1,721,118 research outputs found
An integrated optics pulse shaping device
No full textThesis (M.S.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1981.MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERINGIncludes bibliographical references.by Scott Roger Shepard.M.S
Gough Whitlam at the Labor Party Election Launch, Bankstown, New South Wales, 1993 [picture] /
No full textTitle from inscription on verso.; Inscriptions: "Roger Scott. Gough Whitlam, Labor Party Election Launch, Bankstown, 1993, printed 2007"--In pencil on verso; Signed by photographer lower right on sheet.; Also available online at: http://nla.gov.au/nla.pic-vn6157557; Purchased from Josef Lebovic Gallery, 2012
Bob Hawke during Anti-Vietnam War Demonstation at the Domain, Sydney, 1972 [picture] /
No full textTitle from inscription on verso.; Inscriptions: "Bob Hawke, Anti-Vietnam War Demonstation, Domain, Sydney, 1972, Printed 2007."--In pencil on verso; Signed by photographer lower right on sheet.; Also available online at: http://nla.gov.au/nla.pic-vn6157560; Purchased from Joseph Lebovic Gallery, 2012
Malcolm Fraser addressing the media during the election campaign, Randwick Racecourse, New South Wales, 1975 [picture] /
No full textTitle from inscription on verso and reference source.; Inscriptions: "Roger Scott. Malcolm Fraser Randwick Race Course 1975. Printed 2012."--In pencil on verso; Signed by photographer lower right on sheet.; Also available online at: http://nla.gov.au/nla.pic-vn6157547; Purchased from Josef Lebovic Gallery, 2012
CLE PERCOLATIONS
Get PDFConformal loop ensembles (CLEs) are random collections of loops in a simply connected domain, whose laws are characterized by a natural conformal invariance property. The set of points not surrounded by any loop is a canonical random connected fractal set — a random and conformally invariant analog of the Sierpinski carpet or gasket. In the present paper, we derive a direct relationship between the CLEs with simple loops (CLE κ for κ ∈ ( 8 / 3 , 4 ) , whose loops are Schramm’s SLE κ -type curves) and the corresponding CLEs with nonsimple loops (CLE κ′ with κ′ := 16 /κ ∈ ( 4 , 6 ) , whose loops are SLE κ′ -type curves). This correspondence is the continuum analog of the Edwards–Sokal coupling between the q -state Potts model and the associated FK random cluster model, and its generalization to noninteger q . Like its discrete analog, our continuum correspondence has two directions. First, we show that for each κ ∈ ( 8 / 3 , 4 ) , one can construct a variant of CLE κ as follows: start with an instance of CLE κ ′ , then use a biased coin to independently color each CLE κ′ loop in one of two colors, and then consider the outer boundaries of the clusters of loops of a given color. Second, we show how to interpret CLE κ′ loops as interfaces of a continuum analog of critical Bernoulli percolation within CLE κ carpets — this is the first construction of continuum percolation on a fractal planar domain. It extends and generalizes the continuum percolation on open domains defined by SLE₆ and CLE 6 . These constructions allow us to prove several conjectures made by the second author and provide new and perhaps surprising interpretations of the relationship between CLEs and the Gaussian free field. Along the way, we obtain new results about generalized SLEκ (ρ) curves for ρ < − 2, such as their decomposition into collections of SLE κ-type ‘loops’ hanging off of SLE κ′-type ‘trunks’, and vice versa (exchanging κ and κ′ ). We also define a continuous family of natural CLE variants called boundary conformal loop ensembles (BCLEs) that share some (but not all) of the conformal symmetries that characterize CLEs, and that should be scaling limits of critical models with special boundary conditions. We extend the CLE κ /CLE κ′ correspondence to a BCLE κ /BCLE κ′ correspondence that makes sense for the wider range κ ∈ ( 2 , 4 ] and κ′ ∈[ 4 , 8
Quantum gravity and inventory accumulation
Get PDFWe begin by studying inventory accumulation at a LIFO (last-in-first-out) retailer with two products. In the simplest version, the following occur with equal probability at each time step: first product ordered, first product produced, second product ordered, second product produced. The inventory thus evolves as a simple random walk on Z². In more interesting versions, a p fraction of customers orders the “freshest available” product regardless of type. We show that the corresponding random walks scale to Brownian motions with diffusion matrices depending on p.We then turn our attention to the critical Fortuin–Kastelyn random planar map model, which gives, for each q > 0, a probability measure on random (discretized) two-dimensional surfaces decorated by loops, related to the q-state Potts model. A longstanding open problem is to show that as the discretization gets finer, the surfaces converge in law to a limiting (loop-decorated) random surface. The limit is expected to be a Liouville quantum gravity surface decorated by a conformal loop ensemble, with parameters depending on q. Thanks to a bijection between decorated planar maps and inventory trajectories (closely related to bijections of Bernardi and Mullin), our results about the latter imply convergence of the former in a particular topology. A phase transition occurs at p = 1/2, q = 4.National Science Foundation (U.S.) (Grant DMS 064558
Conformal weldings of random surfaces: SLE and the quantum gravity zipper
No full textWe construct a conformal welding of two Liouville quantum gravity random surfaces and show that the interface between them is a random fractal curve called the Schramm–Loewner evolution (SLE), thereby resolving a variant of a conjecture of Peter Jones. We also demonstrate some surprising symmetries of this construction, which are consistent with the belief that (path-decorated) random planar maps have (SLE-decorated) Liouville quantum gravity as a scaling limit. We present several precise conjectures and open questions
Deterministic Approximations of Random Reflectors
No full textWithin classical optics, one may add microscopic "roughness'' to a macroscopically flat mirror so that parallel rays of a given angle are reflected at different outgoing angles. Taking the limit (as the roughness becomes increasingly microscopic) one obtains a flat surface that reflects randomly, i.e., the transition from incoming to outgoing ray is described by a probability kernel (whose form depends on the nature of the microscopic roughness).
We consider two-dimensional optics (a.k.a. billiards) and show that every random reflector on a line that satisfies a necessary measure-preservation condition (well established in the theory of billiards) can be approximated by deterministic reflectors in this way.National Science Foundation (U.S.) (Grant DMS 0645585
A contour line of the continuum Gaussian free field
Get PDFOriginal manuscript August 14, 2010Consider an instance h of the Gaussian free field on a simply connected planar domain D with boundary conditions −λ on one boundary arc and λ on the complementary arc, where λ is the special constant √π/8 . We argue that even though h is defined only as a random distribution, and not as a function, it has a well-defined zero level line γ connecting the endpoints of these arcs, and the law of γ is SLE(4) . We construct γ in two ways: as the limit of the chordal zero contour lines of the projections of h onto certain spaces of piecewise linear functions, and as the only path-valued function on the space of distributions with a natural Markov property. We also show that, as a function of h, γ is “local” (it does not change when h is modified away from γ ) and derive some general properties of local sets
Duality and the Knizhnik-Polyakov-Zamolodchikov Relation in Liouville Quantum Gravity
No full textWe present a (mathematically rigorous) probabilistic and geometrical proof of the Knizhnik-Polyakov-Zamolodchikov relation between scaling exponents in a Euclidean planar domain D and in Liouville quantum gravity. It uses the properly regularized quantum area measure dμ[subscript γ]=ε[superscript γ[superscript 2]/2]e[superscript γh [subscript ε](z)]dz, where dz is the Lebesgue measure on D, γ is a real parameter, 0≤γ2 is shown to be related to the quantum measure dμγ′, γ′<2, by the fundamental duality γγ′=4
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