ISSN 1439-8516 CN 11-2039/O1
We study a Khovanov type homology close to the original Khovanov homology theory from Frobenius system. The homology is an invariant for oriented links up to isotopy by applying a tautological functor on the geometric complex. The homology has also geometric descriptions by introducing the genus generating operations. We prove that Jones Polynomial is equal to a suitable Euler characteristic of the homology groups. As an application, we compute the homology groups of(2, k)-torus knots for every k∈N.
The old result due to [Ozaki, S.:On the theory of multivalent functions Ⅱ. Sci. Rep. Tokyo Bunrika Daigaku Sect. A, 45-87(1941)], says that if f(z)=zp+Σn=p+1∞anzn is analytic in a convex domain D and for some real α we have Re{exp(iα)f(p)(z)}>0 in D, then f(z) is at most p-valent in D. In this paper, we consider similar problems in the unit disc D={z∈C:|z|<1}.
In this paper, we first introduce Lσ1-(log L)σ2 conditions satisfied by the variable kernels Ω(x, z) for 0≤σ1≤1 and σ2≥0. Under these new smoothness conditions, we will prove the boundedness properties of singular integral operators TΩ, fractional integrals TΩ,α and parametric Marcinkiewicz integrals μΩρ with variable kernels on the Hardy spaces Hp(Rn) and weak Hardy spaces WHp(Rn). Moreover, by using the interpolation arguments, we can get some corresponding results for the above integral operators with variable kernels on Hardy-Lorentz spaces Hp,q(Rn) for all p< q< ∞.
Let A be a small abelian category. For a closed subbifunctor F of ExtA1(-,-), Buan has generalized the construction of Verdier's quotient category to get a relative derived category, where he localized with respect to F-acyclic complexes. In this paper, the homological properties of relative derived categories are discussed, and the relation with derived categories is given. For Artin algebras, using relative derived categories, we give a relative version on derived equivalences induced by F-tilting complexes. We discuss the relationships between relative homological dimensions and relative derived equivalences.
In this paper, we prove that a product F1×F2 of saturated fusion systems is exotic if and only if at least one of the factors is exotic. This result provides a method to construct new exotic fusion systems by known exotic fusion systems. We also investigate central products of saturated fusion systems.
We consider the nonlinear difference equations of the form
Lu=f(n, u), n∈Z,
where L is a Jacobi operator given by(Lu)(n)=a(n)u(n+1)+a(n-1)u(n-1)+b(n)u(n) for n∈Z, {a(n)} and {b(n)} are real valued N-periodic sequences, and f(n, t) is superlinear on t. Inspired by previous work of Pankov [Discrete Contin. Dyn. Syst., 19, 419-430(2007)] and Szulkin and Weth [J. Funct. Anal., 257, 3802-3822(2009)], we develop a non-Nehari manifold method to find ground state solutions of Nehari-Pankov type under weaker conditions on f. Unlike the Nehari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold by using the diagonal method.
Motivated by the theory of isoparametric hypersurfaces, we study submanifolds whose tubular hypersurfaces have some constant higher order mean curvatures. Here a k-th order mean curvature Qkν(k≥1) of a submanifold Mn is defined as the k-th power sum of the principal curvatures, or equivalently, of the shape operator with respect to the unit normal vector ν. We show that if all nearby tubular hypersurfaces of M have some constant higher order mean curvatures, then the submanifold M itself has some constant higher order mean curvatures Qkν independent of the choice of ν. Many identities involving higher order mean curvatures and Jacobi operators on such submanifolds are also obtained. In particular, we generalize several classical results in isoparametric theory given by E. Cartan, K. Nomizu, H. F. Münzner, Q. M. Wang, et al. As an application, we finally get a geometrical filtration for the focal submanifolds of isoparametric functions on a complete Riemannian manifold.
In this paper, we try to describe the relationship between the differentiability of a quasisymmetric homeomorphism and the local Hausdorff dimension of the quasiline at a point.
In this article, we consider the entropy-expansiveness of geodesic flows on closed Riemannian manifolds without conjugate points. We prove that, if the manifold has no focal points, or if the manifold is bounded asymptote, then the geodesic flow is entropy-expansive. Moreover, for the compact oriented surfaces without conjugate points, we prove that the geodesic flows are entropy-expansive. We also give an estimation of distance between two positively asymptotic geodesics of an uniform visibility manifold.
