We express the Chern-Schwartz-MacPherson class of a possibly singular variety in terms of the total Chern class of a bundle of di erential forms with logarithmic poles. As an application, we obtain a formula for the Chern-Schwartz-MacPherson class of a hypersurface of a nonsingular variety, in terms of the Chern-Mather class of a suitable sheaf. x

The Eremenko-Lyubich class of transcendental entire functions with a bounded set of singular values has been much studied. We give a new characterisation of this class of functions. We also give a new result regarding direct singularities which are not logarithmic.

For the magnetic Hamiltonian with singular vector potentials, we analytically continue resolvent to a logarithmic neighborhood of positive real axis and prove estimates there. As applications, obtain asymptotic locations resonances local smoothing estimate for solutions corresponding Schr\"odinger equation.

We consider the reaction-diffusion equation with discontinues coefficients and singular sources in one dimension. In this work, we construct ε-uniformly convergent High Order Compact (HOC) monotone finite difference schemes defined on a priori Shishkin meshes, which have order two, three and four except for a logarithmic factor. Numerical experiments are presented and discussed.

We analyze copulas with a non-trivial singular component by using their Markov kernel representation. In particular, we provide existence results for copulas with a prescribed singular component. The constructions do not only help to deal with problems related to multivariate stochastic systems of lifetimes when joint defaults can occur with a non-zero probability, but even provide a copula max...

A high-order accurate, explicit kernel-split, panel-based, Fourier–Nyström discretization scheme is developed for integral equations associated with the Helmholtz equation in axially symmetric domains. Extensive incorporation of analytic information about singular integral kernels and on-the-fly computation of nearly singular quadrature rules allow for very high achievable accuracy, also in the...

Using the Kaczmarz algorithm, we obtain a Fourier series formulation for functions in the L2 space of singular measures on the unit circle. This formula is applied to the problem of finding reproducing kernel Hilbert spaces inside the classical Hardy space, where the norm is instead that of boundary integration with respect to a singular measure. We also give some conditions ensuring that these...

Direct boundary limit algorithms for evaluating hypersingular Galerkin surface integrals have been successful in identifying and removing the divergent terms, leaving finite integrals to be evaluated. This paper is concerned with the numerical computation of these multi-dimensional integrals. The integrands contain a weakly singular logarithmic term that is difficult to evaluate directly using ...

We show that over any field, the solution set to a system of n linear equations in n unknowns can be computed in parallel with randomization simultaneously in poly-logarithmic time in n and with only as many processors as are utilized to multiply two n×n matrices. A time unit represents an arithmetic operation in the field. For singular systems our parallel timings are asymptotically as fast as...