arXiv subjects
Publications and source records attributed to Paolo Roselli.
This work analyzes the validity of Tonelli's theorem, overcoming the traditional assumption of $\sigma$-finiteness. We prove that the existence and equality of iterated integrals for indicator functions is a necessary and sufficient condition to define a product measure that allow to extend Tonelli's Theorem to more general measure spaces, including $s$-finite measures. Finally, we characterize the semi-finiteness of such a product measure.
This paper presents a new general formulation of the Radon-Nikodym theorem in the setting of abstract measure theory. We introduce the notion of weak localizability for a measure and show that this property is both necessary and sufficient for the validity of a Radon-Nikodym-type representation under a natural compatibility relation between measures. The proof relies solely on elementary tools, such as Markov's inequality and the monotone convergence theorem. In addition to establishing the main result, we provide a constructive approach to envelope functions for families of non-negative measurable functions supported on sets of finite measure.
If a one-variable function is sufficiently smooth, then the limit position of secant lines its graph is a tangent line. By analogy, one would expect that the limit position of secant planes of a two-variable smooth function is a plane tangent to its graph. Amazingly, this is not necessarily true, even when the function is a simple polynomial. Despite this paradox, we show that some analogies with the one-variable case still hold in the multi-variable context, provided we use a particular vector product: the Clifford's geometric one.
We show that the gradient of a multi-variable strongly differentiable function at a point is the limit of a single coordinate-free Clifford quotient between a multi-difference pseudo-vector and a pseudo-scalar, or of a sum of Clifford quotients between scalars (as numerators) and vectors (as denominators), both built on a same non degenerate simplex contracting to that point. Then, we provide some conjectures implied by the foregoing result.
In this work we develop the mathematical framework of !FTL, a new gesture recognition algorithm and we prove its convergence. Such convergence suggests to adopt a notion of shape for smooth gestures as a complex valued function. However, the idea inspiring that notion came to us from Clifford numbers and not from complex numbers. Moreover, the Clifford vector algebra can be used to extend to higher dimensions the notion of shape of a gesture, while complex numbers are useless to that purpose.
In this work we provide an algorithm approximating the tangent bivector at a point of a smooth surface through inscribed triangles converging to the point, regardless their form or position with respect to the tangent plane. This result is obtained approximating Jacobian determinants of smooth plane transformations at a point x through nondegenerate triangles converging to x. We can also approximate the area of a portion of a smooth surface, through a slightly modified notion of area of inscribed triangular polyhedra approaching the surface (without any kind of constraint due to the Schwarz paradox).