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Journal articles on the topic 'Transverse invariants'

Author: Grafiati

Published: 10 March 2023

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1

Collari, Carlo. "Transverse invariants from Khovanov-type homologies." Journal of Knot Theory and Its Ramifications 28, no. 01 (January 2019): 1950012. http://dx.doi.org/10.1142/s0218216519500123.

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In this paper, we introduce a family of transverse invariants arising from the deformations of Khovanov homology. This family includes the invariants introduced by Plamenevskaya and by Lipshitz, Ng, and Sarkar. Then, we investigate the invariants arising from Bar-Natan’s deformation. These invariants, called [Formula: see text]-invariants, are essentially equivalent to Lipshitz, Ng, and Sarkar’s invariants [Formula: see text]. From the [Formula: see text]-invariants, we extract two non-negative integers which are transverse invariants (the [Formula: see text]-invariants). Finally, we give several conditions which imply the non-effectiveness of the [Formula: see text]-invariants, and use them to prove several vanishing criteria for the Plamenevskaya invariant [Formula: see text], and the non-effectiveness of the vanishing of [Formula: see text], for all prime knots with less than 12 crossings.

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2

López Machí, Rafael, and José Martínez Alfaro. "Invariants of transverse foliations." Topology and its Applications 159, no. 2 (February 2012): 519–25. http://dx.doi.org/10.1016/j.topol.2011.09.027.

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3

Lisca, Paolo, and András I. Stipsicz. "Contact surgery and transverse invariants." Journal of Topology 4, no. 4 (October 25, 2011): 817–34. http://dx.doi.org/10.1112/jtopol/jtr022.

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4

GRANT, MARK. "ON SELF-INTERSECTION INVARIANTS." Glasgow Mathematical Journal 55, no. 2 (August 2, 2012): 259–73. http://dx.doi.org/10.1017/s0017089512000481.

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AbstractWe prove that the Hatcher–Quinn and Wall invariants of a self-transverse immersion f: Nn ↬ M2n coincide. That is, we construct an isomorphism between their target groups, which carries one onto the other. We also employ methods of normal bordism theory to investigate the Hatcher–Quinn invariant of an immersion f: Nn ↬ M2n−1.

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5

DING, FAN, and HANSJÖRG GEIGES. "LEGENDRIAN KNOTS AND LINKS CLASSIFIED BY CLASSICAL INVARIANTS." Communications in Contemporary Mathematics 09, no. 02 (April 2007): 135–62. http://dx.doi.org/10.1142/s0219199707002381.

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It is shown that Legendrian (respectively transverse) cable links in S3 with its standard tight contact structure, i.e. links consisting of an unknot and a cable of that unknot, are classified by their oriented link type and the classical invariants (Thurston–Bennequin invariant and rotation number in the Legendrian case, self-linking number in the transverse case). The analogous result is proved for torus knots in the 1-jet space J1(S1) with its standard tight contact structure.

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6

Plamenevskaya, Olga. "Braid monodromy, orderings and transverse invariants." Algebraic & Geometric Topology 18, no. 6 (October 18, 2018): 3691–718. http://dx.doi.org/10.2140/agt.2018.18.3691.

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7

Lipshitz, Robert, Lenhard Ng, and Sucharit Sarkar. "On transverse invariants from Khovanov homology." Quantum Topology 6, no. 3 (2015): 475–513. http://dx.doi.org/10.4171/qt/69.

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8

Ito, Tetsuya. "Braids, chain of Yang–Baxter like operations, and (transverse) knot invariants." Journal of Knot Theory and Its Ramifications 27, no. 11 (October 2018): 1843009. http://dx.doi.org/10.1142/s0218216518430095.

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We introduce a notion of a chain of Yang–Baxter like operations. This is a sequence of solutions of an asymmetric variant of the Yang–Baxter equation and is a multi-operator generalization of (bi)rack/quandles. We discuss knot and link invariants coming from a chain of Yang–Baxter like operations, and give potential applications. Among them, we define a cocycle invariant for transverse links.

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9

CALSAMIGLIA, GABRIEL, and YOHANN GENZMER. "Classification of regular dicritical foliations." Ergodic Theory and Dynamical Systems 37, no. 5 (March 23, 2016): 1443–79. http://dx.doi.org/10.1017/etds.2015.123.

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In this paper we give complete analytic invariants for the set of germs of holomorphic foliations in $(\mathbb{C}^{2},0)$ that become regular after a single blow-up. Some of the invariants describe the holonomy pseudogroup of the germ and are called transverse invariants. The other invariants lie in a finite dimensional complex vector space. Such singularities admit separatrices tangentially to any direction at the origin. When enough separatrices are leaves of a radial foliaton (a condition that can always be attained if the multiplicity of the germ at the origin is at most four) we are able to describe and realize all the analytical invariants geometrically and provide analytic normal forms. As a consequence, we prove that any two such germs sharing the same transverse invariants are conjugated by a very particular type of birational transformation. We also provide explicit examples of universal equisingular unfoldings of foliations that cannot be produced by unfolding functions. With these at hand we are able to explicitly parametrize families of analytically distinct foliations that share the same transverse invariants.

