Related papers: Relating Catlin and D'Angelo $q$-types
When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask: which curves' lengths do we really need to know? It is a result of Duchin--Leininger--Rafi that any flat metric induced by a unit-norm…
We introduce orthogonal ring patterns in the 2-sphere and in the hyperbolic plane, consisting of pairs of concentric circles, which generalize circle patterns. We show that their radii are described by a discrete integrable system. This is…
Ballico proved that a smooth projective variety $X$ of degree $d$ over a finite field of $q$ elements admits a smooth hyperplane section if $q\geq d(d-1)^{\dim X}$. In this paper, we refine this criterion for higher codimensional linear…
We discuss three topics relevant to testing the Standard Model to high precision: commensurate scale relations, which relate observables to each other in pertubation theory without renormalization scale or scheme ambiguity, the relationship…
An integrable deformation of the known integrable model of two interacting p-dimensional and q-dimensional spherical tops is considered. After reduction this system gives rise to the generalized Lagrange and the Kowalevski tops. The…
We present a simple hidden variable model for the singlet state of a pair of qubits, characterized by two kinds, hierarchically ordered, of hidden variables. We prove that, averaging over both types of variables, one reproduces all the…
For every $d\geq 2$, we construct a subset $D\subseteq \{1,2,\dots,n\}^d$ of size $n-o(n)$ such that every affine hyperplane of $\mathbb{R}^d$ intersects $D$ in at most $d$ points, and every hypersphere of $\mathbb{R}^n$ intersects $D$ in…
Traditional algebraic geometric invariants lose some of their potency in positive characteristic. For instance, smooth projective hypersurfaces may be covered by lines despite being of arbitrarily high degree. The purpose of this…
The paper is an extended version of the talk which I gave at the XIX Congresso dell'UMI in Bologna in September 2011. The aim of this paper is twofold: first, to give an overview on the recent development in the classification of surfaces…
The dipole-coupled two-level atoms(qubits) in a single-mode resonant cavity is studied by extended bosonic coherent states. The numerically exact solution is presented. For finite systems, the first-order quantum phase transitions occur at…
In this paper we generalize the well-known notions of affine arclength and affine hypersurface measure to submanifolds of any dimension $d$ in $\mathbb R^n$ , $1 \leq d \leq n-1$. We show that a canonical affine invariant measure exists and…
Let $A$ be an annular end of a complete minimal surface $S$ in $\mathbb R^m$ and let $V$ be a $k$-dimension projective subvariety of $\mathbb P^n(\mathbb C)\ (n=m-1)$. Let $g$ be the generalized Gauss map of $S$ into $V\subset\mathbb…
We provide an overview of a canonical formalism that describes mixed quantum-classical systems in terms of statistical ensembles on configuration space, and discuss applications to measurement theory. It is shown that the formalism allows a…
The paper explores scaling properties of bubbles -- a complex analogue of Arnold tongues, associated to a one-dimensional family of analytic circle diffeomorphisms. Bubbles are smooth loops in the upper half-plane attached at all rational…
Umbral extensions of the stirling numbers of the second kind are considered and the resulting dobinski-like various formulas including new ones are presented. These extensions naturally encompass the two well known q-extensions. The further…
Nonlinear interactions are recognized as potential resources for quantum metrology, facilitating parameter estimation precisions that scale as the exponential Heisenberg limit of $2^{-N}$. We explore such nonlinearity and propose an…
We provide a natural characterisation for the sets of elliptic and hyperbolic hyperplanes of the parabolic quadric Q(2n,q) when q is even. This characterisation is based on the number of elements of these sets through points and codimension…
We give a normal form for pseudo-Einstein contact forms and apply it to construct intrinsic CR normal coordinates parametrized by the structure group of CR geometry. The proof is based on the construction of parabolic normal coordinates by…
We examine quantum statistics of optical photons emitted from atomic ensembles which are classically driven and simultaneously coupled to a two-level atom via microwave photon exchange. Quantum statistics and correlations are analyzed by…
Regular $A_n$-crystals are certain edge-colored directed graphs which are related to representations of the quantized universal enveloping algebra $U_q(\mathfrak{sl}_{n+1})$. For such a crystal $K$ with colors $1,2,...,n$, we consider its…