Elementary Calculus Research Articles

An Information-Theoretic Proof of a Hypercontractive Inequality

The famous hypercontractive estimate discovered independently by Gross, Bonami and Beckner has had a great impact on combinatorics and theoretical computer science since it was first used in this setting in a seminal paper by Kahn, Kalai and Linial. The usual proofs of this inequality begin with two-point space, where some elementary calculus is used and then generalised immediately by introducing another dimension using submultiplicativity (Minkowski’s integral inequality). In this paper, we prove this inequality using information theory. We compare the entropy of a pair of correlated vectors in {0,1}n to their separate entropies, analysing them bit by bit (not as a figure of speech, but as the bits are revealed) using the chain rule of entropy.

A local approach to stability groups

In this short note we prove a local version of Philip Hall’s theorem on the nilpotency of the stability group of a chain of subgroups by only using elementary commutator calculus (Hall’s theorem is a direct consequence of our result). This provides a new way of dealing with stability groups.

The energy-balance method for optimal control in renewable energy applications

A theoretical method is presented, called the energy-balance method, for maximising the energy extracted from a renewable energy converter in terms of determination of an optimal control. The method applies to control systems specified by linear graphs, and graph-theoretic techniques are employed. The method simplifies a number of optimal control problems by essentially expressing the performance objective — maximising energy extraction — in terms of an equivalent objective involving fewer variables, thereby reducing the complexity of the optimisation. As illustrated, in certain cases the optimal control problem may be reduced to one solvable by elementary calculus techniques. The theory is illustrated with examples from solar, wave and wind applications.

On Liouville’s Theorem for Conformal Maps

A theorem of Liouville asserts that the simplest angle-preserving transformations on Euclidean space—translations, dilations, reflections, and inversions—generate all angle-preserving transformations when the dimension is at least 3. This note gives a proof which uses only elementary multivariable calculus and simplifies a differential-geometric argument of Flanders.

On a Family of Polyhedral Singular Vertices

We provide a local description of the curves with minimal length based at singularities in a family of polyhedral surfaces. These singularities are accumulation points of vertices with conical angles equal to π and 4π (or 3π, in a variation). While a part of the minimizing curves behaves quite like the ones reaching conical vertices, the singularities present features such as being connected to points arbitrarily close to them by exactly two minimizing curves. The spaces containing such singularities are constructed as metric quotients of an euclidean half-disk by certain identification patterns along its edge. These patterns are examples of what is known as paper-folding schemes, and we provide the foundational aspects about them which are necessary for our analysis. The arguments are based on elementary metric geometry and calculus.

A connection between Gompertz diffusion model and Vasicek Interest Rate model

The main goal of this paper is to establish a new connection between the Gompertz diffusion model and the Vasicek Interest Rate model. These connection focus on elementary stochastic calculus and Itô's calculus. Firstly, we prove that the exponential of the Vasicek Interest Rate model is a Gompertz diffusion process. Secondly, we prove that the logarithm of the Gompertz diffusion process is a Vasicek Interest Rate model. New computations of the probability transition density function and the mean functions of the processes have quite simple formulations.

Some remarks on the links between the VIR and Gompertz diffusion models and the links between the CIR and Rayleigh diffusion models

The main purpose of this work is to establish new links between the stochastic Cox-Ingersoll-Ross (CIR) model and the stochastic Rayleigh diffusion process (SRDP) and the links between the Vasicek Interest Rate (VIR) process and the stochastic Gompertz diffusion process (SGDP). These links focus on elementary stochastic calculus and Itô’s calculus. Firstly, we prove that the square root of the CIR model is a SRDP. Secondly, we prove that the square of the SRDP is a CIR model. Thirdly, we prove that the exponential of the VIR model is a SGDP. Finally, we prove that the logarithm of the SGDP is a VIR model. New computations of the probability transition density function (PTDF) and the trend functions of the processes have quite simple formulations.

