Fourier Analysis Research Articles

Adjoint Brascamp–Lieb inequalities

AbstractThe Brascamp–Lieb inequalities are a generalization of the Hölder, Loomis–Whitney, Young, and Finner inequalities that have found many applications in harmonic analysis and elsewhere. In this paper, we introduce an “adjoint” version of these inequalities, which can be viewed as an version of the entropic Brascamp–Lieb inequalities of Carlen and Cordero–Erausquin. As applications, we reprove a log‐convexity property of the Gowers uniformity norms, and establish some reverse inequalities for various tomographic transforms. We conclude with some open questions.

A systematic review of crosswalks for converting patient-reported outcome measure scores in hip, knee, and shoulder replacement surgery.

We aimed to systematically review studies of crosswalks for converting patient-reported outcome measure (PROM) scores used in joint replacement, and develop a database of published crosswalks. 4 electronic databases were searched from January 2000 to May 2023 to identify studies reporting the development and/or validation of crosswalks to convert PROM scores in patients undergoing elective hip, knee, or shoulder replacement surgery. Data on study and sample characteristics, source and target PROMs, and crosswalk development and validation methods were extracted from eligible studies. Study reporting was evaluated using the Mapping onto Preference-based measures reporting Standards (MAPS) checklist. 17 studies describing 35 crosswalks were eligible for inclusion. Unidirectional crosswalks were available to convert hip-specific (Oxford Hip Score [OHS]) and knee-specific (Oxford Knee Score [OKS]) scores to the EQ-5D-3L/EQ-5D-5L. Similar crosswalks to convert disease-specific scores (WOMAC) to the EQ-5D-3L, EQ-5D-5L, and ICECAP-O Capability Index were identified. Bidirectional crosswalks for converting OHS and OKS to the HOOS-JR/HOOS-12 and KOOS-JR/KOOS-12, for converting WOMAC to the HOOS-JR/KOOS-JR, and for converting HOOS-Function/KOOS-Function to the PROMIS-Physical Function were also available. Additionally, crosswalks to convert generic PROM scores from the UCLA Activity Scale to the Lower Extremity Activity Scale in both directions were available. No crosswalks were identified for converting scores in shoulder replacement. Development methods varied with the type of target score; most studies used regression, item response theory, or equipercentile equating approaches. Reporting quality was variable, particularly for methods and results items, impacting crosswalk application. This is the first synthesis of published crosswalks for converting joint-specific (OHS, OKS, HOOS, KOOS), disease-specific (WOMAC), and generic PROMs scores (PROMIS-Physical Function, UCLA Activity Scale, Lower Extremity Activity Scale) used to assess joint replacement outcomes, providing a resource for data harmonization and pooled analysis. Crosswalks were developed using regression methods (9 studies), equipercentile equating methods (5 studies), a combination of equipercentile equating and item response theory methods (2 studies), and a combination of regression and equipercentile equating methods (1 study). A range of crosswalk validation approaches were adopted, including the use of external datasets, separate samples or subsets, follow-up data from additional time points, or bootstrapped samples. Efforts are needed to standardize crosswalk methodology and achieve consistent reporting.

The Cyclical Genealogy of Schoenberg’s Second Chamber Symphony

In 1939 Schoenberg was facing the challenge to complete his Second Chamber Symphony, which was started 33 years earlier. Through a harmonic study of his sketches, I propose that Schoenberg made an implicit application of Jewish cyclical repetitions as a creative concept to overcome the long compositional block. This article is divided into two parts. I begin with a discussion on the use of cyclical form in the modern history of Western classical music before laying out some of Schoenberg’s ideas on repetition. Thereafter, harmonic analyses of the two complete movements of the Second Chamber Symphony as well as fragments from the third movement are used to support the argument that an extended cyclic form could have been adopted by Schoenberg as a method of musical organization during the first few years after his conversion back to Judaism. The cyclical formation of Schoenberg’s Second Chamber Symphony could critically be part of an unconscious and anxious process that reveals itself as part of the Semitic musical genealogy from Mendelssohn to Mahler.

