Metric and Complete Metric σ-Frames

A -frame is a lattice in which countable joins exist and binary meets distribute over countable joins. In this paper, the category MFrm, of metric -frames, is introduced, and it is shown to be equivalent to the category MLFrm _u, of metric Lindelöf frames.Finally, it is shown that the complete metric -frames are exactly the cozero parts of complete metric Lindelöf frames.     

The Equivalence of the Complete Einstein-Kahler Metric and the Bergman Metric on YIII

In this paper we give the proof about the equivalence of the complete Einstein- Kahler metric and the Bergman metric on Cartan-Hartogs domain of the third type. And we obtain the method of getting the equivalence of two metrics.     

De Morgan Algebra of Metric Ⅲ

ＤｅＭｏｒｇａｎＡｌｇｅｂｒａｏｆＭｅｔｒｉｃⅢＤｅＭｏｒｇａｎＡｌｇｅｂｒａｏｆＭｅｔｒｉｃⅢＷｅｉＪｕｎ（ＹａｎｃｈｅｎｇＩｎｓｔｉｔｕｔｅｏｆＴｅｃｈｎｏｌｏｇｙ，Ｙａｎｃｈｅｎｇ，２２４００３）ＷａｎｇＺｈｕｄｅｎｇ（ＹａｎｃｈｅｎｇＴｅｃｈ...     

The Equivalence of the Complete Einstein-K■hler Metric and the Bergman Metric on Y_Ⅲ

In this paper we give the proof about the equivalence of the complete Einstein- K■hler metric and the Bergman metric on Cartan-Hartogs domain of the third type. And we obtain the method of getting the equivalence of two metrics.     

Directional Hölder Metric Regularity

This paper studies the first-order behavior of the value function of a parametric optimal control problem with nonconvex cost functions and control constraints. By establishing an abstract result on the Fréchet subdifferential of the value function of a parametric minimization problem, we derive a formula for computing the Fréchet subdifferential of the value function to a parametric optimal control problem. The obtained results improve and extend some previous results.     

On σ-images of Metric Spaces

In this paper the relation between the σ-images of metrical spaces and spaces with σ-locally finite cs-network, or spaces with σ-locally finite cs^*-network, or spaces with σ-locally finite sequence neighborhood network, or spaces with σ-locally finite sequence open network are established by use of σ-mapping.     

Multiplicity mod 2 as a Metric Invariant

We study the multiplicity modulo 2 of real analytic hypersurfaces. We prove that, under some assumptions on the singularity, the multiplicity modulo 2 is preserved by subanalytic bi-Lipschitz homeomorphisms of ℝ ⁿ . In the first part of the paper, we find a subset of the tangent cone which determines the multiplicity mod 2 and prove that this subset of S ⁿ is preserved by the antipodal map. The study of such subsets of S ⁿ enables us to deduce the subanalytic metric invariance of the multiplicity modulo 2 under some extra assumptions on the tangent cone. We also prove a real version of a theorem of Comte, and yield that the multiplicity modulo 2 is preserved by arc-analytic bi-Lipschitz homeomorphisms.     

The Strong Compact-covering ss-images of Metric Spaces

ＴｈｅＳｔｒｏｎｇＣｏｍｐａｃｔ┐ｃｏｖｅｒｉｎｇｓｓ┐ｉｍａｇｅｓｏｆＭｅｔｒｉｃＳｐａｃｅｓＬｉＫｅｄｉａｎ（李克典）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｈａｎｇｑｉｕＴｅａｃｈｅｒｓ＇Ｃｏｌｅｇｅ，Ｈｅｎａｎ，４７６０００）Ａｂ...     

The Lipschitz Continuous Selection of Metric Projections

It has shown that, for Lipschitz continuous metrie projections, it has a Lipschitz continuous selection for 1-Chebyshev subspace and, for each subspace M which is ismorphism to a Hilbert space,P_M admits a selection s which is locally Lipschitz continuous with index 1/2.     

Sylvester–Gallai Theorem and Metric Betweenness

Sylvester conjectured in 1893 and Gallai proved some 40 years later that every finite set S of points in the plane includes two points such that the line passing through them includes either no other point of S or all other points of S. There are several ways of extending the notion of lines from Euclidean spaces to arbitrary metric spaces. We present one of them and conjecture that, with lines in metric spaces defined in this way, the Sylvester--Gallai theorem generalizes as follows: in every finite metric space there is a line consisting of either two points or all the points of the space. Then we present meagre evidence in support of this rash conjecture and finally we discuss the underlying ternary relation of metric betweenness.     

