Starting from the initial triality of physics, namely mathematical philosophy, transfinite set theory and number theory we drive the inevitability of a topological quantum vacuum fluctuation of spacetime resulting in the fundamental reality of pair creation and annihilation. Subsequently we give a simple but strong mathematical proof of Dvoretzky‘s marvellous theorem on measure concentration, thus making dark energy and accelerated cosmic expansion not only an astrophysical measurement and observational reality, but also a plausible topological-geometrical fact of a pointless Cantorian actual universe akin to the Penrose fractal tiling space. This space is described accurately via the von Neumann-Conne noncommutative geometry using their golden mean dimensional function and the corresponding bijection of E-infinity theory. The said theory was developed by the authors of the present paper and their group and is based on and starts from the pioneering efforts of the Canadian physicist G. Ord and the French astrophysicist L. Nottale.

The metric Ramsey problem asks for the largest subset S of a metric space that can be embedded into an ultrametric (more generally into a Hilbert space) with a given distortion. Study of this problem was motivated as a non-linear version of Dvoretzky theorem. Mendel and Naor [29] devised the so-called Ramsey Partitions to address this problem, and showed the algorithmic applications of their techniques to approximate distance oracles and ranking problems. In this article, we study the natural extension of the metric Ramsey problem to graphs, and introduce the notion of Ramsey Spanning Trees . We ask for the largest subset S ⊆ V of a given graph G =( V , E ), such that there exists a spanning tree of G that has small stretch for S . Applied iteratively, this provides a small collection of spanning trees, such that each vertex has a tree providing low stretch paths to all other vertices . The union of these trees serves as a special type of spanner, a tree-padding spanner . We use this spanner to devise the first compact stateless routing scheme with O (1) routing decision time, and labels that are much shorter than in all currently existing schemes. We first revisit the metric Ramsey problem and provide a new deterministic construction. We prove that for every k , any n -point metric space has a subset S of size at least n 1−1/ k that embeds into an ultrametric with distortion 8 k . We use this result to obtain the state-of-the-art deterministic construction of a distance oracle. Building on this result, we prove that for every k , any n -vertex graph G =( V , E ) has a subset S of size at least n 1−1/ k , and a spanning tree of G , that has stretch O ( k log log n ) between any point in S and any point in V .

On Dvoretzky's theorem for subspaces of Lp

Superconcentration, and randomized Dvoretzky's theorem for spaces with 1-unconditional bases

We show that any $n\times m$ matrix $A$ can be approximated in operator norm by a submatrix with a number of columns of order the stable rank of $A$. This improves on existing results by removing an extra logarithmic factor in the size of the extracted matrix. Our proof uses the recent solution of the Kadison-Singer problem. We also develop a sort of tensorization technique to deal with constraint approximation problems. As an application, we provide a sparsification result with equal weights and an optimal approximate John's decomposition for non-symmetric convex bodies. This enables us to show that any convex body in $\mathbb{R}^n$ is arbitrary close to another one having $O(n)$ contact points and fills the gap left in the literature after the results of Rudelson and Srivastava by completely answering the problem. As a consequence, we also show that the method developed by Gu\'edon, Gordon and Meyer to establish the isomorphic Dvoretzky theorem yields to the best known result once we inject our improvements.

The well-known Horodecki criterion asserts that a state $\rho$ on $\mathbf{C}^d \otimes \mathbf{C}^d$ is entangled if and only if there exists a positive map $\Phi : \mathsf{M}_d \to \mathsf{M}_d$ such that the operator $(\Phi \otimes \mathrm{Id})(\rho)$ is not positive semi-definite. We show that the number of such maps needed to detect all the robustly entangled states (i.e., states $\rho$ which remain entangled even in the presence of substantial randomizing noise) exceeds $\exp(c d^3 / \log d)$. The proof is based on the 1977 inequality of Figiel--Lindenstrauss--Milman, which ultimately relies on Dvoretzky's theorem about almost spherical sections of convex bodies. We interpret that inequality as a statement about approximability of convex bodies by polytopes with few vertices or with few faces and apply it to the study of fine properties of the set of quantum states and that of separable states. Our results can be thought of as geometrical manifestations of the complexity of entanglement detection.

Consideration was given to the Robbins---Monro procedure for which the Dvoretzky theorem of its convergence rate was generalized. The rate of convergence is still the most important problem of the theory of stochastic approximation.

