Let $e$ and $q$ be fixed co-prime integers satisfying $1\lt e\lt q$ . Let $\mathscr {C}$ be a certain family of deformations of the curve $y^e=x^q$ . That family is called the $(e,q)$ -curve and is one of the types of curves called plane telescopic curves. Let $\varDelta$ be the discriminant of $\mathscr {C}$ . Following pioneering work by Buchstaber and Leykin (BL), we determine the canonical basis $\{ L_j \}$ of the space of derivations tangent to the variety $\varDelta =0$ and describe their specific properties. Such a set $\{ L_j \}$ gives rise to a system of linear partial differential equations (heat equations) satisfied by the function $\sigma (u)$ associated with $\mathscr {C}$ , and eventually gives its explicit power series expansion. This is a natural generalisation of Weierstrass’ result on his sigma function. We attempt to give an accessible description of various aspects of the BL theory. Especially, the text contains detailed proofs for several useful formulae and known facts since we know of no works which include their proofs.

Systematic reviews rely on identification of studies, initially through electronic searches yielding potentially thousands of studies, and then reviewer-led screening studies for inclusion. This standard method is time- and resource-intensive. We designed and applied an algorithm written in Python involving computer-aided identification of keywords within each paper for an exemplar systematic review of arm impairment after stroke. The standard method involved reading each abstract searching for these keywords. We compared the methods in terms of accuracy in identification of keywords, abstracts' eligibility, and time taken to make a decision about eligibility. For external validation, we adapted the algorithm for a different systematic review, and compared eligible studies using the algorithm with those included in that review. For the exemplar systematic review, the algorithm failed on 72 out of 2,789 documents retrieved (2.6%). Both methods identified the same 610 studies for inclusion. Based on a sample of 21 randomly selected abstracts, the standard screening took 1.58 ± 0.26 min per abstract. Computer output screening took 0.43 ± 0.14 min per abstract. The mean difference between the two methods was 1.15 min ( P  < 0.0001), saving 73% per abstract. For the other systematic review, use of the algorithm resulted in the same studies being identified. One study was excluded based on the interpretation of the comparison intervention. Our purpose-built software was an accurate and significantly time-saving method for identifying eligible abstracts for inclusion in systematic reviews. This novel method could be adapted for other systematic reviews in future for the benefit of authors, reviewers and editors.

The Davydov model describes amide I energy transfer in proteins without dispersion or dissipation. In spite of five decades of study, there are few exact analytical results, especially for the discrete version of this model. Here we develop two methods to determine the exact orthonormal, multi-quanta, eigenstates of the Davydov dimer. The first method involves the integration of a system of ordinary differential equations and the second method applies purely algebraic methods to this problem. We obtain the general expression of the eigenvalues for any number of quanta and also, as examples, apply the methods to the detailed derivation of the eigenvectors for one to four quanta, plus a brief example in the case of $n=5$ and $n=6$.

Temperatūras un dislokāciju blīvuma sadalījumu modelēšana Si

We describe the full two-dimensional scattering of long-lived breathers in a model hexagonal lattice of atoms. The chosen system, representing an idealized model of mica, combines a Lennard-Jones interatomic potential with an "egg-box" harmonic potential well surface. We investigate the dependence of breather properties on the ratio of the well depths associated with the interaction and on-site potentials. High values of this ratio lead to large spatial displacements in adjacent chains of atoms and thus enhance the two-dimensional character of the quasi-one-dimensional breather solutions. This effect is further investigated during breather-breather collisions by following the constrained energy density function in time for a set of randomly excited mobile breather solutions. Certain collisions lead to 60^{∘} scattering, and collisions of mobile and stationary breathers can generate a rich variety of states.

We investigate the hypothesis that protein folding is a kinetic, non-equilibrium process, in which the structure of the nascent chain is crucial. We compare actual amino acid frequencies in loops, α-helices and β-sheets with the frequencies that would arise in the absence of any amino acid bias for those secondary structures. The novel analysis suggests that while specific amino acids exist to drive the formation of loops and sheets, none stand out as drivers for α-helices. This favours the idea that the α-helix is the initial structure of most proteins before the folding process begins.

