We study second to fourth order numerical methods to a type of delta function integrals in one to three dimensions. These delta function integrals arise from recent efficient level set methods for computing the multivalued solutions of nonlinear PDEs. We show that the natural quadrature approach with usual discrete delta functions and support size formulas to the two dimensional delta function ...

One way of characterizing Atle Selberg’s mathematical genius is that he had a “golden touch”. In those domains he thought about in depth, he saw further than generations before him, repeatedly uncovering truths lying below the surface. His breakthroughs on long-standing problems were based on imaginative and novel ideas which, once digested, were appreciated as simple and decisive. The impact o...

Motivation. Among the molecular integrals, the two-center overlap integrals play a major role in any accurate molecular structure calculation. They are central to the calculation of multicenter overlap integrals when using the series expansion formulae for Slater type functions about a new center. Consequently, these integrals require an accurate and fast numerical evaluation. Recently, we show...

We study the asymptotic behavior of zeros of the Selberg zeta-function for the congruence subgroup Γ0(4) as a function of a one-parameter family of characters tending to the trivial character. The motivation for the study comes from observations based on numerical computations. Some of the observed phenomena lead to precise theorems that we prove and compare with the original numerical results....

More general results, including arbitrary Fuchsian groups, can be found in the paper [Bo2] of Borcherds, Sect. 7. Most of them have been proved by the Selberg trace formula, see [Iv] and also [Fi]. The Selberg trace formula in its standard form causes the restriction that the weight is > 2. Borcherds mentions that “with a bit more care this also works for weight 2”. As we mentioned, this bit mo...

We verify the Selberg eigenvalue conjecture for congruence groups of small squarefree conductor, improving on a result of Huxley [20]. The main tool is the Selberg trace formula which, unlike previous geometric methods, allows for treatment of cases where the eigenvalue 1 4 is present. We present a few other sample applications, including the classification of even 2-dimensional Galois represen...