A basically topological interpretation of the Casimir effect is given as a natural intrinsic property of the geometrical topological structure of the quantum-Cantorian micro spacetime. This new interpretation compliments the earlier conventional interpretation as vacuum fluctuation or as a Schwinger source and links the Casimir energy to the so called missing dark energy density of the cosmos. We start with a general outline of the theoretical principle and basic design concepts of a proposed Casimir dark energy nano reactor. In a nutshell the theory and consequently the actual design depends crucially upon the equivalence between the dark energy density of the cosmos and the faint local Casimir effect produced by two sides boundary condition quantum waves. This Casimir effect is then colossally amplified as a one sided quantum wave pushing from the inside on the one sided M?bius-like boundary with nothing balancing it from the non-existent outside. In view of the present theory, this one sided M?bius-like boundary of the holographic boundary of the universe is essentially what leads to the observed accelerated expansion of the cosmos. Thus in principle we will restructure the local topology of space using material nanoscience technology to create an artificial local high dimensionality with a Dvoretzky theorem like volume measure concentration. Needless to say the entire design is based completely on the theory of quantum wave dark energy proposed by the present author. The quintessence of the present theory is easily explained as the intrinsic Casimir topological energy where produced from the zero set of the quantum particle when we extract the empty set quantum wave from it and find by restructuring space via plates similar to that of the classical Casimir experiments but with some modification.

A phase one design of a new free energy nano reactor is presented. The design is based on a basically topological interpretation of the Casimir effect as a natural intrinsic property of the geometrical topological structure of the quantum-Cantorian micro spacetime. In particular we view dark energy, Hawking negative energy, Unruh temperature and zero point vacuum energy as being different sides of the same multi-dimensional coin. This new interpretation compliments the earlier conventional interpretation as vacuum fluctuation or as a Schwinger source and links the Casimir energy to the so-called missing dark energy density of the cosmos. We start with a general outline of the theoretical principle and basic design concepts of a proposed Casimir dark energy nano reactor. In a nutshell the theory and consequently the actual design depend crucially upon the equivalence between the dark energy density of the cosmos and the faint local Casimir effect produced by two sides boundary condition quantum waves. This Casimir effect is then colossally amplified as a one internal quantum wave representing a Hartle-Hawking state vector of the universe pushing from the inside against the boundary of the universe with nothing balancing it from the non-existent outside. This strange situation becomes completely natural and logical when we remember that the boundary of the universe is a one sided M&#246bius like manifold. In view of the present theory, this is essentially what leads to the observed accelerated expansion of the cosmos. As in any reactor, the basic principle in the present design is to produce a gradient so that the excess energy on one side flows to the other side. Thus in principle we will restructure the local topology of space using material nanoscience technology to create an artificial local high dimensionality with a Dvoretzky theorem like 96 percent volume measure concentration. Without going into the intricate nonlinear dynamics and technological detail, it is fair to say that this would lead us to pure, clean, free energy obtained directly from the topology of spacetime via an artificial singularity. Needless to say, the entire design is based completely on the theory of quantum wave dark energy proposed by the present author for the first time in 2011 in a conference held in the Bibliotheca Alexandrina, Egypt and a little later in Shanghai, Republic of China. The quintessence of the present theory is easily explained as the Φ3 intrinsic Casimir topological energy where Φ= (√5-1)/2 is produced from the zero set Φ of the quantum particle when we extract the empty set quantum wave Φ2 from it and find Φ-Φ2=Φ3 by restructuring space via conducting but uncharged plates similar to that of the classical Casimir experiments. Our proposed preliminary design of this Casimir-spacetime artificial singularity reactor follows in a natural way from the above.

We study the way in which the Euclidean subspaces of a Banach space fit together, somewhat in the spirit of the Ka\v{s}in decomposition. The main tool that we introduce is an estimate regarding the convex hull of a convex body in John's position with a Euclidean ball of a given radius, which leads to a new and simplified proof of the randomized isomorphic Dvoretzky theorem. Our results also include a characterization of spaces with nontrivial cotype in terms of arrangements of Euclidean subspaces.

We prove additivity violation of minimum output entropy of quantum channels by straightforward application of \epsilon-net argument and L\'evy's lemma. The additivity conjecture was disproved initially by Hastings. Later, a proof via asymptotic geometric analysis was presented by Aubrun, Szarek and Werner, which uses Dudley's bound on Gaussian process (or Dvoretzky's theorem with Schechtman's improvement). In this paper, we develop another proof along Dvoretzky's theorem in Milman's view showing additivity violation in broader regimes than the existing proofs. Importantly, Dvoretzky's theorem works well with norms to give strong statements but these techniques can be extended to functions which have norm-like structures - positive homogeneity and triangle inequality. Then, a connection between Hastings' method and ours is also discussed. Besides, we make some comments on relations between regularized minimum output entropy and classical capacity of quantum channels.