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10

Freed, Alan D. "Transverse-Isotropic Elastic and Viscoelastic Solids." Journal of Engineering Materials and Technology 126, no. 1 (January 1, 2004): 38–44. http://dx.doi.org/10.1115/1.1631030.

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A set of invariants are presented for transverse-isotropic materials whose gradients produce strain fields, instead of deformation fields as is typically the case. Finite-strain theories for elastic and K-BKZ-type viscoelastic solids are derived. Shear-free and simple shearing deformations are employed to illustrate the constitutive theory.

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11

Juhász, András, Maggie Miller, and Ian Zemke. "Transverse invariants and exotic surfaces in the 4–ball." Geometry & Topology 25, no. 6 (November 30, 2021): 2963–3012. http://dx.doi.org/10.2140/gt.2021.25.2963.

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12

Epstein, Judith, Dmitry Fuchs, and Maike Meyer. "Chekanov–Eliashberg invariants and transverse approximations of Legendrian knots." Pacific Journal of Mathematics 201, no. 1 (November 1, 2001): 89–106. http://dx.doi.org/10.2140/pjm.2001.201.89.

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13

Collari, Carlo. "Transverse link invariants from the deformations of Khovanov 𝔰𝔩3–homology." Algebraic & Geometric Topology 20, no. 4 (July 20, 2020): 1729–68. http://dx.doi.org/10.2140/agt.2020.20.1729.

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14

Plamenevskaya, Olga. "Transverse knots, branched double covers and Heegaard Floer contact invariants." Journal of Symplectic Geometry 4, no. 2 (2006): 149–70. http://dx.doi.org/10.4310/jsg.2006.v4.n2.a2.

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15

Baldwin, John A., and Steven Sivek. "Invariants of Legendrian and transverse knots in monopole knot homology." Journal of Symplectic Geometry 16, no. 4 (2018): 959–1000. http://dx.doi.org/10.4310/jsg.2018.v16.n4.a3.

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16

Etnyre, John, David Vela-Vick, and Rumen Zarev. "Sutured Floer homology and invariants of Legendrian and transverse knots." Geometry & Topology 21, no. 3 (May 10, 2017): 1469–582. http://dx.doi.org/10.2140/gt.2017.21.1469.

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17

Chmutov, S., S. Jablan, K. Karvounis, and S. Lambropoulou. "On the link invariants from the Yokonuma–Hecke algebras." Journal of Knot Theory and Its Ramifications 25, no. 09 (August 2016): 1641004. http://dx.doi.org/10.1142/s0218216516410042.

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In this paper, we study properties of the Markov trace tr[Formula: see text] and the specialized trace [Formula: see text] on the Yokonuma–Hecke algebras, such as behavior under inversion of a word, connected sums and mirror imaging. We then define invariants for framed, classical and singular links through the trace [Formula: see text] and also invariants for transverse links through the trace tr[Formula: see text]. In order to compare the invariants for classical links with the Homflypt polynomial, we develop computer programs and we evaluate them on several Homflypt-equivalent pairs of knots and links. Our computations lead to the result that these invariants are topologically equivalent to the Homflypt polynomial on knots. However, they do not demonstrate the same behavior on links.

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18

Little, Robert D. "Rational Integer Invariants of Regular Cyclic Actions." Canadian Mathematical Bulletin 47, no. 1 (March 1, 2004): 60–72. http://dx.doi.org/10.4153/cmb-2004-008-2.

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AbstractLet g : M2n → M2n be a smooth map of period m ≥ 2 which preserves orientation. Suppose that the cyclic action defined by g is regular and that the normal bundle of the fixed point set F has a g-equivariant complex structure. Let F ⋔ F be the transverse self-intersection of F with itself. If the g-signature Sign(g, M) is a rational integer and n < ϕ(m), then there exists a choice of orientations such that Sign(g, M) = Sign F = Sign(F ⋔ F).

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19

MESÓN, ALEJANDRO M., and FERNANDO VERICAT. "ENTROPIES OF GROUPS AS KNOT INVARIANTS." Journal of Knot Theory and Its Ramifications 12, no. 01 (February 2003): 53–65. http://dx.doi.org/10.1142/s0218216503002305.

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We use the Ghys, Langevin, Walczak-entropy to associate an invariant hK to a knot K. This is done by taking the knot group ΓK, or the orbifold group of K, acting on its Cayley graph [Formula: see text] . Different variants may also be considered: a) if the knot group is hyperbolic the action can be taken on the boundary of [Formula: see text], b) If E (K) is the exterior manifold of K in S3, and if F (K) is a fibration on E (K), then ΓK acts on F (K) by the global holonomy of a foliation transverse to the fibres. We perform some estimations of hK in the first case (ΓK acting on [Formula: see text]) and when a) and b) are considered.

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20

Fuchs, Dmitry, and Serge Tabachnikov. "Invariants of Legendrian and transverse knots in the standard contact space." Topology 36, no. 5 (September 1997): 1025–53. http://dx.doi.org/10.1016/s0040-9383(96)00035-3.

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21

Sabloff, Joshua M. "Invariants of Legendrian Knots in Circle Bundles." Communications in Contemporary Mathematics 05, no. 04 (August 2003): 569–627. http://dx.doi.org/10.1142/s0219199703001075.

