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Peskin's Immersed Boundary (IB) method is one of the most popular numerical methods for many years and has been applied to problems in mathematical biology, fluid mechanics, material sciences, and many other areas. Peskiness IB method is associated with discrete delta functions. It is believed that the IB method is first order accurate in the $L^{\infty}$ norm. But almost no rigorous proof could be found in the literature until recently [Mori, Comm. Pure. Appl. Math: 61:2008] in which the author showed that the velocity is indeed first order accurate for the Stokes equations with a periodic boundary condition. In this paper, we show first order convergence with a $\log h$ factor of the IB method for elliptic interface problems with Dirichlet boundary conditions.

We investigate the energy landscape of the mixed even p-spin model with Ising spin configurations. We show that for any given energy level between zero and the maximal energy, with overwhelming probability there exist exponentially many distinct spin configurations such that their energies stay near this energy level. Furthermore, their magnetizations and overlaps are concentrated around some fixed constants. In particular, at the level of maximal energy, we prove that the Hamiltonian exhibits exponentially many orthogonal peaks. This improves the results of Chatterjee [20] and Ding-Eldan-Zhai [29], where the former established a logarithmic size of the number of the orthogonal peaks, while the latter proved a polynomial size. Our second main result obtains disorder chaos at zero temperature and at any external field. As a byproduct, this implies that the fluctuation of the maximal energy is superconcentrated when the external field vanishes and obeys a Gaussian limit law when the external field is present.

We derive bounds for eigenvalues of the Laplacian of graphs using discrete versions of the Sobolev inequalities and heat kernel estimates.

A vast class of exponential functions are shown to be deterministic. This class includes functions whose exponents are polynomial-like or ``piece-wise'' close to polynomials after differentiation. Many of these functions are proved to be disjoint from the M\"obius function.