Crystallization of random trigonometric polynomials

We give a precise measure of the rate at which repeated differentiation of a random trigonometric polynomial causes the roots of the function to approach equal spacing. This can be viewed as a toy model of crystallization in one dimension. In particular we determine the asymptotics of the distribution of the roots around the crystalline configuration and find that the distribution is not Gaussian.Comment: 10 pages, 3 figure

Fine structure of the zeros of orthogonal polynomials, I. A tale of two pictures

Mhaskar-Saff found a kind of universal behavior for the bulk structure of the zeros of orthogonal polynomials for large n. Motivated by two plots, we look at the finer structure for the case of the Verblunsky coefficients and for what we call the BLS condition: αn = Cb^n + O ((bΔ)^n). In the former case, we describe the results of Stoiciu. In the latter case, we prove asymptotically equal spacing for the bulk of zeros

A back-to-front derivation: the equal spacing of quantum levels is a proof of simple harmonic oscillator physics

The dynamical behaviour of simple harmonic motion can be found in numerous natural phenomena. Within the quantum realm of atomic, molecular and optical systems, two main features are associated with harmonic oscillations: a finite ground-state energy and equally spaced quantum energy levels. Here it is shown that there is in fact a one-to-one mapping between the provision of equally spaced quantum energy levels and simple harmonic motion. The analysis establishes that the Hamiltonian of any system featuring quantized energy levels in an evenly spaced, infinite set must have a quadratic dependence on a pair of canonically conjugate variables. Moreover, specific physical inferences can be drawn. For example, exploiting this 'back-to-front' derivation, and based on the known existence of photons, it can be proved that an electromagnetic energy density is quadratic in both the electric and magnetic fields

Pilot tone design for dispersion estimation in coherent optical fast OFDM systems

We show that inserting pilot tones with frequency intervals inversely proportional to the subcarrier index exhibits greatly improved dispersion estimation performance when compared to the equal spacing design in optical fast orthogonal frequency division multiplexing (F-OFDM). With the proposed design, a 20-Gbit/s four amplitude shift keying optical F-OFDM system with 840-km transmission without optical dispersion compensation is experimentally demonstrated. It is shown that a single F-OFDM symbol with six pilot tones can achieve near-optimal estimation performance for the 840-km dispersion. This is in contrast to the minimum of ten pilot tones using an equal spacing design with either cubic or Fourier-transform-based interpolation

Baryon Masses at Second Order in Chiral Perturbation Theory

We analyze the baryon mass differences up to second order in chiral perturbation theory, including the effects of decuplet intermediate states. We show that the Coleman--Glashow relation has computable corrections of order $(m_d - m_u) m_s$. These corrections are numerically small, and in agreement with the data. We also show that corrections to the $\Sigma$ equal-spacing rule are dominated by electromagnetic contributions, and that the Gell-Mann--Okubo formula has non-analytic corrections of order $m_s^2 \ln m_s$ which cannot be computed from known matrix elements. We also show that the baryon masses cannot be used to extract model-independent information about the current quark masses.Comment: 11 pages, 1 uu-encoded figure, LBL-34779, UCB-PTH-93/2

2D shape classification and retrieval

We present a novel correspondence-based technique for efficient shape classification and retrieval. Shape boundaries are described by a set of (ad hoc) equally spaced points – avoiding the need to extract “landmark points”. By formulating the correspondence problem in terms of a simple generative model, we are able to efficiently compute matches that incorporate scale, translation, rotation and reflection invariance. A hierarchical scheme with likelihood cut-off provides additional speed-up. In contrast to many shape descriptors, the concept of a mean (prototype) shape follows naturally in this setting. This enables model based classification, greatly reducing the cost of the testing phase. Equal spacing of points can be defined in terms of either perimeter distance or radial angle. It is shown that combining the two leads to improved classification/retrieval performance.