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Two closely related topics: higher order Bohr sets and higher order almost automorphy are investigated in this paper. Both of them are related to nilsystems. In the first part, the problem which can be viewed as the higher order version of an old question concerning Bohr sets is studied: for any In the second part, the notion of

This paper explores the Fourier decomposition method to approximate the decomposition of electrocardiogram (ECG) signals into their component waveforms, such as the QRS-complex and T-wave. We compute expansion coefficients using the ℓ1 Fourier transform and the traditional ℓ2 Fourier transform. Numerical examples are presented, and the analysis focuses on ECG signals as a real-world application, comparing the performance of the ℓ1 and ℓ2 Fourier transforms. Our results demonstrate that the ℓ1 Fourier transform significantly enhances the separation of ECG signal components, such as the QRS-complex and T-wave. This improvement is attributed to a notable reduction in the Gibbs phenomenon introduced by the Fourier-series expansion when using the ℓ1 Fourier transform, as opposed to the traditional ℓ2 Fourier transform.

In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: The key tools behind our results are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier transforms for functions such that their Fourier transforms are of monotone type or lacunary series.

In dismemberment cases, forensic anthropologists evaluate bony cut surfaces and estimate saw class characteristics, which can aid in investigative and legal proceedings. Previous publications indicate that saw class characteristics, such as tooth shape, saw set, and power, can be deduced from the kerf profile shape and size. However, these studies are based on subjective visual categorizations, at times with limited statistical assessments. This study used elliptical Fourier analysis to quantitatively assess relationships between kerf shapes and saw class characteristics. Incomplete kerf profiles (n = 133) made with 19 saws in anatomically gifted, macerated human limbs (n = 19) were assessed. Kerf profiles were captured with a stereomicroscope and closed outlines were created and subjected to elliptical Fourier and principal component analyses. PerMANOVAs and Kruskal-Wallis analyses were performed on the resultant principal components to assess the effects of saw set, power, and tooth shape on kerf shape. Cross-validated stepwise discriminant function analyses (DFA) were performed to evaluate classification accuracy. There was no significant difference in entrance and exit morphology (p = 0.31). Significant results were obtained for all saw class characteristics. DFA classified tooth shape with 88.0 % accuracy. Flat and U-shaped kerfs were associated with rip saws while W-shaped kerfs were indicative of crosscut saws. DFA classified saw power with 89.5 % accuracy. On average, mechanical saws produced kerfs with larger widths compared to hand saws. Relationships between kerf floor morphology and saw set, however, were more complex. These quantitative analyses of kerf shape generally support anecdotal relationships established in the literature and its utility in forensic assessment. •Quantitative outline analyses reveal statistical relationships in saw mark analyses.•Entrance and exit kerf profile shapes do not differ significantly.•A W-shaped kerf indicates a crosscut saw.•Flat or U-shaped kerfs can be either crosscut or rip saws.•Morphometric analyses provide objective accuracy rates for saw class predictions.

Let f be a real-valued function of degree d defined on the n -dimensional Boolean cube { ± 1 } n , and f ( x ) = ∑ S ⊂ { 1 , … , n } f ^ ( S ) ∏ k ∈ S x k its Fourier-Walsh expansion. The main result states that there is an absolute constant C > 0 such that the ℓ 2 d / ( d + 1 ) -sum of the Fourier coefficients of f : { ± 1 } n ⟶ [ - 1 , 1 ] is bounded by C d log d . It was recently proved that a similar result holds for complex-valued polynomials on the n -dimensional polytorus T n , but that in contrast to this, a replacement of the n -dimensional torus T n by the n -dimensional cube [ - 1 , 1 ] n leads to a substantially weaker estimate. This in the Boolean case forces us to invent novel techniques which differ from the ones used in the complex or real case. We indicate how our result is linked with several questions in quantum information theory.

