The Moduli Space of Points in the Boundary of Complex Hyperbolic Space

We consider the space M(n,m)\mathcal{M}(n,m) of ordered m-tuples of distinct points in the boundary of complex hyperbolic n-space, H_\mathbbCⁿ\mathbf{H}_{\mathbb{C}}^{n}, up to its holomorphic isometry group PU(n,1). An important problem in complex hyperbolic geometry is to construct and describe the moduli space for M(n,m)\mathcal{M}(n,m). In particular, this is motivated by the study of the deformation space of complex hyperbolic groups generated by loxodromic elements. In the present paper, we give the complete solution to this problem.     

The Moduli Space of Complex Geodesics in the Complex Hyperbolic Plane

We consider the space M\mathcal{M} of ordered m-tuples of distinct complex geodesics in complex hyperbolic 2-space, H_\mathbbC²{\rm\bf H}_{\mathbb{C}}^{2}, up to its holomorphic isometry group PU(2,1). One of the important problems in complex hyperbolic geometry is to construct and describe the moduli space for M\mathcal{M}. This is motivated by the study of the deformation space of groups generated by reflections in complex geodesics. In the present paper, we give the complete solution to this problem.     

The Equivalence of Ishikawa-Mann and Multistep Iterations in Banach Space

Let E be a real Banach space and T be a continuous Φ-strongly accretive operator. By using a new analytical method, it is proved that the convergence of Mann, Ishikawa and three-step iterations are equivalent to the convergence of multistep iteration. The results of this paper extend the results of Rhoades and Soltuz in some aspects.     

Pruitt’s Estimates in Banach Space

Pruitt’s estimates on the expectation and the distribution of the time taken by a random walk to exit a ball of radius r are extended to the infinite-dimensional setting. It is shown that they separate into two pairs of estimates depending on whether the space is type 2 or cotype 2. It is further shown that these estimates characterize type 2 and cotype 2 spaces.     

Convergence criterion and convergence ball of the Newton-type method in Banach space

The paper is concerned with the convergence problem of third-order Newton-type methods for finding zeros of nonlinear operations in Banach spaces. Under the hypothesis that the derivative of f satisfies the weak Lipschitz condition with L-average, the convergence criterion and convergence ball are given. Furthermore, some corollaries are obtained by applying the main results to some special functions L.     

Cohomology of Moduli Space of Cubic Fourfolds Ⅰ

In this paper we compute the cohomology of moduli space of cubic fourfolds with ADE type singularities relying on Kirwan’s blowup and Laza’s GIT construction. More precisely, we obtain the Betti numbers of Kirwan’s resolution of the moduli space. Furthermore, by applying decomposition theorem we obtain the Betti numbers of the intersection cohomology of Baily-Borel compactification of the moduli space.     

Applications of Automata and Graphs: Labeling-Operators in Hilbert Space I

We show that certain representations of graphs by operators on Hilbert space have uses in signal processing and in symbolic dynamics. Our main result is that graphs built on automata have fractal characteristics. We make this precise with the use of Representation Theory and of Spectral Theory of a certain family of Hecke operators. Let G be a directed graph. We begin by building the graph groupoid $\Bbb{G}$ induced by G, and representations of  $\Bbb{G}$ . Our main application is to the groupoids defined from automata. By assigning weights to the edges of a fixed graph G, we give conditions for $\Bbb{G}$ to acquire fractal-like properties, and hence we can have fractaloids or G-fractals. Our standing assumption on G is that it is locally finite and connected, and our labeling of G is determined by the “out-degrees of vertices”. From our labeling, we arrive at a family of Hecke-type operators whose spectrum is computed. As applications, we are able to build representations by operators on Hilbert spaces (including the Hecke operators); and we further show that automata built on a finite alphabet generate fractaloids. Our Hecke-type operators, or labeling operators, come from an amalgamated free probability construction, and we compute the corresponding amalgamated free moments. We show that the free moments are completely determined by certain scalar-valued functions.     

