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Articles de revues sur le sujet « Graded manifold »

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Auteur : Grafiati
Publié le 10 mars 2023
Mis à jour le 11 mars 2023

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1

Sardanashvily, G., et W. Wachowski. « Differential Calculus onN-Graded Manifolds ». Journal of Mathematics 2017 (2017) : 1–19. http://dx.doi.org/10.1155/2017/8271562.

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The differential calculus, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus, overN-graded commutative rings and onN-graded manifolds is developed. This is a straightforward generalization of the conventional differential calculus over commutative rings and also is the case of the differential calculus over Grassmann algebras and onZ2-graded manifolds. We follow the notion of anN-graded manifold as a local-ringed space whose body is a smooth manifoldZ. A key point is that the graded derivation module of the structure ring of graded functions on anN-graded manifold is the structure ring of global sections of a certain smooth vector bundle over its bodyZ. Accordingly, the Chevalley–Eilenberg differential calculus on anN-graded manifold provides it with the de Rham complex of graded differential forms. This fact enables us to extend the differential calculus onN-graded manifolds to formalism of nonlinear differential operators, by analogy with that on smooth manifolds, in terms of graded jet manifolds ofN-graded bundles.
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2

Ševera, Pavol, et Michal Širaň. « Integration of Differential Graded Manifolds ». International Mathematics Research Notices 2020, no 20 (15 février 2019) : 6769–814. http://dx.doi.org/10.1093/imrn/rnz004.

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Abstract We consider the problem of integration of $L_\infty $-algebroids (differential non-negatively graded manifolds) to $L_\infty $-groupoids. We first construct a “big” Kan simplicial manifold (Fréchet or Banach) whose points are solutions of a (generalized) Maurer–Cartan equation. The main analytic trick in our work is an integral transformation sending the solutions of the Maurer–Cartan equation to closed differential forms. Following the ideas of Ezra Getzler, we then impose a gauge condition that cuts out a finite-dimensional simplicial submanifold. This “smaller” simplicial manifold is (the nerve of) a local Lie $\ell $-groupoid. The gauge condition can be imposed only locally in the base of the $L_\infty $-algebroid; the resulting local $\ell $-groupoids glue up to a coherent homotopy, that is, we get a homotopy coherent diagram from the nerve of a good cover of the base to the (simplicial) category of local $\ell $-groupoids. Finally, we show that a $k$-symplectic differential non-negatively graded manifold integrates to a local $k$-symplectic Lie $\ell$-groupoid; globally, these assemble to form an $A_\infty$-functor. As a particular case for $k=2$, we obtain integration of Courant algebroids.
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Bruce, Andrew James, et Janusz Grabowski. « Riemannian Structures on Z 2 n -Manifolds ». Mathematics 8, no 9 (1 septembre 2020) : 1469. http://dx.doi.org/10.3390/math8091469.

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Very loosely, Z2n-manifolds are ‘manifolds’ with Z2n-graded coordinates and their sign rule is determined by the scalar product of their Z2n-degrees. A little more carefully, such objects can be understood within a sheaf-theoretical framework, just as supermanifolds can, but with subtle differences. In this paper, we examine the notion of a Riemannian Z2n-manifold, i.e., a Z2n-manifold equipped with a Riemannian metric that may carry non-zero Z2n-degree. We show that the basic notions and tenets of Riemannian geometry directly generalize to the setting of Z2n-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry.
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De Nicola, Antonio, et Ivan Yudin. « Generalized Goldberg Formula ». Canadian Mathematical Bulletin 59, no 3 (1 septembre 2016) : 508–20. http://dx.doi.org/10.4153/cmb-2016-007-4.

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AbstractIn this paper we prove a useful formula for the graded commutator of the Hodge codifferential with the left wedge multiplication by a fixed p-form acting on the de Rham algebra of a Riemannian manifold. Our formula generalizes a formula stated by Samuel I. Goldberg for the case of 1-forms. As first examples of application we obtain new identities on locally conformally Kähler manifolds and quasi-Sasakian manifolds. Moreover, we prove that under suitable conditions a certain subalgebra of differential forms in a compact manifold is quasi-isomorphic as a CDGA to the full de Rham algebra.
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5

Kotov, Alexei, et Thomas Strobl. « Characteristic classes associated to Q-bundles ». International Journal of Geometric Methods in Modern Physics 12, no 01 (28 décembre 2014) : 1550006. http://dx.doi.org/10.1142/s0219887815500061.

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A Q-manifold is a graded manifold endowed with a vector field of degree 1 squaring to zero. We consider the notion of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds. To each homotopy class of "gauge fields" (sections in the category of graded manifolds) and each cohomology class of a certain subcomplex of forms on the fiber we associate a cohomology class on the base. As any principal bundle yields canonically a Q-bundle, this construction generalizes Chern–Weil classes. Novel examples include cohomology classes that are locally de Rham differential of the integrands of topological sigma models obtained by the AKSZ-formalism in arbitrary dimensions. For Hamiltonian Poisson fibrations one obtains a characteristic 3-class in this manner. We also relate the framework to equivariant cohomology and Lecomte's characteristic classes of exact sequences of Lie algebras.
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6

Tahora, Saraban, et Khondokar M. Ahmed. « Study on De Rham Cohomology Algebra of Manifolds ». Dhaka University Journal of Science 64, no 2 (31 juillet 2016) : 109–13. http://dx.doi.org/10.3329/dujs.v64i2.54484.

