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Large N (large + n)
Selected AbstractsHigher spins and stringy AdS5 × S5,FORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 7-8 2005M. Bianchi Abstract In this lecture I review recent work done in collaboration with Beisert et al. [1-3] (For a concise summary see [4].). After a notational flash on the AdS/CFT correspondence, I will discuss higher spin (HS) symmetry enhancement at small radius and how this is holographically captured by free N = 4 SYM theory. I will then derive the spectrum of perturbative superstring excitations on AdS in this particular limit and successfully compare it with the spectrum of single-trace operators in free ,, = 4 SYM at large N, obtained by means of Polya(kov)'s counting. Decomposing the spectrum into HS multiplets allows one to precisely identify the 'massless' HS doubleton and the lower spin Goldstone multiplets which participate in the pantagruelic Higgs mechanism, termed "La Grande Bouffe". After recalling some basic features of Vasiliev's formulation of HS gauge theories, I will eventually sketch how to describe mass generation in the AdS bulk à la Sückelberg and its holographic implications such as the emergence of anomalous dimensions in the boundary ,, = 4 SYM theory. [source] Exact results in a non-supersymmetric gauge theoryFORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 6-7 2004A. Armoni We consider non-supersymmetric large N orientifold field theories. Specifically, we discuss a gauge theory with a Dirac fermion in the anti-symmetric tensor representation. We argue that, at large N and in a large part of its bosonic sector, this theory is non-perturbatively equivalent to ,, = 1 SYM, so that exact results established in the latter (parent) theory also hold in the daughter orientifold theory. In particular, the non-supersymmetric theory has an exactly calculable bifermion condensate, exactly degenerate parity doublets, and a vanishing cosmological constant (all this to leading order in 1 / N). [source] Field theory on a non-commutative plane: a non-perturbative studyFORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 5 2004F. Hofheinz Abstract The 2d gauge theory on the lattice is equivalent to the twisted Eguchi,Kawai model, which we simulated at N ranging from 25 to 515. We observe a clear large N scaling for the 1- and 2-point function of Wilson loops, as well as the 2-point function of Polyakov lines. The 2-point functions agree with a universal wave function renormalization. The large N double scaling limit corresponds to the continuum limit of non-commutative gauge theory, so the observed large N scaling demonstrates the non-perturbative renormalizability of this non-commutative field theory. The area law for the Wilson loops holds at small physical area as in commutative 2d planar gauge theory, but at large areas we find an oscillating behavior instead. In that regime the phase of the Wilson loop grows linearly with the area. This agrees with the Aharonov-Bohm effect in the presence of a constant magnetic field, identified with the inverse non-commutativity parameter. Next we investigate the 3d ,,4 model with two non-commutative coordinates and explore its phase diagram. Our results agree with a conjecture by Gubser and Sondhi in d = 4, who predicted that the ordered regime splits into a uniform phase and a phase dominated by stripe patterns. We further present results for the correlators and the dispersion relation. In non-commutative field theory the Lorentz invariance is explicitly broken, which leads to a deformation of the dispersion relation. In one loop perturbation theory this deformation involves an additional infrared divergent term. Our data agree with this perturbative result. We also confirm the recent observation by Ambjø rn and Catterall that stripes occur even in d = 2, although they imply the spontaneous breaking of the translation symmetry. [source] Lectures on the plane-wave string/gauge theory dualityFORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 2-3 2004J.C. Plefka Abstract These lectures give an introduction to the novel duality relating type IIB string theory in a maximally supersymmetric plane-wave background to ,, = 4, d = 4, U(N) super Yang-Mills theory in a particular large N and large R-charge limit due to Berenstein, Maldacena and Nastase. In the first part of these lectures the duality is derived from the AdS/CFT correspondence by taking a Penrose limit of the AdS5 × S5 geometry and studying the corresponding double-scaling limit on the gauge theory side. The resulting free plane-wave superstring is then quantized in light-cone gauge. On the gauge theory side of the correspondence the