3-Trees in polyhedral maps

We show that the vertices of the graph of any polyhedral map on the projective plane, torus or Klein bottle can be covered by a subgraph that is a tree of maximum valence 3. This extends a theorem of the author, who previously proved this theorem for the graphs of 3-dimensional polytopes. Several theorems dealing with paths in polyhedral maps are a consequence of these theorems.     

Projective equivalence of Finsler and Riemannian surfaces

In this paper we ask when a Finsler surface is projectively equivalent to a given Riemannian surface and when is a Finsler surface projectively equivalent to some Riemannian surface in general. We obtain a necessary and sufficient condition for projective equivalence in both cases. We then consider the latter condition in terms of the Christoffel symbols of the Riemannian metric and investigate when six functions of two variables are the Christoffel symbols of a Riemannian metric. We employ an exterior differential system to analyze when four functions of two variables are the four projective quantities of a Riemannian metric. We end the paper with a theorem which applies the necessary and sufficient condition to 2-dimensional Randers metrics.     

On a characterization of convexity-preserving maps, Davidon’s collinear scalings and Karmarkar’s projective transformations

In a recent paper, the authors have proved results characterizing convexity-preserving maps defined on a subset of a not-necessarily finite dimensional real vector space as projective maps. The purpose of this note is three-fold. First, we state a theorem characterizing continuous, injective, convexity-preserving maps from a relatively open, connected subset of an affine subspace of ℝ ^m into ℝ ⁿ as projective maps. This result follows from the more general results stated and proved in a coordinate-free manner in the above paper, and is intended to be more accessible to researchers interested in optimization algorithms. Second, based on that characterization theorem, we offer a characterization theorem for collinear scalings first introduced by Davidon in 1977 for deriving certain algorithms for nonlinear optimization, and a characterization theorem for projective transformations used by Karmarkar in 1984 in his linear programming algorithm. These latter two theorems indicate that Davidon’s collinear scalings and Karmarkar’s projective transformations are the only continuous, injective, convexity-preserving maps possessing certain features that Davidon and Karmarkar respectively desired in the derivation of their algorithms. The proofs of these latter two theorems utilize our characterization of continuous, injective, convexity-preserving maps in a way that has implications to the choice of scalings and transformations in the derivation of optimization algorithms in general. The third purpose of this note is to point this out. Received: January 2000 / Accepted: November 2000?Published online January 17, 2001     

Revisiting the foundations of Barbilian?s metrization procedure

In the present work we prove that one of Barbilian?s theorems from 1960 regarding the metrization procedure in the plane admits a natural extension depending on a bilinear form and the relative position of two Apollonian hyperspheres. This result allows us to pursue two fundamental ideas. First, that all the distances with constant curvature can be described by Barbilian?s metrization principle. Secondly, that all the Riemannian metric corresponding to these distances can be obtained with the same unique procedure derived from the main theorem in the text (Theorem 2.5). We show how the hyperbolic metric of the disk, the hyperbolic metric on the exterior of the disk and the hyperbolic metric on the half-plane can be obtained in the same way using Theorem 2.5, which appears here for the first time and is an extension of a Barbilian classical result (Barbilian, 1960 [7]). Furthermore, we obtain metrics corresponding to quadratic forms with signature that includes minus. By considering the norms provided by either Lorentz or Minkowski (pseudo-)inner product as influence functions, two oscillant distances can be generated in some subsets of Lorentz or Minkowski plane. The extension of 1960 Barbilian?s theorem mentioned above allow us to obtain the metrics attached to these two Barbilian distances on corresponding subsets of Lorentz and Minkowski 2-dimensional spaces. The geometric study concludes that these metrics are generalized Lagrange metrics. A result concerning the distance induced by a Riemannian metric as a local Barbilian distance is also proved.     

A Note on Circular Decomposable Metrics

A metric d on a finite set X is called a Kalmanson metric if there exists a circular ordering of points of X, such that d(y, u) + d(z, v) d(y, z) + d(u, v) for all crossing pairs yu and zv of . We prove that any Kalmanson metric d is an l₁-metric, i.e. d can be written as a nonnegative linear combination of split metrics. The splits in the decomposition of d can be selected to form a circular system of splits in the sense of Bandelt and Dress.     