Assignment and modeling of 13CH4 spectrum at 298 K in the lower part of the Tetradecad in the 4970–5300 cm−1 range

The absorption spectrum of the 13CH4 methane isotopologue was recorded on a Bruker IFS-125HR Fourier transform spectrometer at 298 K in the 4970–6200 cm−1 range. In this paper we report the results of assignment and modelling of the line positions and intensities of 13CH4 in the range of weaker absorption between 4970 and 5300 cm−1, corresponding to the lower part of Tetradecad, dominated by the 4ν4 band system near 5180 cm−1. The empirical list in this spectral range contains 1642 lines. Using an effective Hamiltonian initially derived from the analyses of cold Fourier transform (4970–5853 cm−1) and laser direct absorption (5853–6200 cm−1) spectra of the 13CH4, we assigned 1600 lines belonging to four bands of the Tetradecad up to Jmax=16. The 1548 line positions were fitted with an rms deviation of 1.7 × 10−3 cm−1. Measured line intensities were modeled for 674 transitions using the effective dipole transition moments to an rms deviation of about 7 %. The new data were used for the simultaneous global fit of the 13CH4 Hamiltonian parameters for the {Ground state / Dyad / Pentad / Octad / Tetradecad} system and the dipole moment parameters for the {Ground state - Tetradecad} system.

Super-resolution techniques to simulate electronic spectra of large molecular systems

An accurate treatment of electronic spectra in large systems with a technique such as time-dependent density functional theory is computationally challenging. Due to the Nyquist sampling theorem, direct real-time simulations must be prohibitively long to achieve suitably sharp resolution in frequency space. Super-resolution techniques such as compressed sensing and MUSIC assume only a small number of excitations contribute to the spectrum, which fails in large molecular systems where the number of excitations is typically very large. We present an approach that combines exact short-time dynamics with approximate frequency space methods to capture large narrow features embedded in a dense manifold of smaller nearby peaks. We show that our approach can accurately capture narrow features and a broad quasi-continuum of states simultaneously, even when the features overlap in frequency. Our approach is able to reduce the required simulation time to achieve reasonable accuracy by a factor of 20-40 with respect to standard Fourier analysis and shows promise for accurately predicting the whole spectrum of large molecules and materials.

Theory and Measurement of Heat Transport in Solids: How Rigidity and Spectral Properties Govern Behavior.

Models of heat transport in solids, being based on idealized elastic collisions of gas molecules, are flawed because heat and mass diffuse independently in solids but together in gas. To better understand heat transfer, an analytical, theoretical approach is combined with data from laser flash analysis, which is the most accurate method available. Dimensional analysis of Fourier's heat equation shows that thermal diffusivity (D) depends on length-scale, which has been confirmed experimentally for metallic, semiconducting, and electrically insulating solids. A radiative diffusion model reproduces measured thermal conductivity (K = DρcP = D × density × specific heat) for thick solids from ~0 to >1200 K using idealized spectra represented by 2-4 parameters. Heat diffusion at laboratory temperatures (conduction) proceeds by absorption and re-emission of infrared light, which explains why heat flows into, through, and out of a material. Because heat added to matter performs work, thermal expansivity is proportional to ρcP/Young's modulus (i.e., rigidity or strength), which is confirmed experimentally over wide temperature ranges. Greater uptake of applied heat (e.g., cP generally increasing with T or at certain phase transitions) reduces the amount of heat that can flow through the solid, but because K = DρcP, the rate (D) must decrease to compensate. Laser flash analysis data confirm this proposal. Transport properties thus depend on heat uptake, which is controlled by the interaction of light with the material under the conditions of interest. This new finding supports a radiative diffusion mechanism for heat transport and explains behavior from ~0 K to above melting.