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Let M be a circle bundle over a Riemann surface that supports a contact structure transverse to the fibers. This paper presents a combinatorial definition of a differential graded algebra (DGA) that is an invariant of Legendrian knots in M. The invariant generalizes Chekanov's combinatorial DGA invariant of Legendrian knots in the standard contact 3-space using ideas from Eliashberg, Givental, and Hofer's contact homology. The main difficulty lies in dealing with what are ostensibly 1-parameter families of generators for the DGA; these are solved using "Morse–Bott" techniques. As an application, the invariant is used to distinguish two Legendrian knots that are smoothly isotopic, realize a nontrivial homology class, but are not Legendrian isotopic.

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22

Zhu, Jing, Lian Cun Zheng, and Xue Hui Chen. "Similarity Solutions of the MHD Stagnation-Point Flow over a Power-Law Stretching Sheet." Applied Mechanics and Materials 52-54 (March 2011): 1895–900. http://dx.doi.org/10.4028/www.scientific.net/amm.52-54.1895.

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A similarity analysis is performed for a steady laminar boundary layer stagnation-point flow of an electrically conducting fluid in a porous medium subject to a transverse non-uniform magnetic field past a non-linear stretching sheet. A scaling group of transformations is applied to get the invariants. Using the invariants, a third order ordinary differential equation corresponding to the momentum is obtained. We show the existence and uniqueness of convex and concave solutions for the power law exponent, according to the values of magnetic parameter, permeability parameter and velocity ratio parameter.

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23

Yang, Kun, Zhenhua Xia, Yipeng Shi, and Shiyi Chen. "Effect of Oscillation Structures on Inertial-Range Intermittence and Topology in Turbulent Field." Communications in Computational Physics 19, no. 1 (January 2016): 251–72. http://dx.doi.org/10.4208/cicp.120515.280815a.

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AbstractUsing the incompressible isotropic turbulent fields obtained from direct numerical simulation and large-eddy simulation, we studied the statistics of oscillation structures based on local zero-crossings and their relation with inertial-range intermittency for transverse velocity and passive scalar. Our results show that for both the velocity and passive scalar, the local oscillation structures are statistically scale-invariant at high Reynolds number, and the inertial-range intermittency of the overall flow region is determined by the most intermittent structures characterized by one local zero-crossing. Local flow patterns conditioned on the oscillation structures are characterized by the joint probability density function of the invariants of the filtered velocity gradient tensor at inertial range. We demonstrate that the most intermittent regions for longitudinal velocity tend to lay at the saddle area, while those for the transverse velocity tend to locate at the vortex-dominated area. The connection between the ramp-cliff structures in passive scalar field and the corresponding saddle regions in the velocity field is also verified by the approach of oscillation structure classification.

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24

Tchernov, Vladimir. "Vassiliev invariants of Legendrian, transverse, and framed knots in contact three-manifolds." Topology 42, no. 1 (January 2003): 1–33. http://dx.doi.org/10.1016/s0040-9383(01)00039-8.

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25

Baldwin, John A., David Vela-Vick, and Vera Vértesi. "On the equivalence of Legendrian and transverse invariants in knot Floer homology." Geometry & Topology 17, no. 2 (April 30, 2013): 925–74. http://dx.doi.org/10.2140/gt.2013.17.925.

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26

Vance, Katherine. "Tau invariants for balanced spatial graphs." Journal of Knot Theory and Its Ramifications 29, no. 09 (August 2020): 2050066. http://dx.doi.org/10.1142/s0218216520500662.

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In 2003, Ozsváth and Szabó defined the concordance invariant [Formula: see text] for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of [Formula: see text] for knots in [Formula: see text] and a combinatorial proof that [Formula: see text] gives a lower bound for the slice genus of a knot. Recently, Harvey and O’Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory for transverse spatial graphs in [Formula: see text], extending HFK for knots. We define a [Formula: see text]-filtered chain complex for balanced spatial graphs whose associated graded chain complex has homology determined by Harvey and O’Donnol’s graph Floer homology. We use this to show that there is a well-defined [Formula: see text] invariant for balanced spatial graphs generalizing the [Formula: see text] knot concordance invariant. In particular, this defines a [Formula: see text] invariant for links in [Formula: see text]. Using techniques similar to those of Sarkar, we show that our [Formula: see text] invariant is an obstruction to a link being slice.

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27

ZHOU, T., and R. A. ANTONIA. "Reynolds number dependence of the small-scale structure of grid turbulence." Journal of Fluid Mechanics 406 (March 10, 2000): 81–107. http://dx.doi.org/10.1017/s0022112099007296.