New cone fixed point theorems for nonlinear multivalued maps with their applications

In this paper, we first establish some new types of fixed point theorems for nonlinear multivalued maps in cone metric spaces. From those results, we obtain new fixed point theorems for nonlinear multivalued maps in metric spaces and the generalizations of Mizoguchi–Takahashi’s fixed point theorem and Berinde–Berinde’s fixed point theorem. Some applications to the study of metric fixed point theory are given.     

On dually flat Finsler metrics

In this paper, we study the locally dually flat Finsler metrics which arise from information geometry. An equivalent condition of locally dually flat Finsler metrics is given. We find a new method to construct locally dually flat Finsler metrics by using a projectively flat Finsler metric under the condition that the projective factor is also a Finsler metric. Finally, we find that many known Finsler metrics are locally dually flat Finsler metrics determined by some projectively flat Finsler metrics.     

Finsler空间上的Weyl曲率

The Weyl curvature of a Finsler metric is investigated. This curvature constructed from Riemannain curvature. It is an important projective invariant of Finsler metrics. The author gives the necessary conditions on Weyl curvature for a Finsler metric to be Randers metric and presents examples of Randers metrics with non-scalar curvature. A global rigidity theorem for compact Finsler manifolds satisfying such conditions is proved. It is showed that for such a Finsler manifold,if Ricci scalar is negative,then Finsler metric is of Randers type.     

Binary clustering

In many clustering systems (hierarchies, pyramids and more generally weak hierarchies) clusters are generated by two elements only.This paper is devoted to such clustering systems (called binary clustering systems). It provides some basic properties, links with (closed) weak hierarchies and some qualitative versions of bijection theorems that occur in Numerical Taxonomy. Moreover, a way to associate a binary clustering system to every clustering system is discussed.Finally, introducing the notion of weak ultrametrics, a bijection between indexed weak hierarchies and weak ultrametrics is obtained (the standard theorem involves closed weak hierarchies and quasi-ultrametrics).     

非线性Lipschitz算子的定量性质（Ⅰ）─Lip数

本文定义了非线性算子的Ｌｉｐ数，它从数值上刻画了在强等价距离意义下非线性算子的最小Ｌｉｐｓｃｈｉｔｚ常数．基于所引进的Ｌｉｐ数，我们证明了线性算子Ｎｅｕｍａｎｎ引理及扰动引理的非线性推广．我们也给出了Ｌｉｐ数的两个极有意义的估值定理．     

Planarity Algorithms via PQ-Trees (Extended Abstract)

We give an abstract vertex-addition method for planarity testing that encompasses the algorithms of Lempel, Even, and Cederbaum, Shih and Hsu, and Boyer and Myrvold. The main difference between the former and the latter two is the order of vertex addition; the latter two differ only in implementation details. For the general method we give a direct proof of correctness that avoids the use of Kuratowski's theorem. We give a linear-time implementation that simplifies and unifies the Shih-Hsu and Boyer-Myrvold methods. Our algorithm extends to generate embeddings uniformly at random, to count embeddings, to represent all embeddings, and to produce a Kuratowski subgraph of a non-planar graph. Our algorithm keeps track of all possible embeddings by reinterpreting Booth and Lueker's PQ-tree data structure to represent circular instead of linear orders. This interpretation of PQ-trees gives the PC-trees of Shih and Hsu and leads to a simpler, more-symmetric form of PQ-tree reduction.     

On Angles and The Fundamental Theorems of Metric Geometry

In this paper we investigate the abstract angle measure for affine metric spaces. Common features and differences between orthogonal angles and angles with measure ≠ 0 are examined. It turns out that an affine collineation which maps angles with a certain fixed measure α ≠ 0,4 to angles with another fixed measure β is already a metric collineation in nearly all cases (fundamental theorem). An analogous result is stated for projective metric spaces. Some applications concerning minimal conditions for metric collineations are given.     

