Algebraic System Research Articles

Tropical Modeling of Battery Swapping and Charging Station

We propose and investigate a queueing model of a battery swapping and charging station (BSCS) for electric vehicles (EVs). A new approach to the analysis of the queueing model is developed, which combines the representation of the model as a stochastic dynamic system with the use of the methods and results of tropical algebra, which deals with the theory and applications of algebraic systems with idempotent operations. We describe the dynamics of the queueing model by a system of recurrence equations that involve random variables (RVs) to represent the interarrival time of incoming EVs. A performance measure for the model is defined as the mean operation cycle time of the station. Furthermore, the system of equations is represented in terms of the tropical algebra in vector form as an implicit linear state dynamic equation. The performance measure takes on the meaning of the mean growth rate of the state vector (the Lyapunov exponent) of the dynamic system. By applying a solution technique of vector equations in tropical algebra, the implicit equation is transformed into an explicit one with a state transition matrix with random entries. The evaluation of the Lyapunov exponent reduces to finding the limit of the expected value of norms of tropical matrix products. This limit is then obtained using results from the tropical spectral theory of deterministic and random matrices. With this approach, we derive a new exact formula for the mean cycle time of the BSCS, which is given in terms of the expected value of the RVs involved. We present the results of the Monte Carlo simulation of the BSCS’s operation, which show a good agreement with the exact solution. The application of the obtained solution to evaluate the performance of one BSCS and to find the optimal distribution of battery packs between stations in a network of BSCSs is discussed. The solution may be of interest in the case when the details of the underlying probability distributions are difficult to determine and, thus, serves to complement and supplement other modeling techniques with the need to fix a distribution.

The exact solutions to the generalized (2+1)-dimensional nonlinear wave equation

Due to the importance of the nonlinear partial differential equations in applied physics and engineering, many mathematicians and physicists are interesting to the nonlinear partial differential equations. One of the main tasks of studying the nonlinear partial differential equations is to obtain the exact solutions for the nonlinear partial differential equations. In this paper, we study the exact solutions of a generalized (2+1)-dimensional nonlinear partial differential equation. According to the modified hyperbolic function method and the traveling transformation method, the generalized (2+1)-dimensional nonlinear partial differential equation is changed into an ordinary differential equation. By taking the homogeneous balance between the highest order nonlinear terms and the highest order derivative terms, we change the differential equation into an algebraic system. By solving the algebraic system, we obtain the solutions for the nonlinear partial differential equation. At last, some solutions and their figures are given by specific parameters. All the solutions conclude the trigonometric function solutions, the positive hyperbolic function solutions, and the hyperbolic trigonometric function solutions. These solutions exhibit the dynamics of nonlinear waves and the solutions are useful for the study on interaction behavior of nonlinear waves in shallow water, plasma, nonlinear optics and Bose–Einstein condensates.

Analytic solution to functional differential equations via Bell’s polynomials

Abstract It is shown how to approximate the solution of functional differential equations in terms of Bell’s polynomials. Some numerical checks are shown, by using the computer algebra system Mathematica © {{}^{\copyright}} .

Nonlinear Dynamics of a New Class of Micro-Electromechanical Oscillators—Open Problems

In this paper, we propose a new class of micro-electromechanical oscillators. Some investigations based on Melnikov’s approach are applied for identifying some chaotic possibilities. We demonstrate also some specialized modules for investigating the dynamics of these oscillators. This will be included as an integral part of a planned much more general Web-based application for scientific computing. It turns out that the theoretical apparatus for studying the circuit implementation (design, fabricating, etc.) of the considered differential model for large parameter values is extremely complex and requires a serious investigation. This is the reason to offer this model to the attention of specialists working in this scientific direction. Some open problems related to the use of existing computer algebraic systems for the study of this class of oscillators for large values of n,m and N are also posed. In general, the entire article is subordinated to this frank conversation with the readers with the sole purpose being the professional upgrading of the specialized modules provided for this purpose in subsequent licensed versions of CAS.

