A new two-component integrable system with peakon solutions

A new two-component system with cubic nonlinearity and linear dispersion: m t = b u x + 1 2 [ m ( u v − u x v x ) ] x − 1 2 m ( u v x − u x v ) , n t = b v x + 1 2 [ n ( u v − u x v x ) ] x + 1 2 n ( u v x − u x v ) , m = u − u x x , n = v − v x x , where b is an arbitrary real constant, is proposed in this paper. This system is shown integrable with its Lax pair, bi-Hamiltonian structure and infinitely many conservation laws. Geometrically, this system describes a non-trivial one-parameter family of pseudo-spherical surfaces. In the case b =0, the peaked soliton (peakon) and multi-peakon solutions to this two-component system are derived. In particular, the two-peakon dynamical system is explicitly solved and their interactions are investigated in details. Moreover, a new integrable cubic nonlinear equation with linear dispersion m t = b u x + 1 2 [ m ( | u | 2 − | u x | 2 ) ] x − 1 2 m ( u u x ∗ − u x u ∗ ) , m = u − u x x , is obtained by imposing the complex conjugate reduction v = u * to the two-component system. The complex-valued N -peakon solution and kink wave solution to this complex equation are also derived.

Stability of extremes with random sample size

A non-degenerate distribution function F is called maximum stable with random sample size if there exist positive integer random variables Nn, n = 1, 2, ···, with P(Nn = 1) less than 1 and tending to 1 as n → ∞ and such that F and the distribution function of the maximum value of Nn independent observations from F (and independent of Nn ) are of the same type for every index n. By proving the converse of an earlier result of the author, it is shown that the set of all maximum stable distribution functions with random sample size consists of all distribution functions F satisfying where c 2, c 3, · ·· are arbitrary non-negative constants with 0 < c2 + c3 + · ·· <∞, and all distribution functions G and H defined by F(x)= G(c + exp(x)) and F(x) = H(c – exp(–x)), –∞ < x <∞, where c is an arbitrary real constant.

Stability of extremes with random sample size

A non-degenerate distribution function F is called maximum stable with random sample size if there exist positive integer random variables Nn, n = 1, 2, ···, with P(Nn = 1) less than 1 and tending to 1 as n → ∞ and such that F and the distribution function of the maximum value of Nn independent observations from F (and independent of Nn) are of the same type for every index n. By proving the converse of an earlier result of the author, it is shown that the set of all maximum stable distribution functions with random sample size consists of all distribution functions F satisfying where c2, c3, · ·· are arbitrary non-negative constants with 0 < c2 + c3 + · ·· <∞, and all distribution functions G and H defined by F(x)= G(c + exp(x)) and F(x) = H(c – exp(–x)), –∞ < x <∞, where c is an arbitrary real constant.

On a functional equation for the exponential function of a complex variable

The following result is well known in the theory of analytic functions; see [1].Theorem A. Suppose that f(z) is an entire function of a complex variable z. Then f(z) satisfies the functional equationwhere z = x + iy (x, y real), if and only if f(z) = aexp(sz), where a is an arbitrary complex constant and s is an arbitrary real constant.