Radioactive Decay and the Structure of e (∞) Quantum Spacetime

Some theoretical arguments and some experimental evidence are given for the hypothesis that the form of the spectra of the Beta radioactive decay, as well as the micro background radiation and related phenomena, can be traced back to the geometrical–topological structure of spacetime itself. It is argued that this structure obeys the same Aufbau principle upon which Plancks black body radiation law is based. The main conclusion here is that the universe is four-dimensional but every point in this space, however small, is again also four-dimensional and that this accounts for the form of classical and quantum behaviour observed.     

On the 26-Dimensional Leech Lattice and E(∞) Nuclear Space

The aim of the present communication is to draw attention to a connection noticed recently between the KAM theorem, the signature of four manifolds and the dimensions of a nuclear spacetime, E ^(∞), relevant to quantum physics.     

On the Uniqueness of the ADS Spacetime

The uniqueness of the ADS spacetime among all static vacuum spacetimes with the same conformal infinity is proved for dimension n≤7. For dimension n 〉 7, the same result is established under the spin assumption.     

The Fine Structure Constant,Vacuum Polarization and the Geometry of e (∞) Space

This paper is a first attempt to derive the fine structure constant from the geometrical properties of the Cantorian manifold E ^(∞) which is assumed to model actual micro spacetime.     

On the P-Torsion Subgrup of Homotopy Group

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Sporadic randomness,Maxwell's Demon and the Poincaré recurrence times

In the case of fully chaotic systems, the distribution of the Poincaré recurrence times is an exponential whose decay rate is the Kolmogorov–Sinai (KS) entropy. We address the discussion of the same problem, the connection between dynamics and thermodynamics, in the case of sporadic randomness, using the Manneville map as a prototype of this class of processes. We explore the possibility of relating the distribution of Poincaré recurrence times to “thermodynamics”, in the sense of the KS entropy, also in the case of an inverse power-law. This is the dynamic property that Zaslavsky [Physics Today 52 (8) (1999) 39] finds to be responsible for a striking deviation from ordinary statistical mechanics under the form of Maxwell's Demon effect. We show that this way of establishing a connection between thermodynamics and dynamics is valid only in the case of strong chaos, where both the sensitivity to initial conditions and the distribution of the Poincaré recurrence times are exponential. In the case of sporadic randomness, resulting at long times in the Lévy diffusion processes, the sensitivity to initial conditions is initially a power-law, but it becomes exponential again in the long-time scale, whereas the distribution of Poincaré recurrence times keeps, or gets, its inverse power-law nature forever, including the long-time scale where the sensitivity to initial condition becomes exponential. We show that a non-extensive version of thermodynamics would imply the Maxwell's Demon effect to be determined by memory, and thus to be temporary, in conflict with the dynamic approach to Lévy statistics. The adoption of heuristic arguments indicates that this effect is possible, as a form of genuine equilibrium, after completion of the process of memory erasure.     

On the Characterization of the Full p-Defective Finite Group

A finite group G is called a full p-defective group if every p-block of G isof the highest defect.In this paper, we shall mainly study the full p-defectivefinite groups and characterize them.     

On Octonionic Regular Functions and the Szegö Projection on the Octonionic Heisenberg Group

We discuss the octonionic regular functions and the octonionic regular operator on the octonionic Heisenberg group. This is the octonionic version of CR function theory in the theory of several complex variables and regular function theory on the quaternionic Heisenberg group. By identifying the octonionic algebra with \(\mathbb{R }^{8}\) , we can write the octonionic regular operator and the associated Laplacian operator as real \((8\times 8)\) -matrix differential operators. Then we use the group Fourier transform on the octonionic Heisenberg group to analyze the associated Laplacian operator and to construct its kernel. This kernel is exactly the Szegö kernel of the orthonormal projection from the space of \(L^{2}\) functions to the space of \(L^{2}\) regular functions on the octonionic Heisenberg group.     

A Note on the Penrose–Mackay Tiling,HirzbruchSignature of M 4 and Knots in e (∞) Space

Mackays generalization of Penrose tiling is shown to be related to the Hirzbruchsignature, δ, of four manifolds in the case of a Murray–von Neumann continuousdimension in E ^(∞) space. It is further shown that σ is numerically identicalto the Jones knot invariant, V_L, for the right-handed trefoil. Finally, aconjecture regarding the theorem of local isomorphism is presented.     

On the Group of ρ-endotrivial κG-modules

In this paper, we define a group T_ρ(G) of p-endotrivial κG-modules and a generalized Dade group D_ρ(G) for a finite group G. We prove that T_ρ(G)? T_ρ(H)whenever the subgroup H contains a normalizer of a Sylow p-subgroup of G, in this case, K(G)? K(H). We also prove that the group D_ρ(G) can be embedded into T_ρ(G) as a subgroup.     

On Pure Subgroups of the Baer–Specker Group and Weak Crawley Groups

In this work we show how endomorphisms of certain pure subgroups of the Baer–Specker group can be extended to endomorphisms of the whole group. This allows us to establish the existence of a large family of essentially indecomposable slender groups with prescribed endomorphism rings. The extension technique is then applied to show that the Baer–Specker group is a countably-free weak Crawley group which is not a Crawley group.     

On the Characterizability of the Automorphism Groups of Sporadic Simple Groups by Their Element Orders

For G a finite group,π_e(G) denotes the set of orders of elements in G.If Ω is a subsetof the set of natural numbers,h(Ω) stands for the number of isomorphism classes of finite groups withthe stone set Ω of element orders.We say that G is k-distinguishable if h(π_e(G))=k<∞,otherwiseG is called non-distinguishable.Usually,a 1-distinguishable group is called a characterizable group.Itis shown that if M is a sporadic simple group different from M_(12),M_(22),J_2,He,Suz,M~cL and O'N,then Aut(M) is characterizable by its element orders.It is also proved that if M is isomorphic toM_(12),M_(22),He,Suz or O'N,then h(π_e(Aut(M)))∈{1,∞}.