Rational Approximation with Locally Geometric Rates

AbstractWe investigate the rate of pointwise rational approximation of functions from two classes. The distinguishing feature of these classes is the essentially faster convergence of the best uniform rational approximants versus best uniform polynomial approximants. It is known that for piecewise analytic functions “near best” polynomials converging geometrically fast at every point of analyticity of the function exist. Here we construct rational approximants enjoying similar properties. We also show that our construction yields rates of convergence that are, in a certain sense, best possible

Parton Densities in a Nucleon

In this paper we re-analyse the situation with the shadowing corrections (SC) in QCD for the proton deep inelastic structure functions. We reconsider the Glauber - Mueller approach for the SC in deep inelastic scattering (DIS) and suggest a new nonlinear evolution equation. We argue that this equation solves the problem of the SC in the wide kinematic region where \as \kappa = \as \frac{3 \pi \as}{2 Q^2R^2} x G(x,Q^2) \leq 1. Using the new equation we estimate the value of the SC which turn out to be essential in the gluon deep inelastic structure function but rather small in $F_2(x,Q^2)$. We claim that the SC in $xG(x,Q^2)$ is so large that the BFKL Pomeron is hidden under the SC and cannot be seen even in such "hard" processes that have been proposed to test it. We found that the gluon density is proportional to $\ln(1/x)$ in the region of very small $x$. This result means that the gluon density does not reach saturation in the region of applicability of the new evolution equation. It should be confronted with the solution of the GLR equation which leads to saturation.Comment: latex file 53 pages, 27 figures in eps file

Christoffel functions and orthogonal polynomials for Erd¨os weights on (–∞,∞)

We establish bounds on orthonormal polynomials and Christoffel functions associated with weights on IR of the form W2 = e−2Q, where Q : IR → IR is even, and is of faster than polynomial growth at ∞ (so-called Erd¨os weights). Typical examples are Q(x) := expk |x| α1, α > 1, , where expk = expk (exp (... exp(·))) denotes the kth iterated exponential. Further, we obtain uniform estimates on the spacing of all the zeros and on the Christoffel functions. These results complement earlier ones for the case where Q is of polynomial growth at ∞ (so-called Freud weights) and for exponential weights on (−1, 1)

Smoothed Complexity Theory

Smoothed analysis is a new way of analyzing algorithms introduced by Spielman and Teng (J. ACM, 2004). Classical methods like worst-case or average-case analysis have accompanying complexity classes, like P and AvgP, respectively. While worst-case or average-case analysis give us a means to talk about the running time of a particular algorithm, complexity classes allows us to talk about the inherent difficulty of problems. Smoothed analysis is a hybrid of worst-case and average-case analysis and compensates some of their drawbacks. Despite its success for the analysis of single algorithms and problems, there is no embedding of smoothed analysis into computational complexity theory, which is necessary to classify problems according to their intrinsic difficulty. We propose a framework for smoothed complexity theory, define the relevant classes, and prove some first hardness results (of bounded halting and tiling) and tractability results (binary optimization problems, graph coloring, satisfiability). Furthermore, we discuss extensions and shortcomings of our model and relate it to semi-random models.Comment: to be presented at MFCS 201

Froissart boundary for deep inelastic structure functions

In this letter we derive the Froissart boundary in QCD for the deep inelastic structure function in low $x$ kinematic region. We show that the comparison of the Froissart boundary with the new HERA experimental data gives rise to a challenge for QCD to explain the matching between the deep inelastic scattering and real photoproduction process.Comment: 10 pages,7 figure

1/f Noise in Electron Glasses

We show that 1/f noise is produced in a 3D electron glass by charge fluctuations due to electrons hopping between isolated sites and a percolating network at low temperatures. The low frequency noise spectrum goes as \omega^{-\alpha} with \alpha slightly larger than 1. This result together with the temperature dependence of \alpha and the noise amplitude are in good agreement with the recent experiments. These results hold true both with a flat, noninteracting density of states and with a density of states that includes Coulomb interactions. In the latter case, the density of states has a Coulomb gap that fills in with increasing temperature. For a large Coulomb gap width, this density of states gives a dc conductivity with a hopping exponent of approximately 0.75 which has been observed in recent experiments. For a small Coulomb gap width, the hopping exponent approximately 0.5.Comment: 8 pages, Latex, 6 encapsulated postscript figures, to be published in Phys. Rev.

From Fractional Chern Insulators to a Fractional Quantum Spin Hall Effect

We investigate the algebraic structure of flat energy bands a partial filling of which may give rise to a fractional quantum anomalous Hall effect (or a fractional Chern insulator) and a fractional quantum spin Hall effect. Both effects arise in the case of a sufficiently flat energy band as well as a roughly flat and homogeneous Berry curvature, such that the global Chern number, which is a topological invariant, may be associated with a local non-commutative geometry. This geometry is similar to the more familiar situation of the fractional quantum Hall effect in two-dimensional electron systems in a strong magnetic field.Comment: 8 pages, 3 figure; published version with labels in Figs. 2 and 3 correcte

Electrical transport studies of quench condensed Bi films at the initial stage of film growth: Structural transition and the possible formation of electron droplets

The electrical transport properties of amorphous Bi films prepared by sequential quench deposition have been studied in situ. A superconductor-insulator (S-I) transition was observed as the film was made increasingly thicker, consistent with previous studies. Unexpected behavior was found at the initial stage of film growth, a regime not explored in detail prior to the present work. As the temperature was lowered, a positive temperature coefficient of resistance (dR/dT > 0) emerged, with the resistance reaching a minimum before the dR/dT became negative again. This behavior was accompanied by a non-linear and asymmetric I-V characteristic. As the film became thicker, conventional variable-range hopping (VRH) was recovered. We attribute the observed crossover in the electrical transport properties to an amorphous to granular structural transition. The positive dR/dT found in the amorphous phase of Bi formed at the initial stage of film growth was qualitatively explained by the formation of metallic droplets within the electron glass.Comment: 7 pages, 6 figure

QCD evolution of the gluon density in a nucleus

The Glauber approach to the gluon density in a nucleus, suggested by A. Mueller, is developed and studied in detail. Using the GRV parameterization for the gluon density in a nucleon, the value as well as energy and $Q^2$ dependence of the gluon density in a nucleus is calculated. It is shown that the shadowing corrections are under theoretical control and are essential in the region of small $x$. They change crucially the value of the gluon density as well as the value of the anomalous dimension of the nuclear structure function, unlike of the nucleon one. The systematic theoretical way to treat the correction to the Glauber approach is developed and a new evolution equation is derived and solved. It is shown that the solution of the new evolution equation can provide a selfconsistent matching of ``soft" high energy phenomenology with ``hard" QCD physics.Comment: 63 pages,psfig.sty,25 pictures in eps.file