Lower and upper orientable strong diameters of graphs satisfying the Ore condition

AbstractLet D be a strong digraph. The strong distance between two vertices u and v in D, denoted by sdD(u,v), is the minimum size (the number of arcs) of a strong sub-digraph of D containing u and v. For a vertex v of D, the strong eccentricity se(v) is the strong distance between v and a vertex farthest from v. The minimum strong eccentricity among all vertices of D is the strong radius, denoted by srad(D), and the maximum strong eccentricity is the strong diameter, denoted by sdiam(D). The lower (resp. upper) orientable strong radius srad(G) (resp. SRAD(G)) of a graph G is the minimum (resp. maximum) strong radius over all strong orientations of G. The lower (resp. upper) orientable strong diameter sdiam(G) (resp. SDIAM(G)) of a graph G is the minimum (resp. maximum) strong diameter over all strong orientations of G. In this work, we determine a bound of the lower orientable strong diameters and the bounds of the upper orientable strong diameters for graphs G=(V,E) satisfying the Ore condition (that is, σ2(G)=min{d(x)+d(y)|∀xy∉E(G)}≥n), in terms of girth g and order n of G

Algebraic properties of profinite groups

Recently there has been a lot of research and progress in profinite groups. We survey some of the new results and discuss open problems. A central theme is decompositions of finite groups into bounded products of subsets of various kinds which give rise to algebraic properties of topological groups.Comment: This version has some references update

Actions and Invariants of Residually Finite Groups: Asymptotic Methods

The workshop brought together experts in finite group theory, L2-cohomology, measured group theory, the theory of lattices in Lie groups, probability and topology. The common object of interest was residually finite groups, that each field investigates from a different angle

Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes, Application to the Minkowski problem in the Minkowski space

We study the existence of surfaces with constant or prescribed Gauss curvature in certain Lorentzian spacetimes. We prove in particular that every (non-elementary) 3-dimensional maximal globally hyperbolic spatially compact spacetime with constant non-negative curvature is foliated by compact spacelike surfaces with constant Gauss curvature. In the constant negative curvature case, such a foliation exists outside the convex core. The existence of these foliations, together with a theorem of C. Gerhardt, yield several corollaries. For example, they allow to solve the Minkowski problem in the 3-dimensional Minkowski space for datas that are invariant under the action of a co-compact Fuchsian group

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

Mildly dissipative diffeomorphisms of the disk with zero entropy

We discuss the dynamics of smooth diffeomorphisms of the disc with vanishing topological entropy which satisfy the mild dissipation property introduced in [CP]. In particular it contains the H\'enon maps with Jacobian up to 1/4. We prove that these systems are either (generalized) Morse Smale or infinitely renormalizable. In particular we prove for this class of diffeomorphisms a conjecture of Tresser: any diffeomorphism in the interface between the sets of systems with zero and positive entropy admits doubling cascades. This generalizes for these surface dynamics a well known consequence of Sharkovskii's theorem for interval maps

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2