Equivalent Separating Curves on Surfaces and Its Applications to Haken Spheres

1 Introduction Let A be a collection of n pairwise disjoint simple closed curves on an orientable closed surface F of genus n ≥2. We say that A is a complete system of F if the surface obtained by cutting F along A is a 2n-punctured sphere. Let A1, A2 be two non-empty subsets of A. We say that (A1,A2) is a partition of A if A1 ∩A2 = (?) and A1 ∪A2 = A. Let (A1, A2) be a partition of A on F, and C a simple closed curve on F. We say that C is separating with respect to (A1, A2) if it is disjoint from A and it cuts F into two pieces F1,F2 with A1(?) F1,A2 (?) F2.     

Boundary reducible handle additions on simple 3-manifolds

Let M be a simple manifold, and F be a component of δM of genus two. For a slope γ on F, we denote by M（γ） the manifold obtained by attaching a 2-handle to M along a regular neighborhood of γon F. In this paper, we shall prove that there is at most one separating slope γ on F such that M（γ） is δ-reducible.     

On n-relator 3-manifolds Which Are Handlebodies

Let H be a handlebody, y = |∫1,……,∫n| a collection of 2-sided pairwise disjoint simple closed curves on eH. The 3-manifold obtained from H by attaching 2-handles to H .along the curves in y is called an n-relator 3-manifold, and is denoted as Hy. In tiffs paper, a sufficient and necessary condition for Hy to be a handlebody is described.     

A Lower Bound of the Genus of a Self-amalgamated 3-manifolds

Let M be a compact connected oriented 3-manifold with boundary, Q1, Q2 C 0M be two disjoint homeomorphic subsurfaces of cgM, and h ： Q1 → Q2 be an orientation-reversing homeomorphism. Denote by Mh or MQ1=Q2 the 3-manifold obtained from M by gluing Q1 and Q2 together via h. Mh is called a self-amalgamation of M along Q1 and Q2. Suppose Q1 and Q2 lie on the same component F1 of δM1, and F1 - Q1 ∪ Q2 is connected. We give a lower bound to the Heegaard genus of M when M＇ has a Heegaard splitting with sufficiently high distance.     

On the Mean Absolute Geodesic Curvature of Closed Curves in 2-Dimensional Space Forms

In this paper,we establish some formulas on closed curves in 2-dimensional space forms.Mean absolute geodesic curvature is introduced to describe the average curving of a closed curve.Inthis sense,a closed curve could be compared with a geodesic circle that is the boundary of a convex geodesic circular disk containing the closed curve.The comparison can be used to show some properties of space forms only on themselves.     

A Sufficient Condition for the Genus of an Annulus Sum of Two 3-manifolds to Be Non-degenerate

Let Mi be a compact orientable 3-manifold, and Ai a non-separating incompressible annulus on a component of δMi, say Fi, i = 1, 2. Let h ： A1 → A2 be a homeomorphism, and M→M1 ∪h M2, the annulus sum of Mi and M2 along A1 and A2. Suppose that Mi has a Heegaard splitting Vi ∪Si Wi with distance d（Si） ≥ 2g（Mi） ＋ 2g（F3-i） ＋ 1, i = 1, 2. Then g（M） = g（M1） ＋ g（M2）, and the minimal Heegaard splitting of M is unique, which is the natural Heegaard splitting of M induced from Vi∪S1 Wi and V2 ∪S2 W2.     

Holomorphic Curves into ■~N(C) That Share a Set of Moving Hypersurfaces

Let F be a family of holomorphic curves of a domain D in C into a closed subset X in ■~N(C). Let Q_1(z),…, Q_(2t+1)(z) be moving hypersurfaces in ■~N(C) located in pointwise t-subgeneral position with respect to X. If each pair of curves f and g in F share the set {Q_1(z),…, Q_(2t+1)(z)}, then F is normal on D. This result greatly extend some earlier theorems related to Montel's criterion.     

Symmetry of the point spectrum of infinite dimensional hamiltonian operators and its applications

This paper studies the symmetry, with respect to the real axis, of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H. Note that the point spectrum of H can be described as σp(H) = σp(A) ∪σp1(-A*). Using the characteristic of the set σp1(-A*), we divide the point spectrum σp(A) of A into three disjoint parts. Then, a necessary and sufficient condition is obtained under which σp1(-A*) and one part of σp(A) are symmetric with respect to the real axis each other. Based on this result, the symmetry of σp(H) is completely given. Moreover, the above result is applied to thin plates on elastic foundation, plane elasticity problems and harmonic equations.     