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The small-scale structure of grid turbulence is studied primarily using data obtained with a transverse vorticity (ω3) probe for values of the Taylor-microscale Reynolds number Rλ in the range 27–100. The measured spectra of the transverse vorticity component agree within ±10% with those calculated using the isotropic relation over nearly all wavenumbers. Scaling-range exponents of transverse velocity increments are appreciably smaller than exponents of longitudinal velocity increments. Only a small fraction of this difference can be attributed to the difference in intermittency between the locally averaged energy dissipation rate and enstrophy fluctuations. The anisotropy of turbulence structures in the scaling range, which reflects the small values of Rλ, is more likely to account for most of the difference. All four fourth-order rotational invariants Iα (α = 1 to 4) proposed by Siggia (1981) were evaluated. For any particular value of α, the magnitude of the ratio Iα / I1 is approximately constant, independently of Rλ. The implication is that the invariants are interdependent, at least in isotropic and quasi-Gaussian turbulence, so that only one power-law exponent may be sufficient to describe the Rλ dependence of all fourth-order velocity derivative moments in this type of flow. This contrasts with previous suggestions that at least two power-law exponents are needed, one for the rate of strain and the other for vorticity.

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28

Gurrampati, Venkata Ramana Reddy, S. Mohammed Ibrahim, and V. S. Bhagavan. "Similarity transformations of heat and mass transfer effects on steady MHD free convection dissipative fluid flow past an inclined porous surface with chemical reaction." Journal of Naval Architecture and Marine Engineering 11, no. 2 (December 24, 2014): 157–66. http://dx.doi.org/10.3329/jname.v11i2.18313.

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This paper concerns with a steady two-dimensional flow of an electrically conducting incompressible dissipating fluid over an inclined semi-infinite porous surface with heat and mass transfer in presence of chemical reaction. The flow is permeated by a uniform transverse magnetic field. A scaling group of transformations is applied to the governing equations. The system remains invariant due to some relations among the parameters of the transformations. After finding three absolute invariants, a third-order ordinary differential equation corresponding to the momentum equation, and two second-order ordinary differential equations corresponding to energy and diffusion equations are derived. The coupled ordinary differential equations along with the boundary conditions are solved numerically. The effects of various parameters on velocity, temperature and concentration fields as well as skin-friction, Nusselt number and Sherwood number are presented graphically and in tabulated form. DOI: http://dx.doi.org/10.3329/jname.v11i2.18313

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29

Voevodin, V. G., A. N. Morozov, and V. E. Stepanov. "Ray invariants and wave equations for transverse modes in three-dimensional graded-index waveguides." Soviet Journal of Quantum Electronics 22, no. 9 (September 30, 1992): 843–46. http://dx.doi.org/10.1070/qe1992v022n09abeh003611.

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30

Li, Qian, David A. Dillard, and Romesh C. Batra. "Constitutive Relation for Large Deformations of Fiber-Reinforced Rubberlike Materials with Different Response in Tension and Compression." Tire Science and Technology 44, no. 1 (January 1, 2016): 51–72. http://dx.doi.org/10.2346/tire.16.440104.

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ABSTRACT Fiber-reinforced rubberlike materials commonly used in tires undergo large deformations and exhibit different responses in tension and compression along the fiber direction. Assuming that the response of a fiber-reinforced rubberlike material can be modeled as transversely isotropic with the fiber direction as the axis of transverse isotropy, we express the stored energy function in terms of the five invariants of the right Cauchy-Green strain tensor and account for different response in tension and compression along the fiber direction. The constitutive relation accounts for both material and geometric nonlinearities and incorporates effects of the fifth strain invariant, I5. It has been shown by Merodio and Ogden that in shear dominated deformations, I5 makes a significant contribution to the stress-strain curve. We have implemented the proposed constitutive relation in the commercial software, LS-DYNA. The numerical solutions of a few boundary value problems studied here agree with their analytical solutions derived by using Ericksen's inverse approach, in which part of the solution is assumed and unknowns in the presumed solution are found by analyzing the pertinent boundary value problem. However, computed results have not been compared with experimental findings. When test data become available, one can modify the form of the strain energy density and replace the proposed constitutive relation by the new one in LS-DYNA.

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31

Taghizadeh, D. M., and H. Darijani. "Mechanical Behavior Modeling of Hyperelastic Transversely Isotropic Materials Based on a New Polyconvex Strain Energy Function." International Journal of Applied Mechanics 10, no. 09 (November 2018): 1850104. http://dx.doi.org/10.1142/s1758825118501041.

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In this paper, the mechanical behavior of incompressible transversely isotropic materials is modeled using a strain energy density in the framework of Ball’s theory. Based on this profound theory and with respect to physical and mathematical aspects of deformation invariants, a new polyconvex constitutive model is proposed for the mechanical behavior of these materials. From the physical viewpoint, it is assumed that the proposed model is additively decomposed into three parts nominally representing the energy contributions from the matrix, fiber and fiber–matrix interaction where each of the parts should be presented in terms of the invariants consistent with the physics of the deformation. From the mathematical viewpoint, the proposed model satisfies the fundamental postulates on the form of strain energy density, specially polyconvexity and coercivity constraints. Indeed, polyconvexity ensures ellipticity condition, which in turn provides material stability and in combination with coercivity condition, guarantees the existence of the global minimizer of the total energy. In order to evaluate the performance of the proposed strain energy density function, some test data of incompressible transverse materials with pure homogeneous deformations are used. It is shown that there is a good agreement between the test data and the obtained results from the proposed model. At the end, the performance of the proposed model in the prediction of the material behavior is evaluated rather than other models for two representative problems.