Increasing the efficiency of coagulation in resonant gaps due to acoustic flow formation

Relevance. The urgent need to eliminate environmental pollution from industrial emissions of various solid particles. At the same time, maximum attention is paid to cleaning exhaust gases from particles of 2.5 microns in size or less. One of the most promising ways to increase the efficiency of existing gas purification equipment in capturing such particles is their coagulation by exposing the gas flow to high-intensity acoustic vibrations of ultrasonic frequency. However, at low concentrations, even at the maximum permissible sound pressure level, the coagulation efficiency of particles smaller than 2.5 microns is insufficient to increase the recovery rate of gas cleaning equipment. Therefore, there is an urgent need to find new ways to further improve the efficiency of ultrasonic coagulation of particles smaller than 2.5 μm. Aim. To determine the conditions for the formation of vortex flows in ultrasonic fields with the maximum ultrasonic influence in terms of sound pressure level. Conducting comparative studies of the coagulation of particles with a size of 2.5 microns with and without vortex flows. This will make it possible to determine the real values of increasing the efficiency of ultrasonic coagulation during turbulization of a gas-dispersed flow by acoustic flows in comparison with coagulation in a uniform ultrasonic field and without it. Objects. Coagulation of particles under the influence of homogeneous and inhomogeneous ultrasonic fields. Methods. Computer modeling of the formed ultrasonic field by the finite element method using harmonic acoustic analysis. The paper considers the experimental method for studying the process of combining particles under the influence of ultrasonic vibrations. To determine the characteristics of an aerosol during experimental studies, a TIPAS-1 meter based on the small-angle scattering method and the spectral transparency method was used. Results. The paper introduces the results of studies of coagulation of particles with a size of 2.5 microns or less in an ultrasonic field formed in resonant gaps by oscillating disk emitters. The authors proposed to increase the efficiency of coagulation in resonant gaps through the use of ultrasonic disk emitters capable of forming alternating zones of maximum and minimum amplitude oscillations in the resonant gaps. The creation of such zones ensured the formation of vortex-type acoustic flows capable of moving particles within the nodal regions of a standing wave and between them. The involvement of small particles in the formed flows made it possible to increase the probability of their collision. It was established that more effective ultrasonic coagulation provides an increase in the degree of inertial capture for particles of 2.5 microns in size by 6% – from 89 to 95%, for particles of 1.5 microns in size by 7% – from 85 to 92%, and for particles of 0.5 microns by 9% – from 76 to 85%.

Vibrational and Rovibrational Spectroscopic Data for the Ground and First-Excited States of Phosgene (COCl2), Formic Acid (HCOOH), and Chloroformic Acid (ClCOOH).

While "black box" quantum chemical computations for the determination of rovibronic spectral data are not quite at hand, the present work utilizes the titular molecules to showcase how excited-state quantum chemical methods can be conjoined to quartic force field (QFF) anharmonic rovibrational treatments to provide novel and useful predictions for such data. This work employs hybrid QFFs with explicitly correlated coupled cluster theory along with the equation-of-motion formalism to generate harmonic force constants and time-dependent density functional theory (TD-DFT) to produce anharmonic force constants for the generation of electronically excited-state rovibrational spectral data, in effect, rovibronic spectral data. Specific spectroscopic results from this work show that the fundamental C═O stretch in phosgene as well as in cis- and trans-formic acid drop from the region of around 1800 cm-1 to close to 1100 cm-1 or less in the first excited states of each molecule. While such is expected for these n → π* excitations, this work provides quantitative predictions for these fundamental vibrational frequencies. The most notable theoretical result is that the TD-DFT-based QFFs can experience unexpected failures, and their inclusion in excited-state hybrid QFFs should require at least two functionals to be employed. The computation of DFT QFFs is relatively fast, and such a "doubling up" of the QFFs will not greatly increase the computational time.

