Narrow Results

We prove that any knot of ${ℝ}^3$ is isotopic to a Fourier knot of type $(1,1,2)$ obtained by deformation of a Lissajous knot.

We establish sharp convolution and multiplication estimates in weighted Lebesgue, Fourier Lebesgue and modulation spaces. We cover, especially some results in [L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations (Springer, Berlin, 1997); S. Pilipović, N. Teofanov and J. Toft, Micro-local analysis in Fourier Lebesgue and modulation spaces, II, J. Pseudo-Differ. Oper. Appl.1 (2010) 341–376]. The results are also related to some results by Iwabuchi in [T. Iwabuchi, Navier–Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices, J. Differential Equations248 (2010) 1972–2002].

Metacontrast is a form of visual masking in which the target and mask are non-overlapping. In metacontrast, the masking effect is typically largest when the mask is presented some time after the target. This is known as Type-B masking. The present report examines to what extent Type-B metacontrast masking can be explained based on the stimuli involved. The assumption is made that the visibility of the target is, at least in part, determined by the correlation between the amplitude spectrum of the target-and-mask combination and that of the target alone. It is found that the correlation is higher when the stimuli are presented at the same time relative to when they are presented at different times. This relationship follows from the stimuli alone. Thus, one would expect the masking to be weakest when the two stimuli are simultaneous. Type-B correlation functions, in which the largest reductions occur only when the mask is presented after the target, can be obtained by further assuming a temporal integration window with a rapid onset and a shallow decline. In agreement with psychophysical masking studies; the analyses yield functions that are most similar to Type-B masking for moderate mask intensities and become less Type-B like at higher mask intensities. The effects of dark adaptation and spatial separation of target and mask are also modeled.

We discuss the performance of the Search and Fourier Transform algorithms on a hybrid computer constituted of classical and quantum processors working together. We show that this semi-quantum computer would be an improvement over a pure classical architecture, no matter how few qubits are available and, therefore, it suggests an easier implementable technology than a pure quantum computer with arbitrary number of qubits.

A new polarization–interference biomedical diagnostic three-dimensional (3D) Jones-matrix technology with digital Fourier reconstruction of layered maps of optical anisotropy (thesiograms) of dehydrated films (facies) of biological fluids of human organs is presented and experimentally tested. An original model of layered phase scanning of polycrystalline architectonics of supramolecular networks of biological fluid facies is proposed for the purpose of theoretical justification and prognostic use of the obtained results. On its basis, algorithms of Jones-matrix reconstruction of thesiograms of birefringence and dichroism of facies of synovial fluid, bile and blood are found. As a result, layered thesiograms of linear and circular birefringence and dichroism of facies with different spatial–angular architectonics of supramolecular networks are experimentally obtained for the first time. Within the framework of statistical analysis of experimental data, new objective markers (asymmetry and excess of optical anisotropy parameter distributions) for diagnostics of pathological changes in the optical anisotropy of biological fluid facies were defined and clinically tested. As a result, an excellent level of balanced accuracy of the developed polarization–interference Jones-matrix method of layer-by-layer reconstruction of thesiograms of polycrystalline supramolecular networks in differential diagnostics of bile facies (cholelithiasis), synovial fluid (reactive synovitis–septic arthritis) and whole blood (follicular adenoma–papillary thyroid cancer) was achieved.

Frequency channelization is a fundamental signal processing operation employed across various domains, including communications and radio astronomy. The polyphase filterbank (PFB) represents an efficient and versatile means of channelization. When strict constraints are placed on the allowable spectral leakage between neighboring channels, an oversampled PFB design is advantageous. A helpful consequence of the oversampling is that inversion of the PFB to recover high temporal resolution is simplified and can be accomplished accurately using Fourier transforms. We describe this inversion approach and identify key design considerations. We examine the residual error and spectral/temporal leakage behavior when a channelizer and its corresponding inverter are cascaded, concluding that near-perfect reconstruction can be approached with appropriate selection of PFB and inverter design parameters.

Herein, commonly used quantitative bioengineering methods that have been developed to analyze fractionated electrograms recorded from the surface of the atria during atrial fibrillation (AF) are described. Techniques were categorized as time-domain and frequency-domain methods. The main time-domain method is peak counting. Its variations based on preprocessing and thresholding are discussed. The main frequency-domain method is spectral analysis. Two spectral estimators, the discrete Fourier transform (DFT) and the new spectral estimator (NSE) are described. The ability of each estimator to detect the main periodic component of fractionated atrial electrograms is compared. Several spectral parameters that are used for analysis of atrial electrograms including the dominant frequency (DF), dominant amplitude (DA) and mean spectral profile (MP) are defined. Mean values of these parameters are compared in paroxysmal versus persistent AF fractionated electrograms based upon the results of several studies. Time-domain methods are shown to work best for analysis with deterministic, not fractionated atrial electrograms. For fractionated atrial electrograms, frequency-domain methods are often used. The DF, DA and MP spectral parameters are significantly different in paroxysmal versus longstanding persistent AF recordings. The DF and the DA are significantly higher, and the MP is significantly lower, in persistent AF electrogram recordings. The higher DF and DA parameter values reflect substrate remodeling in persistent AF, which increases the stability of the electrical activation pattern. The lower MP value in persistent AF reflects the lower spectral noise floor, indicative of a less complex and more periodic pattern of electrical activity.