Map Preserving Local Properties of a Digital Image

In order to study digital topological properties of a k-surface in Z ⁿ , we generalize the topological number in Bertrand (Pattern Recogn. Lett. 15:1003–1011, 1994). Furthermore, we show that a local (k ₀,k ₁)-isomorphism preserves some digital-topological properties, such as a generalized topological number and a simple k ₀-point, and prove that a local (k ₀,k ₁)-isomorphism takes a simple k ₀-surface in into a simple k ₁-surface in .      

Additive Maps Preserving Similarity of Operators on Banach Spaces

Let X be an infinite-dimensional complex Banach space and denote by B（X） the algebra of all bounded linear operators acting on X. It is shown that a surjective additive map Φ from B（X） onto itself preserves similarity in both directions if and only if there exist a scalar c, a bounded invertible linear or conjugate linear operator A and a similarity invariant additive functional ψ on B（X） such that either Φ（T） = cATA^-1 ＋ ψ（T）I for all T, or Φ（T） = cAT＊A^-1 ＋ ψ（T）I for all T. In the case where X has infinite multiplicity, in particular, when X is an infinite-dimensional Hilbert space, the above similarity invariant additive functional ψ is always zero.     

Revisit of the Theorem of Image Trajectories in the Earth-Moon Space

On the occasion of the 50th anniversary of the theorem of the image trajectories in the Earth-Moon space, the author revisits the theorem and clarifies the relation between the class of image trajectories and the class of symmetric free-return trajectories, which were employed in the Apollo program. In a nutshell, the symmetric free-return trajectories are those image trajectories that intersect the Earth-Moon axis orthogonally at some point above the far side of the Moon. Optimization implications are pointed out.     

Modified Extragradient Methods for a System of Variational Inequalities in Banach Spaces

In this paper, we introduce a new system of general variational inequalities in Banach spaces. We establish the equivalence between this system of variational inequalities and fixed point problems involving the nonexpansive mapping. This alternative equivalent formulation is used to suggest and analyze a modified extragradient method for solving the system of general variational inequalities. Using the demi-closedness principle for nonexpansive mappings, we prove the strong convergence of the proposed iterative method under some suitable conditions.     

Determinant Bundle Over the Universal Moduli Space of Principal Bundles Over the Teichmüller Space

For a Riemann surface X and the moduli of regularly stable G-bundles M, there is a naturally occuring “adjoint” vector bundle over X × M. One can take the determinant of this vector bundle with respect to the projection map onto M. Our aim here is to study the curvature of the determinant bundle as the conformal structure on X varies over the Teichmüller space.     

On Formal Maps Between Generic Submanifolds in Complex Space

Let H:(M,p)→(M ^′,p ^′) be a formal mapping between two germs of real-analytic generic submanifolds in ? ^N with nonvanishing Jacobian. Assuming M to be minimal at p and M ^′ holomorphically nondegenerate at p ^′, we prove the convergence of the mapping H. As a consequence, we obtain a new convergence result for arbitrary formal maps between real-analytic hypersurfaces when the target does not contain any holomorphic curve. In the case when both M and M ^′ are hypersurfaces, we also prove the convergence of the associated reflection function when M is assumed to be only minimal. This allows us to derive a new Artin type approximation theorem for formal maps of generic full rank.     

Growth and Integrability of Fourier Transforms on Euclidean Space

A fundamental theme in classical Fourier analysis relates smoothness properties of functions to the growth and/or integrability of their Fourier transform. By using a suitable class of \(L^{p}\) -multipliers, a rather general inequality controlling the size of Fourier transforms for large and small argument is obtained. As consequences, quantitative Riemann–Lebesgue estimates are obtained and an integrability result for the Fourier transform is developed extending ideas used by Titchmarsh in the one dimensional setting.     

