Calculus Algebra Research Articles

Analysis of the multi-term fractional Bateman equations in radioactive decay by means of Mikusiński algebraic calculus

Fractional Bateman equations model the concentrations of nuclear species through a radioactive decay chain, by considering non-Markovian behavior. Cruz-López and Espinosa-Paredes (Nuclear Engineering and Technology, 2022) solved the mathematical problem with a single memory coefficient, based on the Laplace-transform technique. Cruz-López, Espinosa-Paredes and François (Computer Physics Communications, 2022) extended the analysis to several fractional orders, with Laplace transforms and series expansions checked by induction. In this paper, we solve such a case of distinct memory effects, via the more general method of Mikusiński operational calculus. We do not distinguish between equal and different decay rates. Multivariate Mittag-Leffler functions appear in the closed-form expression.

Matemáticas en el mundo árabe: Un recorrido por la edad de oro del saber

The history of Arabic mathematics is crucial to the development of mathematics. During the Islamic Golden Age (8th-14th centuries), Arab mathematicians made important contributions in algebra, geometry, trigonometry, and arithmetic, applying their knowledge in astronomy, navigation, and engineering. A significant advance was the introduction of the Hindu-Arabic numeral system, which facilitated arithmetic operations. Prominent mathematicians such as Al-Khwarizmi, who developed methods for solving equations and wrote about algebra, and Omar Khayyam, known for his work in algebraic calculus, made fundamental contributions. Thabit ibn Qurra also contributed to geometry and number theory. In addition, Arabic mathematics preserved and translated works of ancient civilizations, which allowed the transmission of this knowledge to Europe and its influence on the Renaissance.

A concrete model for a typed linear algebraic lambda calculus

AbstractWe give an adequate, concrete, categorical-based model for Lambda- ${\mathcal S}$ , which is a typed version of a linear-algebraic lambda calculus, extended with measurements. Lambda- ${\mathcal S}$ is an extension to first-order lambda calculus unifying two approaches of non-cloning in quantum lambda-calculi: to forbid duplication of variables and to consider all lambda-terms as algebraic linear functions. The type system of Lambda- ${\mathcal S}$ has a superposition constructor S such that a type A is considered as the base of a vector space, while SA is its span. Our model considers S as the composition of two functors in an adjunction relation between the category of sets and the category of vector spaces over $\mathbb C$ . The right adjoint is a forgetful functor U, which is hidden in the language, and plays a central role in the computational reasoning.

A Tutorial on Visual Representations of Relational Queries

Query formulation is increasingly performed by systems that need to guess a user's intent (e.g. via spoken word interfaces). But how can a user know that the computational agent is returning answers to the "right" query? More generally, given that relational queries can become pretty complicated, how can we help users understand existing relational queries , whether human-generated or automatically generated? Now seems the right moment to revisit a topic that predates the birth of the relational model: developing visual metaphors that help users understand relational queries. This lecture-style tutorial surveys the key visual metaphors developed for visual representations of relational expressions. We will survey the history and state-of-the art of relationally-complete diagrammatic representations of relational queries, discuss the key visual metaphors developed in over a century of investigating diagrammatic languages, and organize the landscape by mapping their used visual alphabets to the syntax and semantics of Relational Algebra (RA) and Relational Calculus (RC).

Some basic concepts of algebraic calculus in tensors

Some basic concepts of algebraic calculus in tensors

A Novel Proof of the Desnanot-Jacobi Determinant Identity

Summary We present an alternate proof for the Desnanot-Jacobi identity using induction, basic matrix algebra and elementary differential calculus. It is an instructive demonstration of how it is occasionally necessary to utilize results from different fields of mathematics to solve a specific problem.

Fuzzy Observer-Based Consensus Tracking Control for Fractional-Order Multi-Agent Systems Under Cyber-Attacks and Its Application to Electronic Circuits

Consensus control of multi-agent systems (MASs) has applications in various domains. As MASs work in networked environments, their security control becomes critically desirable in response to cyber-attacks. In this paper, the observer-based consensus tracking control problem is investigated for a class of Takagi-Sugeno fuzzy fractional-order multi-agent systems (FOMASs) under cyber-attacks. The malicious cyber attacks can impact the security of topologies of the communication networks of both controllers and observers. To estimate unmeasurable system states, a fuzzy observer is built. It is found that the topology of contact for observer states may be different from that of the feedback signals. A novel mathematical model for T-S fuzzy FOMASs with cyber-attacks is proposed. By using algebraic graph theory, Lyapunov functional, and fractional calculus theory, a distributed feed-back controller is developed for each agent, which guarantee the secure performance of tracking consensus error and observer error. Finally, two numerical examples demonstrate the effectiveness of the suggested control scheme, and the controller design for electronic network circuits shows the applicability of the proposed theoretical results. Simulations results for different differential-orders and coupling strength scenarios are given.

