In 2001, S. Bulman-Fleming et al. initiated the study of three flatness properties (weakly kernel flat, principally weakly kernel flat, translation kernel flat) of right acts $A_{S}$ over a monoid $S$ that can be described by means of when the functor $A_{S} otimes -$ preserves pullbacks. In this paper, we extend these results to $S$-posets and present equivalent descriptions of weakly kernel p...

In this work, we explain a new numerical schemes of collocation methods based on the adapted quadratic approximation of singular integral with logarithmic kernel. This approximation leads to obtain the numerical solution of singular integral equations with logarithmic kernel on an oriented smooth contour.

Second-kind Volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration. Using an adaptation of the analysis originally developed for nonlinear weakly singular Fredholm integral equations, we present a complete discussion of the optimal (global and local) order of convergence of piecewise polynomial ...

We prove a local T (b) theorem for singular integral operators in the spirit of the one for perfect dyadic singular integral operators obtained by Hofmann, Muscalu, Thiele, Tao and the first author. The method consists in reducing one situation to the other by representing via the Beylkin-Coifman-Rokhlin algorithm any singular integral operator with truncated kernel as the sum of a bounded oper...