•Monte Carlo integration of derivatives of oscillatory functions is very expensive.•For Lebesgue integrals, integration by parts gives rise to a density gradient g.•g is defined as a directional ...derivative of density implied by a coordinate chart.•Computation of g requires both the Jacobian and Hessian of the chart.•First- and second-order tangent equations must be solved to find g on trajectories.
Lebesgue integration of derivatives of strongly-oscillatory functions is a recurring challenge in computational science and engineering. Integration by parts is an effective remedy for huge computational costs associated with Monte Carlo integration schemes. In case of Lebesgue integrals over a smooth manifold, however, integration by parts gives rise to a derivative of the density implied by charts describing the domain manifold. This paper focuses on the computation of that derivative, which we call the density gradient function, on general smooth manifolds. We analytically derive formulas for the density gradient and present examples of manifolds determined by popular differential equation-driven systems. We highlight the significance of the density gradient by demonstrating a numerical example of Monte Carlo integration involving oscillatory integrands.
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Lebesgue integration of derivatives of strongly-oscillatory functions is a recurring challenge in computational science and engineering. Integration by parts is an effective remedy for huge ...computational costs associated with Monte Carlo integration schemes. In case of Lebesgue integrals over a smooth manifold, however, integration by parts gives rise to a derivative of the density implied by charts describing the domain manifold. This paper focuses on the computation of that derivative, which we call the density gradient function, on general smooth manifolds. We analytically derive formulas for the density gradient and present examples of manifolds determined by popular differential equation-driven systems. Furthermore, we highlight the significance of the density gradient by demonstrating a numerical example of Monte Carlo integration involving oscillatory integrands.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In this paper, we were able to produce certain coincidence point results for g-nondecreasing selfmappings fulfilling certain rational type contractions in a Hausdorff rectangular metric space ...utilizing C-functions and generalized (θ, φ)-contractive mappings obeying an admissibility-type assumption.
Decomposition integrals provide a framework for non-linear integrals that include Choquet, Shilkret, the PAN, and the concave integrals. All of these integrals found their applications in ...mathematics, notably in decision-making and economy. An important class of decomposition integrals is the class of integrals extending Lebesgue integral in the sense that the decomposition integral with respect to classical measures coincides with Lebesgue integral. In this paper, we consider finite spaces X only and discuss some necessary and sufficient conditions for this property. Also, some construction methods are given and exemplified.
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This timely monograph collects all the basic properties of variable exponent Lebesgue and Sobolev spaces. It provides a much-needed accessible reference work utilizing consistent notation and ...terminology. Many results have new and improved proofs.
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In this paper, we continue the ongoing research on lineability related questions. On this occasion, we shall consider (among others) the classes of integrable functions (in the sense of Riemann, ...Lebesgue, Denjoy, and Khintchine), improving some already known results and expanding the study of lineability to other famous integrable classes never considered before.
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In recent years, we have seen a growing interest in various aspects of approximation theory. This happened due to the increasing complexity of mathematical models that require computer calculations ...and the development of the theoretical foundations of the approximation theory. Approximation theory has broad and important applications in many areas of mathematics, including functional analysis, differential equations, dynamical systems theory, mathematical physics, control theory, probability theory and mathematical statistics, and others. Approximation theory is also of great practical importance, as approximate methods and estimation of approximation errors are used in physics, economics, chemistry, signal theory, neural networks and many other areas. This book presents the works published in the Special Issue "Approximation Theory and Related Applications". The research of the world’s leading scientists presented in this book reflect new trends in approximation theory and related topics.
Here, in this article, we introduce and systematically investigate the ideas of deferred weighted statistical Riemann integrability and statistical deferred weighted Riemann summability for sequences ...of functions. We begin by proving an inclusion theorem that establishes a relation between these two potentially useful concepts. We also state and prove two Korovkin-type approximation theorems involving algebraic test functions by using our proposed concepts and methodologies. Furthermore, in order to demonstrate the usefulness of our findings, we consider an illustrative example involving a sequence of positive linear operators in conjunction with the familiar Bernstein polynomials. Finally, in the concluding section, we propose some directions for future research on this topic, which are based upon the core concept of statistical Lebesgue-measurable sequences of functions.
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The principal goal of this paper is to express the existence and uniqueness
of the best proximity point for a comprehensive contractive non-self mapping
in partially ordered metric spaces. The main ...result covers a lot of former
well-known theorems in related to best proximity point. Moreover, as an
interesting application, integral versions of main theorem are obtained.
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