On the Quotient Compact Images of Metric Spaces

This paper gives an internal characterization of the quotient compact images of metric spaces, which answers a question posed by Arhangel'skii; and an example is constructed to show that the quotient compact images of metric spaces are not preserved by perfect maps.     

Some Notes About the Cardinals of Metric Space

Theorem 1 Let X be a nonempty countable set, K={: is a discrete metric space}, define ≌ iff((?)f) (f is an equilong isomorphism from to , for a given ∈K, define = { ∈K: ≌}. Let C={: ∈K},then |C|=|K|=|{d:d is a metric on X}|=2~((?)0) The Theorem 2 illustrates that there exists a nonempty countable set X on which we can define 2~((?)0) nondiscrete metric spaces such that they are not isomorphic each other.     

Bounding Diameter of Conical Kähler Metric

In this paper we research the differential geometric and algebro-geometric properties of the noncollapsing limit in the conical continuity equation which generalize the theory in La Nave et al. in Bounding diameter of singular Kähler metric, arXiv:1503.03159v1 [23].     

Two Results on Metric Addition in Spherical Space

We establish the concept of metric addition in spherical space and obtain two related results.     

Totally Geodesic Invariant Subm anifolds of Contact Metric Manifold

§１．　ＰｒｅｌｉｍｉｎａｒｉｅｓＬｅｔＭｂｅａ（２ｎ＋１）－ｄｉｍｅｎｓｉｏｎａｌｃｏｎｔａｃｔｍｅｔｒｉｃｍａｎｉｆｏｌｄｗｉｔｈｓｔｒｕｃｔｕｒｅｔｅｎｓｏｒｓ（Φ－，ξ－，η－，ｇ）．ＴｈｅｎｔｈｅｙｓａｔｉｓｆｙΦ－ξ－＝０，η－（ξ－）＝１，Φ－２＝－Ｉ＋η－ξ－，η－（Ｘ）＝ｇ（Ｘ，ξ－），　　　ｇ（Φ－Ｘ，Φ－Ｙ）＝ｇ（Ｘ，Ｙ）－η－（Ｘ）η－（Ｙ），ｇ（Ｘ，Φ－Ｙ）＝ｄη－（Ｘ，Ｙ）（１．１）ＦｏｒａｎｙｖｅｃｔｏｒｆｉｅｌｄｓＸａｎｄＹ…     

Completeness of the Metric Space Induced by Finite Measur

ＣｏｍｐｌｅｔｅｎｅｓｓｏｆｔｈｅＭｅｔｒｉｃＳｐａｃｅＩｎｄｕｃｅｄｂｙＦｉｎｉｔｅＭｅａｓｕｒＹｕＪｉａｎｌｉ；（禹建丽）ＪｉｎＪｉａｑｉ；（金家琦）ＷａｎｇＷｅｉ（（王伟）（ＬｕｏｇａｎｇＩｎｓｔｉｔｕｔｅｏｆＴｅｃｈｎｏｌｏｇｙ）（Ｋａｉｆｅ...     

Metric Diophantine approximation and ‘absolutely friendly’ measures

Let W(ψ) denote the set of ψ-well approximable points in and let K be a compact subset of which supports a measure μ. In this short article, we show that if μ is an ‘absolutely friendly’ measure and a certain μ-volume sum converges then The result obtained is in some sense analogous to the convergence part of Khintchine’s classical theorem in the theory of metric Diophantine approximation. The class of absolutely friendly measures is a subclass of the friendly measures introduced in [2] and includes measures supported on self-similar sets satisfying the open set condition. We also obtain an upper bound result for the Hausdorff dimension of      

Common Fixed Point Theorems in Intuitionistic Fuzzy Metric Spaces

The purpose of this paper is to introduce some types of compatibility of maps, and we prove some common fixed point theorems for mappings satisfying some conditions on intuitionistic fuzzy metric spaces which defined by J.H.Park.     

Canonical Foliations of Certain Classes of Almost Contact Metric Structures

The purpose of this paper is to study the canonical foliations of an almost cosymplectic or almost Kenmotsu manifold M in a unified way. We prove that the canonical foliation F defined by the contact distribution is Riemannian and tangentially almost Kahler of codimension 1 and that F is tangentially Kahler if the manifold M is normal. Furthermore, we show that a semi-invariant submanifold N of such a manifold M admits a canonical foliation FN which is defined by the antiinvariant distribution and a canonical cohomology class c（N） generated by a transversal volume form for FN. In addition, we investigate the conditions when the even-dimensional cohomology classes of N are non-trivial. Finally, we compute the Godbillon Vey class for FN.     

Eigenfunction Expansions for Schrödinger Operators on Metric Graphs

We construct an expansion in generalized eigenfunctions for Schr?dinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.