Related papers: A Simple Formula for Generating Chern Characters b…
In this paper we address Fano manifolds with positive higher Chern characters. They are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. For instance, they should be covered by higher dimensional…
An unusual formula for the Euler characteristics of even dimensional triangulated manifolds is deduced from the generalized Dehn-Sommerville equations.
In this article we give a necessary and sufficient condition to characterize projective submanifolds in ${\mathbb P}^N$ with codimensions 2 and 3. The conditions involve the Chern classes of the manifold and a very ample line bundle on the…
We study limiting lines on degenerations of generic hypersurfaces in $P^n$.
For every positive integer $n\in \mathbb{Z}_+$ we define an `Euler polynomial' $\mathscr{E}_n(t)\in \mathbb{Z}[t]$, and observe that for a fixed $n$ all Chern numbers (as well as other numerical invariants) of all smooth hypersurfaces in…
A permutation is called layered if it consists of the disjoint union of substrings (layers) so that the entries decrease within each layer, and increase between the layers. We find the generating function for the number of permutations on…
Repetitive patterns are ubiquitous in natural and human-made objects, and can be created with a variety of tools and methods. Manual authoring provides unmatched degree of freedom and control, but can require significant artistic expertise…
In this paper, we investigate Fano manifolds whose Chern characters satisfy some positivity conditions. We prove that such manifolds admit long chains of higher order minimal families of rational curves and are covered by higher rational…
Form an $n \times n$ matrix by drawing entries independently from $\{\pm1\}$ (or another fixed nontrivial finitely supported distribution in $\mathbf{Z}$) and let $\phi$ be the characteristic polynomial. Conditionally on the extended…
There is an explicit formula expressing the Chern-Schwartz-MacPherson class of a hypersurface in a nonsingular variety (in characteristic $0$) in terms of the Segre class of its jacobian subscheme; this has been known for a number of years.…
Recent deep learning based approaches have achieved great success on handwriting recognition. Chinese characters are among the most widely adopted writing systems in the world. Previous research has mainly focused on recognizing handwritten…
In this notes we classify toric Fano 4-folds having positive second Chern Character.
An expression for the coefficients of a linear iterative equation in terms of the parameters of the source equation is given both for equations in standard form and for equations in reduced normal form. The operator generating an iterative…
Chinese calligraphy is the writing of Chinese characters as an art form performed with brushes so Chinese characters are rich of shapes and details. Recent studies show that Chinese characters can be generated through image-to-image…
This paper continues the authors' work on the question of unitary equivalence of matrices with entries in the complex-valued functions of a topological space (matrices over spaces). Specifically, we here consider the question of unitary…
We present higher Chern-Simons theories based on (2-)crossed modules. We start from the generalized differential forms in Generalized Differential Calculus and define the corresponding generalized connections which consist of higher…
Chen's iterated integrals are treated within synthetic differential geometry. The main result is that iterated integrals produce a subcomplex of the de Rham complex on the free path space as well as based path spaces.
We study intersection theory and Chern classes of reflexive sheaves on normal varieties. In particular, we define generalization of Mumford's intersection theory on normal surfaces to higher dimensions. We also define and study the second…
For even dimensional manifolds, we prove some twisted anomaly cancellation formulas which generalize some well-known cancellation formulas. For odd dimensional manifolds, we obtain some modularly invariant characteristic forms by the…
The Chern-Simons perturbation theory gives an invariant $d(M,\rho)$ for a pair of a closed oriented 3-manifold $M$ and a representation $\rho$ of the fundamental group. We generalize $d(M,\rho)$ for compact oriented 3-manifolds with torus…