Averages over Surfaces with Infinitely Flat Points

AbstractLet S be a hypersurface in Rn , n ≥ 2, and dμ = ψ dσ, where ψ ∈ C∞0 (Rn) and σ denotes the surface area measure on S. Define the maximal function M associated to S and μ by [formula] It was shown by Stein that when S is the sphere in Rn, n ≥ 3, M (the spherical maximal function) is bounded on Lp(Rn) if and only if p > n/(n − 1). It has also been shown that if S is of finite type, i.e., the curvature vanishes to at most a finite order m at every point of S, then there exists some number pm < ∈ such that M is bounded on Lp(Rn) (n ≥ 3) for all p ∈ (pm, ∈]. On the other hand it is well known that if S is flat, that is, S contains a point at which the curvature vanishes to infinite order, then M may not be bounded on any Lp(Rn), p < ∞. We show that under some hypotheses the maximal functions M associated to flat surfaces S ⊂ R3 are bounded on certain Orlicz spaces LΦ(R3) defined naturally in terms of S

Multi-machine scaling of the main SOL parallel heat flux width in tokamak limiter plasmas

As in many of today’s tokamaks, plasma start-up in ITER will be performed in limiter configuration on either the inner or outer midplane first wall (FW). The massive, beryllium armored ITER FW panels are toroidally shaped to protect panel-to-panel misalignments, increasing the deposited power flux density compared with a purely cylindrical surface. The chosen shaping should thus be optimized for a given radial profile of parallel heat flux, q in the scrape-off layer (SOL) to ensure optimal power spreading. For plasmas limited on the outer wall in tokamaks, this profile is commonly observed to decay exponentially as q q = − exp ( / r λ ) 0 q omp , or, for inner wall limiter plasmas with the double exponential decay comprising a sharp near-SOL feature and a broader main SOL width, λq omp. The initial choice of λq omp , which is critical in ensuring that current ramp-up or down will be possible as planned in the ITER scenario design, was made on the basis of an extremely restricted L-mode divertor dataset, using infra-red thermography measurements on the outer divertor target to extrapolate to a heat flux width at the main plasma midplane. This unsatisfactory situation has now been significantly improved by a dedicated multi-machine ohmic and L-mode limiter plasma study, conducted under the auspices of the International Tokamak Physics Activity, involving 11 tokamaks covering a wide parameter range with R = = 0.4–2.8 m, 1 B I 0 p .2–7.5 T, = 9–2500 kA. Measurements of λq omp in the database are made exclusively on all devices using a variety of fast reciprocating Langmuir probes entering the plasma at a variety of poloidal locations, but with the majority being on the low field side. Statistical analysis of the database reveals nine reasonable engineering and dimensionless scalings. All yield, however, similar predicted values of λq omp mapped to the outside midplane. The engineering scaling with the highest statistical significance, λ = ( / ( )) ( / /κ) − − q 10 P V W m a R omp tot 3 0.38 1.3 , dependent on input power density, aspect ratio and elongation, yields λq omp = [7, 4, 5] cm for Ip = [2.5, 5.0, 7.5] MA, the three reference limiter plasma currents specified in the ITER heat and nuclear load specifications. Mapped to the inboard midplane, the worst case (7.5 MA) corresponds to λq ~ 57 1 ± 4 imp mm, thus consolidating the 50mm width used to optimize the FW panel toroidal shape.EURATOM 633053Czech Science Foundation GA CR P205/12/2327, GA15-10723S, MSMT LM2011021US Department of Energy DE-FG02- 07ER54917, DE-AC02-09CH11466, DE-FC02-04ER5469

Frequency of occurrence of numbers in the World Wide Web

The distribution of numbers in human documents is determined by a variety of diverse natural and human factors, whose relative significance can be evaluated by studying the numbers' frequency of occurrence. Although it has been studied since the 1880's, this subject remains poorly understood. Here, we obtain the detailed statistics of numbers in the World Wide Web, finding that their distribution is a heavy-tailed dependence which splits in a set of power-law ones. In particular, we find that the frequency of numbers associated to western calendar years shows an uneven behavior: 2004 represents a `singular critical' point, appearing with a strikingly high frequency; as we move away from it, the decreasing frequency allows us to compare the amounts of existing information on the past and on the future. Moreover, while powers of ten occur extremely often, allowing us to obtain statistics up to the huge 10^127, `non-round' numbers occur in a much more limited range, the variations of their frequencies being dramatically different from standard statistical fluctuations. These findings provide a view of the array of numbers used by humans as a highly non-equilibrium and inhomogeneous system, and shed a new light on an issue that, once fully investigated, could lead to a better understanding of many sociological and psychological phenomena.Comment: 5 pages, 4 figure

