We generalize Cartan's logarithmic derivative of a smooth map from manifold into Lie group $G$ to maps homogeneous space $M=G/H$, and determine the global monodromy obstruction reconstructing such infinitesimal data. The embedding submanifold $\Sigma \subset M$ becomes an invariant $ under symmetries "Klein geometry" $M$ whose analysis is taken up in [SIGMA 14 (2018), 062, 36 pages, arXiv:1703....

We find a general class of nontrivial stationary states in inelastic gases where, due to dissipation, energy is transferred from large velocity scales to small velocity scales. These steady states exist for arbitrary collision rules and arbitrary dimension. Their signature is a stationary velocity distribution f(v) with an algebraic high-energy tail, f(v) approximately v(-sigma). The exponent s...

The Gauss map g of a surface $$\Sigma $$ in $${\mathbb {R}}^4$$ takes its values the Grassmannian oriented 2-planes : $$G^+(2,4) . We give geometric criteria stability for minimal surfaces terms g. show particular that if area |g(\Sigma ) | is smaller than 2\pi then stable by deformations which fix boundary surface. This answers question Barbosa and Do Carmo (Math Z 173:13–28, 1980)

The well known Haldane map from spin chains into the O(3) non linear sigma model is generalized to the case of spin ladders. This map allows us to explain the different qualitative behaviour between even and odd ladders, exactly in the same way it explains the difference between integer and half-integer spin chains. Namely, for even ladders the topological term in the sigma model action is abse...

We show that the boundary map in periodic cyclic cohomology introduced by Cuntz and Quillen satisfies properties similar to the properties of the boundary map in simplicial homology. We also prove that the boundary map is compatible with the boundary map in algebraic and topological K-Theory, which leads to index theorems. As an application we obtain a new proof of the Connes-Moscovici index th...

We introduce two algebraic completely integrable analogues of the Mumford systems which we call hyperelliptic Prym systems, because every hyperelliptic Prym variety appears as a fiber of their momentum map. As an application we show that the general fiber of the momentum map of the periodic Volterra lattice ȧi = ai(ai−1 − ai+1), i = 1, . . . , n, an+1 = a1, is an affine part of a hyperelliptic ...

A mathematical framework of cohomological field theories (CohFTs) is formulated in the language bigraded manifolds. Algebraic properties operators CohFTs are studied. Methods constructing CohFTs, with or without gauge symmetries, discussed. In particular, a generalization Mathai-Quillen formalism given. Examples such as topological quantum mechanics, sigma model, M-theory, and Yang-Mills theory...

We investigate velocity statistics of homogeneous inelastic gases using the Boltzmann equation. Employing an approximate uniform collision rate, we obtain analytic results valid in arbitrary dimension. In the freely evolving case, the velocity distribution is characterized by an algebraic large-velocity tail, P(v,t) approximately v(-sigma). The exponent sigma(d,epsilon), a nontrivial root of an...

The squared correlation coefficient r(2) (sometimes denoted Delta(2)) is a measure of linkage disequilibrium that is widely used, but computing its expectation E[r(2)] in the population has remained an intriguing open problem. The expectation E[r(2)] is often approximated by the standard linkage deviation sigma(d)(2), which is a ratio of two expectations amenable to analytic computation. In thi...

A number of examples have demonstrated the failure of the Landau-Ginzburg-Wilson LGW paradigm in describing the competing phases and phase transitions of two-dimensional quantum magnets. In this paper, we argue that such magnets possess field theoretic descriptions in terms of their slow fluctuating orders provided certain topological terms are included in the action. These topological terms ma...