Early diagnosis of sepsis enables timely resuscitation and antibiotics and prevents subsequent morbidity and mortality. Clinical approaches relying on point-in-time analysis of vital signs or lab values are often insensitive, non-specific and late diagnostic markers of sepsis. Exploring otherwise hidden information within intervals-in-time, heart rate variability (HRV) has been documented to be both altered in the presence of sepsis, and correlated with its severity. We hypothesized that by continuously tracking individual patient HRV over time in patients as they develop sepsis, we would demonstrate reduced HRV in association with the onset of sepsis. We monitored heart rate continuously in adult bone marrow transplant (BMT) patients (n = 21) beginning a day before their BMT and continuing until recovery or withdrawal (12+/-4 days). We characterized HRV continuously over time with a panel of time, frequency, complexity, and scale-invariant domain techniques. We defined baseline HRV as mean variability for the first 24 h of monitoring and studied individual and population average percentage change (from baseline) over time in diverse HRV metrics, in comparison with the time of clinical diagnosis and treatment of sepsis (defined as systemic inflammatory response syndrome along with clinically suspected infection requiring treatment). Of the 21 patients enrolled, 4 patients withdrew, leaving 17 patients who completed the study. Fourteen patients developed sepsis requiring antibiotic therapy, whereas 3 did not. On average, for 12 out of 14 infected patients, a significant (25%) reduction prior to the clinical diagnosis and treatment of sepsis was observed in standard deviation, root mean square successive difference, sample and multiscale entropy, fast Fourier transform, detrended fluctuation analysis, and wavelet variability metrics. For infected patients (n = 14), wavelet HRV demonstrated a 25% drop from baseline 35 h prior to sepsis on average. For 3 out of 3 non-infected patients, all measures, except root mean square successive difference and entropy, showed no significant reduction. Significant correlation was present amongst these HRV metrics for the entire population. Continuous HRV monitoring is feasible in ambulatory patients, demonstrates significant HRV alteration in individual patients in association with, and prior to clinical diagnosis and treatment of sepsis, and merits further investigation as a means of providing early warning of sepsis.

In this paper, the authors found a transformation that is valid for any arbitrary signal. This transformation is strictly periodical and, therefore, it allows to apply the ordinary -trans- formation for the fitting of the transformed signal. The most interesting application (in accordance with the author’s opinion) is the fitting of the frequency-phase modulated signals that located actually inside the found transformation. This new transformation will be useful for application of the responses of different complex systems when a particular model is absent. As available data, we consider cosmic microwave background data (CMB) associated with the background temperature fluctuations near K. These electro-magnetic (EM) fluctuations of the early Universe were measured at the wide frequency range 30–857 GHz. In this paper, we analyzed the measured data at 353 GHz corresponding to the taken zero pixels. Other details are described in the second section of the paper. This squared matrix corresponding to the measured data contains 2047 lines 2047 columns. If one considers each column as frequency-phase modulated signal, then amplitude-frequency response can be evaluated with the help of -transformation that has the period equals that is valid for any analyzed random signal. These ‘‘universal’’ behavior allows to fit a wide set of random signals and compare them with each other in terms of their amplitude-frequency responses (AFR). Concluding the abstract, one can say that these new possibilities of the traditional -analysis will serve as a common tool in the armory of the methods used by researchers in the data processing area.

Large sparse linear algebraic systems can be found in a variety of scientific and engineering fields and many scientists strive to solve them in an efficient and robust manner. In this paper, we propose an interpretable neural solver, the Fourier neural solver (FNS), to address them. FNS is based on deep learning and a fast Fourier transform. Because the error between the iterative solution and the ground truth involves a wide range of frequency modes, the FNS combines a stationary iterative method and frequency space correction to eliminate different components of the error. Local Fourier analysis shows that the FNS can pick up on the error components in frequency space that are challenging to eliminate with stationary methods. Numerical experiments on the anisotropic diffusion equation, convection–diffusion equation, and Helmholtz equation show that the FNS is more efficient and more robust than the state-of-the-art neural solver.

We consider the overdetermined problems in terms of the Riesz and Bessel potentials on hyperbolic space H n . Taking advantage of the Helgason–Fourier analysis on the hyperbolic space, we apply the moving plane method in integral form to the corresponding integral equations and show that the solution is constant on the boundary of the domain if and only if the domain is a geodesic ball, and therefore, the solution is radially symmetric. Moreover, fractional-order equations involving the Laplace–Beltrami operator on the hyperbolic space are also considered by using their Green’s function estimates. Our operators also include the well-known GJMS operators on the hyperbolic space.