Scaling fields in the two-dimensional abelian sandpile model

We consider the isotropic two-dimensional abelian sandpile model from a perspective based on two-dimensional (conformal) field theory. We compute lattice correlation functions for various cluster variables (at and off criticality), from which we infer the field-theoretic description in the scaling limit. We find a perfect agreement with the predictions of a c=-2 conformal field theory and its massive perturbation, thereby providing direct evidence for conformal invariance and more generally for a description in terms of a local field theory. The question of the height 2 variable is also addressed, with however no definite conclusion yet.Comment: 22 pages, 1 figure (eps), uses revte

Avalanche dynamics, surface roughening and self-organized criticality - experiments on a 3 dimensional pile of rice

We present a two-dimensional system which exhibits features of self-organized criticality. The avalanches which occur on the surface of a pile of rice are found to exhibit finite size scaling in their probability distribution. The critical exponents are $\tau$ = 1.21(2) for the avalanche size distribution and $D$ = 1.99(2) for the cut-off size. Furthermore the geometry of the avalanches is studied leading to a fractal dimension of the active sites of $d_B$ = 1.58(2). Using a set of scaling relations, we can calculate the roughness exponent $\alpha = D - d_B$ = 0.41(3) and the dynamic exponent $z = D(2 - \tau)$ = 1.56(8). This result is compared with that obtained from a power spectrum analysis of the surface roughness, which yields $\alpha$ = 0.42(3) and $z$ = 1.5(1) in excellent agreement with those obtained from the scaling relations.Comment: 7 pages, 8 figures, accepted for publication in PR

Constrained Gauge Fields from Spontaneous Lorentz Violation

Spontaneous Lorentz violation realized through a nonlinear vector field constraint of the type $A_{\mu}A^{\mu}=M^{2}$ ($M$ is the proposed scale for Lorentz violation) is shown to generate massless vector Goldstone bosons, gauging the starting global internal symmetries in arbitrary relativistically invariant theories. The gauge invariance appears in essence as a necessary condition for these bosons not to be superfluously restricted in degrees of freedom, apart from the constraint due to which the true vacuum in a theory is chosen by the Lorentz violation. In the Abelian symmetry case the only possible theory proves to be QED with a massless vector Goldstone boson naturally associated with the photon, while the non-Abelian symmetry case results in a conventional Yang-Mills theory. These theories, both Abelian and non-Abelian, look essentially nonlinear and contain particular Lorentz (and $CPT$) violating couplings when expressed in terms of the pure Goldstone vector modes. However, they do not lead to physical Lorentz violation due to the simultaneously generated gauge invariance.Comment: 15 pages, minor corrections, version to be published in Nucl. Phys.

Toroidal mode number estimation of the edge-localized modes using the KSTAR 3-D electron cyclotron emission imaging system

A new and more accurate technique is presented for determining the toroidal mode number n of edge-localized modes (ELMs) using two independent electron cyclotron emission imaging (ECEI) systems in the Korea Superconducting Tokamak Advanced Research (KSTAR) device. The technique involves the measurement of the poloidal spacing between adjacent ELM filaments, and of the pitch angle ?? O of filaments at the plasma outboard midplane. Equilibrium reconstruction verifies that ?? O is nearly constant and thus well-defined at the midplane edge. Estimates of n obtained using two ECEI systems agree well with n measured by the conventional technique employing an array of Mirnov coils.open3

Orbiting Membranes in M-theory on AdS_7 x S^4 Background

We study classical solutions describing rotating and boosted membranes on AdS_7 x S^4 background in M-theory. We find the dependence of the energy on the spin and R-charge of these solutions. In the flat space limit we get E ~ S^{2/3}, while for AdS at leading order E-S grows as S^{1/3}. The membranes on AdS_4 x S^7 background have briefly been studied as well.Comment: 13 pages, latex, v2: a note and refs. added, some typos correcte

Penrose limit and string quantization in AdS_3 \times S^3

We consider corrections to the Penrose limit of AdS_3 \times S^3 with NS-NS flux which are due to the terms next to leading order in inverse radius expansion. The worldsheet theory of a lightcone string is interacting due to the presence of quartic terms in the action. Perturbative corrections to the spectrum are shown to agree with the results from the exact quantization in AdS_3 \times S^3.Comment: 18 pages v2: typos fixed, reference added, to appear in JHE