytical computation of stress intensity factor for multi-physics problems

This work presents a methodology for the analytical calculation of the stress intensity factor when the stress distribution on the crack surfaces is non-homogeneous. At first, a polynomial function is used to express the non-homogenous stress distribution. Subsequently, the principle of superposition of effects is applied, and the stress intensity factor is computed by multiplying each polynomial term by its respective geometric factor. Finite element fracture model is used to compute the geometric factor of the single polynomial grade. To explain the method, a spherical body is considered, with central and superficial cracks. Each geometric factor depends on a normalized geometrical parameter (the ratio between the crack length and sphere radius). The proposed methodology is applied to determine the stress intensity factor in the case of a crack driving force caused by diffusive fields, such as the concentration gradient in particles of electrodes active material in lithium-ion batteries. The methodology allows to speed up the fracture computation, then it is used to give electrode design guidelines to limit the fracture likeliness and mechanical degradation in lithium-ion batteries, as well as it is the basis for the development of algorithms assessing the capacity loss and the remaining useful life of lithium-ion batteries in real-time.

A Flexible Hardware Accelerator for Booth Polynomial Multiplier

This article presents a parameterized/flexible hardware accelerator design tailored for the Booth polynomial multiplication method. The flexibility is achieved by allowing users to compute multiplication operations across various operand lengths, reaching up to 212 or 4096 bits. Our optimization strategy involves resource reuse, effectively minimizing the overall area cost of the Booth accelerator design. A comprehensive evaluation compares the proposed multiplier design with several non-digitized bit-serial polynomial multiplication accelerators. Implementation is realized in Verilog HDL using the Vivado IDE tool, featuring diverse operand sizes, and post-place and route assessments are performed on the Xilinx Virtex-7 field-programmable gate array device. For the largest considered operand size of 1024 × 1024, our Booth accelerator utilizes 1434 slices and can operate on a maximum frequency of 523.56 MHz. A single polynomial multiplication operation requires 0.977 μs and the total power consumption is 927 mW. Moreover, a comparison to state-of-the-art accelerators reveals that the proposed flexible accelerator is 1.34× faster in computation time and 1.05× more area-efficient than the recent dedicated polynomial multiplication design. Therefore, the implementation results and comparison to the state of the art show that the proposed accelerator is suitable for a wide range of cryptographic applications.

Compact and efficient KEMs over NTRU lattices

The NTRU lattice is a promising candidate to construct practical cryptosystems, in particular key encapsulation mechanism (KEM), resistant to quantum computing attacks. Nevertheless, there are still some inherent obstacles to NTRU-based KEM schemes when considering integrated performance, taking security, bandwidth, error probability, and computational efficiency as a whole, that is as good as and even better than their {R,M}LWE-based counterparts. In this work, we address the challenges by presenting a new family of NTRU-based KEM schemes, denoted as CTRU and CNTR. By bridging low-dimensional lattice codes and high-dimensional NTRU-lattice-based cryptography with careful design and analysis, to the best of our knowledge, CTRU and CNTR are the first NTRU-based KEM schemes featuring scalable ciphertext compression via only one single ciphertext polynomial, and are the first that can outperform {R,M}LWE-based KEM schemes in terms of integrated performance. For instance, when compared to Kyber, the only KEM scheme currently standardized by NIST, our recommended parameter set CNTR-768 exhibits approximately 12% smaller ciphertext size, when its security is strengthened by (8,7) bits for classical and quantum security respectively, with a significantly lower error probability (2−230 for CNTR-768 vs. 2−164 for Kyber-768). In terms of the state-of-the-art AVX2 implementation of Kyber-768, CNTR-768,achieves a speedup of 2.7X in KeyGen, 3.3X in Encaps, and 1.6X in Decaps, respectively. When compared to the NIST Round 3 finalist NTRU-HRSS, CNTR-768,features 15% smaller ciphertext size, coupled with an improvement of (55,49) bits for classical and quantum security respectively. As for the AVX2 implementation, CNTR-768,outperforms NTRU-HRSS by 26X in KeyGen, 3.0X in Encaps, and 2.2X in Decaps, respectively. Along the way, we develop new techniques for more accurate error probability analysis, and a unified number theoretic transform (NTT) implementation for multiple parameter sets, which may be of independent interest.

