A&zeta;x- and Open <i>C_D^*</i>-Filters Process of Compactifications and Any Hausdorff Compactification

By means of a characterization of compact spaces in terms of open CD*-filters induced by a , a - and open CD*-filters process of compactifications of an arbitrary topological space Y is obtained in Sec. 3 by embedding Y as a dense subspace of , YS = {ε |ε is an open CD*-filter that does not converge in Y}, YT = {A|A is a basic open CD*-filter that does not converge in Y}, is the topology induced by the base B = {U*|U is open in Y, U ≠φ} and U* = {F∈Ysw (or YTw)|U∈F}. Furthermore, an arbitrary Hausdorff compactification (Z, h) of a Tychonoff space X?can be obtained from a by the?similar process in Sec.3.

On the Inverse Limit of Higson Compactifications

Higson have introduced the conception of "Higson’s corona" (see [1]). For a given metric space X, it is a kind of compactification of X related to the metric d on it. Denote by BR(X) the set {y ∈ X\d(x,y) < R}. Recall that a slowly oscillating function on X is a function f G C*(X) satisfying the following condition:

Resolvable Spaces and Compactifications

This paper deals with spaces such that their compactification is a resolvable space. A characterization of space such that its one point compactification (resp. Wallman compactification) is a resolvable space is given.

An Expository on Some Nonstandard Compactifications

Three classical compactification procedures are presented with nonstandard flavour. This is to illustrate the applicability of Nonstandard analytic tool to beginners interested in Nonstandard analytic methods. The general procedure is as follows: A suitable equivalence relation is defined on an enlargement *X of the space X which is a completely regular space or a locally compact Hausdorff space or a locally compact Abelian group. Accordingly, every f in C(X,R) (the space of bounded continuous real valued functions on X) or Cc(X,R) (the space of continuous real valued functions on X with compact support) or the dual group Γ of the locally compact Abelian group G is extended to the set of the above mentioned equivalence classes. A compact topology on is obtained as the weak topology generated by these extensions of f. Then X is naturally imbedded densely in .

Equivariant R-Test Configurations and Semistable Limits of Q-Fano Group Compactifications

Let G be a connected,complex reductive group.In this paper,we classify G×G equivariant normal R-test configurations of a polarized G-compactification.Then,for Q-Fano G-compactifications,we express the H-invariants of their equivariant normal R-test configurations in terms of the combinatory data.Based onHan and Li“Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties”,we compute the semistable limit of aK-unstable FanoG-compactification.As an application,we show that for the two smooth K-unstable Fano SO4(C)-compactifications,the corresponding semistable limits are indeed the limit spaces of the normalized Kähler-Ricci flow.

COMPACTIFICATIONS OF BANACH SPACES AND CONSTRUCTION OF DIFFUSION PROCESSES

Let E be a separable Banach space and μ be a probability measure on E. We consider Dirichlet forms εon L2(E,m).A special compactification MГ of E is studied in order to give a simple sufficient condition which ensures that the complement MГ-E has zero ε-capacity.As an application we prove that the classical Dirichlet forms introduced in Albeverio-Rockner[1]satisfy this sufficient condition.

PREORDER RELATIONS IN FUZZY COMPACTIFICATIONS

We have proved that L-fuzzy unit interval I(L) is N-compact, and then applying fuzzy imbedding theory to the general case that the value domain is

Effects of Compactification on Free Massive Scalar Fields in Five-Dimensional Space-Time with an Extra Time Dimension: Analysing Some Results

This paper deals with some aspects of two-time physics (i.e., 2T + 3S five-dimensional space) for a Minkowski-like space with distinct speeds of causality for the time dimensions. Detailed calculations are provided to obtain results of Kaluza-Klein type compactification for free massive scalar fields and abelian free gauge fields. As already indicated in the literature, a tower of massive fields results from the compactification with mass terms having signs opposite to those of the ones appearing in other five-dimensional theories with an extra space dimension. We perform elaborate numerical calculations to highlight the magnitude of the imaginary masses and ask if we need to explore alternative compactification techniques.