We describe, for the first time, the full 2D scattering of long-lived breathers in a model hexagonal lattice of atoms. The chosen system, representing an idealized model of mica, combines a Lennard-Jones interatomic potential with an "egg-box" harmonic potential well surface. We investigate the dependence of breather properties on the ratio of the well depths associated to the interaction and on-site potentials. High values of this ratio lead to large spatial displacements in adjacent chains of atoms and thus enhance the two dimensional character of the quasi-one-dimensional breather solutions. This effect is further investigated during breather-breather collisions by following the constrained energy density function in time for a set of randomly excited mobile breather solutions. Certain collisions lead to 60^∘ scattering, and collisions of mobile and stationary breathers can generate a rich variety of states.

The main aim of this paper is an exposition of the theory of Buchstaber and Leykin on the heat equations for the multivariate sigma functions. We treat only the elliptic curve case, but keeping the most general elliptic curve equation, which may be useful for number theoretic applications. Introduction Let (ω′, ω′′) be a fixed basis of a lattice in C satisfying ω′′/ω′ > 0. Let ℘(u) be the Weierstrass function associated to this lattice, which satisfies the equation (0.1) ℘′(u)2 = 4℘(u) − g2℘(u)− g3, where

We consider the heat equations satisfied by the sigma function associated with a planar curve, extending and developing earlier pioneering work of Buchstaber and Leykin. These heat equations lead to useful {\em linear} recursive relations for the coefficients of power series expansion of the sigma function. In particular we exhibit explicit results for curves of genus 3, and give a new constructive proof of an explicit expression for the main matrix in the theory for {\em any} hyperelliptic curve. We also state and prove a new explicit formula for the eigenvalues of the linear operators associated with this matrix, as well as other practical formulae.

We quantize the β-Fermi–Pasta–Ulam (FPU) model with nearest and next-nearest neighbor interactions using a number conserving approximation and a numerically exact diagonalization method. Our numerical mean field bi-phonon spectrum shows excellent agreement with the analytic mean field results of Ivić and Tsironis (2006), except for the wave vector at the midpoint of the Brillouin zone. We then relax the mean field approximation and calculate the eigenvalue spectrum of the full Hamiltonian. We show the existence of multi-phonon bound states and analyze the properties of these states by varying the system parameters. From the calculation of the spatial correlation function we then show that these multi-phonon bound states are particle like states with finite spatial correlation. Accordingly we identify these multi-phonon bound states as the quantum equivalent of the breather solutions of the corresponding classical FPU model. The four-phonon spectrum of the system is then obtained and its properties are studied. We then generalize the study to an extended range interaction and consider the quantization of the β-FPU model with next-nearest-neighbor interactions. We analyze the effect of the next-nearest-neighbor interactions on the eigenvalue spectrum and the correlation functions of the system.

In this paper we consider a 2D hexagonal crystal lattice model first proposed by Mann, Eilbeck and Russell in 1998. We perform a detailed numerical study of nonlinear propagating localized modes, that is, propagating discrete breathers and kinks. The original model is extended to allow for arbitrary atomic interactions, and to allow atoms to travel out of the unit cell. A new on-site potential is considered with a periodic smooth function with hexagonal symmetry. We are able to confirm the existence of long-lived propagating discrete breathers. Our simulations show that, as they evolve, breathers appear to localize in frequency space, i.e. the energy moves from sidebands to a main frequency band. Our numerical findings shed light on the open question of whether exact moving breather solutions exist in 2D hexagonal layers in physical crystal lattices. (C) 2015 Elsevier B.V. All rights reserved.

We review research on the role of nonlinear coherent phenomena (e.g. breathers and kinks) in the formation of linear decorations in mica crystal. The work is based on a new model for the motion of the mica hexagonal K layer, which allows displacement of the atoms from the unit cell. With a simple piece-wise polynomial inter-particle potential, we verify the existence of localized long-lived breathers in an idealized lattice at 0 K. Moreover, our model allows us to observe long-lived localized kinks. We study the interactions of such localized modes along a lattice direction, and in addition demonstrate fully two dimensional scattering of such pulses for the first time. For large interatomic forces we observe a spreading horseshoe-shaped wave, a type of shock wave but with a breather profile.