Transient Characteristics of the Low-Temperature PEMFC: A Comparative Analysis Using a Physics-Based Model

Low-temperature proton exchange membrane fuel cells (LT-PEMFCs) have experienced a considerable technological leap in the last few decades for transportation and stationary applications. Specifically, the precious metal loading in the catalyst layers (CLs) is notably decreased while enhancing the overall cell performance.1 However, the durability and cost remain major factors that affect the widespread commercialization of the LT-PEMFC. Though this can be achieved with further material development, some existing issues may be resolved by implementing effective design and control strategies. A good understanding of fuel cell water dynamics is required to improve durability and performance. Although some dynamic characteristics can be observed from the experiments, a physics-based model of the LT-PEMFC is necessary to study in-depth transient responses under various conditions like current profiles, operating temperatures, pressure, and humidification levels. Lately, the transient analysis of the fuel cell has gained more attention to elucidate the complex transport phenomena.2 Several transient models of the LT-PEMFC are reported to capture the two-phase behavior inside the cell. Usually, these models are 1D, 2D, or single channel 3D with assumptions like isothermal operation, vapor-equilibrated membrane, simplified reaction kinetics, and more.3–5 Hence, they do not provide spatio-temporal insights into the cell performance containing water dynamics with various flow configurations.The current study focuses on a transient physics-based model of the LT-PEMFC for dynamic analysis across its thickness and flow channels. The membrane used in LT-PEMFC requires humidification to enhance the ionic conductivity and function properly.6 This study considers Schroeder’s paradox for the membrane, which exhibits a different water uptake behavior in the presence of saturated water vapor and liquid water.7 The water uptake property is strongly dependent on the temperature. Therefore, we have developed a single polynomial expression for the membrane water uptake, including temperature and water activity dependency. Using the governing equations & empirical correlations from the numerical and experimental studies, we develop a comprehensive 3D, multiphase, non-isothermal, transient physics-based model of the LT-PEMFC that can closely approximate the liquid water transport under various operating conditions. It is a Multiphysics model that couples electrochemical reactions, porous media flow, transport of reactant species, liquid water transport, heat generation, charge transfer, and ionomer water transport. COMSOL Multiphysics is used to solve the model. It provides a spatio-temporal water distribution inside the membrane, CL, microporous layer, gas diffusion layer, and flow channels. Experiments are performed on 32 cm2 LT-PEMFC with the hybrid flow configuration at 75 °C and 80% RH to validate the physics-based model in the steady state condition. The validation study suggests that the present multiphase model estimates the LT-PEMFC performance with an acceptable RMS current density error of 15 mA/cm2. The simulations are operated in the Galvano-dynamic mode, and the corresponding LT-PEMFC response for the hybrid configuration is shown in Figure 1(a) at various operating pressures. Also, the preliminary results of the liquid water saturation under the channel and rib are incorporated in Figure 1(b). It shows that the water accumulation under the rib is more than the channel due to lower local temperature and weaker convection under the rib, promoting water vapor condensation. Further, the comparison between hybrid and parallel configurations is performed and illustrated in this presentation. Another objective is to study how quickly the liquid water saturation and membrane water content react upon changing the load cycle, RH, operating temperature, and flow configuration. This work is an important step toward understanding the impact of various flow configurations and operating conditions on the water dynamics and membrane water content inside the LT-PEMFC. The developed physics-based model would be extended further by considering the membrane deformation due to the shrinking-swelling under the humidification and thermal cycling, which is critical for the performance and degradation.

The Mathieu conjecture for [formula omitted] reduced to an abelian conjecture

We reduce the Mathieu conjecture for SU(2) to a conjecture about moments of Laurent polynomials in two variables with single variable polynomial coefficients.

