相关论文: Notes on Lebesgue integration
The top-down approach of engineering software integration is considered in this parer. A set of advantages of this approach are presented, by examples. All examples are supplied by open source code.
We introduce a notion being a $k$-fold Lebesgue function for measure preserving transformations, where any $2$-fold Lebesgue function is just ordinary Lebesgue. We discuss how this new metrical isomorphisms invariant of dynamical systems is…
This short note contains elementary evaluations of some Euler sums.
We consider Choquet integrals with respect to dyadic Hausdorff content of non-negative functions which are not necessarily Lebesgue measurable. We study the theory of Lebesgue points. The studies yield convergence results and also a density…
We obtain some new inequalities of Chebyshev Type.
An existing solvability result for relaxed one-sided Lipschitz algebraic inclusions is substantially improved. This enhanced solvability result allows the design of a very robust numerical method for the approximation of a solution of the…
We introduce a symbolic method for the evaluation of definite integrals containing combinations of various functions, including exponentials, logarithm and products of Bessel functions of different types. The method we develop is naturally…
In the theory of time scales, given $\mathbb{T}$ a time scale with at least two distinct elements, an integration theory is developed using ideas already well known as Riemann sums. Another, more daring, approach is to treat an integration…
The paper presents a counterexample to the Hodge conjecture.
A construction of integration, function calculus, and exterior calculus is made, allowing for integration of unital magma valued functions against (compactified) unital magma valued measures over arbitrary topological spaces. The Riemann…
Sesquilinear forms which are not necessarily positive may have a different behavior, with respect to a positive form, on each side. For this reason a Lebesgue-type decomposition on one side is provided for generic forms satisfying a…
A version of the Lebesgue differentiation theorem is offered, where the $L^p$ norm is replaced with any rearrangement-invariant norm. Necessary and sufficient conditions for a norm of this kind to support the Lebesgue differentiation…
In this short note a new proof of the monotone con- vergence theorem of Lebesgue integral on \sigma-class is given.
An algorithm for integration of polynomial functions with variable weight is considered. It provides extension of the Gaussian integration, with appropriate scaling of the abscissas and weights. Method is a good alternative to usually…
New index transforms, involving the square of Bessel functions of the first kind as the kernel are considered. Mapping properties such as the boundedness and invertibility are investigated for these operators in the Lebesgue spaces.…
Symmetries and reductions of some algebraic equations are considered. Transformations that preserve the form of several algebraic equations, as well as transformations that reduce the degree of these equations, are described. Illustrative…
In this note we obtain a new convergence result for the Adomian decomposition method.
Further formulas are presented involving quantum mechanics, thermodynamics, and integrable systems. Modifications of dispersionless theory are developed.
Estimates of some integrals related to variations of smooth functions are presented.
In this work we obtain a transference theorem for Lebesgue spaces with $A_{\infty }$ weights, namely, starting from some uniform-norm inequalities it is possible to obtain similar inequalities in Lebesgue spaces with $A_{\infty }$ weights.…