Article
Mathematical & Computational Biology
Joseph E. Carroll
Summary: A deterministic model is proposed to describe the interaction between an immune system and an invading virus. The long-term behavior of the solution is investigated, and two simple functions of the model parameters are found to determine the persistence or extinction of the virus.
MATHEMATICAL BIOSCIENCES AND ENGINEERING
(2022)
Article
Computer Science, Interdisciplinary Applications
H. Hassani, J. A. Tenreiro Machado, E. Naraghirad, Z. Avazzadeh
Summary: This paper introduces a general class of nonlinear system of fractional partial differential equations with initial and boundary conditions. A hybrid method based on the transcendental Bernstein series and the generalized shifted Chebyshev polynomials is proposed for finding the optimal solution of the nonlinear system of fractional partial differential equations. The solution of the nonlinear system of fractional partial differential equations is expanded in terms of the transcendental Bernstein series and the generalized shifted Chebyshev polynomials, as basis functions with unknown free coefficients and control parameters. The corresponding operational matrices of fractional derivatives are then derived for the basis functions. These basis functions, with their operational matrices of fractional order derivatives and the Lagrange multipliers, transform the problem into a nonlinear system of algebraic equations. By means of Darbo's fixed point theorem and Banach contraction principle, an existence result and a unique result for the solution of the nonlinear system of fractional partial differential equations are obtained, respectively. The convergence analysis is discussed and several illustrative experiments illustrate the efficiency and accuracy of the proposed method.
ENGINEERING WITH COMPUTERS
(2023)
Article
Mathematics
Christer Oscar Kiselman
Summary: The paper introduces a generalization of Joseph Liouville's concept of elementary functions by Ramon Edgar Moore and Alexander M. Gofen. Gofen further defines two variants of this concept, namely scalar generalized elementary functions and vector generalized elementary functions, and formulates a conjecture regarding them. The authors prove that, under certain modified conjectures, the two classes are different.
COMPLEX VARIABLES AND ELLIPTIC EQUATIONS
(2023)
Article
Mathematics, Applied
Peter E. Kloeden
Summary: This paper discusses an elementary inequality for autonomous Caputo fractional differential equation (FDE) of order a ? (0, 1) in R-d with a dissipativity condition. This inequality is fundamental for investigating qualitative and dynamical properties of such equations. The paper illustrates its use and effectiveness in demonstrating the global existence and uniqueness of solutions of such equations when the vector field is only locally Lipschitz. It also establishes the existence of an absorbing set that is positively invariant. Finally, it shows that an equilibrium solution of a nonlinear Caputo FDE is locally asymptotically stable when the matrix of its linear part is negative definite.
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
(2023)
Article
Mathematics, Applied
Jeffrey S. Ovall, Samuel E. Reynolds
Summary: H-1-conforming Galerkin methods on polygonal meshes employ local finite element functions to solve Poisson problems with polynomial source and boundary data. These methods have recently been extended to handle curvilinear polygons in mesh cells. We propose an integration approach that reduces integrals on cells to integrals along their boundaries, and demonstrate the practical performance of our methods through numerical experiments.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Physics, Mathematical
P. -l. Giscard, A. Tamar
Summary: This paper presents integral series representations of solutions for all equations of the Heun class, with specific examples and Python implementations provided. Of particular note is the demonstration of convergence for the new solutions to the Teukolsky radial equation.
JOURNAL OF MATHEMATICAL PHYSICS
(2022)
Article
Computer Science, Information Systems
S. Emmy Wei
Summary: This paper introduces a new method and algorithms to solve the problem of aliasing induced by nonlinear functions. Upsampling and downsampling are employed to avoid aliasing, and theoretical analyses and exemplary algorithms are provided.
Article
Mathematics
Lucian-Miti Ionescu, Cristina-Liliana Pripoae, Gabriel-Teodor Pripoae
Summary: The study focuses on Polya vector fields associated with holomorphic functions, using techniques of differential geometry to refine the analysis. Special Mobius transformations of a certain form are shown to have unique properties in the canonical geometry of the plane, and various types of affine connections are characterized. A program for classifying holomorphic functions based on curvature and torsion tensor fields is proposed.
