Gibrat's law for cities: uniformly most powerful unbiased test of the Pareto against the lognormal

We address the general problem of testing a power law distribution versus a log-normal distribution in statistical data. This general problem is illustrated on the distribution of the 2000 US census of city sizes. We provide definitive results to close the debate between Eeckhout (2004, 2009) and Levy (2009) on the validity of Zipf's law, which is the special Pareto law with tail exponent 1, to describe the tail of the distribution of U.S. city sizes. Because the origin of the disagreement between Eeckhout and Levy stems from the limited power of their tests, we perform the {\em uniformly most powerful unbiased test} for the null hypothesis of the Pareto distribution against the lognormal. The $p$-value and Hill's estimator as a function of city size lower threshold confirm indubitably that the size distribution of the 1000 largest cities or so, which include more than half of the total U.S. population, is Pareto, but we rule out that the tail exponent, estimated to be $1.4 \pm 0.1$, is equal to 1. For larger ranks, the $p$-value becomes very small and Hill's estimator decays systematically with decreasing ranks, qualifying the lognormal distribution as the better model for the set of smaller cities. These two results reconcile the opposite views of Eeckhout (2004, 2009) and Levy (2009). We explain how Gibrat's law of proportional growth underpins both the Pareto and lognormal distributions and stress the key ingredient at the origin of their difference in standard stochastic growth models of cities \cite{Gabaix99,Eeckhout2004}.Comment: 7 pages + 2 figure

Turbulence and Multiscaling in the Randomly Forced Navier Stokes Equation

We present an extensive pseudospectral study of the randomly forced Navier-Stokes equation (RFNSE) stirred by a stochastic force with zero mean and a variance $\sim k^{4-d-y}$, where $k$ is the wavevector and the dimension $d = 3$. We present the first evidence for multiscaling of velocity structure functions in this model for $y \geq 4$. We extract the multiscaling exponent ratios $\zeta_p/\zeta_2$ by using extended self similarity (ESS), examine their dependence on $y$, and show that, if $y = 4$, they are in agreement with those obtained for the deterministically forced Navier-Stokes equation ($3d$NSE). We also show that well-defined vortex filaments, which appear clearly in studies of the $3d$NSE, are absent in the RFNSE.Comment: 4 pages (revtex), 6 figures (postscript

Adiabaticity Conditions for Volatility Smile in Black-Scholes Pricing Model

Our derivation of the distribution function for future returns is based on the risk neutral approach which gives a functional dependence for the European call (put) option price, C(K), given the strike price, K, and the distribution function of the returns. We derive this distribution function using for C(K) a Black-Scholes (BS) expression with volatility in the form of a volatility smile. We show that this approach based on a volatility smile leads to relative minima for the distribution function ("bad" probabilities) never observed in real data and, in the worst cases, negative probabilities. We show that these undesirable effects can be eliminated by requiring "adiabatic" conditions on the volatility smile

Vascular abnormalities visualized by multislice computed tomography of the abdomen: accidental findings or immediate causes of pain syndrome? (topic review)

Abnormalities of the abdominal aorta branches may cause chronic mesenteric ischemia, progressive pathological changes of the internal organs in this setting, and chronic pain syndrome. The causes of chronic mesenteric ischemia may be divided into atherosclerotic and non-atherosclerotic. Atherosclerosis of the unpaired branches of the abdominal aorta involves stenosis or occlusion. Other causes include fibromuscular dysplasia, vasculitis (Takayasu, segmental mediolytic arteriopathy), and median arcuate ligament syndrome. These syndromes, the pathogenesis of some of which remains controversial, lead to nonspecific complaints such as abdominal pain, weight loss, and others. Digital subtraction angiography or duplex ultrasound may provide hemodynamic information in cases of vascular disease in this area. However, multislice spiral computed tomography is in many cases the first choice because it allows for a comprehensive assessment of the state of blood vessels and associated morphological changes of internal organs. Structural changes accompanying these syndromes can also occur in patients who are undergoing a medical examination for other reasons. However, these syndromes should not be diagnosed solely on the basis of imaging; instead, the findings should be compared with the clinical presentation, which implies collaboration of radiologists and clinicians

Inertial- and Dissipation-Range Asymptotics in Fluid Turbulence

We propose and verify a wave-vector-space version of generalized extended self similarity and broaden its applicability to uncover intriguing, universal scaling in the far dissipation range by computing high-order (\leq 20\/) structure functions numerically for: (1) the three-dimensional, incompressible Navier Stokes equation (with and without hyperviscosity); and (2) the GOY shell model for turbulence. Also, in case (2), with Taylor-microscale Reynolds numbers 4 \times 10^{4} \leq Re_{\lambda} \leq 3 \times 10^{6}\/, we find that the inertial-range exponents (\zeta_{p}\/) of the order - p\/ structure functions do not approach their Kolmogorov value p/3\/ as Re_{\lambda}\/ increases.Comment: RevTeX file, with six postscript figures. epsf.tex macro is used for figure insertion. Packaged using the 'uufiles' utilit

Dragon-kings: mechanisms, statistical methods and empirical evidence

This introductory article presents the special Discussion and Debate volume "From black swans to dragon-kings, is there life beyond power laws?" published in Eur. Phys. J. Special Topics in May 2012. We summarize and put in perspective the contributions into three main themes: (i) mechanisms for dragon-kings, (ii) detection of dragon-kings and statistical tests and (iii) empirical evidence in a large variety of natural and social systems. Overall, we are pleased to witness significant advances both in the introduction and clarification of underlying mechanisms and in the development of novel efficient tests that demonstrate clear evidence for the presence of dragon-kings in many systems. However, this positive view should be balanced by the fact that this remains a very delicate and difficult field, if only due to the scarcity of data as well as the extraordinary important implications with respect to hazard assessment, risk control and predictability.Comment: 20 page

New Genetic Markers for Molecular Typing of Bacillus anthracis Strains

Objective: Identification of new markers for the molecular typing of Bacillus anthracis. Materials and methods. The genomes of 16 B. anthracis strains from the collection of the Stavropol Research Anti-Plague Institute, 11 B. anthracis strains and 5 strains of Bacillus cereus from GenBank were investigated. The methods of in vitro and in silico analysis of canonical and whole-genome single nucleotide polymorphisms (SNP), genome regions with variable number of tandem repeats (VNTR) were used for the analysis. Results and discussion. It has been established that there are deletions and (or) SNPs in some of B. anthracis strains of the main genetic lineage B, within the homologous genes of the tri-cistronic operon gerH, which encodes spore germination proteins. gerA genes contain the Bams34 VNTR locus, the sizes of genes in different strains vary due to the different number of tandem repeats and the presence of indels, which suggests the variability of GerA spore germination proteins. In the area of reverse primer annealing, some of them have several SNPs or deletions, which makes impossible PCR amplification of the Bams34 locus. Previously not described VNTR locus, SNPs and indels in sequences of plasmids pXO1 and pXO2, as well as SNP in chromosomal gene of glycerol-3-phosphate transporter were identified. Two pairs of PCR primers for the variable regions of the plasmids were designed. VNTR-locus, SNP and indels in sequences of plasmids pXO1 and pXO2 are suitable genetic markers for the differentiation of typical virulent diplasmid strains belonging to the main genetic lineages of B. anthracis A, B and C. The allele T of SNP within chromosomal glpT gene is specific for one of two strains isolated during the outbreak of anthrax and distinguishes it from all other strains of B. anthracis