Deformation of Vect(R)\mathbb{R})-Modules of Symbols

We consider the action of the Lie algebra of polynomial vector fields, $\mathfrak{vect}(1)$, by the Lie derivative on the space of symbols $\mathcal{S}_\delta^n=\bigoplus_{j=0}^n \mathcal{F}_{\delta-j}$. We study deformations of this action. We exhibit explicit expressions of some 2-cocycles generating the second cohomology space $\mathrm{H}^2_{\rm diff}(\mathfrak{vect}(1),{\cal D}_{\nu,\mu})$ where ${\cal D}_{\nu,\mu}$ is the space of differential operators from $\mathcal{F}_\nu$ to $\mathcal{F}_\mu$. Necessary second-order integrability conditions of any infinitesimal deformations of $\mathcal{S}_\delta^n$ are given. We describe completely the formal deformations for some spaces $\mathcal{S}_\delta^n$ and we give concrete examples of non trivial deformations

Couplings of gravity to antisymmetric gauge fields

We classify all the first-order vertices of gravity consistently coupled to a system of 2-form gauge fields by computing the local BRST cohomology H(s|d) in ghost number 0 and form degree n. The consistent deformations are at most linear in the undifferentiated two-form, confirming the previous results of [1] that geometrical theories constructed from a nonsymmetric gravity theory are physically inconsistent or trivial. No assumption is made here on the degree of homogeneity in the derivatives nor on the form of the gravity action.Comment: 11 pages, no figures, Latex2.0

On the existence of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds

Let (M,w) be a compact symplectic 2n-manifold, and g a Riemannian metric on M compatible with w. For instance, g could be Kahler, with Kahler form w. Consider compact Lagrangian submanifolds L of M. We call L Hamiltonian stationary, or H-minimal, if it is a critical point of the volume functional under Hamiltonian deformations. It is called Hamiltonian stable if in addition the second variation of volume under Hamiltonian deformations is nonnegative. Our main result is that if L is a compact, Hamiltonian stationary Lagrangian in C^n satisfying the extra condition of being Hamiltonian rigid, then for any M,w,g as above there exist compact Hamiltonian stationary Lagrangians L' in M contained in a small ball about some p in M and locally modelled on tL for small t>0, identifying M near p with C^n near 0. If L is Hamiltonian stable, we can take L' to be Hamiltonian stable. Applying this to known examples L in C^n shows that there exist families of Hamiltonian stable, Hamiltonian stationary Lagrangians diffeomorphic to T^n, and to (S^1 x S^{n-1})/{1,-1}, and with other topologies, in every compact symplectic 2n-manifold (M,w) with compatible metric g.Comment: 25 pages. (v2) minor changes, final version to appear in American Journal of Mathematic

On four-dimensional 2-handlebodies and three-manifolds

We show that for any n > 3 there exists an equivalence functor from the category of n-fold connected simple coverings of B^3 x [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, to the category Chb^{3+1} of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a consequence, we obtain an equivalence theorem for simple coverings of S^3 branched over links, which provides a complete solution to the long-standing Fox-Montesinos covering moves problem. This last result generalizes to coverings of any degree results by the second author and Apostolakis, concerning respectively the case of degree 3 and 4. We also provide an extension of the equivalence theorem to possibly non-simple coverings of S^3 branched over embedded graphs. Then, we factor the functor above through an equivalence functor from H^r to Chb^{3+1}, where H^r is a universal braided category freely generated by a Hopf algebra object H. In this way, we get a complete algebraic description of the category Chb^{3+1}. From this we derive an analogous description of the category Cob^{2+1} of 2-framed relative 3-dimensional cobordisms, which resolves a problem posed by Kerler.Comment: 213 pages, 272 figures, 15 table

