Asymptotic Distribution of the Eigenvalues of the Vibrating-membrane Problem |
Common terms and phrases
A¹(N ASYMPTOTIC DISTRIBUTION boundary condition 1.6 boundary condition 20 boundary of G c² H v₂ classical minimum definition congruent squares conjunction continuous function decrease defines the nth density function differential equation 1.4 dxdy eigen eigenfunction belonging eigenfunctions and eigenvalues eigenvalues of G eligible functions ential equation 1.4 equal to zero figure 2.b function v₁ functions eligible further eligibility condition genvalue gion G given finite number greatest lower bound lattice points lemma limit A(N linearly independent maximum maximum-minimum definition maximum-minimum problem minimizing function minimum problem minimum value nth eigenvalue belonging nth minimum number of conditions NUMBER OF CONGRUENT number of eigenvalues number of lattice partial derivatives paxdy pdxdy pdxdy=0 permissible functions problem which defines PROOF proved quarter-circle region G satisfy the boundary solution square region STANFORD STANFORD UNIVERSITY sufficiently large tion unit square variational problem vibrating membrane ᎩᎩ