Fast theoretical predictions for spherical Fourier analysis of large-scale structures

Fast theoretical predictions for spherical Fourier analysis of large-scale structures

Gaussian estimates vs. elliptic regularity on open sets

AbstractGiven an elliptic operator $$L= - {{\,\textrm{div}\,}}(A \nabla \cdot )$$ L = - div ( A ∇ · ) subject to mixed boundary conditions on an open subset of $$\mathbb {R}^d$$ R d , we study the relation between Gaussian pointwise estimates for the kernel of the associated heat semigroup, Hölder continuity of L-harmonic functions and the growth of the Dirichlet energy. To this end, we generalize an equivalence theorem of Auscher and Tchamitchian to the case of mixed boundary conditions and to open sets far beyond Lipschitz domains. Yet, we prove the consistency of our abstract result by encompassing operators with real-valued coefficients and their small complex perturbations into one of the aforementioned equivalent properties. The resulting kernel bounds open the door for developing a harmonic analysis for the associated semigroups on rough open sets.

Process deviation detection method for dry‐type air‐core reactor based on power factor frequency characteristics

AbstractTo address the problem that the process deviation of dry‐type air‐core reactors cannot be effectively detected, electrical parameters of reactor are numerically analysed at different frequencies. A new method of process deviation detection by the power factor frequency characteristic is proposed. Considering on‐site power frequency interference, a harmonic analysis method is proposed, which is based on the integer multiple of the power frequency cycle, to calculate the power factor. The results show that the resistance loss remains relatively constant at power frequency and gradually increases at higher frequencies with the number of deviation turns increases. The eddy current loss remains basically unchanged at different turn deviations and frequencies. The change laws of resistance loss and the resistance loss power factor are the same. In a double logarithmic coordinate system, resistance loss power factor changes from linear to non‐linear. The non‐linearity degree can be used to detect the process deviation. The test results are consistent with the theory, and the correctness of the detection method is verified.

Elastic metasurfaces for Scholte-Stoneley wave control.

In this work, we investigate the dynamics of Scholte-Stoneley waves (SSWs) travelling along elastic metasurfaces, e.g. thin resonant structures embedding mechanical oscillators, placed at the interface between solid and fluid. To this purpose, an analytical dispersion law, valid in the long-wavelength regime, is derived and used to reveal the hybridization of SSWs with the collective resonance of the mechanical oscillators and the conversion of SSWs into leaky modes within the fluid. The analytical predictions are validated through numerical simulations that include both dispersive and harmonic analysis. Our findings disclose the capabilities of elastic metasurfaces in filtering, trapping and converting SSWs along fluid-solid interfaces, thus supporting the design of novel devices for solid-fluid interaction across various engineering applications, including microfluidics. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'.

Design and Modelling of MEMS Vibrating Internal Ring Gyroscopes for Harsh Environments.

This paper presents a design, model, and comparative analysis of two internal MEMS vibrating ring gyroscopes for harsh environmental conditions. The proposed design investigates the symmetric structure of the vibrating ring gyroscopes that operate at the identical shape of wine glass mode resonance frequencies for both driving and sensing purposes. This approach improves the gyroscope's sensitivity and precision in rotational motion. The analysis starts with an investigation of the dynamic behaviour of the vibrating ring gyroscope with the detailed derivation of motion equations. The design geometry, meshing technology, and simulation results were comprehensively evaluated on two internal vibrating ring gyroscopes. The two designs are distinguished by their support spring configurations and internal ring structures. Design I consists of eight semicircular support springs and Design II consists of sixteen semicircular support springs. These designs were modelled and analyzed using finite element analysis (FEA) in Ansys 2023 R1 software. This paper further evaluates static and dynamic performance, emphasizing mode matching and temperature stability. The results reveal that Design II, with additional support springs, offers better mode matching, higher resonance frequencies, and better thermal stability compared to Design I. Additionally, electrostatic, modal, and harmonic analyses highlight the gyroscope's behaviour under varying DC voltages and environmental conditions. Furthermore, this study investigates the impact of temperature fluctuations on performance, demonstrating the robustness of the designs within a temperature range from -100 °C to 100 °C. These research findings suggest that the internal vibrating ring gyroscopes are highly suitable for harsh conditions such as high temperature and space applications.

