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Correspondences Of Hypersurfaces In Hyperbolic Poincaré Manifolds And Conformally Invariant Pdes, Vincent Bonini, José M. Espinar, Jie Qing
Correspondences Of Hypersurfaces In Hyperbolic Poincaré Manifolds And Conformally Invariant Pdes, Vincent Bonini, José M. Espinar, Jie Qing
Mathematics
On a hyperbolic Poincaré manifold, we derive an explicit relationship between the eigenvalues of Weyl-Schouten tensor of a conformal representative of the conformal infinity and the principal curvatures of the level sets of the associated geodesic defining function. This considerably simplifies the arguments and generalizes the results of Gálvez, Mira and the second author. In particular, we obtain the equivalence between Christoffel-type problems for hypersurfaces in a hyperbolic Poincar´e manifold and scalar curvature problems on the conformal infinity.
Consecutive Patterns: From Permutations To Column-Convex Polyominoes And Back, Don Rawlings, Mark Tiefenbruck
Consecutive Patterns: From Permutations To Column-Convex Polyominoes And Back, Don Rawlings, Mark Tiefenbruck
Mathematics
We expose the ties between the consecutive pattern enumeration problems associated with permutations, compositions, column-convex polyominoes, and words. Our perspective allows powerful methods from the contexts of compositions, column-convex polyominoes, and of words to be applied directly to the enumeration of permutations by consecutive patterns. We deduce a host of new consecutive pattern results,including a solution to the (2m+1)-alternating pattern problem on permutations posed by Kitaev.
U-Singularity And T-Topos Theoretic Entropy, Goro C. Kato
U-Singularity And T-Topos Theoretic Entropy, Goro C. Kato
Mathematics
We will give descriptions of u-singularities as we introduce the notion of t-topos theoretic entropies. The unifying methodology for a u-singularity is the universal mapping property of an inverse or direct limit. The qualitative, conceptual, and structural analyses of u-singularities are given in terms of inverse and direct limits of micro decompositions of a presheaf and coverings of an object in t-site in the theory of temporal topos.
Urcohomologies And Cohomologies Of N -Complexes, Naoya Hiramatsu, Goro Kato
Urcohomologies And Cohomologies Of N -Complexes, Naoya Hiramatsu, Goro Kato
Mathematics
For a general sequence of objects and morphisms, we construct two N-complexes. Then we can define cohomologies (i, k)-type of the N-complexes not only on a diagonal region but also in the triangular region. We obtain an invariant defined on a general sequence of objects and morphisms. For a short exact sequence of N-complexes, we get the associated long exact sequence generalizing the classical long exact sequence. Lastly, several properties of the vanishing cohomologies of N-complexes are given.