We prove some fixed point theorems in partially ordered sets, providing an extension of the Banach contractive mapping theorem. Having studied previously the nondecreasing case, we consider in this paper nonincreasing mappings as well as non monotone mappings. We also present some applications to first–order ordinary differential equations with periodic boundary conditions, proving the existence of a unique solution admitting the existence of a lower solution.
We give a summary on the recent development of chaos theory in topological dynamics, focusing on Li-Yorke chaos, Devaney chaos, distributional chaos, positive topological entropy, weakly mixing sets and so on, and their relationships.
In this paper, we continue to discuss the normality concerning omitted holomorphic function and get the following result. Let $$\mathcal{F}$$ F be a family of meromorphic functions on a domain D, k ≥ 4 be a positive integer, and let a(z) and b(z) be two holomorphic functions on D, where a(z) ≢ 0 and f(z) ≢ ∞ whenever a(z) = 0. If for any $$f \in \mathcal{F}$$ f ∈ F , f′(z) − a(z)f k (z) ≠ b(z), then $$\mathcal{F}$$ F is normal on D.
In this paper, we construct random two-faced families of matrices with non-Gaussian entries to approximate a bi-free central limit distribution with a positive definite covariance matrix. We prove that, under modest conditions weaker than independence, a family of random two-faced families of matrices with non-Gaussian entries is asymptotically bi-free from a two-faced family of constant diagonal matrices.
Let D be a finite and simple digraph with vertex set V(D). The minimum degree δ of a digraph D is defined as the minimum value of its out-degrees and its in-degrees. If D is a digraph with minimum degree δ and edge-connectivity λ, then λ ≤ δ. A digraph is maximally edge-connected if λ = δ. A digraph is called super-edge-connected if every minimum edge-cut consists of edges incident to or from a vertex of minimum degree. In this note we show that a digraph is maximally edge-connected or super-edge-connected if the number of arcs is large enough.
Basing upon the recent development of the Patterson-Sullivan measures with a Holder continuous nonzero potential function, we use tools of both dynamics of geodesic flows and geometric properties of negatively curved manifolds to present a new formula illustrating the relation between the exponential decay rate of Patterson-Sullivan measures with a Holder continuous potential function and the corresponding critical exponent.
The set of self-mapping degrees of S 3-geometry 3-manifolds in [2] have some mistakes when the fundamental group of the 3-manifold is D*4n , O*48, D′ n⋅2q or T′8⋅3q . So we need to make this errata.The self-mapping degrees of all closed and oriented 3-manifolds are listed in [1] and [5]. The table of spherical case in [5] is mostly quoted from [2]. The results in [5] which do not involve the corrections made here still valid. The results in [5] involving the corrections made here should be also changed.
In this paper we give a classification of special endomorphisms of nil-manifolds: Let f : N/Γ → N/Γ be a covering map of a nil-manifold and denote by A: N/Γ → N/Γ the nil-endomorphism which is homotopic to f. If f is a special TA-map, then A is a hyperbolic nil-endomorphism and f is topologically conjugate to A.
In this paper, we study the complete f-moment convergence for widely orthant dependent (WOD, for short) random variables. A general result on complete f-moment convergence for arrays of rowwise WOD random variables is obtained. As applications, we present some new results on complete f-moment convergence for WOD random variables. We also give an application to nonparametric regression models based onWOD errors by using the complete convergence that we established. Finally, the choice of the fixed design points and the weight functions for the nearest neighbor estimator are proposed, and a numerical simulation is provided to verify the validity of the theoretical result.
In this paper, we prove quasi-modularity property for the twisted Gromov-Witten theory of $$\mathcal{O}(3)$$ O ( 3 ) over ℙ2. Meanwhile, we derive its holomorphic anomaly equation.
We consider a fourth order nonlinear PDE involving the critical Sobolev exponent on a bounded domain of ℝ n , n ≥ 5 with Navier condition on the boundary. We study the lack of compactness of the problem and we provide an existence theorem through a new index formula.
