We study geometric properties of the integral curves of an implicit differential equation in a neighbourhood of a codimension ≤ 1 singularity. We also deal with the way these singularities bifurcate in generic families of equations and the changes in the associated geometry. The main tool used

for its image sets: If N ≤ αd, then with probability one, dimHB 〈α〉(E) = 1 α dimHE for all Borel sets E ⊂ (0,∞)N. (3) We provide sufficient conditions for the image BH(E) to be a Salem set or to have interior points. The results in (2) and (3) describe the geometric and Fourier analytic properties of BH

We present measurements of the fractal dimensions associated to the spin clusters for Z4 and Z5 spin models. We also attempted to measure similar fractal dimensions for the generalised Fortuin Kastelyn (FK) clusters in these models but we discovered that these clusters do not percolate at the critical point of the model under consideration. These results clearly mark a difference in the behaviour of these non local objects compared to the Ising model or the 3-state Potts model which corresponds to the simplest cases of ZN spin models with N = 2 and N = 3 respectively. We compare these fractal dimensions with the ones obtained for SLE interfaces.

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We give an overview of some new and old results on geometric properties of eigenfunctions of Laplacians on Riemannian manifolds. We discuss the properties of nodal sets and critical points, the number of nodal domains, as well as asymptotic properties of eigenfunctions in the high energy limit

In this paper we study property testing for basic geometric properties. We aim at developing efficient algorithms which determine whether a given (geometrical) object has a predetermined property Q or is “far ” from any object having the property. We show that many basic geometric properties have

This paper studies algebraic and geometric properties of curve–curve, curve– surface, and surface–surface bisectors. The computation is in general difficult since the bisector is determined by solving a system of nonlinear equations. Geometric considerations will help us to determine several

In our previous paper, we presented the combinatorial theory for minimal isostatic pinned frameworks- Assur graphs- which arise in the analysis of mechanical linkages. In this paper we further explore the geometric properties of Assur graphs, with a focus on singular realizations which have static

Abstract—In this letter we discuss geometric properties of the multiuser decorrelator receiver. In particular we show that the prevalent geometric interpretation of the correlating vectors comprising the decorrelator receiver in terms of orthogonal projections of the signature vectors is incorrect

-similarity characteristic are considered to be an intrinsic aspects of nuclear structure properties. For the description of nuclear geometric properties, nuclear fractal dimension is an irreplaceable variable similar to the nuclear radius. In order to determine these two variables, a new nuclear potential energy formula