composite super Yang-Mills operators dual to string excitations are identified, and it is shown how the string spectrum can be mapped to the planar scaling dimensions of these operators. In the second part of these lectures we study the correspondence at the interacting respectively non-planar level. On the gauge theory side it is demonstrated that the large N large R-charge limit in question preserves contributions from Feynman graphs of all genera through the emergence of a new genus counting parameter , in agreement with the string genus expansion for non-zero gs. Effective quantum mechanical tools to compute higher genus contributions to the scaling dimensions of composite operators are developed and explicitly applied in a genus one computation. We then turn to the interacting string theory side and give an elementary introduction into light-cone superstring field theory in a plane-wave background and point out how the genus one prediction from gauge theory can be reproduced. Finally, we summarize the present status of the plane-wave string/gauge theory duality. [source] Testing linear-theory predictions of galaxy formationMONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 4 2000Ben Sugerman The angular momentum of galaxies is routinely ascribed to a process of tidal torques acting during the early stages of gravitational collapse, and is predicted from the initial mass distribution using second-order perturbation theory and the Zel'dovich approximation. We test this theory for a flat hierarchical cosmogony using a large N -body simulation with sufficient dynamic range to include tidal fields, allow resolution of individual galaxies, and thereby expand on previous studies. The predictions of linear collapse, linear tidal torque, and biased-peaks galaxy formation are applied to the initial conditions and compared with results for evolved bound objects. We find relatively good correlation between the predictions of linear theory and actual galaxy evolution. Collapse is well described by an ellipsoidal model within a shear field, which results primarily in triaxial objects that do not map directly to the initial density field. While structure formation from early times is a complex history of hierarchical merging, salient features are well described by the simple spherical-collapse model. Most notably, we test several methods for determining the turnaround epoch, and find that turnaround is successfully described by the spherical-collapse model. The angular momentum of collapsing structures grows linearly until turnaround, as predicted, and continues quasi-linearly until shell crossing. The predicted angular momentum for well-resolved galaxies at turnaround overestimates the true turnaround and final values by a factor of ,3, with a scatter of ,70 per cent, and only marginally yields the correct direction of the angular momentum vector. We recover the prediction that final angular momentum scales as mass to the 5/3 power. We find that mass and angular momentum also vary proportionally with peak height. In view of the fact that the observed galaxy collapse is a stochastic hierarchical and non-linear process, it is encouraging that the linear theory can serve as an effective predictive and analytic tool. [source] The Polls: State-Level Presidential Approval: Results from the Job Approval ProjectPRESIDENTIAL STUDIES QUARTERLY, Issue 1 2003JEFFREY E. COHEN In this article, the author reports on the state job approval data set. Although the data set includes state-level job approval for governors and U. S. senators, it also contains similar data on the president. These data allow us to test propositions about public opinion toward the president at a level of aggregation lower than the national level but higher than the individual level. More than one thousand state-level presidential job approval readings exist in this data set. Such a large N allows us to test ideas and bypotheses that other limited N designs cannot accomplish. In the first part of this article, the author discusses the properties of these data. In the second part, he uses these data to test two competing theories of the impact of economic perceptions on presidential approval, the retrospection versus the prospection model, which provides one example of how these data can be put to use. [source] Out-of-core compression and decompression of large n -dimensional scalar fieldsCOMPUTER GRAPHICS FORUM, Issue 3 2003Lawrence Ibarria We present a simple method for compressing very large and regularly sampled scalar fields. Our method is particularlyattractive when the entire data set does not fit in memory and when the sampling rate is high relative to thefeature size of the scalar