Extra Time Dimension: Deriving Relativistic Space-Time Transformations, Kinematics, and Example of Dimensional Compactification Using Time-Dependent Non-Relativistic Quantum Mechanics

We consider a five-dimensional Minkowski space with two time dimensions characterized by distinct speeds of causality and three space dimensions. Formulas for relativistic coordinate and velocity transformations are derived, leading to a new expression for the speed limit. Extending the ideas of Einstein’s Theory of Special Relativity, concepts of five-velocity and five-momenta are introduced. We get a new formula for the rest energy of a massive object. Based on a non-relativistic limit, a two-time dependent Schrödinger-like equation for infinite square-well potential is developed and solved. The extra time dimension is compactified on a closed loop topology with a period matching the Planck time. It generates interference of additional quantum states with an ultra-small period of oscillation. Some cosmological implications of the concept of four-dimensional versus five-dimensional masses are briefly discussed, too.

A Modified Wallman Method of Compactification

Closed and basic closed C*D-filters are used in a process similar to Wallman method for compactifications of the topological spaces Y, of which, there is a subset of C*(Y) containing a non-constant function, where C*(Y) is the set of bounded real continuous functions on Y. An arbitrary Hausdorff compactification (Z,h) of a Tychonoff space X can be obtained by using basic closed C*D-filters from in a similar way, where C(Z) is the set of real continuous functions on Z.

PHASE PORTRAITS OF THE LESLIE-GOWER SYSTEM

In this paper we characterize the phase portraits of the Leslie-Gower model for competition among species.We give the complete description of their phase portraits in the Poincarédisc(i.e.,in the compactification of R^(2) adding the circle S1 of the infinity)modulo topological equivalence.It is well-known that the equilibrium point of the Leslie-Gower model in the interior of the positive quadrant is a global attractor in this open quadrant,and in this paper we characterize where the orbits attracted by this equilibrium born.

METRIC ENTROPY OF HOMEOMORPHISM ON NON-COMPACT METRIC SPACE

Let T ： X → X be a uniformly continuous homeomorphism on a non-compact metric space （X, d）. Denote by X＊ = X ∪ {x＊} the one point compactification of X and T ＊ ： X＊ → X＊ the homeomorphism on X＊ satisfying T ＊｜X = T and T ＊x＊ = x＊. We show that their topological entropies satisfy hd（T, X） ≥ h（T ＊, X＊） if X is locally compact. We also give a note on Katok’s measure theoretic entropy on a compact metric space.

FREDHOLM OPERATORS ON THE SPACE OF BOUNDED SEQUENCES

Necessary and sufficient conditions are studied that a bounded operator Tx =（x1^＊x, x2^＊x,…） on the space e∞, where xn^＊∈e∞^＊, is lower or upper semi-Fredholm; in particular, topological properties of the set {x1^＊, x2^＊ …} are investigated. Various estimates of the defect d（T） = codim R（T）, where R（T） is the range of T, are given. The case of xn^＊ = dnxtn^＊,where dn ∈ R and xtn^＊ 〉 0 are extreme points of the unit ball Be∞^＊, that is, tn ∈ βN, is considered. In terms of the sequence {tn}, the conditions of the closedness of the range R（T） are given and the value d（T） is calculated. For example, the condition {n ： 0 〈 ｜da｜ 〈 δ}= θ for some 5 is sufficient and if for large n points tn are isolated elements of the sequence {tn}, then it is also necessary for the closedness of R（T） （tn0 is isolated if there is a neighborhood U of tno satisfying tn ∈ U for all n ≠ n0）. If {n ： ｜dn｜ 〈 δ} = θ, then d（T） is equal to the defect δ{tn} of {tn}. It is shown that if d（T） = ∞ and R（T） is closed, then there exists a sequence {An} of pairwise disjoint subsets of N satisfying XAn ∈ R（T）.