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In the present paper some aspects of exterior derivative, graded algebra, cohomology algebra, de Rham cohomology algebra, singular homology, cohomology class are studied. Graded subspace, smooth map, a singular P- - simplex in a manifold M, oriented n- manifold M, the space of P- cycles and P- boundaries, Pth singular homology and homology class are treated in our paper. A theorem 3.03 is established which is related to orientable manifold. Dhaka Univ. J. Sci. 64(2): 109-113, 2016 (July)
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7

GILLE, CATHERINE. « ON THE LE-MURAKAMI-OHTSUKI INVARIANT IN DEGREE 2 FOR SEVERAL CLASSES OF 3-MANIFOLDS ». Journal of Knot Theory and Its Ramifications 12, no 01 (février 2003) : 17–45. http://dx.doi.org/10.1142/s0218216503002287.

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The 3-manifolds invariant of Le, Murakami and Ohtsuki is the universal finite type invariant for integral homology spheres. It takes values in the graded algebra of trivalent graphs and it is known that its degree one part is essentially the Casson-Walker-Lescop invariant. Here we compute the degree two term for several classes of 3-manifolds. In particular, we give an expression of ω (ML) up to order 2 when MLis the 3-manifold obtained by Dehn surgery along a framed link L with one or two components.
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8

IKEDA, NORIAKI, et KOZO KOIZUMI. « CURRENT ALGEBRAS AND QP-MANIFOLDS ». International Journal of Geometric Methods in Modern Physics 10, no 06 (30 avril 2013) : 1350024. http://dx.doi.org/10.1142/s0219887813500242.

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Generalized current algebras introduced by Alekseev and Strobl in two dimensions are reconstructed by a graded manifold and a graded Poisson brackets. We generalize their current algebras to higher dimensions. QP-manifolds provide the unified structures of current algebras in any dimension. Current algebras give rise to structures of Leibniz/Loday algebroids, which are characterized by QP-structures. Especially, in three dimensions, a current algebra has a structure of a Lie algebroid up to homotopy introduced by Uchino and one of the authors, which has a bracket of a generalization of the Courant–Dorfman bracket. Anomaly cancellation conditions are reinterpreted as generalizations of the Dirac structure.
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9

Paepe, Karl De. « Graded subalgebras of the Lie algebra of a smooth manifold ». International Journal of Mathematics and Mathematical Sciences 27, no 3 (2001) : 141–48. http://dx.doi.org/10.1155/s0161171201010511.

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10

Gualtieri, Marco, Mykola Matviichuk et Geoffrey Scott. « Deformation of Dirac Structures via L∞ Algebras ». International Mathematics Research Notices 2020, no 14 (22 juin 2018) : 4295–323. http://dx.doi.org/10.1093/imrn/rny134.

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Abstract The deformation theory of a Dirac structure is controlled by a differential graded Lie algebra that depends on the choice of an auxiliary transversal Dirac structure; if the transversal is not involutive, one obtains an $L_\infty $ algebra instead. We develop a simplified method for describing this $L_\infty $ algebra and use it to prove that the $L_\infty $ algebras corresponding to different transversals are canonically $L_\infty $–isomorphic. In some cases, this isomorphism provides a formality map, as we show in several examples including (quasi)-Poisson geometry, Dirac structures on Lie groups, and Lie bialgebras. Finally, we apply our result to a classical problem in the deformation theory of complex manifolds; we provide explicit formulas for the Kodaira–Spencer deformation complex of a fixed small deformation of a complex manifold, in terms of the deformation complex of the original manifold.
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11

Sharygin, G., et D. Talalaev. « On the Lie-formality of Poisson manifolds ». Journal of K-Theory 2, no 2 (4 mars 2008) : 361–84. http://dx.doi.org/10.1017/is008001011jkt030.

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12

Kuribayashi, Katsuhiko, et Toshihiro Yamaguchi. « The Vanishing problem of the string class with degree 3 ». Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 65, no 1 (août 1998) : 129–42. http://dx.doi.org/10.1017/s1446788700039446.

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AbstractLet ξbe anSO(n)-bundle over a simple connected manifoldMwith a spin structureQ → M. The string class is an obstruction to h1 the structure groupLSpin(n)of the loop group bundleLQ → LMto the universal central extension ofLSpain(n)by the circle. We prove that the string class vanishes if and only if 1/2 the first Pontrjagin clsss of values whenMis a compact simply connected homogeneous space of rank one, a simpiy connected 4 dimensional manifold or a finite product space of those manifolds. This result is deduced by using the Eclesberg spectral sequence converging to the modpcohomology ofLMwhoseE2-term to the Hochschild homology of the modpcohomology algebra ofM. The key to the consideration is existence of a morphism of algebras, which is injective below degree 3, from an important graded commutator algebra into the Hochschild homology of a certain graded commutative algebra.
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13

Wang, Qiu Shi. « Hochschild Cohomology of the Cohomology Algebra of Closed Orientable Three- Manifolds ». McGill Science Undergraduate Research Journal 16, no 1 (15 avril 2021) : 57–60. http://dx.doi.org/10.26443/msurj.v16i1.61.

