An operator $I$ on a real Lie algebra $A$ is called complex structure if $I^2=-Id$ and the $\sqrt{-1}$-eigenspace $A^{1,0}$ subalgebra in complexification of $A$. A hypercomplex triple structures $I,J$ $K$ satisfying quaternionic relations. We call nilpotent quaternionic-solvable there exists finite filtration by quaternionic-invariant subalgebras with commutative subquotients which converges t...

In this paper, we classify the indecomposable non-nilpotent solvable Lie algebras with $N(R_n,m,r)$ nilradical,by using the derivation algebra and the automorphism group of $N(R_n,m,r)$.We also prove that these solvable Lie algebras are complete and unique, up to isomorphism.

We consider here the Byzantine Agreement problem (BA) in synchronous systems with homonyms in the case where some identifiers may be forgeable. More precisely, the n processes share a set of l (1 ≤ l ≤ n) identifiers. Assuming that at most t processes may be Byzantine and at most k (t ≤ k ≤ l) of these identifiers are forgeable in the sense that any Byzantine process can falsely use them, we pr...

When studying non-linear (higher order MUSCL type) nite volume approximation of a linear convection diiusion problem, one is confronted with a question whether the corresponding nonlinear discrete problem is solvable. In this contribution we note that the corresponding discrete problem L h u h = f h in two dimensions in general need not to possess a solution. However, we show that to every righ...

We present a deterministic, log-space algorithm that solves st-connectivity in undirected graphs. The previous bound on the space complexity of undirected st-connectivity was log(·) obtained by Armoni, Ta-Shma, Wigderson and Zhou [ATSWZ00]. As undirected st-connectivity is complete for the class of problems solvable by symmetric, non-deterministic, log-space computations (the class SL), this al...

We show that there is a field F complete with respect to a discrete valuation whose residue field is perfect and there is a finite Galois extension K|F such that there is no solvable Galois extension L|F such that the extension KL|K is unramified, where KL is the composite of K and L. As an application we deduce that that there is a field F as above and there is a smooth, projective, geometrica...

We present a deterministic, log-space algorithm that solves st-connectivity in undirected graphs. The previous bound on the space complexity of undirected st-connectivity was log(·) obtained by Armoni, Ta-Shma, Wigderson and Zhou [ATSWZ00]. As undirected st-connectivity is complete for the class of problems solvable by symmetric, non-deterministic, log-space computations (the class SL), this al...

Characterizing the universal coverings of smooth projective varieties is an old and hard question. Central to the subject is a conjecture of Shafarevich according to which the universal cover X̃ of a smooth projective variety is holomorphically convex, meaning that for every infinite sequence of points without limit points in X̃ there exists a holomorphic function unbounded on this sequence. In t...

We study the half-integral k-Directed Disjoint Paths Problem ( 2kDDPP) in highly strongly connected digraphs. The integral kDDPP is NP-complete even when restricted to instances where k = 2, and the input graph is L-strongly connected, for any L ≥ 1. We show that when the integrality condition is relaxed to allow each vertex to be used in two paths, the problem becomes efficiently solvable in h...