Riemann Integral Research Articles

Strong derivative and the essentially Riemann integral

We characterize the primitives of Riemann integrable functions using the concept of strong differentiability. We also introduce the notion of essentially Riemann integrable function and characterize it descriptively using its primitive. We also give a characterization of this function that parallels Lebesgue's criterion for Riemann integrability.

Calculus for fuzzy functions with strongly linearly independent fuzzy coefficients

This paper establishes a theory of calculus for fuzzy-number-valued functions whose range is embedded in a specific Banach space of fuzzy numbers. These spaces are generated by strongly linearly independent (SLI) fuzzy numbers whose operations are induced by an isomorphism with Rn. We use the notion of Fréchet differentiability and Riemann integrability for these functions and present a fundamental theorem of calculus. Lastly, we develop a theory of fuzzy differential equations and provide an existence and uniqueness theorem.

Some Integral Inequalities for Generalized Convex Fuzzy-Interval-Valued Functions via Fuzzy Riemann Integrals

In this study, we introduce the new concept of h-convex fuzzy-interval-valued functions. Under the new concept, we present new versions of Hermite–Hadamard inequalities (H–H inequalities) are called fuzzy-interval Hermite–Hadamard type inequalities for h-convex fuzzy-interval-valued functions (h-convex FIVF) by means of fuzzy order relation. This fuzzy order relation is defined level wise through Kulisch–Miranker order relation defined on fuzzy-interval space. Fuzzy order relation and inclusion relation are two different concepts. With the help of fuzzy order relation, we also present some H–H type inequalities for the product of h-convex FIVFs. Moreover, we have also established strong relationship between Hermite–Hadamard–Fej´er (H–H–Fej´er) type inequality and h-convex FIVF. There are also some special cases presented that can be considered applications. There are useful examples provided to demonstrate the applicability of the concepts proposed in this study. This paper's thoughts and methodologies could serve as a springboard for more research in this field.

Statistical Riemann and Lebesgue Integrable Sequence of Functions with Korovkin-Type Approximation Theorems

In this work we introduce and investigate the ideas of statistical Riemann integrability, statistical Riemann summability, statistical Lebesgue integrability and statistical Lebesgue summability via deferred weighted mean. We first establish some fundamental limit theorems connecting these beautiful and potentially useful notions. Furthermore, based upon our proposed techniques, we establish the Korovkin-type approximation theorems with algebraic test functions. Finally, we present two illustrative examples under the consideration of positive linear operators in association with the Bernstein polynomials to exhibit the effectiveness of our findings.

Connection between the Riemann integrability of a multi-valued function and of its convex hull

For a Banach space X we demonstrate the equivalence of the following two properties:(1) X is B-convex (that is, possesses a nontrivial infratype), and(2) if F:[0,1]→2X∖{∅} is a multifunction with bounded values, convF denotes the multifunction t↦conv(F(t)), then the Riemann integrability of convF is equivalent to the Riemann integrability of F.For multifunctions with relatively norm compact values the Riemann integrability of convF is equivalent to the Riemann integrability of F without any restrictions on the Banach space X.

Neutrosophic Riemann integration and its properties

Neutrosophic Riemann integration and its properties

A Remark on the Change of Variable Theorem for the Riemann Integral

In 1961, Kestelman first proved the change in the variable theorem for the Riemann integral in its modern form. In 1970, Preiss and Uher supplemented his result with the inverse statement. Later, in a number of papers (Sarkhel, Výborný, Puoso, Tandra, and Torchinsky), the alternative proofs of these theorems were given within the same formulations. In this note, we show that one of the restrictions (namely, the boundedness of the function f on its entire domain) can be omitted while the change of variable formula still holds.

Implementation of computation formulas for certain classes of Apostol-type polynomials and some properties associated with these polynomials

The main purpose of this paper is to present various identities and computation formulas for certain classes of Apostol-type numbers and polynomials. The results of this paper contain not only the $\lambda$-Apostol-Daehee numbers and polynomials, but also Simsek numbers and polynomials, the Stirling numbers of the first kind, the Daehee numbers, and the Chu-Vandermonde identity. Furthermore, we derive an infinite series representation for the $\lambda$-Apostol-Daehee polynomials. By using functional equations containing the generating functions for the Cauchy numbers and the Riemann integrals of the generating functions for the $\lambda$-Apostol-Daehee numbers and polynomials, we also derive some identities and formulas for these numbers and polynomials. Moreover, we give implementation of a computation formula for the $\lambda$-Apostol-Daehee polynomials in Mathematica by Wolfram language. By this implementation, we also present some plots of these polynomials in order to investigate their behaviour some randomly selected special cases of its parameters. Finally, we conclude the paper with some comments and observations on our results.