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Let F be a field of characteristic other than 2. We show that the zeroth Hochschild cohomology vector space HH0(A) of a degree 3 graded commutative Frobenius F-algebra A = iAi, where we will always assume A0 = F, is isomorphic to the direct sum of the degree 0, 2 and 3 graded components and the kernel of a certain natural evaluation map ιμ : A1 Λ2(A1). In particular, this holds forA = H∗(M; F) the cohomology algebra of a closed orientable 3-manifoldM. In Theorem A of [1], Charette proves the vanishing of a certain discriminantΔassociated to a closed orientable 3-manifold L with vanishing cup product 3-form. It turns out that if we could show that HH2,−2(A) = 0for A = H∗(L;C), we would have found a more elementary proof of this part of Charette’s theorem. We show that for any β 3, the degree 3 graded commutative Frobenius algebra A with μA = 0and dim(A1) = β satisfiesHH2,−2(A) = 0. Thus Charette’s theorem is not simplified.
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14

Iacono, Donatella, et Marco Manetti. « On Deformations of Pairs (Manifold, Coherent Sheaf) ». Canadian Journal of Mathematics 71, no 5 (9 janvier 2019) : 1209–41. http://dx.doi.org/10.4153/cjm-2018-027-8.

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AbstractWe analyse infinitesimal deformations of pairs $(X,{\mathcal{F}})$ with ${\mathcal{F}}$ a coherent sheaf on a smooth projective variety $X$ over an algebraically closed field of characteristic 0. We describe a differential graded Lie algebra controlling the deformation problem, and we prove an analog of a Mukai–Artamkin theorem about the trace map.
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15

Stavracou, T. « Theory of Connections on Graded Principal Bundles ». Reviews in Mathematical Physics 10, no 01 (janvier 1998) : 47–79. http://dx.doi.org/10.1142/s0129055x98000033.

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The geometry of graded principal bundles is discussed in the framework of graded manifold theory of Kostant–Berezin–Leites. We first review the basic elements of this theory establishing at the same time supplementary properties of graded Lie groups and their actions. Particular emphasis is given in introducing and studying free actions in the graded context. Next, we investigate the geometry of graded principal bundles; we prove that they have several properties analogous to those of ordinary principal bundles. In particular, we show that the sheaf of vertical derivations on a graded principal bundle coincides with the graded distribution induced by the action of the structure graded Lie group. This result leads to a natural definition of the graded connection in terms of graded distributions; its relation with Lie superalgebra-valued graded differential forms is also exhibited. Finally, we define the curvature for the graded connection and we prove that the curvature controls the involutivity of the horizontal graded distribution corresponding to the graded connection.
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16

Ben-Bassat, Oren, et Jonathan Block. « Milnor descent for cohesive dg-categories ». Journal of K-theory 12, no 3 (20 août 2013) : 433–59. http://dx.doi.org/10.1017/is013007003jkt236.

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AbstractWe show that the functor from curved differential graded algebras to differential graded categories, defined by the second author, sends Cartesian diagrams to homotopy Cartesian diagrams, under certain reasonable hypotheses. This is an extension to the arena of dg-categories of a construction of projective modules due to Milnor. As an example, we show that the functor satisfies descent for certain partitions of a complex manifold.
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17

Grabowski, Janusz. « Z-Graded Extensions of Poisson Brackets ». Reviews in Mathematical Physics 09, no 01 (janvier 1997) : 1–27. http://dx.doi.org/10.1142/s0129055x97000026.

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A Z-graded Lie bracket { , }P on the exterior algebra Ω(M) of differential forms, which is an extension of the Poisson bracket of functions on a Poisson manifold (M,P), is found. This bracket is simultaneously graded skew-symmetric and satisfies the graded Jacobi identity. It is a kind of an 'integral' of the Koszul–Schouten bracket [ , ]P of differential forms in the sense that the exterior derivative is a bracket homomorphism: [dμ, dν]P=d{μ, ν}P. A naturally defined generalized Hamiltonian map is proved to be a homomorphism between { , }P and the Frölicher–Nijenhuis bracket of vector valued forms. Also relations of this graded Poisson bracket to the Schouten–Nijenhuis bracket and an extension of { , }P to a graded bracket on certain multivector fields, being an 'integral' of the Schouten–Nijenhuis bracket, are studied. All these constructions are generalized to tensor fields associated with an arbitrary Lie algebroid.
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18

Lee, Jae Min, et Byungdo Park. « A Superbundle Description of Differential K-Theory ». Axioms 12, no 1 (12 janvier 2023) : 82. http://dx.doi.org/10.3390/axioms12010082.

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We construct a model of differential K-theory using superbundles with a Z/2Z-graded connection and a differential form on the base manifold and prove that our model is isomorphic to the Freed–Lott–Klonoff model of differential K-theory.
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19

Giovannardi, Gianmarco. « Higher Dimensional Holonomy Map for Rules Submanifolds in Graded Manifolds ». Analysis and Geometry in Metric Spaces 8, no 1 (1 juillet 2020) : 68–91. http://dx.doi.org/10.1515/agms-2020-0105.

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AbstractThe deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion.
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Zhang, H. H., et G. W. Ma. « Fracture modeling of isotropic functionally graded materials by the numerical manifold method ». Engineering Analysis with Boundary Elements 38 (janvier 2014) : 61–71. http://dx.doi.org/10.1016/j.enganabound.2013.10.006.

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Zhang, Huihua, Simin Liu et Shangyu Han. « The numerical manifold method for transient moisture diffusion in 2D functionally graded materials ». IOP Conference Series : Earth and Environmental Science 189 (6 novembre 2018) : 032017. http://dx.doi.org/10.1088/1755-1315/189/3/032017.