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32

Bischoff-Beiermann, B., and O. T. Bruhns. "A physically motivated set of invariants and tensor generators in the case of transverse isotropy." International Journal of Engineering Science 32, no. 10 (October 1994): 1531–52. http://dx.doi.org/10.1016/0020-7225(94)90162-7.

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33

Nashed, Gamal G. L. "Rotating black holes in the teleparallel equivalent of general relativity." International Journal of Modern Physics D 25, no. 07 (June 2016): 1650079. http://dx.doi.org/10.1142/s0218271816500796.

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We derive set of solutions with flat transverse sections in the framework of a teleparallel equivalent of general relativity which describes rotating black holes. The singularities supported from the invariants of torsion and curvature are explained. We investigate that there appear more singularities in the torsion scalars than in the curvature ones. The conserved quantities are discussed using Einstein–Cartan geometry. The physics of the constants of integration is explained through the calculations of conserved quantities. These calculations show that there is a unique solution that may describe true physical black hole.

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34

Di Colandrea, Francesco, Alessio D’Errico, Maria Maffei, Hannah M. Price, Maciej Lewenstein, Lorenzo Marrucci, Filippo Cardano, Alexandre Dauphin, and Pietro Massignan. "Linking topological features of the Hofstadter model to optical diffraction figures." New Journal of Physics 24, no. 1 (January 1, 2022): 013028. http://dx.doi.org/10.1088/1367-2630/ac4126.

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Abstract In two, three and even four spatial dimensions, the transverse responses experienced by a charged particle on a lattice in a uniform magnetic field are fully controlled by topological invariants called Chern numbers, which characterize the energy bands of the underlying Hofstadter Hamiltonian. These remarkable features, solely arising from the magnetic translational symmetry, are captured by Diophantine equations which relate the fraction of occupied states, the magnetic flux and the Chern numbers of the system bands. Here we investigate the close analogy between the topological properties of Hofstadter Hamiltonians and the diffraction figures resulting from optical gratings. In particular, we show that there is a one-to-one relation between the above mentioned Diophantine equation and the Bragg condition determining the far-field positions of the optical diffraction peaks. As an interesting consequence of this mapping, we discuss how the robustness of diffraction figures to structural disorder in the grating is a direct analogue of the robustness of transverse conductance in the quantum Hall effect.

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35

Li, Shuguang. "The Maximum Stress Failure Criterion and the Maximum Strain Failure Criterion: Their Unification and Rationalization." Journal of Composites Science 4, no. 4 (October 22, 2020): 157. http://dx.doi.org/10.3390/jcs4040157.

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The maximum strain failure criterion is unified with the maximum stress failure criterion, after exploring the implications of two considerations responsible for this: (1) the failure strains for the direct strain components employed in the maximum strain criterion are all defined under uniaxial stress states, not uniaxial strain states, and (2) the contributions to the strain in a direction as a result of the Poisson effect do not contribute to the failure of the material in that direction. Incorporating these considerations into the maximum strain criterion, the maximum stress criterion is reproduced. For 3D stress/strain state applications primarily, the unified maximum stress/strain criterion is then subjected to further rationalization in the context of transversely isotropic materials by eliminating the treatments that undermine the objectivity of the failure criterion. The criterion is then applied based on the maximum and minimum direct stresses, the maximum transverse shear stress and the maximum longitudinal shear stress as the invariants of the stress state, instead of the conventional stress components directly.

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36

Webb, G. M., M. Brio, and G. P. Zank. "Symmetries of the triple degenerate DNLS equations for weakly nonlinear dispersive MHD waves." Journal of Plasma Physics 54, no. 2 (October 1995): 201–44. http://dx.doi.org/10.1017/s0022377800018456.

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Lie symmetries, conservation laws, and Lagrangian and Hamiltonian formulations of the triple degenerate, derivative nonlinear Schrödinger (TDNLS) equations for weakly nonlinear dispersive, magnetohydrodynamic (MHD) waves are derived. The equations describe how Alfvén waves propagating parallel to the background magnetic field B interact with the magneto-acoustic modes near the triple umbilic point where the fast, slow and Alfvén mode phase speeds coincide. The Lie point symmetries are used to derive classical similarity solutions of the equations. In particular, the similarity solutions corresponding to time translation, space translation and rotational invariance symmetries are reduced to quadrature. The dispersionless TDNLS system is of hydrodynamic type, and has three families of characteristics analogous to the slow, intermediate and fast modes of MilD. The Riemann invariants corresponding to each of these families are obtained in closed analytic form. Examples of solitary wave and periodic travelling wave solutions are investigated by plotting the contours of the Hamiltonian H(v, w) in the (v, w) phase plane, where the canonical variables v and w correspond to the normalized transverse magnetic field perturbations. An analysis of the prolongation Lie algebra is carried out in order to investigate the integrability of the equations.

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37

Charmetant, Adrien, Emmanuelle Vidal-Sallé, and Philippe Boisse. "3D Hyperelastic Constitutive Model for Yarn Behaviour Description." Key Engineering Materials 504-506 (February 2012): 267–72. http://dx.doi.org/10.4028/www.scientific.net/kem.504-506.267.

