Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations

In this article, we study the self-similar solutions of the 2-component Camassa-Holm equations% \begin{equation} \left\{ \begin{array} [c]{c}% \rho_{t}+u\rho_{x}+\rho u_{x}=0 m_{t}+2u_{x}m+um_{x}+\sigma\rho\rho_{x}=0 \end{array} \right. \end{equation} with \begin{equation} m=u-\alpha^{2}u_{xx}. \end{equation} By the separation method, we can obtain a class of blowup or global solutions for $\sigma=1$ or $-1$. In particular, for the integrable system with $\sigma=1$, we have the global solutions:% \begin{equation} \left\{ \begin{array} [c]{c}% \rho(t,x)=\left\{ \begin{array} [c]{c}% \frac{f\left( \eta\right) }{a(3t)^{1/3}},\text{ for }\eta^{2}<\frac {\alpha^{2}}{\xi} 0,\text{ for }\eta^{2}\geq\frac{\alpha^{2}}{\xi}% \end{array} \right. ,u(t,x)=\frac{\overset{\cdot}{a}(3t)}{a(3t)}x \overset{\cdot\cdot}{a}(s)-\frac{\xi}{3a(s)^{1/3}}=0,\text{ }a(0)=a_{0}% >0,\text{ }\overset{\cdot}{a}(0)=a_{1} f(\eta)=\xi\sqrt{-\frac{1}{\xi}\eta^{2}+\left( \frac{\alpha}{\xi}\right) ^{2}}% \end{array} \right. \end{equation} where $\eta=\frac{x}{a(s)^{1/3}}$ with $s=3t;$ $\xi>0$ and $\alpha\geq0$ are arbitrary constants.\newline Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems.Comment: 5 more figures can be found in the corresponding journal paper (J. Math. Phys. 51, 093524 (2010) ). Key Words: 2-Component Camassa-Holm Equations, Shallow Water System, Analytical Solutions, Blowup, Global, Self-Similar, Separation Method, Construction of Solutions, Moving Boundar

Mexico's Health System: More Comprehensive Reform Needed

Jason Lakin discusses and critiques a Policy Forum that reviews 25 years of reform to the Mexican health system and argues that more comprehensive reform is needed

First evidence for a latitudinal body mass effect in extant Crocodylia and the relationships of their reproductive characters

Relationships between distribution patterns and body size have been documented in many endothermic taxa. However, the evidence for these trends in ectotherms generally is equivocal, and there have been no studies of effects in crocodylians specifically. Here, we examine the relationship between latitudinal distribution and body mass in 20 extant species of crocodylians, as well as the relationships between seven important reproductive variables. Using phylogenetically independent contrasts to inform generalized linear models, we provide the first evidence of a latitudinal effect on adult female body mass in crocodylians. In addition, we explore the relationships between reproductive variables including egg mass, hatchling mass and clutch size. We report no correlation between egg mass and clutch size, upholding previously reported within-species trends. We also find no evidence of a correlation between measures of latitudinal range and incubation temperature, contrasting with the trends found in turtles

Experimental Biological Protocols with Formal Semantics

Both experimental and computational biology is becoming increasingly automated. Laboratory experiments are now performed automatically on high-throughput machinery, while computational models are synthesized or inferred automatically from data. However, integration between automated tasks in the process of biological discovery is still lacking, largely due to incompatible or missing formal representations. While theories are expressed formally as computational models, existing languages for encoding and automating experimental protocols often lack formal semantics. This makes it challenging to extract novel understanding by identifying when theory and experimental evidence disagree due to errors in the models or the protocols used to validate them. To address this, we formalize the syntax of a core protocol language, which provides a unified description for the models of biochemical systems being experimented on, together with the discrete events representing the liquid-handling steps of biological protocols. We present both a deterministic and a stochastic semantics to this language, both defined in terms of hybrid processes. In particular, the stochastic semantics captures uncertainties in equipment tolerances, making it a suitable tool for both experimental and computational biologists. We illustrate how the proposed protocol language can be used for automated verification and synthesis of laboratory experiments on case studies from the fields of chemistry and molecular programming

Amplified biochemical oscillations in cellular systems

We describe a mechanism for pronounced biochemical oscillations, relevant to microscopic systems, such as the intracellular environment. This mechanism operates for reaction schemes which, when modeled using deterministic rate equations, fail to exhibit oscillations for any values of rate constants. The mechanism relies on amplification of the underlying stochasticity of reaction kinetics within a narrow window of frequencies. This amplification allows fluctuations to beat the central limit theorem, having a dominant effect even though the number of molecules in the system is relatively large. The mechanism is quantitatively studied within simple models of self-regulatory gene expression, and glycolytic oscillations.Comment: 35 pages, 6 figure

Electromagnetically Induced Transparency and Slow Light with Optomechanics

Controlling the interaction between localized optical and mechanical excitations has recently become possible following advances in micro- and nano-fabrication techniques. To date, most experimental studies of optomechanics have focused on measurement and control of the mechanical subsystem through its interaction with optics, and have led to the experimental demonstration of dynamical back-action cooling and optical rigidity of the mechanical system. Conversely, the optical response of these systems is also modified in the presence of mechanical interactions, leading to strong nonlinear effects such as Electromagnetically Induced Transparency (EIT) and parametric normal-mode splitting. In atomic systems, seminal experiments and proposals to slow and stop the propagation of light, and their applicability to modern optical networks, and future quantum networks, have thrust EIT to the forefront of experimental study during the last two decades. In a similar fashion, here we use the optomechanical nonlinearity to control the velocity of light via engineered photon-phonon interactions. Our results demonstrate EIT and tunable optical delays in a nanoscale optomechanical crystal device, fabricated by simply etching holes into a thin film of silicon (Si). At low temperature (8.7 K), we show an optically-tunable delay of 50 ns with near-unity optical transparency, and superluminal light with a 1.4 microseconds signal advance. These results, while indicating significant progress towards an integrated quantum optomechanical memory, are also relevant to classical signal processing applications. Measurements at room temperature and in the analogous regime of Electromagnetically Induced Absorption (EIA) show the utility of these chip-scale optomechanical systems for optical buffering, amplification, and filtering of microwave-over-optical signals.Comment: 15 pages, 9 figure

