Standard Euclidean Metric Research Articles

Stable minimal hypersurfaces in ℝ^N+1+ℓ with singular set an arbitrary closed K⊂{0}×ℝ^ℓ

With respect to a $C^\infty$ metric which is close to the standard Euclidean metric on $\mathbb{R}^{N+1+\ell}$, where $N\ge 7$ and $\ell\ge 1$ are given, we construct a class of embedded $(N+\ell)$-dimensional hypersurfaces (without boundary) which are minimal and strictly stable, and which have singular set equal to an arbitrary preassigned closed subset $K\subset \{0\}\times \mathbb{R}^{\ell}$. Thus the question is settled, with a strong affirmative, as to whether there can be " gaps" or even fractional dimensional parts in the singular set. Such questions, for both stable and unstable minimal submanifolds, remain open in all dimensions in the case of real analytic metrics and, in particular, for the standard Euclidean metric. The construction used here involves the analysis of solutions $u$ of the symmetric minimal surface equation on domains $\Omega\subset\mathbb{R}^{n}$ whose symmetric graphs (i.e., $\{(x,\xi)\in\Omega\times \mathbb{R}^{m}: |\xi| = u(x)\}$) lie on one side of a cylindrical minimal cone including, in particular, a Liouville type theorem for complete solutions (i.e., the case $\Omega=\mathbb{R}^{n}$).

The Wigner distribution and 2D classical maps

The Wigner spacing distribution has a long and illustrious history in nuclear physics and in the quantum mechanics of classically chaotic systems. In this paper, a novel connection between the Wigner distribution and 2D classical mechanics is introduced. Based on a well-known correspondence between the Wigner distribution and the 2D Poisson point process, the hypothesis that typical pseudo-trajectories of a 2D ergodic map have a Wignerian nearest-neighbor spacing distribution (NNSD) is put forward and numerically tested. The standard Euclidean metric is used to compute the interpoint spacings. In all test cases, the hypothesis is upheld, and the range of validity of the hypothesis appears to be robust in the sense that it is not affected by the presence or absence of: (i) mixing; (ii) time-reversal symmetry; and/or (iii) dissipation.

New r-Minimal Hypersurfaces via Perturbative Methods

A hypersurface in a Riemannian manifold is r-minimal if its (r+1)-curvature, the (r+1)th elementary symmetric function of its principal curvatures, vanishes identically. If n>2(r+1) we show that the rotationally invariant r-minimal hypersurfaces in ℝn+1 are nondegenerate in the sense that they carry no nontrivial Jacobi fields decaying rapidly enough at infinity. Combining this with a computation of the (r+1)-curvature of normal graphs and the deformation theory in weighted Holder spaces developed by Mazzeo, Pacard, Pollack, Uhlenbeck and others, we produce new infinite dimensional families of r-minimal hypersurfaces in ℝn+1 obtained by perturbing noncompact portions of the catenoids. We also consider the moduli space \({\mathcal{M}}_{r,k}(g)\) of elliptic r-minimal hypersurfaces with k≥2 ends of planar type in ℝn+1 endowed with an ALE metric g, and show that \({\mathcal{M}}_{r,k}(g)\) is an analytic manifold of formal dimension k(n+1), with \({\mathcal{M}}_{r,k}(g)\) being smooth for a generic g in a neighborhood of the standard Euclidean metric.

On the chromatic number of Euclidean space and the Borsuk problem

The present paper is devoted to two problems: the problem of the chromatic number of a metric space going back to Nelson, Erdős, and Hadwiger and the Borsuk problem of the partition of a bounded set of positive diameter into part of smaller diameter. By the chromatic number χ(X, ρ,A) of a metric space X with metric ρ for a set of forbidden distances A we mean the minimal number of colors in which one can paint X so that no two points of one color are not at a distance (in the metric ρ) belonging to A. In what follows, we shall consider the case (X, ρ) = (Rn, | · |2), where | x− y |2 is the standard Euclidean metric. By the Borsuk number f(n) we mean the minimal number of parts into which one can divide an arbitrary bounded set of positive diameter so that the diameter of each part will be less than that of the original set. The first problem dates back to the middle of the last century, while the second one, to the thirties. In the meantime, new and diverse results were obtained [1], [2] for the chromatic number and the Borsuk number. We are concerned with asymptotic lower bounds for the chromatic number χ(Rn, | · |2, {1}) and the Borsuk number as n → ∞. Now it is known that (see, respectively, [3], [4], and [2], [5]) (1.239 . . . + o(1)) ≤ χ(R, | · |2, {1}) ≤ (3 + o(1)), (1.225 . . . + o(1)) √ n ≤ f(n) ≤ (1.224 . . . + o(1)).

The Sphere Is Not Flat

(2006). The Sphere Is Not Flat. The American Mathematical Monthly: Vol. 113, No. 2, pp. 171-173.

The ultrametric Hilbert-space description of quantum measurements with a finite exactness

We provide a mathematical description of quantum measurements with a finite exactness. The exactness of a quantum measurement is used as a new metric on the space of quantum states. This metric differs very much from the standard Euclidean metric. This is the so-called ultrametric. We show that a finite exactness of a quantum measurement cannot he described by real numbers. Therefore, we must change the basic number field. There exist nonequivalent ultrametric Hilbert space representations already in the finite-dimensional case (compare with ideas of L. de Broglie). Different preparation procedures could generate nonequivalent representations. The Heisenberg uncertainty principle is a consequence of properties of a preparation procedure. The uncertainty principle “time-energy” is a consequence of the Schrodinger dynamics.