field in all dimensions. Although we report results foranddata sets, the proposedapproach may be applied to higher dimensions. The method is based on the new Lorenzo predictor, introducedhere, which estimates the value of the scalar field at each sample from the values at processed neighbors. The predictedvalues are exact when the n-dimensional scalar field is an implicit polynomial of degreen, 1. Surprisingly,when the residuals (differences between the actual and predicted values) are encoded using arithmetic coding,the proposed method often outperforms wavelet compression in anL,sense. The proposed approach may beused both for lossy and lossless compression and is well suited for out-of-core compression and decompression,because a trivial implementation, which sweeps through the data set reading it once, requires maintaining only asmall buffer in core memory, whose size barely exceeds a single (n,1)- dimensional slice of the data. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Compression, scalar fields,out-of-core. [source] OH concentration time histories in n -alkane oxidationINTERNATIONAL JOURNAL OF CHEMICAL KINETICS, Issue 12 2001D. F. Davidson OH radical concentration time histories were measured behind reflected shocks in the oxidation of four n -alkanes: propane, n -butane, n -heptane, and n -decane. Initial reflected shock conditions of these measurements were 1357,1784 K, 2.02,3.80 atm, with fuel concentrations of 300,2000 ppm, and equivalence ratios from 0.8 to 1.2. OH concentrations were measured using narrow-linewidth ring-dye laser absorption of the R1(5) line of the A,X (0,0) transition at 306.5 nm. These concentration time-history measurements were compared to the modeled predictions of eight large n -alkane oxidation mechanisms currently available in the literature and the kinetic implications of these measurements are discussed. These data, in conjunction with recent measurements of n -alkane ignition times and ethylene yields in n -alkane pyrolysis experiments, also performed in this laboratory, provide a unique database of species concentration time histories for n -alkane mechanism validation. © 2001 John Wiley & Sons, Inc. Int J Chem Kinet 33: 775,783, 2001 [source] On k -detour subgraphs of hypercubesJOURNAL OF GRAPH THEORY, Issue 1 2008Nana Arizumi Abstract A spanning subgraph G of a graph H is a k - detour subgraph of H if for each pair of vertices , the distance, , between x and y in G exceeds that in H by at most k. Such subgraphs sometimes also are called additive spanners. In this article, we study k -detour subgraphs of the n -dimensional cube, , with few edges or with moderate maximum degree. Let denote the minimum possible maximum degree of a k -detour subgraph of . The main result is that for every and On the other hand, for each fixed even and large n, there exists a k -detour subgraph of with average degree at most . © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 55,64, 2008 [source] DFT study and NBO analysis of the mutual interconversion of cumulene compoundsJOURNAL OF PHYSICAL ORGANIC CHEMISTRY, Issue 5 2007Davood Nori-Shargh Abstract The B3LYP/6-31G* method was used to investigate the configurational properties of allene (1,2-propadiene) (1), 1,2,3-butatriene (2), 1,2,3,4-pentateriene (3), 1,2,3,4,5-hexapentaene (4), 1,2,3,4,5,6-heptahexaene (5), 1,2,3,4,5,6,7-octaheptaene (6), 1,2,3,4,5,6,7,8-nonaoctaene (7), and 1,2,3,4,5,6,7,8,9-decanonaene (9). The calculations at the B3LYP/6-31G* level of theory showed that the mutual interconversion energy barrier in compounds 1,8 are: 209.73, 131.77, 120.34, 85.00, 80.91, 62.19, 55.56, and 46.83,kJ,mol,1, respectively. The results showed that the difference between the average CC double bond lengths () values in cumulene compounds 1 and 2, is larger than those between 7 and 8, which suggest that with large n (number of carbon atoms in cumulene chain), the values approach a limiting value. Accordingly, based on the plotted data, the extrapolation to n,=,,, gives nearly the same limiting (i. e., ). Also, NBO results revealed that the sum of , -bond occupancies, , decrease from 1 to 8, and inversely, the sum of , -antibonding orbital occupancies, , increase from compound 1 to compound 8. The decrease of values for compounds 1,8, is found to follow the same trend as the barrier heights of mutual interconversion in compounds 1,8, while the decrease of the barrier height of mutual interconversion in compounds 1,8 is found to follow the opposite trend as the increase in the number of carbon atom. Accordingly, besides the previously reported allylic resonant stabilization effect in the transition state structures, the results reveal that the values, , ,(EHOMO,,,ELUMO), and