Functional data analysis is a relatively new and rapidly growing area of statistics. This is partly due to technological advancements which have made it possible to generate new types of data that are in the form of curves. Because the data are functions, they lie in function spaces, which are of infinite dimension. To analyse functional data, one way, which is widely used, is to employ princip...

In his comment [1] on our paper [2], Pikovsky contrasts two definitions for the phase of a stochastic oscillator by way of an analytically solvable model system. In [3] the phase is defined in terms of a system of isochrons Σθ, analogous to Poincaré sections, with the property that for any initial condition on one isochron Σa, the mean first passage time (MFPT) to a second isochron Σb, b > a, w...

We analyze the Ritz–Galerkin method for symmetric eigenvalue problems and prove a priori eigenvalue error estimates. For a simple eigenvalue, we prove an error estimate that depends mainly on the approximability of the corresponding eigenfunction and provide explicit values for all constants. For a multiple eigenvalue we prove, in addition, apparently the first truly a priori error estimates th...

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 ...

We study the spectra and eigenvectors of the adjacency matrices of scale-free networks when bidirectional interaction is allowed, so that the adjacency matrix is real and symmetric. The spectral density shows an exponential decay around the center, followed by power-law long tails at both spectrum edges. The largest eigenvalue lambda1 depends on system size N as lambda1 approximately N1/4 for l...

In this paper we formulate the equilibrium equation for a beam made of graphene subjected to some boundary conditions and acted upon by axial compression and nonlinear lateral constrains as a fourth-order nonlinear boundary value problem. We first study the nonlinear eigenvalue problem for buckling analysis of the beam. We show the solvability of the eigenvalue problem as an asymptotic expansio...

1 Edelman-Sutton Conjecture (cont.) 1.1 Existence of eigenvalue-eigenfunction pair for the smallest eigenvalue of cH β In the last lecture, we proved ∃{f n } and a nonnegative random variable K with ||f n || 2 2 = 1, ||f n || 2 * K a.s. ∀n, so that f n , H β f n → Λ 0 = inf ||f || 2 2 =1 f ∈L * f, H β f a.s. Also, we proved the lemma: If {f n }is bounded in L * , then we can find a subsequence ...

For bounded smooth domains the number λ1 is the so-called principal eigenvalue. It has a unique eigenfunction, which is positive, and this eigenfunction is the only positive one (up to normalization). Two main references for this type of results, which are usually called maximum principles, are the books by Walter (1964) and by Protter and Weinberger (1967). Extensions to general bounded non-sm...

We study an eigenvalue problem associated with a reaction-diffusionadvection equation of the KPP type in a cellular flow. We obtain upper and lower bounds on the eigenvalues in the regime of a large flow amplitude A 1. It follows that the minimal pulsating traveling front speed c∗(A) satisfies the upper and lower bounds C1A ≤ c∗(A) ≤ C2A. Physically, the speed enhancement is related to the boun...

The classical eigenfunction method for the solution of contact problems involving wear is formulated in the context of the finite element method. Static reduction is used to reduce the full stiffness matrix to the N contact nodes, after which the assumption of a separated variable solution leads to a linear eigenvalue problem with N eigenvalues and eigenfunctions. A general solution to the tran...