518 research outputs found
Rational solutions of the discrete time Toda lattice and the alternate discrete Painleve II equation
The Yablonskii-Vorob'ev polynomials , which are defined by a second
order bilinear differential-difference equation, provide rational solutions of
the Toda lattice. They are also polynomial tau-functions for the rational
solutions of the second Painlev\'{e} equation (). Here we define
two-variable polynomials on a lattice with spacing , by
considering rational solutions of the discrete time Toda lattice as introduced
by Suris. These polynomials are shown to have many properties that are
analogous to those of the Yablonskii-Vorob'ev polynomials, to which they reduce
when . They also provide rational solutions for a particular
discretisation of , namely the so called {\it alternate discrete}
, and this connection leads to an expression in terms of the Umemura
polynomials for the third Painlev\'{e} equation (). It is shown that
B\"{a}cklund transformation for the alternate discrete Painlev\'{e} equation is
a symplectic map, and the shift in time is also symplectic. Finally we present
a Lax pair for the alternate discrete , which recovers Jimbo and Miwa's
Lax pair for in the continuum limit .Comment: 23 pages, IOP style. Title changed, and connection with Umemura
polynomials adde
Spectral characteristics for a spherically confined -1/r + br^2 potential
We consider the analytical properties of the eigenspectrum generated by a
class of central potentials given by V(r) = -a/r + br^2, b>0. In particular,
scaling, monotonicity, and energy bounds are discussed. The potential is
considered both in all space, and under the condition of spherical confinement
inside an impenetrable spherical boundary of radius R. With the aid of the
asymptotic iteration method, several exact analytic results are obtained which
exhibit the parametric dependence of energy on a, b, and R, under certain
constraints. More general spectral characteristics are identified by use of a
combination of analytical properties and accurate numerical calculations of the
energies, obtained by both the generalized pseudo-spectral method, and the
asymptotic iteration method. The experimental significance of the results for
both the free and confined potential V(r) cases are discussed.Comment: 16 pages, 4 figure
Introductory clifford analysis
In this chapter an introduction is given to Clifford analysis and the underlying Clifford algebras. The functions under consideration are defined on Euclidean space and take values in the universal real or complex Clifford algebra, the structure and properties of which are also recalled in detail. The function theory is centered around the notion of a monogenic function, which is a null solution of a generalized Cauchy–Riemann operator, which is rotation invariant and factorizes the Laplace operator. In this way, Clifford analysis may be considered as both a generalization to higher dimension of the theory of holomorphic functions in the complex plane and a refinement of classical harmonic analysis. A notion of monogenicity may also be associated with the vectorial part of the Cauchy–Riemann operator, which is called the Dirac operator; some attention is paid to the intimate relation between both notions. Since a product of monogenic functions is, in general, no longer monogenic, it is crucial to possess some tools for generating monogenic functions: such tools are provided by Fueter’s theorem on one hand and the Cauchy–Kovalevskaya extension theorem on the other hand. A corner stone in this function theory is the Cauchy integral formula for representation of a monogenic function in the interior of its domain of monogenicity. Starting from this representation formula and related integral formulae, it is possible to consider integral transforms such as Cauchy, Hilbert, and Radon transforms, which are important both within the theoretical framework and in view of possible applications
Integrable semi-discretization of the coupled nonlinear Schr\"{o}dinger equations
A system of semi-discrete coupled nonlinear Schr\"{o}dinger equations is
studied. To show the complete integrability of the model with multiple
components, we extend the discrete version of the inverse scattering method for
the single-component discrete nonlinear Schr\"{o}dinger equation proposed by
Ablowitz and Ladik. By means of the extension, the initial-value problem of the
model is solved. Further, the integrals of motion and the soliton solutions are
constructed within the framework of the extension of the inverse scattering
method.Comment: 27 pages, LaTeX2e (IOP style
Artificial drainage of peatlands: hydrological and hydrochemical process and wetland restoration
Peatlands have been subject to artificial drainage for centuries. This drainage has been in response to agricultural demand, forestry, horticultural and energy properties of peat and alleviation of flood risk. However, the are several environmental problems associated with drainage of peatlands. This paper describes the nature of these problems and examines the evidence for changes in hydrological and hydrochemical processes associated with these changes. Traditional black-box water balance approaches demonstrate little about wetland dynamics and therefore the science of catchment response to peat drainage is poorly understood. It is crucial that a more process-based approach be adopted within peatland ecosystems. The environmental problems associated with peat drainage have led, in part, to a recent reversal in attitudes to peatlands and we have seen a move towards wetland restoration. However, a detailed understanding of hydrological, hydrochemical and ecological process-interactions will be fundamental if we are to adequately restore degraded peatlands, preserve those that are still intact and understand the impacts of such management actions at the catchment scale
Phase 1 dose-ranging study of ezatiostat hydrochloride in combination with lenalidomide in patients with non-deletion (5q) low to intermediate-1 risk myelodysplastic syndrome (MDS)
Fighting Corruption When Existing Corruption-Control Levels Count: What Do Wealth-Effects Tell Us in Africa?
Teachers’ appraisals of adjectives relating to mathematics tasks
Curricular implementations are unlikely to deliver the anticipated benefits for mathematics learners if written guidance to teachers is interpreted and enacted differently from the ways that policymakers and curriculum designers intend. One way in which this could happen is in relation to the mathematics tasks that teachers deploy in the classroom. Teachers and curriculum designers have developed an extensive vocabulary for describing tasks, using adjectives such as ‘rich’, ‘open’, ‘real-life’, ‘engaging’ and so on. But do teachers have a shared understanding of what these adjectives mean when they are applied to mathematics tasks? In Study 1 we investigated teachers’ appraisals of adjectives used to describe mathematics tasks, finding that task appraisals vary on seven dimensions, which we termed engagement, demand, routineness, strangeness, inquiry, context and interactivity. In Study 2, focusing on the five most prominent dimensions, we investigated whether teachers have a shared understanding of the meaning of adjectives when applied to mathematics tasks. We found that there was some agreement about inquiry and context, some disagreement about routineness, and clear disagreement about engagement and demand. We conclude that at least some adjectives commonly used to describe tasks are interpreted very differently by different teachers. Implications for how tasks might be discussed meaningfully by teachers, teacher educators and curriculum designers are highlighted
The Political Economy of Development Assistance: Peril to Government Quality Dynamics in Africa
Unreliable numbers: error and harm induced by bad design can be reduced by better design
Number entry is a ubiquitous activity and is often performed in safety- and mission-critical procedures, such as healthcare, science, finance, aviation and in many other areas. We show that Monte Carlo methods can quickly and easily compare the reliability of different number entry systems. A surprising finding is that many common, widely used systems are defective, and induce unnecessary human error. We show that Monte Carlo methods enable designers to explore the implications of normal and unexpected operator behaviour, and to design systems to be more resilient to use error. We demonstrate novel designs with improved resilience, implying that the common problems identified and the errors they induce are avoidable
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