Applications of Squeeze Theorem to Limiting Processes Involving Riemann Integration

The goal of this Classroom Capsule is to illustrate applications of the Squeeze Theorem, specifically when calculating limits of Riemann Integrals of sequences of functions. Students are first expo...

A Bibliometric Analysis for Lebesgue Measure Integration in Optimization

In solving mathematical problems so far, Riemann's integral theory is quite adequate for solving pure mathematics and applications problems. But not all problems can be solved using this integration, such as a discontinuous function that is not Riemann's integration. Lebesgue integral is an integration concept based on measure and can solve finite and unlimited function problems and be solved in a more general set domain. One of the bases of this integration is the Lebesgues measure includes the set of real numbers, where the length of the interval is the endpoints. The alternative use of this integral is widely used in various studies such as partial differential equations, quantum mechanics, and probabilistic analysis, requiring the integration of arbitrary set functions. This paper will show a comprehensive bibliometric survey of peer-reviewed articles referring to Lebesgue measure in integration. Search results are obtained 832 papers in the google scholar database and 997 papers using Lebesgue measure integration in optimization. It can also be seen that the research have 4 clusters and 3 clusters respectively with scattered keywords for each cluster. Finally, using bibliographic data can be obtained Lebesgues measure in integration and optimization supports many of the research and provides productive citations to citing the study.

Fractionally integrated Gauss-Markov processes and applications

We investigate the stochastic processes obtained as the fractional Riemann-Liouville integral of order α∈(0,1) of Gauss-Markov processes. The general expressions of the mean, variance and covariance functions are given. Due to the central role, for the fractional integral of standard Brownian motion and of the non-stationary/stationary Ornstein-Uhlenbeck processes, the covariance functions are carried out in closed-form. In order to clarify how the fractional order parameter α affects these functions, their numerical evaluations are shown and compared also with those of the corresponding processes obtained by ordinary Riemann integral. The results are useful for fractional neuronal models with long range memory dynamics and involving correlated input processes. The simulation of these fractionally integrated processes can be performed starting from the obtained covariance functions. A suitable neuronal model is proposed. Graphical comparisons are provided and discussed.

A NEW HARMONIC MEAN DERIVATIVE-BASED SIMPSON’S 1/3-TYPE SCHEME FOR RIEMANN- STIELTJES INTEGRAL

In this research paper, a new harmonic mean derivative-based Simpson’s 1/3 scheme has been presented for the Riemann-Stieltjes integral (RS-integral). The basic and composite forms of the proposed scheme with local and global error terms have been derived for the RS-integral. The proposed scheme has been reduced using g(t) = t for Riemann integral. Experimental work has been discussed to verify the theoretical results of the new proposed scheme against existing schemes using MATLAB. The order of accuracy, computational cost and average CPU time (in seconds) of the new proposed scheme have been computed. Finally, it is observed from computational results that the proposed scheme is better than existing schemes

EFFICIENT DERIVATIVE-BASED SIMPSON’S 1/3-TYPE SCHEME USING CENTROIDAL MEAN FOR RIEMANN-STIELTJES INTEGRAL

In this paper, a new efficient derivative-based quadrature scheme of Simpson’s 1/3-type is proposed using the centroidal mean for the approximation of Riemann-Stieltjes integral (RS-integral). Theorems are proved related to the basic form, composite form, local and global errors of the new scheme for the RS-integral. The reduction of the new proposed scheme is verified using g(t) = t for Riemann integral. The theoretical results of new proposed scheme have been proved by experimental work using programming in MATLAB against existing schemes. The order of accuracy, computational cost and average CPU time (in seconds) of the new proposed scheme are determined. The results obtained show the effectiveness of the proposed scheme compared to the existing schemes.

HERONIAN MEAN DERIVATIVE-BASED SIMPSON’S-TYPE SCHEME FOR RIEMANN-STIELTJES INTEGRAL

In this paper, a new heronian mean derivative-based quadrature scheme of Simpson’s 1/3-type is proposed for the approximation of the Riemann-Stieltjes integral (RS-integral). Theorems are proved related to the basic form, composite form, local and global errors of the new scheme for the RS-integral. The reduction of the new proposed scheme is verified using g(t) = t for Riemann integral. The theoretical results of the new proposed scheme have been proved by experimental work using programming in MATLAB against existing schemes. The order of accuracy, computational cost and average CPU time (in seconds) of the new proposed scheme are determined. The results obtained show the effectiveness of the proposed scheme compared to the existing schemes.

