Hyperbolic-like estimates for higher order equations

The main goal of this paper is to derive long time estimates of the energy for the higher order hyperbolic equations with time-dependent coefficients. in particular, we estimate the energy in the hyperbolic zone of the extended phase space by means of a function f (t) which depends on the principal part and on the coefficients of the terms of order m - 1. Then we look for sufficient conditions that guarantee the same energy estimate from above in all the extended phase space. We call this class of estimates hyperbolic-like since the energy behavior is deeply depending on the hyperbolic structure of the equation. In some cases, these estimates produce a dissipative effect on the energy. (C) 2012 Elsevier Inc. All rights reserved.

On the generalized canonical equations of Hamilton for a time-dependent mass particle

Fundamental principles of mechanics were primarily conceived for constant mass systems. Since the pioneering works of Meshcherskii (see historical review in Mikhailov (Mech. Solids 10(5):32-40, 1975), efforts have been made in order to elaborate an adequate mathematical formalism for variable mass systems. This is a current research field in theoretical mechanics. In this paper, attention is focused on the derivation of the so-called 'generalized canonical equations of Hamilton' for a variable mass particle. The applied technique consists in the consideration of the mass variation process as a dissipative phenomenon. Kozlov's (Stek. Inst. Math 223:178-184, 1998) method, originally devoted to the derivation of the generalized canonical equations of Hamilton for dissipative systems, is accordingly extended to the scenario of variable mass systems. This is done by conveniently writing the flux of kinetic energy from or into the variable mass particle as a 'Rayleigh-like dissipation function'. Cayley (Proc. R Soc. Lond. 8:506-511, 1857) was the first scholar to propose such an analogy. A deeper discussion on this particular subject will be left for a future paper.

Growth standards of patients with Noonan and Noonan-like syndromes with mutations in the RAS/MAPK pathway

Noonan syndrome (NS) and Noonan-like syndromes (NLS) are autosomal dominant disorders caused by heterozygous mutations in genes of the RAS/MAPK pathway. The aim of the study was to construct specific growth charts for patients with NS and NLS. Anthropometric measurements (mean of 4.3 measurements per patient) were obtained in a mixed cross-sectional and longitudinal mode from 127 NS and 10 NLS patients with mutations identified in PTPN11 (n?=?90), SOS1 (n?=?14), RAF1 (n?=?10), KRAS (n?=?8), BRAF (n?=?11), and SHOC2 (n?=?4) genes. Height, weight, and body mass index (BMI) references were constructed using the lambda, mu, sigma (LMS) method. Patients had birth weight and length within normal ranges for gestational age although a higher preterm frequency (16%) was observed. Mean final heights were 157.4?cm [-2.4 standard deviation score (SDS)] and 148.4?cm (-2.2?SDS) for adult males and females, respectively. BMI SDS was lower when compared to Brazilian standards (BMI SDS of -0.9 and -0.5 SDS for males and females, respectively). Patients harboring mutations in RAF1 and SHOC2 gene were shorter than other genotypes, whereas patients with SOS1 and BRAF mutations had more preserved postnatal growth. In addition, patients with RAF1 and BRAF had the highest BMI whereas patients with SHOC2 and KRAS mutations had the lowest BMI. The present study established the first height, weight, and BMI reference curves for NS and NLS patients, based only on patients with a proven molecular cause. These charts can be useful for the clinical follow-up of patients with NS and NLS. (c) 2012 Wiley Periodicals, Inc.

A Skyrme-like model with an exact BPS bound

We propose a new Skyrme-like model with fields taking values on the sphere S3 or, equivalently, on the group SU(2). The action of the model contains a quadratic kinetic term plus a quartic term which is the same as that of the Skyrme-Faddeev model. The novelty of the model is that it possess a first order Bogomolny type equation whose solutions automatically satisfy the second order Euler-Lagrange equations. It also possesses a lower bound on the static energy which is saturated by the Bogomolny solutions. Such Bogomolny equation is equivalent to the so-called force free equation used in plasma and solar Physics, and which possesses large classes of solutions. An old result due to Chandrasekhar prevents the existence of finite energy solutions for the force free equation on the entire three- dimensional space R3. We construct new exact finite energy solutions to the Bogomolny equations for the case where the space is the three-sphere S3, using toroidal like coordinates.