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Zhang, Huihua, Simin Liu et Shangyu Han. « Modelling steady moisture diffusion in functionally graded materials with the numerical manifold method ». IOP Conference Series : Earth and Environmental Science 189, no 4 (6 novembre 2018) : 042018. http://dx.doi.org/10.1088/1755-1315/189/4/042018.

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23

Zhang, Zizhen, et Limin Song. « Dynamics of a Computer Virus Propagation Model with Delays and Graded Infection Rate ». Advances in Mathematical Physics 2017 (2017) : 1–13. http://dx.doi.org/10.1155/2017/4514935.

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A four-compartment computer virus propagation model with two delays and graded infection rate is investigated in this paper. The critical values where a Hopf bifurcation occurs are obtained by analyzing the distribution of eigenvalues of the corresponding characteristic equation. In succession, direction and stability of the Hopf bifurcation when the two delays are not equal are determined by using normal form theory and center manifold theorem. Finally, some numerical simulations are also carried out to justify the obtained theoretical results.
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Zhao, Yan, Guo Xin Zhang et Hai Feng Li. « Simulation of Numerical Test of Concrete Microscopic Structure by Second-Order Manifold Method ». Applied Mechanics and Materials 477-478 (décembre 2013) : 968–71. http://dx.doi.org/10.4028/www.scientific.net/amm.477-478.968.

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To simulate the numerical test of concrete, the random aggregate model according to the Monte Carlo method and Fuller Graded Formula is carried out based on the assumption that the concrete is a multi-phases composite material composed of matrix. By adding the function of tracing the propagation of cracks,the Numerical Manifold Method proposed by Shi Genhua is developed which can simulate both the discontinuity of block system and the tensile or shear failure of intact block. The random aggregate model according to the Monte Carlo method and Fuller Graded Formula is carried out, and the concrete fracture process is simulated by the NMM. The strength and failure pattern are in good agreement with the experimental data, which shows that the method put forward and the program developed in this paper can effectively simulate the fracture process of concrete composed of multi-cracks.
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Zhang, H. H., S. Y. Han, L. F. Fan et D. Huang. « The numerical manifold method for 2D transient heat conduction problems in functionally graded materials ». Engineering Analysis with Boundary Elements 88 (mars 2018) : 145–55. http://dx.doi.org/10.1016/j.enganabound.2018.01.003.

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26

DE BARTOLOMEIS, PAOLO, et ANDREI IORDAN. « DEFORMATIONS OF LEVI FLAT STRUCTURES IN SMOOTH MANIFOLDS ». Communications in Contemporary Mathematics 16, no 02 (avril 2014) : 1350015. http://dx.doi.org/10.1142/s0219199713500156.

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We study intrinsic deformations of Levi flat structures on a smooth manifold. A Levi flat structure on a smooth manifold L is a couple (ξ, J) where ξ ⊂ T(L) is an integrable distribution of codimension 1 and J : ξ → ξ is a bundle automorphism which defines a complex integrable structure on each leaf. A deformation of a Levi flat structure (ξ, J) is a smooth family {(ξt, Jt)}t∈]-ε,ε[ of Levi flat structures on L such that (ξ0, J0) = (ξ, J). We define a complex whose cohomology group of order 1 contains the infinitesimal deformations of a Levi flat structure. In the case of real analytic Levi flat structures, this cohomology group is [Formula: see text] where (𝒵*(L), δ, {⋅,⋅}) is the differential graded Lie algebra associated to ξ.
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Ida, Cristian, et Sabinşan Mercheşan. « A Note on Coeffective 1–Differentiable Cohomology ». Analele Universitatii "Ovidius" Constanta - Seria Matematica 22, no 1 (10 décembre 2014) : 127–39. http://dx.doi.org/10.2478/auom-2014-0011.

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AbstractAfter a brief review of some basic notions concerning 1-differentiable cohomology, named here ď-cohomology, we introduce a Lichnerowicz ď– cohomology in a classical way. Next, following the classical study of coeffective cohomology, a special attention is paid to the study of some problems concerning coeffective cohomology in the graded algebra of 1– differentiable forms. Also, the case of an almost contact metric (2n+1)– dimensional manifold is considered and studied in our context.
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28

Zhao, Tao, et Dianjie Bi. « Delay Induced Hopf Bifurcation of an Epidemic Model with Graded Infection Rates for Internet Worms ». Mathematical Problems in Engineering 2017 (2017) : 1–10. http://dx.doi.org/10.1155/2017/9563862.

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A delayed SEIQRS worm propagation model with different infection rates for the exposed computers and the infectious computers is investigated in this paper. The results are given in terms of the local stability and Hopf bifurcation. Sufficient conditions for the local stability and the existence of Hopf bifurcation are obtained by using eigenvalue method and choosing the delay as the bifurcation parameter. In particular, the direction and the stability of the Hopf bifurcation are investigated by means of the normal form theory and center manifold theorem. Finally, a numerical example is also presented to support the obtained theoretical results.
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29

Abramov, Viktor. « Matrix 3-Lie superalgebras and BRST supersymmetry ». International Journal of Geometric Methods in Modern Physics 14, no 11 (23 octobre 2017) : 1750160. http://dx.doi.org/10.1142/s0219887817501602.

