摘要:

The geometric, algebraic, and combinatorial explanations of Dobinski's formula are presented by mixed volumes of compact convex sets, Möbius inversion, difference operator, and species. The employed method may be useful in proving some other combinatorial identities

ISSN:0022-2526    

DOI:10.1002/sapm1994923191

年代:1994

数据来源: WILEY

摘要:

Consider equations of the formwhereqis in general a complex vector and the functionFdepends nontrivially both onqand onqy. We show that a familySof such equations can be investigated by the inverse scattering method. If an equation (*) belongs toS, the functionFdepends linearly onqand algebraically onqy. We show that the familyScontains a subfamily in which each equation can be obtained from the two dimensional Toda lattice equations by a Bäcklund transformation

ISSN:0022-2526    

DOI:10.1002/sapm1994923201

年代:1994

数据来源: WILEY

摘要:

Trigonometric Fourier series are, in general, difficult to sum to high accuracy. An example is given by the seriesin whichαandβ(>0) are rational numbers satisfying 0<β/α≤1, whereλis an independent variable and j is a positive integer or zero. This paper presents a method for the efficient evaluation of the sum of such series. Fourier series which are the real or the imaginary part of , but which are not explicitly expressible as simple polynomials inλ, are obtained as the sum of a logarithic term and an infinite series in powers ofλ, whose expansion is valid when 0<λ≤(2π/α)and is exact. When the Fourier series is expressible as a polynomial inλ, the method identifies t

ISSN:0022-2526    

DOI:10.1002/sapm1994923213

年代:1994

数据来源: WILEY

摘要:

In this paper, we study the oblique interaction of weakly, nonlinear, long internal gravity waves in both shallow and deep fluids. The interaction is classified as weak when where Δ1=|cm/cn−cosδ|, Δ2=|cn/cm−cosδ|,cm,n, are the linear, long wave speeds for waves with mode numbersm, n, δis the angle between the respective propagation directions, andαmeasures the wave amplitude. In this case, each wave is governed by its own Kortweg‐de Vries (KdV) equation for a shallow fluid, or intermediate long‐wave (ILW) equation for a deep fluid, and the main effect of the interaction is an 0(α) phase shift. A strong interaction (I) occurs when Δ1,2are 0(α), and this case is governed by two coupled Kadomtsev‐Petviashvili (KP) equations for a shallow fluid, or two coupled two‐dimensional ILW equations for deep fluids. A strong interaction (II) occurs when Δ1is 0(α), and (or vice versa), and in this case, each wave is governed by its own KdV equation for a shallow fluid, or ILW equation for a deep fluid. The main effect of the interaction is that the phase shift associated with Δ1leads to a local distortion of the wave speed of the moden. When the interacting waves belong to the same mode (i.e.,m = n) the general results simplify and we show that for a weak interaction the phase shift for obliquely interacting waves is always negative (positive) for (1/2+cosδ)>0(<0), while the interaction term always has the same polarity

ISSN:0022-2526    

DOI:10.1002/sapm1994923249

年代:1994

数据来源: WILEY