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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.

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.

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.

Plants undergo various natural changes that dramatically modify their genomes. One is polyploidization and the second is hybridization. Both are regarded as key factors in plant evolution and result in phenotypic differences in different plant organs. In , we can find both examples in nature, and this genus has a seed shape diversity that has long been recognized as a valuable source of information for infrageneric classification. Morphometric analysis is a statistical study of shape and size and their covariations with other variables. Traditionally, seed shape description was limited to an approximate comparison with geometric figures (rounded, globular, reniform, or heart-shaped). Seed shape quantification has been based on direct measurements, such as area, perimeter, length, and width, narrowing statistical analysis. We used seed images and processed them to obtain silhouettes. We performed geometric morphometric analyses, such as similarity to geometric models and elliptic Fourier analysis, to study the hybrid offspring of and . We generated synthetic tetraploids of and performed controlled crosses between diploid and to analyze seed morphology. After imaging capture and post-processing, statistical analysis revealed differences in seed size, but not in shape, between diploids and tetraploids, as well as some differences in shape among the parentals and hybrids. A detailed inspection using fluorescence microscopy allowed for the identification of shape differences in the cells of the seed coat. In the case of hybrids, differences were found in circularity and solidity. Overal seed shape is maternally regulated for both species, whereas cell shape cannot be associated with any of the sexes. Our results provide additional tools useful for the combination of morphology with genetics, ecology or taxonomy. Seed shape is a robust indicator that can be used as a complementary tool for the genetic and phylogenetic analyses of hybrid populations.

White sharks undergo pronounced ecological and dietary shifts across ontogeny, and their teeth play a central role in mediating these changes. Understanding the complexity within shark tooth and jaw mechanics, plus the fine‐scale tooth morphology, can provide insights into how feeding strategies and, hence, dietary niches and ecological function evolve with age and size. These morphological changes underpin ontogenetic niche shifts, revealing how functional adaptations in dentition enable white sharks to exploit different prey resources throughout development. This study provides novel insights into the ontogenetic and positional variation in C. carcharias dentition, integrating both Elliptic Fourier Analysis (EFA) and traditional morphometric approaches. We reveal significant patterns of tooth morphology that vary with jaw position and ontogenetic stage, reflecting functional adaptations to changing dietary and biomechanical demands. A key ontogenetic shift was identified as teeth transitioned from narrow, cuspidate forms with accessory cusplets in juveniles to broader, serrated teeth in larger individuals. We found no significant differences in tooth morphology between sexes, aligning with known similarities in diet and body shape in the eastern Australian white shark population. Significant anterior‐to‐posterior variation in tooth form was observed within the jaw, with lateral teeth becoming more compressed and recurved, suggesting functional transitions in prey handling throughout the jaw. Additionally, we documented structural changes in jaw morphology at approximately 210 cm PCL, corresponding to broader teeth and increased bite capacity. These shifts likely reflect developmental milestones in feeding capability, supporting the transition from a solely piscivorous diet to the inclusion of marine mammal prey. White sharks exhibit distinct ontogenetic shifts in tooth morphology that underpin changes in diet and ecological function. Using Elliptic Fourier Analysis and traditional morphometrics, we reveal significant variation in dentition across jaw position and developmental stage, with a major transition occurring at 210 cm PCL and at the sixth tooth within the jaw. These findings highlight functional adaptations in feeding strategies that support shifts from piscivory to marine mammal predation.