Some classes of homeomorphisms that preserve multiplicity and tangent cones

In this paper we present some applications of A’Campo-Lˆe’s Theorem and we study some relations between Zariski’s Questions A and B. It is presented some classes of homeomorphisms that preserve multiplicity and tangent cones of complex analytic sets. Moreover, we present a class of homeomorphisms that has the multiplicity as an invariant when we consider right equivalence and this class contains many known classes of homeomorphisms that preserve tangent cones. In particular, we present some effective approaches to Zariski’s Question A. We show a version of these results looking at infinity. Additionally, we present some results related with Nash modification and Lipschitz Geometry

On Zariski’s multiplicity problem at infinity

We address a metric version of Zariski's multiplicity conjecture at infinity that says that two complex algebraic affine sets which are bi-Lipschitz homeomorphic at infinity must have the same degree. More specifically, we prove that the degree is a bi-Lipschitz invariant at infinity when the bi-Lipschitz homeomorphism has Lipschitz constants close to 1. In particular, we have that a family of complex algebraic sets bi-Lipschitz equisingular at infinity has constant degree. Moreover, we prove that if two polynomials are weakly rugose equivalent at infinity, then they have the same degree. In particular, we obtain that if two polynomials are rugose equivalent at infinity or bi-Lipschitz contact equivalent at infinity or bi-Lipschitz right-left equivalent at infinity, then they have the same degree

Some classes of homeomorphisms that preserve multiplicity and tangent cones

In this paper it is presented some classes of homeomorphisms that preserve multiplicity and tangent cones of complex analytic sets. Moreover, we present a class of homeomorphisms that has the multiplicity as an invariant when we consider right equivalence and this class contains many known classes of homeomorphisms that preserve tangent cones. Finally, we show versions of these results looking at infinity

Multiplicity, regularity and blow-spherical equivalence of complex analytic sets

This paper is devoted to study multiplicity and regularity of complex analytic sets. We present an equivalence for complex analytical sets, named blow-spherical equivalence and we obtain several applications with this new approach. For example, we reduce to homogeneous complex algebraic sets a version of Zariski's multiplicity conjecture in the case of blow-spherical homeomorphism, we give some partial answers to the Zariski's multiplicity conjecture, we show that a blow-spherical regular complex analytic set is smooth and we give a complete classification of the complex analytic curves.The author was also partially supported by CNPq-Brazil grant 303811/2018-8 and Gobierno Vasco Grant IT1094-16

A proof of the differentiable invariance of the multiplicity using spherical blowing-up

In this paper we use some properties of spherical blowing-up to give an alternative and more geometric proof of Gau-Lipman Theorem about the differentiable invariance of the multiplicity of complex analytic sets. Moreover, we also provide a generalization of the Ephraim-Trotman Theorem.This is a post-peer-review, pre-copyedit version of an article published in Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas. The final authenticated version is available online at: http://doi.org/10.1007/s13398-018-0537-

On Lipschitz rigidity of complex analytic sets

We prove that any complex analytic set in $\mathbb{C}^n$ which is Lipschitz normally embedded at infinity and has tangent cone at infinity that is a linear subspace of $\mathbb{C}^n$ must be an affine linear subspace of $\mathbb{C}^n$ itself. No restrictions on the singular set, dimension nor codimension are required. In particular, any complex algebraic set in $\mathbb{C}^n$ which is Lipschitz regular at infinity is an affine linear subspace.The first named author was partially supported by CNPq-Brazil grant 302764/2014-

Multiplicity of singularities is not a bi-Lipschitz invariant

It was conjectured that multiplicity of a singularity is bi-Lipschitz invariant. We disprove this conjecture constructing examples of bi-Lipschitz equivalent complex algebraic singularities with different values of multiplicity.Lev Birbrair: Partially supported by CNPq grant 302655/2014-0. Alexandre Fernandes: Partially supported by CNPq grant grant304221/2017-9 and by CAPES-BRASIL Finance Code 001. J. Edson Sampaio: Partially supported by CNPq-Brazil grant 303811/2018-8, by the ERCEA 615655 NMST Consolidator Grant and also by the Basque Government through the BERC 2018-2021 program and Gobierno Vasco Grant IT1094-16, by the Spanish Ministry of Science, Innovation and Universities: BCAM Severo Ochoa accreditation SEV-2017-0718. Misha Verbitsky: Partially supported by the Russian Academic Excellence Project ‘5-100’, FAPERJ E-26/202.912/2018 and CNPq - Process 313608/2017-2