Constraint solving in non-permutative nominal abstract syntax

Nominal abstract syntax is a popular first-order technique for encoding, and reasoning about, abstract syntax involving binders. Many of its applications involve constraint solving. The most commonly used constraint solving algorithm over nominal abstract syntax is the Urban-Pitts-Gabbay nominal unification algorithm, which is well-behaved, has a well-developed theory and is applicable in many cases. However, certain problems require a constraint solver which respects the equivariance property of nominal logic, such as Cheney's equivariant unification algorithm. This is more powerful but is more complicated and computationally hard. In this paper we present a novel algorithm for solving constraints over a simple variant of nominal abstract syntax which we call non-permutative. This constraint problem has similar complexity to equivariant unification but without many of the additional complications of the equivariant unification term language. We prove our algorithm correct, paying particular attention to issues of termination, and present an explicit translation of name-name equivariant unification problems into non-permutative constraints

Muon - proton inelastic scattering

This experiment will examine muon-proton inelastic scattering for virtual-photon energies of 10 to 110 GeV and for |q{sup 2}| values of 0.2 to 20.0 (GeV/c){sup 2}. The virtual-photon total cross sections {sigma}{sub t} + {epsilon}{sigma}{sub s}, or the equivalent expression in W{sub 1} and W{sub 2}, will be measured over this range of virtual-photon energies and q{sup 2} values. Some separation of {sigma}{sub T} and {sigma}{sub S}, or equivalently W{sub 1} and W{sub 2}, will be made. The multiplicity, momentum spectra and angular spectra of the charged hadrons produced in this reaction will be measured. Some channels such as {mu} + P {yields} {mu} + P + P{sup 0} will be isolated and completely analyzed. The experiment uses a hydrogen target, wire spark chambers and an analyzing magnet of conventional design

Detection of C3_{3} in Diffuse Interstellar Clouds

The smallest polyatomic carbon chain, C$_{3}$, has been identified in interstellar clouds (A$_{v}\sim$1 mag) towards $\zeta$ Ophiuchi, 20 Aquilae, and $\zeta$ Persei by detection of the origin band in its $A^{1}\Pi_{u}-X^{1}\Sigma^{+}_{g}$ electronic transition, near 4052\AA. Individual rotational lines were resolved up to $J$=30 enabling the rotational level column densities and temperature distributions to be determined. The inferred limits for the total column densities ($\sim$1 to 2$\times10^{12}$ cm$^{-2}$) offer a strong incentive to laboratory and astrophysical searches for the longer carbon chains. Concurrent searches for C$_2^{+}$, C$_2^{-}$ and C$_3^{-}$ were negative but provide sensitive estimates for their maximum column densities.Comment: 14 pages, 5 figures, 3 tables. To appear in the Astrophysical Journa

Identification of the first ATRIP-deficient patient and novel mutations in ATR define a clinical spectrum for ATR-ATRIP Seckel Syndrome

A homozygous mutational change in the Ataxia-Telangiectasia and RAD3 related (ATR) gene was previously reported in two related families displaying Seckel Syndrome (SS). Here, we provide the first identification of a Seckel Syndrome patient with mutations in ATRIP, the gene encoding ATR-Interacting Protein (ATRIP), the partner protein of ATR required for ATR stability and recruitment to the site of DNA damage. The patient has compound heterozygous mutations in ATRIP resulting in reduced ATRIP and ATR expression. A nonsense mutational change in one ATRIP allele results in a C-terminal truncated protein, which impairs ATR-ATRIP interaction; the other allele is abnormally spliced. We additionally describe two further unrelated patients native to the UK with the same novel, heterozygous mutations in ATR, which cause dramatically reduced ATR expression. All patient-derived cells showed defective DNA damage responses that can be attributed to impaired ATR-ATRIP function. Seckel Syndrome is characterised by microcephaly and growth delay, features also displayed by several related disorders including Majewski (microcephalic) osteodysplastic primordial dwarfism (MOPD) type II and Meier-Gorlin Syndrome (MGS). The identification of an ATRIP-deficient patient provides a novel genetic defect for Seckel Syndrome. Coupled with the identification of further ATR-deficient patients, our findings allow a spectrum of clinical features that can be ascribed to the ATR-ATRIP deficient sub-class of Seckel Syndrome. ATR-ATRIP patients are characterised by extremely severe microcephaly and growth delay, microtia (small ears), micrognathia (small and receding chin), and dental crowding. While aberrant bone development was mild in the original ATR-SS patient, some of the patients described here display skeletal abnormalities including, in one patient, small patellae, a feature characteristically observed in Meier-Gorlin Syndrome. Collectively, our analysis exposes an overlapping clinical manifestation between the disorders but allows an expanded spectrum of clinical features for ATR-ATRIP Seckel Syndrome to be define