1,720,978 research outputs found
Multiscale Weak Compactness in Metric Spaces
No full textThe aim of this paper is to provide a useful tool for a better understanding of the approach proposed in [11] to extend to the setting of metric spaces profile decomposition theorems. To this aim we shall deal with a less general context which has the advantage of making the analogies with the linear case more evident
Elementary properties of optimal irrigation patterns
No full textIn this paper we follow the approach in Maddalena et al. (Interfaces and1
Free Boundaries 5, 391–415, 2003) to the study of the ramified structures and we
identify some geometrical properties enjoyed by optimal irrigation patterns. These
properties are “elementary” in the sense that they are not concerned with the regularity
at the ending points of such structures, where the presumable selfsimilarity
properties should take place. This preliminary study already finds an application in
G. Devillanova and S. Solimini (Math. J. Univ. Padua, to appear), where it is used in
order to discuss the irrigability of a given measure
On the dimension of an irrigable measure
No full textIn this paper the problem of determining if a given measure is irrigable, in the sense of [4], or not is addressed. A notion of irrigability dimension of a measure is given and lower and upper bounds are proved in terms of the minimal Hausdorff and respectively Minkowski dimension of a set on which the measure is concentrated. A notion of resolution dimension of a measure based on its discrete approximations is also introduced and its relation with the irrigation dimension is studie
Min-Max Solutions to Some Scalar Field Equations
No full textWe show the variational structure of a multiplicity result of positive solutions u is an element of H(1) (R(N)) to the equation -Delta u + a(x)u = u(p), where N >= 2, p > 1 with p = 3 and the potential a(x) is a positive function enjoying a planar symmetry. We require suitable decay assumptions which are widely implied by those in [6], in which Wei and Yan have obtained an analogous multiplicity result by using different techniques
Concentration estimates and multiple solutions to elliptic problems at critical growth
No full textIn this paper, we consider the problem -Δu = |u|2*-2u+λu in ω, u = 0 on ∂ω, where ω is an open regular bounded subset of RN (N ≥3), 2* = 2N/N-2 is the critical Sobolev exponent and λ> 0. Our main result asserts that, if N ≥7, the problem has infinitely many solutions and, from the point of view of the compactness arguments employed here, the restriction on the dimension N cannot be weakened
Elements of set theory and recursive arguments
No full textIn this paper we provide a self-contained introduction to some of the basic topics of
Mathematical Analysis, comprising natural and unrestricted set theoretic
methods. The note reflects partially the contents of a lecture given by the second author during the International Workshop on New Horizons in Teaching Science in Messina on June 2018. More precisely, following a quite new didactic approach, we recall here some basic facts on the Generalized Induction Principle as well as the Recursion Theorem, which plays a crucial role in the foundation of Mathematical Logic. Some elements of von Neumann, Gödel and Bernays (NGB) set theory are given in the last section. The note provides the preliminary tools that are essential in order to study the classical notion of Dedekind completeness
The fabulous destiny of Richard Dedekind
No full textBy using the preliminary results given in a previous divulgative note, we present here a concise and self--contained introduction to the construction of the real field as the unique, up to increasing isomorphism, Dedekind complete totally ordered field. Moreover, we also show the equivalence between the Dedekind completeness property on totally ordered fields and some meaningful well--known notions present in the literature, such as the Cauchy completeness on totally ordered Archimedean fields. This characterization result allows us to correctly encode the Dedekind completeness for totally ordered fields in the general abstract setting of metric spaces.
We believe that the essential parts of the paper can be easily accessed by anyone with some experience in abstract mathematical thinking. The paper completes the lecture given by the second author during the International Workshop on New Horizons in Teaching Science in Messina on June 2018
On weak convergence in metric spaces
No full textThis note gives an exposition of various extensions of the notion
of weak convergence to metric spaces. They are motivated by applications,
such as existence of xed points of non-expansive maps, and analysis of the
defect of compactness relative to gauge groups in Banach spaces, where weak
convergence is generally less useful than respectively, asymptotic centers in [14]
and polar convergence in a preliminary version of [35]). The note compares
notions of convergence of weak type found in literature, in particular the notion
of -convergence introduced by Lim in [25], polar convergence introduced by
the authors, and the modes of convergence of weak type introduced by Jost
[20], Sosov [36] and Monod [28] in Hadamard spaces. Some applications of
polar convergence, such as the existence of xed points for nonexpansive maps
and a suitable variant of the Brezis-Lieb Lemma are produced
Surface instabilities in graded tubular tissues induced by volumetric growth
No full textGrowth-induced pattern formation in tubular tissues is intimately correlated to normal physiological functions.
Moreover, either the microstructure or certain diseases can give rise to material inhomogeneity, which can
lead to a change of shape in the tissue. Therefore, it is of fundamental importance to understand surface
instabilities and pattern transitions of graded tubular tissues. In this paper we perform such analysis by the
use of a mechanical model of a graded tube which grows with a fixed outer boundary by focusing on a planestrain
problem within the framework of nonlinear elasticity. A theoretical model is established to determine the
uniform growth state, the critical growth factor, and the critical wavenumber for a general material model and
for a general material gradient. For a case study, the material is specified by the neo-Hookean model, and the
shear modulus is assumed to decay linearly or exponentially from the inner surface. Then, a parametric study is
carried out to unravel the effects of material and geometrical parameters on the bifurcation threshold and the
associated wrinkled pattern. In addition, a finite element model, which is validated by the theoretical one, is
developed to trace the post-buckling evolution. It is found that wrinkled pattern will evolve into an arch mode
and then into a creasing mode if the modulus decays linearly. However, the typical creasing mode may give
way to a period-doubling mode when applying an exponentially decaying modulus, and there is a co-existence
of the creasing mode and the wrinkling mode. As a result, different modulus gradients can generate diverse
pattern formations. The obtained results are useful to supply insight into the effects of material inhomogeneity
and different modulus gradients on surface instabilities and morphology evolutions in graded tubular tissues
- …
