Asymptotic expansions of the solutions of the Cauchy problem for nonlinear parabolic equations

Let $u$ be a solution of the Cauchy problem for the nonlinear parabolic equation $$\partial_t u=\Delta u+F(x,t,u,\nabla u) \quad in \quad{\bf R}^N\times(0,\infty), \quad u(x,0)=\varphi(x)\quad in \quad{\bf R}^N,$$ and assume that the solution $u$ behaves like the Gauss kernel as $t\to\infty$. In this paper, under suitable assumptions of the reaction term $F$ and the initial function $\varphi$, we establish the method of obtaining higher order asymptotic expansions of the solution $u$ as $t\to\infty$. This paper is a generalization of our previous paper, and our arguments are applicable to the large class of nonlinear parabolic equations

Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source

A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by families of point transformations. A class of variable coefficient (1+1)-dimensional semilinear reaction-diffusion equations of the general form $f(x)u_t=(g(x)u_x)_x+h(x)u^m$ ($m\ne0,1$) is studied from the symmetry point of view in the framework of the approach proposed. The singular subclass of the equations with $m=2$ is singled out. The group classifications of the entire class, the singular subclass and their images are performed with respect to both the corresponding (generalized extended) equivalence groups and all point transformations. The set of admissible transformations of the imaged class is exhaustively described in the general case $m\ne2$. The procedure of classification of nonclassical symmetries, which involves mappings between classes of differential equations, is discussed. Wide families of new exact solutions are also constructed for equations from the classes under consideration by the classical method of Lie reductions and by generation of new solutions from known ones for other equations with point transformations of different kinds (such as additional equivalence transformations and mappings between classes of equations).Comment: 40 pages, this is version published in Acta Applicanda Mathematica

Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source

This paper deals with the long-time behavior of solutions of nonlinear reaction-diffusion equations describing formation of morphogen gradients, the concentration fields of molecules acting as spatial regulators of cell differentiation in developing tissues. For the considered class of models, we establish existence of a new type of ultra-singular self-similar solutions. These solutions arise as limits of the solutions of the initial value problem with zero initial data and infinitely strong source at the boundary. We prove existence and uniqueness of such solutions in the suitable weighted energy spaces. Moreover, we prove that the obtained self-similar solutions are the long-time limits of the solutions of the initial value problem with zero initial data and a time-independent boundary source

Flower Bats (Glossophaga soricina) and Fruit Bats (Carollia perspicillata) Rely on Spatial Cues over Shapes and Scents When Relocating Food

Natural selection can shape specific cognitive abilities and the extent to which a given species relies on various cues when learning associations between stimuli and rewards. Because the flower bat Glossophaga soricina feeds primarily on nectar, and the locations of nectar-producing flowers remain constant, G. soricina might be predisposed to learn to associate food with locations. Indeed, G. soricina has been observed to rely far more heavily on spatial cues than on shape cues when relocating food, and to learn poorly when shape alone provides a reliable cue to the presence of food.Here we determined whether G. soricina would learn to use scent cues as indicators of the presence of food when such cues were also available. Nectar-producing plants fed upon by G. soricina often produce distinct, intense odors. We therefore expected G. soricina to relocate food sources using scent cues, particularly the flower-produced compound, dimethyl disulfide, which is attractive even to G. soricina with no previous experience of it. We also compared the learning of associations between cues and food sources by G. soricina with that of a related fruit-eating bat, Carollia perspicillata. We found that (1) G. soricina did not learn to associate scent cues, including dimethyl disulfide, with feeding sites when the previously rewarded spatial cues were also available, and (2) both the fruit-eating C. perspicillata and the flower-feeding G. soricina were significantly more reliant on spatial cues than associated sensory cues for relocating food.These findings, taken together with past results, provide evidence of a powerful, experience-independent predilection of both species to rely on spatial cues when attempting to relocate food

Ultrastructural and functional fate of recycled vesicles in hippocampal synapses

Efficient recycling of synaptic vesicles is thought to be critical for sustained information transfer at central terminals. However, the specific contribution that retrieved vesicles make to future transmission events remains unclear. Here we exploit fluorescence and time-stamped electron microscopy to track the functional and positional fate of vesicles endocytosed after readily releasable pool (RRP) stimulation in rat hippocampal synapses. We show that most vesicles are recovered near the active zone but subsequently take up random positions in the cluster, without preferential bias for future use. These vesicles non-selectively queue, advancing towards the release site with further stimulation in an actin-dependent manner. Nonetheless, the small subset of vesicles retrieved recently in the stimulus train persist nearer the active zone and exhibit more privileged use in the next RRP. Our findings reveal heterogeneity in vesicle fate based on nanoscale position and timing rules, providing new insights into the origins of future pool constitution