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The preforming stage of the LCM composite manufacturing processes lead to fibrous reinforcement deformations which may be very large especially for double curvature shapes. Those deformations have significant influence on the second stage of the process, i.e. the injection of the resin. A way to predict accurately the spatial distribution of the permeability tensor consists in simulating for various configurations, the deformed shape of the reinforcement at the scale of the yarns. Mesoscopic scale analyses of textile reinforcements generally consider the yarns as a continuous material despite their fibrous nature. In order to have an accurate simulation tool, it is necessary to build up a constitutive law which accounts for the physical specificities linked to the microstructure of the yarns. Several models exist with reasonable accuracy. The present paper proposes a new approach in the hyperelasticity framework. The proposed model is based on the definition of mathematical invariants linked to the four main deformation modes of the yarn material: tension, compaction, longitudinal shear and transverse shear. The strain energy potential build up with those invariants is identified using classical fabric material tests: uni- and bi-axial tension and compression. The model has been validated on laboratory tests such as bias extension tests and gives promising results.

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38

Rosen, Joseph, and Roy Kelner. "Modified Lagrange invariants and their role in determining transverse and axial imaging resolutions of self-interference incoherent holographic systems." Optics Express 22, no. 23 (November 13, 2014): 29048. http://dx.doi.org/10.1364/oe.22.029048.

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39

Micunovic, Milan, and Ljudmila Kudrjavceva. "A low order viscoplasticity of transversely isotropic quasi-rate independent materials." Theoretical and Applied Mechanics 41, no. 3 (2014): 233–45. http://dx.doi.org/10.2298/tam1403233m.

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As found by experiments quasi rate independent materials (QRI) describe very well behavior of steels in very wide range of strains and strain rates ([3],[4]). This property has been combined with tensor representation modeling using a generalized associative flow rule based not on the yield function but on a more general loading function. Seemingly rate independent QRI producing incremental evolution equations show rate sensitivity by means of variability of yield stress with stress rate. On the other hand transverse isotropy appears in metal forming issues like in rolled car body sheets [18]. Here an extension of tensor generators and invariants is needed to include the preferred anisotropy direction. Such a procedure has been made here. In addition we believe that the results of this paper are applicable to dynamic deformation of orthogneiss rocks treated recently in [5].

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Usal, Melek. "On Continuum Damage Modeling of Fiber Reinforced Viscoelastic Composites with Microcracks in terms of Invariants." Mathematical Problems in Engineering 2015 (2015): 1–15. http://dx.doi.org/10.1155/2015/624750.

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A continuum damage model is developed for the linear viscoelastic behavior of composites with microcracks consisting of an isotropic matrix reinforced by two arbitrarily independent and inextensible fiber families. Despite the fact that the matrix material is isotropic, the model in consideration bears the characteristic of directed media included in the transverse isotropy symmetry group solely due to its fibers distributions and the existence of microcracks. Using the basic laws of continuum damage mechanics and equations belonging to kinematics and deformation geometries of fibers, the constitutive functions have been obtained. It has been detected as a result of the thermodynamic constraints that the stress potential function is dependent on two symmetric tensors and two vectors, whereas the dissipative stress function is dependent on four symmetric tensors and two vectors. To determine arguments of the constitutive functionals, findings relating to the theory of invariants have been used as a method because of the fact that isotropy constraint is imposed on the material. As a result the linear constitutive equations of elastic stress, dissipative stress, and strain energy density release rate have been written in terms of material coordinate description. Using these expressions, total stress has been found.

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Li, Xuefeng, Weinan Lu, and Yuqi He. "3D Mechanical Characters and Their Fabric Evolutions of Granular Materials by DEM Simulation." Mathematical Problems in Engineering 2022 (August 9, 2022): 1–15. http://dx.doi.org/10.1155/2022/4765887.

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To quantitatively describe the influence of the contact characteristics of granular materials on their mechanical response, the true-triaxial tests with different particle shapes are simulated by the discrete element method (DEM), and the connection between the evolutions of particle contact fabric and the anisotropic mechanical responses is studied. The contact normal vector of the particle in 3D space is described by two independent angles, by which the contact fabric tensor is defined. The amplitude parameters in three orthogonal directions are defined by the invariants of the plane fabric tensor, which are scalars and describe the degree of anisotropy of the contact fabric in each plane. The expression of orthotropic fabric tensor is derived from the amplitude parameters, with the change of geometric space of contact normal vector, which is reduced to the different tensor of transverse isotropic naturally. The fabric tensor can be directly applied to the constitutive equation to describe the effects of the particle contact on the mechanical response. For verifying the rationality of contact characteristics described by fabric tensor, four particle shapes are clumped by PFC3D. The mechanical properties of specimens with different particles are simulated under the true-triaxial loading path, and the data of contact normal vector is extracted in real time. The simulation results showed that the particle shapes have a significant effect on the 3D stress-strain relationship and strength, which showed apparent anisotropy, and the invariants of fabric tensor can be used to describe the evolution of particle contact in the loading process.

