Limit Theorems and Fluctuations for Point Vortices of Generalized Euler Equations

We prove a mean field limit, a law of large numbers and a central limit theorem for a system of point vortices on the 2D torus at equilibrium with positive temperature. The point vortices are formal solutions of a class of equations generalising the Euler equations, and are also known in the literature as generalised inviscid SQG. The mean-field limit is a steady solution of the equations, the CLT limit is a stationary distribution of the equations.

A scaling limit for the stochastic mSQG equations with multiplicative transport noises

We consider on the 2D torus the modified Surface Quasi-Geostrophic (mSQG) equation with [Formula: see text]-initial data and perturbed by multiplicative transport noise. Under a suitable scaling of the noises, we show that the solutions converge weakly to the unique solution of the dissipative mSQG equation.

Prescribed Hamiltonian Connectedness of 2D Torus

Tori are important fundamental interconnection networks for multiprocessor systems. Hamiltonian paths are important in information communication of multiprocessor systems, and Hamiltonian path embedding capability is an important aspect to determine if a network topology is suitable for a real application. In real systems, some links may have better performance. Therefore, when embedding Hamiltonian path into interconnection networks, it is desirable that these Hamiltonian paths would pass through the links with better performance. Given a two two-dimensional torus T (m, n) with m, n ≥ 5 odd, let L be a linear forest with at most two edges in T (m, n) and let u and v be two distinct vertices in T (m, n) such that none of the paths in L has u or v as internal node or both of them as end nodes. In this paper, we construct a hamiltonian path of T (m, n) between u and v passing through L.

Eigenfunctions with Infinitely Many Isolated Critical Points

We construct a Riemannian metric on the 2D torus, such that for infinitely many eigenvalues of the Laplace–Beltrami operator, a corresponding eigenfunction has infinitely many isolated critical points. A minor modification of our construction implies that each of these eigenfunctions has a level set with infinitely many connected components (i.e., a linear combination of two eigenfunctions may have infinitely many nodal domains).

Evaluation of Over Looped 2d Mesh Topology for Network on Chip

The arrangements of nodes in the network identifies the complexity of the network. To reduce the complexity, a structural arrangements of nodes has to be taken care. The mesh topology yields attraction than the other traditional topologies. Making the opposite corner nodes to communicate with less hops and avoiding the centre of the networks traffic, Over-Looped 2D Mesh Topology is proposed. For a homogeneous systems the proposed work can be deployed without altering any of the switch component compositions. By making the flits, travel in the outer corner nodes with the help of looping nodes will make the journey from source to destination with less hops. For smaller network below 4x4 the looping is less responsive. For odd or even number of columns and rows the looping can be done. The number of columns and number of rows need not to be equal. The left over nodes will be looped accordingly. The hop count of the Over-Looped 2D Mesh Topology compared to 2D mesh decreases the journey by 25%. The wiring segmentation and the wiring length of the system more than 10 % from 2D mesh and less than 20% from 2D Torus