Riemann Integral Research Articles

Random Numbers And Monte Carlo Approximation In Fuzzy Riemann Integral

In this paper, we want to improve Monte Carlo approximation in fuzzy Riemann integral it means that calculate exact amount of fuzzy Riemann integral based on \(\alpha\)-level sets with partition of generating function of random numbers' RAND' in commercial software MATLAB.

An Analytical Formula for Potential Water Vapor in an Atmosphere of Constant Lapse Rate

Accurate calculation of precipitable water vapor (PWV) in the atmosphere has always been a matter of importance for meteorologists. Potential water vapor (POWV) or maximum precipitable water vapor can be an appropriate base for estimation of probable maximum precipitation (PMP) in an area, leading to probable maximum flood (PMF) and flash flood management systems. PWV and POWV have miscellaneously been estimated by means of either discrete solutions such as tables, diagrams or empirical methods; however, there is no analytical formula for POWV even in a particular atmospherical condition. In this article, fundamental governing equations required for analytical calculation of POWV are first introduced. Then, it will be shown that this POWV calculation relies on a Riemann integral solution over a range of altitude whose integrand is merely a function of altitude. The solution of the integral gives rise to a series function which is bypassed by approximation of saturation vapor pressure in the range of -55 to 55 degrees Celsius, and an analytical formula for POWV in an atmosphere of constant lapse rate is proposed. In order to evaluate the accuracy of the suggested equation, exact calculations of saturated adiabatic lapse rate (SALR) at different surface temperatures were performed. The formula was compared with both the diagrams from the US Weather Bureau and SALR. The results demonstrated unquestionable capability of analytical solutions and also equivalent functions.

Weaker forms of continuity and vector-valued Riemann integration

It was proved by Kadets that a weak$^{*}$-continuous function on $[0,1]$ taking values in the dual of a Banach space $X$ is Riemann-integrable precisely when $X$ is finite-dimensional. In this note, we prove a Fréchet-space analogue of this result by

Mean Value Integral Inequalities

Let \(F:[a,b]\longrightarrow \R\) have zero derivative in a dense subset of \([a,b]\). What else we need to conclude that \(F\) is constant in \([a,b]\)? We prove a result in this direction using some new Mean Value Theorems for integrals which are the real core of this paper. These Mean Value Theorems are proven easily and concisely using Lebesgue integration, but we also provide alternative and elementary proofs to some of them which keep inside the scope of the Riemann integral.

Riemann Integral of Functions from R into n-dimensional Real Normed Space

Riemann Integral of Functions from R into n-dimensional Real Normed Space

Henstock-Kurzweil Integral on [a,b

The theory of the Riemann integral was not fully satisfactory. Many important functions do not have a Riemann integral. So, Henstock and Kurzweil make the new theory of integral. From the background, the writer will be research about Henstock-Kurzweil integral and also theorems of Henstock- Kurzweil Integral. Henstock- Kurzweil Integral is generalized from Riemann integral. In this case the writer uses research methods literature or literature study carried out by way explore, observe, examine and identify the existing knowledge in the literature. In this thesis explain about partition which used in Henstock- Kurzweil Integral, definition and some property of Henstock- Kurzweil Integral. And some properties of Henstock- Kurzweil integral as follows: value of the Henstock- Kurzweil integral is unique, linearity of the Henstock-Kurzweil integral, Additivity of the Henstock-Kurzweil integral, Cauchy criteria, nonnegativity of Henstock-Kurzweil integral and primitive function.

Calculus and analysis: a combined approach

Preface. 0.1 Short Introduction. 0.2 Introduction. Acknowledgments. 1 Calculus I. 1.1 A Sketch of the Development of Rigor in Calculus and Analysis. 1.2 Basics. 1.3 Limits and Continuous Functions. 1.4 Differentiation. 1.5 Applications of Differentiation. 1.6 Riemann Integration. 2 Calculus II. 2.1 Techniques of Integration. 2.2 Improper Integrals. 2.3 Series of Real Numbers. 2.4 Series of Functions. 2.5 Analytical Geometry and Elementary Vector Calculus. 3 Calculus III. 3.1 Vector-Valued Functions of Several Variables. 3.2 Derivatives of Vector-Valued Functions of Several Variables. 3.3 Applications of Differentiation. 3.4 Integration of Functions of Several Variables. 3.5 Vector Calculus. 3.6 Generalizations of the Fundamental Theorem of Calculus. Appendix A. A.1 Construction of the Real-Number System. A.2 The Lebesgue Criterion for Riemann Integrability. A.3 Properties of the Determinant. A.4 The Inverse Mapping Theorem. References. Index of Notation. Index of Terminology. Problem Solutions.