Extended Error Expansion of Classical Midpoint Rectangle Rule for Cauchy Principal Value Integrals on an Interval

The classical composite midpoint rectangle rule for computing Cauchy principal value integrals on an interval is studied. By using a piecewise constant interpolant to approximate the density function, an extended error expansion and its corresponding superconvergence results are obtained. The superconvergence phenomenon shows that the convergence rate of the midpoint rectangle rule is higher than that of the general Riemann integral when the singular point coincides with some priori known points. Finally, several numerical examples are presented to demonstrate the accuracy and effectiveness of the theoretical analysis. This research is meaningful to improve the accuracy of the collocation method for singular integrals.

Simpson’s rule for the Riemann-Stieltjes integral

This paper utilizes the concept of algebraic precision to construct Simpson’s rule for the Riemann-Stieltjes integral. This kind of quadrature rule obtains two orders of algebraic precision. And the error term of the Simpson’s rule is investigated. At last, it is noted that the convergence of this method is 2, worse than the Simpson’s rule for the Riemann integral, which has a convergence of order 3.

Homotopy analysis Shehu transform method for solving fuzzy differential equations of fractional and integer order derivatives

In this paper, we propose the fuzzy Shehu transform method (FSTM) using Zadeh’s decomposition theorem and fuzzy Riemann integral of real-valued functions on finite intervals. As an alternative to standard fuzzy Laplace transform and the fuzzy Sumudu integral transform, we established some potential useful (new or known) properties of the FSTM and validate their applications. Furthermore, the FSTM is coupled with the well-known homotopy analysis method to obtain the approximate and exact solutions of fuzzy differential equations of integer and non-integer order derivatives. The convergence analysis and the error analysis of the suggested technique are provided and supported by graphical solutions. Comparison of the numerical simulations of exact and approximate solutions of two fuzzy fractional partial differential equations are tabulated to further justify the reliability and efficiency of the proposed method.

The Black–Scholes Equation with Impulses at Random Times Via Generalized Riemann Integral

The classical pricing theory requires that the simple sets of outcomes are extended, using the Kolmogorov Extension Theorem, to a sigma-algebra of measurable sets in an infinite-dimensional sample space whose elements are continuous paths; the process involved are represented by appropriate stochastic differential equations (using Itô calculus); a suitable measure for the sample space can be found by means of the Girsanov and Radon–Nikodym Theorems; the derivative asset valuation is determined by means of an expression using Lebesgue integration. It is known that if we replace Lebesgue’s by the generalized Riemann integration to obtain the expectation, the same result can be achieved by elementary methods. In this paper, we consider the Black–Scholes PDE subject to impulse action. We replace the process which follows a geometric Brownian motion by a process which has additional impulsive displacements at random times. Instead of constants, the volatility and the risk-free interest rate are considered as continuous functions which can vary in time. Using the Feynman–Ka[Formula: see text] formulation based on generalized Riemann integration, we obtain a pricing formula for a European call option which copes with many discontinuities. This paper seeks to develop techniques of mathematical analysis in derivative pricing theory which are less constrained by the standard assumption of lognormality of prices. Accordingly, the paper is aimed primarily at analysis rather than finance. An example is given to illustrate the main results.

Calculus for linearly correlated fuzzy number-valued functions

This paper establishes the calculus for linearly correlated fuzzy number-valued functions. In particular, a definition of derivative is introduced by using representation functions and a linear isomorphism when the basic fuzzy number is non-symmetric. Our definition is more general than that leading to concept of Fréchet derivative. Derivative and Riemann integral are introduced using the canonical form of a linearly correlated fuzzy number-valued function when the basic fuzzy number is symmetric. Some relevant properties of derivatives and integrals of linearly correlated fuzzy number-valued functions are further investigated for symmetric and non-symmetric basic fuzzy number, respectively.

Riemann intergability versus continuity for vector-valued functions

The interplay between Riemann integrability and continuity is an interesting topic of modern analysis. In this paper, Riemann integrability of vector-valued continuous functions, property of Lebesgue and weak property of Lebesgue are surveyed and discussed. We also prove that `1(N, X) has the property of Lebesgue.