Calculus Algebra Research Articles

Generalized interval vector spaces and interval optimization

This paper presents a method for endowing the generalized interval space with some different structures, such as vector spaces, order relations and an algebraic calculus. With these concepts we formulate interval optimization problems and relate them to classic multi-objective optimization problems. We also present a version of the Von Neumann’s Mini-max Theorem in the interval context.

Mathematical Methodologies in Physics and Their Applications in Derivation of Velocity and Acceleration Theories

The principles to use variables and mathematical methodologies in physics are addressed. A set of refined definitions with designated variables are used to derive the Velocity and Acceleration Theories in Distance Field and Vector Space. Mathematical methodologies such as Linear Algebra and Vector Calculus are used systematically in a step by step derivation process. The proof of the theories can be easily achieved by substitution of the designated variables with a set of parameters that matches the same assumptions and conditions in every step of the derivation process.

Simulation as a Teaching Tool in Finance

An excessive use of algebra and stochastic calculus in undergraduate finance courses is an impediment to learning because it shifts our attention from finance to mathematics; and also because, for most of our students, algebra serves to obfuscate rather than to clarify the basic concepts in finance that we are trying to teach. In this paper we demonstrate with examples, particularly on the subject of risk, that finance lectures may be given without algebra by replacing equations on a blackboard with simple interactive simulation computer screens integrated with the lecture.

DNA algorithms of implementing biomolecular databases on a biological computer.

In this paper, DNA algorithms are proposed to perform eight operations of relational algebra (calculus), which include Cartesian product, union, set difference, selection, projection, intersection, join, and division, on biomolecular relational databases.

Organization of interdisciplinary connections as a condition of modernization of mathematical education of technical university students

The article considers the problem of mastering of mathematics by engineering students. One of these conditions is the use of integrative interdisciplinary connections. Examples of such connections between linear algebra and vector calculus are considered as independent mathematics course modules.

The topology of macro financial flows: An application of stochastic flow diagrams

A large portion of Macroeconomic and Financial research is built upon classical applications of Linear Algebra (such as regression analysis) and Stochastic Calculus (such as valuation models). As a result, most Macroeconomic and Financial research has inherited a focus on geometric locations rather than logical relations. Ideally, Econometric models could be complemented with Topological and Graph-Theoretical tools that recognize the hierarchy and relationships between system constituents. Stochastic Flow Diagrams (SFDs) are topological representations of complex dynamic systems. We construct a network of financial instruments and show how SFDs allow researchers to monitor the flow of capital across the financial system. Because our approach is dynamic, it models how and for how long a financial shock propagates through the system. Practical applications include stress-testing of investment portfolios under user-defined scenarios, and the discovery of Macro trading opportunities. SFDs add Topology to the Econometric toolkit used by Macroeconomists, and may enlighten perennial controversies, such as the one involving Keynesians and Austrian-school economists. Our findings have important implications for regulators, market designers and Macro investors.

A note on Boole polynomials

Boole polynomials play an important role in the area of number theory, algebra and umbral calculus. In this paper, we investigate some properties of Boole polynomials and consider Witt-type formulas for the Boole numbers and polynomials. Finally, we derive some new identities of those polynomials from the Witt-type formulas which are related to Euler polynomials.

Cartooning in Algebra and Calculus

This article discusses how teachers can create cartoons for undergraduate math classes, such as college algebra and basic calculus. The practice of cartooning for teaching can be helpful for communication with students and for students’ conceptual understanding.

Algebraic approach to the Lane–Emden equation

The algebraic solution of the Lane–Emden equation is obtained by means of an algebraic operational calculus and the Adomian polynomials.

The Topology of Macro Financial Flows: An Application of Stochastic Flow Diagrams

A large portion of Macroeconomic and Financial research is built upon classical applications of Linear Algebra (such as regression analysis) and Stochastic Calculus (such as valuation models). As a result, most Macroeconomic and Financial research has inherited a focus on geometric locations rather than logical relations. Ideally, Econometric models could be complemented with Topological and Graph-Theoretical tools that recognize the hierarchy and relationships between system constituents.Stochastic Flow Diagrams (SFDs) are topological representations of complex dynamic systems. We construct a network of financial instruments and show how SFDs allow researchers to monitor the flow of capital across the financial system. Because our approach is dynamic, it models how and for how long a financial shock propagates through the system. Practical applications include stress-testing of investment portfolios under user-defined scenarios, and the discovery of Macro trading opportunities. SFDs add Topology to the Econometrics toolkit used by Macroeconomists, and may enlighten long-standing controversies, such as the one involving Keynesians and Austrian-school economists. Our findings have important implications for regulators, market designers and Macro investors.

Mikusiński's operational calculus with algebraic foundations and applications to Bessel functions

We construct an operational calculus supported on the algebraic operational calculus introduced by Bengochea and Verde [Linear algebraic foundations of the operational calculi. Adv Appl Math. 2011;47:330–351]. With this operational calculus we study the solution of certain Bessel-type equations.

Using MS Excel to solve and simulate the Life-Cycle/Permanent-Income Model of Consumption and Saving

The objective of this manuscript is to provide a simple guide to instructors and students on how to solve and simulate a discrete-time specification of the Life-Cycle/Permanent-Income Model of Consumption and Saving (LCPI Model) using only algebra and basic calculus. The solution and simulation procedure makes use of the simple computing environment in MS Excel. Understanding this process will enable students to operate and experiment with one of the preeminent models that is used in modern economic analysis. The dynamic LCPI Model in this manuscript includes a social security program to demonstrate some relevant applications of the solution technique and methods.

