المؤلفون: Muñoz-Cabello, C., Nuño-Ballesteros, J. J., Sinha, R. Oset

مصطلحات موضوعية: Mathematics - Algebraic Geometry, Mathematics - Differential Geometry, 32S30 (Primary) 32S25, 58K25 (Secondary)

الوصف: We develop a Thom-Mather theory of frontals analogous to Ishikawa's theory of deformations of Legendrian singularities but at the frontal level, avoiding the use of the contact setting. In particular, we define concepts like frontal stability, versality of frontal unfoldings or frontal codimension. We prove several characterizations of stability, including a frontal Mather-Gaffney criterion, and of versality. We then define the method of reduction with which we show how to construct frontal versal unfoldings of plane curves and show how to construct stable unfoldings of corank 1 frontals with isolated instability which are not necessarily versal. We prove a frontal version of Mond's conjecture in dimension 1. Finally, we classify stable frontal multigerms and give a complete classification of corank 1 stable frontals from $\mathbb C^3$ to $\mathbb C^4$.
Comment: 25 pages

الوصول الحر: http://arxiv.org/abs/2302.13621Test

المؤلفون: Muñoz-Cabello, C., Nuño-Ballesteros, J. J., Sinha, R. Oset

مصطلحات موضوعية: Mathematics - Algebraic Geometry, Mathematics - Differential Geometry, 32S30 (Primary) 32S25, 58S25 (Secondary)

الوصف: We consider singularities of frontal surfaces of corank one and finite frontal codimension. We look at the classification under left-right-equivalence and introduce the notion of frontalisation for singularities of fold type. We define the cuspidal and the transverse double point curves and prove that the frontal has finite codimension if and only if both curves are reduced. Finally, we also discuss about the frontal versions of the Marar-Mond formulas and the Mond's conjecture.
Comment: 22 pages, 16 figures

الوصول الحر: http://arxiv.org/abs/2205.02097Test

المؤلفون: Ribes, I. Breva, Sinha, R. Oset

مصطلحات موضوعية: Mathematics - Algebraic Geometry, Mathematics - Complex Variables

الوصف: We study the simplicity of map-germs obtained by the operation of augmentation and describe how to obtain their versal unfoldings. When the augmentation comes from an $\mathscr{A}_e$-codimension 1 germ or the augmenting function is a Morse function, we give a complete characterisation for simplicity. These characterisations yield all the simple augmentations in all explicitly obtained classifications of $\mathscr{A}$-simple monogerms except for one ($F_4$ in Mond's list from $\mathbb{C}^2$ to $\mathbb{C}^3$). Moreover, using our results we produce a list of simple augmentations from $\mathbb{C}^4$ to $\mathbb{C}^4$.

الوصول الحر: http://arxiv.org/abs/2203.09223Test

المؤلفون: Nishimura, T., Sinha, R. Oset, Ruas, M. A. S., Atique, R. Wik

المصدر: Mathematische Annalen, 366 (2016), 573-611

مصطلحات موضوعية: Mathematics - Algebraic Geometry, Mathematics - Differential Geometry, Mathematics - Geometric Topology, 58K40 (primary), 57R45, 58K20 (secondary)

الوصف: In this paper, a systematic method is given to construct all liftable vector fields over an analytic multigerm $f: (\mathbb{K}^n, S)\to (\mathbb{K}^p,0)$ of corank at most one admitting a one-parameter stable unfolding.
Comment: 34 pages. In ver. 2, several careless mistakes for calculations in Section 6 were corrected

الوصول الحر: http://arxiv.org/abs/1408.3825Test

المؤلفون: Sinha, R. Oset, Ruas, M. A. S., Atique, R. Wik

مصطلحات موضوعية: Mathematics - Complex Variables, Mathematics - Algebraic Geometry, 58K40, 32S05, 32S70