Multiplicity and degree as bi‐Lipschitz invariants for complex sets

We study invariance of multiplicity of complex analytic germs and degree of complex affine sets under outer bi-Lipschitz transformations (outer bi-Lipschitz homeomorphims of germs in the first case and outer bi-Lipschitz homeomorphims at infinity in the second case). We prove that invariance of multiplicity in the local case is equivalent to invariance of degree in the global case. We prove invariance for curves and surfaces. In the way we prove invariance of the tangent cone and relative multiplicities at infinity under outer bi-Lipschitz homeomorphims at infinity, and that the abstract topology of a homogeneous surface germ determines its multiplicity.The first named author is partially supported by IAS and by ERCEA 615655 NMST Consolidator Grant, MINECO by the project reference MTM2013-45710-C2-2-P, by the Basque Government through the BERC 2014-2017 program, by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323, by Bolsa Pesquisador Visitante Especial (PVE) - Ciencias sem Fronteiras/CNPq Project number: 401947/2013-0 and by Spanish MICINN project MTM2013-45710-C2-2-P. The second named author was partially supported by CNPq-Brazil grant 302764/2014-7. The third named author was partially supported by the ERCEA 615655 NMST Consolidator Grant and also by the Basque Government through the BERC 2014-2017 program and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323

Profile of female swimmers competing in the 50 m events at the 2021 LEN European Championships

This study aimed to understand whether there are significant differ- ences in stroke kinematics between tiers in female swimmers com- peting in the four 50 m events of the 2021 European Championships and to understand the speed-time relationship in the four race events per tier. Participants were all female swimmers (backstroke: 78 swim- mers; breaststroke: 75 swimmers; butterfly: 74 swimmers; freestyle: 87 swimmers) who participated in the 50 m events at the 2021 LEN European Championships held in Budapest (i.e. heats, semi-finals, and final). For each swimming stroke, swimmers were divided into three tiers (best-performing swimmers, intermedium-performing swimmers, and poorest-performing swimmers). Swimming speed revealed a significant tier effect (p < 0.05) in all race sections for all swimming strokes. The other stroke kinematic variables revealed divergent findings, but the stroke frequency presented an overall tier effect (p < 0.05) across all four swimming strokes. Curve fitting for all swimming strokes and tiers revealed a cubic relationship. Thus, it should be considered that female swimmers who compete in 50 m events in major competitions adopt an all-out strategy. The present data provide coaches with insightful information about the main trend in 50 m sprint events, specifically in each section of the race.The work was supported by the Fundação para a Ciência e a Tecnologia [UIDB/DTP/04045/2020].info:eu-repo/semantics/publishedVersio

Normative data of the start in the 50 m events at the 2021 LEN European Championships and understanding its relationship with the final race

This study aimed to: (i) present normative data of the variables related to the start in the four swim strokes by tier and sex, and; (ii) understand the relationship between the 15th meter mark time and the final race time of the male and female swimmers competing in the four 50 m events at the 2021 European Championships. Participants were all male and female swimmers who competed in the 50 m events at the 2021 LEN European Championships held in Budapest. The official race times and block times were retrieved from the official competition website. All starting variables were analyzed in a dedicated software for race analysis. The 15th meter mark time was used as the start main outcome. For all events by sex, the 15th meter mark time was the variable presenting the highest and largest tier effect (p < 0.001) besides the final race time. Overall, despite the swim stroke, the variables related to the underwater phase were also responsible for the significant tier effect (p < 0.001). The 15th meter mark time presented a high to very-high relationship with the final race time in all four swim strokes. This relationship was stronger in freestyle (both sexes). That is, swimmers who achieve the 15th meter mark sooner are more likely to deliver better performances. Coaches must be aware that the underwater phase plays a key-role on the swimmers’ (both sexes) start performance. Nonetheless, different strategies can be used based on the swimmers’ strength and weaknesses. Moreover, the start performance in all four swim strokes and in both sexes can strongly predict the final race time.This research was supported by national funds (FCT - Portuguese Foundation for Science and Technology) under the project UIDB/DTP/04045/2020info:eu-repo/semantics/publishedVersio