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Given a matrix Lie algebra one can construct the 3-Lie algebra by means of the trace of a matrix. In the present paper, we show that this approach can be extended to the infinite-dimensional Lie algebra of vector fields on a manifold if instead of the trace of a matrix we consider a differential 1-form which satisfies certain conditions. Then we show that the same approach can be extended to matrix Lie superalgebras [Formula: see text] if instead of the trace of a matrix we make use of the supertrace of a matrix. It is proved that a graded triple commutator of matrices constructed with the help of the graded commutator and the supertrace satisfies a graded ternary Filippov–Jacobi identity. In two particular cases of [Formula: see text] and [Formula: see text], we show that the Pauli and Dirac matrices generate the matrix 3-Lie superalgebras, and we find the non-trivial graded triple commutators of these algebras. We propose a Clifford algebra approach to 3-Lie superalgebras induced by Lie superalgebras. We also discuss an application of matrix 3-Lie superalgebras in BRST-formalism.
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COQUEREAUX, R., R. HÄUβLING et F. SCHECK. « ALGEBRAIC CONNECTIONS ON PARALLEL UNIVERSES ». International Journal of Modern Physics A 10, no 01 (10 janvier 1995) : 89–98. http://dx.doi.org/10.1142/s0217751x95000048.

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For any manifold M we introduce a ℤ-graded differential algebra Ξ, which, in particular, is a bimodule over the associative algebra C(M⋃M). We then introduce the corresponding covariant differentials and show how this construction can be interpreted in terms of Yang-Mills and Higgs fields. This is a particular example of noncommutative geometry. It differs from the prescription of Connes in the following way: the definition of Ξ does not rely on a given Dirac-Yukawa operator acting on a space of spinors.
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de Bartolomeis, Paolo, et Andrei Iordan. « Maurer-Cartan equation in the DGLA of graded derivations ». Complex Manifolds 8, no 1 (1 janvier 2021) : 183–95. http://dx.doi.org/10.1515/coma-2020-0113.

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Abstract Let M be a smooth manifold and D = ℒΨ+𝒥Ψ a solution of the Maurer-Cartan equation in the DGLA of graded derivations D* (M) of differential forms on M, where Ψ, Ψ are differential 1-form on M with values in the tangent bundle TM and ℒΨ, 𝒥Ψ are the d * and i * components of D. Under the hypothesis that IdT ( M ) + Ψ is invertible we prove that Ψ = b ( Ψ ) = - 1 2 _ ( I d T M + Ψ ) - 1 ∘ [ Ψ , Ψ ] ℱ 𝒩 {\rm{\Psi = }}b\left( {\rm{\Psi }} \right) = - {1 \over {}}{\left( {I{d_{TM}} + {\rm{\Psi }}} \right)^{ - 1}} \circ {\left[ {{\rm{\Psi }},{\rm{\Psi }}} \right]_{\mathcal{F}\mathcal{N}}} , where [·, ·]𝒡𝒩 is the Frölicher-Nijenhuis bracket. This yields to a classification of the canonical solutions e Ψ = ℒ Ψ +𝒥b ( Ψ ) of the Maurer-Cartan equation according to their type: e Ψ is of finite type r if there exists r∈ 𝒩 such that Ψr∘ [Ψ, Ψ]𝒡𝒩 = 0 and r is minimal with this property, where [·, ·]𝒡𝒩 is the Frölicher-Nijenhuis bracket. A distribution ξ ⊂TM of codimension k ⩾ 1 is integrable if and only if the canonical solution e Ψ associated to the endomorphism Ψ of TM which is trivial on ξ and equal to the identity on a complement of ξ in TM is of finite type ⩽ 1, respectively of finite type 0 if k = 1.
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32

Zhang, Limei, Fei Guo et Hong Zheng. « The MLS-based numerical manifold method for nonlinear transient heat conduction problems in functionally graded materials ». International Communications in Heat and Mass Transfer 139 (décembre 2022) : 106428. http://dx.doi.org/10.1016/j.icheatmasstransfer.2022.106428.

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Zhang, H. H., S. M. Liu, S. Y. Han et L. F. Fan. « The numerical manifold method for crack modeling of two-dimensional functionally graded materials under thermal shocks ». Engineering Fracture Mechanics 208 (mars 2019) : 90–106. http://dx.doi.org/10.1016/j.engfracmech.2019.01.002.

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Le Donne, Enrico, et Séverine Rigot. « Besicovitch Covering Property on graded groups and applications to measure differentiation ». Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no 750 (1 mai 2019) : 241–97. http://dx.doi.org/10.1515/crelle-2016-0051.

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Abstract We give a complete answer to which homogeneous groups admit homogeneous distances for which the Besicovitch Covering Property (BCP) holds. In particular, we prove that a stratified group admits homogeneous distances for which BCP holds if and only if the group has step 1 or 2. These results are obtained as consequences of a more general study of homogeneous quasi-distances on graded groups. Namely, we prove that a positively graded group admits continuous homogeneous quasi-distances satisfying BCP if and only if any two different layers of the associated positive grading of its Lie algebra commute. The validity of BCP has several consequences. Its connections with the theory of differentiation of measures is one of the main motivations of the present paper. As a consequence of our results, we get for instance that a stratified group can be equipped with some homogeneous distance so that the differentiation theorem holds for each locally finite Borel measure if and only if the group has step 1 or 2. The techniques developed in this paper allow also us to prove that sub-Riemannian distances on stratified groups of step 2 or higher never satisfy BCP. Using blow-up techniques this is shown to imply that on a sub-Riemannian manifold the differentiation theorem does not hold for some locally finite Borel measure.
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35

Zhao, Yan, Guo Xin Zhang, Xiao Chuan Wu et Fu Xin Chai. « Simulation Study of Fracture Process of Concrete Mesoscopic Structure by Manifold Method ». Advanced Materials Research 243-249 (mai 2011) : 875–78. http://dx.doi.org/10.4028/www.scientific.net/amr.243-249.875.

