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Interval-valued maps adjust integral inequalities using different types of ordering relations, including inclusion and center-radius, both of which behave differently. Our purpose was to develop various novel bounds and refinements for weighted Hermite-Hadamard inequalities as well as their product form by employing new types of fractional integral operators under a cr-order relation. Mostly authors have used inclusion order to adjust inequalities in interval maps, but they have some flaws, specifically they lack the property of comparability between intervals. However, we show that under cr-order, it satisfies all relational properties of intervals, including reflexivity, antisymmetry, transitivity, and comparability and preserves integrals as well. Furthermore, we provide numerous interesting remarks, corollaries, and examples in order to demonstrate the accuracy of our findings. [ABSTRACT FROM AUTHOR]

In this paper, we study the singularities of Feynman integrals by compactifying the integration domain as well as the ambient space of these integrals, by embedding them in higher-dimensional space. In this compactified space, the singularities occur due to the meeting of compactified propagators at non-general position. The present analysis, which had been previously used only for the singularities of second type, is used to study other kinds of singularities viz threshold, pseudo-threshold and anomalous threshold singularities. We study various one-loop and two-loop examples and obtain their singularities. We also present observations based on results obtained, that allow us to determine whether the singularities lie on the physical sheet or not for some simple cases. Thus, this work at the frontier of our knowledge of Feynman integral calculus sheds insight into the analytic structure. [ABSTRACT FROM AUTHOR]

This work was developed to present constructive steps of multiple integrals using the open-source software Geogebra. The main focus was directed towards creating three-dimensional graphs of integrals through Riemann sums in two variables. Some practical examples are developed to demonstrate the reliability of the presented results, which are compared using traditional algebraic methods and computed in Geogebra. With this, we aim to highlight the potential of Geogebra in teaching integral calculus and make the graphical visualization process less exhaustive. [ABSTRACT FROM AUTHOR]

In this paper we study the everywhere Hölder continuity of the minima of the following class of vectorial integral funcionals: with some general conditions on the density . We make the following assumptions about the function . Let be a bounded open subset of , with , and let be a Carathéodory function, where and with . We make the following growth conditions on : there exists a constant such that for a.e. , for every and every with , and with , for a.e. , , and . Assuming that the previous growth hypothesis holds, we prove the following regularity result. If is a local minimizer of the previous functional, then for every , with . The regularity of minimizers is obtained by proving that each component stays in a suitable De Giorgi class and, from this, we conclude Hölder continuity. [ABSTRACT FROM AUTHOR]

Following Faber–Pandharipande, we use the virtual localization formula for the moduli space of stable maps to $\mathbb {P}^{1}$ to compute relations between Hodge integrals. We prove that certain generating series of these integrals are polynomials. [ABSTRACT FROM AUTHOR]

Differential and Integral Calculus (CDI) is an essential subject for teaching Mathematics and other sciences. Despite this importance, we observed student failure and high rates of failure or dropout, which justifies the relevance of considering epistemological aspects that make it possible to understand different phenomena in the teaching of this discipline. In this sense, we propose a study of epistemological elements of density and mass knowledge, through multivariational integrals, since integral is essential knowledge for the area of exact sciences. The objective of this study is to investigate the development and implementation of an intervention proposal, centered on study and research activities, that offers CDI students opportunities to explore the concept of the integral of one and more variables. To this end, we carried out an investigation through the creation and implementation of an intervention based on work with task-solving episodes, in order to analyze generalization movement(s) that students performed to define a multivariational defined integral from defined integrals of a variable. The results showed that the students mobilized a set of knowledge of multiple integrals, from the context of calculating mass in one, two and three dimensions. Expansive generalization was used to expand procedural issues in the calculation of an integral, while reconstructive generalization was used to understand structural aspects of the Riemann integral of more than one variable. [ABSTRACT FROM AUTHOR]

We study Γ-convergence of nonconvex integrals of the calculus of variations in strongly connected sets when the integrands do not have polynomial growth and can take infinite values. Applications to homogenization of unbounded integrals in strongly perforated sets are also developed. [ABSTRACT FROM AUTHOR]