Amending the Traditional ‘ACI Commentary Jc Method’ and Other Sources

This article aims to amend the traditional formulas for polar moment of inertia suggested in Section R8.4.4.2.3 of ACI 318-19 and other sources. The authors claim that these formulas have been incorrectly derived as far back as 1960s due to an incorrect implementation of Steiner’s theorem (parallel axis theorem) for sections spanning in three-dimensional space. To support the claim, a formal proof using elementary calculus is presented in order to obtain the correct formulas with mathematical rigor. Then, the implications of using the traditional formulas versus the correct ones are investigated with the solution of a design problem related to a combined footing. Doi: 10.28991/CEJ-2023-09-11-015 Full Text: PDF

Odd-Period Cycles of the Logistic Map

We locate the onset of odd-period cycles of the logistic map using only elementary algebra and calculus.

On Unsteady Internal Flows of Incompressible Fluids Characterized by Implicit Constitutive Equations in the Bulk and on the Boundary

Long-time and large-data existence of weak solutions for initial- and boundary-value problems concerning three-dimensional flows of incompressible fluids is nowadays available not only for Navier–Stokes fluids but also for various fluid models where the relation between the Cauchy stress tensor and the symmetric part of the velocity gradient is nonlinear. The majority of such studies however concerns models where such a dependence is explicit (the stress is a function of the velocity gradient), which makes the class of studied models unduly restrictive. The same concerns boundary conditions, or more precisely the slipping mechanisms on the boundary, where the no-slip is still the most preferred condition considered in the literature. Our main objective is to develop a robust mathematical theory for unsteady internal flows of implicitly constituted incompressible fluids with implicit relations between the tangential projections of the velocity and the normal traction on the boundary. The theory covers numerous rheological models used in chemistry, biorheology, polymer and food industry as well as in geomechanics. It also includes, as special cases, nonlinear slip as well as stick–slip boundary conditions. Unlike earlier studies, the conditions characterizing admissible classes of constitutive equations are expressed by means of tools of elementary calculus. In addition, a fully constructive proof (approximation scheme) is incorporated. Finally, we focus on the question of uniqueness of such weak solutions.

Multigrid Methods for the Solution of Nonlinear Variational Inequalities

In this research, we investigate the numerical solution of second member problems that depend on the solution obtained through a multigrid method. Specifically, we focus on the application of multigrid techniques for solving nonlinear variational inequalities. The main objective is to establish the uniform convergence of the multigrid algorithm. To achieve this, we employ elementary subdifferential calculus and draw insights from the convergence theory of nonlinear multigrid methods.

Application of Vedic Formulas in Elementary Calculus

In this paper, applications using Vedic mathematics in Calculas is covered. Discovering the Vedic Mathematics' Contribution to Advanced Calculus is the scientist's main objective. Jagadguru Shree Bharti Krishna Tirth ji created Vedic Mathematics between 1884 and 1960, which was descended from the Vedas between 1911 and 1918. Our mathematics are based on sixteen sutras from the Atharva Vedas, according to Shree Bharti Krishna Tirth ji. It is used in a variety of topics, including science, engineering, astrology, astronomy, and others, in addition to basic arithmetic.

A symmetric version of the Euler equations by using Generalized Bernoulli Method

The aim of this article is to show a way to extend the usefulness of the Generalized Bernoulli Method (GBM) with the purpose to apply it for the case of variational problems with functionals that depend explicitly of all the variables. Moreover, after expressing the Euler equations in terms of this extension of GBM, we will see that the resulting equations acquire a symmetric form, which is not shared by the known Euler equations. We will see that this symmetry is useful because it allows us to recall these equations with ease. The presentation of three examples shows that by applying GBM, the Euler equations are obtained just as well as it does the known Euler formalism but with much less effort, which makes GBM ideal for practical applications. In fact, given a variational problem, GBM establishes the corresponding Euler equations by means of a systematic procedure, which is easy to recall, based in both elementary calculus and algebra without having to memorize the known formulas. Finally, in order to extend the practical applications of the proposed method, this work will employ GBM with the purpose to apply it for the case of solving isoperimetric problems.

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.