Stationary Markov perfect equilibria in risk sensitive stochastic overlapping generations models

The new class of Gaussian processes which are obtained by a compact perturbation in the reproducing kernel of Wiener process is in- troduced. The nite families of increments of our processes on small time intervals behave as increments of Wiener process. Consequently a lot of asymptotical properties of Wiener process are inherited. The law of iterated logarithm, the analogue of the Levy modulus of continuity and almost unifor- mity of hitting distribution on the small circles are proved. The renormalized Fourier{Wiener transform of the self-intersection local time for compactly perturbed Wiener process is constructed. In this article we consider self-intersection local times for one class of planar Gaussian processes. The interest to the self-intersection local times of planar Brownian motion has a long history since the theorem of Dvoretzky, Erdos, Kaku- tani (3) which established the existence of multiple self-intersections. The various kinds of renormalization were proposed for the self-intersection local times of pla- nar Brownian motion and certain Levy processes in the articles (4, 5, 10, 15, 16). Most of these articles essentially use the Markov property of the considered pro- cess. The aim of this paper is to present an approach to investigation of planar Gaussian processes which does not use the Markov property. We introduce a new class of Gaussian processes which are obtained with the help of compact pertur- bation in the reproducing kernel of Wiener process. Such processes inherit many properties of Wiener process. As an example we prove here the law of iterated logarithm, the analogue of the Levy modulus of continuity and almost uniformity of hitting distribution on the small circles. The nite families of small increments of our processes behave like the increments of Wiener process. This allows us to prove that compactly perturbed Wiener process has strong local nondeterminism property, which is a generalization of local nondeterminism property introduced by S.Berman (1). Finally we present the renormalization for Fourier{Wiener trans- form of the self-intersection local times for compactly perturbed Wiener process.

What singles out quantum mechanics as the fundamental theory of nature? Here we study local measurements in generalized probabilistic theories (GPTs) and investigate how observational limitations affect the production of correlations. We find that if only a subset of typical local measurements can be made then all the bipartite correlations produced in a GPT can be simulated to a high degree of accuracy by quantum mechanics. Our result makes use of a generalization of Dvoretzky's theorem for GPTs. The tripartite correlations can go beyond those exhibited by quantum mechanics, however.

Two infinite versions of the nonlinear Dvoretzky theorem

We introduce a randomized iterative fragmentation procedure for finite metric spaces, which is guaranteed to result in a polynomially large subset that is $D$-equivalent to an ultrametric, where $D\in (2,\infty)$ is a prescribed target distortion. Since this procedure works for $D$ arbitrarily close to the nonlinear Dvoretzky phase transition at distortion 2, we thus obtain a much simpler probabilistic proof of the main result of Bartel, Linial, Mendel, and Naor, answering a question from Mendel and Naor, and yielding the best known bounds in the nonlinear Dvoretzky theorem. Our method utilizes a sequence of random scales at which a given metric space is fragmented. As in many previous randomized arguments in embedding theory, these scales are chosen irrespective of the geometry of the metric space in question. We show that our bounds are sharp if one utilizes such a "scale-oblivious" fragmentation procedure.

Abstract This paper is inspired by a counter example of J. Kurzweil published in [5], whose intention was to demonstrate that a certain property of linear operators on finite-dimensional spaces need not be preserved in infinite dimension. We obtain a stronger result, which says that no infinite-dimensional Banach space can have the given property. Along the way, we will also derive an interesting proposition related to Dvoretzky's theorem.

We explore possibilities and limitations of a purely topological approach to the Dvoretzky theorem.

The goal of this note is to show that Hastings' counterexample to the additivity of minimal output von Neumann entropy can be readily deduced from a sharp version of Dvoretzky's theorem on almost spherical sections of convex bodies.

We obtain explicit estimates of the constants related to the concentrations of measures and distance, deviation and Dvoretzky's theorem for the finite-dimensional subspaces of a wide class of function and other spaces including, in particular, various anisotropic spaces of Besov, Lizorkin–Triebel and Sobolev types endowed with geometrically friendly norms defined in terms of averaged differences, local polynomial approximations, functional calculus, wavelets and other means. New approaches are shown to be providing better estimates in the abstract setting as well.