Lie Symmetries and Exact Solutions of First Order Difference Schemes

We show that any first order ordinary differential equation with a known Lie point symmetry group can be discretized into a difference scheme with the same symmetry group. In general, the lattices are not regular ones, but must be adapted to the symmetries considered. The invariant difference schemes can be so chosen that their solutions coincide exactly with those of the original differential equation.Comment: Minor changes and journal-re

Marginal cost-based pricing of distribution: a case study

This paper presents results of a software development project carried out by the “Electricity North West” (ENW) and “TNEI” to find economic use-of-system charges for the extra high-voltage (EHV) network. Several cost-based charging models which satisfy principles set by the Regulator, such as cost reflectivity, predictability, stability and transparency were developed. In this paper, the emphasis is put on the developed software and the comparison of nodal marginal charges obtained from the proposed pricing models

Multiscale expansion and integrability properties of the lattice potential KdV equation

We apply the discrete multiscale expansion to the Lax pair and to the first few symmetries of the lattice potential Korteweg-de Vries equation. From these calculations we show that, like the lowest order secularity conditions give a nonlinear Schroedinger equation, the Lax pair gives at the same order the Zakharov and Shabat spectral problem and the symmetries the hierarchy of point and generalized symmetries of the nonlinear Schroedinger equation.Comment: 10 pages, contribution to the proceedings of the NEEDS 2007 Conferenc

Lie point symmetries of difference equations and lattices

A method is presented for finding the Lie point symmetry transformations acting simultaneously on difference equations and lattices, while leaving the solution set of the corresponding difference scheme invariant. The method is applied to several examples. The found symmetry groups are used to obtain particular solutions of differential-difference equations

Continuous Symmetries of Difference Equations

Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program: the use of Lie groups to study difference equations. We show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners. One is to use generalized symmetries acting on solutions of difference equations, but leaving the lattice invariant. The other is to restrict to point symmetries, but to allow them to also transform the lattice.Comment: Review articl

Discrete derivatives and symmetries of difference equations

We show on the example of the discrete heat equation that for any given discrete derivative we can construct a nontrivial Leibniz rule suitable to find the symmetries of discrete equations. In this way we obtain a symmetry Lie algebra, defined in terms of shift operators, isomorphic to that of the continuous heat equation.Comment: submitted to J.Phys. A 10 Latex page

Multiple-scale analysis of discrete nonlinear partial difference equations: the reduction of the lattice potential KdV

We consider multiple lattices and functions defined on them. We introduce slow varying conditions for functions defined on the lattice and express the variation of a function in terms of an asymptotic expansion with respect to the slow varying lattices. We use these results to perform the multiple--scale reduction of the lattice potential Korteweg--de Vries equation.Comment: 17 pages. 1 figur

Neurodegeneration with brain iron accumulation: update on pathogenic mechanisms.

Perturbation of iron distribution is observed in many neurodegenerative disorders, including Alzheimer's and Parkinson's disease, but the comprehension of the metal role in the development and progression of such disorders is still very limited. The combination of more powerful brain imaging techniques and faster genomic DNA sequencing procedures has allowed the description of a set of genetic disorders characterized by a constant and often early accumulation of iron in specific brain regions and the identification of the associated genes; these disorders are now collectively included in the category of neurodegeneration with brain iron accumulation (NBIA). So far 10 different genetic forms have been described but this number is likely to increase in short time. Two forms are linked to mutations in genes directly involved in iron metabolism: neuroferritinopathy, associated to mutations in the FTL gene and aceruloplasminemia, where the ceruloplasmin gene product is defective. In the other forms the connection with iron metabolism is not evident at all and the genetic data let infer the involvement of other pathways: Pank2, Pla2G6, C19orf12, COASY, and FA2H genes seem to be related to lipid metabolism and to mitochondria functioning, WDR45 and ATP13A2 genes are implicated in lysosomal and autophagosome activity, while the C2orf37 gene encodes a nucleolar protein of unknown function. There is much hope in the scientific community that the study of the NBIA forms may provide important insight as to the link between brain iron metabolism and neurodegenerative mechanisms and eventually pave the way for new therapeutic avenues also for the more common neurodegenerative disorders. In this work, we will review the most recent findings in the molecular mechanisms underlining the most common forms of NBIA and analyze their possible link with brain iron metabolism

Assessing reservoir operations risk under climate change

Risk-based planning offers a robust way to identify strategies that permit adaptive water resources management under climate change. This paper presents a flexible methodology for conducting climate change risk assessments involving reservoir operations. Decision makers can apply this methodology to their systems by selecting future periods and risk metrics relevant to their planning questions and by collectively evaluating system impacts relative to an ensemble of climate projection scenarios (weighted or not). This paper shows multiple applications of this methodology in a case study involving California\u27s Central Valley Project and State Water Project systems. Multiple applications were conducted to show how choices made in conducting the risk assessment, choices known as analytical design decisions, can affect assessed risk. Specifically, risk was reanalyzed for every choice combination of two design decisions: (1) whether to assume climate change will influence flood-control constraints on water supply operations (and how), and (2) whether to weight climate change scenarios (and how). Results show that assessed risk would motivate different planning pathways depending on decision-maker attitudes toward risk (e.g., risk neutral versus risk averse). Results also show that assessed risk at a given risk attitude is sensitive to the analytical design choices listed above, with the choice of whether to adjust flood-control rules under climate change having considerably more influence than the choice on whether to weight climate scenarios

Integrability of Differential-Difference Equations with Discrete Kinks

In this article we discuss a series of models introduced by Barashenkov, Oxtoby and Pelinovsky to describe some discrete approximations to the \phi^4 theory which preserve travelling kink solutions. We show, by applying the multiple scale test that they have some integrability properties as they pass the A_1 and A_2 conditions. However they are not integrable as they fail the A_3 conditions.Comment: submitted to the Proceedings of the workshop "Nonlinear Physics: Theory and Experiment.VI" in a special issue di Theoretical and Mathematical Physic