ISSN 1439-8516 CN 11-2039/O1
This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature, and in which rational homogeneous spaces play a prominent role. This selection is largely arbitrary and mainly reflects the interests of the author.
In this paper, we present the singular supercritical Trudinger-Moser inequalities on the unit ball B in Rn, where n ≥ 2. More precisely, we show that for any given α > 0 and 0 < t < n, then the following two inequalities hold for ∀ u ∈ W0,r1,n(B),
We also consider the problem of the sharpness of the constant αn,t. Furthermore, by employing the method of estimating the lower bound and using the concentration-compactness principle, we establish the existence of extremals. These results extend the known results when t = 0 to the singular version for 0 < t < n.
In this paper, first we introduce n-polygroups and characterize 2-polygroups of order 4 up to isomorphism. Then using 2-polygroups we introduce 2-Krasner hyperfields and we show that there exactly exists one 2-Krasner hyperfield of order 4. Moreover, we propose a hyperfield of order 4 which is not as a quotient hyperfield F/G. Finally, some programs written in MATLAB which are based on obtained results compute the number of polygroups, weak polygroups and Krasner hyperfields of order 4 up to isomorphism.
Let A be a symmetric and positive definite (1, 1) tensor on a bounded domain Ω in an ndimensional metric measure space (Rn, < , > , e-ψdv). In this paper, we investigate the Dirichlet eigenvalue problem of a system of equations of elliptic operators in weighted divergence form
where LA,ψ = div(A∇(·))-<Aψ, ∇(·)>, α is a nonnegative constant and u is a vector-valued function. Some universal inequalities for eigenvalues of this problem are established. Moreover, as applications of these results, we give some estimates for the upper bound of ςk+1 and the gap of ςk+1 -ςk in terms of the first k eigenvalues. Our results contain some results for the Lamé system and a system of equations of the drifting Laplacian.
A path factor of G is a spanning subgraph of G such that its each component is a path. A path factor is called a P≥n-factor if its each component admits at least n vertices. A graph G is called P≥n-factor covered if G admits a P≥n-factor containing e for any e ∈ E(G), which is defined by [Discrete Mathematics, 309, 2067-2076 (2009)]. We first define the concept of a (P≥n, k)-factor-critical covered graph, namely, a graph G is called (P≥n, k)-factor-critical covered if G-D is P≥n-factor covered for any D ⊆ V (G) with |D| = k. In this paper, we verify that (i) a graph G with κ(G) ≥ k + 1 is (P≥2, k)-factor-critical covered if bind(G) > (2+k)/3; (ii) a graph G with |V (G)| ≥ k + 3 and κ(G) ≥ k + 1 is (P≥3, k)-factor-critical covered if bind(G) ≥ (4+k)/3.
We know that in Ringel-Hall algebra of Dynkin type, the set of all skew commutator relations between the iso-classes of indecomposable modules forms a minimal Gröbner-Shirshov basis, and the corresponding irreducible elements forms a PBW type basis of the Ringel-Hall algebra. We aim to generalize this result to the derived Hall algebra DH(An) of type An. First, we compute all skew commutator relations between the iso-classes of indecomposable objects in the bounded derived category Db(An) using the Auslander-Reiten quiver of Db(An), and then we prove that all possible compositions between these skew commutator relations are trivial. As an application, we give a PBW type basis of DH(An).
Let H2(D2) be the Hardy space over the bidisk D2, and let Mψ,φ = [(ψ(z) - φ(w))2] be the submodule generated by (ψ(z) - φ(w))2, where ψ(z) and φ(w) are nonconstant inner functions. The related quotient module is denoted by Nψ,φ = H2(D2) ? Mψ,φ. In this paper, we give a complete characterization for the essential normality of Nψ,φ. In particular, if ψ(z) = z, we simply write Mψ,φ and Nψ,φ as Mφ and Nφ respectively. This paper also studies compactness of evaluation operators L(0)|Nφ and R(0)|Nφ, essential spectrum of compression operator Sz on Nφ, essential normality of compression operators Sz and Sw on Nφ.
In this study, by using moving frame along frontal of Legendre curve, we define frontal partner curves on unit sphere S2. We give the relationships between curvatures of Legendre curves and frontal partner curves are strengthen by an example.
