Weakening Assumptions for Deterministic Subexponential Time Non-Singular Matrix Completion

In (Kabanets, Impagliazzo, 2004) it is shown how to decide the circuit polynomial identity testing problem (CPIT) in deterministic subexponential time, assuming hardness of some explicit multilinear polynomial family for arithmetical circuits. In this paper, a special case of CPIT is considered, namely low-degree non-singular matrix completion (NSMC). For this subclass of problems it is shown how to obtain the same deterministic time bound, using a weaker assumption in terms of determinantal complexity. Hardness-randomness tradeoffs will also be shown in the converse direction, in an effort to make progress on Valiant's VP versus VNP problem. To separate VP and VNP, it is known to be sufficient to prove that the determinantal complexity of the m-by-m permanent is $m^{\omega(\log m)}$. In this paper it is shown, for an appropriate notion of explicitness, that the existence of an explicit multilinear polynomial family with determinantal complexity m^{\omega(\log m)}$is equivalent to the existence of an efficiently computable generator$G_n$for multilinear NSMC with seed length$O(n^{1/\sqrt{\log n}})$. The latter is a combinatorial object that provides an efficient deterministic black-box algorithm for NSMC. ``Multilinear NSMC'' indicates that$G_n$only has to work for matrices$M(x)$of$poly(n)$size in$n$variables, for which$det(M(x))$ is a multilinear polynomial

Deterministic Black-Box Identity Testing π\pi-Ordered Algebraic Branching Programs

In this paper we study algebraic branching programs (ABPs) with restrictions on the order and the number of reads of variables in the program. Given a permutation $\pi$ of $n$ variables, for a $\pi$-ordered ABP ($\pi$-OABP), for any directed path $p$ from source to sink, a variable can appear at most once on $p$, and the order in which variables appear on $p$ must respect $\pi$. An ABP $A$ is said to be of read $r$, if any variable appears at most $r$ times in $A$. Our main result pertains to the identity testing problem. Over any field $F$ and in the black-box model, i.e. given only query access to the polynomial, we have the following result: read $r$ $\pi$-OABP computable polynomials can be tested in \DTIME[2^{O(r\log r \cdot \log^2 n \log\log n)}]. Our next set of results investigates the computational limitations of OABPs. It is shown that any OABP computing the determinant or permanent requires size $\Omega(2^n/n)$ and read $\Omega(2^n/n^2)$. We give a multilinear polynomial $p$ in $2n+1$ variables over some specifically selected field $G$, such that any OABP computing $p$ must read some variable at least $2^n$ times. We show that the elementary symmetric polynomial of degree $r$ in $n$ variables can be computed by a size $O(rn)$ read $r$ OABP, but not by a read $(r-1)$ OABP, for any $0 < 2r-1 \leq n$. Finally, we give an example of a polynomial $p$ and two variables orders $\pi \neq \pi'$, such that $p$ can be computed by a read-once $\pi$-OABP, but where any $\pi'$-OABP computing $p$ must read some variable at least $2^n

Beirut Blast: A port city in crisis

On 4th of August 2020, the Lebanese capital and port city, Beirut, was rocked by a massive explosion that has killed hundreds and injured thousands more, ravaging the heart of the city’s nearby downtown business district and neighbouring housing areas, where more than 750,000 people live. The waterfront neighbourhood and a number of dense residential neighbourhoods in the city’s eastern part were essentially flattened. Lebanese Government officials believe that the blast was caused by around 2,700 tonnes of ammonium nitrate stored near the city’s cargo port without proper control for six years. The disaster devastating Beirut’s port and city shows the latent danger of safe storage of potentially dangerous goods in modern ports, particularly ones located close to the heart of the city. The huge blast tore through major grain silos, stoking fears of shortages in a nation that imports nearly all its food and is already reeling from economic crisis. As the WFP (World Food Programme) said in a statement: the blast will “exacerbate the grim economic and food-security situation.” The Beirut blast also reminds us of the importance of ports in the contemporary globalized world. It calls our attention to safety and security, of governance and collaboration between port and city or region and of accessibility to the hinterland. What do local governments and port cities need to do to enhance the safety and security issues in port terminals? Or, to put it differently, how do we reconcile this challenge between the ports we need to feed us, serve us, provide us with medicines, equipment, etc. and the ports that threatens us

Prospects for high-resolution microwave spectroscopy of methanol in a Stark-deflected molecular beam

Recently, the extremely sensitive torsion-rotation transitions in methanol have been used to set a tight constraint on a possible variation of the proton-to-electron mass ratio over cosmological time scales. In order to improve this constraint, laboratory data of increased accuracy will be required. Here, we explore the possibility for performing high-resolution spectroscopy on methanol in a Stark-deflected molecular beam. We have calculated the Stark shift of the lower rotational levels in the ground torsion-vibrational state of CH3OH and CD3OH molecules, and have used this to simulate trajectories through a typical molecular beam resonance setup. Furthermore, we have determined the efficiency of non-resonant multi-photon ionization of methanol molecules using a femtosecond laser pulse. The described setup is in principle suited to measure microwave transitions in CH3OH at an accuracy below 10^{-8}

Stronger Lower Bounds and Randomness-Hardness Trade-Offs Using Associated Algebraic Complexity Classes

We associate to each Boolean language complexity class C the algebraic class a·C consisting of families of polynomials {fn} for which the evaluation problem over Z is in C. We prove the following lower bound and randomness-to-hardness results: 1. If polynomial identity testing (PIT) is in NSUBEXP then a·NEXP does not have poly size constant-free arithmetic circuits. 2. a·NEXP RP does not have poly size constant-free arithmetic circuits. 3. For every fixed k, a·MA does not have arithmetic circuits of size nk. Items 1 and 2 strengthen two results due to Kabanets and Impagliazzo [7]. The third item improves a lower bound due to Santhanam [11]. We consider the special case low-PIT of identity testing for (constant-free) arithmetic circuits with low formal degree, and give improved hardness-to-randomness trade-offs that apply to this case. Combining our results for both directions of the hardness-randomness connection, we demonstrate a case where derandomization of PIT and proving lower bounds are equivalent. Namely, we show that low-PIT ∈ i.o-NTIME[2no(1)]/no(1) if and only if there exists a family of multilinear polynomials in a·NE/lin that requires constant-free arithmetic circuits of super-polynomial size and formal degree