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On the assumption that the concrete is a multi-phases composite material composed of matrix, aggregate and the bonding interface, the failure progress of concrete is simulated. It requires that numerical method not only can simulate forces and stresses, the failure growth and contacts, but also can simulate the discontinuities such as joints and cracks，the large deformation after the failure and the propagation of multi-cracks. By adding the function of tracing the propagation of cracks，the Numerical Manifold Method proposed by Shi Genhua is developed which can simulate both the discontinuity of block system and the tensile or shear failure of intact block. The random aggregate model according to the Monte Carlo method and Fuller Graded Formula is carried out, and the concrete fracture process is simulated by the NMM. The strength and failure pattern are in good agreement with the experimental data, which shows that the method put forward and the program developed in this paper can effectively simulate the fracture process of concrete composed of multi-cracks.
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Oikonomou, V. K. « Low dimensional supersymmetries in SUSY Chern–Simons systems and geometrical implications ». International Journal of Geometric Methods in Modern Physics 13, no 06 (15 juin 2016) : 1650083. http://dx.doi.org/10.1142/s0219887816500833.

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We study in detail the underlying graded geometric structure of abelian [Formula: see text] supersymmetric Chern–Simons theory in (2 + 1)-dimensions. This structure is an attribute of the hidden unbroken one-dimensional [Formula: see text] supersymmetries that the system also possesses. We establish the result that the geometric structures corresponding to the bosonic and to the fermionic sectors are equivalent fiber bundles over the (2 + 1)-dimensional manifold. Moreover, we find a geometrical answer to the question why some and not all of the fermionic sections are related to a [Formula: see text] supersymmetric algebra. Our findings are useful for the quantum theory of Chern–Simons vortices.
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Zhao, Yan, Guo Xin Zhang, Song Nan Ru et Fu Xin Chai. « Simulation of Numerical Test of Concrete Mechanical Properties Base on Numerical Manifold Method ». Advanced Materials Research 217-218 (mars 2011) : 1739–42. http://dx.doi.org/10.4028/www.scientific.net/amr.217-218.1739.

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In order to reflect the heterogeneity of concrete at mesoscopic level and simulate the mechanics response from the mesoscopic to macroscopical in the course of load, the random aggregate model according to the Monte Carlo method and Fuller Graded Formula is carried out based on the assumption that the concrete is a multi-phases composite material composed of matrix, and the material parameters are defined based on the test. In this paper, the Numerical Manifold Method introduced not only can correctly simulate stress, deformation and failure of concrete, but can simulate propagation of multi-cracks in the concrete, and failure plane growth can be searched by stress results automatically, so it can simulate the kind high discontinuous problem very well. On this basis, NMM is adopted to simulate the tests on concrete mechanical performance. The creation, propagation and fracture process of cracks in compression of concrete are present. The computational results are in good agreement with the experimental data.
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38

Zhang, H. H., X. L. Ji, S. Y. Han et L. F. Fan. « Determination of T-stress for thermal cracks in homogeneous and functionally graded materials with the numerical manifold method ». Theoretical and Applied Fracture Mechanics 113 (juin 2021) : 102940. http://dx.doi.org/10.1016/j.tafmec.2021.102940.

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Poncin, Norbert. « On the Cohomology of the Nijenhuis–Richardson Graded Lie Algebra of the Space of Functions of a Manifold ». Journal of Algebra 243, no 1 (septembre 2001) : 16–40. http://dx.doi.org/10.1006/jabr.2001.8827.

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SARDANASHVILY, G. « SUPERMETRICS ON SUPERMANIFOLDS ». International Journal of Geometric Methods in Modern Physics 05, no 02 (mars 2008) : 271–86. http://dx.doi.org/10.1142/s021988780800276x.

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By virtue of the well-known theorem, a structure Lie group K of a principal bundle P → X is reducible to its closed subgroup H iff there exists a global section of the quotient bundle P/K → X. In gauge theory, such sections are treated as Higgs fields, exemplified by pseudo-Riemannian metrics on a base manifold X. Under some conditions, this theorem is extended to principal superbundles in the category of G-supermanifolds. Given a G-supermanifold M and a graded frame superbundle over M with a structure general linear supergroup, a reduction of this structure supergroup to an orthogonal-symplectic supersubgroup is associated to a supermetric on a G-supermanifold M.
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41

Davenport, P. W., D. J. Dalziel, B. Webb, J. R. Bellah et C. J. Vierck. « Inspiratory resistive load detection in conscious dogs ». Journal of Applied Physiology 70, no 3 (1 mars 1991) : 1284–89. http://dx.doi.org/10.1152/jappl.1991.70.3.1284.