A novel homomorphic polynomial public key encapsulation algorithm

Background: One of the primary drivers in development of novel quantum-safe cryptography techniques is the ongoing National Institute of Standards and Technology (NIST) Post-Quantum Cryptography (PQC) competition, which aims to identify quantum-safe algorithms for standardization. Although NIST has recently announced candidates to be standardized, the development of novel PQC algorithms remains desirable to address the challenges of quantum computing. Furthermore, to enhance security and improve performance. Methods: This paper introduces a novel public key encapsulation algorithm that incorporates an additional layer of encryption during key construction procedure, through a hidden ring. This encryption involves modular multiplication over the hidden ring using a homomorphism operator that is closed under addition and scalar multiplication. The homomorphic encryption key is comprised of two values - one used to create the hidden ring and the other to form an encryption operator. This homomorphic encryption can be applied to any polynomials during key construction over a finite field with their coefficients considered private. Particularly, the proposed homomorphic encryption operator can be applied to the public key of the Multivariate Public Key Cryptography schemes (MPKC) to hide the structure of its central map construction. Results: This paper presents a new variant of the MPKC with its public key encrypted using the proposed homomorphic operator. This novel scheme is called the Homomorphic Polynomial Public Key (HPPK) algorithm, which simplifies MPKC central map to two multivariate polynomials constructed from polynomial multiplications. The HPPK algorithm employs a single polynomial vector for the plaintext and a multi-variate noise vector associated with the central map. In contrast, in MPKC, a single multivariate vector is created by segmenting the secret plaintext over a small finite field. The HPPK algorithm is Indistinguishability Under Chosen-Plaintext Attack (IND-CPA) secure, and its classical complexity for cracking is exponential in the size of the prime field GF(p).

Active learning-based domain adaptive localized polynomial chaos expansion

The paper presents a novel methodology to build surrogate models of complicated functions by an active learning-based sequential decomposition of the input random space and construction of localized polynomial chaos expansions, referred to as domain adaptive localized polynomial chaos expansion (DAL-PCE). The approach utilizes sequential decomposition of the input random space into smaller sub-domains approximated by low-order polynomial expansions. This allows the approximation of functions with strong nonlinearities, discontinuities, and/or singularities that often appear in dynamical systems. Decomposition of the input random space and local approximations alleviates the Gibbs phenomenon for these types of problems and confines error to a very small vicinity near the non-linearity. The global behavior of the surrogate model is therefore significantly better than existing methods, as shown in numerical examples, including an engineering dynamical system exhibiting discontinuous response. The whole process is driven by an active learning routine that uses the recently proposed Θ criterion to assess local variance contributions (Novák et al., 2021). The proposed approach balances both exploitation of the surrogate model and exploration of the input random space and thus leads to efficient and accurate approximation of the original mathematical model. The numerical results show the superiority of the DAL-PCE in comparison to (i) a single global polynomial chaos expansion and (ii) the recently proposed stochastic spectral embedding (SSE) method (Marelli et al., 2021) developed as an accurate surrogate model and which is based on a similar domain decomposition process. This method represents a general framework upon which further extensions and refinements can be based and which can be combined with any technique for non-intrusive polynomial chaos expansion construction.

Analytical computation of stress intensity factor for active material particles of lithium ion batteries

The stress intensity factor is a widely used parameter in linear elastic fracture mechanics to assess the stress field near the crack tip, and it is usually defined as the product between the remote stress, the square root of the crack length, and a geometric factor depending on the geometry of the test specimen, the size and location of the crack. However, this well-established expression cannot be used in case of non-constant stress distribution on crack surfaces, typically resulting from diffusive field. This work presents an analytical procedure to compute the stress intensity factor due to any kind of stress distribution that can be expressed as a polynomial. Firstly, the non-constant stress distribution is expressed as a polynomial. Then, the stress intensity factor is computed according to the principle of superposition of effects, as the sum of the stress components of each single polynomial grade and the corresponding geometric factors. The geometric factors for sphere with central and superficial cracks are determined using a finite element model, and closed-form expressions for these geometric factors are provided as functions of a normalized geometric parameter. These functions can be used in case of any stress loading and spherical geometry. The analytical procedure is specifically applied to assess the stress intensity factor caused by stress resulting from lithium diffusion in active material particles of lithium ions batteries electrodes. The results are compared with those obtained using a multiphysics finite element fracture model, showing good agreement and demonstrating the accuracy of the proposed analytical procedure. Then, the procedure presented in this work enables avoiding expensive multiphysics simulations and can be used to develop fast, accurate, and computationally efficient lithium ion batteries degradation models for the online estimation of the capacity decay with charge/discharge cycles.

