ISSN:0022-2526    

DOI:10.1002/sapm1973524291

年代:1973

数据来源: WILEY

摘要:

It is shown that π is the infinium gap between the consecutive square roots of the eigenvalues of the wave equation in a hypespherical domain for both the Neumann (free) and the full range of mixed (elastic) homogeneous boundary conditions. Previous literature contains the same information apparently only for the Dirichlet (fixed) boundary condition. These square roots of the eigenvalues are the zeros of solutions of a differential equation in Bessel functions (first kind) and their first derivatives. The infinium gap is uniform for Bessel functions of ordersx≥ ½ (as well as forx= 0). The intervals between the roots are actually monotone decreasing in length. These results are obtained by interlacing zeros of Bessel and associated functions and comparing their relative displacements with oscillation theory. IfWldenotes thelth positive root for some fixed orderx, the minimum gap property assures that {exp(±iwlt|l= 1, 2,...} form a Riesz basis inL2(0, τ) for τ>2. This has application to the problem of controlling solutions of the wave equation by controlling the boundary

ISSN:0022-2526    

DOI:10.1002/sapm1973524329

年代:1973

数据来源: WILEY

摘要:

We are given the question: how many phone calls are needed fornpeople to pool all their information in a succession ofk‐person party line phone calls? The question was proposed by Erdös for the special caseK= 2. We prove here the result that [n− 2/K− 1] + [n− 1 /k] + 1 calls are required if 1 ≤n≤k2, while 2[n− 2/k− 1] are

ISSN:0022-2526    

DOI:10.1002/sapm1973524345

年代:1973

数据来源: WILEY

摘要:

A study is made of the way in which small random inhomogeneities in a transmission medium affect the statistical properties of a system of waves. It is shown that provided the spectral cumulants are sufficiently smooth at some initial time, a sequence of closures for the zeroth order spectral functions can be deduced which describe asymptotically the transfer of energy between wave numbers. Of particular importance is the fact that the closure equations are derived without the need to resort to ad hoc statistical assumptions. The general theory is applied to the problem of the propagation of water waves over an irregular bottom topography.

ISSN:0022-2526    

DOI:10.1002/sapm1973524359

年代:1973

数据来源: WILEY

ISSN:0022-2526    

DOI:10.1002/sapm1973524377

年代:1973

数据来源: WILEY

ISSN:0022-2526    

DOI:10.1002/sapm1973524389

年代:1973

数据来源: WILEY