The main purpose of this paper is to establish, using the Littlewood-Paley-Stein theory (in particular, the Littlewood-Paley-Stein square functions), a Calderón-Torchinsky type theorem for the following Fourier multipliers on anisotropic Hardy spaces H p (ℝ n ; A) associated with expensive dilation A: $${T_m}f(x) = \int_{{\mathbb{R}^n}} {m(\xi)\hat f(\xi){{\rm{e}}^{{\rm{2}}\pi {\rm{i}}x \cdot \xi}}d\xi}.$$ T m f ( x ) = ∫ ℝ n m ( ξ ) f ^ ( ξ ) e 2 π i x ⋅ ξ d ξ . Our main Theorem is the following: Assume that m(ξ) is a function on ℝ n satisfying $$\mathop {\sup}\limits_{j \in \mathbb{Z}} {\left\| {{m_j}} \right\|_{{W^s}({A^ *})}} \zeta _ - ^{- 1}\left({{1 \over p} - {1 \over 2}} \right)$$ s > ζ − − 1 ( 1 p − 1 2 ) Then T m is bounded from H p (ℝ n ; A) to H p (ℝ n ; A) for all 0 < p ≤ 1 and $${\left\| {{T_m}} \right\|_{H_A^p \to H_A^p}} \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle\sim}\vphantom{_x}}$}} \mathop {\sup}\limits_{j \in \mathbb{Z}} {\left\| {{m_j}} \right\|_{{W^s}({A^ *})}},$$ ‖ T m ‖ H A p → H A p ≲ sup j ∈ ℤ ‖ m j ‖ W s ( A ∗ ) , where A* denotes the transpose of A. Here we haveusedthe notations m j (ξ)= m(A* j ξ)φ(ξ)and is a suitable cut-off function on ℝ n , and W s (A*) is an anisotropic Sobolev space associated with expansive dilation A* on ℝ n .
We study property T for an action α of a discrete group Γ on a unital C *-algebra $$\mathscr{A}$$ A . Our main results improve some well-known results about property T for groups. Moreover, we introduce Hilbert $$\mathscr{A}$$ A -module property T and show that the action α has property T if and only if the reduced crossed product $$\mathscr{A}\;{\rtimes_{\alpha, r}}$$ A ⋊ α , r Γ has Hilbert $$\mathscr{A}$$ A -module property T.
In this note, we study a rich operator class denoted by $${\cal P}{{\cal B}_n}({\rm{\Omega}})$$ P B n ( Ω ) which includes all homogeneous operators and quasi-homogeneous operators in the Cowen-Douglas class. A complete unitarily classification theorem is given. Furthermore, we also concern the curvature and similarity of operators in $${\cal P}{{\cal B}_n}({\rm{\Omega}})$$ P B n ( Ω ) .
Let G be a graph with vertex set V(G), edge set E(G) and maximum degree Δ respectively. G is called degree-magic if it admits a labelling of the edges by integers {1, 2, …, |E(G)|} such that for any vertex v the sum of the labels of the edges incident with v is equal to $${{1 + \left| {E(G)} \right|} \over 2} \cdot d(v)$$ 1 + | E ( G ) | 2 ⋅ d ( v ) , where d(v) is the degree of v. Let f be a proper edge coloring of G such that for each vertex v ∈ V(G), |{e : e ∈ E v , f(e) ≤ Δ/2}| = |{e : e ∈ E v , f(e) > Δ/2}|, and such an f is called a balanced edge coloring of G. In this paper, we show that if G is a supermagic even graph with a balanced edge coloring and m ≥ 1, then (2m + 1)G is a supermagic graph. If G is a d-magic even graph with a balanced edge coloring and n ≥ 2, then nG is a d-magic graph. Results in this paper generalise some known results.
This paper studies the a[sup.3] .sub.0-shadowing property for the dynamics of diffeomorphisms defined on closed manifolds. The C .sup.1 interior of the set of all two dimensional diffeomorphisms with the a[sup.3] .sub.0-shadowing property is described by the set of all Anosov diffeomorphisms. The C .sup.1-stably a[sup.3] .sub.0-shadowing property on a non-trivial transitive set implies the diffeomorphism has a dominated splitting.
In this paper, we generalize the concept of asymptotic Hankel operators on $$H^2(\mathbb{D})$$ H 2 ( D ) to the Hardy space $$H^2(\mathbb{D}^n)$$ H 2 ( D n ) (over polydisk) in terms of asymptotic Hankel and partial asymptotic Hankel operators and investigate some properties in case of its weak and strong convergence. Meanwhile, we introduce i th-partial Hankel operators on $$H^2(\mathbb{D}^n)$$ H 2 ( D n ) and obtain a characterization of its compactness for n > 1. Our main results include the containment of Toeplitz algebra in the collection of all strong partial asymptotic Hankel operators on $$H^2(\mathbb{D}^n)$$ H 2 ( D n ) . It is also shown that a Toeplitz operator with symbol ϕ is asymptotic Hankel if and only if ϕ is holomorphic function in $$L^\infty(\mathbb{T}^n)$$ L ∞ ( T n ) .
The Gelfand-Kirillov dimension is an invariant which can measure the size of infinite-dimensional algebraic structures. In this article, we show that it can also measure the reducibility of scalar generalized Verma modules. In particular, we use it to determine the reducibility of scalar generalized Verma modules associated with maximal parabolic subalgebras in the Hermitian symmetric case.