CALCULUS ON FRACTAL SUBSETS OF REAL LINE — II: CONJUGACY WITH ORDINARY CALCULUS

Calculus on fractals, or Fα-calculus, developed in a previous paper, is a calculus based fractals F ⊂ R, and involves Fα-integral and Fα-derivative of orders α, 0 < α ≤ 1, where α is the dimension of F. The Fα-integral is suitable for integrating functions with fractal support of dimension α, while the Fα-derivative enables us to differentiate functions like the Cantor staircase. Several results in Fα-calculus are analogous to corresponding results in ordinary calculus, such as the Leibniz rule, fundamental theorems, etc. The functions like the Cantor staircase function occur naturally as solutions of Fα-differential equations. Hence the latter can be used to model processes involving fractal space or time, which in particular include a class of dynamical systems exhibiting sublinear behaviour. In this paper we show that, as operators, the Fα-integral and Fα-derivative are conjugate to the Riemann integral and ordinary derivative respectively. This is accomplished by constructing a map ψ which takes Fα-integrable functions to Riemann integrable functions, such that the corresponding integrals on appropriate intervals have equal values. Under suitable conditions, a restriction of ψ also takes Fα-differentiable functions to ordinarily differentiable functions such that their values at appropriate points are equal. Further, this conjugacy is generalized to one between Sobolev spaces in ordinary calculus and Fα-calculus. This conjugacy is useful, among other things, to find solutions to Fα-differential equations: they can be mapped to ordinary differential equations, and the solutions of the latter can be transformed back to get those of the former. This is illustrated with a few examples.

CALCULUS ON FRACTAL CURVES IN Rn

A new calculus on fractal curves, such as the von Koch curve, is formulated. We define a Riemann-like integral along a fractal curve F, called Fα-integral, where α is the dimension of F. A derivative along the fractal curve called Fα-derivative, is also defined. The mass function, a measure-like algorithmic quantity on the curves, plays a central role in the formulation. An appropriate algorithm to calculate the mass function is presented to emphasize its algorithmic aspect. Several aspects of this calculus retain much of the simplicity of ordinary calculus. We establish a conjugacy between this calculus and ordinary calculus on the real line. The Fα-integral and Fα-derivative are shown to be conjugate to the Riemann integral and ordinary derivative respectively. In fact, they can thus be evalutated using the corresponding operators in ordinary calculus and conjugacy. Sobolev Spaces are constructed on F, and Fα-differentiability is generalized. Finally we touch upon an example of absorption along fractal paths, to illustrate the utility of the framework in model making.

Factors associated with success in a calculus course: an examination of personal variables

This study examined relationships between students’ personal variables (gender, prior achievements, age and academic major) and their success in the first year undergraduate calculus course. The study sample consisted of 59 first year undergraduate students taking Math 154 Calculus II course. A written test about integral, sequence and series including demographic survey items was used to gather data. The test was administered prior to and upon the completion of the calculus course. Multiple regression analysis result indicated that there is relationship between students’ personal variables (gender and prior achievements) and their success. Gender differences favouring males typically occurred on Riemann sum and Riemann integral.

Riemann Integral of Functions from R into Real Normed Space

Riemann Integral of Functions from R into Real Normed Space In this article, we define the Riemann integral on functions from R into real normed space and prove the linearity of this operator. As a result, the Riemann integration can be applied to a wider range of functions. The proof method follows the [16].

