Black probes of Schrödinger spacetimes

We consider black probes of Anti-de Sitter and Schrödinger spacetimes embedded in string theory and M-theory and construct perturbatively new black hole geometries. We begin by reviewing black string configurations in Anti-de Sitter dual to finite temperature Wilson loops in the deconfined phase of the gauge theory and generalise the construction to the confined phase. We then consider black strings in thermal Schrödinger, obtained via a null Melvin twist of the extremal D3-brane, and construct three distinct types of black string configurations with spacelike as well as lightlike separated boundary endpoints. One of these configurations interpolates between the Wilson loop operators, with bulk duals defined in Anti-de Sitter and another class of Wilson loop operators, with bulk duals defined in Schrödinger. The case of black membranes with boundary endpoints on the M5-brane dual to Wilson surfaces in the gauge theory is analysed in detail. Four types of black membranes, ending on the null Melvin twist of the extremal M5-brane exhibiting the Schrödinger symmetry group, are then constructed. We highlight the differences between Anti-de Sitter and Schrödinger backgrounds and make some comments on the properties of the corresponding dual gauge theories.

Nasolabial symmetry and aesthetics in children with complete unilateral cleft lip and palate

The objective of this study was to evaluate the association between nasolabial symmetry and aesthetics in children with complete unilateral cleft lip and palate (CUCLP). Frontal and basal photographs of 60 consecutively treated children with CUCLP (cleft group: 41 boys and 19 girls, mean (SD) age 11 (2) years) and 44 children without clefts (control group: 16 boys and 28 girls, mean (SD) age 11(2) years), were used for evaluation of nasolabial symmetry and aesthetics. Nasal and labial measurements were made to calculate the coefficient of asymmetry (CA). The 5-grade aesthetic index described by Asher-McDade et al. was used to evaluate nasolabial appearance. Correlation and regression analysis were used to identify an association between aesthetics and CA, sex, and the presence of CUCLP. Ten measurements in the cleft, and 2 in the control, group differed significantly between the cleft and non-cleft (or right and left) sides, respectively. The significantly higher values of 9 of 11 CA in the children with CUCLP indicated that they had more asymmetrical nasolabial areas than children without clefts. However, the regression analyses showed that only a few CA were associated with nasolabial aesthetics. In conclusion, nasolabial aesthetics and nasolabial symmetry seem to be only weakly associated in patients with CUCLP.

Lions-type compactness and Rubik actions on the Heisenberg group

In this paper we prove a Lions-type compactness embedding result for symmetric unbounded domains of the Heisenberg group. The natural group action on the Heisenberg group TeX is provided by the unitary group U(n) × {1} and its appropriate subgroups, which will be used to construct subspaces with specific symmetry and compactness properties in the Folland-Stein’s horizontal Sobolev space TeX. As an application, we study the multiplicity of solutions for a singular subelliptic problem by exploiting a technique of solving the Rubik-cube applied to subgroups of U(n) × {1}. In our approach we employ concentration compactness, group-theoretical arguments, and variational methods.

Symmetry and the polar state of condensed molecular matter

Polar molecular crystals seem to contradict a quantum mechanical statement, according to which no stationary state of a system features a permanent electrical polarization. By stationary we understand here an ensemble for which thermal averaging applies. In the language of statistical mechanics we have thus to ask for the thermal expectation value of the polarization in molecular crystals. Nucleation aggregates and growing crystal surfaces can provide a single degree of freedom for polar molecules required to average the polarization. By means of group theoretical reasoning and Monte Carlo simulations we show that such systems thermalize into a bi-polar state featuring zero bulk polarity. A two domain, i.e. bipolar state is obtained because boundaries are setting up opposing effective electrical fields. Described phenomena can be understood as a process of partial ergodicity-restoring. Experimentally, a bi-polar state of molecular crystals was demonstrated using phase sensitive second harmonic generation and scanning pyroelectric microscopy