Lin F. Yang

The hierarchical nature of the spin alignment of dark matter haloes in filaments

Author(s): Aragon-Calvo, MA; Yang, LF | Abstract: Dark matter haloes in cosmological filaments and walls have (in average) their spin vector aligned with their host structure. While haloes in walls are aligned with the plane of the wall independently of their mass, haloes in filaments present a mass-dependent two-regime orientation. Here, we show that the transition mass determining the change in the alignment regime (from parallel to perpendicular) depends on the hierarchical level in which the halo is located, reflecting the hierarchical nature of the Cosmic Web. By explicitly exposing the hierarchical structure of the CosmicWeb, we are able to identify the contributions of different components of the filament network to the alignment signal. We propose a unifying picture of angular momentum acquisition that is based on the results presented here and previous results found by other authors. In order to do a hierarchical characterization of the Cosmic Web, we introduce a new implementation of the multiscale morphology filter, the MMF-2, that significantly improves the identification of structures and explicitly describes their hierarchy.

New Time-Space Upperbounds for Directed Reachability in High-genus and H-minor-free Graphs.

We obtain the following new simultaneous time-space upper bounds for the directed reachability problem. (1) A polynomial-time, O(n^{2/3} * g^{1/3})-space algorithm for directed graphs embedded on orientable surfaces of genus g. (2) A polynomial-time, O(n^{2/3})-space algorithm for all H-minor-free graphs given the tree decomposition, and (3) for K_{3,3}-free and K_5-free graphs, a polynomial-time, O(n^{1/2 + epsilon})-space algorithm, for every epsilon > 0. For the general directed reachability problem, the best known simultaneous time-space upper bound is the BBRS bound, due to Barnes, Buss, Ruzzo, and Schieber, which achieves a space bound of O(n/2^{k * sqrt(log(n))}) with polynomial running time, for any constant k. It is a significant open question to improve this bound for reachability over general directed graphs. Our algorithms beat the BBRS bound for graphs embedded on surfaces of genus n/2^{omega(sqrt(log(n))}, and for all H-minor-free graphs. This significantly broadens the class of directed graphs for which the BBRS bound can be improved.

Streaming symmetric norms via measure concentration

We characterize the streaming space complexity of every symmetric norm l (a norm on ℝn invariant under sign-flips and coordinate-permutations), by relating this space complexity to the measure-concentration characteristics of l. Specifically, we provide nearly matching upper and lower bounds on the space complexity of calculating a (1 ± ε)-approximation to the norm of the stream, for every 0 < ε ≤ 1/2. (The bounds match up to (ε-1 logn) factors.) We further extend those bounds to any large approximation ratio D≥ 1.1, showing that the decrease in space complexity is proportional to D2, and that this factor the best possible. All of the bounds depend on the median of l(x) when x is drawn uniformly from the l2 unit sphere. The same median governs many phenomena in high-dimensional spaces, such as large-deviation bounds and the critical dimension in Dvoretzkys Theorem. The family of symmetric norms contains several well-studied norms, such as all lp norms, and indeed we provide a new explanation for the disparity in space complexity between p ≤ 2 and p > 2. In addition, we apply our general results to easily derive bounds for several norms that were not studied before in the streaming model, including the top-k norm and the k-support norm, which was recently employed for machine learning tasks. Overall, these results make progress on two outstanding problems in the area of sublinear algorithms (Problems 5 and 30 in http://sublinear.info.

Streaming Space Complexity of Nearly All Functions of One Variable on Frequency Vectors

A central problem in the theory of algorithms for data streams is to determine which functions on a stream can be approximated in sublinear, and especially sub-polynomial or poly-logarithmic, space. Given a function g, we study the space complexity of approximating ∑i=1n g(|fi|), where f ∈ Zn is the frequency vector of a turnstile stream. This is a generalization of the well-known frequency moments problem, and previous results apply only when g is monotonic or has a special functional form. Our contribution is to give a condition such that, except for a narrow class of functions g, there is a space-efficient approximation algorithm for the sum if and only if g satisfies the condition. The functions g that we are able to characterize include all convex, concave, monotonic, polynomial, and trigonometric functions, among many others, and is the first such characterization for non-monotonic functions. Thus, for nearly all functions of one variable, we answer the open question from the celebrated paper of Alon, Matias and Szegedy (1996).

New Bounds for the CLIQUE-GAP Problem Using Graph Decomposition Theory

Halldorsson, Sun, Szegedy, and Wang (ICALP 2012) [16] investigated the space complexity of the following problem CLIQUE-GAP(r, s): given a graph stream G, distinguish whether \(\omega (G) \ge r\) or \(\omega (G) \le s\), where \(\omega (G)\) is the clique-number of G. In particular, they give matching upper and lower bounds for CLIQUE-GAP(r, s) for any r and \(s =c\log (n)\), for some constant c. The space complexity of the CLIQUE-GAP problem for smaller values of s is left as an open question. In this paper, we answer this open question. Specifically, for \(s=\tilde{O}(\log (n))\) and for any \(r>s\), we prove that the space complexity of CLIQUE-GAP problem is \(\tilde{\varTheta }(\frac{ms^2}{r^2})\). Our lower bound is based on a new connection between graph decomposition theory (Chung, Erdos, and Spencer [11], and Chung [10]) and the multi-party set disjointness problem in communication complexity.

Approximate Convex Hull of Data Streams

Given a finite set of points

Scalable streaming tools for analyzing N-body simulations: Finding halos and investigating excursion sets in one pass

P \subseteq \mathbb{R}^d

Dark matter contribution to Galactic diffuse gamma ray emission

, we would like to find a small subset

Warmth Elevating the Depths: Shallower Voids with Warm Dark Matter

S \subseteq P

Ringing the initial Universe: the response of overdensity and transformed-density power spectra to initial spikes

such that the convex hull of