Mixed Bruce-Roberts numbers

[EN] We extend the notions of mu*- sequences and Tjurina numbers of functions to the framework of Bruce-Roberts numbers, that is, to pairs formed by the germ at 0 of a complex analytic variety X. Cn and a finitely R( X)-determined analytic function germ f : (Cn, 0). (C, 0). We analyze some fundamental properties of these numbers.Part of this work was developed during the stay of the first author at the Departamento de Matematica of ICMC, Sao Carlos, Universidade de Sao Paulo (Brazil), in February and July 2018. The first author wishes to thank this institution for their hospitality and working conditions and to FAPESP for financial support. The first author was partially supported by MICINN Grant PGC2018-094889-B-I00 and FAPESP Grant 2014/00304-2. The second author was partially supported by CNPq Grant 306306/2015-8 and FAPESP Grant 2014/00304-2.Bivià-Ausina, C.; Ruas, M. (2020). Mixed Bruce-Roberts numbers. Proceedings of the Edinburgh Mathematical Society. 63(2):456-474. https://doi.org/10.1017/S0013091519000543S456474632Damon, J. (1996). Higher multiplicities and almost free divisors and complete intersections. Memoirs of the American Mathematical Society, 123(589), 0-0. doi:10.1090/memo/0589Wahl, J. M. (1983). Derivations, automorphisms and deformations of quasihomogeneous singularities. Proceedings of Symposia in Pure Mathematics, 613-624. doi:10.1090/pspum/040.2/713285De Goes Grulha, N. (2008). THE EULER OBSTRUCTION AND BRUCE-ROBERTS’ MILNOR NUMBER. The Quarterly Journal of Mathematics, 60(3), 291-302. doi:10.1093/qmath/han011Greuel, G.-M. (1975). Der Gau�-Manin-Zusammenhang isolierter Singularit�ten von vollst�ndigen Durchschnitten. Mathematische Annalen, 214(3), 235-266. doi:10.1007/bf01352108Gaffney, T. (1996). Multiplicities and equisingularity of ICIS germs. Inventiones Mathematicae, 123(1), 209-220. doi:10.1007/bf01232372Damon, J. (2002). On the freeness of equisingular deformations of plane curve singularities. Topology and its Applications, 118(1-2), 31-43. doi:10.1016/s0166-8641(01)00040-2Bruce, J. W., & Roberts, R. M. (1988). Critical points of functions on analytic varieties. Topology, 27(1), 57-90. doi:10.1016/0040-9383(88)90007-9Decker, W. , Greuel, G.-M. , Pfister, G. and Schönemann, H. , Singular 4-0-2. A computer algebra system for polynomial computations. Available at http://www.singular.uni-kl.de (2015).Looijenga, E. J. N. (1984). Isolated Singular Points on Complete Intersections. doi:10.1017/cbo9780511662720AHMED, I., RUAS, M. A. S., & TOMAZELLA, J. N. (2013). Invariants of topological relative right equivalences. Mathematical Proceedings of the Cambridge Philosophical Society, 155(2), 307-315. doi:10.1017/s0305004113000297Aleksandrov, A. G. (1986). COHOMOLOGY OF A QUASIHOMOGENEOUS COMPLETE INTERSECTION. Mathematics of the USSR-Izvestiya, 26(3), 437-477. doi:10.1070/im1986v026n03abeh001155Briançon, J., & Maynadier-Gervais, H. (2002). Sur le nombre de Milnor d’une singularité semi-quasi-homogène. Comptes Rendus Mathematique, 334(4), 317-320. doi:10.1016/s1631-073x(02)02256-2Giusti, M., & Henry, J.-P.-G. (1980). Minorations de nombres de Milnor. Bulletin de la Société mathématique de France, 79, 17-45. doi:10.24033/bsmf.1907Hauser, H., & Müller, G. (1993). Affine varieties and lie algebras of vector fields. Manuscripta Mathematica, 80(1), 309-337. doi:10.1007/bf03026556Liu, Y. (2018). Milnor and Tjurina numbers for a hypersurface germ with isolated singularity. Comptes Rendus Mathematique, 356(9), 963-966. doi:10.1016/j.crma.2018.07.004Nuno-Ballesteros, J. J., Orefice, B., & Tomazella, J. N. (2011). THE BRUCE-ROBERTS NUMBER OF A FUNCTION ON A WEIGHTED HOMOGENEOUS HYPERSURFACE. The Quarterly Journal of Mathematics, 64(1), 269-280. doi:10.1093/qmath/har032Ohmoto, T., Suwa, T., & Yokura, S. (1997). A remark on the Chern classes of local complete intersections. Proceedings of the Japan Academy, Series A, Mathematical Sciences, 73(5), 93-95. doi:10.3792/pjaa.73.93Lê Tráng, D. (1974). Calculation of Milnor number of isolated singularity of complete intersection. Functional Analysis and Its Applications, 8(2), 127-131. doi:10.1007/bf0107859

