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Journal articles on the topic 'Lagrangian families'

Author: Grafiati

Published: 7 July 2024

Last updated: 31 July 2025

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1

Golovko, Roman. "On topologically distinct infinite families of exact Lagrangian fillings." Archivum Mathematicum, no. 5 (2022): 287–93. http://dx.doi.org/10.5817/am2022-5-287.

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2

Paoletti, Roberto. "On families of Lagrangian submanifolds." manuscripta mathematica 107, no. 2 (2002): 145–50. http://dx.doi.org/10.1007/s002290100229.

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3

Chen, Bang-Yen. "CLASSIFICATION OF LAGRANGIAN SURFACES OF CONSTANT CURVATURE IN THE COMPLEX EUCLIDEAN PLANE." Proceedings of the Edinburgh Mathematical Society 48, no. 2 (2005): 337–64. http://dx.doi.org/10.1017/s0013091504000203.

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AbstractOne of the most fundamental problems in the study of Lagrangian submanifolds from a Riemannian geometric point of view is the classification of Lagrangian immersions of real-space forms into complex-space forms. In this article, we solve this problem for the most basic case; namely, we classify Lagrangian surfaces of constant curvature in the complex Euclidean plane $\mathbb{C}^2$. Our main result states that there exist 19 families of Lagrangian surfaces of constant curvature in $\mathbb{C}^2$. Twelve of the 19 families are obtained via Legendre curves. Conversely, Lagrangian surfaces

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4

Matessi, Diego. "Some families of special Lagrangian tori." Mathematische Annalen 325, no. 2 (2003): 211–28. http://dx.doi.org/10.1007/s00208-002-0360-2.

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5

Doria, R. M., and S. Machado. "Yang-Mills Families." JOURNAL OF ADVANCES IN PHYSICS 13, no. 4 (2017): 4927–55. http://dx.doi.org/10.24297/jap.v13i6.6173.

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The Yang-Mills theory structure is based on group theory. It rules the symmetry relationship where the number of potential fields transforming under a same group must be equal to the number of group generators. It defines the group valued expression from where the corresponding non-abelian symmetry properties are derived. Nevertheless based on different origins as Kaluza-Klein, fibre bundles, supersymmetry, s-model , BRST and anti-BRST algorithm, counting degrees of freedom leads to a Yang-Mills extension under the existence of different potential fields rotating under a same single grou

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6

Consul, P. C. "Some bivariate families of lagrangian probability distributions." Communications in Statistics - Theory and Methods 23, no. 10 (1994): 2895–906. http://dx.doi.org/10.1080/03610929408831423.

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7

Kamenova, Ljudmila, and Misha Verbitsky. "Families of Lagrangian fibrations on hyperkähler manifolds." Advances in Mathematics 260 (August 2014): 401–13. http://dx.doi.org/10.1016/j.aim.2013.10.033.

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8

Cariñena, José F., and José Fernández-Núñez. "Some Applications of Affine in Velocities Lagrangians in Two-Dimensional Systems." Symmetry 14, no. 12 (2022): 2520. http://dx.doi.org/10.3390/sym14122520.

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The two-dimensional inverse problem for first-order systems is analysed and a method to construct an affine Lagrangian for such systems is developed. The determination of such Lagrangians is based on the theory of the Jacobi multiplier for the system of differential equations. We illustrate our analysis with several examples of families of forces that are relevant in mechanics, on one side, and of some relevant biological systems, on the other.

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9

Janeczko, Stanisław. "Generating families for images of Lagrangian submanifolds and open swallowtails." Mathematical Proceedings of the Cambridge Philosophical Society 100, no. 1 (1986): 91–107. http://dx.doi.org/10.1017/s0305004100065889.

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SummaryIn this paper we study the symplectic relations appearing as the generalized cotangent bundle liftings of smooth stable mappings. Using this class of symplectic relations the classification theorem for generic (pre) images of lagrangian submanifolds is proved. The normal forms for the respective classified puilbacks and pushforwards are provided and the connections between the singularity types of symplectic relation, mapped lagrangian submanifold and singular image, are established. The notion of special symplectic triplet is introduced and the generic local models of such triplets are

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10

JOYCE, DOMINIC. "RULED SPECIAL LAGRANGIAN 3-FOLDS IN [Copf ]3." Proceedings of the London Mathematical Society 85, no. 1 (2002): 233–56. http://dx.doi.org/10.1112/s0024611502013485.

