Many combinatorial configurations (for example, perfect codes, latin squares and hypercubes, combinatorial designs and their q-ary generalizations — subspace designs) can be defined as an eigenfunction on a graph with some discrete restrictions. The study of these configurations often leads to the question about the minimum possible difference between two configurations from the same class (it ...

1. Introduction. The main purpose of this article is to give a stronger counterexample to the " hot spots " conjecture than the one presented in Burdzy and Werner [7]. Along the way we define and partly analyze a new process, which we call fiber Brownian motion, and which may have some interest of its own. Consider a Euclidean domain that has a discrete spectrum for the Laplacian with Neumann b...

This paper develops an eigenfunction expansion approach to solve discretely monitored first passage time problems for a rich class of Markov processes, including diffusions and subordinate diffusions with jumps, whose transition or Feynman-Kac semigroups possess eigenfunction expansions in L spaces. Many processes important in finance are in this class, including OU, CIR, (JD)CEV diffusions and...

We investigate the mKdV hierarchy with integral type of source (mKdVHWS), which consist of the reduced AKNS eigenvalue problem with r = q and the mKdV hierarchy with extra term of the integration of square eigenfunction. First we propose a method to find the explicit evolution equation for eigenfunction of the auxiliary linear problems of the mKdVHWS. Then we determine the evolution equations o...

We study the critical points of Laplace eigenfunctions on polygonal domains with a focus second Neumann eigenfunction. show that if each convex quadrilaterals has no eigenfunction an interior point, then there exists quadrilateral unstable point. also point second-Neumann Lip-1 polygon orthogonal sides is acute vertex.

In this article we develop new techniques for studying concentration of Laplace eigenfunctions $\phi_\lambda$ as their frequency, $\lambda$, grows. The method consists controlling $\phi_\lambda(x)$ by decomposing into a superposition geodesic beams that run through the point $x$. Each beam is localized in phase-space on tube centered around whose radius shrinks slightly slower than $\lambda^{-\...

Direct simulation methods of Electron Paramagnetic Resonance (EPR) spectra are an alternative to standard eigenfunction-based approaches. Both methods can be employed to find numerical solutions of the Stochastic Liouville Equation (SLE) in order to compute spectral lineshapes. It can be shown rigorously that the domain of validity of eigenfunction expansion solutions of the SLE covers only a l...