Critical Behaviour of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom distribution

We exploit a relation between the mean number $N_{m}$ of minima of random Gaussian surfaces and extreme eigenvalues of random matrices to understand the critical behaviour of $N_{m}$ in the simplest glass-like transition occuring in a toy model of a single particle in $N$-dimensional random environment, with $N\gg 1$. Varying the control parameter $\mu$ through the critical value $\mu_c$ we analyse in detail how $N_{m}(\mu)$ drops from being exponentially large in the glassy phase to $N_{m}(\mu)\sim 1$ on the other side of the transition. We also extract a subleading behaviour of $N_{m}(\mu)$ in both glassy and simple phases. The width $\delta{\mu}/\mu_c$ of the critical region is found to scale as $N^{-1/3}$ and inside that region $N_{m}(\mu)$ converges to a limiting shape expressed in terms of the Tracy-Widom distribution

Restricted Isometries for Partial Random Circulant Matrices

In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampling matrix are small. Many potential applications of compressed sensing involve a data-acquisition process that proceeds by convolution with a random pulse followed by (nonrandom) subsampling. At present, the theoretical analysis of this measurement technique is lacking. This paper demonstrates that the $s$th order restricted isometry constant is small when the number $m$ of samples satisfies $m \gtrsim (s \log n)^{3/2}$, where $n$ is the length of the pulse. This bound improves on previous estimates, which exhibit quadratic scaling

Mathematics of Gravitational Lensing: Multiple Imaging and Magnification

The mathematical theory of gravitational lensing has revealed many generic and global properties. Beginning with multiple imaging, we review Morse-theoretic image counting formulas and lower bound results, and complex-algebraic upper bounds in the case of single and multiple lens planes. We discuss recent advances in the mathematics of stochastic lensing, discussing a general formula for the global expected number of minimum lensed images as well as asymptotic formulas for the probability densities of the microlensing random time delay functions, random lensing maps, and random shear, and an asymptotic expression for the global expected number of micro-minima. Multiple imaging in optical geometry and a spacetime setting are treated. We review global magnification relation results for model-dependent scenarios and cover recent developments on universal local magnification relations for higher order caustics.Comment: 25 pages, 4 figures. Invited review submitted for special issue of General Relativity and Gravitatio

Benign external hydrocephalus: a review, with emphasis on management

Benign external hydrocephalus in infants, characterized by macrocephaly and typical neuroimaging findings, is considered as a self-limiting condition and is therefore rarely treated. This review concerns all aspects of this condition: etiology, neuroimaging, symptoms and clinical findings, treatment, and outcome, with emphasis on management. The review is based on a systematic search in the Pubmed and Web of Science databases. The search covered various forms of hydrocephalus, extracerebral fluid, and macrocephaly. Studies reporting small children with idiopathic external hydrocephalus were included, mostly focusing on the studies reporting a long-term outcome. A total of 147 studies are included, the majority however with a limited methodological quality. Several theories regarding pathophysiology and various symptoms, signs, and clinical findings underscore the heterogeneity of the condition. Neuroimaging is important in the differentiation between external hydrocephalus and similar conditions. A transient delay of psychomotor development is commonly seen during childhood. A long-term outcome is scarcely reported, and the results are varying. Although most children with external hydrocephalus seem to do well both initially and in the long term, a substantial number of patients show temporary or permanent psychomotor delay. To verify that this truly is a benign condition, we suggest that future research on external hydrocephalus should focus on the long-term effects of surgical treatment as opposed to conservative management

Exome sequencing identifies germline variants in DIS3 in familial multiple myeloma

[Excerpt] Multiple myeloma (MM) is the third most common hematological malignancy, after Non-Hodgkin Lymphoma and Leukemia. MM is generally preceded by Monoclonal Gammopathy of Undetermined Significance (MGUS) [1], and epidemiological studies have identified older age, male gender, family history, and MGUS as risk factors for developing MM [2]. The somatic mutational landscape of sporadic MM has been increasingly investigated, aiming to identify recurrent genetic events involved in myelomagenesis. Whole exome and whole genome sequencing studies have shown that MM is a genetically heterogeneous disease that evolves through accumulation of both clonal and subclonal driver mutations [3] and identified recurrently somatically mutated genes, including KRAS, NRAS, FAM46C, TP53, DIS3, BRAF, TRAF3, CYLD, RB1 and PRDM1 [3,4,5]. Despite the fact that family-based studies have provided data consistent with an inherited genetic susceptibility to MM compatible with Mendelian transmission [6], the molecular basis of inherited MM predisposition is only partly understood. Genome-Wide Association (GWAS) studies have identified and validated 23 loci significantly associated with an increased risk of developing MM that explain ~16% of heritability [7] and only a subset of familial cases are thought to have a polygenic background [8]. Recent studies have identified rare germline variants predisposing to MM in KDM1A [9], ARID1A and USP45 [10], and the implementation of next-generation sequencing technology will allow the characterization of more such rare variants. [...]French National Cancer Institute (INCA) and the Fondation Française pour la Recherche contre le Myélome et les Gammapathies (FFMRG), the Intergroupe Francophone du Myélome (IFM), NCI R01 NCI CA167824 and a generous donation from Matthew Bell. This work was supported in part through the computational resources and staff expertise provided by Scientific Computing at the Icahn School of Medicine at Mount Sinai. Research reported in this paper was supported by the Office of Research Infrastructure of the National Institutes of Health under award number S10OD018522. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. The authors thank the Association des Malades du Myélome Multiple (AF3M) for their continued support and participation. Where authors are identified as personnel of the International Agency for Research on Cancer / World Health Organization, the authors alone are responsible for the views expressed in this article and they do not necessarily represent the decisions, policy or views of the International Agency for Research on Cancer / World Health Organizatio