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This is the fifth in a series of papers constructing explicit examples of special Lagrangian submanifolds in ${\mathbb C}^m$. A submanifold of ${\mathbb C}^m$ is ruled if it is fibred by a family of real straight lines in ${\mathbb C}^m$. This paper studies ruled special Lagrangian 3-folds in ${\mathbb C}^3$, giving both general theory and families of examples. Our results are related to previous work of Harvey and Lawson, Borisenko, and Bryant. Special Lagrangian cones in ${\mathbb C}^3$ are automatically ruled, and each ruled special Lagrangian 3-fold is asymptotic to a unique special Lagran

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11

Bourgeois, Frédéric, Joshua M. Sabloff, and Lisa Traynor. "Lagrangian cobordisms via generating families: Construction and geography." Algebraic & Geometric Topology 15, no. 4 (2015): 2439–77. http://dx.doi.org/10.2140/agt.2015.15.2439.

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12

Bernard, Patrick, and Gonzalo Contreras Barandarián. "A generic property of families of Lagrangian systems." Annals of Mathematics 167, no. 3 (2008): 1099–108. http://dx.doi.org/10.4007/annals.2008.167.1099.

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13

Amerik, Ekaterina, and Frédéric Campana. "On families of lagrangian tori on hyperkähler manifolds." Journal of Geometry and Physics 71 (September 2013): 53–57. http://dx.doi.org/10.1016/j.geomphys.2013.04.004.

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14

Markushevich, D. "Lagrangian families of jacobians of genus 2 curves." Journal of Mathematical Sciences 82, no. 1 (1996): 3268–84. http://dx.doi.org/10.1007/bf02362472.

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15

Matessi, Diego. "Isometric Embeddings of Families of Special Lagrangian Submanifolds." Annals of Global Analysis and Geometry 29, no. 3 (2006): 197–220. http://dx.doi.org/10.1007/s10455-005-9008-2.

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16

Chen, Bang-Yen. "Jacobi's elliptic functions and Lagrangian immersions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 126, no. 4 (1996): 687–704. http://dx.doi.org/10.1017/s0308210500023003.

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First, we establish a sharp inequality between the squared mean curvature and the scalar curvature for a Lagrangian submanifold in a nonflat complex-space-form. Then, by utilising the Jacobi's elliptic functions en and dn, we introduce three families of Lagrangian submanifolds and two exceptional Lagrangian submanifolds Fn, Ln in nonflat complex-space-forms which satisfy the equality case of the inequality. Finally, we obtain the complete classification of Lagrangian submanifolds in nonflat complex-space-forms which satisfy this basic equality.

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17

Chauca, J., R. Doria, and L. S. Mendes. "Abelian Constructivist Lagrangian." JOURNAL OF ADVANCES IN PHYSICS 21 (October 13, 2023): 139–99. http://dx.doi.org/10.24297/jap.v21i.9530.

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Constructivist lagrangian propiates a diverse approach to field theory. Introduce the set action. Consider fields families under a same symmetry group. The resulting fields set extends the standard atomist field theory to a whole field theory. An associative physics is proposed. The grouping physics. The relationship between the part and the whole is considered. A third quantum type beyond Planck granularity and quantum mechanics wave-particle is obtained. The quantum inserted in the whole. Differentiated energy packets are formed. A quantum system is constituted. The abelian grouping physics

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18

SABLOFF, JOSHUA M., and LISA TRAYNOR. "OBSTRUCTIONS TO THE EXISTENCE AND SQUEEZING OF LAGRANGIAN COBORDISMS." Journal of Topology and Analysis 02, no. 02 (2010): 203–32. http://dx.doi.org/10.1142/s179352531000029x.

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Capacities that provide both qualitative and quantitative obstructions to the existence of a Lagrangian cobordism between two (n - 1)-dimensional submanifolds in parallel hyperplanes of ℝ2n are defined using the theory of generating families. Qualitatively, these capacities show that, for example, in ℝ4 there is no Lagrangian cobordism between two ∞-shaped curves with a negative crossing when the lower end is "smaller". Quantitatively, when the boundary of a Lagrangian ball lies in a hyperplane of ℝ2n, the capacity of the boundary gives a restriction on the size of a rectangular cylinder into

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19

Sabloff, Joshua M., and Lisa Traynor. "Obstructions to Lagrangian cobordisms between Legendrians via generating families." Algebraic & Geometric Topology 13, no. 5 (2013): 2733–97. http://dx.doi.org/10.2140/agt.2013.13.2733.

