Shallow Circuits with High-Powered Inputs

A polynomial identity testing algorithm must determine whether an input polynomial (given for instance by an arithmetic circuit) is identically equal to 0. In this paper, we show that a deterministic black-box identity testing algorithm for (high-degree) univariate polynomials would imply a lower bound on the arithmetic complexity of the permanent. The lower bounds that are known to follow from derandomization of (low-degree) multivariate identity testing are weaker. To obtain our lower bound it would be sufficient to derandomize identity testing for polynomials of a very specific norm: sums of products of sparse polynomials with sparse coefficients. This observation leads to new versions of the Shub-Smale tau-conjecture on integer roots of univariate polynomials. In particular, we show that a lower bound for the permanent would follow if one could give a good enough bound on the number of real roots of sums of products of sparse polynomials (Descartes' rule of signs gives such a bound for sparse polynomials and products thereof). In this third version of our paper we show that the same lower bound would follow even if one could only prove a slightly superpolynomial upper bound on the number of real roots. This is a consequence of a new result on reduction to depth 4 for arithmetic circuits which we establish in a companion paper. We also show that an even weaker bound on the number of real roots would suffice to obtain a lower bound on the size of depth 4 circuits computing the permanent.Comment: A few typos correcte

Log-concavity and lower bounds for arithmetic circuits

One question that we investigate in this paper is, how can we build log-concave polynomials using sparse polynomials as building blocks? More precisely, let $f = \sum\_{i = 0}^d a\_i X^i \in \mathbb{R}^+[X]$ be a polynomial satisfying the log-concavity condition a\_i^2 \textgreater{} \tau a\_{i-1}a\_{i+1} for every $i \in \{1,\ldots,d-1\},$ where \tau \textgreater{} 0. Whenever $f$ can be written under the form $f = \sum\_{i = 1}^k \prod\_{j = 1}^m f\_{i,j}$ where the polynomials $f\_{i,j}$ have at most $t$ monomials, it is clear that $d \leq k t^m$. Assuming that the $f\_{i,j}$ have only non-negative coefficients, we improve this degree bound to $d = \mathcal O(k m^{2/3} t^{2m/3} {\rm log^{2/3}}(kt))$ if \tau \textgreater{} 1, and to $d \leq kmt$ if $\tau = d^{2d}$. This investigation has a complexity-theoretic motivation: we show that a suitable strengthening of the above results would imply a separation of the algebraic complexity classes VP and VNP. As they currently stand, these results are strong enough to provide a new example of a family of polynomials in VNP which cannot be computed by monotone arithmetic circuits of polynomial size

Arithmetic circuits: the chasm at depth four gets wider

In their paper on the "chasm at depth four", Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2^o(m) also admit arithmetic circuits of depth four and size 2^o(m). This theorem shows that for problems such as arithmetic circuit lower bounds or black-box derandomization of identity testing, the case of depth four circuits is in a certain sense the general case. In this paper we show that smaller depth four circuits can be obtained if we start from polynomial size arithmetic circuits. For instance, we show that if the permanent of n*n matrices has circuits of size polynomial in n, then it also has depth 4 circuits of size n^O(sqrt(n)*log(n)). Our depth four circuits use integer constants of polynomial size. These results have potential applications to lower bounds and deterministic identity testing, in particular for sums of products of sparse univariate polynomials. We also give an application to boolean circuit complexity, and a simple (but suboptimal) reduction to polylogarithmic depth for arithmetic circuits of polynomial size and polynomially bounded degree

Energy Efficient Engine (E3) controls and accessories detail design report

An Energy Efficient Engine program has been established by NASA to develop technology for improving the energy efficiency of future commercial transport aircraft engines. As part of this program, a new turbofan engine was designed. This report describes the fuel and control system for this engine. The system design is based on many of the proven concepts and component designs used on the General Electric CF6 family of engines. One significant difference is the incorporation of digital electronic computation in place of the hydromechanical computation currently used

Integrated design for integrated photonics: from the physical to the circuit level and back

Silicon photonics is maturing rapidly on a technology basis, but design challenges are still prevalent. We discuss these challenges and explain how design of photonic integrated circuits needs to be handled on both the circuit as on the physical level. We also present a number of tools based on the IPKISS design framework

An Autonomous Surface Vehicle for Long Term Operations

Environmental monitoring of marine environments presents several challenges: the harshness of the environment, the often remote location, and most importantly, the vast area it covers. Manual operations are time consuming, often dangerous, and labor intensive. Operations from oceanographic vessels are costly and limited to open seas and generally deeper bodies of water. In addition, with lake, river, and ocean shoreline being a finite resource, waterfront property presents an ever increasing valued commodity, requiring exploration and continued monitoring of remote waterways. In order to efficiently explore and monitor currently known marine environments as well as reach and explore remote areas of interest, we present a design of an autonomous surface vehicle (ASV) with the power to cover large areas, the payload capacity to carry sufficient power and sensor equipment, and enough fuel to remain on task for extended periods. An analysis of the design and a discussion on lessons learned during deployments is presented in this paper.Comment: In proceedings of MTS/IEEE OCEANS, 2018, Charlesto

The implementation and operation of a variable-response electronic throttle control system for a TF-104G aircraft

During some flight programs, researchers have encountered problems in the throttle response characteristics of high-performance aircraft. To study and to help solve these problems, the National Aeronautics and Space Administration Ames Research Center's Dryden Flight Research Facility (Ames-Dryden) conducted a study using a TF-104G airplane modified with a variable-response electronic throttle control system. Ames-Dryden investigated the effects of different variables on engine response and handling qualities. The system provided transport delay, lead and lag filters, second-order lags, command rate and position limits, and variable gain between the pilot's throttle command and the engine fuel controller. These variables could be tested individually or in combination. Ten research flights were flown to gather data on engine response and to obtain pilot ratings of the various system configurations. The results should provide design criteria for engine-response characteristics. The variable-response throttle components and how they were installed in the TF-104G aircraft are described. How the variable-response throttle was used in flight and some of the results of using this system are discussed

The Limited Power of Powering: Polynomial Identity Testing and a Depth-four Lower Bound for the Permanent

Polynomial identity testing and arithmetic circuit lower bounds are two central questions in algebraic complexity theory. It is an intriguing fact that these questions are actually related. One of the authors of the present paper has recently proposed a "real {\tau}-conjecture" which is inspired by this connection. The real {\tau}-conjecture states that the number of real roots of a sum of products of sparse univariate polynomials should be polynomially bounded. It implies a superpolynomial lower bound on the size of arithmetic circuits computing the permanent polynomial. In this paper we show that the real {\tau}-conjecture holds true for a restricted class of sums of products of sparse polynomials. This result yields lower bounds for a restricted class of depth-4 circuits: we show that polynomial size circuits from this class cannot compute the permanent, and we also give a deterministic polynomial identity testing algorithm for the same class of circuits.Comment: 16 page