POSITIVELY CURVED COMPLETE NONCOMPACT KHLER MANIFOLDS

In this paper we give a partial affirmative answer to a conjecture of Greene-Wu and Yau. Among other things, we prove that a complete noncompact Kahler surface with positive and bounded sectional curvature and with finite analytic Chern number c1（M）^2 is biholomorphic to C2.

NO₂Excited State Properties Revisited: An Effect of Extra Compactified Dimensions

Experiments on NO2 reveal a substructure underlying the optically excited isolated hyperfine structure (hfs) levels of the molecule. This substructure is seen in a change of the symmetry of the excited molecule and is represented by the two “states” and of a hfs-level. Optical excitation induces a transition from the ground state of the molecule to the excited state . However, the molecule evolves from to in a time τ0 ≈ 3 μs. Both and have the radiative lifetime τR ≈ 40 μs, but and differ in the degree of polarization of the fluorescence light. Zeeman coherence in the magnetic sublevels is conserved in the transition &rarr;, and optical coherence of and is able to affect (inversion effect) the transition &rarr;. This substructure, which is not caused by collisions with baryonic matter or by intramolecular dynamics in the molecule, contradicts our knowledge on an isolated hfs-level. We describe the experimental results using the assumption of extra dimensions with a compactification space of the size of the molecule, in which dark matter affects the nuclei by gravity. In , all nuclei of NO2 are confined in a single compactification space, and in , the two O nuclei of NO2 are in two different compactification spaces. Whereas and represent stable configurations of the nuclei,represents an unstable configuration because the vibrational motion in shifts one of the two O nuclei periodically off the common compactification space, enabling dark matter interaction to stimulate the transition &rarr;with the rate (τ0)&minus;1. We revisit experimental results, which were not understood before, and we give a consistent description of these results based on the above assumption.

Proof That Spatial Transitions Release/Absorb Energy That Compactification Necessarily Leads to Changes in Volume, Energy and Entropy

Using simple box quantization, we demonstrate explicitly that a spatial transition will release or absorb energy, and that compactification releases latent heat with an attendant change in volume and entropy. Increasing spatial dimension for a given number of particles costs energy while decreasing dimensions supplies energy, which can be quantified, using a generalized version of the Clausius-Clapyeron relation. We show this explicitly for massive particles trapped in a box. Compactification from N -dimensional space to (N - 1) spatial dimensions is also simply demonstrated and the correct limit to achieve a lower energy result is to take the limit, Lw → 0, where Lw is the compactification length parameter. Higher dimensional space has more energy and more entropy, all other things being equal, for a given cutoff in energy.

More Compactification for Differential Systems

This article is a review and promotion of the study of solutions of differential equations in the “neighborhood of infinity” via a non traditional compactification. We define and compute critical points at infinity of polynomial autonomuos differential systems and develop an explicit formula for the leading asymptotic term of diverging solutions to critical points at infinity. Applications to problems of completeness and incompleteness (the existence and nonexistence respectively of global solutions) of dynamical systems are provided. In particular a quadratic competing species model and the Lorentz equations are being used as arenas where our technique is applied. The study is also relevant to the Painlevé property and to questions of integrability of dynamical systems.

A Inherent Characterization of L-fuzzy Compactification

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INTRINSIC TOPOLOGY AND REFINEMENT OF HUTTON UNIT INTERVAL

<正> This paper introduces the theory of continuous lattices to the study of the Hutton unit interval I(L). some theorems related to I(L) are pithily proved. A kind of intrinsic topologies is applied to refining the topology of I(L),and a new fuzzy unit interval,called the H(λ) unit interval,is defined.Based on the H(λ) unit interval the H(λ)-complete regularity is introduced.Also,the theory of. H(λ)-stone-ech compactifications is established

Strongly Zero-Dimensional Locales

New kinds of strongly zero-dimensional locales are introduced and characterized, which are different from Johnstone's, and almost all the topological properties for strongly zero-dimensional spaces have the pointloss localic forms. Particularly. the Stone-Cech compactification of a strongly zero-diluensional locale is stongly zero-dimensional.