the C atom number could be considered as significant criteria for the mutual interconversion in cumulene compounds 1,8. This work reports also useful predictive linear relationships between mutual interconversion energy barriers () in cumulene compounds and the following four parameters: , , ,(EHOMO,,,ELUMO), and CNumber. Copyright © 2007 John Wiley & Sons, Ltd. [source] Continuum limits for classical sequential growth modelsRANDOM STRUCTURES AND ALGORITHMS, Issue 2 2010Graham Brightwell Abstract A random graph order, also known as a transitive percolation process, is defined by taking a random graph on the vertex set {0,,,n , 1} and putting i below j if there is a path i = i1,ik = j in the graph with i1 < , < ik. Rideout and Sorkin Phys. Rev. D 63 (2001) 104011 provide computational evidence that suitably normalized sequences of random graph orders have a "continuum limit." We confirm that this is the case and show that the continuum limit is always a semiorder. Transitive percolation processes are a special case of a more general class called classical sequential growth models. We give a number of results describing the large-scale structure of a general classical sequential growth model. We show that for any sufficiently large n, and any classical sequential growth model, there is a semiorder S on {0,,,n - 1} such that the random partial order on {0,,,n - 1} generated according to the model differs from S on an arbitrarily small proportion of pairs. We also show that, if any sequence of classical sequential growth models has a continuum limit, then this limit is (essentially) a semiorder. We give some examples of continuum limits that can occur. Classical sequential growth models were introduced as the only models satisfying certain properties making them suitable as discrete models for spacetime. Our results indicate that this class of models does not contain any that are good approximations to Minkowski space in any dimension , 2. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010 [source] Properly colored subgraphs and rainbow subgraphs in edge-colorings with local constraintsRANDOM STRUCTURES AND ALGORITHMS, Issue 4 2003Noga Alon We consider a canonical Ramsey type problem. An edge-coloring of a graph is called m-good if each color appears at most m times at each vertex. Fixing a graph G and a positive integer m, let f(m, G) denote the smallest n such that every m -good edge-coloring of Kn yields a properly edge-colored copy of G, and let g(m, G) denote the smallest n such that every m -good edge-coloring of Kn yields a rainbow copy of G. We give bounds on f(m, G) and g(m, G). For complete graphs G = Kt, we have c1mt2/ln t , f(m, Kt) , c2mt2, and cmt3/ln t , g(m, Kt) , cmt3/ln t, where c1, c2, c, c are absolute constants. We also give bounds on f(m, G) and g(m, G) for general graphs G in terms of degrees in G. In particular, we show that for fixed m and d, and all sufficiently large n compared to m and d, f(m, G) = n for all graphs G with n vertices and maximum degree at most d. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2003 [source] Dyson's nonintersecting Brownian motions with a few outliersCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 3 2009Mark Adler Consider n nonintersecting Brownian particles on , (Dyson Brownian motions), all starting from the origin at time t = 0 and forced to return to x = 0 at time t = 1. For large n, the average mean density of particles has its support, for each 0 < t < 1, on the interval ±,2nt(1 , t). The Airy process ,,(,) is defined as the motion of these nonintersecting Brownian motions for large n but viewed from the curve ,, : y = ,2nt(1 , t) with an appropriate space-time rescaling. Assume now a finite number r of these particles are forced to a different target point, say a = ,0,n/2 > 0. Does it affect the Brownian fluctuations along the curve ,, for large n? In this paper, we show that no new process appears as long as one considers points (y, t) , ,, such that 0 < t < (1 + ,),1, which is the t -coordinate of the point of tangency of the tangent to the curve passing through (,0,n/2, 1). At this point the fluctuations obey a new statistics, which we call the Airy process with r outliers ,,(r)(,) (in short, r-Airy process). The log of the probability that at time , the cloud does not exceed x is given by the Fredholm determinant of a new kernel (extending the Airy kernel), and it satisfies a nonlinear PDE in x and ,, from which the asymptotic behavior of the process can be deduced for , , ,,. This kernel is closely related to one found by Baik, Ben Arous, and Péché in the context of multivariate statistics. © 2008 Wiley Periodicals, Inc. [source] | ||||||||||||||||