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20

Mikhailov, Andrei. "Integration over families of Lagrangian submanifolds in BV formalism." Nuclear Physics B 928 (March 2018): 107–59. http://dx.doi.org/10.1016/j.nuclphysb.2018.01.006.

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21

Ionel, Marianty. "Second Order Families of Special Lagrangian Submanifolds in ℂ4". Journal of Differential Geometry 65, № 2 (2003): 211–72. http://dx.doi.org/10.4310/jdg/1090511687.

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22

Chen, Bang-Yen, and Luc Vrancken. "Lagrangian submanifolds satisfying a basic equality." Mathematical Proceedings of the Cambridge Philosophical Society 120, no. 2 (1996): 291–307. http://dx.doi.org/10.1017/s0305004100074867.

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AbstractIn [3], B. Y. Chen proved that, for any Lagrangian submanifold M in a complex space-form Mn(4c) (c = ± 1), the squared mean curvature and the scalar curvature of M satisfy the following inequality:He then introduced three families of Riemannian n-manifolds and two exceptional n-spaces Fn, Ln and proved the existence of a Lagrangian isometric immersion pa from into ℂPn(4) and the existence of Lagrangian isometric immersions f, l, ca, da from Fn, Ln, , into ℂHn(− 4), respectively, which satisfy the equality case of the inequality. He also proved that, beside the totally geodesie ones, th

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23

Simion FILIP. "Counting special lagrangian fibrations in twistor families of K3 surfaces." Annales scientifiques de l'École normale supérieure 53, no. 3 (2020): 713–50. http://dx.doi.org/10.24033/asens.2432.

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24

Castaño Bernard, Ricardo. "Symplectic invariants of some families of Lagrangian $T^3$-fibrations." Journal of Symplectic Geometry 2, no. 3 (2004): 279–308. http://dx.doi.org/10.4310/jsg.2004.v2.n3.a1.

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25

Fatima, Aeeman, Fazal M. Mahomed, and Chaudry Masood Khalique. "Noether symmetries and exact solutions of an Euler–Bernoulli beam model." International Journal of Modern Physics B 30, no. 28n29 (2016): 1640011. http://dx.doi.org/10.1142/s0217979216400117.

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In this paper, a Noether symmetry analysis is carried out for an Euler–Bernoulli beam equation via the standard Lagrangian of its reduced scalar second-order equation which arises from the standard Lagrangian of the fourth-order beam equation via its Noether integrals. The Noether symmetries corresponding to the reduced equation is shown to be the inherited Noether symmetries of the standard Lagrangian of the beam equation. The corresponding Noether integrals of the reduced Euler–Lagrange equations are deduced which remarkably allows for three families of new exact solutions of the static beam

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26

Chauca, J., and R. Doria. "Non-Abelian Constructivist Lagrangian." JOURNAL OF ADVANCES IN PHYSICS 21 (October 13, 2023): 200–230. http://dx.doi.org/10.24297/jap.v21i.9531.

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A whole Yang-Mills symmetry is proposed. A grouping physics is constituted. It consists in inserting a given Yang-Mills field Aaμ in a fields set {Aa μI } constituted by other fields families, I = 1, . . . , N. Each field becomes part of a whole. A set action physics happens preserving the Yang-Mills symmetry. However the usual properties of an isolated field are extended to antireductionist properties. An associative physics is formed. A Yang-Mills whole quantum system is constituted. A whole Yang-Mills physics isobtained. The quantum corresponding to a specific Aa μI field inserted in a whol

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27

Esen, Oğul, Manuel Lainz Valcázar, Manuel de León, and Juan Carlos Marrero. "Contact Dynamics: Legendrian and Lagrangian Submanifolds." Mathematics 9, no. 21 (2021): 2704. http://dx.doi.org/10.3390/math9212704.