Galois groups and prime divisors in random quadratic sequences

AbstractGiven a set $S=\{x^2+c_1,\dots,x^2+c_s\}$ defined over a field and an infinite sequence $\gamma$ of elements of S, one can associate an arboreal representation to $\gamma$ , generalising the case of iterating a single polynomial. We study the probability that a random sequence $\gamma$ produces a “large-image” representation, meaning that infinitely many subquotients in the natural filtration are maximal. We prove that this probability is positive for most sets S defined over $\mathbb{Z}[t]$ , and we conjecture a similar positive-probability result for suitable sets over $\mathbb{Q}$ . As an application of large-image representations, we prove a density-zero result for the set of prime divisors of some associated quadratic sequences. We also consider the stronger condition of the representation being finite-index, and we classify all S possessing a particular kind of obstruction that generalises the post-critically finite case in single-polynomial iteration.

Hamiltonian Cycle Parameterized by Treedepth in Single Exponential Time and Polynomial Space

Hamiltonian Cycle Parameterized by Treedepth in Single Exponential Time and Polynomial Space

A new approach for designing the Shewhart-type control charts with generalized sensitizing rules

Two commonly used approaches for developing a Shewhart-type control chart with sensitizing rules are; the Markov Chain approach (MCA) and the single polynomial approach (SPA). The MCA for designing a control chart involves linear equations based on the choice of sensitizing rules. The MCA often becomes complex as the system of linear equations increases in size. The existing SPA for designing a control chart with sensitizing rules results in inconsistent in-control run length properties (RLP). To overcome the limitations of existing approaches, we present a new SPA for designing the Shewhart-type control charts with sensitizing rules. Furthermore, the new SPA has an interesting relationship with classical geometric distribution, generalized geometric distribution, and MCA. To show the significance of the new SPA, we have implemented it in designing the two-sided and one-sided mean control charts with sensitizing rules. The performance of the proposed and existing control charts is evaluated and compared using various RLP. The results reveal that the in-control RLP of the new SPA based mean control charts are sustained at their desired level. The out-of-control RLP is elaborated to highlight an optimal choice of sensitizing rules and other factors. Extensive comparative analysis shows that the new SPA based mean control charts have advantages over sustained behavior and computational reduction. Lastly, a real-life example of proposed methods is included using data on turnaround time of complete blood count (CBC) test results.

A new scheme for approximating the weakly efficient solution set of vector rational optimization problems

In this paper, we provide a new scheme for approximating the weakly efficient solution set for a class of vector optimization problems with rational objectives over a feasible set defined by finitely many polynomial inequalities. More precisely, we present a procedure to obtain a sequence of explicit approximations of the weakly efficient solution set of the problem in question. Each approximation is the intersection of the sublevel set of a single polynomial and the feasible set. To this end, we make use of the achievement function associated with the considered problem and construct polynomial approximations of it over the feasible set from above. Remarkably, the construction can be converted to semidefinite programming problems. Several nontrivial examples are designed to illustrate the proposed new scheme.

Inverse Kinematics of a Class of 6R Collaborative Robots with Non-Spherical Wrist

The spread of cobotsin common industrial practice has led constructors to prefer the development of collaborative features that are necessary to prevent injuries to operators over the realization of simple kinematic structures for which the joints-to-workspace mapping is well known. An example is given by the replacement in serial robots of spherical wrists with safer solutions, where the danger of crushing and shearing is intrinsically avoided. Despite this tendency, the kinematic map between actuated joints and the Cartesian workspace remains of paramount importance for robot analysis and programming, deserving the attention of the research community. This paper proposes a closed-form solution for the inverse kinematics of a class of 6R robotic arms with six degrees of freedom and non-spherical wrists. The solutions are worked out by a single polynomial, of minimum degree, in terms of one of the positioning parameters chosen for the description of the robot posture. The roots of such a polynomial are then back-substituted to determine all the remaining unknowns. A numerical example is finally shown to verify the validity of the proposed implementation for a commercial collaborative robot.

