Article
On bounded continuous solutions of the archetypal equation with rescaling

This paper aims at unifying and clarifying the recent advances in the analysis of the fractional and generalized fractional Partial Diﬀerential Equations of Caputo and RiemannLiouville type arising essentially from the probabilistic point of view. This point of view leads to the path integral representation for the solutions of these equations, which is seen to be stable with respect to the initial data and key parameters and is directly amenable to numeric calculations (MonteCarlo simulation). In many cases these solutions can be compactly presented via the wide class of operatorvalued analytic functions of the MittagLeﬄer type, which are proved to be expressed as the Laplace transforms of the exit times of monotone Markov processes.
We consider a stochastic model of clock synchronization in a wireless network of N sensors interacting with one dedicated accurate time server. For large N we find an estimate of the final time sychronization error for global and relative synchronization. The main results concern the behavior of the network on different timescales tN→∞ , N→∞ . We discuss the existence of phase transitions and find the exact timescales for which an effective clock synchronization of the system takes place.
We consider Markov models of multicomponent systems with synchronizing interaction. Under natural regularity assumptions about the message routing graph, they have nice longtime behavior. We are interested in limit probability laws related to the steady state viewed from the centerofmass coordinate system.
Functional classes on a curve in a plane (a partial case of a spatial curve) can be described by the approximation speed by functions that are harmonic in threedimensional neighbourhoods of the curve. No constructive description of functional classes on rather general surfaces in R 3 and R 4 has been presented in literature so far. The main result of the paper is Theorem 1.
Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible crosssection of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a crosssection exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a crosssection in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational crosssection in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational Wequivariant map T   >G/T where T is a maximal torus of G and W the Weyl group.