Global Geometry of 3-Body Motions with Vanishing Angular Momentum Ⅰ

Following Jacobi＇s geometrization of Lagrange＇s least action principle, trajectories of classical mechanics can be characterized as geodesics on the configuration space M with respect to a suitable metric which is the conformal modification of the kinematic metric by the factor （U ＋ h）, where U and h are the potential function and the total energy, respectively. In the special case of 3-body motions with zero angular momentum, the global geometry of such trajectories can be reduced to that of their moduli curves, which record the change of size and shape, in the moduli space of oriented m-triangles, whose kinematic metric is, in fact, a Riemannian cone over the shape space M^＊≌S^2 （1/2）.
In this paper, it is shown that the moduli curve of such a motion is uniquely determined by its shape curve （which only records the change of shape） in the case of h≠0, while in the special case of h = 0 it is uniquely determined up to scaling. Thus, the study of the global geometry of such motions can be further reduced to that of the shape curves, which are time-parametrized curves on the 2-sphere characterized by a third order ODE. Moreover, these curves have two remarkable properties, namely the uniqueness of parametrization and the monotonieity, that constitute a solid foundation for a systematic study of their global geometry and naturally lead to the formulation of some pertinent problems.     

A REMARK ON IMPLICITIZING RATIONAL CURVES WITH BASE POINTS

A simple relationship between the Bezout matrix corresponding to a rational curve with base points and the Bezout matrix corresponding to the same rational curve except that whose base points are eliminated is clarified. Based on this relationship,the author proves that the implicit equation of a rational curve with base points is the largest rton-zero leading principal minor of the gezout resultant corresponding to the rational curve assuming that the rational curve doesn‘t have triva/base point 0,and thus provides a simple approach to Jmplicitze rational curves with base points. Furthermore，as a by-product ，art algorithm is presented to compute the base points of a rational curve.     

Instructions for Authors

<正>Submission Authors must use LaTeX for typewriting,and visit our website www.actamath.com to submit your paper.Our address is Editorial Office of Acta Mathematica Sinica,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,P.R.China.     

THE 8~(th) INTERNATIONAL CONGRESS ON INDUSTRIAL AND APPLIED MATHEMATICS

正August 10-14,2015Beijin,China The International Congress on Industrial and Applied Mathematics(ICIAM)is the premier international congress in the field of applied mathematics held every four years under the auspices of the International Council for Industrial and Applied Mathematics.From August 10 to 14,2015,mathematicians,scientists     

FRACTIONAL INTEGRALS OF THE WEIERSTRASS FUNCTIONS： THE EXACT BOX DIMENSION

The present paper investigates the fractal structure of fractional integrals of Weierstrass functions. The ezact box dimension for such functions many important cases is established. We need to point out that, although the result itself achieved in the present paper is interesting, the new technique and method should be emphasized. These novel ideas might be useful to establish the box dimension or Hausdorff dimension (especially for the lower bounds) for more general groups of functions.     

English Series

正1 Aims and Scope Acta Mathematicae Applicatae Sinica(English Series)is a quarterly journal established by the Chinese Mathematical Society.The journal publishes high quality research papers from all branches of applied mathematics,particularly welcomes those from partial differential equations,computational mathematics,applied probability,mathematical finance,statistics,dynamical systems,optimization and management science.     

CHINESE ANNALS OF MATHEMATICS SERIES B Volume 35 · 2014 TOTAL CONTENTS

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Congruences,ideals and annihilators in standard QBCC-algebras

We characterize congruence lattices of standard QBCC-algebras and their connection with the congruence lattices of congruence kernels. Work on the paper was supported by Council of Czech Government No J14/98:153100011.     

A decomposition of continuity via ideals

A new class of sets in ideal topological spaces is introduced and using these sets, a decomposition of continuity is given.      

When a compact (countably compact) set is closed. II

We obtain (a) necessary and sufficient conditions and (b) sufficient conditions for a compact (countably compact) set to be closed in products (sequential products) and subspaces (sequential subspaces) of normal spaces. As a consequence of these, sufficient conditions are obtained for (i) the closedness of arbitrary (countable) union of closed sets and (ii) the equality of the union of the closures and the closure of the union of arbitrary (countable) families of sets in these spaces. It is also shown that these results do not hold for quotients of even T ₄,-spaces.     

ASYMPTOTIC BEHAVIOR OF ECKHOFF＇S METHOD FOR FOURIER SERIES CONVERGENCE ACCELERATION

The current paper considers the problem of recovering a function using a limited number of its Fourier coefficients. Specifically, a method based on Bernoulli-like polynomials suggested and developed by Krylov, Lanczos, Gottlieb and Eckhoff is examined. Asymptotic behavior of approximate calculation of the so-called ＂jumps＂ is studied and asymptotic L2 constants of the rate of convergence of the method are computed.