Article
Mathematics
Timothy Ferguson
Summary: In this passage, Andre demonstrates the properties of minimal differential equations for (sic)-functions and E-functions, as well as transcendence results for E-function values. These results are further used to strengthen the Siegel-Shidlovskii theorem for E-functions.
JOURNAL OF NUMBER THEORY
(2021)
Article
Mathematics, Applied
V. Kavitha, Dumitru Baleanu, Soumya George, J. Grayna
Summary: This article focuses on establishing the measure pseudo-almost automorphic solution of an integro-differential equation with impulses, using Banach contraction principle mapping and fixed point theorems. Examples are provided to illustrate the significance of the theoretical findings.
Article
Mathematics, Applied
Sedef Emin, Arran Fernandez
Summary: In this paper, the authors solve a multi-term fractional differential equation with continuous variable coefficients and incommensurate fractional orders by using a direct method of successive approximations. They establish an explicit solution by rigorously checking the solution function via substitution. This is the first time that an explicit analytical solution has been found for this general problem, and connections with previous results in the literature are discussed.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Thai Son Doan, Peter E. Kloeden
Summary: This paper investigates the attractors of autonomous Caputo fractional differential equations and proves that they have a certain triangular structure and satisfy the same smooth condition and dissipativity condition as ordinary differential equations. Based on this, several bifurcations of scalar fractional differential equations, including saddle-node and pitchfork bifurcations, are established.
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
(2022)
Article
Mathematics
Andrei D. Polyanin, Alexei Zhurov
Summary: This study explores a nonlinear multi-parameter reaction-diffusion system with delays, considering both cases of different and identical diffusion coefficients. Various techniques are used to solve the system and numerous new exact solutions are obtained. These solutions can be applied to model delay processes in biology, ecology, biochemistry, and medicine.
Article
Physics, Multidisciplinary
Gregg Jaeger
Summary: The elementary particles in relativistic quantum field theory are not simple field quanta, but rather a supplement to quantum fields, providing consistency and unity between particle physics practice and its basis in quantum field theory.
Article
Mathematics
Eugene Stepanov, Dario Trevisan
Summary: We study the validity of an extension of Frobenius theorem on integral manifolds for some classes of Pfaff-type systems of partial differential equations involving multidimensional rough signals.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Computer Science, Theory & Methods
Robert H. C. Moir, Robert M. Corless, David J. Jeffrey
Summary: This paper introduces a method for correcting discontinuous integrals by extending the concept of unwinding numbers for complex functions to handle the passage of integration paths through branch points. By treating the codomain of a complex function as a pair of 2-dimensional real manifolds, the approach computes the intersection of paths and branch boundaries to identify where discontinuities appear.
JOURNAL OF SYMBOLIC COMPUTATION
(2021)
Article
Computer Science, Theory & Methods
Erika abraham, James H. Davenport, Matthew England, Gereon Kremer
Summary: The paper introduces a new algorithm for determining the satisfiability of conjunctions of nonlinear polynomial constraints over the reals, based on a variant of Cylindrical Algebraic Decomposition. The algorithm incrementally constructs solution candidates guided by input constraints and previous conflicts, demonstrating differences and benefits compared to other existing methods.
JOURNAL OF LOGICAL AND ALGEBRAIC METHODS IN PROGRAMMING
(2021)
Article
Mathematics
Neil J. Calkin, Eunice Y. S. Chan, Robert M. Corless, David J. Jeffrey, Piers W. Lawrence
Summary: This article examines the eigenvalues and singular value decomposition of the Mandelbrot matrix family, and explores the characteristics and fractal structure of the dominant eigenvalues and singular vectors.
