Kinetically constrained spin models on trees

We analyze kinetically constrained 0-1 spin models (KCSM) on rooted and unrooted trees of finite connectivity. We focus in particular on the class of Friedrickson-Andersen models FA-jf and on an oriented version of them. These tree models are particularly relevant in physics literature since some of them undergo an ergodicity breaking transition with the mixed first-second order character of the glass transition. Here we first identify the ergodicity regime and prove that the critical density for FA-jf and OFA-jf models coincide with that of a suitable bootstrap percolation model. Next we prove for the first time positivity of the spectral gap in the whole ergodic regime via a novel argument based on martingales ideas. Finally, we discuss how this new technique can be generalized to analyze KCSM on the regular lattice $\mathbb{Z}^d$.Comment: Published in at http://dx.doi.org/10.1214/12-AAP891 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Polynomial maps with invertible sums of Jacobian matrices and of directional Derivatives

Let $F: C^n \rightarrow C^m$ be a polynomial map with $degF=d \geq 2$. We prove that $F$ is invertible if $m = n$ and $\sum^{d-1}_{i=1} JF(\alpha_i)$ is invertible for all $i$, which is trivially the case for invertible quadratic maps. More generally, we prove that for affine lines $L = \{\beta + \mu \gamma | \mu \in C\} \subseteq C^n$ ($\gamma \ne 0$), $F|_L$ is linearly rectifiable, if and only if $\sum^{d-1}_{i=1} JF(\alpha_i) \cdot \gamma \ne 0$ for all $\alpha_i \in L$. This appears to be the case for all affine lines $L$ when $F$ is injective and $d \le 3$. We also prove that if $m = n$ and $\sum^{n}_{i=1} JF(\alpha_i)$ is invertible for all $\alpha_i \in C^n$, then $F$ is a composition of an invertible linear map and an invertible polynomial map $X+H$ with linear part $X$, such that the subspace generated by $\{JH(\alpha) | \alpha \in C^n\}$ consists of nilpotent matrices

Note on Hermitian Jacobi Forms

We compare the spaces of Hermitian Jacobi forms (HJF) of weight $k$ and indices $1,2$ with classical Jacobi forms (JF) of weight $k$ and indices $1,2,4$. Using the embedding into JF, upper bounds for the order of vanishing of HJF at the origin is obtained. We compute the rank of HJF as a module over elliptic modular forms and prove the algebraic independence of the generators in case of index 1. Some related questions are discussed.Comment: 24 pages; title changed, abstract changed, some proofs expanded and new results adde

Scalar Field Cosmologies With Inverted Potentials

Regular bouncing solutions in the framework of a scalar-tensor gravity model were found in a recent work. We reconsider the problem in the Einstein frame (EF) in the present work. Singularities arising at the limit of physical viability of the model in the Jordan frame (JF) are either of the Big Bang or of the Big Crunch type in the EF. As a result we obtain integrable scalar field cosmological models in general relativity (GR) with inverted double-well potentials unbounded from below which possess solutions regular in the future, tending to a de Sitter space, and starting with a Big Bang. The existence of the two fixed points for the field dynamics at late times found earlier in the JF becomes transparent in the EF.Comment: 18 pages, 4 figure

The jugular foramen - a morphometric study

The jugular foramen (JF) varies in shape and size from side to side in the same cranium, and in different crania, racial groups and sexes. Side dominance is also said to be common. The foramen’s irregular shape, its formation by two bones and the numerous nerves and venous channels that pass through it further compound its anatomy. A morphometric study of 20 (40 JF) adult male Nigerian dry skulls was carried out. A bony bridge completely partitioned the JF in 3 (7.5%) of the JF. There was no tripartite JF. The JF mean length on the right and left were 13.90 mm (11.6–17.0 mm) and 14.11 mm (9.2–20.2 mm), while their widths measured 10.22 mm (6.8 –14.4 mm) and 9.57 mm (7.4–12.8 mm) on the right and left respectively. The mean JF area on the right was 437.49 mm (265.35–669.54 mm) and that on the left was 419.48 mm (276.46–634.60 mm). Side predominance of one of the JF appeared in 80% of cases. When present, the predominance of the right side was 55%, with 25% on the left. There was a difference in the length and width on each side but no significant difference in the length, width and area of the JF between the two sides. There was a positive correlation between skull width/ length and height/length ratio and JF area and length on each side. In conclusion, complete bony subdivision of the JF was not common among our study population and although the JF was generally larger on the right in our population, this is not statistically significant. A higher skull width/length and height/length ratio is associated with a greater JF length and area