Residue Theorem and Its Application

Sometimes it can be frustrating to get the anti-derivatives of some functions. Some definite integrals are difficult, or even impossible, to calculate using the elementary calculus method, especially in the field of complex analysis. In such cases, Cauchy's Residue Theorem might be proper. In this article, we state the difference between Cauchy's Integral Theorem and the Residue Theorem, followed by the definitions of isolated singular point and residue with some examples given after. These are prerequisites for understanding the theorem. A short statement of the content of Cauchy's Residue Theorem has been made, according to which the integral of a function along a positively oriented simple closed contour is related to the residues, which can be obtained through Laurent expansion, of all the limited isolated singular points inside the contour. In addition, using the Cauchy-Goursat theorem, we developed the derivation of CR Theorem. It denotes that the value of a function around a simple closed contour is zero if the function is analytic at all points interior to and on that contour. Then we launch an application and accomplish it in two ways. The first method makes use of the Fourier Matrix, while the second is related to power series. In both of these approaches, we feel the efficacy of Cauchy's Residue Theorem.

Towards the Centenary of Sheffer Polynomial Sequences: Old and Recent Results

Sheffer’s work is about to turn 100 years after its publication. In reporting this important event, we recall some interesting old and recent results, aware of the incompleteness of the wide existing literature. Particularly, we recall Sheffer’s approach, the theory of Rota and his collaborators, the isomorphism between the group of Sheffer polynomial sequences and the so-called Riordan matrices group. This inspired the most recent approaches based on elementary matrix calculus. The interesting problem of orthogonality in the context of Sheffer sequences is also reported, recalling the results of Sheffer, Meixner, Shohat, and the very recent one of Galiffa et al., and of Costabile et al.

Deforming $||.||_{1}$ into $||.||_{\infty}$ via Polyhedral Norms: A Pedestrian Approach

We consider, and study with elementary calculus, the polyhedral norms $||x||_{(k)}=$ sum of the $\mathit{k}$ largest among the $|x_{i}|$'s. Besides their basic properties, we provide various expressions of the unit balls associated with them and determine all the facets and vertices of these balls. We do the same with the dual norm $||.||_{(k)}^{\ast }$ of $||.||_{(k)}$. The study of these polyhedral norms is motivated, among other reasons, by the necessity of handling sparsity in some modern optimization problems, as is explained at the end of the paper.

Maximum Liquefaction of Gases Revisited

Joule-Thomson liquefiers are the commonest machines to liquefy gases. Over the years, countless number of articles have been published on the subject. Dozens of 1st and 2nd law analyses were carried out on Joule-Thomson liquefaction cycles. And yet an aspect of purely theoretical interest seems to have passed unnoticed, namely: for a given volume of gas, what conditions should be fulfilled to achieve maximum liquefaction without considering engineering details of design equipment and the highly irreversible character of work-consuming devices, heat exchangers, heat leaks and the throttling process. This work addressed this issue by applying the 1st law analysis and elementary calculus prescriptions to a simple Linde-Hampson liquefying process. The same approach could be applied to other liquefying cycles. As is well-known, for a given mass flow rate of a gas, maximum fraction liquefied occurs when the pre-cooling temperature, Ti , and initial pressure, Pi , lie on the inversion curve. It has been proved that this is only true if an additional condition is fulfilled. Expressions for it were derived for the van der Waals, RKS and PR equations of state.

Back to basics: fundamental principles of system dynamics and queueing theory

Back to basics: fundamental principles of system dynamics and queueing theory

O uso de jogos matemáticos no trabalho com o cálculo mental

O cálculo mental consiste em diferentes estratégias que visamàobtenção de resultados por meio de métodos mais simples de resolução, tendo comoauxílioo cálculo escrito. Uma das possibilidades para desenvolvê-lo é a utilização dos jogos matemáticos, pois eles propiciam a observaçãoeo trabalho individual e coletivo. Neste artigo, buscou-seinvestigar como estudantes dos anos finais do ensino fundamental utilizam o cálculo mental no trabalho com jogos matemáticos. Para tal,procedeu-se auma pesquisa com um grupo de 16 alunos de uma escola da rede estadual de ensino localizada no município de Castelo, Espírito Santo. Fundamentou-seteoricamente em Parra (1996), Carvalho (2011), Lorenzato (2012), Kishimoto (2017), entre outros. As atividadesrealizadas permitiram concluir que,durante a aplicação dos jogos, os alunos tiveram maior comprometimento em realizar as tarefas. Identificou-seousodos algoritmos da adição da multiplicação como principais estratégias de cálculo mental.