A Complete Metric in the Function Space D α[0,∞)

Subspaces D of D[0,1] and of D[0,) are defined. With constructed metrics d , D are shown to be complete separable metric spaces. Conditions for tightness of probability measures in D are given.     

A Remark on the Dimension of the Bergman Space of Some Hartogs Domains

Let D be a Hartogs domain of the form \(D=\{(z,w)\in\mathbb{C}\times\mathbb{C}^{N}: \|w\| where u is a subharmonic function on ?. We prove that the Bergman space of holomorphic and square integrable functions on D is either trivial or infinite-dimensional.     

Stability Analysis for maxwell’s Equation with a Thermal Effect in one Space Dimension

In this paper we study the asymptotic behavior of a system modeling heating of material by microwaves. Various assumptions have been made, concerning complexity (nonhomogeneous structure) and the two-phase state of the material. The mathematical model includes Maxwell’s and heat–transfer equations. Stability of solutions of the system is shown.     

Propagation of Singularities for Schrödinger Equations on the Euclidean Space with a Scattering Metric

Given a scattering metric on the Euclidean space. We consider the Schrödinger equation corresponding to the metric, and study the propagation of singularities for the solution in terms of the “homogeneous wavefront set”. We also prove that the notion of the homogeneous wavefront set is essentially equivalent to that of the quadratic scattering wavefront set introduced by Wunsch (1999 Wunsch , J. ( 1999 ). Propagation of singularities and growth for Schrödinger operators . Duke Math. J. 98 : 137 – 186 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). One of the main results in Wunsch (1999 Wunsch , J. ( 1999 ). Propagation of singularities and growth for Schrödinger operators . Duke Math. J. 98 : 137 – 186 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) follows on the Euclidean space with a weaker, almost optimal condition on the potential.     

Irregular C1,β-Surfaces with an Analytic Metric

We prove that in the class C ^1, with <1/13 it is possible to continuously deform an analytic convex surface of positive Gaussian curvature (or a plane) so as to lose boundedness of the extrinsic curvature in the Pogorelov sense. We demonstrate how to replace the bound <1/13 with <1/7.     

A Geometrically Aberrant Banach Space with Uniformly Normal Structure

A.L. Browu [2] devised an equivalent norm ||·||_B on (l_2, ||·||) such that itsrestriction to M = { (α_1,α_2,…) ∈l_2, α_1 = 0} remains the original l_2-norm ||·||_2, and itsrestriction to M_k = span(e_l ,e_k), k≥3 is an l_p (k) -norm, where (e_k) is a naturalbasis on l_2 and p(k)→∞ as k-∞, p(3) = 16. J.R. Giles, B .Sims & S. Swamina-     

A Note on the BMO Space with Non-doubling Measure

Let μi be a positive Radon measure which only satisfies an appropriate growth condition, and RBMO(μ) be the BMO space associated with μ which was introduced by Tolsa. In this paper, it is proved that the definition of RBMO(ρ) can be weakened sufficiently.     

A Quasilinear System Related with the Asymptotic Equation of the Nematic Liquid Crystal’s Director Field*

In this paper, the author studies the local existence of strong solutions and their possible blow-up in time for a quasilinear system describing the interaction of a short wave induced by an electron field with a long wave representing an extension of the motion of the director field in a nematic liquid crystal’s asymptotic model introduced in [Saxton, R. A., Dynamic instability of the liquid crystal director. In: Current Progress in Hyperbolic Systems (Lindquist, W. B., ed.), Contemp. Math., Vol.100, Amer. Math.Soc., Providence, RI, 1989, pp.325–330] and [Hunter, J. K. and Saxton, R. A., Dynamics of director fields, SIAM J. Appl. Math., 51, 1991, 1498–1521] and studied in [Hunter, J. K. and Zheng, Y., On a nonlinear hyperbolic variational equation I, Arch. Rat. Mech. Anal.,129, 1995, 305–353], [Hunter, J. K. and Zheng, Y., On a nonlinear hyperbolic variational equation II, Arch. Rat. Mech. Anal., 129, 1995, 355–383] and in [Zhang, P. and Zheng,Y., On oscillation of an asymptotic equation of a nonlinear variational wave equation,Asymptotic Anal., 18, 1998, 307–327] and, more recently, in [Bressan, A., Zhang, P. and Zheng, Y., Asymptotic variational wave equations, Arch. Rat. Mech. Anal., 183, 2007,163–185].     

Fixed Points of a φ-G-Contractive Mapping with Respect to a c-Distance on an Abstract Metric Space Endowed with a Graph

Mathematical Notes - In this paper, we prove the existence of fixed points for any mapping that is a Banach G-contraction with respect to a c-distance on a cone metric space endowed with a graph.     

