Related papers: Rectifiability; a survey
We establish the rectifiability of measures satisfying a linear PDE constraint. The obtained rectifiability dimensions are optimal for many usual PDE operators, including all first-order systems and all second-order scalar operators. In…
Motivated by recent interest concerning "puncture repair" in the conformal geometry of compact Riemannian manifolds, a brief exposition on generalisation to the setting of quasiconformal mappings on certain metric measure spaces is…
Our purpose is to state quantitative conditions ensuring the rectifiability of a $d$--varifold $V$ obtained as the limit of a sequence of $d$--varifolds $(V_i)_i$ which need not to be rectifiable. More specifically, we introduce a sequence…
The discrete wave equation in a multidimensional uniform space with local defects and sources is considered. The characterization of all possible defect configurations corresponding to given amplitudes of waves at the receivers (detectors)…
Harmonic maps are nonlinear extensions of harmonic functions. They are critical points of natural energy functionals between Riemannian manifolds. Such type of problems appear in Physics, Geometry of Finance and the study of regularity and…
We define rectifiability in $\mathbb{R}^{n}\times\mathbb{R}$ with a parabolic metric in terms of $C^1$ graphs and Lipschitz graphs with small Lipschitz constants and we characterize it in terms of approximate tangent planes and tangent…
This is a survey of the inverse spectral problem on (mainly compact) Riemannian manifolds, with or without boundary. The emphasis is on wave invariants: on how wave invariants have been calculated and how they have been applied to concrete…
One goal of geometric measure theory is to understand how measures in the plane or higher dimensional Euclidean space interact with families of lower dimensional sets. An important dichotomy arises between the class of rectifiable measures,…
This survey contains a selection of topics unified by the concept of positive semi-definiteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or…
We present a comprehensive survey on removability of compact plane sets with respect to various classes of holomorphic functions. We also discuss some applications and several open questions, some of which are new.
Results involving various mean value properties are reviewed for harmonic, biharmonic and metaharmonic functions. It is also considered how the standard mean value property can be weakened to imply harmonicity and belonging to other classes…
This article surveys the mathematics of the cut and project method as applied to point sets, called here {\em model sets}. It covers the geometric, arithmetic, and analytical sides of this theory as well as diffraction and the connection…
The paper has two main goals. The first is to take a new approach to rearrangements on certain classes of measurable real-valued functions on $\mathbb{R}^n$. Rearrangements are maps that are monotonic (up to sets of measure zero) and…
Primitive recursion, mu-recursion, universal object and universe theories, complexity controlled iteration, code evaluation, soundness, decidability, G\"odel incompleteness theorems, inconsistency provability for set theory, constructive…
The status of precision electroweak data, tests of the standard model, determination of its parameters, and constraints on new physics, are surveyed.
The survey provides an overview of the developing area of parameterized algorithms for graph modification problems. We concentrate on edge modification problems, where the task is to change a small number of adjacencies in a graph in order…
The dissertation provides a comparative analysis of a number of variability tools currently in use. It serves as a catalogue for practitioners interested in the topic. We compare a range of modelling, configuring, and management tools for…
Each complex network (or class of networks) presents specific topological features which characterize its connectivity and highly influence the dynamics of processes executed on the network. The analysis, discrimination, and synthesis of…
Complexity is a multi-faceted phenomenon, involving a variety of features including disorder, nonlinearity, and self-organisation. We use a recently developed rigorous framework for complexity to understand measures of complexity. We…
We study some fundamental properties of real rectifiable currents and give a generalization of King's theorem in characterizing currents defined by positive real holomorphic chains. Our proof uses Siu's semicontinuity theorem and largely…