It has been shown that the findings of d -metric spaces may be deduced from S -metric spaces by considering d ϖ , ϰ = Λ ϖ , ϖ , ϰ . In this study, no such concepts that translate to the outcomes of metric spaces are considered. We establish standard fixed point theorems for integral type contractions involving rational terms in the context of complete S -metric spaces and discuss their implications. We also provide examples to illustrate the work. This paper's findings generalize and expand a number of previously published conclusions. In addition, the abstract conclusions are supported by an application of the Riemann-Liouville calculus to a fractional integral problem and a supportive numerical example. [ABSTRACT FROM AUTHOR]

This paper proves an equality for the case of twice-differentiable convex functions involving conformable fractional integrals. Using the established equality, we give new Simpson-type inequalities for the case of twice-differentiable convex functions via conformable fractional integrals. We also consider some special cases which can be deduced from the main results. [ABSTRACT FROM AUTHOR]

We investigate the optimal rate of convergence in the multidimensional normal approximation of vector-valued Wiener-Itô integrals whose components all belong to a fixed Wiener chaos with coinciding orders. By combining Malliavin calculus, Stein's method for normal approximation and the method of cumulants, we obtain the optimal rate of convergence with respect to a suitable smooth distance. As applications, we derive the optimal rates of convergence for complex Wiener-Itô integrals, vector-valued Wiener-Itô integrals with kernels of step functions and vector-valued Toeplitz quadratic functionals. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some reverses of Callebaut and Hölder inequalities for isotonic functionals via a reverse of Young's inequality we have established recently. Applications for integrals and n-tuples of real numbers are provided as well. [ABSTRACT FROM AUTHOR]

In this paper, we study the functorial descent from self-contragredient cuspidal automorphic representations π of GL7(A) with LS (s; π; ^ ³ / having a pole at s=1 to the split exceptional group G2(A), using Fourier coefficients associated to two nilpotent orbits of E7. We show that one descent module is generic, and under suitable local conditions, it is cuspidal and π is a weak functorial lift of each of its irreducible summands. This establishes the first functorial descent involving the exotic exterior cube L-function. However, we show that the other descent module supports not only the nondegenerate Whittaker–Fourier integral on G2(A) but also every degenerate Whittaker– Fourier integral. Thus it is generic, but not cuspidal. [ABSTRACT FROM AUTHOR]

In this paper, we study the functorial descent from self-contragredient cuspidal automorphic representations π of GL7(A) with LS (s; π; ^ ³ / having a pole at s=1 to the split exceptional group G2(A), using Fourier coefficients associated to two nilpotent orbits of E7. We show that one descent module is generic, and under suitable local conditions, it is cuspidal and π is a weak functorial lift of each of its irreducible summands. This establishes the first functorial descent involving the exotic exterior cube L-function. However, we show that the other descent module supports not only the nondegenerate Whittaker–Fourier integral on G2(A) but also every degenerate Whittaker– Fourier integral. Thus it is generic, but not cuspidal. [ABSTRACT FROM AUTHOR]

We consider a classical case of integrals containing an irrational integrand in the form of a square root of a quadratic polynomial. It is known that such "irrational integrals" can be expressed in terms of elementary functions by one of three of Euler's substitutions. It is less well known that the Euler substitutions have a geometric interpretation. In the framework of this interpretation, one can see that the number 3 is not the most suitable. We show that it is natural to introduce a fourth Euler substitution. In his original treatise, Leonhard Euler used two substitutions which are sufficient to cover all cases. [ABSTRACT FROM AUTHOR]

We study the mapping properties of singular integral operators along surfaces of revolutions on product domains. For several classes of surfaces, we prove sharp Lp bounds (1< p< ∞) for these singular integral operators as well as their corresponding maximal operators. By using these Lp bounds and an extrapolation argument we obtain the Lp boundedness of these operators under optimal conditions on the singular kernels. Our results extend and improve several results previously obtained by many authors. [ABSTRACT FROM AUTHOR]

