Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients

Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach

A solution of a problem of Sophus Lie: Normal forms of 2-dim metrics admitting two projective vector fields

We give a complete list of normal forms for the 2-dimensional metrics that admit a transitive Lie pseudogroup of geodesic-preserving transformations and we show that these normal forms are mutually non-isometric. This solves a problem posed by Sophus Lie.Comment: This is an extended version of the paper that will appear in Math. Annalen. Some typos were corrected, references were updated, title was changed (as in the journal version). 31 page

Deformation quantization of linear dissipative systems

A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of non-stationary Poisson brackets, i.e. corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation we develop a consistent deformation quantization procedure involving a non-stationary star-product $*_t$ and an ``extended'' operator of time derivative $D_t=\partial_t+...$, differentiating the $\ast_t$-product. As in the usual case, the $\ast_t$-algebra of physical observables is shown to admit an essentially unique (time dependent) trace functional $\mathrm{Tr}_t$. Using these ingredients we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.Comment: 14 pages, typos correcte

Harnessing hypoxic adaptation to prevent, treat, and repair stroke

The brain demands oxygen and glucose to fulfill its roles as the master regulator of body functions as diverse as bladder control and creative thinking. Chemical and electrical transmission in the nervous system is rapidly disrupted in stroke as a result of hypoxia and hypoglycemia. Despite being highly evolved in its architecture, the human brain appears to utilize phylogenetically conserved homeostatic strategies to combat hypoxia and ischemia. Specifically, several converging lines of inquiry have demonstrated that the transcription factor hypoxia-inducible factor-1 (HIF1-1) mediates the activation of a large cassette of genes involved in adaptation to hypoxia in surviving neurons after stroke. Accordingly, pharmacological or molecular approaches that engage hypoxic adaptation at the point of one of its sensors (e.g., inhibition of HIF prolyl 4 hydroxylases) leads to profound sparing of brain tissue and enhanced recovery of function. In this review, we discuss the potential mechanisms that could subserve protective and restorative effects of augmenting hypoxic adaptation in the brain. The strategy appears to involve HIF-dependent and HIF-independent pathways and more than 70 genes and proteins activated transcriptionally and post-transcriptionally that can act at cellular, local, and system levels to compensate for oxygen insufficiency. The breadth and depth of this homeostatic program offers a hopeful alternative to the current pessimism towards stroke therapeutics