A Counterexample for the Change of Variable Formula in KH Integrals

It was shown in \cite{B} and \cite{Ke} that if \(F(x)\) and \(\Psi(x)\) are Riemann integrals of the form \(\int_a^x f\text{dx}\) and \(\int_b^x \psi \text{dx}\) resp., then \(\psi\cdot f\circ\Psi\), if defined, is Riemann integrable. Furthermore, the change of variable formula applies, giving \(\int_b^x \psi\cdot f\circ\Psi \, \text{dx} = F(\psi(x))-F(\psi(b))\). It is natural to try to generalize this theorem to the Kurzweil-Henstock integral (this question was also dealt with in a paper by the author \cite{Be}); in other words, assuming that \(F\) and \(\Psi\) are KH integrals of \(f\) and \(\psi\) resp., one would expect that \(\psi\cdot f\circ\Psi\) be KH integrable. We show in this paper that this is false, and produce a counterexample based on the middle-third Cantor set and some rudiments of fractal geometry. In other words, by a well known theorem, we prove that the composition of two ACG functions needs not be ACG (in fact, we prove more generally that the composition of two absolutely continuous functions needs not be ACG). Of course, examples that show that the composition of two absolutely continuous functions needs not be absolutely continuous exist in the context of the Lebesgue integral, but since KH integrals need not be absolutely continuous, one cannot infer from these examples the validity of the above claim in the context of KH integration. On the other hand, the subtle method developed in this paper seems to be new, is entirely constructive, and we believe it could be applied to other interesting constructions. \vskip 0pt plus 10pt

A Feynman-Kac Solution to a Random Impulsive Equation of Schrödinger Type

If a force is applied to a particle undergoing Brownian motion, the resulting motion has a state function which satisfies a diffusion or Schrödinger-type equation. We consider a process in which Brownian motion is replaced by a process which has Brownian transitions at all times other than random times at which the transitions have an additional "impulsive" displacement. Using a Feynman-Kac formulation based on generalized Riemann integration, we examine the resulting equation of motion.

A Streamlined Proof of an Essential Calculus Fact

We present a short, classroom-friendly proof of a standard theorem of calculus: continuity on an interval implies Riemann integrability on that interval.

A TWO POINTS TAYLOR'S FORMULA FOR THE GENERALISED RIEMANN INTEGRAL

A two points Taylor’s formula for the generalised Riemann integral and various bounds for the remainder are established. Moreover, particular instances of interest are given.

Generalizing a limit description of the natural logarithm

If f is a continuous positive-valued function defined on the closed interval from a to x and if k 0 > 0, then . This generalizes a result of Finlayson describing ln(x) as a certain limit. Applications include new descriptions of arcsin(x) and arctan(x) as limits. This note could find use as an enrichment material in an introductory calculus course in the unit(s) introducing the Riemann integral and its applications.

A computer-verified monadic functional implementation of the integral

We provide a computer-verified exact monadic functional implementation of the Riemann integral in type theory. Together with previous work by O’Connor, this may be seen as the beginning of the realization of Bishop’s vision to use constructive mathematics as a programming language for exact analysis.

Monotone Convergence Theorem for the Riemann Integral

The monotone convergence theorem holds for the Riemann integral, provided (of course) it is assumed that the limit function is Riemann integrable. It might be thought, though, that this would be difficult to prove and inappropriate for an undergraduate course. In fact the identity is elementary: in the Lebesgue theory it is only the integrability of the limit function that is deep. This article shows how to prove the monotone convergence theorem for Riemann integrals using a simple compactness argument (i.e., invoking Cousin's lemma). This material could reasonably and appropriately be used in classroom presentations where the students are indoctrinated on this antiquated, but still popular, integration theory.

Exact solutions of KdV equation with time-dependent coefficients

In this paper, we study the Korteweg-de Vries (KdV) equation having time dependent coefficients from the Lie symmetry point of view. We obtain Lie point symmetries admitted by the equation for various forms for the time-dependent coefficients. We use the symmetries to construct the group-invariant solutions for each of the cases of the arbitrary coefficients. Subsequently, the 1-soliton solution is obtained by the aid of solitary wave ansatz method. It is observed that the soliton solution will exist provided that these time-dependent coefficients are all Riemann integrable.

A geometric note on integration

We propose a decomposition principle for the Riemann–Stieltjes integration and study some of its implications which lead to three distinct forms for the Riemann integral for monotonically varying differentiable functions.