This paper is devoted to studing Bergman spaces induced by regular-weight A_{ω1,2}^{p}(M) (1 < p < ∞) on annular and positive Toeplitz operators on these spaces. The dual spaces of Bergman spaces induced by regular-weight are characterized. We also obtain equivalent conditions for boundedness and compactness of positive Toeplitz operators between these regular-weighted Bergman spaces.
In this paper, the numerical radius of bounded block operator matrices on Hilbert space is studied. First, the generalized form of numerical radius inequalities of off-diagonal block operator matrix is studied, and taking advantage of the unitary similarity invariance of numerical radius and the generalized mixed Schwarz inequality, the inequalities of the numerical radius of sum of two bounded linear operators are considered. Then, numerical radius inequalities for 2×2 bounded block operator matrices are given. Finally, the conclusion is applied in the bounded infinite dimensional Hamiltonian operator and the inequalities of its numerical radius are obtained.
Image registration is fundamental to image processing. The vector field regularization model performs relatively well among a large number of registration methods. However, it still can't correspond to all interested regions across images correctly. Therefore, we hope to study the theory of the vector field regularization model to see whether there are some problems with the design of the model. Moreover, as there are two unknowns which are related by an initial value problem in the regularization model, it is novel in mathematics. The vector field regularization model takes the form min_{v} {α||v||_{H}^{2} + ρ(T (y^{v}(τ)), S)}, where T is a template image, S is a reference image, y^{v}(τ):x ? y^{v}(τ;0, x) is a transformation determined by the solution y^{v}(s;0, x) of the initial value problem dy/ds=v(s, y), y(0)=x, ρ is a similarity functional, α> 0 is a regularization parameter and H is a Hilbert space. In this paper, we firstly show the vector field regularization model has stable solutions and then demonstrate its convergence. The above results can be obtained by the standard arguments of regularized problems together with the convergence relation of y^{v}(τ) and v. However, the requirements for ρ, S and T are relatively strong under the existing regularization theory. We give relatively weak conditions for ρ, S and T by taking full advantage of the good properties of y^{v}(τ). In addition, we verify that three commonly used similarity functionals in image registration satisfy the given conditions.
Let ?_{n} ∈ C (D), ?_{j} ∈ C (T) and K ≥ 1, where n ≥ 2 is an integer, j ∈ {1,..., n -1}. In this paper, we establish a Schwarz-Pick type inequality for the K-quasiconformal self-mapping f of the unit disk D satisfying the inhomogeneous polyharmonic equation Δ^{n}f=?_{n} with the associated Dirichlet boundary value condition:Δ^{n-1}f|_{T}=?_{n-1},..., Δ^{1}f|_{T}=?_{1} and f(0)=0. Furthermore, we prove that this result is asymptotically sharp in the sense that||?_{j}||_{∞} → 0 (j=1,..., n) and K → 1^{+}, where||?_{n}||_{∞}:=sup_{z∈D}|?_{n}(z)|and||?_{j}||_{∞}:=sup_{z∈T}|?_{j}(z)|(j=1, 2,..., n -1).
This paper investigates the initial value problem for a class of set differential equations in Fréchet space F. Based on that the set K_{c}(F) of all compact convex subsets of a Fréchet space F is considered as a projective limit of semilinear metric spaces K_{c}(E^{i}), and the properties of projective limit, we introduce the notions of the Fréchet partial derivative, hyperconcave and hyperconvex of set-valued functions. By using the method of quasilinearization and comparison principle, we construct two monotone iterative sequences in K_{c}(F), and obtain the sequences of approximate solutions which converge uniformly and rapidly to the unique solution of the problem. The obtained results enrich and develop the theory of set-valued differential equations in Fréchet space F.
A non-increasing sequence π=(d_{1},...,d_{n} of nonnegative integers is said to be graphic if it is realizable by a simple graph G on n vertices. A graphic sequence π=(d_{1},...,d_{n} is said to be potentially _{3}C_{l}-graphic if there is a realization of π containing cycles of every length r, 3 ≤ r ≤ l. It is well-known that if the nonincreasing degree sequence (d_{1},..., d_{l}) of a graph G on l vertices satisfies the Pósa condition that d_{l +1-i} ≥ i + 1 for every i with 1 ≤ i < l/2, then G is either pancyclic or bipartite. In this paper, we obtain a Pósa-type condition of potentially _{3}C_{l}-graphic sequences, that is, we prove that if l ≥ 5 is an integer, n ≥ l and π=(d_{1},...,d_{n} is a graphic sequence with d_{l +1-i} ≥ i + 1 for every i with 1 ≤ i < l/2, then π is potentially _{3}C_{l}-graphic. We show that this result is an asymptotic solution to a problem due to Li et al.[Adv. Math. (China), 2004, 33(3):273-283]. As an application, we also show that this result completely implies the value σ(C_{l}, n) for l ≥ 5 and n ≥ l due to Lai[J. Combin. Math. Combin. Comput., 2004, 49:57-64].
Let P be an isolated singularity of multiplicity 4 of a complex surface Y. It is well-known that there is a locally irreducible finite covering π:(Y, P) → (X, p) with π^{-1}(p)=P, and a Jung's resolution f:? → Y. Let W_{p} be the exceptional divisor of (π?f)^{-1}(p). We will prove that W_{p} has a unique decomposition into fundamental cycles W_{p}=2Z_{1} or W_{p}=Σ_{α=1}^{l} Z_{α} satisfying some conditions. We will define a local index w_{p} for π at p and compute it by the above decomposition of W_{p}. In particular, we will show that (Y, P) is singular iff w_{p} ≥ 1. As another application of the decomposition of W_{p}, we also compute the number of blown-downs needed to get the minimal resolution from ?.
An optimal planar drawing of a graph is an embedding in the plane so that the number of crossings is as small as possible. The number of crossings in an optimal planar drawing of a graph G is the crossing number cr(G) of G. A graph is k-planar if it can be embedded in the planar so that each edge is crossed at most k times. Zhang et al. (2012) proved that the crossing number of any 1-planar graph on n vertices is at most n -2, and this upper bound is best possible. Czap, Harant and Hudák (2014) proved that the crossing number of any 2-planar graph on n vertices is at most 5(n-2). In this paper, we give a better upper bound for the crossing number of 2-planar graphs and show from the point of view of combinatorics that K_{n} is 2-planar if and only if n ≤ 7 (surprisedly, this was an open problem until 2019, in when Angelini solved it with computer assistance).
We use analytic methods and properties of the classical Gauss sums to study the computational problems of some certain special symmetric Gauss sums, and give some new and interesting identities and second-order linear recurrence formulae for them.
This note is concerned with the effect of small Lipschitz perturbations of a discrete dynamical system in Banach spaces. Let f, g be continuous map from a Banach space X into itself. If f has regular nondegenrate snap-back repellers or heteroclinic cycles and g is a small Lipschitz perturbations of f, then g has regular nondegenrate snap-back repellers or heteroclinic cycles. In addition, the regular nondegenrate heteroclinic cycles implying the snap-back repellers is studied in complete metric spaces.
Suppose f, g, u ∈ ∩_{q>1} H^{q}, H_{f}, H_{g}, H_{u} are Hankel operators from the usual Hardy space of unit disk H^{2} to H^{2}. In this paper, we completely characterize when the product of three Hankel operators i>H_{f}H_{g}H_{u} on Hardy space has finite rank property, and we also give two nontrivial examples. Moreover, we describe the finite rank property of truncated Toeplitz operators defined on the model space.