The spin contribution to the form factor of quantum graphs

Following the quantisation of a graph with the Dirac operator (spin-1/2) we explain how additional weights in the spectral form factor K(\tau) due to spin propagation around orbits produce higher order terms in the small-\tau asymptotics in agreement with symplectic random matrix ensembles. We determine conditions on the group of spin rotations sufficient to generate CSE statistics.Comment: 9 page

1-(2,6-Dichlorobenzoyl)-3-(3-methoxyphenyl)thiourea

The two aromatic rings in the title compound, C15H12Cl2N2O2S, enclose a dihedral angle of 37.49 (6)°. The molecule exists in the solid state in its thione form with typical thiourea C-S and C-O bonds lengths, as well as shortened C-N bonds. An intramolecular N-H...O hydrogen bond stabilizes the molecular conformation. In the crystal, molecules are connected by N-H...O and N-H...S hydrogen bonds, forming chains running along the alpha axis. Key indicators: single-crystal X-ray study; T = 173 K; mean σ (C–C) = 0.002 Å; disorder in main residue; R factor = 0.035; wR factor = 0.087; data-to-parameter ratio = 18.9

Clarke subgradients of stratifiable functions

We establish the following result: if the graph of a (nonsmooth) real-extended-valued function $f:\mathbb{R}^{n}\to \mathbb{R}\cup\{+\infty\}$ is closed and admits a Whitney stratification, then the norm of the gradient of $f$ at $x\in{dom}f$ relative to the stratum containing $x$ bounds from below all norms of Clarke subgradients of $f$ at $x$. As a consequence, we obtain some Morse-Sard type theorems as well as a nonsmooth Kurdyka-\L ojasiewicz inequality for functions definable in an arbitrary o-minimal structure

Crystal structure of the N-benzyloxycarbonyl-Alanyl-Phenylalanyl-methyl ester: the importance of the H-bonding pattern

Large crystals of the methyl ester of the N-a-benzyloxycarbonyl protected Ala-Phe dipeptide (Z-AF-OMe) were obtained after the very slow evaporation of a solution of the corresponding carboxylic acid (Z-AF-OH) in methanol containing an excess of HCl. The structure was confirmed by single crystal X-ray diffraction data. It crystallizes in the orthorhombic space group P212121 with unit cell dimensions a = 5.0655(6) Å, b = 8.4614(8) Å, c = 46.856(5) Å, V = 2008.3(4) Å3, Z = 4. In the crystal, the molecules form hydrogen bonded chains running along the a axis of the unit cell. Other secondary interactions are also discussed

Spectral Statistics for the Dirac Operator on Graphs

We determine conditions for the quantisation of graphs using the Dirac operator for both two and four component spinors. According to the Bohigas-Giannoni-Schmit conjecture for such systems with time-reversal symmetry the energy level statistics are expected, in the semiclassical limit, to correspond to those of random matrices from the Gaussian symplectic ensemble. This is confirmed by numerical investigation. The scattering matrix used to formulate the quantisation condition is found to be independent of the type of spinor. We derive an exact trace formula for the spectrum and use this to investigate the form factor in the diagonal approximation

4-(4-Nitrophenoxy)biphenyl

The two phenyl rings of the biphenyl unit of the title compound, C18H13NO3, are almost coplanar [dihedral angle 6.70 (9)°]. The nitrophenyl ring, on the other hand, is significantly twisted out of the plane of the these two rings, making dihedral angles of 68.83 (4)° with the middle ring and 62.86 (4)° with the end ring. The nitro group is twisted by 12.1 (2)° out of the plane of the phenyl ring to which it is attached. Key indicators: single-crystal X-ray study; T = 173 K; mean σ(C–C) = 0.002 A° ; R factor = 0.040; wR factor = 0.118; data-to-parameter ratio = 12.8

Quantum graphs with singular two-particle interactions

We construct quantum models of two particles on a compact metric graph with singular two-particle interactions. The Hamiltonians are self-adjoint realisations of Laplacians acting on functions defined on pairs of edges in such a way that the interaction is provided by boundary conditions. In order to find such Hamiltonians closed and semi-bounded quadratic forms are constructed, from which the associated self-adjoint operators are extracted. We provide a general characterisation of such operators and, furthermore, produce certain classes of examples. We then consider identical particles and project to the bosonic and fermionic subspaces. Finally, we show that the operators possess purely discrete spectra and that the eigenvalues are distributed following an appropriate Weyl asymptotic law

An empirical initial-final mass relation from hot, massive white dwarfs in NGC 2168 (M35)

The relation between the zero-age main sequence mass of a star and its white-dwarf remnant (the initial-final mass relation) is a powerful tool for exploration of mass loss processes during stellar evolution. We present an empirical derivation of the initial-final mass relation based on spectroscopic analysis of seven massive white dwarfs in NGC 2168 (M35). Using an internally consistent data set, we show that the resultant white dwarf mass increases monotonically with progenitor mass for masses greater than 4 solar masses, one of the first open clusters to show this trend. We also find two massive white dwarfs foreground to the cluster that are otherwise consistent with cluster membership. These white dwarfs can be explained as former cluster members moving steadily away from the cluster at speeds of <~0.5 km/s since their formation and may provide the first direct evidence of the loss of white dwarfs from open clusters. Based on these data alone, we constrain the upper mass limit of WD progenitors to be >=5.8 solar masses at the 90% confidence level for a cluster age of 150 Myr.Comment: 14 pages, 3 figures. Accepted for publication in the Astrophysical Journal Letters. Contains some acknowledgements not in accepted version (for space reasons), otherwise identical to accepted versio

Partner orbits and action differences on compact factors of the hyperbolic plane. Part I: Sieber-Richter pairs

Physicists have argued that periodic orbit bunching leads to universal spectral fluctuations for chaotic quantum systems. To establish a more detailed mathematical understanding of this fact, it is first necessary to look more closely at the classical side of the problem and determine orbit pairs consisting of orbits which have similar actions. In this paper we specialize to the geodesic flow on compact factors of the hyperbolic plane as a classical chaotic system. We prove the existence of a periodic partner orbit for a given periodic orbit which has a small-angle self-crossing in configuration space which is a `2-encounter'; such configurations are called `Sieber-Richter pairs' in the physics literature. Furthermore, we derive an estimate for the action difference of the partners. In the second part of this paper [13], an inductive argument is provided to deal with higher-order encounters.Comment: to appear on Nonlinearit