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Ozsváth, Peter, and András I. Stipsicz. "Contact surgeries and the transverse invariant in knot Floer homology." Journal of the Institute of Mathematics of Jussieu 9, no. 3 (May 11, 2010): 601–32. http://dx.doi.org/10.1017/s1474748010000095.

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AbstractWe study naturality properties of the transverse invariant in knot Floer homology under contact (+1)-surgery. This can be used as a calculational tool for the transverse invariant. As a consequence, we show that the Eliashberg–Chekanov twist knots En are not transversely simple for n odd and n > 3.

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Chai, Zhen, Weiyi Wang, Zhongxing Tian, and Yuting Xu. "Edge-mode polariton chains in the dielectric whispering gallery modes and two-dimensional material’s topological system." Journal of Optics 24, no. 6 (April 19, 2022): 064001. http://dx.doi.org/10.1088/2040-8986/ac60bb.

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Abstract Topological polaritons are a new topological phenomenon that has a combination of advantages such as strong nonlinearity, low effective mass, and topological invariants. However, the actual configurations of topological polaritons based on two-dimensional materials have not been discussed yet. In this paper, we theoretically demonstrate a Su-Schrieffer-Heeger (SSH) model (Sinev et al 2015 Nanoscale 7 11904; Dobrykh et al 2018 Phys. Rev. Lett. 121 163901; Qi et al 2020 Phys. Rev. A 102 022404) with topological polaritons by TiO2 whispering gallery modes and nanodisk arrays coupled with two-dimensional transition metal dichalcogenides (TMDCs) exciton materials. The transverse electric-polarized strong evanescent fields around the TiO2 nanodisks coupled with the TMDCs excitons for each atom can construct strong coupling between exciton-polaritons with a distinct resonant Rabi splitting dispersion of approximately 14 nm. By changing the distance between adjacent nanodisks, we observed obvious edge states for the SSH polariton chain at the wavelength of polaritons. This study paves the way toward new topological polaritons in large nonlinear devices with an edge transport based on two-dimensional materials.

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Shariff, M. H. B. M., Roger Bustamante, Mokarram Hossain, and Paul Steinmann. "A novel spectral formulation for transversely isotropic magneto-elasticity." Mathematics and Mechanics of Solids 22, no. 5 (December 29, 2015): 1158–76. http://dx.doi.org/10.1177/1081286515618999.

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Classical invariants, despite most of them having unclear physical interpretation and not having experimental advantages, have been extensively used in modeling nonlinear magneto-elastic materials. In this paper, a new set of spectral invariants, which have some advantages over classical invariants, is proposed to model the behavior of transversely isotropic nonlinear magneto-elastic bodies. The novel spectral invariant formulation, which is shown to be more general, is used to analytically solve some simple magneto-mechanical boundary value problems. With the aid of the proposed spectral invariants it is possible to study, in a much simpler manner, the effect of different types of deformations on the response of the magneto-elastic material.

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Bucur, Voichita, Simone Garros, and Claire Y. Barlow. "The Effect of Hydrostatic Pressure on Physical Properties and Microstructure of Spruce and Cherry." Holzforschung 54, no. 1 (January 28, 2000): 83–92. http://dx.doi.org/10.1515/hf.2000.013.

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Summary The effect of hydrostatic pressure on the density, the ultrasonic velocities and the microstructure of spruce and cherry wood has been studied. Generally speaking, under hydrostatic pressure wood becomes less heterogeneous and less anisotropic than natural wood. In spruce, crushing and buckling of the thin-walled cells in the earlywood takes place. This also has the effect of disrupting the medullary rays, which assume a zig-zag path through the structure. Cherry has a much more homogeneous structure, and the main effect of the hydrostatic pressure is compaction of the vessels by buckling of the walls. The fibres are scarcely affected by the treatment. The width of the earlywood zone decreased after the application of pressure by 26% in spruce, and by 11% in cherry. The average density was increased by the hydrostatic pressure by 26% for spruce and by 46% for cherry. The densitometric profile of spruce demonstrates significant changes following the pressure treatment, with the minimum density DMin increasing and the maximum density DMax decreasing. For cherry, the densitometric profile is shifted rather uniformly towards higher densities, and the annual ring profile is spatially slightly compacted but otherwise similar to that of untreated wood. The anisotropy of wood (expressed by the ratio of acoustic invariants) decreased by 56% for spruce and by 33% for cherry. The structural damage in spruce is predominantly found in the radial (R) direction, and this corresponds to a reduction of 73% in the velocity of the longitudinal ultrasonic waves in the radial direction, VRR. In cherry, the structural damage is mainly in the transverse, T direction. The velocity of the longitudinal ultrasonic waves in the transverse direction, VTT is reduced by 44%. The medullary rays in cherry seem to be the most important anatomical feature influencing the propagation of ultrasonic waves.

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Forte, Sandra, and Maurizio Vianello. "Restricted Invariants on the Space of Elasticity Tensors." Mathematics and Mechanics of Solids 11, no. 1 (February 2006): 48–82. http://dx.doi.org/10.1177/1081286505046483.