AMERICAN MATHEMATICAL MONTHLY
(2022)
Article
Mathematics, Applied
Eunice Y. S. Chan, Robert M. Corless
Summary: The chaos game representation (CGR) is a method to visualize one-dimensional sequences. This paper demonstrates how to construct CGR and its applications in biology, where it can uncover unknown patterns in DNA or proteins. Moreover, it suggests introducing CGR in the classroom for modeling or dynamical systems courses. The sequences tested include those from the On-line Encyclopedia of Integer Sequences and experimental mathematics.
Article
Education & Educational Research
Rupert Ward, Tom Crick, James H. Davenport, Paul Hanna, Alan Hayes, Alastair Irons, Keith Miller, Faron Moller, Tom Prickett, Julie Walters
Summary: Employers are shifting towards selecting and developing employees based on skills rather than qualifications. Governments are also focusing on skills-based development to improve productivity. This has led to increased interest in digital badging and micro-credentialing for granular, skills-based development. The use of an online skills profiling tool is explored to incorporate badges and micro-credentials within existing qualifications and align skills developed in learning with job roles.
JOURNAL OF INTERACTIVE MEDIA IN EDUCATION
(2023)
Proceedings Paper
Computer Science, Theory & Methods
James H. Davenport, Akshar Nair, Gregory Sankaran, Ali K. Uncu
Summary: McCallum-style CAD is an improvement on the original Collins version, but suffers from nullification issues. The recently-justified Lazard-style CAD does not have this problem, but reintroduces nullification issues when transporting equational constraints. This paper explains the problem and solutions, and shows that our approach achieves similar improvements in Lazard-style CAD without failing due to nullification as McCallum does. The case of multiple equational constraints is also considered.
PROCEEDINGS OF THE INTERNATIONAL SYMPOSIUM ON SYMBOLIC & ALGEBRAIC COMPUTATION, ISSAC 2023
(2023)
Proceedings Paper
Engineering, Multidisciplinary
Siyuan Deng, Greg Reid, D. J. Jeffrey
Summary: Parametric Linear Algebra involves the analysis of matrices with symbolic elements, where properties like the rank can vary based on subsequently assigned values. This epitomizes the specialization problem, which can be addressed through the development of specialized software like Maple for handling symbolic coefficients.
2021 23RD INTERNATIONAL SYMPOSIUM ON SYMBOLIC AND NUMERIC ALGORITHMS FOR SCIENTIFIC COMPUTING (SYNASC 2021)
(2021)
Proceedings Paper
Engineering, Multidisciplinary
D. J. Jeffrey
Summary: The paper discusses the two cognate functions Tree T and Lambert W. The varying opinions on the branch structure of the Lambert W function in the complex plane are reviewed. The proposal to give independent branch structures for the two functions is put forward to allow users convenience in accessing them in computer algebra systems.
2021 23RD INTERNATIONAL SYMPOSIUM ON SYMBOLIC AND NUMERIC ALGORITHMS FOR SCIENTIFIC COMPUTING (SYNASC 2021)
(2021)
Proceedings Paper
Engineering, Multidisciplinary
Gereon Kremer, Erika Abraham, Matthew England, James H. Davenport
Summary: Cylindrical algebraic coverings are a new method based on the theory of cylindrical algebraic decomposition for reasoning in nonlinear real arithmetic theory. The more careful implementation within cvc5 simplifies proof generation for nonlinear real arithmetic problems and positions cvc5 at the forefront of currently available SMT solvers for QF_NRA. Exciting experimental results have been announced, highlighting the increasing practical benefits of this new approach.
2021 23RD INTERNATIONAL SYMPOSIUM ON SYMBOLIC AND NUMERIC ALGORITHMS FOR SCIENTIFIC COMPUTING (SYNASC 2021)
(2021)
Article
Mathematics, Applied
Chris Brimacombe, Robert M. Corless, Mair Zamir
Summary: This paper surveys the historical development of Mathieu functions and modified Mathieu functions, identifies gaps in current software capability related to double eigenvalues of the Mathieu equation, demonstrates techniques for computing Puiseux expansions and generalized eigenfunctions, and provides short biographies of major mathematical researchers in the history of Mathieu functions.
