Why the general Zakharov-Shabat equations form a hierarchy?

The totality of all Zakharov-Shabat equations (ZS), i.e., zero-curvature equations with rational dependence on a spectral parameter, if properly defined, can be considered as a hierarchy. The latter means a collection of commuting vector fields in the same phase space. Further properties of the hierarchy are discussed, such as additional symmetries, an analogue to the string equation, a Grassmannian related to the ZS hierarchy, and a Grassmannian definition of soliton solutions.Comment: 13p

Communication practices of the Karen in Sheffield: Seeking to navigate their three zones of displacement

This study investigates communication practices of a newly arrived Karen refugee community in the UK who, as well as establishing themselves in a strange country, seek to keep in touch, campaign politically and maintain identity collectively through communication and contact with their global diaspora. We look at the technologies, motivations and inhibiting factors applying to the communication by adult members of this community and construct the idea of three zones of displacement which help to model the particular contexts, challenges and methods of their communication. We find that overall, they are using a wide range of internet-based technologies, with the aim to 'keep-in-touch' (personal contacts) and to 'spread the word' (political communication). This also includes archaic, traditional and hybrid methods to achieve extended communication with contacts in other 'zones'. We also identify the importance of the notion of ‘village’ as metaphor and entity in their conceptualisation of diasporic and local community cohesion. We identify the key inhibitors to their communication as cost, education, literacy and age. Finally, we speculate on the uncertain outcomes of their approach to digital media in achieving their political aims

The symplectic Deligne-Mumford stack associated to a stacky polytope

We discuss a symplectic counterpart of the theory of stacky fans. First, we define a stacky polytope and construct the symplectic Deligne-Mumford stack associated to the stacky polytope. Then we establish a relation between stacky polytopes and stacky fans: the stack associated to a stacky polytope is equivalent to the stack associated to a stacky fan if the stacky fan corresponds to the stacky polytope.Comment: 20 pages; v2: To appear in Results in Mathematic

Amphiphilic monomers bridge hydrophobic polymers and water

Water dissolves a hydrophilic polymer, but not a hydrophobic polymer. Many monomers of hydrophilic polymers, however, are amphiphilic, with a hydrophobic vinyl group for radical polymerization, as well as a hydrophilic group. Consequently, such an amphiphilic monomer may form solutions with both water and hydrophobic polymers. Ternary mixtures of amphiphilic monomer, hydrophobic polymer, and water have recently been used as precursors for interpenetrating polymer networks of hydrophilic polymers and hydrophobic polymers of unusual properties. However, the phase behavior of the ternary mixtures of amphiphilic monomer, hydrophobic polymer, and water themselves has not been studied. Here we mix the amphiphilic monomer acrylic acid, the hydrophobic polymer poly(methyl methacrylate), and water. In the mixture, the hydrophobic polymer can form various morphologies, including solution, micelle, gel, and polymer glass. We interpret these findings by invoking that the hydrophobic and hydrophilic groups of the amphiphilic monomer enable it to function as a bridge. That is, the hydrophobic functional group binds with the hydrophobic polymer, and the hydrophilic functional group binds with water. This picture leads to a simple modification to the Flory-Huggins theory, which agrees well with our experimental data. Amphiphilic monomers offer a rich area for further study for scientific insight, as well as for expanding opportunities to develop materials of self-assembled structures with unusual properties.</p

Finite Size Scaling and ``perfect'' actions: the three dimensional Ising model

Using Finite-Size Scaling techniques, we numerically show that the first irrelevant operator of the lattice $\lambda\phi^4$ theory in three dimensions is (within errors) completely decoupled at $\lambda=1.0$. This interesting result also holds in the Thermodynamical Limit, where the renormalized coupling constant shows an extraordinary reduction of the scaling-corrections when compared with the Ising model. It is argued that Finite-Size Scaling analysis can be a competitive method for finding improved actions.Comment: 13 pages, 3 figure

Housing Sales in Urban Beijing

In the housing market, new properties sometimes experience delays before they are sold. Such delays reflect the preferences of buyers in respect of the homes’ characteristics. Therefore, it is important for managerial purposes to identify the causes of housing sales delays. After analyzing the delays in sales of housing in Beijing City, China, the principal finding of this study is that delays are largely explained by the dwellings’ characteristics and location. Policy implications of the research findings, particularly those related to means of reducing the delays, are discussed

Approximating multi-dimensional Hamiltonian flows by billiards

Consider a family of smooth potentials $V_{\epsilon}$, which, in the limit $\epsilon\to0$, become a singular hard-wall potential of a multi-dimensional billiard. We define auxiliary billiard domains that asymptote, as $\epsilon\to0$ to the original billiard, and provide asymptotic expansion of the smooth Hamiltonian solution in terms of these billiard approximations. The asymptotic expansion includes error estimates in the $C^{r}$ norm and an iteration scheme for improving this approximation. Applying this theory to smooth potentials which limit to the multi-dimensional close to ellipsoidal billiards, we predict when the separatrix splitting persists for various types of potentials

Gr\"obner-Shirshov bases for Lie algebras over a commutative algebra

In this paper we establish a Gr\"{o}bner-Shirshov bases theory for Lie algebras over commutative rings. As applications we give some new examples of special Lie algebras (those embeddable in associative algebras over the same ring) and non-special Lie algebras (following a suggestion of P.M. Cohn (1963) \cite{Conh}). In particular, Cohn's Lie algebras over the characteristic $p$ are non-special when $p=2,\ 3,\ 5$. We present an algorithm that one can check for any $p$, whether Cohn's Lie algebras is non-special. Also we prove that any finitely or countably generated Lie algebra is embeddable in a two-generated Lie algebra

Renormalization group and isochronous oscillations

We show how the condition of isochronicity can be studied for two dimensional systems in the renormalization group (RG) context. We find a necessary condition for the isochronicity of the Cherkas and another class of cubic systems. Our conditions are satisfied by all the cases studied recently by Bardet et al \cite{bard} and Ghose Choudhury and Guh