Lane-Emden equations perturbed by nonhomogeneous potential in the super critical case

Our purpose of this paper is to study positive solutions of Lane-Emden equation − Δ u = V u p i n R N ∖ { 0 } $$\begin{array}{} -{\it\Delta} u = V u^p\quad {\rm in}\quad \mathbb{R}^N\setminus\{0\} \end{array}$$ (0.1) perturbed by a non-homogeneous potential V when p ∈ [ p c , N + 2 N − 2 ) , $\begin{array}{} p\in [p_c, \frac{N+2}{N-2}), \end{array}$ where pc is the Joseph-Ludgren exponent. When p ∈ ( N N − 2 , p c ) , $\begin{array}{} p\in (\frac{N}{N-2}, p_c), \end{array}$ the fast decaying solution could be approached by super and sub solutions, which are constructed by the stability of the k-fast decaying solution wk of −Δ u = up in ℝ N ∖ {0} by authors in [9]. While the fast decaying solution wk is unstable for p ∈ ( p c , N + 2 N − 2 ) , $\begin{array}{} p\in (p_c, \frac{N+2}{N-2}), \end{array}$ so these fast decaying solutions seem not able to disturbed like (0.1) by non-homogeneous potential V. A surprising observation that there exists a bounded sub solution of (0.1) from the extremal solution of − Δ u = u N + 2 N − 2 $\begin{array}{} -{\it\Delta} u = u^{\frac{N+2}{N-2}} \end{array}$ in ℝ N and then a sequence of fast decaying solutions and slow decaying solutions could be derived under appropriated restrictions for V.

Protein folding while chaperone bound is dependent on weak interactions

It is generally assumed that protein clients fold following their release from chaperones instead of folding while remaining chaperone-bound, in part because binding is assumed to constrain the mobility of bound clients. Previously, we made the surprising observation that the ATP-independent chaperone Spy allows its client protein Im7 to fold into the native state while continuously bound to the chaperone. Spy apparently permits sufficient client mobility to allow folding to occur while chaperone bound. Here, we show that strengthening the interaction between Spy and a recently discovered client SH3 strongly inhibits the ability of the client to fold while chaperone bound. The more tightly Spy binds to its client, the more it slows the folding rate of the bound client. Efficient chaperone-mediated folding while bound appears to represent an evolutionary balance between interactions of sufficient strength to mediate folding and interactions that are too tight, which tend to inhibit folding.

Backwards or forwards? Direction of working and success in problem solving

We pose ourselves the question: What can one infer from the direction of working when solvers work on the same task for a second time? This is discussed on the basis of 44 problem solving processes of the TIMSS task K10. A natural hypothesis is that working forwards can be taken as evidence that the task is recognized and a solution path is recalled. This can be confirmed by our analysis. A surprising observation is that when working backwards, pivotal for success is (in case of K10) to change to working forwards soon after reaching the barrier.

Endemic Behaviour of SIS Epidemics with General Infectious Period Distributions

We study the endemic behaviour of a homogeneously mixing SIS epidemic in a population of size N with a general infectious period, Q, by introducing a novel subcritical branching process with immigration approximation. This provides a simple but useful approximation of the quasistationary distribution of the SIS epidemic for finite N and the asymptotic Gaussian limit for the endemic equilibrium as N → ∞. A surprising observation is that the quasistationary distribution of the SIS epidemic model depends on Q only through

Endemic Behaviour of SIS Epidemics with General Infectious Period Distributions

We study the endemic behaviour of a homogeneously mixing SIS epidemic in a population of size N with a general infectious period, Q, by introducing a novel subcritical branching process with immigration approximation. This provides a simple but useful approximation of the quasistationary distribution of the SIS epidemic for finite N and the asymptotic Gaussian limit for the endemic equilibrium as N → ∞. A surprising observation is that the quasistationary distribution of the SIS epidemic model depends on Q only through