A Type-2 fuzzy hybrid preference optimization methodology for electric vehicle charging station location

This work focuses on a hybrid preference-based electric vehicle charging station location problem, which considers multiple optimization preferences of distribution network operators, charge station owners, and electric vehicle users. The problem is formulated by an uncertain mixed-integer programming model. Due to the multi-fold uncertainty of the charging process, the uncertain model parameters are expressed as Type-2 fuzzy variables (T2-FVs). The critical value-based type reduction method is adopted to handle the high computational complexity. The proposed uncertain model is converted to its equivalent deterministic chance-constrained programming model. The deterministic counterpart is solved by General Algebraic Modeling System (GAMS). At last, numerical simulations are performed to demonstrate the proposed location strategy as well as some sensitivity analyses. The results indicate that for any given parameters, the equivalent deterministic model follows the general form of mixed-integer programming one that can be easily solved by GAMS. The proposed methodology can effectively handle the multi-fold uncertainty of the charging process. Compared with crisp models, the proposed location strategy can provide more robust location decisions for electric vehicle charging stations (EVCSs). In addition, we also found that different interest groups have conflicting preferences for the locations of EVCSs, so considering multiple hybrid optimization preferences is necessary.

New quadratic/serendipity finite volume element solutions on arbitrary triangular/quadrilateral meshes

AbstractBy postprocessing quadratic and eight‐node serendipity finite element solutions on arbitrary triangular and quadrilateral meshes, we obtain new quadratic/serendipity finite volume element solutions for solving anisotropic diffusion equations. The postprocessing procedure is implemented in each element independently, and we only need to solve a 6‐by‐6 (resp. 8‐by‐8) local linear algebraic system for triangular (resp. quadrilateral) element. The novelty of this paper is that, by designing some new quadratic dual meshes, and adding six/eight special constructed element‐wise bubble functions to quadratic/serendipity finite element solutions, we prove that the postprocessed solutions satisfy local conservation property on the new dual meshes. In particular, for any full anisotropic diffusion tensor, arbitrary triangular and quadrilateral meshes, we present a general framework to prove the existence and uniqueness of new quadratic/serendipity finite volume element solutions, which is better than some existing ones. That is, the existing theoretical results are improved, especially we extend the traditional rectangular assumption to arbitrary convex quadrilateral mesh. As a byproduct, we also prove that the new solutions converge to exact solution with optimal convergence rates under and norms on primal arbitrary triangular/quasi–parallelogram meshes. Finally, several numerical examples are carried out to validate the theoretical findings.

Dimension results for extremal-generic polynomial systems over complete toric varieties

We study polynomial systems with prescribed monomial supports in the Cox ring of a toric variety built from a complete polyhedral fan. We present combinatorial formulas for the dimension of their associated subvarieties under genericity assumptions on the coefficients of the polynomials. Using these formulas, we identify at which degrees generic systems in polytopal algebras form regular sequences. Our motivation comes from sparse elimination theory, where knowing the expected dimension of these subvarieties leads to specialized algorithms and to large speed-ups for solving sparse polynomial systems. As a special case, we classify the degrees at which regular sequences defined by weighted homogeneous polynomials can be found, answering an open question in the Gröbner bases literature. We also show that deciding whether a sparse system is generically a regular sequence in a polytopal algebra is hard from the point of view of theoretical computational complexity.

Computational analysis of a class of singular nonlinear fractional multi-order heat conduction model of the human head

The subject of the article is devoted to the development of a matrix collocation technique based upon the combination of the fractional-order shifted Vieta–Lucas functions (FSVLFs) and the quasilinearization method (QLM) for the numerical evaluation of the fractional multi-order heat conduction model related to the human head with singularity and nonlinearity. The fractional operators are adopted in accordance with the Liouville–Caputo derivative. The quasilinearization method (QLM) is first utilized in order to defeat the inherent nonlinearity of the problem, which is converted to a family of linearized subequations. Afterward, we use the FSVLFs along with a set of collocation nodes as the zeros of these functions to reach a linear algebraic system of equations at each iteration. In the weighted L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document} norm, the convergence analysis of the FSVLFs series solution is established. We especially assert that the expansion series form of FSVLFs is convergent in the infinity norm with order O(1K3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}(\\frac{1}{K^3})}$$\\end{document}, where K represents the number of FSVLFs used in approximating the unknown solution. Diverse computational experiments by running the presented combined QLM-FSVLFs are conducted using various fractional orders and nonlinearity parameters. The outcomes indicate that the QLM-FSVLFs produces efficient approximate solutions to the underlying model with high-order accuracy, especially near the singular point. Furthermore, the methodology of residual error functions is employed to measure the accuracy of the proposed hybrid algorithm. Comparisons with existing numerical models show the superiority of QLM-FSVLFs, which also is straightforward in implementation.