Riemannian Calculus of Variations Using Strongly Typed Tensor Calculus

In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for finding natural formulations of various objects. A tensor bundle formalism, crucially relying on the notion of pullback bundle, will be used to create a rich type system with which to distinguish objects. The type system and relevant notation is designed to “telescope” to accommodate a level of detail appropriate to a set of calculations. Various techniques using this formalism will be developed and demonstrated with the goal of providing a relatively complete and uniform method of coordinate-free computation. The calculus of variations pertaining to maps between Riemannian manifolds will be formulated using the strongly typed tensor formalism and associated techniques. Energy functionals defined in terms of first-order Lagrangians are the focus of the second half of this paper, in which the first variation, the Euler–Lagrange equations, and the second variation of such functionals will be derived.

Use of assignment mode for effective assessment of students to achieve course outcomes

To ensure the quality learning of students, course outcome assessment is vital element of outcome based education. To achieve course outcomes, various modes of pedagogy and assessment are essential. There are different modes of assessment for In-semester examination like multiple choice questions, time bound pen and paper, presentations, seminars, assignment etc. This paper explains the use of assignment mode for effective assessment of students. This paper highlights how assignment mode helps students to apply concepts taught in classroom to solve various problems on applications of engineering mathematics. We had done assessment of 610+ F. Y. B. Tech students by giving different assignment to each student which included different set of questions for the course Linear Algebra and Univariate Calculus with time bound submission. This ensured that students have to solve the assignment independently without taking help of peers. This paper describes how assignment mode has improved various competencies of students. This paper is illustrated with responses of F. Y. B. Tech students. The proposed method is useful for online and offline mode of examinations.

Characterization of three-dimensional fractional viscoelastic models through complex modulus analysis and polar decomposition

Soft materials such as gels, elastomers, and biological tissues have diverse applications in nature and technology due to their viscoelastic nature. These soft materials often exhibit complex rheology and display elastic and viscous characteristics when undergoing deformation. In recent years, fractional calculus has emerged as a promising tool to explain the viscoelastic behavior of soft materials. Scalar constants are primarily used to quantify viscoelastic elements such as springs and dashpots. However, in three-dimensional (3D) space, not all materials show the same elastic or viscoelastic properties in all directions, especially under elastic/viscoelastic wave propagation (or anisotropy). Though previously reported studies on viscoelastic models have explained a power-law decay of the memory functions, none of them explicitly explained the 3D complex modulus through a matrix notation. In this paper, we present a mathematical formulation that employs tensor algebra and fractional calculus to derive the 3D complex modulus of Kelvin–Voigt, Maxwell, and other arrangements of viscoelastic models. The 3D complex modulus provides information about the elastic wave propagation in a media and can be used to explain anisotropy in different viscoelastic materials. Additionally, an advanced formulation of the moduli can improve the modeling in finite element analysis of 3D viscoelastic materials where discretization is vital for studying media of asymmetric shapes. Finally, we demonstrated a polar decomposition method to visualize viscoelastic tensors using the Green–Christoffel tensor and surface plots to represent the degrees of anisotropy and viscoelasticity in the Fourier domain when the medium is probed by a time-harmonic homogeneous plane wave.

Formal verification of multiplier circuits using computer algebra

Abstract Digital circuits are widely utilized in computers, because they provide models for various digital components and arithmetic operations. Arithmetic circuits are a subclass of digital circuits that are used to execute Boolean algebra. To avoid problems like the infamous Pentium FDIV bug, it is critical to ensure that arithmetic circuits are correct. Formal verification can be used to determine the correctness of a circuit with respect to a certain specification. However, arithmetic circuits, particularly integer multipliers, represent a challenge to current verification methodologies and, in reality, still necessitate a significant amount of manual labor. In my dissertation we examine and develop automated reasoning approaches based on computer algebra, where the word-level specification, modeled as a polynomial, is reduced by a Gröbner basis inferred by the gate-level representation of the circuit. We provide a precise formalization of this reasoning process, which includes soundness and completeness arguments and adds to the mathematical background in this field. On the practical side we present an unique incremental column-wise verification algorithm and preprocessing approaches based on variable elimination that simplify the inferred Gröbner basis. Furthermore, we provide an algebraic proof calculus in this thesis that allows obtaining certificates as a by-product of circuit verification in order to boost confidence in the outcomes of automated reasoning tools. These certificates can be efficiently verified with independent proof checking tools.

