Search for Wieferich Primes through the use of Periodic Binary Strings

The result of the distributed computing projectWieferich@Home is presented: the binary periodic numbers of bit pseudo-length j ≤ 3500 obtained by replication of a bit string of bit pseudo-length k ≤ 24 and increased by one are Wieferich primes only for the cases of 1092 or 3510.

Anti-Periodic Boundary Value Problem for Impulsive Fractional Integro Differential Equations

MSC 2010: 34A37, 34B15, 26A33, 34C25, 34K37

Automorphisms of the Planar Tree Power Series Algebra and the Non-Associative Logarithm

2000 Mathematics Subject Classification: 17A50, 05C05.

A Note on the Asymptotic Behaviour of a Periodic Multitype Galton-Watson Branching Process

2000 Mathematics Subject Classification: 60J80.

Direct laser writing of a channel waveguide in Al2O3:Nd thin film

The femtosecond laser modification of refractive index in amorphous Al2O3:Nd thin film prepared by rf magnetron sputtering is investigated. Modifications of the refractive index in a sample with a single Al2O3:Nd layer and in a sample composed of the Al2O3:Nd layer and SiO2 layer on the top were compared. Advantages arising from addition of the SiO2 layer are shown. The film was patterned in order to form an active waveguide. Waveguide loss and mode composition were investigated experimentally and theoretically. Spectrum and kinetics of luminescence in the region of 1.06 μm were measured.

Lower Bounds on the Directed Sweepwidth of Planar Shapes

We investigate a recently introduced width measure of planar shapes called sweepwidth and prove a lower bound theorem on the sweepwidth.

On the Approximation by Convolution Operators in Homogeneous Banach Spaces of Periodic Functions

AMS Subject Classification 2010: 41A25, 41A27, 41A35, 41A36, 41A40, 42Al6, 42A85.

Local Energy Decay in Even Dimensions for the Wave Equation with a Time-Periodic Non-Trapping Metric and Applications to Strichartz Estimates

2000 Mathematics Subject Classification: 35B40, 35L15.

Thermalized polarization dynamics of a discrete optical-waveguide system with four-wave mixing

Statistical mechanics of two coupled vector fields is studied in the tight-binding model that describes propagation of polarized light in discrete waveguides in the presence of the four-wave mixing. The energy and power conservation laws enable the formulation of the equilibrium properties of the polarization state in terms of the Gibbs measure with positive temperature. The transition line T=∞ is established beyond which the discrete vector solitons are created. Also in the limit of the large nonlinearity an analytical expression for the distribution of Stokes parameters is obtained, which is found to be dependent only on the statistical properties of the initial polarization state and not on the strength of nonlinearity. The evolution of the system to the final equilibrium state is shown to pass through the intermediate stage when the energy exchange between the waveguides is still negligible. The distribution of the Stokes parameters in this regime has a complex multimodal structure strongly dependent on the nonlinear coupling coefficients and the initial conditions.

Assembly accuracy analysis for small components with a planar surface in large-scale metrology

Large-scale mechanical products, such as aircraft and rockets, consist of large numbers of small components, which introduce additional difficulty for assembly accuracy and error estimation. Planar surfaces as key product characteristics are usually utilised for positioning small components in the assembly process. This paper focuses on assembly accuracy analysis of small components with planar surfaces in large-scale volume products. To evaluate the accuracy of the assembly system, an error propagation model for measurement error and fixture error is proposed, based on the assumption that all errors are normally distributed. In this model, the general coordinate vector is adopted to represent the position of the components. The error transmission functions are simplified into a linear model, and the coordinates of the reference points are composed by theoretical value and random error. The installation of a Head-Up Display is taken as an example to analyse the assembly error of small components based on the propagation model. The result shows that the final coordination accuracy is mainly determined by measurement error of the planar surface in small components. To reduce the uncertainty of the plane measurement, an evaluation index of measurement strategy is presented. This index reflects the distribution of the sampling point set and can be calculated by an inertia moment matrix. Finally, a practical application is introduced for validating the evaluation index.

Nonlinear refraction in optical lattices induced by the wave-soliton interference

In the framework of 1D Nonlinear Shrödinger Equation (NSE) we demonstrate how one can control the refractive angle of a fundamental soliton beam passing through an optical lattice, by adjusting either the shape of an individual waveguide or the relative positions of waveguides. Even for a single scatterer its shape has a nontrivial effect on the refraction direction. In the case of shallow modulation we provide an analytical description based of the effect on the soliton perturbation theory. When one considers a lattice of scatterers, there emanates an additional form factor in the radiation density (RD) of emitted waves referring to the wave-soliton beating and interference inside the lattice. We concentrate on the results for two cases: periodic lattice and disordered lattice of scattering shapes. © 2011 IEEE.

Optical communication based on the periodic nonlinear Fourier transform signal processing

In this work we introduce the periodic nonlinear Fourier transform (PNFT) and propose a proof-of-concept communication system based on it by using a simple waveform with known nonlinear spectrum (NS). We study the performance (addressing the bit-error-rate (BER), as a function of the propagation distance) of the transmission system based on the use of the PNFT processing method and show the benefits of the latter approach. By analysing our simulation results for the system with lumped amplification, we demonstrate the decent potential of the new processing method.

Strongly localized moving discrete dissipative breather-solitons in Kerr nonlinear media supported by intrinsic gain

We investigate the mobility of nonlinear localized modes in a generalized discrete Ginzburg-Landau-type model, describing a one-dimensional waveguide array in an active Kerr medium with intrinsic, saturable gain and damping. It is shown that exponentially localized, traveling discrete dissipative breather-solitons may exist as stable attractors supported only by intrinsic properties of the medium, i.e., in the absence of any external field or symmetry-breaking perturbations. Through an interplay by the gain and damping effects, the moving soliton may overcome the Peierls-Nabarro barrier, present in the corresponding conservative system, by self-induced time-periodic oscillations of its power (norm) and energy (Hamiltonian), yielding exponential decays to zero with different rates in the forward and backward directions. In certain parameter windows, bistability appears between fast modes with small oscillations and slower, large-oscillation modes. The velocities and the oscillation periods are typically related by lattice commensurability and exhibit period-doubling bifurcations to chaotically "walking" modes under parameter variations. If the model is augmented by intersite Kerr nonlinearity, thereby reducing the Peierls-Nabarro barrier of the conservative system, the existence regime for moving solitons increases considerably, and a richer scenario appears including Hopf bifurcations to incommensurately moving solutions and phase-locking intervals. Stable moving breathers also survive in the presence of weak disorder. © 2014 American Physical Society.

Mode-locking intracavity dynamics:modeling nonlinear power and energy fluctuations

Modern high-power, pulsed lasers are driven by strong intracavity fluctuations. Critical in driving the intracavity dynamics is the nontrivial phase profiles generated and their periodic modification from either nonlinear mode-coupling, spectral filtering or dispersion management. Understanding the theoretical origins of the intracavity fluctuations helps guide the design, optimization and construction of efficient, high-power and high-energy pulsed laser cavities. Three specific mode-locking component are presented for enhancing laser energy: waveguide arrays, spectral filtering and dispersion management. Each component drives a strong intracavity dynamics that is captured through various modeling and analytic techniques.