Continuation Theory for Contractions on Spaces with Two Vector-Valued Metrics

We develop a continuation theory for contractive maps on spaces with two vector-valued metrics. Applications are presented for systems of operator equations in Banach spaces and, in particular, for systems of abstract Hammerstein integral equations. The use of vector-valued metrics makes it possible for each equation of a system to have its own Lipschitz property, while the use of two such metrics makes it possible for the Lipschitz condition to be expressed with respect to an incomplete metric.     

Subvarieties of generic hypersurfaces in a nonsingular projective toric variety

We investigate the subvarieties contained in generic hypersurfaces of projective toric varieties and prove two main theorems. The first generalizes Clemens’ famous theorem on the genus of curves in hypersurfaces of projective spaces to curves in hypersurfaces of toric varieties and the second improves the bound in the special case of toric varieties in a theorem of Ein on the positivity of subvarieties contained in sufficiently ample generic hypersurfaces of projective varieties. Both depend on a hypothesis which deals with the surjectivity of multiplication maps of sections of line bundles on the toric variety. We also obtain an infinitesimal Torelli theorem for hypersurfaces of toric varieties.     

Statistical properties for nonhyperbolic maps with finite range structure

We establish the central limit theorem and non-central limit theorems for maps admitting indifferent periodic points (the so-called intermittent maps). We also give a large class of Darling-Kac sets for intermittent maps admitting infinite invariant measures. The essential issue for the central limit theorem is to clarify the speed of -convergence of iterated Perron-Frobenius operators for multi-dimensional maps which satisfy Renyi's condition but fail to satisfy the uniformly expanding property. Multi-dimensional intermittent maps typically admit such derived systems. There are examples in section 4 to which previous results on the central limit theorem are not applicable, but our extended central limit theorem does apply.

     

The best approximation theorems and variational inequalities for discontinuous mappings in Banach spaces

We discuss Ky Fan's theorem and the variational inequality problem for discontinuous mappings f in a Banach space X. The main tools of analysis are the variational characterizations of the metric projection operator and the order-theoretic fixed point theory. Moreover, we derive some properties of the metric projection operator in Banach spaces. As applications of our best approximation theorems, three fixed point theorems for non-self maps are established and proved under some conditions. Our results are generalizations and improvements of various recent results obtained by many authors.     

On vector bundles of finite order

We define Finsler metrics of finite order on a holomorphic vector bundle by imposing estimates on the holomorphic bisectional curvature. We generalize the vanishing theorem of Griffiths and Cornalba regarding Hermitian bundles of finite order to the Finsler context. We develop a value distribution theory for holomorphic maps from the projectivization of a vector bundle to projective space. We show that the projectivization of a Finsler bundle of finite order can be immersed into a projective space of sufficiently large dimension via a map of finite order.     

Common fixed point results for noncommuting mappings without continuity in generalized metric spaces

Using the setting of a generalized metric space, a fixed point theorem is proved for one map, and several fixed point theorems are proved for two maps. These results generalize several well known comparable results in the literature.     

Indefinite Kähler submanifolds with positive index of relative nullity

We deal with complex submanifolds in indefinite space forms. In particular, submanifolds with large index of relative nullity are emphasized. In that context, we prove cylinder theorems in terms of indefinite metrics. We also give a systematic way of constructing a family of new complete and closed indefinite complex submanifolds in the projective setting.In the appendix, we show that the method used for complex cases can be applied to real indefinite geometry. We prove real cylinder theorems including B-scrolls in the general signature. We also show two decomposition lemmas which clarify the relationships between the Hartman-Nirenberg cylinder theorem and slanted cylinder theorems in indefinite geometry.     

Real Projective Iterated Function Systems

This paper contains four main results associated with an attractor of a projective iterated function system (IFS). The first theorem characterizes when a projective IFS has an attractor which avoids a hyperplane. The second theorem establishes that a projective IFS has at most one attractor. In the third theorem the classical duality between points and hyperplanes in projective space leads to connections between attractors that avoid hyperplanes and repellers that avoid points, as well as hyperplane attractors that avoid points and repellers that avoid hyperplanes. Finally, an index is defined for attractors which avoid a hyperplane. This index is shown to be a nontrivial projective invariant.