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We are proposing Tulczyjew’s triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a special contact manifold and a Morse family) for a Legendrian submanifold. Contact Hamiltonian and Lagrangian Dynamics are recast as Legendrian submanifolds of the tangent contact manifold. In this picture, the Legendre transformation is determined to be a passage between two d

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28

Karimov, S. R., and A. G. Sokol'skii. "A method for continuing families of periodic solutions of Lagrangian systems." Journal of Applied Mathematics and Mechanics 53, no. 1 (1989): 1–10. http://dx.doi.org/10.1016/0021-8928(89)90125-1.

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29

Izumiya, Shyuichi, Donghe Pei, and Masatomo Takahashi. "SINGULARITIES OF EVOLUTES OF HYPERSURFACES IN HYPERBOLIC SPACE." Proceedings of the Edinburgh Mathematical Society 47, no. 1 (2004): 131–53. http://dx.doi.org/10.1017/s0013091503000312.

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AbstractWe study the differential geometry of hypersurfaces in hyperbolic space. As an application of the theory of Lagrangian singularities, we investigate the contact of hypersurfaces with families of hyperspheres or equidistant hyperplanes.AMS 2000 Mathematics subject classification: Primary 53A35. Secondary 58C27

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30

Joyce, Dominic. "Special Lagrangian Submanifolds with Isolated Conical Singularities. IV. Desingularization, Obstructions and Families." Annals of Global Analysis and Geometry 26, no. 2 (2004): 117–74. http://dx.doi.org/10.1023/b:agag.0000031067.19776.15.

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31

Goldstein, Edward. "A Construction of New Families of Minimal Lagrangian Submanifolds via Torus Actions." Journal of Differential Geometry 58, no. 2 (2001): 233–61. http://dx.doi.org/10.4310/jdg/1090348326.

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32

Glazebrook, K. D., D. Ruiz-Hernandez, and C. Kirkbride. "Some indexable families of restless bandit problems." Advances in Applied Probability 38, no. 3 (2006): 643–72. http://dx.doi.org/10.1239/aap/1158684996.

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In 1988 Whittle introduced an important but intractable class of restless bandit problems which generalise the multiarmed bandit problems of Gittins by allowing state evolution for passive projects. Whittle's account deployed a Lagrangian relaxation of the optimisation problem to develop an index heuristic. Despite a developing body of evidence (both theoretical and empirical) which underscores the strong performance of Whittle's index policy, a continuing challenge to implementation is the need to establish that the competing projects all pass an indexability test. In this paper we employ Git

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33

Glazebrook, K. D., D. Ruiz-Hernandez, and C. Kirkbride. "Some indexable families of restless bandit problems." Advances in Applied Probability 38, no. 03 (2006): 643–72. http://dx.doi.org/10.1017/s000186780000121x.

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In 1988 Whittle introduced an important but intractable class of restless bandit problems which generalise the multiarmed bandit problems of Gittins by allowing state evolution for passive projects. Whittle's account deployed a Lagrangian relaxation of the optimisation problem to develop an index heuristic. Despite a developing body of evidence (both theoretical and empirical) which underscores the strong performance of Whittle's index policy, a continuing challenge to implementation is the need to establish that the competing projects all pass an indexability test. In this paper we employ Git

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34

Chen, Hongsen, Zhangxin Chen, and Baoyan Li. "Thehpversion of Eulerian-Lagrangian mixed discontinuous finite element methods for advection-diffusion problems." International Journal of Mathematics and Mathematical Sciences 2003, no. 53 (2003): 3385–411. http://dx.doi.org/10.1155/s0161171203112215.

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We study thehpversion of three families of Eulerian-Lagrangian mixed discontinuous finite element (MDFE) methods for the numerical solution of advection-diffusion problems. These methods are based on a space-time mixed formulation of the advection-diffusion problems. In space, they use discontinuous finite elements, and in time they approximately follow the Lagrangian flow paths (i.e., the hyperbolic part of the problems). Boundary conditions are incorporated in a natural and mass conservative manner. In fact, these methods are locally conservative. The analysis of this paper focuses on advect

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35

Gadbled, Agnes. "Families of monotone symplectic manifolds constructed via symplectic cut and their Lagrangian submanifolds." Algebraic & Geometric Topology 11, no. 4 (2011): 2319–68. http://dx.doi.org/10.2140/agt.2011.11.2319.