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The physiological mechanisms mediating the detection of mechanical loads are unknown. This is, in part, due to the lack of an animal model of load detection that could be used to investigate specific sensory systems. We used American Foxhounds with tracheal stomata to behaviorally condition the detection of inspiratory occlusion and graded resistive loads. The resistive loads were presented with a loading manifold connected to the inspiratory port of a non-rebreathing valve. The dogs signaled detection of the load by lifting their front paw off a lever. Inspiratory occlusion was used as the initial training stimulus, and the dogs could reliably respond within the first or second inspiratory effort to 100% of the occlusion presentations after 13 trials. Graded resistances that spanned the 50% detection threshold were then presented. The detection threshold resistances (delta R50) were 0.96 and 1.70 cmH2O.l-1.s. Ratios of delta R50 to background resistance were 0.15 and 0.30. The near-threshold resistive loads did not significantly change expired PCO2 or breathing patterns. These results demonstrate that dogs can be conditioned to reliably and specifically signal the detection of graded inspiratory mechanical loads. Inspiration through the tracheal stoma excludes afferents in the upper extrathoracic trachea, larynx, pharynx, nasal passages, and mouth from mediating load detection in these dogs. It is unknown which remaining afferents (vagal or respiratory muscle) are responsible for load detection.
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CRANE, LOUIS. « RELATIONAL SPACETIME, MODEL CATEGORIES AND QUANTUM GRAVITY ». International Journal of Modern Physics A 24, no 15 (20 juin 2009) : 2753–75. http://dx.doi.org/10.1142/s0217751x0904614x.

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We propose a mathematically concrete way of modelling the suggestion that in quantum gravity the spacetime manifold disappears. We replace the underlying point set topological space with several apparently different models, which are actually related by pairs of adjoint functors from rational homotopy theory. One is a discrete approximation to the causal null path space derived from the multiple images in the spacetime theory of gravitational lensing, described as an object in the model category of differential graded Lie algebras. Another of our models appears as a thickening of spacetime, which we interpret as a formulation of relational geometry. This model is produced from the finite dimensional differential graded algebra of differential forms which can be transmitted out of a finite region consistent with the Bekenstein bound by another functor, called geometric realisation. The thickening of spacetime, which we propose as a version of relational spacetime, has a surprizingly rich structure. Information which would make up a spin bundle over spacetime is contained in it, making it possible to include fermionic fields in a geometric state sum over it. Avenues toward constructing an actual quantum theory of gravity on our models are given a preliminary exploration.
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Deeley, Robin J., Magnus Goffeng et Bram Mesland. « The bordism group of unbounded KK-cycles ». Journal of Topology and Analysis 10, no 02 (juin 2018) : 355–400. http://dx.doi.org/10.1142/s1793525318500012.

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We consider Hilsum’s notion of bordism as an equivalence relation on unbounded [Formula: see text]-cycles and study the equivalence classes. Upon fixing two [Formula: see text]-algebras, and a ∗-subalgebra dense in the first [Formula: see text]-algebra, a [Formula: see text]-graded abelian group is obtained; it maps to the Kasparov [Formula: see text]-group of the two [Formula: see text]-algebras via the bounded transform. We study properties of this map both in general and in specific examples. In particular, it is an isomorphism if the first [Formula: see text]-algebra is the complex numbers (i.e. for [Formula: see text]-theory) and is a split surjection if the first [Formula: see text]-algebra is the continuous functions on a compact manifold with boundary when one uses the Lipschitz functions as the dense ∗-subalgebra.
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44

Abouzaid, Mohammed. « Homological mirror symmetry without correction ». Journal of the American Mathematical Society 34, no 4 (24 mai 2021) : 1059–173. http://dx.doi.org/10.1090/jams/973.

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Let X X be a closed symplectic manifold equipped with a Lagrangian torus fibration over a base Q Q . A construction first considered by Kontsevich and Soibelman produces from this data a rigid analytic space Y Y , which can be considered as a variant of the T T -dual introduced by Strominger, Yau, and Zaslow. We prove that the Fukaya category of tautologically unobstructed graded Lagrangians in X X embeds fully faithfully in the derived category of (twisted) coherent sheaves on Y Y , under the technical assumption that π 2 ( Q ) \pi _2(Q) vanishes (all known examples satisfy this assumption). The main new tool is the construction and computation of Floer cohomology groups of Lagrangian fibres equipped with topological infinite rank local systems that correspond, under mirror symmetry, to the affinoid rings introduced by Tate, equipped with their natural topologies as Banach algebras.
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Marrucci, Monica, Gerold Zeilinger, Adriano Ribolini et Wolfgang Schwanghart. « Origin of Knickpoints in an Alpine Context Subject to Different Perturbing Factors, Stura Valley, Maritime Alps (North-Western Italy) ». Geosciences 8, no 12 (28 novembre 2018) : 443. http://dx.doi.org/10.3390/geosciences8120443.

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Natural catchments are likely to show the existence of knickpoints in their river networks. The origin and genesis of the knickpoints can be manifold, considering that the present morphology is the result of the interactions of different factors such as tectonic movements, quaternary glaciations, river captures, variable lithology, and base-level changes. We analyzed the longitudinal profiles of the river channels in the Stura di Demonte Valley (Maritime Alps) to identify the knickpoints of such an alpine setting and to characterize their origins. The distribution and the geometry of stream profiles were used to identify the possible causes of the changes in stream gradients and to define zones with genetically linked knickpoints. Knickpoints are key geomorphological features for reconstructing the evolution of fluvial dissected basins, when the different perturbing factors affecting the ideally graded fluvial system have been detected. This study shows that even in a regionally small area, perturbations of river profiles are caused by multiple factors. Thus, attributing (automatically)-extracted knickpoints solely to one factor, can potentially lead to incomplete interpretations of catchment evolution.
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Kupriyanov, Vladislav G., et Richard J. Szabo. « Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry ». Journal of Physics A : Mathematical and Theoretical 55, no 3 (28 décembre 2021) : 035201. http://dx.doi.org/10.1088/1751-8121/ac411c.

