We prove the existence of Kähler-Einstein metric on a K-stable Fano manifold using the recent compactness result on Kähler-Ricci flows. The key ingredient is an algebro-geometric description of the asymptotic behavior of Kähler-Ricci flow on Fano manifolds. This is in turn based on a general finite dimensional discussion, which is interesting in its own and could potentially apply to other prob...

The notion of a C-ultrahomogeneous graph, due to Isaksen et al., is adapted for digraphs, and then a strongly connected ~ C4-ultrahomogeneous oriented graph on 168 vertices and 126 pairwise arc-disjoint 4-cycles is presented, with regular indegree and outdegree 3 and no circuits of lengths 2 and 3, by altering a definition of the Coxeter graph via pencils of ordered lines of the Fano plane in w...

In their paper “Quantum cohomology of projective bundles over IP” (Trans. Am. Math. Soc. (1998) 350:9 3615-3638) Z. Qin and Y. Ruan introduce interesting techniques for the computation of the quantum ring of Fano manifolds which are projectivized bundles over projective spaces; in particular, in the case of splitting bundles they prove under some restrictions the conjecture of Batyrev about the...

By combining the concepts of graviton and matroid, we outline a new gravitational theory which we call gravitoid theory. The idea of this theory emerged as an attempt to link the mathematical structure of matroid theory with M-theory. Our observations are essentially based on the formulation of matroid bundle due to MacPherson and Anderson-Davis. Also, by considering the oriented matroid theory...

A conjecture of G. Fan and A. Raspaud asserts that every bridgeless cubic graph contains three perfect matchings with empty intersection. We suggest a possible approach to problems of this type, based on the concept of a balanced join in an embedded graph. We use this method to prove that bridgeless cubic graphs of oddness two have Fano colorings using only 5 lines of the Fano plane. This is a ...

Given a 3-uniform hypergraph F , let ex3(n,F) denote the maximum possible size of a 3-uniform hypergraph of order n that does not contain any subhypergraph isomorphic to F . Our terminology follows that of [16] and [10], which are comprehensive survey articles of Turán-type extremal graph and hypergraph problems, respectively. Also see the monograph of Bollobás [2]. There is an extensive litera...

For k-uniform hypergraphs F and H and an integer r ≥ 2, let cr,F (H) denote the number of r-colorings of the set of hyperedges of H with no monochromatic copy of F and let cr,F (n) = maxH∈Hn cr,F (H), where the maximum runs over all k-uniform hypergraphs on n vertices. Moreover, let ex(n, F ) be the usual extremal or Turán function, i.e., the maximum number of hyperedges of an n-vertex k-unifor...

3 In this paper, we give a complete characterization of binary matroids 4 with no P9-minor. A 3-connected binary matroid M has no P9-minor 5 if and only if M is one of the internally 4-connected non-regular minors 6 of a special 16-element matroid Y16, a 3-connected regular matroid, a 7 binary spike with rank at least four, or a matroid obtained by 3-summing 8 copies of the Fano matroid to a 3-...

An ab initio theory for Fano resonances in plasmonic nanostructures and metamaterials is developed using the Feshbach formalism. It reveals the role played by the electromagnetic modes and material losses in the system, and enables the engineering of Fano resonances in arbitrary geometries. A general formula for the asymmetric resonance in a nonconservative system is derived. The influence of t...

This study theoretically investigates Fano resonances and dips of an Au-SiO2-Au nanomatryoshka that is excited by a nearby electric dipole. An analytical solution of dyadic Green's functions is used to analyze the radiative and nonradiative power spectra of a radial dipole in the proximity of a nanomatryoshka. From these spectra, the plasmon modes and Fano resonances that accompany the Fano dip...