TRANSFORMATION OF PRE-SERVICE MATHEMATICS STUDENT'S ALGEBRA AND CALCULUS THINKING IN SOLVING DIFFERENTIAL EQUATION PROBLEMS

This study aims to analyze the transformation of algebraic thinking and calculus of preservice the mathematics students based on SOLO taxonomy in solving differential equations problems. The research subjects were 86 students in the mathematics education study program. Subject selection uses purposive sampling (students who take courses in differential equations). Data were collected using problem-solving tests and interviews which were then analyzed using the descriptive qualitative method with the following stages: (1) transcribing test and interview data, (2) coding segmentation, (3) analyzing student thinking transformations, and (4) concluding. The results showed that the transformation of algebraic and calculus thinking was used by students at each level of thinking to solve problems. The higher the level of thinking achieved, the better and the maximum transformation of algebraic and calculus thinking used by students. These results indicate that students need to be well supported and facilitated in problem-solving to achieve higher levels of thinking, such as the relational and extended abstract levels.

A Sine-Space, Mixed-Coordinate Polarization Representation for Rotated Phased Arrays: Using a linear algebra calculus to transform radiated fields

A Sine-Space, Mixed-Coordinate Polarization Representation for Rotated Phased Arrays: Using a linear algebra calculus to transform radiated fields

Practical algebraic calculus and Nullstellensatz with the checkers Pacheck and Pastèque and Nuss-Checker

AbstractAutomated reasoning techniques based on computer algebra have seen renewed interest in recent years and are for example heavily used in formal verification of arithmetic circuits. However, the verification process might contain errors. Generating and checking proof certificates is important to increase the trust in automated reasoning tools. For algebraic reasoning, two proof systems, Nullstellensatz and polynomial calculus, are available and are well-known in proof complexity. A Nullstellensatz proof captures whether a polynomial can be represented as a linear combination of a given set of polynomials by providing the co-factors of the linear combination. Proofs in polynomial calculus dynamically capture that a polynomial can be derived from a given set of polynomials using algebraic ideal theory. In this article we present the practical algebraic calculus as an instantiation of the polynomial calculus that can be checked efficiently. We further modify the practical algebraic calculus and gain LPAC (practical algebraic calculus + linear combinations) that includes linear combinations. In this way we are not only able to represent both Nullstellensatz and polynomial calculus proofs, but we are also able to blend both proof formats. Furthermore, we introduce extension rules to simulate essential rewriting techniques required in practice. For efficiency we also make use of indices for existing polynomials and include deletion rules too. We demonstrate the different proof formats on the use case of arithmetic circuit verification and discuss how these proofs can be produced as a by-product in formal verification. We present the proof checkers Pacheck, Pastèque, and Nuss-Checker. Pacheck checks proofs in practical algebraic calculus more efficiently than Pastèque, but the latter is formally verified using the proof assistant Isabelle/HOL. The tool Nuss-Checker is used to check proofs in the Nullstellensatz format.

Symmetry group analysis and conservation laws of the potential modified KdV equation using the scaling method

In this paper, using the Lie symmetry method, we obtain the Lie symmetry group of the potential modified Korteweg–de Vries (potential mKdV) equation, and we construct the optimal system of one-dimensional subalgebras for the Lie symmetry algebra of this equation. Furthermore, similarity reductions related to the infinitesimal generators of the Lie symmetry group are obtained. Also, we construct new conservation laws for this equation by the scaling method. This method is algorithmic and based on the linear algebra tools and calculus of variations. The density of the conservation laws is constructed using the Euler operator and scaling symmetry of the equation and also the associated flux is calculated by the homotopy operator. This density-flux pair renders a conservation law for the potential mKdV equation. We specify the density-flux pairs of conservation laws in ranks 4, 8, 12, 16 and 20 for the potential mKdV equation.

Multiple analogical proportions

Analogical proportions are statements of the form “a is to b as c is to d”, denoted a : b : : c : d, that may apply to any type of items a, b, c, d. Analogical proportions, as a building block for analogical reasoning, is then a tool of interest in artificial intelligence. Viewed as a relation between pairs ( a , b ) and ( c , d ), these proportions are supposed to obey three postulates: reflexivity, symmetry, and central permutation (i.e., b and c can be exchanged). The logical modeling of analogical proportions expresses that a and b differ in the same way as c and d, when the four items are represented by vectors encoding Boolean properties. When items are real numbers, numerical proportions – arithmetic and geometric proportions – can be considered as prototypical examples of analogical proportions. Taking inspiration of an old practice where numerical proportions were handled in a vectorial way and where sequences of numerical proportions of the form x 1 : x 2 : ⋯ : x n : : y 1 : y 2 : ⋯ : y n were in use, we emphasize a vectorial treatment of Boolean analogical proportions and we propose a Boolean logic counterpart to such sequences. This provides a linear algebra calculus of analogical inference and acknowledges the fact that analogical proportions should not be considered in isolation. Moreover, this also leads us to reconsider the postulates underlying analogical proportions (since central permutation makes no sense when n ⩾ 3) and then to formalize a weak form of analogical proportion which no longer obeys the central permutation postulate inherited from numerical proportions. But these weak proportions may still be combined in multiple weak analogical proportions.

