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) ...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:展开更多
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.
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...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.展开更多
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. Increasi...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.展开更多
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 gen...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 <sup>*</sup>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 <span style="white-space:nowrap;">Γ of the locally compact Abelian group G is extended to the set <img alt="" src="Edit_b9535172-924d-44f0-bab3-c49db17a3b7a.png" /> of the above mentioned equivalence classes. A compact topology on <img alt="" src="Edit_9d7962a3-b8a3-4693-b62a-078c8c4b4853.png" /> is obtained as the weak topology generated by these extensions of f. Then X is naturally imbedded densely in <img alt="" src="Edit_f7d403b2-eff3-4555-b8e7-1b106e06d2e7.png" />.展开更多
The left multiplicative continuous compactification is the universal semigroup compactification of a semitopological semigroup.In this paper an internal construction of a quotient space of the left multiplicative cont...The left multiplicative continuous compactification is the universal semigroup compactification of a semitopological semigroup.In this paper an internal construction of a quotient space of the left multiplicative continuous compactification of a semitopological semigroup is constructed as a space of z-filters展开更多
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 ...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.展开更多
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 d...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.展开更多
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...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.展开更多
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*(...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.展开更多
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...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 →, and optical coherence of and is able to affect (inversion effect) the transition →. 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 →with the rate (τ0)−1. We revisit experimental results, which were not understood before, and we give a consistent description of these results based on the above assumption.展开更多
文摘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:
文摘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.
文摘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.
文摘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.
文摘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 <sup>*</sup>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 <span style="white-space:nowrap;">Γ of the locally compact Abelian group G is extended to the set <img alt="" src="Edit_b9535172-924d-44f0-bab3-c49db17a3b7a.png" /> of the above mentioned equivalence classes. A compact topology on <img alt="" src="Edit_9d7962a3-b8a3-4693-b62a-078c8c4b4853.png" /> is obtained as the weak topology generated by these extensions of f. Then X is naturally imbedded densely in <img alt="" src="Edit_f7d403b2-eff3-4555-b8e7-1b106e06d2e7.png" />.
文摘The left multiplicative continuous compactification is the universal semigroup compactification of a semitopological semigroup.In this paper an internal construction of a quotient space of the left multiplicative continuous compactification of a semitopological semigroup is constructed as a space of z-filters
文摘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.
文摘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.
文摘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.
文摘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.
文摘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 →, and optical coherence of and is able to affect (inversion effect) the transition →. 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 →with the rate (τ0)−1. We revisit experimental results, which were not understood before, and we give a consistent description of these results based on the above assumption.
