R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

At nanoscale, quantum-dot cellular automata (QCA) defines a new device architecture that permits the innovative design of digital systems. Features of these systems are the allowed crossing of signal lines with different orientation in polarization on a Cartesian plane, the potential of high throughput due to efficient pipelining, fast signal switching, and propagation. However, QCA designs of even modest complexity suffer from the negative impact due to the placement of long lines of cells among clocking zones, thus resulting in increased delay, slow timing, and sensitivity to thermal fluctuations. In this paper, different schemes for clocking and timing of the QCA systems are proposed; these schemes utilize 2D techniques that permit a reduction in the longest line length in each clocking zone. The proposed clocking schemes utilize logic-propagation techniques that have been developed for systolic arrays. Placement of QCA cells is modified to ensure correct signal generation and timing. The significant reduction in the longest line length permits a fast timing and efficient pipelining to occur while guaranteeing a kink-free behavior in switching.

Two-Dimensional Schemes for Clocking/Timing of QCA Circuits

The future of system-on-chip (SoC) technologies, based on the scaling of current FET-based integrated circuitry, is being predicted to reach fabrication limits by the year 2015. Economic limits may be reached before that time. Continued scaling of electronic devices to molecular scales will undoubtedly require a paradigm shift from the FET-based switch to an alternative mechanism of information representation and processing. This paradigm shift will also have to encompass the tools and design culture that have made the current SoC technology possible-the ability to design monolithic integrated circuits with many hundreds of millions of transistors. In this paper, we examine the initial development of a tool to automate the design of one of the promising emerging nanoelectronic technologies, quantum-dot cellular automata, which has been proposed as a computing paradigm based on single electron effects within quantum dots and molecules.

Design Tools for an Emerging SoC Technology: Quantum-Dot Cellular Automata

Hyperbolicity of the complex of free factors

Publisher Summary The aim of this chapter is to review the prospects for the development and practical introduction of ultra small electron devices, including nanoscale field-effect transistors (FETs) and single-electron transistors (SETs), as well as new concepts for nanometer-scalable memory cells. Lithographically defined SETs can hardly be a panacea, since the critical dimension of such transistor for the room temperature operation should be below 1 nm. Due to the finite yield of molecular devices and their sensitivity to randomly charged impurities, this approach requires a substantial revision of integrated circuit architectures, ranging from defect-tolerant versions of memory matrices and number crunching processors to more radical solutions like hardware-implemented neuromorphic networks capable of advanced image recognition and more intelligent information processing tasks.

Electronics Below 10 nm

We define metrics on Culler-Vogtmann space, which are an analogue of the Thurston metric and are constructed using stretching factors. In fact the metrics we study are related, one being a symmetrised version of the other. We investigate the basic properties of these metrics, showing the advantages and pathologies of both choices. We show how to compute stretching factors between marked metric graphs in an easy way and we discuss the behaviour of stretching factors under iterations of automorphisms. We study metric properties of folding paths, showing that they are geodesic for the non-symmetric metric and, if they do not enter the thin part of Outer Space, quasi-geodesic for the symmetric metric.

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Metric properties of outer space

Let W be a compact manifold and let ρ be a representation of its fundamental group into PSL(2, C). Then the volume of ρ is defined by taking any ρ-equivariant map from the universal cover W~ to ℍ 3 and then by integrating the pullback of the hyperbolic volume form on a fundamental domain. It turns out that such a volume does not depend on the choice of the equivariant map. Dunfield extended this construction to the case of a noncompact (cusped) manifold M, but he did not prove that the volume is well defined in all cases. We prove here that the volume of a representation is always well defined and depends only on the representation. Moreover, we show that this volume can be easily computed by straightening any ideal triangulation of M. We show that the volume of a representation is bounded from above by the relative simplicial volume of M. Finally, we prove a rigidity theorem for representations of the fundamental group of a hyperbolic manifold. Namely, we prove that if M is hyperbolic and vol(ρ) = vol(M), then ρ is discrete and faithful.

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Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds

Let W be a compact manifold and let \rho be a representation of its fundamental group into PSL(2,C). The volume of \rho is defined by taking any \rho-equivariant map from the universal cover of W to H^3 and then by integrating the pull-back of the hyperbolic volume form on a fundamental domain. It turns out that such a volume does not depend on the choice of the equivariant map. Dunfield extended this construction to the case of a non-compact (cusped) manifold M, but he did not prove the volume is well-defined in all cases. We prove here that the volume of a representation is always well-defined and depends only on the representation. We show that this volume can be easily computed by straightening any ideal triangulation of M. We show that the volume of a representation is bounded from above by the relative simplicial volume of M. Finally, we prove a rigidity theorem for representations of the fundamental group of a hyperbolic manifold. Namely, we prove that if M is hyperbolic and vol(\rho)=vol(M) then \rho is discrete and faithful.

We prove a volume-rigidity theorem for Fuchsian representations of fundamental groups of hyperbolic k-manifolds into Isom \(\mathbb{H}^n\). Namely, we show that if M is a complete hyperbolic k-manifold with finite volume, then the volume of any representation of π1(M) into isom \(\mathbb{H}^n\), 3 ≤ k ≤ n, is less than the volume of M, and the volume is maximal if and only if the representation is discrete, faithful and ‘k-Fuchsian’

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Maximal Volume Representations are Fuchsian

We investigate the effect of finite temperature on the behavior of logic circuits based on the principle of quantum cellular automata (QCA) and of ground state computation. In particular, we focus on the error probability for a wire of QCA cells that propagates a logic state. A numerical model and an analytical, more approximate model are presented for evaluation of the partition function of such a system and, consequently, of the desired probabilities. We compare the results of the two models, assess the limits of validity of the analytical approach, and show that error probabilities depend on the ratio of the energy splitting between the ground state and first excited state to the thermal energy kT. We then provide estimates of the maximum operating temperature for a few relevant cases, and discuss possible approaches for increasing it.

Stefano Francaviglia

Papers

Metric properties of outer space

Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds

Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds

Maximal Volume Representations are Fuchsian

Thermal behavior of quantum cellular automaton wires