Article
Mathematics, Applied
R. Bradford, J. H. Davenport, M. England, A. Sadeghimanesh, A. Uncu
Summary: The DEWCAD project aims to push back the Doubly Exponential Wall of Cylindrical Algebraic Decomposition by integrating SAT/SMT technology, extending Lazard projection theory, and developing new algorithms. It also focuses on applications in economics and bio-network analysis.
ACM COMMUNICATIONS IN COMPUTER ALGEBRA
(2021)
Proceedings Paper
Computer Science, Information Systems
James H. Davenport, Benjamin Pring
Summary: This paper demonstrates that the overheads involved in implementing quantum oracles for a generic key-recovery attack against block ciphers can be reduced by using fine-grained approach to quantum amplitude amplification and design of the required quantum oracles. The effort in cryptanalysis can be reduced to less than r with respect to the number of plaintext-ciphertext pairs needed to identify a user's key. Additionally, full quantum resource estimations for AES-128/192/256 are provided along with code in the Q# quantum programming language.
SELECTED AREAS IN CRYPTOGRAPHY
(2021)
Proceedings Paper
Education, Scientific Disciplines
Rupert Ward, Oliver Phillips, David Bowers, Tom Crick, James H. Davenport, Paul Hanna, Alan Hayes, Alastair Irons, Tom Prickett
Summary: There is a significant gap between higher education learning outcomes and employer requirements, with traditional accreditation methods struggling to address this gap. The proposal of using a 21st Century skills taxonomy to bridge this divide is discussed, showcasing how it can support a microcredentialing framework and enable more personalized learning within educational institutions and throughout one's career.
PROCEEDINGS OF THE 2021 IEEE GLOBAL ENGINEERING EDUCATION CONFERENCE (EDUCON)
(2021)
Article
Mathematics, Applied
Johannes Middeke, David J. Jeffrey, Christoph Koutschan
Summary: In this study, LU and QR matrix decompositions using exact computations are considered, showing a connection between row factors and the Smith-Jacobson normal form of the matrix. Two types of common factors are identified: systematic factors, which depend on the reduction process, and statistical factors, which are specific to the data. Experimental testing confirms the conclusions drawn regarding the mechanisms and frequency of occurrence of statistical factors.
MATHEMATICS IN COMPUTER SCIENCE
(2021)
Article
Mathematics, Applied
Arman Hashemzadeh Kalvari, Alireza Ansari, Hassan Askari
Summary: In this paper, the inverse Laplace transform of the Volterra mu-function and its evaluation using different complex contours are considered. The generalized Ramanujan's integral representations for the Volterra mu-function with general variations of the parameters are established. The asymptotic analysis of this function with large parameters using the steepest descent method is also discussed. Furthermore, it is shown that the solution of the Volterra integral equation with a differentiated-order fractional integral operator is the Volterra mu-function.
INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS
(2024)
Article
Mathematics, Applied
Yong-Kum Cho, Seok-Young Chung, Young Woong Park
Summary: This article investigates the positivity of an integral transform by using Sturm's theory, where the kernel of the transform arises from an oscillatory solution of a second-order linear differential equation. Positivity criteria for Hankel transforms and trigonometric integrals defined on the positive real line are obtained as applications.
INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS
(2024)
Article
Mathematics, Applied
Neila Ben Romdhane, Hana Boukattaya
Summary: This paper discusses the connection between the interlacing of zeros and the orthogonality of a given sequence of polynomials, focusing on particular cases of d-orthogonal polynomials. The authors characterize the 2-orthogonality of the sequence by the existence of a certain ratio expressed in terms of the zeros. They also study the interlacing of zeros, d-orthogonality, and positivity of the ratio for (d + 1)-fold symmetric polynomials. Necessary and sufficient conditions for a given sequence to satisfy a particular (d + 1)-order recurrence relation are provided, along with illustrative examples.
INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS
(2024)