Solutions of Time-Space Fractional Partial Differential Equations Using Picard's Iterative Method

Abstract In this paper, we present Picard's iterative method (PIM) for solving time–space fractional partial differential equations, where the derivatives are considered in the Caputo sense. We prove the existence and uniqueness of solutions. Additionally, we demonstrate the versatility of our proposed approach by obtaining exact solutions for a diverse set of equations. This method is user-friendly and directly applicable to any computer algebra system. The proposed method avoids intricate computations associated with the Adomian decomposition method, such as calculating Adomian polynomials, or the requirements of other methods like choosing a homotopy in the homotopy perturbation method, identification and manipulation of the invariant subspace in invariant subspace method or constructing a variational function in the variational iteration method. Thus, the proposed method is a versatile and efficient tool for exploring systems that involve both temporal and spatial fractional derivatives.

Enhanced Security in Public Key Cryptography: A Novel Approach Combining Gaussian Graceful Labeling and NTRU Public Key Cryptosystem

This research explores an encryption system that combines the Nth-degree Truncated Polynomial Ring Unit (NTRU) public key cryptosystem with Gaussian Graceful Labeling. This process assigns distinct labels to a graph's vertices, resulting in successive Gaussian integers. The NTRU method offers enhanced security and efficient key exchange. The communication encryption process depends on integers P, a, and b, with P being the largest prime number in the vertex labeling. The original receivers are the vertex labeling with the largest prime number coefficient, while all other receivers receive messages from the sender. A polynomial algebraic mixing system and a clustering principle based on the abecedarian probability proposition are used in NTRU encryption and decryption. The choice of relatively prime integers p and q in NTRU plays a role in the construction of polynomial rings used for encryption and decryption, with specific choices and properties designed to ensure scheme security.

An optimization method for solving a general class of the inverse system of nonlinear fractional order PDEs

In this paper, we introduce a general class of the inverse system of nonlinear fractional order partial differential equations (GCISNF-PDEs) with initial-boundary and two overdetermination conditions. An optimization method is considered based on the generalized shifted Legendre polynomials (GSLPs) for solving GCISNV-FPDEs. The concept of the fractional order derivatives (F-Ds) is utilized in the Caputo type. Operational matrices (OMs) of classical derivatives and F-Ds of GSLPs are extracted. Making use of GSLPs, OMs, and Lagrange multipliers method, we reduce the given GCISNF-PDEs into an algebraic system of equations. The proposed approach achieves satisfactory results simply for a small number of the novel GSLPs. In this work, two mathematical examples are illustrated to analyse the introduced method convergence and test its validity as well as applicability.

Presentations of diquandles and diquandle coloring invariants for solid torus knots and links

A diquandle is a set equipped with two quandle operations interacting via a kind of distributive laws which come from Reidemeister moves on dichromatic links. This algebraic systems provide coloring invariants for dichromatic links. In this paper, we give explicit constructions of free diquandles and diquandle presentations, and then discuss Tietze transformations for the diquandle presentations. We also introduce the fundamental diquandles for dichromatic links. Particularly, we describe the fundamental diquandles and diquandle counting invariants for knots and links in the solid torus via annulus diagrams. We append the tables of diquandles and dikei’s of orders [Formula: see text].

Multi-polar Q-hesitant Fuzzy Soft Implicative and Positive Implicative Ideals in BCK/BCI-algebras.

This paper focuses on exploring restricted mathematical concepts within the domain of BCK/BCI-algebras, specifically delving into the intricate realm of Multi-polar Q-hesitant fuzzy soft implicative and positive implicative ideals. BCK and BCI-algebras are pivotal structures in mathematical logic and algebraic systems, finding widespread applications in fields like computer science and artificial intelligence. Our contribution lies in the introduction and thorough investigation of the innovative notions of multi-polar Q-hesitant fuzzy soft implicative and positive implicative ideals, uniquely tailored for BCK/BCI-algebras. These ideals exhibit exceptional flexibility in managing uncertain and hesitant information, serving as potent tools for modeling and solvingreal-world problems characterized by imprecise or incomplete data. This study rigorously defines and explores the foundational properties of multi-polar Q-hesitant fuzzy soft implicative ideals, underscoring their relevance and applicability within BCK/BCI-algebras. Additionally, we present the concept of positive implicative ideals, establishing their interconnectedness with multi-polar Q-hesitant fuzzy soft implicative ideals. Our investigation delves into these ideals’ algebraic and logical facets, offering valuable insights into their mutual interactions and significance within the context of BCK/BCI-algebras. To facilitate practical implementation, we develop algorithms and methodologies for identifying and characterizing multi-polar Q-hesitant fuzzy soft implicative and positive implicative ideals. These computational tools enable efficient decision-making in scenariosinvolving uncertainty. Through illustrative examples and case studies, we showcase the potential of these ideals in handling complex, uncertain information, demonstrating their efficacy in aiding problem-solving processes. This research contributes significantly to advancing BCK/BCI-algebra theory by introducing innovative mathematical structures that bridge the gap between fuzzy logic, soft computing, and implicative ideals. The proposed multi-polar Q-hesitant fuzzy soft implicativeand positive implicative ideals open new avenues for addressing real-world problems characterized by imprecision and uncertainty. As such, they represent a valuable addition to the field of algebraic structures and their applications.

