we study different completeness definitions for two categories of lattice-valued cauchy spaces and the relations between these definitions. we also show the equivalence of a so-called completion axiom and the existence of a completion.

The Cauchy problem for harmonic maps from Minkowski space with its standard flat metric to a certain non-constant curvature Lorentzian 2-metric is studied. The target manifold is distinguished by the fact that the Euler–Lagrange equation for the energy functional is Darboux integrable. The time evolution of the Cauchy data is reduced to an ordinary differential equation of Lie type associated t...

We use a new construction in the spirit of Gromov’s convex integration to prove a rather surprising result that every affine map is the boundary value of a weakly quasiregular map in any Sobolev space with index p < n/2. Our method is based on constructing special Cauchy sequences by convex integration for certain vectorial Hamilton–Jacobi equations.  2000 Académie des sciences/Éditions scient...

The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces b H s (R) defined by the norm ‖v0‖ b Hr s (R) := ‖〈ξ〉 b v0‖Lr′ ξ , 〈ξ〉 = (1 + ξ) 1 2 , 1 r + 1 r = 1. Local well-posedness for the jth equation is shown in the parameter range 2 ≥ r > 1, s ≥ 2j−1 2r . The proof uses an appropriate variant of the Fourier restriction norm method. A coun...

Let F (z) = z−H(z) with order o(H(z)) ≥ 1 be a formal map from C to C and G(z) the formal inverse map of F (z). We first study the deformation Ft(z) = z − tH(z) of F (z) and its formal inverse Gt(z) = z + tNt(z). (Note that Gt=1(z) = G(z) when o(H(z)) ≥ 2.) We show that Nt(z) is the unique power series solution of a Cauchy problem of a PDE, from which we derive a recurrent formula for Gt(z). Se...

Let F (z) = z−H(z) with order o(H(z)) ≥ 1 be a formal map from C to C and G(z) the formal inverse map of F (z). We first study the deformation Ft(z) = z − tH(z) of F (z) and its formal inverse Gt(z) = z + tNt(z). (Note that Gt=1(z) = G(z) when o(H(z)) ≥ 2.) We show that Nt(z) is the unique power series solution of a Cauchy problem of a PDE, from which we derive a recurrent formula for Gt(z). Se...

in this paper, we present legendre wavelet method to obtain numerical solution of a singular integro-differential equation. the singularity is assumed to be of the cauchy type. the numerical results obtained by the present method compare favorably with those obtained by various galerkin methods earlier in the literature.

We explore properties of Cauchy-Stieltjes families that have no counterpart in exponential families. We relate the variance function of the iterated Cauchy-Stieltjes family to the pseudo-variance function of the initial Cauchy-Stieltjes family. We also investigate when the domain of means can be extended beyond the ”natural domain”.

Γ-ultrametric spaces are spaces which satisfy all the axioms of an ultrametric space except that the distance function takes values in a complete lattice Γ instead of R≥0. Γ-ultrametric spaces have been extensively studied as a way to weaken the notion of an ultrametric space while still providing enough structure to be useful (see for example [17], [18], [8]). The many uses of Γ-ultrametric sp...

Recently, K. Kamano studied sums of products of Bernoulli numbers including poly-Bernoulli numbers. A relation among these sums was given, and an explicit expression of sums of two products was also given, reduced to the famous Euler’s formula. The concept of poly-Cauchy numbers is given by the author as a generalization of the classical Cauchy number and an analogue of poly-Bernoulli number. I...