This note contains a proof of the exponential separation of depth-d circuits from depth(d + 1) circuits due to Håstad [Has89]. The separating functions are the Sipser functions, denoted fd+1,n. We use a simplified proof of Håstad’s second switching lemma due to Neil Thapen [Tha09].

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ERICA FLEISHMAN*, NEIL MCDONAL†, RALPH MAC NALLY‡, DENNIS D. MURPHY§, JACK WALTERS¶ and TED FLOYD**†† * Center for Conservation Biology, Department of Biological Sciences, Stanford University, Stanford, CA 94305– 5020, USA; † 1900 Klamath Falls Way, Las Vegas, NV 89128, USA; ‡ Australian Centre for Biodiversity: Analysis, Policy and Management, School of Biological Sciences, Monash University 3...

Following Labour’s defeat in the 1983 general election, Michael Foot resigned as Party Leader. In the election to choose his successor (contested under the new electoral college format) Neil Kinnock received over seventy-one per cent of the electoral college vote, enjoying the crucial support of all the largest trade unions, and he also emerged as the clear winner in the CLP and PLP sections (R...

In advanced economies, collateralization of …xed assets plays a key role in supporting arms-length trade. But e¤ective collateral requires secure property rights so that assets can be pledged against default. Otherwise, trade is restricted to relational contracting within networks which use social collateral. This paper develops a model of contracts and matching of producers and suppliers where...

Elliptic equations model the behaviour of scalar quantities u, such as temperature or gravitational potential, which are in an equilibrium situation subject to certain imposed boundary conditions. In his first four lectures, John Urbas discussed linear 1 elliptic equations. In his lectures on the minimal surface equation, Graham Williams discussed the minimal surface equation, a quasilinear 2 e...

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