International Journal of Mathematics

We construct the homogeneous Einstein equation for generalized flag manifolds G/K of a compact simple Lie group G whose isotropy representation decomposes into five inequivalent irreducible Ad(K)-submodules. To this end, we apply a new technique which is based on a fibration of a flag manifold over another such space and the theory of Riemannian submersions. We classify all generalized flag manifolds with five isotropy summands, and we use Gröbner bases to study the corresponding polynomial systems for the Einstein equation. For the generalized flag manifolds E₆/(SU(4) × SU(2) × U(1) × U(1)) and E₇/(U(1) × U(6)) we find explicitly all invariant Einstein metrics up to isometry. For the generalized flag manifolds SO(2ℓ + 1)/(U(1) × U(p) × SO(2(ℓ - p - 1) + 1)) and SO(2ℓ)/(U(1) × U(p) × SO(2(ℓ - p - 1))) we prove existence of at least two non-Kähler–Einstein metrics. For small values of ℓ and p we give the precise number of invariant Einstein metrics.

Two link diagrams on compact surfaces are strongly equivalent if they are related by Reidemeister moves and orientation preserving homeomorphisms of the surfaces. They are stably equivalent if they are related by the two previous operations and adding or removing handles. Turaev and Turner constructed a link homology for each stable equivalence class by applying an unoriented topological quantum field theory (TQFT) to a geometric chain complex similar to Bar-Natan's one. In this paper, by using an unoriented homotopy quantum field theory (HQFT), we construct a link homology for each strong equivalence class. Moreover, our homology yields an invariant of links in the oriented I-bundle of a compact surface.

We give a version of the Łojasiewicz inequality for the real polynomials on non-compact domains. The inequality takes in account not only distance to a fiber, but also distance to a polar set. It improves the recent results of [D. S. Tiep, H. H. Vui and N. T. Thao, Łojasiewicz inequality for polynomial functions on non-compact domains, Int. J. Math.23(4) (2012), Article ID: 1250033, 28 pp., doi:10.1142/S0129167X12500334], since we consider a distance to a smaller set. Then we use this new version of the inequality to obtain a sufficient condition for the existence of a vanishing component at infinity for real polynomials in several variables.

We study limiting distribution of the sequence of pull-backs of smooth (1, 1) forms and positive closed currents by meromorphic endomorphisms of compact Kähler manifolds.

We generalize results on the connection between existence and uniqueness of integrals and representation theoretic properties for Hopf algebras and compact groups. For this, given a coalgebra C, we study analogues of the existence and uniqueness properties for the integral functor Hom^C(C, -), which generalizes the notion of integral in a Hopf algebra. We show that the coalgebra C is co-Frobenius if and only if dim(Hom^C(C, M)) = dim(M) for all finite dimensional right (left) comodules M. As applications, we give a few new categorical characterizations of co-Frobenius, quasi-co-Frobenius (QcF) coalgebras and semiperfect coalgebras, and re-derive classical results of Lin, Larson, Sweedler and Sullivan on Hopf algebras. We show that a coalgebra is QcF if and only if the category of left (right) comodules is Frobenius, generalizing results from finite dimensional algebras, and we show that a one-sided QcF coalgebra is two-sided semiperfect. We also construct a class of examples derived from quiver coalgebras to show that the results of the paper are the best possible. Finally, we examine the case of compact groups, and note that algebraic integrals can be interpreted as certain skew-invariant measure theoretic integrals on the group.

Using the geometry of geodesics, we discuss the global aspects of complete real hypersurfaces in hyperbolic spaces of constant holomorphic sectional curvature over any division algebra 𝕂. Our assumption is that the shape operator and the curvature transformation with respect to the normal unit have the same eigenspaces. Note that we do not assume constancy of the principal curvatures. Under this assumption, we give a complete global classification of such hypersurfaces. Since the argument is purely geometric, we need not vary the argument for different base algebras. The foliations of with totally geodesic leaves called 𝕂-lines play an important role.

Let f be a holomorphic germ on ℂ² that fixes the origin and is tangent to the identity. Assume that f has a non-degenerate characteristic direction [v]. Hakim gave conditions that guarantee the existence of attracting domains along [v], however, when f has only one characteristic direction, these conditions are not satisfied. We prove that when [v] is unique, the existence results still hold. In particular, there is a domain Ω whose points converge to the origin along [v] and, on Ω, f is conjugate to a translation. Furthermore, if f is a global automorphism, the corresponding domain of attraction is a Fatou–Bieberbach domain.

In this paper, a discrete version of continuous non-autonomous predator–prey model with delays is investigated. By using Gaines and Mawhin's continuation theorem of coincidence degree theory and the method of Lyapunov function, some sufficient conditions for the existence and globally asymptotically stability of positive periodic solution of difference equations in consideration are established. Finally, some numerical examples are given to verify the theoretical analysis.

Singular Finsler metrics, such as Kropina metrics and m-Kropina metrics, have a lot of applications in the real world. In this paper, we classify a class of singular (α, β)-metrics which are locally projectively flat with constant flag curvature in dimension n = 2 and n ≥ 3 respectively. Further, we determine the local structure of m-Kropina metrics and particularly Kropina metrics which are projectively flat with constant flag curvature and prove that such metrics must be locally Minkowskian but are not necessarily flat-parallel.