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36

Boll, Raphael, Matteo Petrera, and Yuri B. Suris. "Multi-time Lagrangian 1-forms for families of Bäcklund transformations: Toda-type systems." Journal of Physics A: Mathematical and Theoretical 46, no. 27 (2013): 275204. http://dx.doi.org/10.1088/1751-8113/46/27/275204.

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37

Chen, B. Y. "Three additional families of Lagrangian surfaces of constant curvature in complex projective plane." Journal of Geometry and Physics 56, no. 4 (2006): 666–69. http://dx.doi.org/10.1016/j.geomphys.2005.04.011.

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38

Vianna, Renato. "Continuum families of non-displaceable Lagrangian tori in $(\mathbb{C}P^1)^{2m}$." Journal of Symplectic Geometry 16, no. 3 (2018): 857–83. http://dx.doi.org/10.4310/jsg.2018.v16.n3.a8.

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39

POPOV, G., and P. TOPALOV. "Discrete analog of the projective equivalence and integrable billiard tables." Ergodic Theory and Dynamical Systems 28, no. 5 (2008): 1657–84. http://dx.doi.org/10.1017/s014338570700096x.

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AbstractA class of discrete dynamical systems called projectively (or geodesically) equivalent Lagrangian systems is defined. We prove that these systems admit families of integrals. In the case of geodesically equivalent billiard tables, these integrals are pairwise commuting. We describe a family of geodesically equivalent billiard tables on surfaces of constant curvature. This is a special case of the so-called ‘Liouville billiard tables’.

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40

Alfonso-Rodriguez, Ranses, and S. Roy Choudhury. "Novel Lagrangian hierarchies, generalized variational ODE’s, and families of regular and embedded solitary waves." Journal of Physics A: Mathematical and Theoretical 53, no. 37 (2020): 375701. http://dx.doi.org/10.1088/1751-8121/aba4d1.

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41

Boll, Raphael, Matteo Petrera, and Yuri B. Suris. "Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems." Journal of Physics A: Mathematical and Theoretical 48, no. 8 (2015): 085203. http://dx.doi.org/10.1088/1751-8113/48/8/085203.

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42

QURESHI, MUHAMMAD IMRAN. "Constructing projective varieties in weighted flag varieties II." Mathematical Proceedings of the Cambridge Philosophical Society 158, no. 2 (2015): 193–209. http://dx.doi.org/10.1017/s0305004114000620.

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AbstractWe give the construction of weighted Lagrangian GrassmannianswLGr(3,6) and weighted partialA3flag varietywFL1,3coming from the symplectic Lie group Sp(6, ℂ) and the general linear group GL(4, ℂ) respectively. We give general formulas for their Hilbert series in terms of Lie theoretic data. We use them as key varieties (Format) to construct some families of polarized 3-folds in codimension 7 and 9. Finally, we list all the distinct weighted flag varieties in codimension (4 ⩽c⩽ 10.

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43

Sukhov, E. A., and E. V. Volkov. "Numerical Orbital Stability Analysis of Nonresonant Periodic Motions in the Planar Restricted Four-Body Problem." Nelineinaya Dinamika 18, no. 4 (2022): 0. http://dx.doi.org/10.20537/nd221201.

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We address the planar restricted four-body problem with a small body of negligible mass moving in the Newtonian gravitational field of three primary bodies with nonnegligible masses. We assume that two of the primaries have equal masses and that all primary bodies move in circular orbits forming a Lagrangian equilateral triangular configuration. This configuration admits relative equilibria for the small body analogous to the libration points in the three-body problem. We consider the equilibrium points located on the perpendicular bisector of the Lagrangian triangle in which case the bodies c

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44

CARBERRY, EMMA, and IAN MCINTOSH. "MINIMAL LAGRANGIAN 2-TORI IN $\mathbb{CP}^2$ COME IN REAL FAMILIES OF EVERY DIMENSION." Journal of the London Mathematical Society 69, no. 02 (2004): 531–44. http://dx.doi.org/10.1112/s0024610703005039.

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45

CARAVAGLIOS, FRANCESCO, and STEFANO MORISI. "GAUGE BOSON FAMILIES IN GRAND UNIFIED THEORIES OF FERMION MASSES: $E_6^4 \rtimes S_4$." International Journal of Modern Physics A 22, no. 14n15 (2007): 2469–91. http://dx.doi.org/10.1142/s0217751x07036646.

