Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory

We present a study of what may be called an intrinsic metric for a general regular Dirichlet form. For such forms we then prove a Rademacher type theorem. For strongly local forms we show existence of a maximal intrinsic metric (under a weak continuity condition) and for Dirichlet forms with an absolutely continuous jump kernel we characterize intrinsic metrics by bounds on certain integrals. We then turn to applications on spectral theory and provide for (measure perturbation of) general regular Dirichlet forms an Allegretto–Piepenbrink type theorem, which is based on a ground state transform, and a Shnol' type theorem. Our setting includes Laplacian on manifolds, on graphs and α-stable processes.     

Maximizers for Gagliardo–Nirenberg inequalities and related non-local problems

In this paper we study the existence of maximizers for two families of interpolation inequalities, namely a generalized Gagliardo–Nirenberg inequality and a new inequality involving the Riesz energy. Two basic tools in our argument are a generalization of Lieb’s Translation Lemma and a Riesz energy version of the Brézis–Lieb lemma.     

Entropy and the Uncertainty Principle

We generalize, improve and unify theorems of Rumin, and Maassen–Uffink about classical entropies associated with quantum density matrices. These theorems refer to the classical entropies of the diagonals of a density matrix in two different bases. Thus, they provide a kind of uncertainty principle. Our inequalities are sharp because they are exact in the high-temperature or semi-classical limit.     

Method validation for determination of the 15 European-priority polycyclic aromatic hydrocarbons in primary smoke condensates by gas chromatography/mass spectrometry: interlaboratory study

A collaborative study was conducted to validate an analytical method for quantification of the 15 polycyclic aromatic hydrocarbons (PAHs) regarded in 2002 as a health concern by the former Scientific Committee on Food of the European Commission (SCF) in primary smoke condensates. The method is based on gas chromatography/mass spectrometry of a cyclohexane extract with solid-phase cleanup through silica gel. The analytes were detected in the selected-ion monitoring mode and quantified by using 3 isotopically labeled internal standard compounds. Seventeen laboratories participated in the collaborative validation study, of which 12 reported valid results. The data were subjected to Cochran, single Grubbs, and double Grubbs tests for statistical outliers. A maximum of 2 outliers was eliminated before further statistical evaluation of the method performance characteristics. Depending on the analyte, the results showed relative standard deviations for repeatability between 4.2 and 30% and for reproducibility from 9.9 to 60%. The recoveries varied between about 50 and 85%, except those for cyclopenta[cd]pyrene, dibenzo[a,l]pyrene, and dibenzo[a,h]pyrene. Nevertheless, because Commission Directive 2005/10/EC allows for a recovery range of 50-120% for (BaP) benzo[a]pyrene in various foods, it can be concluded that the method performs appropriately within the analytical range between 5 and 25 microg/kg of primary smoke condensate. For BaP the validated analytical range covered 5-20 microg/kg, and for benzo[a]anthracene (BaA) 10-25 microg/kg. The method is suitable for monitoring BaP and BaA at their respective maximum permitted levels of 10 and 20 microg/kg. Three analytes, benzo[b]-, benzo[j]-, and benzo[k]-fluoranthene could not be separated by all of the participants and were therefore treated as the sum. Nevertheless, with this method the pattern of the respective concentrations of these 15 PAHs can be monitored in primary smoke condensate as suggested by the SCF.     

Interactions of designer antibiotics and the bacterial ribosomal aminoacyl-tRNA site

The X-ray crystal structures for the complexes of three designer antibiotics, compounds 1, 2, and 3, bound to two models for the ribosomal aminoacyl-tRNA site (A site) at 2.5-3.0 Angstroms resolution and that of neamine at 2.8 Angstroms resolution are described. Furthermore, the complex of antibiotic 1 bound to the A site in the entire 30S ribosomal subunit of Thermus thermophilus is reported at 3.8 Angstroms resolution. Molecular dynamics simulations revealed that the designer compounds provide additional stability to bases A1492 and A1493 in their extrahelical forms. Snapshots from the simulations were used for free energy calculations, which revealed that van der Waals and hydrophobic effects were the driving forces behind the binding of designer antibiotic 3 when compared to the parental neamine.     

