Emergent Phase Space Description of Unitary Matrix Model

We show that large $N$ phases of a $0$ dimensional generic unitary matrix model (UMM) can be described in terms of topologies of two dimensional droplets on a plane spanned by eigenvalue and number of boxes in Young diagram. Information about different phases of UMM is encoded in the geometry of droplets. These droplets are similar to phase space distributions of a unitary matrix quantum mechanics (UMQM) ($(0 + 1)$ dimensional) on constant time slices. We find that for a given UMM, it is possible to construct an effective UMQM such that its phase space distributions match with droplets of UMM on different time slices at large $N$. Therefore, large $N$ phase transitions in UMM can be understood in terms of dynamics of an effective UMQM. From the geometry of droplets it is also possible to construct Young diagrams corresponding to $U(N)$ representations and hence different large $N$ states of the theory in momentum space. We explicitly consider two examples : single plaquette model with $\text{Tr} U^2$ terms and Chern-Simons theory on $S^3$. We describe phases of CS theory in terms of eigenvalue distributions of unitary matrices and find dominant Young distributions for them.Comment: 52 pages, 15 figures, v2 Introduction and discussions extended, References adde

A vortex method for viscous unsteady flows

A simple method is presented for the creation of vorticity from a solid boundary in the form of discrete vortex blobs. These vortices have uniform-vorticity cores and are released from a number of points distributed around the body surface at a small distance away from the surface. Their strengths are obtained by satisfying the no-slip condition at the midpoints of the corresponding vortex sheets. The dynamics of these vortices is then considered by including the effect of viscous diffusion by random13; walk approach. The method is applied to the starting flow past a circular cylinder. A detailed parametric study shows that viscosity has the most significant effect on13; the computed results and, in order to obtain meaningful results, it must be reduced suitably to compensate for the artificial viscosity arising from numerical integration13; of equations of motion of the discrete vortices. Results are obtained for three typical Reynolds numbers of 9500, 3000 and 550, and are found to be in very good agreement13; with experimental measurements for the centerline velocity distribution. In particular, the case of Re=550 illustrates the usefulness of the present model even at low Reynolds numbers. The vortex patterns, velocity vectors and particle path lines also conform to the experimental flow visualization pictures. Though the method is developed for a simple configuration like a circular cylinder, it can be easily extended to any general13; body shape

When Hashing Met Matching: Efficient Spatio-Temporal Search for Ridesharing

Carpooling, or sharing a ride with other passengers, holds immense potential for urban transportation. Ridesharing platforms enable such sharing of rides using real-time data. Finding ride matches in real-time at urban scale is a difficult combinatorial optimization task and mostly heuristic approaches are applied. In this work, we mathematically model the problem as that of finding near-neighbors and devise a novel efficient spatio-temporal search algorithm based on the theory of locality sensitive hashing for Maximum Inner Product Search (MIPS). The proposed algorithm can find $k$ near-optimal potential matches for every ride from a pool of $n$ rides in time $O(n^{1 + \rho} (k + \log n) \log k)$ and space $O(n^{1 + \rho} \log k)$ for a small $\rho < 1$. Our algorithm can be extended in several useful and interesting ways increasing its practical appeal. Experiments with large NY yellow taxi trip datasets show that our algorithm consistently outperforms state-of-the-art heuristic methods thereby proving its practical applicability