Narrow Results

A derivative nonlinear Schrödinger model is shown to support localized N-body bound states for several ranges (called bands) of the coupling constant η. The ranges of η within each band can be completely determined using number theoretic concepts such as Farey sequences and continued fractions. For N≥3, the N-body bound states can have both positive and negative momenta. For η>0, bound states with positive momentum have positive binding energy, while states with negative momentum have negative binding energy.

Skew correction of a scanned document page is an important preprocessing step in document image analysis. We propose here a fast and robust skew estimation algorithm based on rank analysis in Farey sequence. Our target document class comprises two major Indian scripts with headlines, namely Devnagari and Bangla. At the beginning, straight edge segments from the edge map of the document page are detected by our algorithm using properties of digital straightness. Straight edges derived in this manner are binned by Farey ranks in correspondence with their slopes. The principal bin, identified from these bins using the strength of accumulated edge points, represents the principal direction along the direction of headlines, from which the gross skew angle is estimated. A fast refinement algorithm is then applied with a finer tuning of Farey ranks, to detect the skew up to the desired level of precision. The algorithm has been tested on a diverse set of document images, containing Bangla and Devnagari scripts. Experimental results are quite encouraging in terms of accuracy, sensitivity to non-textual objects, effectiveness in dealing with unrestricted layouts, and computational efficiency.

The Nagumo–Sato (NS) model is a one-dimensional piecewise linear map that describes simplified dynamics of a single neuron. The NS model and its network extension, coupled Nagumo–Sato models, exhibit complex behavior both in their transient dynamics and after converging to periodic orbits. However, the way the period and the transient length change against the parameters is not completely understood. In this study, we numerically investigate the transient and periodic dynamics in single and coupled NS models. Simulation results indicate the following observations. (1) The period of a single NS model shows layered structures associated with the Farey sequence. (2) Two coupled NS models show discontinuity in the transient length, even though the period does not change. (3) In the case of an associative memory model consisting of NS models, there exists a small parameter region where both the period and the transient length increase considerably. The dynamics within the region is much more complex than that outside the region.

This study investigates the behavior of a damped, inelastic, sinusoidally forced impact oscillator which has its barrier placed such that the oscillator always vibrates under compression about its subharmonic resonant frequencies. The Poincaré sections at near subharmonic resonance conditions exhibit finger-shaped chaotic attractors, similar to the strange attractor mapping of Hénon and the ones found by Holmes in his study of chaotic resonances of a buckled beam. The number of such fingers are observed to increase as the barrier distance from the equilibrium is decreased. These chaotic states are interspersed with regimes of periodic behavior, with the periodicity being in accordance with well defined period adding laws. This study also focuses on the ordered behavior of the one-impact period-$p$ orbits around the higher subharmonics of the oscillator.

This paper studies the product ${\overline{G}}_n$ of the binomial coefficients in the $n$th row of Pascal’s triangle, which equals the reciprocal of the product of all the reduced and unreduced Farey fractions of order $n$. It studies its size as a real number, measured by $\mathrm{\text{log}}({\overline{G}}_n)$, and its prime factorization, measured by the order of divisibility ${\nu }_p({\overline{G}}_n)={\mathrm{\text{ord}}}_p({\overline{G}}_n)$ by a fixed prime $p$, each viewed as a function of $n$. It derives three formulas for ${\mathrm{\text{ord}}}_p({\overline{G}}_n)$, two of which relate it to base $p$ radix expansions of integers up to $n$, and which display different facets of its behavior. These formulas are used to determine the maximal growth rate of each ${\mathrm{\text{ord}}}_p({\overline{G}}_n)$ and to explain structure of the fluctuations of these functions. It also defines analogous functions ${\nu }_b({\overline{G}}_n)$ for all integer bases $b\ge 2$ using base $b$ radix expansions replacing base $p$-expansions. A final topic relates factorizations of ${\overline{G}}_n$ to Chebyshev-type prime-counting estimates and the prime number theorem.