الوصف: We generalise the operations of augmentation and concatenations in order to obtain multigerms of analytic (or smooth) maps $(\mathbb K^n,S)\rightarrow(\mathbb K^p,0)$ with $\mathbb K=\mathbb C$ or $\mathbb R$ from monogerms and some special multigerms. We then prove that any corank 1 codimension 2 multigerm in Mather's nice dimensions $(n,p)$ with $n\geq p-1$ can be constructed using augmentations and these operations.
Comment: 34 pages, 2 figures

الوصول الحر: http://arxiv.org/abs/1404.3149Test

المؤلفون: Ichiki, S., Nishimura, T., Sinha, R. Oset, Ruas, M. A. S.

مصطلحات موضوعية: Mathematics - Differential Geometry, 57R45, 58C25, 58K50

الوصف: We define generalized distance-squared mappings, and we concentrate on the plane to plane case. We classify generalized distance-squared mappings of the plane into the plane in a recognizable way.
Comment: 13 pages, 1 figure

الوصول الحر: http://arxiv.org/abs/1404.2841Test

المؤلفون: SINHA, R. OSET, RUAS, M. A. S., ATIQUE, R. WIK

المصدر: Mathematica Scandinavica, 2016 Jan 01. 119(2), 197-222.

الوصول الحر: https://www.jstor.org/stable/45148844Test

المؤلفون: Riul, P. Benedini, Sinha, R. Oset

مصطلحات موضوعية: projections, normal sections, curvature locus, immersed surfaces, immersed $3$-manifolds, singular corank $1$ manifolds, 57R45, 53A05, 58K05

الوصف: We use normal sections to relate the curvature locus of regular (resp. singular corank $1$) $3$-manifolds in $\mathbb{R}^6$ (resp.\ $\mathbb R^5$) with regular (resp.\ singular corank $1$) surfaces in $\mathbb R^5$ (resp. $\mathbb R^4$). For example, we show how to generate a Roman surface by a family of ellipses different to Steiner's way. We also study the relations between the regular and singular cases through projections. We show that there is a commutative diagram of projections and normal sections which relates the curvature loci of the different types of manifolds, and therefore, that the second order geometry of all of them is related. In particular, we define asymptotic directions for singular corank $1$ $3$-manifolds in $\mathbb R^5$ and relate them to asymptotic directions of regular $3$-manifolds in $\mathbb R^6$ and singular corank $1$ surfaces in $\mathbb R^4$.

وصف الملف: application/pdf

العلاقة: https://projecteuclid.org/euclid.pm/1607655922Test; Publ. Mat. 65, no. 1 (2021), 389-407

الإتاحة: https://doi.org/10.5565/PUBLMAT6512114Test
https://projecteuclid.org/euclid.pm/1607655922Test

المؤلفون: Deolindo-Silva, J. L., Sinha, R. Oset

المساهمون: Dirección General de Universidades e Investigación, CAPES/JSPS

المصدر: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas ; volume 115, issue 2 ; ISSN 1578-7303 1579-1505

مصطلحات موضوعية: Applied Mathematics, Computational Mathematics, Geometry and Topology, Algebra and Number Theory, Analysis

الإتاحة: https://doi.org/10.1007/s13398-021-01019-1Test

المؤلفون: Deolindo-Silva, J. L., Sinha, R. Oset

المصدر: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales / RACSAM; Apr2021, Vol. 115 Issue 2, p1-19, 19p

مستخلص: We study the geometry of surfaces in R 5 by relating it to the geometry of regular and singular surfaces in R 4 obtained by orthogonal projections. In particular, we obtain relations between asymptotic directions, which are not second order geometry for surfaces in R 5 but are in R 4 . We also relate the umbilic curvatures of each type of surface and their contact with spheres. We then consider the surfaces as normal sections of 3-manifolds in R 6 and again relate asymptotic directions and contact with spheres by defining an appropriate umbilic curvature for 3-manifolds. [ABSTRACT FROM AUTHOR]

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