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L 2 -estimates for the d -equation and Witten’s proof of the Morse inequalities

Bo Berndtsson (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

This is an introduction to Witten’s analytic proof of the Morse inequalities. The text is directed primarily to readers whose main interest is in complex analysis, and the similarities to Hörmander’s L 2 -estimates for the ∂ ¯ -equation is used as motivation. We also use the method to prove L 2 -estimates for the d -equation with a weight e - t φ where φ is a nondegenerate Morse function.

Large time behaviour of a conservation law regularised by a Riesz-Feller operator: the sub-critical case

Carlota Maria Cuesta, Xuban Diez-Izagirre (2023)

Czechoslovak Mathematical Journal

We study the large time behaviour of the solutions of a nonlocal regularisation of a scalar conservation law. This regularisation is given by a fractional derivative of order 1 + α , with α ∈ ( 0 , 1 ) , which is a Riesz-Feller operator. The nonlinear flux is given by the locally Lipschitz function | u | q - 1 u / q for q > 1 . We show that in the sub-critical case, 1 < q < 1 + α , the large time behaviour is governed by the unique entropy solution of the scalar conservation law. Our proof adapts the proofs of the analogous results for the local...

Lebesgue's Convergence Theorem of Complex-Valued Function

Keiko Narita, Noboru Endou, Yasunari Shidama (2009)

Formalized Mathematics

In this article, we formalized Lebesgue's Convergence theorem of complex-valued function. We proved Lebesgue's Convergence Theorem of realvalued function using the theorem of extensional real-valued function. Then applying the former theorem to real part and imaginary part of complex-valued functional sequences, we proved Lebesgue's Convergence Theorem of complex-valued function. We also defined partial sums of real-valued functional sequences and complex-valued functional sequences and showed their...

Left general fractional monotone approximation theory

George A. Anastassiou (2016)

Applicationes Mathematicae

We introduce left general fractional Caputo style derivatives with respect to an absolutely continuous strictly increasing function g. We give various examples of such fractional derivatives for different g. Let f be a p-times continuously differentiable function on [a,b], and let L be a linear left general fractional differential operator such that L(f) is non-negative over a closed subinterval I of [a,b]. We find a sequence of polynomials Qₙ of degree ≤n such that L(Qₙ) is non-negative over I,...

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