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DCDS-B

This article presents the Euler-Maclaurin expansions of the
hypersingular
integrals $\int_{a}^{b}\frac{g(x)}{|x-t|^{m+1}}dx$ and $\int_{a}^{b}%
\frac{g(x)}{(x-t)^{m+1}}dx$ with arbitrary singular point $t$ and
arbitrary non-negative integer $m$. These general expansions are
applicable to a large range of hypersingular integrals, including
both popular hypersingular integrals discussed in the literature and
other important ones which have not been addressed yet. The
corresponding mid-rectangular formulas and extrapolations, which can
be calculated in fairly straightforward ways, are investigated.
Numerical examples are provided to illustrate the features of the
numerical methods and verify the theoretical conclusions.

CPAA

We study the combined effect of
concave and convex nonlinearities on the number of positive
solutions for a fractional system
involving critical Sobolev exponents. With the help of the
Nehari manifold, we prove that the system admits at least two
positive solutions when the pair of parameters $(\lambda,\mu)$
belongs to a suitable subset of $R^2$.

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