Two dimensional dynamical systems which admit Lie and Noether symmetries

We prove two theorems which relate the Lie point symmetries and the Noether symmetries of a dynamical system moving in a Riemannian space with the special projective group and the homothetic group of the space respectively. The theorems are applied to classify the two dimensional Newtonian dynamical systems, which admit a Lie point/Noether symmetry. Two cases are considered, the non-conservative and the conservative forces. The use of the results is demonstrated for the Kepler - Ermakov system, which in general is non-conservative and for potentials similar to the H\`enon Heiles potential. Finally it is shown that in a FRW background with no matter present, the only scalar cosmological model which is integrable is the one for which 3-space is flat and the potential function of the scalar field is exponential. It is important to note that in all applications the generators of the symmetry vectors are found by reading the appropriate entry in the relevant tables.Comment: 25 pages, 17 table

Shopping centre siting and modal choice in Belgium: a destination based analysis

Although modal split is only one of the elements considered in decision-making on new shopping malls, it remarkably often arises in arguments of both proponents and opponents. Today, this is also the case in the debate on the planned development of three major shopping malls in Belgium. Inspired by such debates, the present study focuses on the impact of the location of shopping centres on the travel mode choice of the customers. Our hypothesis is that destination-based variables such as embeddedness in the urban fabric, accessibility and mall size influence the travel mode choice of the visitors. Based on modal split data and location characteristics of seventeen existing shopping centres in Belgium, we develop a model for a more sustainable siting policy. The results show a major influence of the location of the shopping centre in relation to the urban form, and of the size of the mall. Shopping centres that are part of a dense urban fabric, measured through population density, are less car dependent. Smaller sites will attract more cyclists and pedestrians. Interestingly, our results deviate significantly from the figures that have been put forward in public debates on the shopping mall issue in Belgium

Paratransit: the need for a regulatory revolution in the light of institutional inertia

This chapter begins by defining what is traditionally meant by the term 'paratransit', before exploring why it has remained a relatively niche transport concern. Societal trends have shifted to a pattern of demand that is ill-suited to the system design for conventional public transport. Emerging IT applications offer the potential to introduce a new model of public transport appropriate to the travel needs of the 21st century. Paratransit modes are appealing because they could dynamically match the supply of a service with the level of demand required, unlike conventional models of public transport based on fading historical demand patterns. But the regulatory environment for the local passenger sector has been built incrementally over many years around the institutional frameworks for buses and taxis. Paratransit alternatives often do not fully fit under any of these categorisations with the result that they often do not have an institutional home and thus either upset the status quo (as with Uber currently) or else are still born. A redefinition of paratransit is proposed to facilitate a regulatory change to help address the institutional challenges of paratransit innovation

The nonlinear Schroedinger equation for the delta-comb potential: quasi-classical chaos and bifurcations of periodic stationary solutions

The nonlinear Schroedinger equation is studied for a periodic sequence of delta-potentials (a delta-comb) or narrow Gaussian potentials. For the delta-comb the time-independent nonlinear Schroedinger equation can be solved analytically in terms of Jacobi elliptic functions and thus provides useful insight into the features of nonlinear stationary states of periodic potentials. Phenomena well-known from classical chaos are found, such as a bifurcation of periodic stationary states and a transition to spatial chaos. The relation of new features of nonlinear Bloch bands, such as looped and period doubled bands, are analyzed in detail. An analytic expression for the critical nonlinearity for the emergence of looped bands is derived. The results for the delta-comb are generalized to a more realistic potential consisting of a periodic sequence of narrow Gaussian peaks and the dynamical stability of periodic solutions in a Gaussian comb is discussed.Comment: Enhanced and revised version, to appear in J. Nonlin. Math. Phy

PKMζ is essential for spinal plasticity underlying the maintenance of persistent pain

<p>Abstract</p> <p>Background</p> <p>Chronic pain occurs when normally protective acute pain becomes pathologically persistent. We examined here whether an isoform of protein kinase C (PKC), PKMζ, that underlies long-term memory storage in various brain regions, also sustains nociceptive plasticity in spinal cord dorsal horn (SCDH) mediating persistent pain.</p> <p>Results</p> <p>Cutaneous injury or spinal stimulation produced persistent increases of PKMζ, but not other atypical PKCs in SCDH. Inhibiting spinal PKMζ, but not full-length PKCs, reversed plasticity-dependent persistent painful responses to hind paw formalin and secondary mechanical hypersensitivity and SCDH neuron sensitization after hind paw capsaicin, without affecting peripheral sensitization-dependent primary heat hypersensitivity after hind paw capsaicin. Inhibiting spinal PKMζ, but not full-length PKCs, also reversed mechanical hypersensitivity in the rat hind paw induced by spinal stimulation with intrathecal dihydroxyphenylglycine. Spinal PKMζ inhibition also alleviated allodynia 3 weeks after ischemic injury in rats with chronic post-ischemia pain (CPIP), at a point when allodynia depends on spinal changes. In contrast, spinal PKMζ inhibition did not affect allodynia in rats with chronic contriction injury (CCI) of the sciatic nerve, or CPIP rats early after ischemic injury, when allodynia depends on ongoing peripheral inputs.</p> <p>Conclusions</p> <p>These results suggest spinal PKMζ is essential for the maintenance of persistent pain by sustaining spinal nociceptive plasticity.</p