Biomechanical properties of nasal tissues

Introduction. Plastic rhinosurgery and augmentation rhinoplasty are very relevant today. Especially in relation to patients with congenital saddle deformity of the nasal dorsum, as well as patients with iatrogenic disorders of the nose shape, which dramatically reduce the quality of human life. There are violations of the function of the nose. The purpose of the work. Purpose. Within the framework of the necessity of performing silicone implantation rhinoplasty, the strength, elastic and hyperelastic properties of the soft tissues of the nasal fascia and periosteum were studied, the dynamic viscosity of the tissues of the nasal fascia and periosteum was evaluated, determining their relaxation properties using elastic, hyperelastic and rheological models. Methods. Linear, bilinear (with two Young modules), exponential, hyperelastic (neohookean, Mooney-Rivlin, Ogden, Yeoh, polynomial and Veronda-Westmann) and elastic-viscous (Maxwell) models of biological tissues were used. We used the Mathcad 15.0 computer algebra system and the universal package of interdisciplinary programs ANSYS Multiphysics Software (version 2022 R2). The accuracy of replacing the properties of real tissues with the results of calculations was calculated on the basis of indicators of descriptive statistics (standard deviation, maximum absolute error, maximum relative error and correlation coefficient). Results. It is proved that the bilinear model for the exact reproduction of the stress-strain curve assumes at the molecular level of tissues an initial linear reaction of elastin fibers, passing at ε=εcr into the final linear reaction of the collagen matrix. It was found that the properties of the periosteum (correlation coefficient R=0.9999) and nasal fascia (correlation coefficient R=0.9999) are described most adequately by the 5-parametric polynomial model of the 2nd order and the Yeoh model of the 3rd order (R=0.999 and R=1, respectively), the least accurately by the simple 1-parameter neohookean model (R=0.898 for the periosteum and R=0.905 for the fascia). Among elastic models, the exponential dependence characterizes the behavior of nasal materials quite well. Conclusion. Biomechanically verified that the periosteum is thicker, stronger and more rigid than the fascia. The established very high viscosity values indicate the slowness of relaxation processes in tissues. The periosteum relaxes tension more slowly, which improves the result of implant fixation. On the contrary, the relaxation time of the periosteum is higher than that of the fascia, from which it is concluded that it is preferable not to cut the periosteum during subcostal implantation.

Two Preconditioners for Time-Harmonic Eddy-Current Optimal Control Problems

In this paper, we consider the numerical solution of a large complex linear system with a saddle-point form obtained by the discretization of the time-harmonic eddy-current optimal control problem. A new Schur complement is proposed for this algebraic system, extending it to both the block-triangular preconditioner and the structured preconditioner. A theoretical analysis proves that the eigenvalues of block-triangular and structured preconditioned matrices are located in the interval [1/2, 1]. Numerical simulations show that two new preconditioners coupled with a Krylov subspace acceleration have good feasibility and effectiveness and are superior to some existing efficient algorithms.

A comparative study of discretization techniques for augmented Urysohn type nonlinear functional Volterra integral equations and their convergence analysis

This article considers the nonlinear functional Volterra integral equation, the Urysohn type class of nonlinear integral equations. In an approach based on the Picard iterative method, the solution's existence and uniqueness are demonstrated. The trapezoidal and Euler discretization methods are used for the approximation of a numerical solution, and a nonlinear algebraic system of equations is obtained. The first and second order of convergence for the Euler and the trapezoidal methods respectively to the solution is manifested by employing the Gronwall inequality and its discrete form. A new Gronwall inequality is devised in order to prove the trapezoidal method's convergence. Finally, some numerical examples are provided which attest to the application, effectiveness, and reliability of the methods.

Mean dimension of natural extension of algebraic systems

Mean dimension may decrease after taking the natural extension. In this paper we show that mean dimension is preserved by natural extension for an endomorphism on a compact metrizable abelian group. As an application, we obtain that the mean dimension of an algebraic cellular automaton coincides with the mean dimension of its natural extension, which strengthens a result of Burguet and Shi [Israel J. Math. (to appear).] with a different proof.

