[en] It is shown that, in four dimensions, it is possible to introduce coordinates so that an analytic metric locally takes block diagonal form, i.e. one can find coordinates such that g_αβ = 0 for (α, β) in S where S = {(1, 3), (1, 4), (2, 3), (2, 4)}. We call a coordinate system in which the metric takes this form a 'doubly biorthogonal coordinate system'. We show that all such coordinate systems are determined by a pair of coupled second-order partial differential equations.

[en] In this paper we extend some properties of analytic functions on several complex variables to solutions of overdetermined systems of complex partial differential equations. It is proved that many global properties of analytic functions are true for solutions of the Vekua system in special cases. The relation between analytic functions and solutions of quasi-linear systems is discussed in the paper. (author). 8 refs

[en] A simple proof of Magnus' Freiheitsatz for one-relator groups is given which can be easily applied with slight modifications to prove a generalization of the theorem to any number of relators. (author). 10 refs

[en] We show that one can devise through the symmetry approach a procedure to check the linearizability of a difference equation via a point or a discrete Cole–Hopf transformation. If the equation is linearizable, then the symmetry provides the linearizing transformation. At the end, we present a few examples of applications for equations defined on four lattice points. (paper)

[en] We show that the the integrable subclasses of the equations q_t = f(x, t)q₃ + H(x, t, q, q₁) are the same as the integrable subclasses of the equations q_t = q₃ + F(q, q₁). (orig.)

[en] We introduce a bracket on 1-forms defined on J^∞(S¹,Rⁿ), i.e. the infinite jet extension of the space of loops, and prove that it satisfies the standard properties of a Poisson bracket. Using this bracket, we show that certain hierarchies appearing in the framework of F-manifolds with a compatible flat connection (M, ∇, ○) are Hamiltonian in a generalized sense. Moreover, we show that if a metric g compatible with ∇ is also invariant with respect to ○, then this generalized Hamiltonian setup reduces to the standard one. (paper)

[en] The methodical procedure to obtain the invariants of the Richardson equation is presented. These invariants are the self similar variables that will permit the partial differential equation to be treated as an ordinary differential equation

[en] In this paper we give a short introduction to open problems and recent studies of classes of partial differential equations, which—in contrast to reaction–diffusion systems—describe phenomena with local interactions. Partial differential equations coupled with ordinary differential equations, models of transport type and hyperbolic systems are discussed with respect to their pattern forming behaviour. (open problem)

[en] In this paper, the Cousin Problems for an Overdetermined System of complex Partial Differential Equations are considered. It is proven that, under suitable hypotheses on its coefficients, the Cousin Problems can be solved on the domains of Cⁿ, which satisfy the same conditions as those in Complex Analysis. (author). 6 refs

[en] Fredholm Lie groupoids were introduced by Carvalho, Nistor and Qiao as a tool for the study of partial differential equations on open manifolds. This article extends the definition to the setting of locally compact groupoids and proves that “the Fredholm property is local”. Let $\mathcal{G}\rightrightarrows <!--\; ???\; -->X$ be a topological groupoid and $(U_i{)}_{i\in <!--\; ???\; -->I}$ be an open cover of X. We show that $\mathcal{G}$ is a Fredholm groupoid if, and only if, its reductions ${\mathcal{G}}_{U_i}^{U_i}$ are Fredholm groupoids for all $i\in <!--\; ???\; -->I$. We exploit this criterion to show that many groupoids encountered in practical applications are Fredholm. As an important intermediate result, we use an induction argument to show that the primitive spectrum of $C^{\ast <!--\; ???\; -->}(\mathcal{G})$ can be written as the union of the primitive spectra of all $C^{\ast <!--\; ???\; -->}({\mathcal{G}}_{U_i}^{U_i})$, for $i\in <!--\; ???\; -->I$.