The original problem of geodesic representation, solved by Dini (1869) with reality restrictions and by Lie (1883) in complete generality, may be stated as follows : Find all pairs of surfaces S and Sx whose points may be put into correspondence in such a way that every geodesic on the one surface is pictured by a geodesic on the other. Apart from the trivial case where Sx is isometric with (ap...

The space of directed sets is a Banach space in which convex compact subsets of Rn are embedded. Each directed set is visualized as a (nonconvex) subset of Rn, which is comprised of a convex, a concave and a mixed-type part. Following an idea of A. Rubinov, the directed subdifferential of a difference of convex (DC) functions is defined as the directed difference of the corresponding embedded c...

In this paper we are interested in pointwise regularity of solutions to elliptic equations. In a first result, we prove that if the modulus of mean oscillation of ∆u at the origin is Dini (in Lp average), then the origin is somehow a Lebesgue point of continuity (still in Lp average) for the second derivatives D2u. We extend this pointwise regularity result to the obtacle problem for the Laplac...

3 Armando Cavalcante Franco Dias, Michele de Cassia Pereira e Silva, Simone Raposo 4 Cotta, Francisco Dini-Andreote,Fábio Lino Soares Jr, Joana Falcão Salles, João Lúcio 5 Azevedo, Jan Dirk van Elsas and Fernando Dini Andreote 6 7 8 Center for Nuclear Energy in Agriculture, University of São Paulo, CENA/USP, Piracicaba, 9 Brazil 10 Department of Microbiology Paulo de Góes, URFJ/RJ, University o...

Starting from second-order conditions for C scalar unconstrained optimization problems described in terms of the second-order Dini directional derivative, we pose the problem, whether similar conditions for C vector optimization problems can be derived. We define second-order Dini directional derivatives for vector functions and apply them to formulate such conditions as a Conjecture. The proof...

Levelling is the most precise technique for height difference measurements in geomatics engineering. Various systematic errors affect precise levelling observations and reduce the precision of the observed height differences. This study investigates digital levels residual compensator error and observational method for its elimination. For this purpose the levelling data, which was collected wi...

We present a 2 × 2 Lax representation for discrete circular nets of constant negative Gauß curvature. It is tightly linked to the 4D consistency of the Lax representation of discrete K-nets (in asymptotic line parametrization). The description gives rise to Bäcklund transformations and an associated family. All the members of that family – although no longer circular – can be shown to have cons...

We prove in the general setting of lower semicontinuous functions on Banach spaces the relation between the Rockafellar directional derivative and the mixed lower limit of the lower Dini derivatives. As a byproduct we derive the famous inclusions of tangent cones of closed sets in Banach spaces. The results are established using as principal tool multidirectional mean value inequalities [Aussel...

In this paper, a basic property that a Liapunov-Krasovskii functional must have when used for studying the stability of time delay systems under Carath¶eodory conditions is studied. This property concerns the equality, almost everywhere, of the upper right-hand Dini derivative of the Liapunov-Krasovskii functional evaluated at the solution, and of the Liapunov-Krasovskii derivative (computed by...

We study the problem of accessibility of boundary points for domains in the sub–Riemannian setting of the first Heisenberg group. A sufficient condition for accessibility is given. It is a Dini–type continuity condition for the horizontal gradient of the defining function. The sharpness of this condition is shown by examples.