Towards Checking Bigraphical Context-Aware Models

Context-aware computing is an important aspect of Pervasive Computing, Cyber-physical Systems (CPS), Cloud computing, etc. Recently, few works have been reported on formal approaches for this area. Bigraph was proposed as a mathematical structure that is consist of two graphs, a place graph and a link graph, intended for modeling distributed, mobile systems, etc. Bigraphs and its corresponding Bigraphical Reactive Systems (BRSs) present a modern, graphical structure, algebra calculus for the description of the syntax and semantics of systems in accordance with the orthogonal notions of connectivity and locality. Bigraphs and BRSs have been applied in various areas. Bigraph is used to formalize context-aware models in this paper. Then, we present an approach to check the formal context-aware models. The formalisms and checking are explicitly depicted through a simple but non-trivial example.

A Noncommutative Algebraic Operational Calculus for Boundary Problems

We set up a left ring of fractions over a certain ring of boundary problems for linear ordinary differential equations. The fraction ring acts naturally on a new module of generalized functions. The latter includes an isomorphic copy of the differential algebra underlying the given ring of boundary problems. Our methodology employs noncommutative localization in the theory of integro-differential algebras and operators. The resulting structure allows to build a symbolic calculus in the style of Heaviside and Mikusinski, but with the added benefit of incorporating boundary conditions where the traditional calculi allow only initial conditions. Admissible boundary conditions include multiple evaluation points and nonlocal conditions. The operator ring is noncommutative, containing all integrators initialized at any evaluation point.

Generalized Carleson–Newman inner functions

We study a class of inner functions introduced by Gorkin, Mortini, and Nikolski, and motivated by Banach algebra and functional calculus applications. Answering their question, we produce a singular function that cannot be multiplied into this generalized Carleson–Newman class of inner functions (the class of functions satisfying the so called weak embedding property). Furthermore, we give elementary proofs for some results on these classes that were obtained earlier using arguments related to the Gelfand representation of \(H^\infty \).

The algebra of multirelations

Multirelational semantics are well suited to reasoning about programs involving two kinds of non-determinism. This paper lays the categorical foundations for an algebraic calculus of multirelations.

Robert Murphy: Mathematician and Physicist

Despite his short life, Robert Murphy (1806-1843) was a mathematician and physicist of “true genius,” according to Augustus De Morgan [Venn 2009]. Murphy’s research can be categorized into three areas: Algebraic Equations, Integral Equations, and Operator Calculus [Allaire 2002]. The majority of scholarship on Murphy is centered around his contributions to physics; however, historians such as Petrova [1978] and Bradley and Allaire [2002] have explored Murphy’s linear operator theory. In the paper “Robert Murphy: Mathematician and Physicist,” we provide a unified exposition in which we synthesize and expand on existing accounts of Murphy’s life and mathematical contributions. Additionally, we give an overview of his mathematical papers and accomplishments in hopes of inspiring historians to examine and analyze his original works. This biography can be found in the online journal Convergence, available at the website of the Mathematical Association of America (www.maa.org). As noted in our biography, all but one of Murphy’s papers are available for download from Google Books or the Journal Storage Database (JSTOR). Murphy’s first paper, Refutation of a Pamphlet Written by the Rev. John Mackey Entitled “A Method of Making a Cube a Double of a Cube, Founded on the Principles of Elementary Geometry,” wherein His Principles Are Proved Erroneous and the Required Solution Not Yet Obtained [1824], has not been available in the United States. As a result, the authors here provide a transcript of the paper with commentary. Refutation of a Pamphlet ... 1

The Odd nilHecke Algebra and its Diagrammatics

We introduce an odd version of the nilHecke algebra and develop an odd analogue of the thick diagrammatic calculus for nilHecke algebras. We graphically describe idempotents which give a Morita equivalence between odd nilHecke algebras and the rings of odd symmetric functions in finitely many variables. Cyclotomic quotients of odd nilHecke algebras are Morita equivalent to rings which are odd analogues of the cohomology rings of Grassmannians. Like their even counterparts, odd nilHecke algebras categorify the positive half of quantum sl(2).

Algebraic foundations of split hypercomplex nonlinear adaptive filtering

A split hypercomplex learning algorithm for the training of nonlinear finite impulse response adaptive filters for the processing of hypercomplex signals of any dimension is proposed. The derivation strictly takes into account the laws of hypercomplex algebra and hypercomplex calculus, some of which have been neglected in existing learning approaches (e.g., for quaternions). Already in the case of quaternions, we can predict improvements in performance of hypercomplex processes. The convergence of the proposed algorithms is rigorously analyzed. Copyright © 2012 John Wiley & Sons, Ltd.

Completeness of algebraic CPS simulations

The algebraic lambda calculus and the linear algebraic lambda calculus are two extensions of the classical lambda calculus with linear combinations of terms. They arise independently in distinct contexts: the former is a fragment of the differential lambda calculus, the latter is a candidate lambda calculus for quantum computation. They differ in the handling of application arguments and algebraic rules. The two languages can simulate each other using an algebraic extension of the well-known call-by-value and call-by-name CPS translations. These simulations are sound, in that they preserve reductions. In this paper, we prove that the simulations are actually complete, strengthening the connection between the two languages.