This paper discusses the mathematical and didactical problems of teaching indefinite integral in the context of the ubiquitous availability of online integral calculators. The symbol of indefinite integral introduced by Leibniz, unfortunately, does not contain an indication of the interval on which the antiderivatives should be calculated. This yields mismatched results of calculating the same integral on different online calculators. The paper justifies the expediency of including the interval of integration in the designation of indefinite integral or, at least, in the settings of problems of its calculation. Indication of the integration interval ensures the uniqueness of the solution to the problem of calculating the indefinite integral, the uniqueness that does not occur in the traditional setting of the integration problem. This is important because online calculators provide users with formulas that differ from each other and can represent both different and coincidental antiderivatives of the same function. Analysis of the domains of definition of the obtained by calculators antiderivatives allows users to understand how they relate to each other. [ABSTRACT FROM AUTHOR]

We present the research outcomes of a project in Mathematics Education about the design and implementation of an instrument to learn the integral for the arc length of a function by using differential elements as the strategy for its construction. The research was done via a didactic intervention in a regular Integral Calculus course. The instrument was designed based on historical-epistemological analyses of the works of van Heuraet and Fermat. The main result of this research was that students achieve a more robust conceptualization of the integral for the arc length, supported by its construction with differential elements and its geometric foundation. [ABSTRACT FROM AUTHOR]

Copyright of Revista Integración is the property of Universidad Industrial de Santander and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

An efficient difference method is constructed for solving one-dimensional nonlinear time-space fractional Ginzburg-Landau equation. The discrete method is developed by adopting the - scheme to handle Caputo fractional derivative, while a fourth-order difference method is invoked for space discretization. The well-posedness and a priori bound of the numerical solution are rigorously studied, and we prove that the difference scheme is unconditionally convergent in pointwise sense with the rate of , where and are the time and space steps respectively. In addition, the proposed method is extended to solve two-dimensional problem, and corresponding theoretical analysis is established. Several numerical tests are also provided to validate our theoretical analysis. [ABSTRACT FROM AUTHOR]

Many researchers in recent years have studied fractional integrals and derivatives. Some authors recently introduced generalized fractional integrals, the so-called unified fractional integrals. In this article, we establish certain new integral inequalities by employing the unified fractional integral operators. In fact, for a class of n (n ∈ N); positive continuous and decreasing functions on [v1; v2]; certain new classes of integral inequalities are discussed. The inequalities established in this manuscript are more general forms of the classical inequalities given in the literature. The existing classical inequalities can be rectified by imposing the conditions stated in remarks. By imposing certain conditions on ħ and ᴧ available in the literature, many new forms of fractional integral inequalities can be produced. [ABSTRACT FROM AUTHOR]

Copyright of Paradigma is the property of Universidad Pedagogica Experimental Libertador and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Every application of integration by parts can be done with a tabular method. The trick is to identify and consider each new integral in the table before deciding how to proceed. This paper supplements a classic introduction to integration by parts with a particular tabular method called Row Integration by Parts (RIP). Approaches to tabular methods found online and in the literature often fail to clearly identify new integrals since the focus is on constructing columns, thus new integrals are ignored. With RIP, each integral is identified with a row and tables are built one row at a time. This allows RIP to be an efficient bookkeeping method and to apply to a much wider range of functions than those typically considered with other tabular methods. [ABSTRACT FROM AUTHOR]

The minor problem, namely the study of the spectrum of a principal submatrix of a Hermitian matrix taken at random on its orbit under conjugation, is revisited, with emphasis on the use of orbital integrals and on the connection with branching coefficients in the decomposition of an irreducible representation of U(n), resp. SU(n), into irreps of U(n - 1), resp. SU(n - 1). As is well known, the branching coefficients are trivial (equal to 0 or 1) for the branchings of U(n)⊃ U(n - 1), while they are not for SU(n) ⊃ SU(n - 1), where multiplicities may appear. In the latter case, the problem is shown to be related to the distribution of spacings in the minor problem. An explicit expression is obtained for the multiplicities, in terms of an integral stemming from the minor problem, and an Ansatz is given for a closed form expression for arbitrary n. [ABSTRACT FROM AUTHOR]

