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.     

Counting perfect matchings in n-extendable graphs

The structural theory of matchings is used to establish lower bounds on the number of perfect matchings in n-extendable graphs. It is shown that any such graph on p vertices and q edges contains at least ⌈(n+1)!/4[q-p-(n-1)(2Δ-3)+4]⌉ different perfect matchings, where Δ is the maximum degree of a vertex in G.     

Finding more perfect matchings in leapfrog fullerenes

We use some recent results on the existence of long cycles in leapfrog fullerenes to establish new exponential lower bounds on the number of perfect matchings in such graphs. The new bounds are expressed in terms of Fibonacci numbers.     

Decoherence in quantum mechanics

We study decoherence in a simple quantum mechanical model using two approaches. Firstly, we follow the conventional approach to decoherence where one is interested in solving the reduced density matrix from the perturbative master equation. Secondly, we consider our novel correlator approach to decoherence where entropy is generated by neglecting observationally inaccessible correlators. We show that both methods can accurately predict decoherence time scales. However, the perturbative master equation generically suffers from instabilities which prevents us to reliably calculate the system’s total entropy increase. We also discuss the relevance of the results in our quantum mechanical model for interacting field theories.     

Resolving Curvature Singularities in Holomorphic Gravity

We formulate a holomorphic theory of gravity and study how the holomorphy symmetry alters the two most important singular solutions of general relativity: black holes and cosmology. We show that typical observers (freely) falling into a holomorphic black hole do not encounter a curvature singularity. Likewise, typical observers do not experience Big Bang singularity. Unlike Hermitian gravity (Mantz and Prokopec in , 2008), holomorphic gravity does not respect the reciprocity symmetry and thus it is mainly a toy model for a gravity theory formulated on complex space-times. Yet it is a model that deserves a closer investigation since in many aspects it resembles Hermitian gravity and yet calculations are simpler. Our study of light bending and gravitational waves in weak holomorphic gravitational fields strongly suggests that holomorphic gravity reduces to general relativity at large distance scales.     

On the Reduction of Some Dispersionless Integrable Systems

We study the invariance, reduction, exact solutions and conservation laws of the dispersionless Kadomtsev-Petviashivili and the heavenly equation. The existence of nontrivial conservation laws lead to repeated reductions paving the way for determining exact solutions. A variety of such solutions exist dependent on the subalgebra of Lie point symmetries that is chosen in the reduction procedure.     

Symmetry and transformation properties of iterative ordinary differential equation

Symmetries of linear iterative equations and new conditions on the infinitesimals are obtained. Regarding the expressions of the solutions in terms of the parameters of the source equation, an ansatz is made on the original parameters. We have also obtained an expression for the source parameters of the transformed equation under equivalence transformations. We conducted this work with a special emphasis on second-, third- and fourth-order equations, although some of our results are valid for equations of a general order.     

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.     

Inducing gas flow and swirl in tubes using ionic wind from corona discharges

Air induction and rotation about the axis of a vertical tube is generated by an assembly of corona discharges between pinpoints and earthed electrodes, which induces a swirling ionic wind. The mechanism is elucidated and the geometrical configuration of the electric field lines of force is optimised by studying the deposition of charged particles on the earth electrodes, by numerical modelling of a simplified geometry and ultimately by maximising the measured tangential velocities. Upward convective flows of up to a litre per second are provided by an additional ionic wind pump at the base of the tube. With assemblies of up to three layers of six points each, tangential velocities of up to 3.3 m/s (≈900 rpm) are attained at the periphery, as recorded by small Pitot tubes. The concept, developed particularly for microgravity environments, appears suitable for adding a substantial centrifugal contribution to the operation of electrostatic precipitators and as a basis for further progress on electrical field-controlled burners.