Smoke Rings from Smoke

We give an algorithm which extracts vortex filaments (“smoke rings”) from a given 3D velocity field. Given a filament strength h> 0, an optimal number of vortex filaments, together with their extent and placement, is given by the zero set of a complex valued function over the domain. This function is the global minimizer of a quadratic energy based on a Schrödinger operator. Computationally this amounts to finding the eigenvector belonging to the smallest eigenvalue of a Laplacian type sparse matrix. Turning traditional vector field representations of flows, for example, on a regular grid, into a corresponding set of vortex filaments is useful for visualization, analysis of measured flows, hybrid simulation methods, and sparse representations. To demonstrate our method we give examples from each of these

Ring-LWE Cryptography for the Number Theorist

In this paper, we survey the status of attacks on the ring and polynomial learning with errors problems (RLWE and PLWE). Recent work on the security of these problems [Eisentr\"ager-Hallgren-Lauter, Elias-Lauter-Ozman-Stange] gives rise to interesting questions about number fields. We extend these attacks and survey related open problems in number theory, including spectral distortion of an algebraic number and its relationship to Mahler measure, the monogenic property for the ring of integers of a number field, and the size of elements of small order modulo q.Comment: 20 Page

Efficient quantum processing of ideals in finite rings

Suppose we are given black-box access to a finite ring R, and a list of generators for an ideal I in R. We show how to find an additive basis representation for I in poly(log |R|) time. This generalizes a recent quantum algorithm of Arvind et al. which finds a basis representation for R itself. We then show that our algorithm is a useful primitive allowing quantum computers to rapidly solve a wide variety of problems regarding finite rings. In particular we show how to test whether two ideals are identical, find their intersection, find their quotient, prove whether a given ring element belongs to a given ideal, prove whether a given element is a unit, and if so find its inverse, find the additive and multiplicative identities, compute the order of an ideal, solve linear equations over rings, decide whether an ideal is maximal, find annihilators, and test the injectivity and surjectivity of ring homomorphisms. These problems appear to be hard classically.Comment: 5 page

Variational Hilbert space truncation approach to quantum Heisenberg antiferromagnets on frustrated clusters

We study the spin-$\frac{1}{2}$ Heisenberg antiferromagnet on a series of finite-size clusters with features inspired by the fullerenes. Frustration due to the presence of pentagonal rings makes such structures challenging in the context of quantum Monte-Carlo methods. We use an exact diagonalization approach combined with a truncation method in which only the most important basis states of the Hilbert space are retained. We describe an efficient variational method for finding an optimal truncation of a given size which minimizes the error in the ground state energy. Ground state energies and spin-spin correlations are obtained for clusters with up to thirty-two sites without the need to restrict the symmetry of the structures. The results are compared to full-space calculations and to unfrustrated structures based on the honeycomb lattice.Comment: 22 pages and 12 Postscript figure

Net and Prune: A Linear Time Algorithm for Euclidean Distance Problems

We provide a general framework for getting expected linear time constant factor approximations (and in many cases FPTAS's) to several well known problems in Computational Geometry, such as $k$-center clustering and farthest nearest neighbor. The new approach is robust to variations in the input problem, and yet it is simple, elegant and practical. In particular, many of these well studied problems which fit easily into our framework, either previously had no linear time approximation algorithm, or required rather involved algorithms and analysis. A short list of the problems we consider include farthest nearest neighbor, $k$-center clustering, smallest disk enclosing $k$ points, $k$th largest distance, $k$th smallest $m$-nearest neighbor distance, $k$th heaviest edge in the MST and other spanning forest type problems, problems involving upward closed set systems, and more. Finally, we show how to extend our framework such that the linear running time bound holds with high probability