This note deals with some variants of the integral mean value theorem. Mainly a variant of Sahoo's theorem and a variant of Wayment's theorem were proved. Our approach is rather elementary and does not use advanced techniques from analysis. The simple auxiliary functions were used to prove the results. [ABSTRACT FROM AUTHOR]

Graphical analysis of the relationships between pairs of integrals and their corresponding derivatives resulted in the development of a procedure one may describe as 'pictorial integration'. The arctangent of the derivative of various functions was analysed to confirm that this accurately reports the magnitude of angles made by integral functions from the horizontal, for polynomial, trigonometric, exponential, and transcendental forms. This was used to generate pictures of certain functions that have no known simple algebraic formula. The pictorial representations for the integral of e x 2 and for the ellipse arc length integral are described. The pictures were confirmed through the use of Newton (Taylor) series expressions over appropriate domains, the second derivatives of the functions which are the first derivatives of their known derivatives, and electronic integration. Several powerful computer systems are available that quickly and easily graph such complicated integrals and confirm these results, but the analysis here explains for Calculus students the detailed graphical relationships between integrals and their derivatives. [ABSTRACT FROM AUTHOR]

We derive the general rules of functional integration in the theories of Schwarzian type, thus completing the elaboration of Schwarzian functional integrals calculus initiated in [1, 2]. Our approach is mathematically rigorous and does not contain any unproved conjectures. It is based on the analysis of the properties of the measures on the groups of diffeomorphisms, and does not appeal for the experience from other physical models. Its great merit consists in reducing a problem of functional integration to that of the only functional integral (24) that is calculated explicitly with the result written in the form of the ordinary integral. We evaluate two-point and four-point correlation functions defined as functional integrals over the groups and , and discuss the difference between the results in the two cases. [ABSTRACT FROM AUTHOR]

Integrals of the Calculus of Variations with p , q {p,q} -growth may have not smooth minimizers, not even bounded, for general p , q {p,q} exponents. In this paper we consider the scalar case, which contrary to the vector-valued one allows us not to impose structure conditions on the integrand f ⁢ (x , ξ) {f(x,\xi)} with dependence on the modulus of the gradient, i.e. f ⁢ (x , ξ) = g ⁢ (x , | ξ |) {f(x,\xi)=g(x,|\xi|)}. Without imposing structure conditions, we prove that if q p {\frac{q}{p}} is sufficiently close to 1, then every minimizer is locally Lipschitz-continuous. [ABSTRACT FROM AUTHOR]

In this work we give optimal, i.e., necessary and sufficient, conditions for integrals of the calculus of variations to guarantee the existence of solutions--both weak and variational solutions--to the associated L2-gradient flow. The initial values are merely L2 functions with possibly infinite energy. In this context, the notion of integral convexity, i.e., the convexity of the variational integral and not of the integrand, plays the crucial role; surprisingly, this type of convexity is weaker than the convexity of the integrand. We demonstrate this by means of certain quasi-convex and nonconvex integrands. [ABSTRACT FROM AUTHOR]

The subject of fractional calculus (that is, the calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past over four decades, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of mathematical, physical, engineering and statistical sciences. Various operators of fractional-order derivatives as well as fractional-order integrals do indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables. The main object of this survey-cum-expository article is to present a brief elementary and introductory overview of the theory of the integral and derivative operators of fractional calculus and their applications especially in developing solutions of certain interesting families of ordinary and partial fractional "differintegral" equations. This general talk will be presented as simply as possible keeping the likelihood of non-specialist audience in mind. [ABSTRACT FROM AUTHOR]