Analysis of Five Mathematical Models for Crop Yield Prediction

A review of mathematical models used for the prediction of crop yield has been presented. Though there are many other non-mathematical techniques also available for the purpose, but mathematical modeling used for any real world problem opens many perspectives and provides many possible solu-tions of the problem for the betterment of human race. Five mathematical models (remote sensing followed by mathematical modeling, fuzzy logic based model, multiple linear regression mechanistic model, linear algebra based descriptive model and growth model based on TOMGRO mechanistic model) have been extracted and analyzed critically. These models are based on different mathematical concepts and techniques (non-linear optimization, fuzzy logic, linear predictor functions, linear algebra and differential calculus) covering a wide range of mathematical modeling. The general forms of these models have been derived. Average accuracy of presented models was found to be in the range 90% - 99% that strongly favors the optimum usage of mathematical modeling for crop yield forecasting processes. The section giving gaps and future research prospects presents the comparative analysis of the models. Development of new and moderated mathematical models for more precision and better accuracy has also been suggested using new mathematical techniques and hybridization or modifying the existing models.

Clifford Algebra and Hypercomplex Number as well as Their Applications in Physics

The Clifford algebra is a unification and generalization of real number, complex number, quaternion, and vector algebra, which accurately and faithfully characterizes the intrinsic properties of space-time, providing a unified, standard, elegant, and open language and tools for numerous complicated mathematical and physical theories. So it is worth popularizing in the teaching of undergraduate physics and mathematics. Clifford algebras can be directly generalized to 2n-ary associative algebras. In this generalization, the matrix representation of the orthonormal basis of space-time plays an important role. The matrix representation carries more information than the abstract definition, such as determinant and the definition of inverse elements. Without this matrix representation, the discussion of hypercomplex numbers will be difficult. The zero norm set of hypercomplex numbers is a closed set of special geometric meanings, like the light-cone in the realistic space-time, which has no substantial effect on the algebraic calculus. The physical equations expressed in Clifford algebra have a simple formalism, symmetrical structure, standard derivation, complete content. Therefore, we can hope that this magical algebra can complete a new large synthesis of modern science.

Stratification Methods for an Auxiliary Variable Model-Based Allocation under a Superpopulation Model

In this paper, the problem of optimum stratification of heteroscedastic populations in stratified sampling is considered for a known allocation under Simple Random Sampling With and Without Replacement (SRSWR & SRSWOR) design. The known allocation used in the problem is one of the model-based allocations proposed by Gupt [1,2] under a superpopulation model considered by Hanurav [3], Rao [4], and Gupt and Rao [5] which was modified by the author (Gupt [1,2]) to a more general form. The problem of finding optimum boundary points of stratification (OBPS) in stratifying populations considered here is based on an auxiliary variable which is highly correlated with the study variable. Equations giving the OBPS have been derived by minimizing the variance of estimator of the population mean. Since the equations giving OBPS are implicit and difficult for solving, some methods of finding approximately optimum boundary points of stratification (AOBPS) have also been obtained as the solutions of the equations giving OBPS. While deriving equations giving OBPS and methods of finding AOBPS, basic statistical definitions, tools of calculus, analytic functions and tools of algebra are used. While examining the efficiencies of the proposed methods of stratification, they are tested in a few generated populations and a live population. All the proposed methods of stratification are found to be efficient and suitable for practical applications. In this study, although the proposed methods are obtained under a heteroscedastic superpopulation model for level of heteroscedasticity one, the methods have shown robustness in empirical investigation in varied levels of heteroscedastic populations. The stratification methods proposed here are new as they are derived for an allocation, under the superpopulation model, which has never been used earlier by any researcher in the field of construction of strata in stratified sampling. The proposed methods may be a fascinating piece of work for researchers amidst the vigorously progressing theoretical research in the area of stratified sampling. Besides, by virtue of exhibiting high efficiencies in the performance of the methods, the work may provide a practically feasible solution in the planning of socio-economic survey.

Quantum inspiration to build a neural model based on the Day-Ahead Market of the Polish Power Exchange

The article is an attempt of the methodological approach to the proposed quantum-inspired method of neural modeling of prices quoted on the Day-Ahead Market operating at TGE S.A. In the proposed quantum-inspired neural model it was assumed, inter alia, that it is composed of 12 parallel Perceptron ANNs with one hidden layer. Moreover, it was assumed that weights and biases as processing elements are described by density matrices, and the values flowing through the Artificial Neural Network of Signals are represented by qubits. Calculations checking the correctness of the adopted method and model were carried out with the use of linear algebra and vector-matrix calculus in MATLAB and Simulink environments. The obtained research results were compared to the results obtained from the neural model with the use of a comparative model.