Maximal groups in βS can be trivial

We show that it is consistent that if S is the free semigroup or free group on countably many generators, there exist idempotents in βS whose maximal group is a singleton. The same conclusion holds if S is the semigroup of finite subsets of ω under the operation of union.     

Almost periodic compactifications of group extensions

Let and be groups and let be an extension of by . Given a property of group compactifications, one can ask whether there exist compactifications and of N and K such that the universal -compactification of G is canonically isomorphic to an extension of by . We prove a theorem which gives necessary and sufficient conditions for this to occur for general properties and then apply this result to the almost periodic and weakly almost periodic compactifications of G.     

The Falkner-Skan Equation II: Dynamics and the Bifurcations of P- and Q-Orbits

The Falkner-Skan equation is a reversible three-dimensional system of ordinary differential equations with two distinguished straight-line trajectories which form a heteroclinic loop between fixed points at infinity. We showed in the previous paper (1995, J. Differential Equations119, 336-394) that at positive integer values of the parameter λ there are bifurcations creating large sets of periodic and other interesting trajectories. Here we show that all but two of these trajectories are destroyed in another sequence of bifurcations as λ→∞, and by considering topological invariants and orderings on certain manifolds we obtain unusually detailed information about the sequences of bifurcations which can occur.     

The Moduli Space of Real Abelian Varieties with Level Structure

The moduli space of principally polarized Abelian varieties with real structure and with level N = 4m structure (with m1) is shown to coincide with the set of real points of a quasi-projective algebraic variety defined over , and to consist of finitely many copies of the quotient of the space GL(n, )/O(N) (of positive definite symmetric matrices) by the principal congruence subgroup of level N in GL(n, ).     

Anomalous brightness of quasars resolved by supersymmetry

This article expands the development of the concept of reciprocal symmetry (Ref. 1) and points out that it is (by definition) the supersymmetry of nature. First we derive the relation between the supersymmetric, reciprocal spacetime coordinate transformations of Ref. 1 and the standard Lorentz transformations of relativity. Then we demonstrate or prove the assertion in Ref. 1 that the Robertson-Walker and the Schwarzschild metrics map (exactly) reciprocally. Finally, we derive the relativistic cosmic redshift as a function of distance of the source from the observer in the implied pseudo-dynamic Machian observable universe model. This uniquely consistent physical cosmological model is then applied to interpret the redshifts from quasars. In so doing, we find that this new interpretation puts the quasars considerably closer than does the interpretation of the big-bang theory [see Eq. (36)] and seems to remove the brightness or magnitude anomaly for these objects. As discussed in the Appendix, it also explains why the big-bang interpretation (including the inflationary universe model, etc.) gives good results.     

Spaces N ∪ R and their dimensions

About spaces N∪$\text{R}$ (see [2, Exercise 5I]), the following are proved: (1) dim $\text{N∪}\text{R}\text{= dim β(N∪}\text{R}\text{)?N∪}\text{R}\text{,(2)if|β(N∪}\text{R}\text{)?N∪}\text{R}\text{|<2}{}^{\mathrm{?}}{}^{\mathrm{o}}$, then no real-valued continuous fu ction on N∪$\text{R}$ is onto (and hence, dim $\text{N∪}\text{R}\text{=0}$), (3) any compact metric space without isolated points is homeomorphic to some $\text{β(N∪}\text{R}\text{)?N∪}\text{R}$ and (4)there are spaces X,X₁ and X₂ of the form N∪$\text{R}$ such that X=X₁∪X₂,X₂andX₂ are zero sets of X, and dim X=n, dimX₁=dimX₂=0, where n=1,2,… or ∞.     

Mach's principle and the reciprocal symmetry of nature

This article develops the postulate that spacetime-charge inversion invariance reflects a fundamental reciprocal symmetry in nature between the two long range forces, from which the derivation of Mach's principle (i.e., the principle that the fundamental parameters of the electromagnetically elementary charged particle are related to those describing the electromagnetically observable universe) follows quite easily. Interpreting this result, it is argued that relativity and quantum mechanics can be made conceptually compatible and mathematically consistent by this reciprocal symmetry if one realizes that relativity isboth a macroscopic, semiclassical theory (i.e., the global half of relativity, described by Eq. (1.1), including special and general relativity) and a microscopic theory (i.e., the local half of relativity, described by Eq. (2.1), including relativistic quantum mechanics and field theory). The reciprocal symmetry of nature, then, promises unique (differential and/or integral) relationships between the coordinate variables of the observers of these tworeciprocally related theories, which implies unique, consistent numerical values for the scalar curvatureR, the massM, and the critical density for closure, _c, of the observable universe [derived from the elementary particle parameters (i.e., the electron mass and Coulomb radius)]. With this symmetry we also postulate a plausible mechanism for spontaneous generation of matter from the ubiquitous (zero-mass ether) nothingness of the Dirac sea of filled negative energy states, and can consistently interpret both the positive and negative-energy state solutions of Dirac's equation for massive, spin-1/2 (i.e., fermion) particles and both the advanced and retarded potential solutions of electromagnetic field equations. It is pointed out that, with this interpretation of the advanced potential solutions from electromagnetic field theory, one can actuallyderive causality from electromagnetic theory.     

Compactifications of banach spaces and construction of diffusion processes

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Some new insights into a known Chen‐like system

In the paper entitled ‘A novel chaotic system and its topological horseshoe’ in [Nonlinear Analysis: Modelling and Control 18 (1) (2013) 66–77], proposed the 3D chaotic system, , and discussed some of its dynamics according to theoretical and numerical analysis of its parameters . The present work is devoted to giving some new insights into the system for b≥0. Combining theoretical analysis and numerical simulations, some new results are formulated. On the one hand, after some known errors, mainly the distribution of its equilibrium point which is pointed out, correct results are formulated. On the other hand, some of its more rich dynamical properties hiding and not found previously, such as the stability, fold bifurcation, pitchfork bifurcation, degenerated pitchfork bifurcation, and Hopf bifurcation of its isolated equilibria, the dynamics of non‐isolated equilibria, the singularly degenerate heteroclinic cycle, the heteroclinic orbit, and the dynamics at infinity are clearly revealed. Using these results, one can easily explain those interesting phenomena for invariant Lyapunov exponent spectrum and amplitude control that are presented in the known literature. What is more important, we probably demonstrate a new route to chaos. Copyright © 2015 John Wiley & Sons, Ltd.     

All non-P-points are the limits of nontrivial sequences in supercompact spaces

A Hausdorff topological space is called supercompact if there exists a subbase such that every cover consisting of this subbase has a subcover consisting of two elements. In this paper, we prove that every non-P-point in any continuous image of a supercompact space is the limit of a nontrivial sequence. We also prove that every non-P-point in a closed -subspace of a supercompact space is a cluster point of a subset with cardinal number But we do not know whether this statement holds when replacing by the countable cardinal number. As an application, we prove in ZFC that there exists a countable stratifiable space which has no supercompact compactification.