Hypercomplex analysis and integration of systems of ordinary differential equations

We review the theory of hypercomplex numbers and hypercomplex analysis with the ultimate goal of applying them to issues related to the integration of systems of ordinary differential equations (ODEs). We introduce the notion of hypercomplexification, which allows the lifting of some results known for scalar ODEs to systems of ODEs. In particular, we provide another approach to the construction of superposition laws for some Riccati‐type systems, we obtain invariants of Abel‐type systems, we derive integrable Ermakov systems through hypercomplexification, we address the problem of linearization by hypercomplexification, and we provide a solution to the inverse problem of the calculus of variations for some systems of ODEs. Copyright © 2016 John Wiley & Sons, Ltd.     

<Emphasis Type="Italic">parkin</Emphasis> counteracts symptoms in a <Emphasis Type="Italic">Drosophila</Emphasis> model of Parkinson's disease

Background

Parkinson's disease, a prevalent neurodegenerative disease, is characterized by the reduction of dopaminergic neurons resulting in the loss of motor control, resting tremor, the formation of neuronal inclusions and ultimately premature death. Two inherited forms of PD have been linked to mutations in the α-synuclein and parkin genes. The parkin protein functions as an ubiquitin ligase targeting specific proteins for degradation. Expression of human α-synuclein in Drosophila neurons recapitulates the loss of motor control, the development of neuronal inclusions, degeneration of dopaminergic neurons and the ommatidial array to provide an excellent genetic model of PD.

Results

To investigate the role of parkin, we have generated transgenic Drosophila that conditionally express parkin under the control of the yeast UAS enhancer. While expression of parkin has little consequence, co-expression of parkin with α-synuclein in the dopaminergic neurons suppresses the α-synuclein-induced premature loss of climbing ability. In addition directed expression of parkin in the eye counteracts the α-synuclein-induced degeneration of the ommatidial array. These results show that parkin suppresses the PD-like symptoms observed in the α-synuclein-dependent Drosophila model of PD.

Conclusion

The highly conserved parkin E3 ubiquitin ligase can suppress the damaging effects of human α-synuclein. These results are consistent with a role for parkin in targeting α-synuclein to the proteasome. If this relationship is conserved in humans, this suggests that up-regulation of parkin should suppress α-synucleinopathic PD. The development of therapies that regulate parkin activity may be crucial in the treatment of PD.

     

Liquid-crystalline phase behavior of a colloidal rod-plate mixture

The phase behavior of rod-plate mixtures was investigated using model systems containing unambiguously rod- and plate-shaped colloids. We find that the theoretically disputed biaxial nematic phase is unstable with respect to demixing into an isotropic and two uniaxial nematic phases. The phase behavior at very high densities is exceptionally rich and includes the coexistence of up to four different liquid crystalline phases, which stem from the coupling between the employed particle shapes and polydispersity.     

Symmetries,Conservation Laws and Multipliers via Partial Lagrangians and Noether’s Theorem for Classically Non-Variational Problems

We show how one can construct conservation laws of equations that are not variational but are Euler–Lagrange in part using Noether-type symmetries associated with partial Lagrangians. These Noether-type symmetries are, usually, not symmetries of the system. The resultant construction of the conservation law resorts to a formula equivalent to Noether’s theorem. A variety of examples are given.     

Relationship between Symmetries andConservation Laws

The fundamental relation between Lie-Bäcklund symmetry generators andconservation laws of an arbitrary differential equation is derived without regardto a Lagrangian formulation of the differential equation. This relation is used inthe construction of conservation laws for partial differential equations irrespectiveof the knowledge or existence of a Lagrangian. The relation enables one toassociate symmetries to a given conservation law of a differential equation.Applications of these results are illustrated for a range of examples.     

Noether-Type Symmetries and Conservation Laws Via Partial Lagrangians

We show how one can construct conservation laws of Euler-Lagrange-type equations via Noether-type symmetry operators associated with what we term partial Lagrangians. This is even in the case when a system does not directly have a usual Lagrangian, e.g. scalar evolution equations. These Noether-type symmetry operators do not form a Lie algebra in general. We specify the conditions under which they do form an algebra. Furthermore, the conditions under which they are symmetries of the Euler-Lagrange-type equations are derived. Examples are given including those that admit a standard Lagrangian such as the Maxwellian tail equation, and equations that do not such as the heat and nonlinear heat equations. We also obtain new conservation laws from Noether-type symmetry operators for a class of nonlinear heat equations in more than two independent variables.     

On Solutions of Some Non-Linear Differential Equations Arising in Newtonian and Non-Newtonian Fluids

Some steady as well as unsteady solutions of the equations of motion for an incompressible Newtonian and non-Newtonian (second-grade) fluids are obtained by applying different methods including the Lie symmetry group method. The flows considered are axially symmetric with the swirling motion, and the governing equations for second-grade fluid flow have been modeled. Expressions for streamlines, velocity and vorticity components are constructed explicitly in each case. Exact analytical solutions in second-grade fluid are obtained and compared with the corresponding viscous solutions.     

Linearizability criteria for systems of two second-order differential equations by complex methods

Lie??s linearizability criteria for scalar second-order ordinary differential equations had been extended to systems of second-order ordinary differential equations by using geometric methods. These methods not only yield the linearizing transformations but also the solutions of the nonlinear equations. Here, complex methods for a scalar ordinary differential equation are used for linearizing systems of two second-order ordinary and partial differential equations, which can use the power of the geometric method for writing the solutions. Illustrative examples of mechanical systems including the Lane?CEmden type equations which have roots in the study of stellar structures are presented and discussed.