Related papers: Attainable numbers and the Lagrange spectrum

We give a survey of the Lagrange inversion formula, including different versions and proofs, with applications to combinatorial and formal power series identities.

We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus on phenomena related to unboundedness of the Laplacians. This includes (failure of) essential selfadjointness, absence of essential spectrum and stochastic…

In this paper it is demonstrated how rigorous numerics may be applied to the one-dimensional nonlinear Schr\"odinger equation (NLS); specifically, to determining bound--state solutions and establishing certain spectral properties of the…

We give tight lower and upper bounds on the expected missing mass for distributions over finite and countably infinite spaces. An essential characterization of the extremal distributions is given. We also provide an extension to totally…

We prove a multivariate Lagrange-Good formula for functionals of uncountably many variables and investigate its relation with inversion formulas using trees. We clarify the cancellations that take place between the two aforementioned…

Two numbers are spectral equivalent if they have the same length spectrum. We show how to compute the equivalence classes of this relation. Moreover, we show that these classes can only have either 1,2 or infinitely many elements.

We consider Laplacians on periodic equilateral metric graphs. The spectrum of the Laplacian consists of an absolutely continuous part (which is a union of an infinite number of non-degenerated spectral bands) plus an infinite number of flat…

Orthogonality of eigenstates of different energies and its implications in potential scattering are unlabeled. Scalar products of scattering states of different energies are found to have finite non-orthogonal terms in potentials of finite…

The desired shifts of the boundaries of spectral allowed zones of periodical systems are demonstrated. In particular, the phenomenon of merging neighbor allowed zones is exhibited and its simple explanation is given. It is also shown how to…

Important spectral features, such as the emptiness of the residual spectrum, countability of the point spectrum, provided the space is separable, and a characterization of spectral gap at $0$, known to hold for bounded scalar type spectral…

We discuss some extensions and refinements of the variance bounds for both real and complex numbers. The related bounds for the eigenvalues and spread of a matrix are also derived here.

The set of semialgebraic graphs having countable list-chromatic numbers is characterized. Some other related sets of graphs having countable list-chromatic numbers also are.

In this paper we prove an existence theorem concerning linear forms of a given Diophantine type and apply it to study the structure of the spectrum of lattice exponents.

The numerical range in the quaternionic setting is, in general, a non convex subset of the quaternions. The essential numerical range is a refinement of the numerical range that only keeps the elements that have, in a certain sense,…

In this paper, we present Mie coefficients of double-layered sphere and consider the scattering problem, including the topics on field distribution, electromagnetic cross section, extinction spectra as well as some potential peculiar…

We define the independence ratio and the chromatic number for bounded, self-adjoint operators on an L^2-space by extending the definitions for the adjacency matrix of finite graphs. In analogy to the Hoffman bounds for finite graphs, we…

We introduce a new natural notion of convergence for permutations at any specified scale, in terms of the density of patterns of restricted width. In this setting we prove that limits may be chosen independently at a countably infinite…

For a sequence of self--adjoint operators, which converges in the norm resolvent sense, the formula is derived, which expresses the essential spectrum of the limit through the essential spectrum of the elements of the sequence.

I review recent lattice results on the spectrum and structure of baryons. Limitations due to the quenched approximation and un-physically heavy up and down quarks are discussed, and interfaces between first principles studies of QCD (or…

This is a survey article describing some recent results at the interface of homogeneous dynamics and Diophantine approximation.