Perturbed Three Vortex Dynamics

It is well known that the dynamics of three point vortices moving in an ideal fluid in the plane can be expressed in Hamiltonian form, where the resulting equations of motion are completely integrable in the sense of Liouville and Arnold. The focus of this investigation is on the persistence of regular behavior (especially periodic motion) associated to completely integrable systems for certain (admissible) kinds of Hamiltonian perturbations of the three vortex system in a plane. After a brief survey of the dynamics of the integrable planar three vortex system, it is shown that the admissible class of perturbed systems is broad enough to include three vortices in a half-plane, three coaxial slender vortex rings in three-space, and `restricted' four vortex dynamics in a plane. Included are two basic categories of results for admissible perturbations: (i) general theorems for the persistence of invariant tori and periodic orbits using Kolmogorov-Arnold-Moser and Poincare-Birkhoff type arguments; and (ii) more specific and quantitative conclusions of a classical perturbation theory nature guaranteeing the existence of periodic orbits of the perturbed system close to cycles of the unperturbed system, which occur in abundance near centers. In addition, several numerical simulations are provided to illustrate the validity of the theorems as well as indicating their limitations as manifested by transitions to chaotic dynamics.Comment: 26 pages, 9 figures, submitted to the Journal of Mathematical Physic

Integrability Analysis of a Two-Component Burgers-Type Hierarchy

The Lax integrability of a two-component polynomial Burgers-type dynamical system is analyzed by using a differential-algebraic approach. Its linear adjoint matrix Lax representation is constructed. A related recursive operator and an infinite hierarchy of nonlinear Lax integrable dynamical systems of the Burgers–Korteweg–de-Vries type are obtained by the gradient-holonomic technique. The corresponding Lax representations are presented.Лаксiвську iнтегровнiсть двокомпонентної полiномiальної динамiчної системи типу Бюргерса проаналiзовано за допомогою диференцiально-алгебраїчного пiдходу. Побудовано її лiнiйне спряжене матричне лаксiвське зображення. Вiдповiдний рекурсивний оператор та нескiнченну iєрархiю нелiнiйних динамiчних систем типу Бюргерса – Кортевега – де Фрiза, iнтегровних за Лаксом, отримано за допомогою градiєнтно-голономного методу. Наведено вiдповiднi лаксiвськi зображення

On the minimum number of neighbours for good routing performance in MANETs

In a mobile ad hoc network, where nodes are deployed without any wired infrastructure and communicate via multihop wireless links, the network topology is based on the nodes’ locations and transmission ranges. The nodes communicate through wireless links, with each node acting as a relay when necessary to allow multihop communications. The network topology can have a major impact on network performance. We consider the impact of number and placement of neighbours on mobile network performance. Specifically, we consider how neighbour node placement affects the network overhead and routing delay. We develop an analytical model, verified by simulations, which shows widely varying performance depending on source node speed and, to a lesser extent, number of neighbour nodes

cAMP-dependent Protein Kinase Activation Lowers Hepatocyte cAMP

Rat hepatocyte protein kinase was activated by incubating the cells with various cAMP analogs. Boiled extracts were then prepared and Sephadex G-25 chromatography was carried out. The G-25 procedure separated the analogs from cAMP since the resin had the unexpected property of binding cyclic nucleotides with differing affinities. Separation was necessary because the analogs would otherwise interfere with the sensitive protein kinase activation method developed for assay of cAMP. The cAMP analogs, but not 5\u27-AMP, lowered basal cAMP by 50-70%. The effect was rapid, analog concentration-dependent, and occurred parallel with phosphorylase activation, suggesting that the cAMP analogs act through cAMP-dependent protein kinase activation. A cAMP analog completely blocked the cAMP elevation produced by relatively low concentrations of glucagon, but did not block the phosphorylase response, indicating that the cAMP analog substitutes for cAMP as the intracellular activator of protein kinase. One implication of the results is that elevation of cAMP and protein kinase activity by hormones has a negative feedback effect on the cellular cAMP level

A perspective on vibration-induced size segregation of granular materials

Segregation of particulate mixtures is a problem of great consequence in industries involved with the handling and processing of granular materials in which homogeneity is generally required. While there are several factors that may be responsible for segregation in bulk solids, it is well accepted that nonuniformity in particle size is a fundamental contributor. When the granular material is exposed to vibrations, the question of whether or not convection is an essential ingredient for size segregation is addressed by distinguishing between the situation where vibrations are not sufficiently energetic to promote a mean flow of the bulk solid, and those cases where a convective flow does occur. Based on experimental and simulation results in the literature, as well as dynamical systems analysis of a recent model of a binary granular mixture, it is proposed that "void-filling" beneath large particles is a universal mechanism promoting segregation, while convection essentially provides a means of mixing enhancement

Reduced pre-Lie algebraic structures, the weak and weakly deformed Balinsky-Novikov type symmetry algebras and related Hamiltonian operators

The Lie algebraic scheme for constructing Hamiltonian operators is differential-algebraically recast and an effective approach is devised for classifying the underlying algebraic structures of integrable Hamiltonian systems. Lie–Poisson analysis on the adjoint space to toroidal loop Lie algebras is employed to construct new reduced pre-Lie algebraic structures in which the corresponding Hamiltonian operators exist and generate integrable dynamical systems. It is also shown that the Balinsky–Novikov type algebraic structures, obtained as a Hamiltonicity condition, are derivations on the Lie algebras naturally associated with differential toroidal loop algebras. We study nonassociative and noncommutive algebras and the related Lie-algebraic symmetry structures on the multidimensional torus, generating via the Adler–Kostant–Symes scheme multi-component and multi-dimensional Hamiltonian operators. In the case of multidimensional torus, we have constructed a new weak Balinsky–Novikov type algebra, which is instrumental for describing integrable multidimensional and multicomponent heavenly type equations. We have also studied the current algebra symmetry structures, related with a new weakly deformed Balinsky–Novikov type algebra on the axis, which is instrumental for describing integrable multicomponent dynamical systems on functional manifolds. Moreover, using the non-associative and associative left-symmetric pre-Lie algebra theory of Zelmanov, we also explicate Balinsky–Novikov algebras, including their fermionic version and related multiplicative and Lie structures

The integrability of Lie-invariant geometric objects generated by ideals in the Grassmann algebra

We investigate closed ideals in the Grassmann algebra serving as bases of Lie-invariant geometric objects studied before by E. Cartan. Especially, the E. Cartan theory is enlarged for Lax integrable nonlinear dynamical systems to be treated in the frame work of the Wahlquist Estabrook prolongation structures on jet-manifolds and Cartan-Ehresmann connection theory on fibered spaces. General structure of integrable one-forms augmenting the two-forms associated with a closed ideal in the Grassmann algebra is studied in great detail. An effective Maurer-Cartan one-forms construction is suggested that is very useful for applications. As an example of application the developed Lie-invariant geometric object theory for the Burgers nonlinear dynamical system is considered having given rise to finding an explicit form of the associated Lax type representation