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In third quantization the origin of fermion families is easy to understand: the electron field, the muon field and the tau field are identical fields in precisely the same sense as three electrons are identical and indistinguishable particles of a theory of second quantization. In both cases, the permutation of these fields or particles leaves the Lagrangian invariant. One can also extend the concept of family to gauge bosons. This can be obtained through the semidirect product of the gauge group with the group of permutations of n objects. In this paper we have studied the group [Formula: see

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46

Shen, Wen. "Global Riemann solvers for several 3 × 3 systems of conservation laws with degeneracies." Mathematical Models and Methods in Applied Sciences 28, no. 08 (2018): 1599–626. http://dx.doi.org/10.1142/s0218202518500446.

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We study several [Formula: see text] systems of conservation laws, arising in the modeling of two-phase flow with rough porous media and traffic flow with rough road condition. These systems share several features. The systems are of mixed type, with various degeneracies. Some families are linearly degenerate, while others are not genuinely nonlinear. Furthermore, along certain curves in the domain, the eigenvalues and eigenvectors of different families coincide. Most interestingly, in some suitable Lagrangian coordinate, the systems are partially decoupled, where some unknowns can be solved i

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47

Lim, Cong Han, Jeffrey T. Linderoth, James R. Luedtke, and Stephen J. Wright. "Parallelizing Subgradient Methods for the Lagrangian Dual in Stochastic Mixed-Integer Programming." INFORMS Journal on Optimization 3, no. 1 (2021): 1–22. http://dx.doi.org/10.1287/ijoo.2019.0029.

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The dual decomposition of stochastic mixed-integer programs can be solved by the projected subgradient algorithm. We show how to make this algorithm more amenable to parallelization in a master-worker model by describing two approaches, which can be combined in a natural way. The first approach partitions the scenarios into batches and makes separate use of subgradient information for each batch. The second approach drops the requirement that evaluation of function and subgradient information is synchronized across the scenarios. We provide convergence analysis of both methods. We also evaluat

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48

Arthamonov, S., J. Harnad, and J. Hurtubise. "Tau functions, infinite Grassmannians, and lattice recurrences." Journal of Mathematical Physics 64, no. 2 (2023): 023502. http://dx.doi.org/10.1063/5.0110404.

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The addition formulae for KP τ-functions, when evaluated at lattice points in the KP flow group orbits in the infinite dimensional Sato-Segal-Wilson Grassmannian, give infinite parametric families of solutions to discretizations of the KP hierarchy. The CKP hierarchy may similarly be viewed as commuting flows on the Lagrangian sub-Grassmannian of maximal isotropic subspaces with respect to a suitably defined symplectic form. Evaluating the τ-functions at a sublattice of points within the KP orbit, the resulting discretization gives solutions both to the hyperdeterminantal relations (or Kashaev

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49

Wei, Dongyi, Zhifei Zhang, and Wenbin Zhao. "Global existence of the two-dimensional axisymmetric Euler equations for the Chaplygin gas with large angular velocities." Advanced Nonlinear Studies 22, no. 1 (2022): 635–58. http://dx.doi.org/10.1515/ans-2022-0031.

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Abstract The Chaplygin gas model is both interesting and important in the theory of gas dynamics and conservation laws, all the characteristic families of which are linearly degenerate. Majda conjectured that the shock formation never happens for smooth data. In this article, we prove the conjecture for the two space dimensional axisymmetric case. Different from previous approaches to study wave equations with different speeds, we reformulate the problem in the Lagrangian coordinates and consider a single wave equation with variable coefficients. This not only gives a simpler proof but also en

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50

EPPERSON, DOUGLAS. "A NEW SEARCH FOR LEPTON FLAVOR VIOLATION IN HIGH-Q2 POSITRON-PROTON COLLISIONS." International Journal of Modern Physics A 16, supp01b (2001): 882–84. http://dx.doi.org/10.1142/s0217751x01008382.

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A search for Lepton Flavor Violation in Deeply Inelastic positron-proton scattering was conducted at the HERA Colider, using the ZEUS detector. This process permits the probing of the four-fermion interaction, in particular those terms that involve two quarks and two leptons. Strong limits exist from rare decays and from lower-energy experiments, but they involve the lighter quarks. At HERA, limits can be set that also involve interactions with heavier sea quarks, and all three lepton families. these limits define the energy scale probed in the effective Lagrangian, and, in a more specific mod

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