A limit theorem for products of free unitary operators

This paper establishes necessary and sufficient conditions for the sequence of products of freely independent unitary operators to converge in distribution to the uniform law on the unit circle. The author would like to express his gratitude to Diana Bloom for her help with editing, and to Professor Raghu Varadhan for useful discussions.     

Parameterized reduced order modeling of misaligned stacked disks rotor assemblies

Light and flexible rotating parts of modern turbine engines operating at supercritical speeds necessitate application of more accurate but rather computationally expensive 3D FE modeling techniques. Stacked disks misalignment due to manufacturing variability in the geometry of individual components constitutes a particularly important aspect to be included in the analysis because of its impact on system dynamics. A new parametric model order reduction algorithm is presented to achieve this goal at affordable computational costs. It is shown that the disks misalignment leads to significant changes in nominal system properties that manifest themselves as additional blocks coupling neighboring spatial harmonics in Fourier space. Consequently, the misalignment effects can no longer be accurately modeled as equivalent forces applied to a nominal unperturbed system. The fact that the mode shapes become heavily distorted by extra harmonic content renders the nominal modal projection-based methods inaccurate and thus numerically ineffective in the context of repeated analysis of multiple misalignment realizations. The significant numerical bottleneck is removed by employing an orthogonal projection onto the subspace spanned by first few Fourier harmonic basis vectors. The projected highly sparse systems are shown to accurately approximate the specific misalignment effects, to be inexpensive to solve using direct sparse methods and easy to parameterize with a small set of measurable eccentricity and tilt angle parameters. Selected numerical examples on an industrial scale model are presented to illustrate the accuracy and efficiency of the algorithm implementation.     

Observation of the chaotic spin-wave soliton trains in magnetic films

The transition from stationary to chaotic spin-wave soliton trains has been observed. The experiment utilized cw excitation of envelope solitons through self-modulation instability of spin waves. By increasing the spin-wave power, the secondary self-modulation instability succeeded the primary modulation instability, resulting in after-modulation of the soliton train amplitude. Further increase of the spin-wave power led to development of the higher-order instabilities, resulting in formation of the chaotic soliton train.     

An example of dynamically induced coherent states

We consider an example where coherent states appear naturally (dynamically). We study the interaction between an arbitrary quantized matter with a quantized electromagnetic field. Using an approximation that is similar to the Hartree–Fock one and a variational principle, we demonstrate that the electromagnetic field is created by the charged matter exactly in Glauber coherent states with parameters determined by mean currents of the matter.     

Slow light on a printed circuit board

Slow-light effects induced by stimulated Raman scattering in polymer waveguides on a printed circuit board are shown to enable a widely tunable delay of broadband optical signals, suggesting an advantageous platform for optical information processing and ultrafast optical waveform transformation.     

Subwavelength terahertz spin-flip laser based on a magnetic point-contact array

We present a theoretical design for a single-mode, truly subwavelength terahertz disk laser based on a nanocomposite gain medium comprising an array of normal-metal/ferromagnetic (FM) point contacts embedded in a thin dielectric layer. Stimulated emission of light occurs due to spin-flip relaxation of spin-polarized electrons injected from the FM side of the contacts. Ultrahigh electrical current densities in the contacts and a dielectric material with a large refractive index, neither condition being achievable in conventional semiconductor media, enables the thresholds of lasing to be overcome for the lowest-order modes of the disk, making single-mode operation possible.     

Weak variations of Lipschitz graphs and stability of phase boundaries

In the case of Lipschitz extremals of vectorial variational problems, an important class of strong variations originates from smooth deformations of the corresponding non-smooth graphs. These seemingly singular variations, which can be viewed as combinations of weak inner and outer variations, produce directions of differentiability of the functional and lead to singularity-centered necessary conditions on strong local minima: an equality, arising from stationarity, and an inequality, implying configurational stability of the singularity set. To illustrate the underlying coupling between inner and outer variations, we study in detail the case of smooth surfaces of gradient discontinuity representing, for instance, martensitic phase boundaries in non-linear elasticity.     

Geodesic Webs on a Two-Dimensional Manifold and Euler Equations

We prove that any planar 4-web defines a unique projective structure in the plane in such a way that the leaves of the web foliations are geodesics of this projective structure. We also find conditions for the projective structure mentioned above to contain an affine symmetric connection, and conditions for a planar 4-web to be equivalent to a geodesic 4-web on an affine symmetric surface. Similar results are obtained for planar d-webs, d>4, provided that additional d−4 second-order invariants vanish.