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Abstract We formulate general definitions of semi-classical gauge transformations for noncommutative gauge theories in general backgrounds of string theory, and give novel explicit constructions using techniques based on symplectic embeddings of almost Poisson structures. In the absence of fluxes the gauge symmetries close a Poisson gauge algebra and their action is governed by a P ∞-algebra which we construct explicitly from the symplectic embedding. In curved backgrounds they close a field dependent gauge algebra governed by an L ∞-algebra which is not a P ∞-algebra. Our technique produces new all orders constructions which are significantly simpler compared to previous approaches, and we illustrate its applicability in several examples of interest in noncommutative field theory and gravity. We further show that our symplectic embeddings naturally define a P ∞-structure on the exterior algebra of differential forms on a generic almost Poisson manifold, which generalizes earlier constructions of differential graded Poisson algebras, and suggests a new approach to defining noncommutative gauge theories beyond the gauge sector and the semi-classical limit based on A ∞-algebras.
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Bolognini, Davide, et Ulderico Fugacci. « Betti splitting from a topological point of view ». Journal of Algebra and Its Applications 19, no 06 (27 juin 2019) : 2050116. http://dx.doi.org/10.1142/s0219498820501169.

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A Betti splitting [Formula: see text] of a monomial ideal [Formula: see text] ensures the recovery of the graded Betti numbers of [Formula: see text] starting from those of [Formula: see text] and [Formula: see text]. In this paper, we introduce an analogous notion for simplicial complexes, using Alexander duality, proving that it is equivalent to a recursive splitting condition on links of some vertices. We provide results ensuring the existence of a Betti splitting for a simplicial complex [Formula: see text], relating it to topological properties of [Formula: see text]. Among other things, we prove that orientability for a manifold without boundary is equivalent to the admission of a Betti splitting induced by the removal of a single facet. Taking advantage of our topological approach, we provide the first example of a monomial ideal which admits Betti splittings in all characteristics but with characteristic-dependent resolution. Moreover, we introduce new numerical descriptors for simplicial complexes and topological spaces, useful to deal with questions concerning the existence of Betti splitting.
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48

Mare, Augustin-Liviu. « A Characterization of the Quantum Cohomology Ring of G/B and Applications ». Canadian Journal of Mathematics 60, no 4 (1 août 2008) : 875–91. http://dx.doi.org/10.4153/cjm-2008-037-8.

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AbstractWe observe that the small quantum product of the generalized flag manifold G/B is a product operation ★ on H*(G/B) ⊗ ℝ[q1, . . . , ql] uniquely determined by the facts that it is a deformation of the cup product on H*(G/B); it is commutative, associative, and graded with respect to deg(qi ) = 4; it satisfies a certain relation (of degree two); and the corresponding Dubrovin connection is flat. Previously, we proved that these properties alone imply the presentation of the ring (H*(G/B)⊗ℝ[q1, . . . , ql], ★) in terms of generators and relations. In this paper we use the above observations to give conceptually new proofs of other fundamental results of the quantum Schubert calculus for G/B: the quantumChevalley formula of D. Peterson (see also Fulton andWoodward) and the “quantization by standard monomials” formula of Fomin, Gelfand, and Postnikov for G = SL(n, ℂ). The main idea of the proofs is the same as in Amarzaya–Guest: from the quantum -module of G/B one can decode all information about the quantum cohomology of this space.
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Zhang, H. H., S. M. Liu, S. Y. Han et L. F. Fan. « Corrigendum to “The numerical manifold method for crack modeling of two-dimensional functionally graded materials under thermal shocks” [Eng. Fract. Mech. 208 (2019) 90–106] ». Engineering Fracture Mechanics 230 (mai 2020) : 106975. http://dx.doi.org/10.1016/j.engfracmech.2020.106975.

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50

YAMAMOTO, MINORU. « FIRST ORDER SEMI-LOCAL INVARIANTS OF STABLE MAPS OF 3-MANIFOLDS INTO THE PLANE ». Proceedings of the London Mathematical Society 92, no 2 (20 février 2006) : 471–504. http://dx.doi.org/10.1112/s0024611505015534.

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In the late 1980s, Vassiliev introduced new graded numerical invariants of knots, which are now called Vassiliev invariants or finite-type invariants. Since he made this definition, many people have been trying to construct Vassiliev type invariants for various mapping spaces. In the early 1990s, Arnold and Goryunov introduced the notion of first order (local) invariants of stable maps. In this paper, we define and study {\it first order semi-local invariants} of stable maps and those of stable fold maps of a closed orientable 3-dimensional manifold into the plane. We show that there are essentially eight first order semi-local invariants. For a stable map, one of them is a constant invariant, six of them count the number of singular fibers of a given type which appear discretely (there are exactly six types of such singular fibers), and the last one is the Euler characteristic of the Stein factorization of this stable map. Besides these invariants, for stable fold maps, the Bennequin invariant of the singular value set corresponding to definite fold points is also a first order semi-local invariant. Our study of unstable fold maps with codimension 1 provides invariants for the connected components of the set of all fold maps.
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