An important question that arises in the study of high-dimensional vector representations learned from data are, given a set [Formula: see text] of vectors and a query q, estimate the number of points within a specified distance threshold of q. We develop two estimators, LSH count and multiprobe count that use locality-sensitive hashing to preprocess the data to accurately and efficiently estimate the answers to such questions via importance sampling. A key innovation is the ability to maintain a small number of hash tables via preprocessing data structures and algorithms that sample from multiple buckets in each hash table. We give bounds on the space requirements and sample complexity of our schemes and demonstrate their effectiveness in experiments on a standard word embedding data set.

We study the intercommuting of semilocal strings and Skyrmions, for a wide range of internal parameters, velocities and intersection angles by numerically evolving the equations of motion. We find that the collisions of strings and strings, strings and Skyrmions, and Skyrmions and Skyrmions, all lead to intercommuting for a wide range of parameters. Even the collisions of unstable Skyrmions and strings leads to intercommuting, demonstrating that the phenomenon of intercommuting is very robust, extending to dissimilar field configurations that are not stationary solutions. Even more remarkably, at least for the semilocal U (2) formulation considered here, all intercommutations trigger a reversion to U(1) Nielsen-Olesen strings.Physical systems can have a large number of spontaneously broken symmetries, some of which may be local (gauged) while others may be global. In cases where the symmetries are only partially gauged, or gauged with unequal strengths, semilocal strings may be present [1,2,3,4]. If the gauge couplings are widely disparate, the strings may also be stable. Semilocal string solutions exist in the standard electroweak model, but the SU (2) and U (1) gauge couplings are close enough that the solutions are unstable [5,6]. Recently twisted and current-carrying semilocal string solutions have been discovered that may be stable for other values of parameters [7,8]. Semilocal strings arise in supersymmetric QCD if the number of flavors is larger than the order of the symmetry group [9]. In string theory, they can arise as field theoretic realizations of cosmic D-strings [10], in which case they may play a role in cosmology. The formation of semilocal strings in a phase transition has been studied in Refs. [11,12], in a cosmological setting in Refs. [13,14], and in superstring cosmology [15].In the case of topological gauged [16,17,18] and global [19] U (1) strings, numerical evolution of the field theory equations show that colliding strings intercommute (Fig. 1) for almost [20] any scattering angle and velocity. Thus the probability of intercommuting is unity for topological strings. Qualitative arguments [19,21] have been given in an effort to understand intercommutation of topological strings. In the context of string theory cosmic strings and QCD strings, intercommuting probabilities have been calculated in Refs. [23,24] within various approximations.In this paper we study the interactions of semilocal strings, which are topological structures embedded in the solution space of the partially gauged theory, and the related semilocal Skyrmions which are fat and fuzzy objects (still with one "long", i.e. stringlike dimension) on which strings can terminate, much like strings ending on monopoles. (We shall show below (Figs 1, 2) however, that the termination is dynamic, moving along the string.) The scalar field in a Skyrmion can be arbitrarily close to its vacuum expectation value everywhere, in contrast to a string in which the scalar field magnitude vanishes (which is a nonvacuum value) on some curve. ...

It is a well known analytic result in general relativity that the 2-dimensional area of the apparent horizon of a black hole remains invariant regardless of the motion of the observer, and in fact is independent of the t = constant slice, which can be quite arbitrary in general relativity. Nonetheless the explicit computation of horizon area is often substantially more difficult in some frames (complicated by the coordinate form of the metric), than in other frames. Here we give an explicit demonstration for very restricted metric forms of (Schwarzschild and Kerr) vacuum black holes. In the Kerr-Schild coordinate expression for these spacetimes they have an explicit Lorentzinvariant form. We consider boosted versions with the black hole moving through the coordinate system. Since these are stationary black hole spacetimes, the apparent horizons are two dimensional cross sections of their event horizons, so we compute the areas of apparent horizons in the boosted space with (boosted) t = constant, and obtain the same result as in the unboosted case. Note that while the invariance of area is generic, we deal only with black holes in the Kerr-Schild form, and consider only one particularly simple change of slicing which amounts to a boost. Even with these restrictions we find that the results illuminate the physics of the horizon as a null surface and provide a useful pedagogical tool. As far as we can determine, this is the first explicit calculation of this type demonstrating the area invariance of horizons. Further, these calculations are directly relevant to transformations that arise in computational 123 388 S. Akcay et al.representation of moving black holes. We present an application of this result to initial data for boosted black holes.

An important question that arises in the study of high dimensional vector representations learned from data is: given a set D of vectors and a query q, estimate the number of points within a specified distance threshold of q. We develop two estimators, LSH Count and Multi-Probe Count that use locality sensitive hashing to preprocess the data to accurately and efficiently estimate the answers to such questions via importance sampling. A key innovation is the ability to maintain a small number of hash tables via preprocessing data structures and algorithms that sample from multiple buckets in each hash table. We give bounds on the space requirements and sample complexity of our schemes, and demonstrate their effectiveness in experiments on a standard word embedding dataset.