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A linear function defined on the space of elasticity tensors is a restricted invariant under a group of rotations G if it has an invariant restriction to a proper subspace which is larger than the set left fixed by the action of G itself. A necessary and sufficient condition for a function to be a restricted invariant is given using concepts related with isotypic decomposition, Haar integration and G -dependence. The result is applied to characterize isotropic and transversely isotropic restricted invariants.

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Prescott, Paul R., James P. Kenneally, and Terence J. Hughes. "Relating crevassing to non-linear strain in the floating part of Jakobshavn Isbræ, West Greenland." Annals of Glaciology 36 (2003): 149–56. http://dx.doi.org/10.3189/172756403781816392.

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AbstractJakobshavn Isbræ is a major ice stream that drains the west-central Greenland ice sheet and becomes afloat in Jakobshavn Isfjord (69° N, 49° W), where it has maintained the world’s fastest-known sustained velocity and calving rate (7 kma−1) for at least four decades. The floating portion is approximately 12 km long and 6 km wide. Surface elevations and motion vectors were determined photogrammetrically for about 500 crevasses on the floating ice, and adjacent grounded ice, using aerial photographs obtained 2 weeks apart in July l985. Surface strain rates were computed from a mesh of 399 quadrilateral elements having velocity measurements at each corner. It is shown that heavy crevassing of floating ice invalidates the assumptions of linear strain theory that (i) surface strain in the floating ice is homogeneous in both space and time, (ii) the squares and products of strain components are nil, and (iii) first- and second-order rotation components are small compared to strain components. Therefore, strain rates and rotation rates were also computed using non-linear strain theory. The percentage difference between computed linear and non-linear second invariants of strain rate per element were greatest (mostly in the range 40–70%) where crevassing is greatest. Isopleths of strain rate parallel and transverse to flow and elevation isopleths relate crevassing to known and inferred pinning points.

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BESSE, NICOLAS. "GLOBAL WEAK SOLUTIONS FOR THE RELATIVISTIC WATERBAG CONTINUUM." Mathematical Models and Methods in Applied Sciences 22, no. 01 (January 2012): 1150001. http://dx.doi.org/10.1142/s0218202512005848.

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In this paper we consider the relativistic waterbag continuum which is a useful PDE for collisionless kinetic plasma modeling recently developed in Ref. 11. The waterbag representation of the statistical distribution function of particles can be viewed as a special class of exact weak solution of the Vlasov equation, allowing to reduce this latter into a set of hydrodynamic equations (with the complexity of a multi-fluid model) while keeping its kinetic features (Landau damping and nonlinear resonant wave-particle interaction). These models are very promising because they are very useful for analytical theory and numerical simulations of laser-plasma and gyrokinetic physics.10–16, 56, 57 The relativistic waterbag continuum is derived from two phase-space variable reductions of the relativistic Vlasov–Maxwell equations through the existence of two underlying exact invariants, one coming from physics properties of the dynamics is the canonical transverse momentum, and the second, named the "water-bag" and coming from geometric property of the phase-space is just the direct consequence of the Liouville Theorem. In this paper we prove the existence and uniqueness of global weak entropy solutions of the relativistic waterbag continuum. Existence is based on vanishing viscosity method and bounded variations (BV) estimates to get compactness while proof of uniqueness relies on kinetic formulation of the relativistic waterbag continuum and the associated kinetic entropy defect measure.

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Le Stum, Bernard, and Adolfo Quirós. "The Filtered Poincaré Lemma in Higher Level (With Applications to Algebraic Groups)." Nagoya Mathematical Journal 191 (2008): 79–110. http://dx.doi.org/10.1017/s0027763000025915.

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AbstractWe show that the Poincaré lemma we proved elsewhere in the context of crystalline cohomology of higher level behaves well with regard to the Hodge filtration. This allows us to prove the Poincaré lemma for transversal crystals of level m. We interpret the de Rham complex in terms of what we call the Berthelot-Lieberman construction and show how the same construction can be used to study the conormal complex and invariant differential forms of higher level for a group scheme. Bringing together both instances of the construction, we show that crystalline extensions of transversal crystals by algebraic groups can be computed by reduction to the filtered de Rham complexes. Our theory does not ignore torsion and, unlike in the classical case (m = 0), not all invariant forms are closed. Therefore, close invariant differential forms of level m provide new invariants and we exhibit some examples as applications.

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KARNEVSKIY, M. S., and S. A. PASTON. "FEYNMAN PERTURBATION THEORY FOR GAUGE THEORY ON TRANSVERSE LATTICE." International Journal of Modern Physics A 25, no. 18n19 (July 30, 2010): 3621–40. http://dx.doi.org/10.1142/s0217751x10049244.

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Feynman perturbation theory for non-Abelian gauge theory in light-like gauge is investigated. A lattice along two spacelike directions is used as a gauge invariant ultraviolet regularization. For preservation of the polynomiality of action, we use as independent variables arbitrary (nonunitary) matrices related to the link of the lattice. The action of the theory is selected in such a way to preserve as much as possible the rotational invariance, which remains after an introduction of the lattice, as well as to make superfluous degrees of freedom vanish in the limit of removing the regularization. Feynman perturbation theory is constructed and diagrams which does not contain ultraviolet divergences are analyzed. The scheme of renormalization of this theory is discussed.

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