Steiner’s formula in the Heisenberg group

Steiner’s tube formula states that the volume of an ϵ-neighborhood of a smooth regular domain in Rn is a polynomial of degree n in the variable ϵ whose coefficients are curvature integrals (also called quermassintegrals). We prove a similar result in the sub-Riemannian setting of the first Heisenberg group. In contrast to the Euclidean setting, we find that the volume of an ϵ-neighborhood with respect to the Heisenberg metric is an analytic function of ϵ that is generally not a polynomial. The coefficients of the series expansion can be explicitly written in terms of integrals of iteratively defined canonical polynomials of just five curvature terms.

Frobenius Solution for Legendre's Equation, Rodrigue's Formula and Normalization

The Frobenius solution to Legendre/s equation is developed in detail as is Rodrigue's formula, which is employed to normalize Legendre polynomials.

A Universal Formula for Continuous Utility

A single formula assigns a continuous utility function to every representable preference relation.

Ms. Praed. 31 - Formula novitiorum, p. III

Vorbesitzer: Dominikanerkloster Frankfurt am Main

Publishing Linked Data - There is no One-Size-Fits-All Formula

Publishing Linked Data is a process that involves several design decisions and technologies. Although some initial guidelines have been already provided by Linked Data publishers, these are still far from covering all the steps that are necessary (from data source selection to publication) or giving enough details about all these steps, technologies, intermediate products, etc. Furthermore, given the variety of data sources from which Linked Data can be generated, we believe that it is possible to have a single and uni�ed method for publishing Linked Data, but we should rely on di�erent techniques, technologies and tools for particular datasets of a given domain. In this paper we present a general method for publishing Linked Data and the application of the method to cover di�erent sources from di�erent domains.

A Compact Formula for the Derivative of a 3-D Rotation in Exponential Coordinates

We present a compact formula for the derivative of a 3-D rotation matrix with respect to its exponential coordinates. A geometric interpretation of the resulting expression is provided, as well as its agreement with other less-compact but better-known formulas. To the best of our knowledge, this simpler formula does not appear anywhere in the literature. We hope by providing this more compact expression to alleviate the common pressure to reluctantly resort to alternative representations in various computational applications simply as a means to avoid the complexity of differential analysis in exponential coordinates.

INSTRUCCIO y formula per a la bona execucio de la pragmatica dels processos de absencia [Texto impreso]

Port. con esc. real xil