In a recent communication [V. Jadhao and N. Makri, J. Chem. Phys. 129, 161102 (2008)], we introduced an iterative Monte Carlo (IMC) path integral methodology for calculating complex-time correlation functions. This method constitutes a stepwise evaluation of the path integral on a grid selected by a Monte Carlo procedure, circumventing the exponential growth of statistical error with increasing propagation time, while realizing the advantageous scaling of importance sampling in the grid selection and integral evaluation. In the present paper, we present an improved formulation of IMC, which is based on a bead-adapted sampling procedure; thus leading to grid point distributions that closely resemble the absolute value of the integrand at each iteration. We show that the statistical error of IMC does not grow upon repeated iteration, in sharp contrast to the performance of the conventional path integral approach which leads to exponential increase in statistical uncertainty. Numerical results on systems with up to 13 degrees of freedom and propagation up to 30 times the “thermal” time hβ/2 illustrate these features. [ABSTRACT FROM AUTHOR]

As an application of atomistic simulation methods to heat capacities, path-integral molecular dynamics has been used to calculate the constant-volume heat capacities of light and heavy water in the gas, liquid, and solid phases. While the classical simulation based on conventional molecular dynamics has estimated the heat capacities too high, the quantum simulation based on path-integral molecular dynamics has given reasonable results based on the simple point-charge/flexible potential model. The calculated heat capacities (divided by the Boltzmann constant) in the quantum simulation are 3.1 in the vapor H2O at 300 K, 6.9 in the liquid H2O at 300 K, and 4.1 in the ice Ih H2O at 250 K, respectively, which are comparable to the experimental data of 3.04, 8.9, and 4.1, respectively. The quantum simulation also reproduces the isotope effect. The heat capacity in the liquid D2O has been calculated to be 10% higher than that of H2O, while it is 13% higher in the experiment. The results demonstrate that the path-integral simulation is a promising approach to quantitatively evaluate the heat capacities for molecular systems, taking account of quantum-mechanical vibrations as well as strongly anharmonic motions. [ABSTRACT FROM AUTHOR]

A set of general reweighted random series methods for metric affine spaces is developed. The extension of the theorems to metric affine spaces demands the introduction of a configuration-independent reference metric tensor; this geometric object is used to treat the path expansion coefficients beyond the core path, in both the partial averaging and the reweighted random series approach. Numerical tests are conducted by simulating a particle in a ring. The reweighted random series results show better convergence properties and better statistical quality at a fraction of the cost compared with the related partial averaging simulation. [ABSTRACT FROM AUTHOR]

The Kirkwood-Buff integrals of acetone-water mixtures are determined using two experimental techniques: small-angle neutron scattering and vapor pressure measurements, in order to test the precision and reliability that can be achieved. The data are then compared with those previously reported by different authors, which tend to show considerable variation between them. The various possible sources of inaccuracies are pointed out, both from experimental origins and from the numerical treatment of the data. Comparison with recent simulation results allows to critically compare different models and provide some information about the microstructure of the aqueous mixture. [ABSTRACT FROM AUTHOR]

The quantum instanton approximation for thermal rate constants, a type of quantum transition state theory (QTST), is applied to a model proton transfer reaction in liquid methyl chloride developed by Azzouz and Borgis. Monte Carlo path integral methods are used to carry out the calculations, and two other closely related QTST’s, namely, the centroid-density and Hansen–Andersen QTST, are also evaluated for comparison using the present path integral approach. A technique is then introduced that calculates the kinetic isotope effect directly via thermodynamic integration of the rate with respect to hydrogen mass, which has the practical advantage of avoiding costly evaluation of the activation free energy. The present application to the Azzouz–Borgis problem shows that the above three types of QTST provide very similar results for the rate, within 30% of each other, which is nontrivial considering the totally different derivations of these QTSTs; the latter rates are also in reasonable agreement with some other previous results (e.g., obtained via molecular dynamics with quantum transitions), within a factor of ∼2(7) for the H(D) transfer, thus significantly diminishing the possible range of the exact rates. In addition, it is revealed that a small but nonnegligible inconsistency exists in the parametrization of the Azzouz-Borgis model employed in previous studies, which resulted in the large apparent discrepancy in the calculated rates.© 2005 American Institute of Physics. [ABSTRACT FROM AUTHOR]