We present new addition formulae for the Weierstrass functions associated with a general elliptic curve. We prove the structure of the formulae in n-variables and give the explicit addition formulae for the 2- and 3-variable cases. These new results were inspired by new addition formulae found in the case of an equianharmonic curve, which we can now observe as a specialization of the results here. The new formulae, and the techniques used to find them, also follow the recent work for the generalization of Weierstrass functions to curves of higher genus.

For elliptic curves, expressions for the periods of elliptic integrals of the second kind in terms of theta-constants, have been known since the middle of the 19th century. In this paper we consider the problem of generalizing these results to curves of higher genera, in particular to a special class of algebraic curves, the so-called $(n,s)$-curves. It is shown that the representations required can be obtained by the comparison of two equivalent expressions for the projective connection, one due to Fay-Wirtinger and the other from Klein-Weierstrass. As a principle example, we consider the case of the genus two hyperelliptic curve, and a number of new Thomae and Rosenhain-type formulae are obtained. We anticipate that our analysis for the genus two curve can be extended to higher genera hyperelliptic curves, as well as to other classes of $(n,s)$ non-hyperelliptic curves.

We discuss a family of multi-term addition formulae for Weierstrass functions on specialized curves of low genus with many automorphisms, concentrating mostly on the case of genus 1 and 2. In the genus 1 case, we give addition formulae for the equianharmonic and lemniscate cases, and in genus 2 we find some new addition formulae for a number of curves.

In this paper, interstitial migration generated by scattering with a mobile breather is investigated numerically in a Frenkel-Kontorova one-dimensional lattice. Consistent with experimental results it is shown that interstitial diffusion is more likely and faster than vacancy diffusion. Our simulations support the hypothesis that a long-range energy transport mechanism involving moving nonlinear vibrational excitations may significantly enhance the mobility of point defects in a crystal lattice.

We develop a simple 1D model for the scattering of an incoming particle hitting the surface of mica crystal, the transmission of energy through the crystal by a localized mode, and the ejection of atom(s) at the incident or distant face. This is the first attempt to model the experiment described in Russell and Eilbeck in 2007 (EPL, v. 78, 10004). Although very basic, the model shows many interesting features, for example a complicated energy dependent transition between breather modes and a kink mode, and multiple ejections at both incoming and distant surfaces. In addition, the effect of a heavier surface layer is modelled, which can lead to internal reflections of breathers or kinks at the crystal surface.

We examine tracks in crystals of muscovite of high energy charged particles, and of mobile lattice excitations created by kinetic atomic scattering. The mobile lattice excitations are interpreted as a type of breather, here called a quodon. The typical energy of a quodon can be found from the decay of potassium K 40 atoms in the crystal and supports their interpretation as a type of breather. In turn, this establishes a unique signature for energetic quodons, the 'kinked-line' tracks, allowing discrimination against tracks formed by charged particles. The stability of quodons against crystal defects and thermal motion is considered. Measurements on energetic quodon tracks, with flight paths up to 530mm, show that they can propagate more than 10 9 unit cells with no evidence of energy loss. This suggests that quodons might persist indefinitely in certain crystals of high quality. Evidence is presented for a new type of mobile lattice excitation that is capable of creating energetic quodons, which also is stable against lattice defects. Possible practical applications of quodons are considered briefly. Although quodons can induce fusion in deuterium or tritium, present indications are that the rate is too low to be of practical use. Finally, a nonlinear lattice effect that might increase this rate is suggested.

We compare and contrast three different methods for the construction of the differential relations satisfied by the fundamental Abelian functions associated with an algebraic curve. We realize these Abelian functions as logarithmic derivatives of the associated sigma function. In two of the methods, the use of the tau function, expressed in terms of the sigma function, is central to the construction of differential relations between the Abelian functions.