REKONSTRUKSI MODEL PROGRAM NONLINIER DENGAN FUNGSI POLINOMIAL MENJADI BENTUK PROGRAM KUADRATIK

The purpose of this research to revisits methods that are more effective in nonlinear Optimization with single variable polynomial functions at high degrees. Models with linear objective functions and constraint functions are Polynomials of third, fourth and fifth degree reconstructed into subproblems that are easier to solve, namely quadratic programs, using bilinear Auxiliary Functions and solved by MATLAB simulations. The method used is the development of Tawarmalani & Sahinidis' research regarding relaxation with Auxiliary Functions. Examples of nonlinear Optimization with polynomial functions are also given to illustrate the implementation of this algorithm. The results of the research show that the application of the development reconstruction method produces a global solution that is no better than the solution to the original problem so that it is not an effective alternative method to use.

Planar and spherical four-bar linkage [formula omitted] algebraic input–output equations

The algebraic polynomial input–output (IO) equations relating any two of the relative joint displacement parameters, called vi and vj, between any of the six distinct pairs of rigid links in arbitrary planar and spherical four-bar mechanisms are derived. First, the forward kinematics transformation matrices of the corresponding serial kinematic chains are computed in terms of their Denavit–Hartenberg parameters, but with all angles converted to tangent half-angle parameters. These matrices are mapped to their corresponding Study soma coordinates. The serial kinematic chain is closed by equating the soma coordinates to the identity array. Algebraic polynomial elimination methods are then used to obtain a single polynomial in terms of only the design and the selected IO joint displacement parameters. This yields six independent algebraic IO Equations for each of the planar and spherical 4R linkages; the same techniques are applied to derive six additional algebraic IO equations for each of the RRRP and PRRP planar linkages providing a catalogue of 24. The utility of these IO equation sets is demonstrated via discussion of the associated mobility and design parameter spaces.

Predicting Electricity Consumption in the Kingdom of Saudi Arabia

Forecasting energy consumption in Saudi Arabia for the period from 2020 until 2030 is investigated using a two-part composite model. The first part is the frontier, and the second part is the autoregressive integrated moving average (ARIMA) model that helps avoid the large disparity in predictions in previous studies, which is what this research seeks to achieve. The sample of the study has a size of 30 observations, which are the actual consumption values in the period from 1990 to 2019. The philosophy of this installation is to reuse the residuals to extract the remaining values. Therefore, it becomes white noise and the extracted values are added to increase prediction accuracy. The residuals were calculated and the ARIMA (0, 1, 0) model with a constant was developed both of the residual sum of squares and the root means square errors, which were compared in both cases. The results demonstrate that prediction accuracy using complex models is better than prediction accuracy using single polynomial models or randomly singular models by an increase in the accuracy of the estimated consumption and an improvement of 18.5% as a result of the synthesizing process, which estimates the value of electricity consumption in 2030 to be 575 TWh, compared to the results of previous studies, which were 365, 442, and 633 TWh.

One single polynomial function-based charge voltage curve and its application to estimate the states of lithium-ion batteries in series

Accurately estimating the state of a battery is crucial to its safety and remaining life prediction. This paper proposes a mathematical treatment, which enables us to estimate both the state of charge (SOC) and the state of health (SOH) of each battery in a pack. The charge voltage curve (CVC) of a fresh single battery, charged at 0.2C rate, is numerically fitted by a polynomial function. Since the internal resistance of a battery varies with the charging current and the SOH, the current-interruption method is adopted to determine the amount of the translation and scaling of the reference CVC under different conditions. Our experiment confirms that the generalized CVC can be successfully applied to various scenarios of different charge rates and SOHs. With the CVC of a battery, the adaptive moment estimation, accompanied by a 30-min (or 45-min) charge voltage data, is employed to find out the current SOC and SOH of one single battery or each of the battery in a series pack. The experiment in nine different cases demonstrates that the estimation of the current SOC and SOH by this CVC approach is excellent and the maximum error could be lower than 3.6 %.