In this paper, we are interested in positive solutions of

$$ \begin{align*}\left\{ \begin{array}{@{}ll} -\Delta u = a(x)v^{p-1}, \quad &\text{ in } \Omega,\\ -\Delta v = b(x)u^{q-1}, \quad &\text{ in } \Omega,\\ u,v>0, \quad &\text{ in } \Omega,\\ u=v=0, \quad &\text{ on } \partial\Omega, \end{array} \right. \end{align*} $$

$\Omega $

${\mathbb {R}}^N (N \ge 3)$

$ a(x), b(x)$

$\Bigl \lfloor \frac {N}{2} \Bigr \rfloor $

$\frac {N}{2}$

$ a(x)=b(x)=1$

$\Omega $

$$ \begin{align*}(p-1)(q-1)> \Big(1+\frac{2N}{\lambda_H}\Big)^2\left(\frac{q}{p}\right),\end{align*} $$

$\lambda _H$

$\Omega .$

$\lambda _H$

$q.$

I outline a general thermodynamic condition for the earliest steps in the origin of life based on fluctuation theorems developed in the last two decades. I argue that the exponentially developing loop of asymmetric autocatalysis and thermodynamic tail-wind condition (TTC) in the prebiotic clutter was a key to a particular trajectory of decluttering via a sequence of early symmetry breaking events. Such decluttering was bound to result, most prominently, in homochiral amino acids and homochiral sugars composing nucleotides as the TTC exponentially favoured asymmetric autocatalytic processes over catalytic and symmetric autocatalytic processes in the clutter. I describe the loop's structure, including its chemical and physical properties, and explain that the TTC/asymmetric autocatalysis loop intersected with multiple chemical, geological and climatological feedback loops, thus providing conditions for the propagation of living systems as we know them.

Punctuated equilibrium, in the sense of Eldredge and Gould in 1972, and path dependence (Gould in 2002), dominate evolutionary processes, many of whose dynamics can be expressed in terms of interacting information sources (Wallace in 2010). Argument based on Feynman's (in 2000) characterization of information as a form of free energy leads to a close, if inverse, analogy between evolutionary transitions and economic ratchets. Driven by such a ‘self-referential’ mechanism, increases in available metabolic free energy – via the aerobic transition – led to the eukaryotic transition and to life as we know it. Formal analysis focuses on groupoid symmetries associated with the cognitive processes of gene expression, an extension of the symmetry breaking/making perspectives of physical science into biological phenomena. This suggests that understanding modalities of cognitive gene expression, as opposed to focus on genes themselves, provides the deepest insight into evolutionary phenomena, a perspective at some variance with current simplistic gene-centred views that constrain evolutionary theory.

Bifurcation of non-radial solutions from radial solutions of a semilinear elliptic equation with negative exponent in expanding annuli of ℝ2 is studied. To obtain the main results, we use a blow-up argument via the Morse index of the regular entire solutions of the equation

The main results of this paper can be seen as applications of the results obtained recently for finite Morse index solutions of the equation

with N ⩾ 2 and p > 0.

Tabling has been used for some time to improve efficiency of Prolog programs by memorizing answered queries. The same idea can be naturally used to memorize visited states during search for planning. In this paper we present a planner developed in the Picat language to solve the Petrobras planning problem. Picat is a novel Prolog-like language that provides pattern matching, deterministic and non-deterministic rules, and tabling as its core modelling and solving features. We demonstrate these capabilities using the Petrobras problem, where the goal is to plan transport of cargo items from ports to platforms using vessels with limited capacity. Monte Carlo Tree Search has been so far the best technique to tackle this problem and we will show that by using tabling we can achieve much better runtime efficiency and better plan quality.

In this paper, we consider the following Robin problem:

$$\begin{eqnarray*}\displaystyle \left\{ \begin{array}{ @{}ll@{}} \displaystyle - \Delta u= \mid x{\mathop{\mid }\nolimits }^{\alpha } {u}^{p} , \quad & \displaystyle x\in \Omega , \\ \displaystyle u\gt 0, \quad & \displaystyle x\in \Omega , \\ \displaystyle \displaystyle \frac{\partial u}{\partial \nu } + \beta u= 0, \quad & \displaystyle x\in \partial \Omega , \end{array} \right.&&\displaystyle\end{eqnarray*}$$

$\Omega $

${ \mathbb{R} }^{N} $

$N\geq 3$

$p\gt 1$

$\alpha \gt 0$

$\beta \gt 0$

$\nu $

$\partial \Omega $

$\beta $

We study the destabilization mechanism in a unidirectional ring of identical oscillators,perturbed by the introduction of a long-range connection. It is known that for ahomogeneous, unidirectional ring of identical Stuart-Landau oscillators the trivialequilibrium undergoes a sequence of Hopf bifurcations eventually leading to thecoexistence of multiple stable periodic states resembling the Eckhaus scenario. We showthat this destabilization scenario persists under small non-local perturbations. In thiscase, the Eckhaus line is modulated according to certain resonance conditions. In the casewhen the shortcut is strong, we show that the coexisting periodic solutions split up intotwo groups. The first group consists of orbits which are unstable for all parametervalues, while the other one shows the classical Eckhaus behavior.

The paper analyzes an endogenous mechanism leading perfectly symmetric economies to diverge in the long run after unifying their financial asset markets. The standard growth model with overlapping generations of consumers (OLG) is extended to include uncertainty and a financial asset. In the absence of an international asset market, the two autarkic economies converge to the same globally attracting steady state under rational expectations dynamics. When the two asset markets are unified internationally, additional asymmetric steady states appear, implying that the steady state with equal levels of capital becomes unstable, causing symmetry breaking. The paper derives general sufficient conditions for a saddle node bifurcation of the symmetric steady state. A numerical example shows that these effects occur, in particular when the production function and the function of absolute risk aversion are isoelastic.

Metamaterials made of exclusively metallic cut wire pairs have been experimentally demonstrated to exhibit a negative refractive index at optical frequencies. However, other related works on slightly different wire and plate pairs have not shown a negative index. In this paper, we present the analogy between previously reported S-shaped metamaterials and asymmetric cut wire pairs by a simple unifying approach. These two structures present a negative index for some geometrical configurations. Using simulations and experimental measurements in the microwave domain, we investigate the material properties of the last structure. Applying the inversion method from transmission and reflection responses, we show that a negative index is exhibited due to simultaneous negative permittivity ε and permeability μ. A negative index n is experimentally verified in a bulk prism engineered by stacking several layers of the metamaterial.

In this paper, we analyse a model of a regular tree loss network that supports two types of calls: unicast calls that require unit capacity on a single link, and multicast calls that require unit capacity on every link emanating from a node. We study the behaviour of the distribution of calls in the core of a large network that has uniform unicast and multicast arrival rates. At sufficiently high multicast call arrival rates the network exhibits a ‘phase transition’, leading to unfairness due to spatial variation in the multicast blocking probabilities. We study the dependence of the phase transition on unicast arrival rates, the coordination number of the network, and the parity of the capacity of edges in the network. Numerical results suggest that the nature of phase transitions is qualitatively different when there are odd and even capacities on the links. These phenomena are seen to persist even with the introduction of nonuniform arrival rates and multihop multicast calls into the network. Finally, we also show the inadequacy of approximations such as the Erlang fixed-point approximations when multicasting is present.