Invariant solutions as internal singularities of nonlinear differential equations and their use for qualitative analysis of implicit and numerical solutions

Lie group analysis of nonlinear differential equations reveals existence of singularities provided by invariant solutions and invisible from the form of the equation in question. We call them internal singularities in contrast with external singularities manifested by the form of the equation. It is illustrated by way of examples that internal singularities are useful for analyzing a behaviour of solutions of nonlinear differential equations near external singularities.     

A new conservation theorem

A general theorem on conservation laws for arbitrary differential equations is proved. The theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables. The new theorem does not require existence of a Lagrangian and is based on a concept of an adjoint equation for non-linear equations suggested recently by the author. It is proved that the adjoint equation inherits all symmetries of the original equation. Accordingly, one can associate a conservation law with any group of Lie, Lie-Bäcklund or non-local symmetries and find conservation laws for differential equations without classical Lagrangians.     

Central neurocytoma: proton MR spectroscopy and diffusion weighted MR imaging findings

Purpose

To present proton magnetic resonance spectroscopy and diffusion-weighted imaging (DWI) findings of central neurocytoma (CN).

Methods and Materials

Imaging findings of seven patients with the histopathological diagnosis of CN (five male and two female; age range, 21–28 years of age) were evaluated retrospectively. In addition to conventional magnetic resonance imaging features, we also assessed the metabolite ratios and tumor normalized apparent diffusion coefficient (NADC), which was calculated by dividing the tumor apparent diffusion coefficient (ADC) values by normal ADC. Approval from our institutional review board was obtained for this review.

Results

The tumor choline/creatine ratios were 5.17±2.38, while N-acetyl aspartate/choline and N-acetyl aspartate/creatine ratios were 0.33±0.15 and 1.84±1.38, respectively. On DWI, tumors had heterogeneous hyperintense appearances when compared with the contralateral parietal lobe white matter and tumor NADC values were 0.63±0.05.

Conclusion

Significantly increased choline/creatine and decreased N-acetyl aspartate/choline ratios with lower NADC values in CN resemble high-grade gliomas and complicate the diagnosis. Familarity its physiologic features would help to presurgical diagnosis of ventricular and exraventricular CNs.     

Minimization of the Truncation Error by Grid Adaptation

A new grid adaptation strategy, which minimizes the truncation error of a pth-order finite difference approximation, is proposed. The main idea of the method is based on the observation that the global truncation error associated with discretization on nonuniform meshes can be minimized if the interior grid points are redistributed in an optimal sequence. The method does not explicitly require the truncation error estimate, and at the same time, it allows one to increase the design order of approximation globally by one, so that the same finite difference operator reveals superconvergence properties on the optimal grid. Another very important characteristic of the method is that if the differential operator and the metric coefficients are evaluated identically by some hybrid approximation, then the single optimal grid generator can be employed in the entire computational domain independently of points where the hybrid discretization switches from one approximation to another. Generalization of the present method to multiple dimensions is presented. Numerical calculations of several one-dimensional and one two-dimensional test examples demonstrate the performance of the method and corroborate the theoretical results.     

Zero Point Energy of a Massless Scalar Field in the Cosmic String Space-Time

We consider the ground state energy of a massive scalar field in the space-time of a thick cosmic string in the 2+1 dimensional case for arbitrary angle deficit by using the zeta-function approach. Final numerical calculations were made in the massless case, only. We show that the zero point energy is negative, and for small angle deficit it is proportional to the fourth degree of the deficit.