Fast loops on semi-weighted homogeneous hypersurface singularities

We show the existence of ($1+\frac{w_2}{w_3}$)-fast loops on semi-weighted homogeneous hypersurface singularities with weights $w_1\geq w_2>w_3$. In particular we show that semi-weighted homogeneous hypersurface singularities have metrical conical structure only if its two low weights are equal

Rigidity of bi-Lipschitz equivalence of weighted homogeneous function-germs in the plane

The main goal of this work is to show that if two weighted homogeneous (but not homogeneous) function-germs (\C^2,0)\to(\C,0) are bi-Lipschitz equivalent, in the sense that these function-germs can be included in a strongly bi-Lipschitz trivial family of weighted homogeneous function-germs, then they are analytically equivalent

Inner Metric Geometry of Complex Algebraic Surfaces with Isolated Singularities

We produce examples of complex algebraic surfaces with isolated singularities such that these singularities are not metrically conic, i.e. the germs of the surfaces near singular points are not bi-Lipschitz equivalent, with respect to the inner metric, to cones. The technique used to prove the nonexistence of the metric conic structure is related to a development of Metric Homology. The class of the examples is rather large and it includes some surfaces of Brieskorn.Comment: 12 page

Multiplicity of analytic hypersurface singularities under bi-Lipschitz homeomorphisms

We give partial answers to a metric version of Zariski's multiplicity conjecture. In particular, we prove the multiplicity of complex analytic surface (not necessarily isolated) singularities in $\mathbb{C}^3$ is a bi-Lipschitz invariant.Comment: Accepted for publication by the Journal of Topolog

Tangent cones of Lipschitz normally embedded sets are Lipschitz normally embedded. Appendix by Anne Pichon and Walter D. Neumann

We prove that tangent cones of Lipschitz normally embedded sets are Lipschitz normally embedded. We also extend to real subanalytic sets the notion of reduced tangent cone and we show that subanalytic Lipschitz normally embedded sets have reduced tangent cones. In particular, we get that Lipschitz normally embedded complex analytic sets have reduced tangent cones.Comment: The paper was reorganized and also it was added an appendix by Anne Pichon and Walter D. Neumann. It has 13 pages and 6 figure

A Time-Segmented Consortium Blockchain for Robotic Event Registration

A blockchain, during its lifetime, records large amounts of data, that in a common usage its kept on its entirety. In a robotics environment, the old information is useful for human evaluation, or oracles interfacing with the blockchain but it is not useful for the robots that require only current information in order to continue their work. This causes a storage problem in blockchain nodes that have limited storage capacity, such as in the case of nodes attached to robots that are usually built around embedded solutions. This paper presents a time-segmentation solution for devices with limited storage capacity, integrated in a particular robot-directed blockchain called RobotChain. Results are presented regarding the proposed solution that show that the goal of restricting each node's capacity is reached without compromising all the benefits that arise from the use of blockchains in these contexts, and on the contrary, it allows for cheap nodes to use this blockchain, reduces storage costs and allows faster deployment of new nodes

On the bi-Lipschitz contact equivalence of plane complex function-germs

In this short note, we consider the problem of bi-Lipschitz contact equivalence of complex analytic function-germs of two variables. It is inquiring about the infinitesimal sizes of such function-germs, up to bi-Lipschitz changes of coordinates. We show that this problem is equivalent to the problem of the right topological classification.Comment: The second version is substantially modified. 1) We have dropped section 5 of the first version. 2) The introduction is slightly modified. 3) Section 3 has been augmented with a correct statement of its main result (now referred as) Theorem 3.6. as a consequence of new Lemmas 3.1 and 3.2. Is also now given a proof of Proposition 3.

Collapsing topology of isolated singularities

We proof here the existence of a topological thick and thin decomposition of any closed definable thick isolated singularity germ in the spirit of the recently discovered metric thick and thin decomposition of complex normal surface singularities of [10]. Our thin zone catches exactly the homology of the family of the links collapsing faster than linearly. Simultaneously we introduce a class of rigid homeomorphisms more general than bi-Lipschitz ones, which map the topological thin zone onto the topological thin zone of its image. As a consequence of this point of view for the class of singularities we consider we exhibit an equivalent description of the notion of separating sets in terms of this fast contracting homologyComment: 29 p

On normal embedding of complex algebraic surfaces

We construct examples of complex algebraic surfaces not admitting normal embeddings (in the sense of semialgebraic or subanalytic sets) with image a complex algebraic surface.Comment: 5 pages. Submitted to Proceedings of the 10th International Workshop on Real and Complex Singularitie

Choking horns in Lipschitz Geometry of Complex Algebraic Varieties

We study the Lipschitz Geometry of Complex Algebraic Singularities. For this purpose we introduce the notion of choking horns. A Choking horn is a family of cycles on the family of the sections of an algebraic variety by very small spheres centered at a singular point, such that the cycles cannot be boundaries of nearby chains. The presence of choking horns is an obstruction to metric conicalness as we can see with some classical isolated hypersurfaces singularities which we prove are not metrically conic. We also show that there exist infinitely countably many singular varieties, which are locally homeomorphic, but not locally bi-Lipschitz equivalent with respect to the inner metric.Comment: 11 pages. Appendix by Walter D. Neuman