Coupling of compressible and incompressible flow regions using the multiple pressure variables approach

In many cases, multiphase flows are simulated on the basis of the incompressible Navier–Stokes equations. This assumption is valid as long as the density changes in the gas phase can be neglected. Yet, for certain technical applications such as fuel injection, this is no longer the case, and at least the gaseous phase has to be treated as a compressible fluid. In this paper, we consider the coupling of a compressible flow region to an incompressible one based on a splitting of the pressure into a thermodynamic and a hydrodynamic part. The compressible Euler equations are then connected to the Mach number zero limit equations in the other region. These limit equations can be solved analytically in one space dimension that allows to couple them to the solution of a half‐Riemann problem on the compressible side with the help of velocity and pressure jump conditions across the interface. At the interface location, the flux terms for the compressible flow solver are provided by the coupling algorithms. The coupling is demonstrated in a one‐dimensional framework by use of a discontinuous Galerkin scheme for compressible two‐phase flow with a sharp interface tracking via a ghost‐fluid type method. The coupling schemes are applied to two generic test cases. The computational results are compared with those obtained with the fully compressible two‐phase flow solver, where the Mach number zero limit is approached by a weakly compressible fluid. For all cases, we obtain a very good agreement between the coupling approaches and the fully compressible solver. Copyright © 2014 John Wiley & Sons, Ltd.     

Absolute Continuity of the Spectrum for Periodically Modulated Leaky Wires in $${\mathbb{R}^{3}}$$

We consider a model of leaky quantum wires in three dimensions. The Hamiltonian is a singular perturbation of the Laplacian supported by a line with the coupling which is bounded and periodically modulated along the line. We demonstrate that such a system has a purely absolutely continuous spectrum and its negative part has band structure with an at most finite number of gaps. This result is extended also to the situation when there is an infinite number of the lines supporting the perturbations arranged periodically in one direction. Submitted: August 29, 2005. Accepted: June 9, 2006.     

Uniqueness of non-linear ground states for fractional Laplacians in {\mathbb{R}}

We prove uniqueness of ground state solutions Q = Q(|x|) ≥ 0 of the non-linear equation $$(-\Delta)^s Q+Q-Q^{\alpha+1}= 0 \quad {\rm in} \, \mathbb{R},$$ ( ? Δ ) s Q + Q ? Q α + 1 = 0 i n R , where 0 < s < 1 and 0 < α < 4s/(1?2s) for ${s<\frac{1}{2}}$ s < 1 2 and 0 < α < ∞ for ${s\geq \frac{1}{2}}$ s ≥ 1 2 . Here (?Δ) ^s denotes the fractional Laplacian in one dimension. In particular, we answer affirmatively an open question recently raised by Kenig–Martel–Robbiano and we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for ${s=\frac{1}{2}}$ s = 1 2 and α = 1 in [5] for the Benjamin–Ono equation. As a technical key result in this paper, we show that the associated linearized operator L ₊ = (?Δ) ^s +1?(α+1)Q ^α is non-degenerate; i.e., its kernel satisfies ker L ₊ = span{Q′}. This result about L ₊ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for non-linear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin–Ono (BO) and Benjamin–Bona–Mahony (BBM) water wave equations.     

Algébre de lie de type witt

Let ρ be an abelian group and k a commutative field. Inspired by the example of the Witt algebra, we introduce a family of ρ-graded Lie algebras (called Witt type Lie algebras) V = ⊕α?r Vα. where the dimension of each homogeneous component is 1. We characterise those which are simple and we give a complete classification of these Lie algebras. When P is free and the characteristic of k is zero, we show that the universal central extension of simple Witt type Lie algebras is 1-dimensional, and the corresponding cocycle is similar to the cocycle of the Virasoro algebra.     

Almost-convergent sequences with respect to polynomial hypergroups

Each polynomial hypergroup on ?₀ generates a family of generalized translation operators?T _m on sequence spaces. We introduce the concept of almost convergence for polynomial hypergroups (determined by the operators?T _m ), extending the notion of almost convergence introduced by Lorentz. Our investigations lead to two theorems characterizing almost convergent sequences on polynomial hypergroups.