The weak Frenet frame of non-smooth curves with finite total curvature and absolute torsion

We deal with a notion of weak binormal and weak principal normal for non-smooth curves of the Euclidean space with finite total curvature and total absolute torsion. By means of piecewise linear methods, we first introduce the analogous notation for polygonal curves, where the polarity property is exploited, and then make use of a density argument. Both our weak binormal and normal are rectifiable curves which naturally live in the projective plane. In particular, the length of the weak binormal agrees with the total absolute torsion of the given curve. Moreover, the weak normal is the vector product of suitable parameterizations of the tangent indicatrix and of the weak binormal. In the case of smooth curves with positive curvature, the weak binormal and normal yield (up to a lifting) the classical notions of binormal and normal.Comment: 18 pages, 2 figure

Bounded variation and relaxed curvature of surfaces

We consider a relaxed notion of energy of non-parametric codimension one surfaces that takes account of area, mean curvature, and Gauss curvature. It is given by the best value obtained by approximation with inscribed polyhedral surfaces. The BV and measure properties of functions with finite relaxed energy are studied. Concerning the total mean and Gauss curvature, the classical counterexample by Schwarz-Peano to the definition of area is also analyzed.Comment: 25 page

Strict BV relaxed area of Sobolev maps into the circle: the high dimension case

We deal with the relaxed area functional in the strict $BV$-convergence of non-smooth maps defined in domains of generic dimension and taking values into the unit circle. In case of Sobolev maps, a complete explicit formula is obtained. Our proof is based on tools from Geometric Measure Theory and Cartesian currents. We then discuss the possible extension to the wider class of maps with bounded variation. Finally, we show a counterexample to the locality property in case of both dimension and codimension larger than two

The total intrinsic curvature of curves in Riemannian surfaces

We deal with irregular curves contained in smooth, closed, and compact surfaces. For curves with finite total intrinsic curvature, a weak notion of parallel transport of tangent vector fields is well-defined in the Sobolev setting. Also, the angle of the parallel transport is a function with bounded variation, and its total variation is equal to an energy functional that depends on the "tangential" component of the derivative of the tantrix of the curve. We show that the total intrinsic curvature of irregular curves agrees with such an energy functional. By exploiting isometric embeddings, the previous results are then extended to irregular curves contained in Riemannian surfaces. Finally, the relationship with the notion of displacement of a smooth curve is analyzed.Comment: 28 pages, 1 figur

A variational problem for multifunctions with interaction between leaves

We discuss a variational problem defined on couples of functions that are constrained to take values into the 2-dimensional unit sphere. The energy functional contains, besides standard Dirichlet energies, a non-local interaction term that depends on the distance between the gradients of the two functions. Different gradients are preferred or penalized in this model, in dependence of the sign of the interaction term. In this paper we study the lower semicontinuity and the coercivity of the energy and we find an explicit representation formula for the relaxed energy. Moreover, we discuss the behavior of the energy in the case when we consider multifunctions with two leaves rather than couples of functions

Curvature-dependent energies: The elastic case

We continue our analysis on functionals depending on the curvature of graphs of curves in high codimension Euclidean space. We deal with the “elastic” case, corresponding to a superlinear dependence on the pointwise curvature.We introduce the corresponding relaxed energy functional and prove an explicit representation formula. Different phenomena w.r.t. the “plastic” case, i.e. to the relaxation of the total curvature functional, are observed. A p-curvature functional is well-defined on continuous curves with finite relaxed energy, and the relaxed energy is given by the length plus the p-curvature. The wider class of graphs of one-dimensional BV-functions is treated

Continuum Kinematics with Incompatible-Compatible Decomposition

Abstract. We present a framework for the kinematics of a material body undergoing anelastic deformation. For such processes, the material structure of the body, as reflected by the geometric structure given to the set of body points, changes. The setting we propose may be relevant to phenomena such as plasticity, fracture, discontinuities, and non-injectivity of the deformations. In this framework, we construct an unambiguous decomposition into incompatible and compatible factors which includes the standard elastic-plastic decomposition in plasticity

Performance Assessment in Fingerprinting and Multi Component Quantitative NMR Analyses

An interlaboratory comparison (ILC) was organized with the aim to set up quality control indicators suitable for multicomponent quantitative analysis by nuclear magnetic resonance (NMR) spectroscopy. A total of 36 NMR data sets (corresponding to 1260 NMR spectra) were produced by 30 participants using 34 NMR spectrometers. The calibration line method was chosen for the quantification of a five-component model mixture. Results show that quantitative NMR is a robust quantification tool and that 26 out of 36 data sets resulted in statistically equivalent calibration lines for all considered NMR signals. The performance of each laboratory was assessed by means of a new performance index (named Qp-score) which is related to the difference between the experimental and the consensus values of the slope of the calibration lines. Laboratories endowed with a Qp-score falling within the suitable acceptability range are qualified to produce NMR spectra that can be considered statistically equivalent in terms of relative intensities of the signals. In addition, the specific response of nuclei to the experimental excitation/relaxation conditions was addressed by means of the parameter named NR. NR is related to the difference between the theoretical and the consensus slopes of the calibration lines and is specific for each signal produced by a well-defined set of acquisition parameters