Regularity of Lagrangian flows over RCD*(K, N) spaces

The aim of this note is to provide regularity results for Regular Lagrangian flows of Sobolev vector fields over compact metric measure spaces verifying the Riemannian curvature dimension condition. We first prove, borrowing some ideas already present in the literature, that flows generated by vector fields with bounded symmetric derivative are Lipschitz, providing the natural extension of the standard Cauchy–Lipschitz theorem to this setting. Then we prove a Lusin-type regularity result in the Sobolev case (under the additional assumption that the m.m.s. is Ahlfors regular) therefore extending the already known Euclidean result.

Conserved quantities in the presence of torsion: A generalization of Killing theorem

When spacetime torsion is present, geodesics and autoparallels generically do not coincide. In this work, the well-known method that uses Killing vectors to solve the geodesic equations is generalized for autoparallels. The main definition is that of T-Killing vectors: vector fields such that, when their index is lowered with the metric, have vanishing symmetric derivative when acted upon with a torsionful and metric-compatible derivative. The main property of T-Killing vectors is that their contraction with the autoparallels’ tangents are constant along these curves. As an example, in a static and spherically symmetric situation, the autoparallel equations are reduced to an effective one-dimensional problem. Other interesting properties and extensions of T-Killing vectors are discussed.

On The K-Pseudo Symmetric and Ordinary Differentiation

In 1972, S. Valenti introduced the definition of k-pseudo symmetric derivative and has shown that the set of all points of a continuous function, at which there exists a finite k-pseudo symmetric derivative but the finite ordinary derivative does not exist, is of Lebesgue measure zero. In 1993, L. Zajícek has shown that for a continuous function f, the set of all points, at which f is symmetrically differentiable but no differentiable, is σ-(1 - ε) symmetrically porous for every ε > 0. The question arises: can we transferred the Zajícek’s result to the case of the k-pseudo symmetric derivative?In this paper, we shall show that for each 0 < ε < 1 the set of all points of a continuous function, at which there exists a finite k-pseudo symmetric derivative but the finite ordinary derivative does not exist, is σ-(1 - ε)-porous.

Differentiation of Fourier Series via Orthogonal Derivative

It is very known that if the operator d/dx acts on each term into a convergent Fourier Series (FS), then it may result a divergent series. This situation is remedied applying the symmetric derivative to FS, which implies the existence of the important Fejér-Lanczos Factors. In this paper, we show that the orthogonal derivative also leads to these Factors.Journal of Institute of Science and Technology, 2015, 20(2): 113-114

Self-assembled phthalocyanine derivatives on highly ordered pyrolytic graphite

A novel unsymmetrically substituted hydroxy-functionalized Zn ( II ) phthalocyanine (Pc) 1, bearing long aliphatic chains, namely, dodecyloxy units, has been designed and synthesized to investigate the influence of the terminal hydroxyl group on the formation of self-assembled nanostructures. The symmetric derivative, octadodecyloxy- Zn ( II ) Pc (2) has been also synthesized and used as reference compound for comparison purposes. The supramolecular organization of the Pcs has been carried out by spin-coating on a highly ordered pyrolytic graphite (HOPG) surface and has been investigated by atomic force microscopy (AFM) and scanning electron microscopy (STM). AFM and STM studies showed that unsymmetrically substituted hydroxy-functionalized Zn ( II ) Pc 1 gives rise to the formation of wire-like structures in different lengths from nanometer to micrometer scales, whereas in the case of the symmetrical Zn ( II ) Pc 2 the formation of the wires on HOPG was less pronounced.

Connection formulae for spectral functions associated with singular Sturm–Liouville equations

We consider the Sturm–Liouville equation with the initial condition and suppose that Weyl's limit-point case holds at infinity. Let ρα(μ) be the corresponding spectral function and its symmetric derivative. We show that for almost all μ ∈ R, if exists and is positive for some α ∈ [0, π), then (i) exists and is positive for all β ∈ [0, π), and (ii) for all α1, α2 ∈ (0, π) \ {½ π},