Sharp Estimates of the Constants of Equivalence Between Integral Moduli of Smoothness and K-functionals in the Multivariate Case

The equivalence between various types of moduli of smoothness and respective Peetre K-functionals has been actively explored since the 1960s, in view of the importance of this topic for revealing connections among approximation theory, functional analysis and operator theory. The existence of the embedding constants in this equivalence relation (together with one-sided estimates for these constants) has been established in great generality, but the derived one-sided bounds are rather coarse (see, e.g., Johnen and Scherer, in Constructive Theory of Functions of Several Variables. Proc. Conf., Math. Res. Inst. Oberwolfach, 1976, pp. 119–140, Springer, Berlin, 1977 and the references therein). The problem of finding the sharp embedding constants for this equivalence was posed in Dechevsky, C. R. Acad. Bulg. 42(2), 21–24, 1989 and Int. J. Pure Appl. Math. 33(2), 157–186, 2006, where this problem was solved in the particular case of L ₂-metric, for real-valued and complex-valued functions of one real variable, with definition domain Ω=ℝ or \varOmega = \mathbbT\varOmega =\mathbb{T} (the periodic case). In the present paper we extend the results of Dechevsky to the case of several real variables: Ω=ℝ ⁿ or \varOmega = \mathbbTⁿ\varOmega =\mathbb{T}^{n} , n∈ℕ. We consider two different types of equivalent norms for the Sobolev spaces involved in the K-functional (with and without intermediate mixed partial derivatives) and obtain a separate set of sharp two-sided bounds for the embedding constants in each of these two cases. We also briefly outline how the approach of the present study can be extended to the case of n-dimensional Lie (semi)groups.     

An iterative method of fixed points for closed and quasi-strict pseudo-contractions in Banach spaces

The purpose of this article is to propose a modified hybrid projection algorithm and prove a strong convergence theorem for closed and quasi-strict pseudo-contractions. Its results hold in reflexive, strictly convex and smooth Banach spaces with the property (K). The results of this paper improve and extend the corresponding results of Matsushita and Takahashi (J. Approx. Theory 134:257–266, 2005), Qin and Su (Nonlinear Anal. 67:1958–1965, 2007), Marino and Xu (J. Math. Anal. Appl. 329:336–346, 2007) and others.     

Existence and iterative algorithm of solutions for a system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems in Banach spaces

In this paper, we introduce and study a new system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems and its related auxiliary problems in reflexive Banach spaces. The auxiliary principle technique is applied to study the existence and iterative algorithm of solutions for the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems. Firstly, we prove the existence and uniqueness of solutions of the auxiliary problems for the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems. Secondly, an iterative algorithm for solving the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems is constructed by using this existence and uniqueness result. Finally, we show the existence of solutions of the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems and discuss the convergence analysis of this algorithm. These results improve, unify and generalize many corresponding known results given in literatures.     

The First Exit Time of a Brownian Motion from the Minimum and Maximum Parabolic Domains

Consider a Brownian motion starting at an interior point of the minimum or maximum parabolic domains, namely, D_min={(x,y₁,y₂):||x|| < min{(y₁+1)^1/p₁,(y₂+1)^1/p₂}}D_{\min}=\{(x,y_{1},y_{2}):\|x\|< \min\{(y_{1}+1)^{1/p_{1}},(y_{2}+1)^{1/p_{2}}\}\} and D_max={(x,y₁,y₂):||x|| < max{(y₁+1)^1/p₁,\allowbreak(y₂+1)^1/p₂}}D_{\max}=\{(x,y_{1},y_{2}):\|x\|<\max\{(y_{1}+1)^{1/p_{1}},\allowbreak(y_{2}+1)^{1/p_{2}}\}\} in R ^d+2,d≥1, respectively, where ‖⋅‖ is the Euclidean norm in R ^d , y ₁,y ₂≥−1, and p ₁,p ₂>1. Let t_Dmin\tau_{D_{\min}} and t_Dmax\tau_{D_{\max}} denote the first times the Brownian motion exits from D _min and D _max. Estimates with exact constants for the asymptotics of $\log P(\tau_{D_{\min}}>t)$\log P(\tau_{D_{\min}}>t) and $\log P(\tau_{D_{\max}}>t)$\log P(\tau_{D_{\max}}>t) are given as t→∞, depending on the relationship between p ₁ and p ₂, respectively. The proof methods are based on Gordon’s inequality and early works of Li, Lifshits, and Shi in the single general parabolic domain case.