Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas

We give the best known pseudorandom generators for two touchstone classes in unconditional derandomization: an $\varepsilon$-PRG for the class of size-$M$ depth-$d$ $\mathsf{AC}^0$ circuits with seed length $\log(M)^{d+O(1)}\cdot \log(1/\varepsilon)$, and an $\varepsilon$-PRG for the class of $S$-sparse $\mathbb{F}_2$ polynomials with seed length $2^{O(\sqrt{\log S})}\cdot \log(1/\varepsilon)$. These results bring the state of the art for unconditional derandomization of these classes into sharp alignment with the state of the art for computational hardness for all parameter settings: improving on the seed lengths of either PRG would require breakthrough progress on longstanding and notorious circuit lower bounds. The key enabling ingredient in our approach is a new \emph{pseudorandom multi-switching lemma}. We derandomize recently-developed \emph{multi}-switching lemmas, which are powerful generalizations of H{\aa}stad's switching lemma that deal with \emph{families} of depth-two circuits. Our pseudorandom multi-switching lemma---a randomness-efficient algorithm for sampling restrictions that simultaneously simplify all circuits in a family---achieves the parameters obtained by the (full randomness) multi-switching lemmas of Impagliazzo, Matthews, and Paturi [IMP12] and H{\aa}stad [H{\aa}s14]. This optimality of our derandomization translates into the optimality (given current circuit lower bounds) of our PRGs for $\mathsf{AC}^0$ and sparse $\mathbb{F}_2$ polynomials

Note on symmetric BCJ numerator

We present an algorithm that leads to BCJ numerators satisfying manifestly the three properties proposed by Broedel and Carrasco in [35]. We explicitly calculate the numerators at 4, 5 and 6-points and show that the relabeling property is generically satisfied.Comment: 14 pages, typo in eq.(4.1)is correcte

Suboptimality of Nonlocal Means for Images with Sharp Edges

We conduct an asymptotic risk analysis of the nonlocal means image denoising algorithm for the Horizon class of images that are piecewise constant with a sharp edge discontinuity. We prove that the mean square risk of an optimally tuned nonlocal means algorithm decays according to $n^{-1}\log^{1/2+\epsilon} n$, for an $n$-pixel image with $\epsilon>0$. This decay rate is an improvement over some of the predecessors of this algorithm, including the linear convolution filter, median filter, and the SUSAN filter, each of which provides a rate of only $n^{-2/3}$. It is also within a logarithmic factor from optimally tuned wavelet thresholding. However, it is still substantially lower than the the optimal minimax rate of $n^{-4/3}$.Comment: 33 pages, 3 figure

Signatures of anomalous VVH interactions at a linear collider

We examine, in a model independent way, the sensitivity of a Linear Collider to the couplings of a light Higgs boson to gauge bosons. Including the possibility of CP violation, we construct several observables that probe the different anomalous couplings possible. For an intermediate mass Higgs, a collider operating at a center of mass energy of 500 GeV and with an integrated luminosity of 500 fb$^{-1}$ is shown to be able to constrain the $ZZH$ vertex at the few per cent level, and with even higher sensitivity in certain directions. However, the lack of sufficient number of observables as well as contamination from the $ZZH$ vertex limits the precision with which the $WWH$ coupling can be measured.Comment: Typeset in RevTeX4, 16 pages, 12 figures; V2: minor changes in title and Sec. II and III; V3: version appeared in PRD with minor correctio

Akt1-Inhibitor of DNA binding2 is essential for growth cone formation and axon growth and promotes central nervous system axon regeneration.

Mechanistic studies of axon growth during development are beneficial to the search for neuron-intrinsic regulators of axon regeneration. Here, we discovered that, in the developing neuron from rat, Akt signaling regulates axon growth and growth cone formation through phosphorylation of serine 14 (S14) on Inhibitor of DNA binding 2 (Id2). This enhances Id2 protein stability by means of escape from proteasomal degradation, and steers its localization to the growth cone, where Id2 interacts with radixin that is critical for growth cone formation. Knockdown of Id2, or abrogation of Id2 phosphorylation at S14, greatly impairs axon growth and the architecture of growth cone. Intriguingly, reinstatement of Akt/Id2 signaling after injury in mouse hippocampal slices redeemed growth promoting ability, leading to obvious axon regeneration. Our results suggest that Akt/Id2 signaling is a key module for growth cone formation and axon growth, and its augmentation plays a potential role in CNS axonal regeneration

Employing Helicity Amplitudes for Resummation

Many state-of-the-art QCD calculations for multileg processes use helicity amplitudes as their fundamental ingredients. We construct a simple and easy-to-use helicity operator basis in soft-collinear effective theory (SCET), for which the hard Wilson coefficients from matching QCD onto SCET are directly given in terms of color-ordered helicity amplitudes. Using this basis allows one to seamlessly combine fixed-order helicity amplitudes at any order they are known with a resummation of higher-order logarithmic corrections. In particular, the virtual loop amplitudes can be employed in factorization theorems to make predictions for exclusive jet cross sections without the use of numerical subtraction schemes to handle real-virtual infrared cancellations. We also discuss matching onto SCET in renormalization schemes with helicities in $4$- and $d$-dimensions. To demonstrate that our helicity operator basis is easy to use, we provide an explicit construction of the operator basis, as well as results for the hard matching coefficients, for $pp\to H + 0,1,2$ jets, $pp\to W/Z/\gamma + 0,1,2$ jets, and $pp\to 2,3$ jets. These operator bases are completely crossing symmetric, so the results can easily be applied to processes with $e^+e^-$ and $e^-p$ collisions.Comment: 41 pages + 20 pages in Appendices, 1 figure, v2: journal versio