14,272 research outputs found
Rational G-Circulants Satisfying the Matrix Equation A² = dI + λJ
A g-circulant is a square matrix of rational numbers in which each row is obtained from the preceding row by shifting the elements cyclically g columns to the right. This work studies g-circulants A which satisfy the matrix equation A2 = dI + λJ, where I is the identity matrix and J is the matrix of 1's. Necessary and sufficient conditions are given for the existence of solutions when g = 1. The existence of (0,1) g-circulants satisfying A2 = dI + λJ is shown to be equivalent to the existence of (v, k, λ, g)-addition sets, which are generalizations of difference sets. It is proved that there are no nontrivial (v, k, λ, 1)-addition sets. Some examples of (v, k, λ, g)-addition sets are given and the multiplier theorem for (v, k, λ, g)-addition sets is also proved.</p
Dynamics of a family of piecewise-linear area-preserving plane maps III. Cantor set spectra
This paper studies the behavior under iteration of the maps T_{ab}(x,y) =
(F_{ab}(x)- y, x) of the plane R^2, in which F_{ab}(x)= ax if x>0 and bx if
x<0. These maps are area-preserving homeomorphisms of the plane that map rays
from the origin into rays from the origin. Orbits of the map correspond to
solutions of the nonlinear difference equation x_{n+2}= 1/2(a-b)|x_{n+1}| +
1/2(a+b)x_{n+1} - x_n. This difference equation can be written in an eigenvalue
form for a nonlinear difference operator of Schrodinger type, in which \mu=
1/2(a-b) is viewed as fixed and the energy E=2- 1/2(a+b). The paper studies the
set of parameter values where T_{ab} has at least one nonzero bounded orbit,
which corresponds to an l_{\infty} eigenfunction of the difference operator. It
shows that the for transcendental \mu the set of allowed energy values E for
which there is a bounded orbit is a Cantor set. Numerical simulations suggest
that this Cantor set have positive one-dimensional measure for all real values
of \mu.Comment: v1 21 pages latex, 2 postscript figures; This was former part II in
earlier version. Current part I is math.DS/0301294 and part II is
math.DS/0303007; v2 20 pages latex- revised to reference prior work of
Beardon, Bullett and Rippo
Stokes waves with vorticity
The existence of periodic waves propagating downstream on the surface of a
two-dimensional infinitely deep water under gravity is established for a
general class of vorticities. When reformulated as an elliptic boundary value
problem in a fixed semi-infinite strip with a parameter, the operator
describing the problem is nonlinear and non-Fredholm. A global connected set of
nontrivial solutions is obtained via singular theory of bifurcation. Each
solution on the continuum has a symmetric and monotone wave profile. The proof
uses a generalized degree theory, global bifurcation theory and Wyburn's lemma
in topology, combined with the Schauder theory for elliptic problems and the
maximum principle
Integrable 1D Toda cellular automata
First, we recall the algebro-geometric method of construction of finite field
valued solutions of the discrete KP equation and next we perform a reduction of
the dKP equation to the discrete 1D Toda equation. This gives a method of
construction of solutions of the discrete 1D Toda equation taking values in a
finite field.Comment: 9 pages, 2 figures; Corrected typo
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