Related papers: $M\backslash L$ is not closed

Freiman proved in 1968 that the Lagrange and Markov spectra do not coincide by exhibiting a countable infinite collection $\mathcal{F}$ of isolated points of the Markov spectrum which do not belong the Lagrange spectrum. In this paper, we…

It is well-known from the representation theory of particle physics that the tensor product of two fundamental representation of SU(2) and SU(3) group can be decomposed to obtain the desired spectrum of the physical states. In this paper,…

The triple point numbers and the triple point spectrum of a closed 3-manifold were defined in (R. Vigara, Representaci\'on de 3-variedades por esferas de Dehn rellenantes, PhD Thesis, UNED 2006). They are topological invariants that give a…

This note is about a little extension of Nash's embedding theorem in the case of complete manifolds.

Given a compact oriented 3-manifold M in S^3 with boundary, an (M,2n)-tangle T is a 1-manifold with 2n boundary components properly embedded in M. We say that T embeds in a link L in S^3 if T can be completed to L by a 1-manifold with 2n…

A new sufficient condition for a list of real numbers to be the spectrum of a symmetric doubly stochastic matrix is presented; this is a contribution to the classical spectral inverse problem for symmetric doubly stochastic matrices that is…

We argue that those field theories containing mesons that are dual to weakly curved string backgrounds are non-generic. The spectrum of highly excited mesons in confining field theories behaves as M_n ~ \sqrt{n} (where n is the radial…

If L is a Tonelli Lagrangian defined on the tangent bundle of a compact and connected manifold whose dimension is at least 2, we associate to L the tiered Aubry set and the tiered Mane set (defined in the article). We prove that the tiered…

Gelfand - Na\u{i}mark theorem supplies a one to one correspondence between commutative $C^*$-algebras and locally compact Hausdorff spaces. So any noncommutative $C^*$-algebra can be regarded as a generalization of a topological space.…

A closed 3-manifold $M$ may be described up to some indeterminacy by a Heegaard diagram $\mathcal{D}$. The question "Does $M$ smoothly embed in $\mathbb{R}^4$?'' is equivalent to a property of $\mathcal{D}$ which we call $\textit{doubly…

We call a positive real number $\lambda$ admissible if it belongs to the Lagrange spectrum and there exists an irrational number $\alpha$ such that $\mu(\alpha)=\lambda$. Here $\mu(\alpha)$ denotes the Lagrange constant of $\alpha$ -…

Spectrum is an important numerical invariant of an isolated hypersurface singularity, connecting its topological and analytic structures. The well-known Hertling conjecture tells the relation of range and variance of exponents i.e. elements…

Recently J.Han\v{c}l obtained a result which improves on approximations to real numbers which correspond to the discrete part of Lagrange spectrum. In the present paper we prove a similar result related to the discrete part of Dirichlet…

Combining McShane's identity on a hyperbolic punctured torus with Schmutz's work on the Markov Uniqueness Conjecture (MUC), we find that MUC is equivalent to the identity \begin{equation} \sum_{n=1}^\infty \, \left( 3- L_n \right) \, = \, 4…

It is shown that the numerical range of a linear operator operator in a Hilbert space is a (complete) $(1{+}\sqrt2)$-spectral set. The proof relies, among other things, in the behavior of the Cauchy transform of the conjugates of…

By the m-spectrum of a real number q>1 we mean the set Y^m(q) of values p(q) where p runs over the height m polynomials with integer coefficients. These sets have been extensively investigated during the last fifty years because of their…

We identify a new way to divide the $\delta$-neighborhood of surfaces $\mathcal{M}\subset\mathbb{R}^3$ into a finitely-overlapping collection of rectangular boxes $S$. We obtain a sharp $(l^2,L^p)$ decoupling estimate using this…

We give a presentation of the universal central extension of the three-point loop algebra L over sl_2 by generators and relations. Our presentation arises from the realization of L as the tetrahedron Lie algebra and leads to connections…

A collection of simple closed curves in $\rr^3$ is called a negative slice if it is the intersection of a flat-at-infinity planar Lagrangian surface and $\{y_2 = a \}$ for some $a < 0$. Examples and non-examples of negative slices are…

We present a generalization of Luescher's relation between the finite-volume spectrum and scattering amplitudes to the case of three particles. We consider a relativistic scalar field theory in which the couplings are arbitrary aside from a…