Baby-Step Giant-Step Algorithms for the Symmetric Group

We study discrete logarithms in the setting of group actions. Suppose that $G$ is a group that acts on a set $S$. When $r,s \in S$, a solution $g \in G$ to $r^g = s$ can be thought of as a kind of logarithm. In this paper, we study the case where $G = S_n$, and develop analogs to the Shanks baby-step / giant-step procedure for ordinary discrete logarithms. Specifically, we compute two sets $A, B \subseteq S_n$ such that every permutation of $S_n$ can be written as a product $ab$ of elements $a \in A$ and $b \in B$. Our deterministic procedure is optimal up to constant factors, in the sense that $A$ and $B$ can be computed in optimal asymptotic complexity, and $|A|$ and $|B|$ are a small constant from $\sqrt{n!}$ in size. We also analyze randomized "collision" algorithms for the same problem

Matching for Several Sparse Nominal Variables in a Case-Control Study of Readmission Following Surgery.

Matching for several nominal covariates with many levels has usually been thought to be difficult because these covariates combine to form an enormous number of interaction categories with few if any people in most such categories. Moreover, because nominal variables are not ordered, there is often no notion of a close substitute when an exact match is unavailable. In a case-control study of the risk factors for read-mission within 30 days of surgery in the Medicare population, we wished to match for 47 hospitals, 15 surgical procedures grouped or nested within 5 procedure groups, two genders, or 47 × 15 × 2 = 1410 categories. In addition, we wished to match as closely as possible for the continuous variable age (65-80 years). There were 1380 readmitted patients or cases. A fractional factorial experiment may balance main effects and low-order interactions without achieving balance for high-order interactions. In an analogous fashion, we balance certain main effects and low-order interactions among the covariates; moreover, we use as many exactly matched pairs as possible. This is done by creating a match that is exact for several variables, with a close match for age, and both a near-exact match and a finely balanced match for another nominal variable, in this case a 47 × 5 = 235 category variable representing the interaction of the 47 hospitals and the five surgical procedure groups. The method is easily implemented in R

Does a magnetic field modify the critical behaviour at the metal-insulator transition in 3-dimensional disordered systems?

The critical behaviour of 3-dimensional disordered systems with magnetic field is investigated by analyzing the spectral fluctuations of the energy spectrum. We show that in the thermodynamic limit we have two different regimes, one for the metallic side and one for the insulating side with different level statistics. The third statistics which occurs only exactly at the critical point is {\it independent} of the magnetic field. The critical behaviour which is determined by the symmetry of the system {\it at} the critical point should therefore be independent of the magnetic field.Comment: 10 pages, Revtex, 4 PostScript figures in uuencoded compressed tar file are appende

Electrical properties of isotopically enriched neutron-transmutation-doped ^{70} Ge:Ga near the metal-insulator transition

We report the low temperature carrier transport properties of a series of nominally uncompensated neutron-transmutation doped (NTD) ^{70} Ge:Ga samples very close to the critical concentration N_c for the metal-insulator transition. The concentration of the sample closest to N_c is 1.0004N_c and it is unambiguously shown that the critical conductivity exponent is 0.5. Properties of insulating samples are discussed in the context of Efros and Shklovskii's variable range hopping conduction.Comment: 8 pages using REVTeX, 8 figures, published versio

Secondary school pupils' preferences for different types of structured grouping practices

The aim of this paper is to explore pupils’ preferences for particular types of grouping practices an area neglected in earlier research focusing on the personal and social outcomes of ability grouping. The sample comprised over 5,000 year 9 pupils (aged 13-14 years) in 45 mixed secondary comprehensive schools in England. The schools represented three levels of ability grouping in the lower school (years 7 to 9). Pupils responded to a questionnaire which explored the types of grouping that they preferred and the reasons for their choices. The majority of pupils preferred setting, although this was mediated by their set placement, type of school, socio-economic status and gender. The key reason given for this preference was that it enabled work to be matched to learning needs. The paper considers whether there are other ways of achieving this avoiding the negative social and personal outcomes of setting for some pupils

Path dependent scaling of geometric phase near a quantum multi-critical point

We study the geometric phase of the ground state in a one-dimensional transverse XY spin chain in the vicinity of a quantum multi-critical point. We approach the multi-critical point along different paths and estimate the geometric phase by applying a rotation in all spins about z-axis by an angle $\eta$. Although the geometric phase itself vanishes at the multi-critical point, the derivative with respect to the anisotropy parameter of the model shows peaks at different points on the ferromagnetic side close to it where the energy gap is a local minimum; we call these points `quasi-critical'. The value of the derivative at any quasi-critical point scales with the system size in a power-law fashion with the exponent varying continuously with the parameter $\alpha$ that defines a path, upto a critical value $\alpha = \alpha_{c}=2$. For $\alpha > \alpha_{c}$, or on the paramagnetic side no such peak is observed. Numerically obtained results are in perfect agreement with analytical predictions.Comment: 5 pages, 6 figure

The low-density/high-density liquid phase transition for model globular proteins

The effect of molecule size (excluded volume) and the range of interaction on the surface tension, phase diagram and nucleation properties of a model globular protein is investigated using a combinations of Monte Carlo simulations and finite temperature classical Density Functional Theory calculations. We use a parametrized potential that can vary smoothly from the standard Lennard-Jones interaction characteristic of simple fluids, to the ten Wolde-Frenkel model for the effective interaction of globular proteins in solution. We find that the large excluded volume characteristic of large macromolecules such as proteins is the dominant effect in determining the liquid-vapor surface tension and nucleation properties. The variation of the range of the potential only appears important in the case of small excluded volumes such as for simple fluids. The DFT calculations are then used to study homogeneous nucleation of the high-density phase from the low-density phase including the nucleation barriers, nucleation pathways and the rate. It is found that the nucleation barriers are typically only a few $k_{B}T$ and that the nucleation rates substantially higher than would be predicted by Classical Nucleation Theory.Comment: To appear in Langmui