Einstein-Rosen Bridge (ER), Einstein-Podolsky-Rosen Experiment (EPR) and Zero Measure Rindler-KAM Cantorian Spacetime Geometry (ZMG) Are Conceptually Equivalent

By viewing spacetime as a transfinite Turing computer, the present work is aimed at a generalization and geometrical-topological reinterpretation of a relatively old conjecture that the wormholes of general relativity are behind the physics and mathematics of quantum entanglement theory. To do this we base ourselves on the comprehensive set theoretical and topological machinery of the Cantorian-fractal E-infinity spacetime theory. Going all the way in this direction we even go beyond a quantum gravity theory to a precise set theoretical understanding of what a quantum particle, a quantum wave and quantum spacetime are. As a consequence of all these results and insights we can reason that the local Casimir pressure is the difference between the zero set quantum particle topological pressure and the empty set quantum wave topological pressure which acts as a wormhole “connecting” two different quantum particles with varying degrees of entanglement corresponding to varying degrees of emptiness of the empty set (wormhole). Our final result generalizes the recent conceptual equation of Susskind and Maldacena ER = EPR to become ZMG = ER = EPR where ZMG stands for zero measure Rindler-KAM geometry (of spacetime). These results were only possible because of the ultimate simplicity of our exact model based on Mauldin-Williams random Cantor sets and the corresponding exact Hardy’s quantum entanglement probability P(H) = where is the Hausdorff dimension of the topologically zero dimensional random Cantor thin set, i.e. a zero measure set and . On the other hand the positive measure spatial separation between the zero sets is a fat Cantor empty set possessing a Hausdorff dimension equal while its Menger-Urysohn topological dimension is a negative value equal minus one. This is the mathematical quintessence of a wormhole paralleling multiple connectivity in classical topology. It is both physically there because of the positive measure and not there because of the negative topological dimension.

A Combined Heterotic String and Kähler Manifold Elucidation of Ordinary Energy,Dark Matter,Olbers’s Paradox and Pure Dark Energy Density of the Cosmos

We utilize the topological-geometrical structure imposed by the Heterotic superstring theory on spacetime in conjunction with the K3 Kähler manifold to explain the mysterious nature of dark matter and its coupling to the pure dark energy density of the cosmos. The analogous situations in the case of a Kerr black hole as well as the redundant components of the Riemannian tensor are pointed out and the final result was found to be in complete agreement with all previous theoretical ones as well as all recent accurate measurements and cosmic observations. We conclude by commenting briefly on the Cantorian model of Zitterbewegung and the connection between Olbers’s paradox and dark energy.

ZERO SET OF SOBOLEV FUNCTIONS WITH NEGATIVE POWER OF INTEGRABILITY

Here the authors are interested in the zero set of Sobolev functions and functions of bounded variation with negative power of integrability. The main result is a general Hausdorff dimension estimate on the size of zero set. The research is motivated by the model on van der waal force driven thin film, which is a singular elliptic equation. After obtaining some basic regularity result, the authors get an estimate on the size of singular set; such set corresponds to the thin film rupture set in the thin film model.

Algebraic Construction for Zero-Knowledge Sets

Zero knowledge sets is a new cryptographic primitive introduced by Micali, Rabin, and Kilian in FOCS 2003. It has been intensively studied recently. However all the existing ZKS schemes follow the basic structure by Micali et al. That is, the schemes employ the Merkle tree as a basic structure and mercurial commitments as the commitment units to nodes of the tree. The proof for any query consists of an authentication chain. We propose in this paper a new algebraic scheme that is completely different from all the existing schemes. Our new scheme is computationally secure under the standard strong RSA assumption. Neither mercurial commitments nor tree structure is used in the new construction. In fact, the prover in our construction commits the desired set without any trapdoor information, which is another key important difference from the previous approaches.

A Comparison between the Metric Dimension and Zero Forcing Number of Trees and Unicyclic Graphs

The metric dimension dim（G） of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. The zero forcing number Z（G） of a graph G is the minimum eardinality of a set S of black vertices （whereas vertices in V（G）／S are colored white） such that V（G） is turned black after finitely many applications of ＂the color-change rule＂： a white vertex is converted black if it is the only white neighbor of a black vertex. We show that dim（T） ≤Z（T） for a tree T, and that dim（G）≤Z（G）＋I if G is a unicyclic graph; along the way, we characterize trees T attaining dim（T） = Z（T）. For a general graph G, we introduce the ＂cycle rank conjecture＂. We conclude with a proof of dim（T） - 2 ≤ dim（T ＋ e） ≤dim（T） ＋ 1 for e∈ E（T）.

Level Sets of Certain Subclasses of α-Analytic Functions

For an open set V C Cn, denote by Mα （V） the family of α-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded ＂harmonically fat＂ domain Ω C Cn, a function f ∈M a（Ω／f-1（0）） automatically sat- isfies f ∈M a（Ω）, if it is Caj-1smooth in the z/variable, α ∈ Zn＋ up to the boundary. For a submanifold U C Cn, denote by ma（U）, the set of functions locally approximable by α-analytic functions where each approximating member and its reciprocal （off the singularities） obey the boundary maximum modulus principle. We prove, that for a C3-smooth hypersurface, Ω, a member of ma （Ω）, cannot have constant modulus near a point where the Levi form has a positive eigenvalue, unless it is there the trace of a polyanalytic function of a simple form. The result can be partially generalized to C4-smooth submanifolds of higher codimension, at least near points with a Levi cone condition.