Entanglement entropy of fermions in any dimension and the Widom conjecture

We show that entanglement entropy of free fermions scales faster then area law, as opposed to the scaling $L^{d-1}$ for the harmonic lattice, for example. We also suggest and provide evidence in support of an explicit formula for the entanglement entropy of free fermions in any dimension $d$, $S\sim c(\partial\Gamma,\partial\Omega)\cdot L^{d-1}\log L$ as the size of a subsystem $L\to\infty$, where $\partial\Gamma$ is the Fermi surface and $\partial\Omega$ is the boundary of the region in real space. The expression for the constant $c(\partial\Gamma,\partial\Omega)$ is based on a conjecture due to H. Widom. We prove that a similar expression holds for the particle number fluctuations and use it to prove a two sided estimates on the entropy $S$.Comment: Final versio

Quantum theta functions and Gabor frames for modulation spaces

Representations of the celebrated Heisenberg commutation relations in quantum mechanics and their exponentiated versions form the starting point for a number of basic constructions, both in mathematics and mathematical physics (geometric quantization, quantum tori, classical and quantum theta functions) and signal analysis (Gabor analysis). In this paper we try to bridge the two communities, represented by the two co--authors: that of noncommutative geometry and that of signal analysis. After providing a brief comparative dictionary of the two languages, we will show e.g. that the Janssen representation of Gabor frames with generalized Gaussians as Gabor atoms yields in a natural way quantum theta functions, and that the Rieffel scalar product and associativity relations underlie both the functional equations for quantum thetas and the Fundamental Identity of Gabor analysis.Comment: 38 pages, typos corrected, MSC class change

A Guide to Localized Frames and Applications to Galerkin-like Representations of Operators

This chapter offers a detailed survey on intrinsically localized frames and the corresponding matrix representation of operators. We re-investigate the properties of localized frames and the associated Banach spaces in full detail. We investigate the representation of operators using localized frames in a Galerkin-type scheme. We show how the boundedness and the invertibility of matrices and operators are linked and give some sufficient and necessary conditions for the boundedness of operators between the associated Banach spaces.Comment: 32 page

Approximation of Feynman path integrals with non-smooth potentials

We study the convergence in $L^2$ of the time slicing approximation of Feynman path integrals under low regularity assumptions on the potential. Inspired by the custom in Physics and Chemistry, the approximate propagators considered here arise from a series expansion of the action. The results are ultimately based on function spaces, tools and strategies which are typical of Harmonic and Time-frequency analysis.Comment: 18 page

Understanding Terrorist Organizations with a Dynamic Model

Terrorist organizations change over time because of processes such as recruitment and training as well as counter-terrorism (CT) measures, but the effects of these processes are typically studied qualitatively and in separation from each other. Seeking a more quantitative and integrated understanding, we constructed a simple dynamic model where equations describe how these processes change an organization's membership. Analysis of the model yields a number of intuitive as well as novel findings. Most importantly it becomes possible to predict whether counter-terrorism measures would be sufficient to defeat the organization. Furthermore, we can prove in general that an organization would collapse if its strength and its pool of foot soldiers decline simultaneously. In contrast, a simultaneous decline in its strength and its pool of leaders is often insufficient and short-termed. These results and other like them demonstrate the great potential of dynamic models for informing terrorism scholarship and counter-terrorism policy making.Comment: To appear as Springer Lecture Notes in Computer Science v2: vectorized 4 figures, fixed two typos, more detailed bibliograph

Linear perturbations of the Wigner transform and the Weyl quantization

We study a class of quadratic time-frequency representations that, roughly speaking, are obtained by linear perturbations of the Wigner transform. They satisfy Moyal's formula by default and share many other properties with the Wigner transform, but in general they do not belong to Cohen's class. We provide a characterization of the intersection of the two classes. To any such time-frequency representation, we associate a pseudodifferential calculus. We investigate the related quantization procedure, study the properties of the pseudodifferential operators, and compare the formalism with that of the Weyl calculus.Comment: 38 pages. Contributed chapter for volume on the occasion of Luigi Rodino's 70th birthda

A spoonful of L‐fucose—an efficient therapy for GFUS‐CDG, a new glycosylation disorder

Abstract Congenital disorders of glycosylation are a genetically and phenotypically heterogeneous family of diseases affecting the co‐ and posttranslational modification of proteins. Using exome sequencing, we detected biallelic variants in GFUS (NM_003313.4) c.[632G>A];[659C>T] (p.[Gly211Glu];[Ser220Leu]) in a patient presenting with global developmental delay, mild coarse facial features and faltering growth. GFUS encodes GDP‐L‐fucose synthase, the terminal enzyme in de novo synthesis of GDP‐L‐fucose, required for fucosylation of N‐ and O‐glycans. We found reduced GFUS protein and decreased GDP‐L‐fucose levels leading to a general hypofucosylation determined in patient's glycoproteins in serum, leukocytes, thrombocytes and fibroblasts. Complementation of patient fibroblasts with wild‐type GFUS cDNA restored fucosylation. Making use of the GDP‐L‐fucose salvage pathway, oral fucose supplementation normalized fucosylation of proteins within 4 weeks as measured in serum and leukocytes. During the follow‐up of 19 months, a moderate improvement of growth was seen, as well as a clear improvement of cognitive skills as measured by the Kaufmann ABC and the Nijmegen Pediatric CDG Rating Scale. In conclusion, GFUS‐CDG is a new glycosylation disorder for which oral L‐fucose supplementation is promising

Utility of Whole Blood Thiamine Pyrophosphate Evaluation in TPK1-Related Diseases

TPK1 mutations are a rare, but potentially treatable, cause of thiamine deficiency. Diagnosis is challenging given the phenotypic overlap that exists with other metabolic and neurological disorders. We report a case of TPK1-related disease presenting with Leigh-like syndrome and review the diagnostic utility of thiamine pyrophosphate (TPP) blood measurement. The proband, a 35-year-old male, presented at four months of age with recurrent episodes of post-infectious encephalopathy. He subsequently developed epilepsy, learning difficulties, sensorineural hearing loss, spasticity, and dysphagia. There was a positive family history for Leigh syndrome in an older brother. Plasma lactate was elevated (3.51 mmol/L) and brain MRI showed bilateral basal ganglia hyperintensities, indicative of Leigh syndrome. Histochemical and spectrophotometric analysis of mitochondrial respiratory chain complexes I, II+III, and IV was normal. Genetic analysis of muscle mitochondrial DNA was negative. Whole exome sequencing of the proband confirmed compound heterozygous variants in TPK1: c. 426G>C (p. Leu142Phe) and c. 258+1G>A (p.?). Blood TPP levels were reduced, providing functional evidence for the deleterious effects of the variants. We highlight the clinical and bioinformatics challenges to diagnosing rare genetic disorders and the continued utility of biochemical analyses, despite major advances in DNA sequencing technology, when investigating novel, potentially disease-causing, genetic variants. Blood TPP measurement represents a fast and cost-effective diagnostic tool in TPK1-related diseases

Technological Progress, Employment and the Lifetime of Capital

We study the impact of technological progress on the level of employment in a vintage capital model where: i) capital and labor are gross complementary; ii) labor supply is endogenous and indivisible; iii) there is full employment, and iv) the rate of labor-saving technological progress is endogenous. We characterize the stationary distributions of vintage capital goods and the corresponding equilibrium values for employment and capital lifetime. It is shown that both variables are non-monotonic functions of technological progress indicators. Technological accelerations are found to increase employment provided innovations are not too radical