A Remark on Lower Bounds for the Chromatic Numbers of Spaces of Small Dimension with Metrics ℓ1 and ℓ2

Mathematical Notes - A particular class of estimates related to the Nelson–Erd?s–Hadwiger problem is studied. For two types of spaces, Euclidean and spaces with metric ?1,...     

A family of translation planes of order q 2m+1 with two orbits of length 2 and q 2m+1−1 on l ∞

A class of translation planes of order q ^2m+1, where q is an odd prime power and m1, is constructed. If m=1, then this class is contained in the class of order q ³ constructed by Hiramine [5]. These planes of order q ^2m+1 are of dimension 2m+1 over their kernels. If q ^2m+13³, then the linear translation complements of these planes have two orbits of length 2 and q ^2m+1–1 on l and this class contains many planes which are not generalized André planes. If q ^2m+1= 3³, then each plane of this class is isomorphic to the Hering plane of order 27.Dedicated to Professor Tuyosi Oyama on his 60th birthday     

A Classification of 3-dimensional Paracontact Metric Manifolds with φl=lφ

Let M~3 be a 3-dimensional paracontact metric manifold. Firstly, a classification of M~3 satisfying φQ = Qφ is given. Secondly, manifold M~3 satisfying φl = lφ and having η-parallel Ricci tensor or cyclic η-parallel Ricci tensor is studied.     

求解矩阵方程组A1XB1+C1XD1=E1,A2XB2+C2XD2=E2的迭代算法北大核心CSCD

应用共轭梯度方法和线性投影算子,给出迭代算法求解了线性矩阵方程组A1XB1＋C1XD1=E1,A2XB2＋C2XD2=E2在任意线性子空间上的约束解及其最佳逼近.可以证明,当矩阵方程组A1XB1＋C1XD1=E1,A2XB2＋C2XD2=E2相容时,所给迭代算法经过有限步迭代可得到矩阵方程组的约束解,极小范数解和最佳逼近.文中的数值例子证实了该算法的有效性.     

Completion of ■~2 with a Conformal Metric as a Closed Surface

In this paper, we obtain some asymptotic behavior results for solutions to the prescribed Gaussian curvature equation. Moreover, we prove that under a conformal metric in ■~2, if the total Gaussian curvature is 4π, the conformal area of ■~2 is finite and the Gaussian curvature is bounded, then ■~2 is a compact C~(1,α)surface after completion at ∞, for any α∈(0, 1). If the Gaussian curvature has a H?lder decay at infinity, then the completed surface is C~2. For radial solutions, the same regularity holds if the Gaussian curvature has a limit at infinity.     

Stability of O(p + 1) × O(p + 1)-Invariant Hypersurfaces with Zero Scalar Curvature in Euclidean Space

Equivariant geometry methods are used to study and classify zero scalarcurvature O(p + 1) × O(p + 1)-invariant hypersurfaces in ^{2p + 2}with p > 1. First of all the O(p + 1) × O(p + 1)-invariant hypersurfaces are classified according to their profile curves, and it is shown that there are complete and embedded examples. Next the Morse index of the complete examples is studied, in particular, it is proved that there exist globally stable examples.These stable examples provide counter-examples, in odddimensions greater than or equal to nine, to a Bernstein-type conjecture in the stable class, for immersions with zero scalar curvature.     

Asymptotic 2–periodic difference equations with diagonally self–invertible responses

We provide sufficient conditions which guarantee that all positive solutions of a nonlinear difference equation of third order is asymptotically periodic 2. An estimate of the width of such solutions in terms of the initial values is also provided     

The Arboreal Structure of the Metric Space X ≔ GL 2(ℚ p )/ℚ p ∗GL 2(ℤ p )

In this article we define an arboreal structure on the metric space X: = GL ₂(? _p )/? _p ?GL ₂(? _p ) and give an elementary proof of the fact that X is a (p + 1)-valent tree.     

On the Geometry of the （1, 1）-Tensor Bundle with Sasaki Type Metric

Curvature properties are studied for the Sasaki metric on the (1,1) tensor bundle of a Riemannian manifold. As an application, examples of almost para-Nordenian and para-K(a)hler-Nordenian B-metrics are constructed on the (1,1) tensor bundle by looking at the Sasaki metric. Also, with respect to the para-Nordenian B-structure, paraholomorphic conditions for the complete lifts of vector fields are analyzed.     

关于矩阵方程X+A*X-1A=P的解及其扰动分析

考虑非线性矩阵方程X＋A^＊（X^-1）A=P其中A是n阶非奇异复矩阵,P是n阶Hermite正定矩阵．本文给出了Hermite正定解和最大解的存在性以及获得最大解的一阶扰动界,改进了文[5,6]中的部分结论．