Quantified Derandomization of Linear Threshold Circuits

One of the prominent current challenges in complexity theory is the attempt to prove lower bounds for $TC^0$, the class of constant-depth, polynomial-size circuits with majority gates. Relying on the results of Williams (2013), an appealing approach to prove such lower bounds is to construct a non-trivial derandomization algorithm for $TC^0$. In this work we take a first step towards the latter goal, by proving the first positive results regarding the derandomization of $TC^0$ circuits of depth $d>2$. Our first main result is a quantified derandomization algorithm for $TC^0$ circuits with a super-linear number of wires. Specifically, we construct an algorithm that gets as input a $TC^0$ circuit $C$ over $n$ input bits with depth $d$ and $n^{1+\exp(-d)}$ wires, runs in almost-polynomial-time, and distinguishes between the case that $C$ rejects at most $2^{n^{1-1/5d}}$ inputs and the case that $C$ accepts at most $2^{n^{1-1/5d}}$ inputs. In fact, our algorithm works even when the circuit $C$ is a linear threshold circuit, rather than just a $TC^0$ circuit (i.e., $C$ is a circuit with linear threshold gates, which are stronger than majority gates). Our second main result is that even a modest improvement of our quantified derandomization algorithm would yield a non-trivial algorithm for standard derandomization of all of $TC^0$, and would consequently imply that $NEXP\not\subseteq TC^0$. Specifically, if there exists a quantified derandomization algorithm that gets as input a $TC^0$ circuit with depth $d$ and $n^{1+O(1/d)}$ wires (rather than $n^{1+\exp(-d)}$ wires), runs in time at most $2^{n^{\exp(-d)}}$, and distinguishes between the case that $C$ rejects at most $2^{n^{1-1/5d}}$ inputs and the case that $C$ accepts at most $2^{n^{1-1/5d}}$ inputs, then there exists an algorithm with running time $2^{n^{1-\Omega(1)}}$ for standard derandomization of $TC^0$.Comment: Changes in this revision: An additional result (a PRG for quantified derandomization of depth-2 LTF circuits); rewrite of some of the exposition; minor correction

Axiomatizing Petri Net Concatenable Processes

The concatenable processes of a Petri net $N$ can be characterized abstractly as the arrows of a symmetric monoidal category $P[N]$. Yet, this is only a partial axiomatization, since $P[N]$ is built on a concrete, ad hoc chosen, category of symmetries. In this paper we give a fully equational description of the category of concatenable processes of $N$, thus yielding an axiomatic theory of the noninterleaving behaviour of Petri nets

On the structure of P(n)*P(n) for p=2

We show that P(n)*(P(n)) for p = 2 with its geometrically induced structure maps is not an Hopf algebroid because neither the augmentation Epsilon nor the coproduct Delta are multiplicative. As a consequence the algebra structure of P(n)*(P(n)) is slightly different from what was supposed to be the case. We give formulas for Epsilon(xy) and Delta(xy) and show that the inversion of the formal group of P(n) is induced by an antimultiplicative involution Xi : P(n) -> P(n). Some consequences for multiplicative and antimultiplicative automorphisms of K(n) for p = 2 are also discussed

Measurement of the ratio Gamma(K_L -> gamma gamma)/Gamma(K_L -> pi^0 pi^0 pi^0) with the KLOE detector

We have measured the ratio R=Gamma(K_L -> gamma gamma)/ \Gamma(K_L -> 3 pi^0) using the KLOE detector. From a sample of ~ 10^9 phi-mesons produced at DAFNE, the Frascati phi-factory, we select ~ 1.6 10^8 K_L-mesons tagged by observing K_S -> pi^+ pi^- following the reaction e^+ e^- -> phi -> K_L K_S. From this sample we select 27,375 K_L -> gamma gamma events and obtain R = (2.79 \pm 0.02_{stat} \pm 0.02_{syst}) \times 10^{-3}. Using the world average value for BR(K_{L} -> 3 pi^0), we obtain BR(K_{L} -> gamma gamma) = (5.89 \pm 0.07 \pm 0.08) \times 10^{-4} where the second error is due to the uncertainty on the 3 pi^0 branching fraction.Comment: 14 page

Projectile Δ\Delta Excitations in p(p,n)Nπp(p,n)N\pi Reactions

It has recently been proven from measurements of the spin-transfer coefficients $D_{xx}$ and $D_{zz}$ that there is a small but non-vanishing $\Delta S=0$ component $\sigma_{0}$, in the inclusive $p(p,n)N\pi\,$ reaction cross section $\sigma\,$. It is shown that the dominant part of the measured $\sigma_{0}$ can be explained in terms of the projectile $\Delta$ excitation mechanism. An estimate is further made of contributions to $\sigma_{0}$ from s-wave rescattering process. It is found that s-wave rescattering contribution is much smaller than the contribution coming from projectile $\Delta$ excitation mechanism. The addition of s-wave rescattering contribution to the dominant part, however, improves the fit to the data.Comment: 9 pages, Revtex, figures can be obtained upon reques

N,N'-(p-Phenylene)dibenzamide (PPDB)

C20N202H16, monoclinic, P2~/c, a = 18.065(1), b = 5.247(1), c = 8.027(1) A, fl = 93.99 (1) °, Z = 2. The crystal structure has been refined by least-squares techniques. R w = 7.3%. The structure contains planar phenyl rings which are\ud rotated with respect to the plane of the amide group owing to steric hindrance. The molecules are connected in one dimension by means of hydrogen bonds