Invariance of Spooky Action at a Distance in Quantum Entanglement under Lorentz Transformation

We study the mechanism by which the particle-antiparticle entangled state collapses instantaneously at a distance. By making two key assumptions, we are able to show not only that instantaneous collapse of a wave function at a distance is possible but also that it is an invariant quantity under Lorentz transformation and compatible with relativity. In addition, we will be able to detect in which situation a many-body entangled system exhibits the maximum collapse speed among its entangled particles. Finally we suggest that every force in nature acts via entanglement

EP modular operators and their products

We study first EP modular operators on Hilbert C*-modules and then we provide necessary and sufficient conditions for the product of two EP modular operators to be EP. These enable us to extend some results of Koliha [{\it Studia Math.} {\bf 139} (2000), 81--90.] for an arbitrary C*-algebra and the C*-algebras of compact operators.Comment: 10 pages, accepte

The product of operators with closed range in Hilbert C*-modules

Suppose $T$ and $S$ are bounded adjointable operators with close range between Hilbert C*-modules, then $TS$ has closed range if and only if $Ker(T)+Ran(S)$ is an orthogonal summand, if and only if $Ker(S^*)+Ran(T^*)$ is an orthogonal summand. Moreover, if the Dixmier (or minimal) angle between $Ran(S)$ and $Ker(T) \cap [Ker(T) \cap Ran(S)]^{\perp}$ is positive and $\bar{Ker(S^*)+Ran(T^*)}$ is an orthogonal summand then $TS$ has closed range.Comment: 12 pages, abstract was changed, accepte

Iwasawa theory and the Eisenstein ideal

In this paper, we relate three objects. The first is a particular value of a cup product in the cohomology of the Galois group of the maximal unramified outside p extension of a cyclotomic field containing the pth roots of unity. The second is an Iwasawa module over a nonabelian extension of the rationals, a subquotient of the maximal pro-p abelian unramified completely split at p extension of a certain pro-p Kummer extension of a cyclotomic field that contains all p-power roots of unity. The third is the quotient of an Eisenstein ideal in an ordinary Hecke algebra of Hida by the square of the Eisenstein ideal and the element given by the pth Hecke operator minus one. For the relationship between the latter two objects, we employ the work of Ohta, in which he considered a certain Galois action on an inverse limit of cohomology groups to reestablish the Main Conjecture (for p at least 5) in the spirit of the Mazur-Wiles proof. For the relationship between the former two objects, we construct an analogue to the global reciprocity map for extensions with restricted ramification. These relationships, and a computation in the Hecke algebra, allow us to prove an earlier conjecture of McCallum and the author on the surjectivity of a pairing formed from the cup product for p < 1000. We give one other application, determining the structure of Selmer groups of the modular representation considered by Ohta modulo the Eisenstein ideal.Comment: 37 page

A reciprocity map and the two variable p-adic L-function

For primes p greater than 3, we propose a conjecture that relates the values of cup products in the Galois cohomology of the maximal unramified outside p extension of a cyclotomic field on cyclotomic p-units to the values of p-adic L-functions of cuspidal eigenforms that satisfy mod p congruences with Eisenstein series. Passing up the cyclotomic and Hida towers, we construct an isomorphism of certain spaces that allows us to compare the value of a reciprocity map on a particular norm compatible system of p-units to what is essentially the two-variable p-adic L-function of Mazur and Kitagawa.Comment: 55 page