Converging shocks in elastic-plastic solids

We present an approximate description of the behavior of an elastic-plastic material processed by a cylindrically or spherically symmetric converging shock, following Whitham's shock dynamics theory. Originally applied with success to various gas dynamics problems, this theory is presently derived for solid media, in both elastic and plastic regimes. The exact solutions of the shock dynamics equations obtained reproduce well the results obtained by high-resolution numerical simulations. The examined constitutive laws share a compressible neo-Hookean structure for the internal energy e = e_(s)(I_1)+e_(h)(ρ,ς), where e_(s) accounts for shear through the first invariant of the Cauchy–Green tensor, and e_(h) represents the hydrostatic contribution as a function of the density ρ and entropy ς. In the strong-shock limit, reached as the shock approaches the axis or origin r=0, we show that compression effects are dominant over shear deformations. For an isothermal constitutive law, i.e., e_(h) = e_(h)(ρ), with a power-law dependence e_(h) ∝ ρ_(α), shock dynamics predicts that for a converging shock located at r=R(t) at time t, the Mach number increases as M ∝ [log(1/R)]^α, independently of the space index s, where s=2 in cylindrical geometry and 3 in spherical geometry. An alternative isothermal constitutive law with p(ρ) of the arctanh type, which enforces a finite density in the strong-shock limit, leads to M ∝ R^(−(s−1)) for strong shocks. A nonisothermal constitutive law, whose hydrostatic part eh is that of an ideal gas, is also tested, recovering the strong-shock limit M∝R^(−(s−1)/n(γ)) originally derived by Whitham for perfect gases, where γ is inherently related to the maximum compression ratio that the material can reach, (γ+1)/(γ−1). From these strong-shock limits, we also estimate analytically the density, radial velocity, pressure, and sound speed immediately behind the shock. While the hydrostatic part of the energy essentially commands the strong-shock behavior, the shear modulus and yield stress modify the compression ratio and velocity of the shock far from the axis or origin. A characterization of the elastic-plastic transition in converging shocks, which involves an elastic precursor and a plastic compression region, is finally exposed

Estampagem incremental : influência da rotação e do incremento vertical da ferramenta na estampabilidade do alumínio AA1200-H14

Este trabalho apresenta um estudo sobre o processo de Estampagem Incremental, com o objetivo de avaliar a influência dos parâmetros, rotação (S) e incremento vertical (Δz) da ferramenta na estampabilidade do alumínio AA1200-H14, com espessura de 0,8 mm. A metodologia deste estudo consiste na execução e análise de duas séries de experimentos de Estampagem Incremental do material analisado, o procedimento foi realizado por meio da fabricação de peças com formato de hiperboloide, estampadas a partir de geratrizes com formato hexagonal. O processo foi realizado em um centro de usinagem CNC variando a rotação (S) e o incremento vertical (Δz) da ferramenta, totalizando 18 experimentos. Este estudo apresenta a relação entre a profundidade em que ocorre a trinca (h) na direção Z, incremento vertical (Δz) e rotação da ferramenta (S) em cada ensaio, onde é possível observar que utilizando o conjunto de parâmetros Δz =0,5 mm e N= 800 rpm, se obteve as deformações mais acentuadas ϕ1= 0,814. Para a avaliação da influência dos parâmetros citados na estampabilidade do material foi utilizada a técnica de Projeto e Análise de Experimentos (DoE), aplicando a ferramenta estatistica ANOVA, com auxilio do software Minitab.This work presents a study on the Incremental Sheet Forming process, with the objective of evaluating the influence of the parameters, rotation speed (S) and Step Down (Δz) of the tool on the formability of AA1200-H14 aluminum with a thickness of 0.8 mm. The methodology of this study consists of the execution and analysis of two series of experiments of Incremental Forming of the analyzed material, the procedure is carried out through the manufacture of pieces with the shape of hyperboloids, formed from generatrixes with hexagonal shape. The process is carried out in a CNC machine varying the rotation (S) and the vertical increment (Δz) of the tool, totaling 18 experiments. The study presents the relationship between the depth at which the crack occurs (h) in the Z direction, vertical increment (Δz) and tool rotation (S) in each test, where it is possible to observe that using the set of parameters Δz =0, 5 mm and N= 800 rpm, the most pronounced deformations were obtained ϕ1= 0.814. For the evaluation of the influence of the mentioned parameters on the formability of the material, the Design and Analysis of Experiments (DOE) technique was used, applying the ANOVA statistical tool with the help of the Minitab software