Search results for " Geometry"
showing 10 items of 2294 documents
A blow-up result for a nonlinear wave equation on manifolds: the critical case
2021
We consider a inhomogeneous semilinear wave equation on a noncompact complete Riemannian manifold (Formula presented.) of dimension (Formula presented.), without boundary. The reaction exhibits the combined effects of a critical term and of a forcing term. Using a rescaled test function argument together with appropriate estimates, we show that the equation admits no global solution. Moreover, in the special case when (Formula presented.), our result improves the existing literature. Namely, our main result is valid without assuming that the initial values are compactly supported.
A remark on two notions of flatness for sets in the Euclidean space
2021
In this note we compare two ways of measuring the $n$-dimensional "flatness" of a set $S\subset \mathbb{R}^d$, where $n\in \mathbb{N}$ and $d>n$. The first one is to consider the classical Reifenberg-flat numbers $\alpha(x,r)$ ($x \in S$, $r>0$), which measure the minimal scaling-invariant Hausdorff distances in $B_r(x)$ between $S$ and $n$-dimensional affine subspaces of $\mathbb{R}^d$. The second is an `intrinsic' approach in which we view the same set $S$ as a metric space (endowed with the induced Euclidean distance). Then we consider numbers ${\sf a}(x,r)$'s, that are the scaling-invariant Gromov-Hausdorff distances between balls centered at $x$ of radius $r$ in $S$ and the $n$-dimensi…
On Limits at Infinity of Weighted Sobolev Functions
2022
We study necessary and sufficient conditions for a Muckenhoupt weight $w \in L^1_{\mathrm{loc}}(\mathbb R^d)$ that yield almost sure existence of radial, and vertical, limits at infinity for Sobolev functions $u \in W^{1,p}_{\mathrm{loc}}(\mathbb R^d,w)$ with a $p$-integrable gradient $|\nabla u|\in L^p(\mathbb R^d,w)$. The question is shown to subtly depend on the sense in which the limit is taken. First, we fully characterize the existence of radial limits. Second, we give essentially sharp sufficient conditions for the existence of vertical limits. In the specific setting of product and radial weights, we give if and only if statements. These generalize and give new proofs for results of…
Two examples related to conical energies
2022
In a recent article we introduced and studied conical energies. We used them to prove three results: a characterization of rectifiable measures, a characterization of sets with big pieces of Lipschitz graphs, and a sufficient condition for boundedness of nice singular integral operators. In this note we give two examples related to sharpness of these results. One of them is due to Joyce and M\"{o}rters, the other is new and could be of independent interest as an example of a relatively ugly set containing big pieces of Lipschitz graphs.
A note on Oscar Chisini mean value definition
2012
Mainly on the basis of some notable physical examples reported in a 1929 Oscar Chisini paper, in this brief note it is exposed further possible historic-critical remarks on the definition of statistical mean which lead us towards the realm of Integral Geometry, via the Felix Klein Erlanger Programm.
Geometric Aspects of Thermodynamics
2016
This chapter deals with mathematical aspects of thermodynamics most of which will be seen to be primarily of geometrical nature. Starting with a short excursion to differentiable manifolds we summarize the properties of functions, of vector fields and of one-forms on thermodynamic manifolds. This summary centers on exterior forms over Euclidean spaces and the corresponding differential calculus. In particular, one-forms provide useful tools for the analysis of thermodynamics. A theorem by Caratheodory is developed which is closely related to the second law of thermodynamics. The chapter closes with a discussion of systems which depend on two variables and for which there is an interesting a…
Emotional Seesaw, Compliance, and Mindlessness
2001
Research on emotion conducted so far has ignored situations where the subject experiences a certain emotion, but where the external stimulus that evoked and upholds this emotion suddenly disappears. This kind of situation, however, is relatively common in everyday life. This article attempts to recognize certain consequences of those conditions under which the stimuli justifying our experience of such emotional states as fear or joy suddenly disappear. Research done to date by the author and colleagues indicates increased compliance of the subject when addressed with various requests, commands, or suggestions in the situation termed here “emotional seesaw.” The classical “live” example tha…
Comparison of the Root Canal Curvatures of Human Teeth with the Curvatures of the Resin Blocks Used as Models for Training
2019
In this paper we develop a method to associate a function curvature to human teeth roots.These functions are calculated from radiographs using mathematical software. We use this procedure to compare the curvatures of the roots with the curvatures of the resin blocks that are used for learning. The comparison is made by calculating the distances of the corresponding functions in the \(L_1\) metric.
The number of contacts in random fibre networks
2012
There is a wide range of materials that can be considered as nonwoven random networks of fibres. Such materials include glass-fibre mats, filters, various paper products and structural components of cells and tissues. The mechanical properties of these kinds of networks have been studied extensively for many decades. As many of such networks form more or less two-dimensional structures, they can, to a good approximation, be considered to consist of randomly distributed fibres or filaments connected at their crossings points. Recent development of the resolution of X-ray computed tomography have enabled imaging of the three dimensional structure of such materials with a resolution sufficient…
Real and simulated endoscopy of neurosurgical approaches in an anatomical model
1997
Endoscopy simulation is a new visualisation technique which can be used to visualise patient anatomy as seen through endoscopes. In the following, we describe the application of simulated endoscopy to neurosurgical endoscopic approaches. We assess different aspects of virtual images and their influence on the final result, as judged by clinicians. One of these aspects is the projective geometry which is used to render the 3D images. Rendering using stereographic projection leads to more realistic endoscopic views than perspective projection.