Minimization of hypersurfaces

Let F ∈ Z [ x 0 , … , x n ] F \in \mathbb {Z}[x_0, \ldots , x_n] be homogeneous of degree d d and assume that F F is not a ‘nullform’, i.e., there is an invariant I I of forms of degree d d in n + 1 n+1 variables such that I ( F ) ≠ 0 I(F) \neq 0 . Equivalently, F F is semistable in the sense of Geometric Invariant Theory. Minimizing F F at a prime p p means to produce T ∈ M a t ( n + 1 , Z ) ∩ G L ( n + 1 , Q ) T \in Mat(n+1, \mathbb {Z}) \cap GL(n+1, \mathbb {Q}) and e ∈ Z ≥ 0 e \in \mathbb {Z}_{\ge 0} such that F 1 = p − e F ( [ x 0 , … , x n ] ⋅ T ) F_1 = p^{-e} F([x_0, \ldots , x_n] \cdot T) has integral coefficients and v p ( I ( F 1 ) ) v_p(I(F_1)) is minimal among all such F 1 F_1 . Following Kollár [Electron. Res. Announc. Amer. Math. Soc. 3 (1997), pp. 17–27], the minimization process can be described in terms of applying weight vectors w ∈ Z ≥ 0 n + 1 w \in \mathbb {Z}_{\ge 0}^{n+1} to F F . We show that for any dimension n n and degree d d , there is a complete set of weight vectors consisting of [ 0 , w 1 , w 2 , … , w n ] [0,w_1,w_2,\dots ,w_n] with 0 ≤ w 1 ≤ w 2 ≤ ⋯ ≤ w n ≤ 2 n d n − 1 0 \le w_1 \le w_2 \le \dots \le w_n \le 2 n d^{n-1} . When n = 2 n = 2 , we improve the bound to d d . This answers a question raised by Kollár. These results are valid in a more general context, replacing Z \mathbb {Z} and p p by a PID R R and a prime element of R R . Based on this result and a further study of the minimization process in the planar case n = 2 n = 2 , we devise an efficient minimization algorithm for ternary forms (equivalently, plane curves) of arbitrary degree d d . We also describe a similar algorithm that allows to minimize (and reduce) cubic surfaces. These algorithms are available in the computer algebra system Magma.

A new numerical approach method to solve the Lotka–Volterra predator–prey models with discrete delays

This paper proposes a new approach called Extreme Learning Machine (ELM) to solve the Lotka–Volterra predator–prey models using a novel approximation form. Unlike the traditional methods that involve constructing complex algebraic systems and performing inverse matrix operations, ELM transforms the system of differential equations into a set of nonlinear equations and solves for the parameters through optimization iterations to obtain an approximate solution. Furthermore, we construct a solution form that satisfies the initial conditions, eliminating the need to handle initial conditions during the solving process, making this method more concise. Finally, by comparing with other numerical methods using two sets of models and parameters, ELM can produce high-precision results and further demonstrate the advantages of our method.

Cutting-Edge Monte Carlo Framework: Novel “Walk on Equations” Algorithm for Linear Algebraic Systems

In this paper, we introduce the “Walk on Equations” (WE) Monte Carlo algorithm, a novel approach for solving linear algebraic systems. This algorithm shares similarities with the recently developed WE MC method by Ivan Dimov, Sylvain Maire, and Jean Michel Sellier. This method is particularly effective for large matrices, both real- and complex-valued, and shows significant improvements over traditional methods. Our comprehensive comparison with the Gauss–Seidel method highlights the WE algorithm’s superior performance, especially in reducing relative errors within fewer iterations. We also introduce a unique dominancy number, which plays a crucial role in the algorithm’s efficiency. A pivotal outcome of our research is the convergence theorem we established for the WE algorithm, demonstrating its optimized performance through a balanced iteration matrix. Furthermore, we incorporated a sequential Monte Carlo method, enhancing the algorithm’s efficacy. The most-notable application of our algorithm is in solving a large system derived from a finite-element approximation in constructive mechanics, specifically for a beam structure problem. Our findings reveal that the proposed WE Monte Carlo algorithm, especially when combined with sequential MC, converges significantly faster than well-known deterministic iterative methods such as the Jacobi method. This enhanced convergence is more pronounced in larger matrices. Additionally, our comparative analysis with the preconditioned conjugate gradient (PCG) method shows that the WE MC method can outperform traditional methods for certain matrices. The introduction of a new random variable as an unbiased estimator of the solution vector and the analysis of the relative stochastic error structure further illustrate the potential of our novel algorithm in computational mathematics.