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1.
In this paper, we investigate the evolution of joint invariants under invariant geometric flows using the theory of equivariant moving frames and the induced invariant discrete variational complex. For certain arc length preserving planar curve flows invariant under the special Euclidean group , the special linear group , and the semidirect group , we find that the induced evolution of the discrete curvature satisfies the differential‐difference mKdV, KdV, and Burgers' equations, respectively. These three equations are completely integrable, and we show that a recursion operator can be constructed by precomposing the characteristic operator of the curvature by a certain invariant difference operator. Finally, we derive the constraint for the integrability of the discrete curvature evolution to lift to the evolution of the discrete curve itself. 相似文献
2.
The paper proposes a rational method to derive fairness measures for surfaces. It works in cases where isophotes, reflection lines, planar intersection curves, or other curves are used to judge the fairness of the surface. The surface fairness measure is derived by demanding that all the given curves should be fair with respect to an appropriate curve fairness measure. The method is applied to the field of ship hull design where the curves are plane intersections. The method is extended to the case where one considers, not the fairness of one curve, but the fairness of a one parameter family of curves. Six basic third order invariants by which the fairing measures can be expressed are defined. Furthermore, the geometry of a plane intersection curve is studied, and the variation of the total, the normal, and the geodesic curvature and the geodesic torsion is determined. 相似文献
3.
Kayll Lake 《General Relativity and Gravitation》2004,36(5):1159-1169
The norms associated with the gradients of the two non-differential invariants of the Kerr vacuum are examined. Whereas both locally single out the horizons, their global behavior is more interesting. Both reflect the background angular momentum as the volume of space allowing a timelike gradient decreases with increasing angular momentum becoming zero in the degenerate and naked cases. These results extend directly to the Kerr-Newman geometry. 相似文献
4.
Goryunov proved that the space of local invariants of Vassiliev type for generic maps from surfaces to three-space is three-dimensional.
The basic invariants were the number of pinch points, the number of triple points and one linked to a Rokhlin type invariant.
In this paper we show that, by colouring the complement of the image of the map with a chess board pattern, we can produce
a six-dimensional space of local invariants. These are essentially black and white versions of the above. Simple examples
show how these are more effective.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
5.
In this paper we show that the Boltzmann weights of the three-dimensional Baxter-Bazhanov model give representations of the braid group if some suitable spectral limits are taken. In the trigonometric case we classify all possible spectral limits which produce braid group representations. Furthermore, we prove that for some of them we get cyclotomic invariants of links and for others we obtain tangle invariants generalizing the cyclotomic ones. 相似文献
6.
R. Milson 《Proceedings of the American Mathematical Society》2004,132(10):2825-2831
Chevalley's theorem and its converse, the Sheppard-Todd theorem, assert that finite reflection groups are distinguished by the fact that the ring of invariant polynomials is freely generated. We show that, in the Euclidean case, a weaker condition suffices to characterize finite reflection groups, namely, that a freely-generated polynomial subring is closed with respect to the gradient product.
7.
Alain Lascoux 《Bulletin des Sciences Mathématiques》2006,130(4):354-359
Eliminating the arbitrary coefficients in the equation of a generic plane curve of order n by computing sufficiently many derivatives, one obtains a differential equation. This is a projective invariant. The first one, corresponding to conics, has been obtained by Monge. Sylvester, Halphen, Cartan used invariants of higher order. The expression of these invariants is rather complicated, but becomes much simpler when interpreted in terms of symmetric functions. 相似文献
8.
We find a minimal system of generators and a homogeneous system of parameters for the algebra of invariants of three matrices of third order over a field of an arbitrary characteristic. 相似文献
9.
Field equations of the S2 sigma model (“the A3 model”) with spontaneously broken Z(2) symmetry are presented for (D+1)-dimensional space–time. The A3 model is an extension of the sine-Gordon equation (SGE) and supports kink-like U(1) charged solitons which are a generalization of neutral solitons of the SGE. The natural question arises — is the A3 model completely integrable in (1+1)-dimensional space–time? The Lorentz-invariant scalar A3 field can be viewed as a promising alternative to the Higgs field. 相似文献
10.
The paper is for a general audience and may serve as a preliminary introduction to the theory of finite-type invariants. 相似文献