An exact quantum master equation formalism is constructed for the efficient evaluation of quantum non-Markovian dissipation beyond the weak system-bath interaction regime in the presence of time-dependent external field. A novel truncation scheme is further proposed and compared with other approaches to close the resulting hierarchically coupled equations of motion. The interplay between system-bath interaction strength, non-Markovian property, and required level of hierarchy is also demonstrated with the aid of simple spin-boson systems.© 2005 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Abstract The recently reported (Alcalá Ochoa, 2018) approximation to the Rayleigh–Sommerfeld diffraction integrals (RSDI) is used to obtain a set of expressions to calculate the diffraction properties of non-rotationally symmetric optical systems illuminated with polarization fields. It is proved that the three two-dimensional integrals of RSDI can be expressed accurately from two to eight one-dimensional integrals depending of the type of polarization studied. It is proved also that previous developments based on Taylor's approximations to the RSDI can be described by these new expressions. [ABSTRACT FROM AUTHOR]

Abstract Pseudothermal light by scattering laser light from rotating groundglass has been extensively applied to study optical coherence in both classical and quantum optics since its invention in the 1960s. In this paper, we will show that by replacing the intensity invariant laser light in pseudothermal light source with intensity modulated laser light, superbunching pseudothermal light can be obtained. Two-photon superbunching is experimentally observed by employing common instruments in optical laboratory. Two-photon interference in Feynman's path integral theory is employed to interpret the results. The proposed superbunching pseudothermal light is helpful to understand the physics of two-photon bunching and superbunching. Highlights • Superbunching pseudothermal light is obtained by varying laser light intensity. • Two-photon interference is employed to interpret two-photon superbunching effect. • Superbunching is experimentally observed with sinusoidal and random modulations. [ABSTRACT FROM AUTHOR]

The multiplier for the weak McShane integral which has been introduced by M. Saadoune and R. Sayyad (2014) is characterized. [ABSTRACT FROM AUTHOR]

In the paper, the authors introduce a new concept "strongly extended (s,m)-convex function" and establish some integral inequalities of Simpson's type for strongly extended (s,m)- convex functions. [ABSTRACT FROM AUTHOR]

In this paper, I will review an obstruction for theories of the beginning of the universe which can be formulated as semiclassical path integrals. Hartle and Hawking's no boundary proposal and Vilenkin's tunneling proposal are examples of such theories. Each may be formulated as the quantum amplitude for obtaining a final 3-geometry by integrating over 4-geometries. The result is obtained using a new mathematical tool — Picard–Lefschetz theory — for defining the semiclassical path integral for gravity. The Lorentzian path integral for quantum cosmology with a positive cosmological constant is mathematically meaningful in this approach, but the Euclidean version is not. Framed in this way, the resulting framework and predictions are unique. Unfortunately, the outcome is that primordial gravitational wave fluctuations are unsuppressed. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of determining the stabilizing proportional-integral-derivative controller regions for a general linear system with or without time delay in unity output feedback configuration. When such a problem is solved in the 3D space by a graphical method, it brings in the visualizing difficulties. This paper proposes a new method by projecting the stability boundary in the 3D to the parameterized stability boundary band in the 2D plane of (Kp,Ki) while Kd varies in an interval. The boundary band divides the plane to the regions for which simple and effective stability test is developed and the complete stabilizing regions in the 2D plane of (Kp,Ki) are determined for any given interval on Kd. The rules are presented to find conditionally stable band portions, within which the corresponding stabilizing sub-interval for Kd can be obtained from an analytical formula for the case of no band intersections and from the root locus on Kd for the case of the band intersections. Several examples are designed to cover different cases and illustrate the method. [ABSTRACT FROM AUTHOR]

This article concerns the existence of non-oscillatory solutions to the equation ..., where ak ≥ 0 and hk(t) ≥ t. We generalize existing results and then give an answer to the open question stated in [4]. Moreover we present a new condition based on the integral of (∑ ak)². [ABSTRACT FROM AUTHOR]