Global Protein Demand, Marine Fish Production and Trade Flows in the World of 2050

Economic drivers of change, such as changes in demand and fishing technology, are as important for the future state of fish stocks as factors influencing ecological processes (Quaas et al. 2016) . Fish and seafood (FaS) are among the most globalized commodities, and markets will continue to globalize in the coming decades. Constantly improving fishing technology may lead to increasing fishing pressure unless management effectiveness is significantly improved. To take world-market interactions into account, we quantify a global demand system for wild-caught FaS, resolved at the large marine ecosystem (LME) scale, using FAO Fishstat data. Demand for domestically caught FaS and FaS imports further depends on population size, per-capita incomes, and supply of other protein-rich food (as substitutes for FaS). Based on scenarios for population and income developments, we use the estimated demand system to derive scenarios of future FaS demand. We feed these scenarios into a global bio-economic model to study the effects of FaS consumption levels allocated by globalized markets on catches and stock development. The fisheries model provides biomass development scenarios per functional group taking into account predator-prey interactions in a stylized fashion. We analyze the effect of improving fisheries management, and the scope of expanding the supply of farmed fish for relieving the pressure on wild capture fisheries. Preliminary results suggest that this scope is limited, as farmed FaS is only an imperfect substitute for wild capture fisheries, in particular for the more wealthy consumers, who are likely to grow in number in the coming decades.Proceedings of the Eighteenth Biennial Conference of the International Institute of Fisheries Economics and Trade, held July 11-15, 2016 at Aberdeen Exhibition and Conference Center (AECC), Aberdeen, Scotland, UK

Measurement of high-p_T Single Electrons from Heavy-Flavor Decays in p+p Collisions at sqrt(s) = 200 GeV

The momentum distribution of electrons from decays of heavy flavor (charm and beauty) for midrapidity |y| < 0.35 in p+p collisions at sqrt(s) = 200 GeV has been measured by the PHENIX experiment at the Relativistic Heavy Ion Collider (RHIC) over the transverse momentum range 0.3 < p_T < 9 GeV/c. Two independent methods have been used to determine the heavy flavor yields, and the results are in good agreement with each other. A fixed-order-plus-next-to-leading-log pQCD calculation agrees with the data within the theoretical and experimental uncertainties, with the data/theory ratio of 1.72 +/- 0.02^stat +/- 0.19^sys for 0.3 < p_T < 9 GeV/c. The total charm production cross section at this energy has also been deduced to be sigma_(c c^bar) = 567 +/- 57^stat +/- 224^sys micro barns.Comment: 375 authors from 57 institutions, 6 pages, 3 figures. Submitted to Physical Review Letters. Plain text data tables for the points plotted in figures for this and previous PHENIX publications are (or will be) publicly available at http://www.phenix.bnl.gov/papers.htm

Identified charged hadron production in p+p collisions at sqrt(s)=200 and 62.4 GeV

Transverse momentum distributions and yields for $\pi^{\pm}$, $K^{\pm}$, $p$ and $\bar{p}$ in $p+p$ collisions at $\sqrt{s}$=200 and 62.4 GeV at midrapidity are measured by the PHENIX experiment at the Relativistic Heavy Ion Collider (RHIC). These data provide important baseline spectra for comparisons with identified particle spectra in heavy ion collisions at RHIC. We present the inverse slope parameter $T_{\rm inv}$, mean transverse momentum $$ and yield per unit rapidity $dN/dy$ at each energy, and compare them to other measurements at different $\sqrt{s}$ in $p+p$ and $p+\bar{p}$ collisions. We also present the scaling properties such as $m_T$ scaling, $x_T$ scaling on the $p_T$ spectra between different energies. To discuss the mechanism of the particle production in $p+p$ collisions, the measured spectra are compared to next-to-leading-order or next-to-leading-logarithmic perturbative quantum chromodynamics calculations.Comment: 431 authors from 62 institutions, 32 pages, 23 figures, and 18 tables. Submitted to Physical Review C. Plain text data tables for the points plotted in figures for this and previous PHENIX publications are (or will be) publicly available at http://www.phenix.bnl.gov/papers.htm