JIRAM Observations of Volcanic Flux on Io: Distribution and Comparison to Tidal Heat Flow Models

AbstractJuno has allowed clear, high‐resolution imaging of Io's polar volcanoes using the Jovian Infrared Auroral Mapper (JIRAM) instrument. We have used data from JIRAM's M‐band (4.78 μm) imager from 11 Juno orbits to construct a global map of volcanic flux. This map provides short‐term insight into the spatial distribution of volcanoes and the ways in which high‐ and low‐latitude volcanoes differ. Using spherical harmonic analysis, we quantitatively compare our volcanic flux map to the surface heat flow distribution expected from models of Io's tidal heat deposition (summarized in de Kleer, Park, et al. (2019, https://doi.org/10.26206/d4wc‐6v82). Our observations confirm previously detected systems of bright volcanoes at high latitudes. Our study finds that both poles are comparably active and that the observed flux distribution is inconsistent with an asthenospheric heating model, although the south pole is viewed too infrequently to establish reliable trends.

Confinement Effects of Two-dimensional Surfaces on Water Adsorption and Dissociation over Pt(111).

It has been established that the confined space created by stacking a two dimensional (2D) surface atop a metal catalyst serves as a nano-reactor. According to recent research, when a graphene (Gr) overlayer encloses a Pt catalyst from above, the activation barrier for the water dissociation reaction, a process with major industrial significance, decreases. In order to investigate how the effect of confinement varies among different two-dimensional (2D) materials, we study the adsorption and dissociation barriers of water molecule on Pt(111) under graphene, hexagonal boron nitride (h-BN), and heptazine-based graphitic carbon nitride (g-C3N4) layers using density functional theory calculations. Our findings reveal that the strength of adsorption does not decrease consistently with a reduction in the height of the 2D overlayer. Furthermore, a smaller barrier is not always the consequence of poorer adsorption of the reactant. We also examine the effect of confinement on the shape of the reaction path, on the frequencies of vibrational modes, and on the rate constants derived using the harmonic transition state theory. Overall, all three of the 2D surfaces cause a decrease in barrier height and a weakening of adsorption, though to differing degrees due to a mix of mechanical, geometric and electronic variables.

Influence of Rotation and Viscosity on Parallel Rolls of Electrically Conducting Fluid

Rayleigh–Bénard convection is a fundamental fluid dynamics phenomenon that significantly influences heat transfer in various natural and industrial processes, such as geophysical dynamics in the Earth’s liquid core and the performance of heat exchangers. Understanding the behavior of conductive fluids under the influence of heating, rotation, and magnetic fields is critical for improving thermal management systems. Utilizing the Boussinesq approximation, this study theoretically examines the nonlinear convection of a planar layer of conductive liquid that is heated from below and subjected to rotation about a vertical axis in the presence of a magnetic field. We focus on the onset of stationary convection as the temperature difference applied across the planar layer increases. Our theoretical approach investigates the formation of parallel rolls aligned with the magnetic field under free–free boundary conditions. To analyze the system of nonlinear equations, we expand the dependent variables in a series of orthogonal functions and express the coefficients of these functions as power series in a parameter ϵ. A solution for this nonlinear problem is derived through Fourier analysis of perturbations, extending to O(ϵ8), which allows for a detailed visualization of the parallel rolls. Graphical results are presented to explore the dependence of the Nusselt number on the Rayleigh number (R) and Ekman number (E). We observe that both the local Nusselt number and average Nusselt number increase as the Ekman number decreases. Furthermore, the flow appears to become more deformed as E decreases, suggesting an increased influence of external factors such as rotation. This deformation may enhance mixing within the fluid, thereby improving heat transfer between different regions.

Mapping of morphological coastline changes based on aerial photographs and Discrete Fourier Transform, Hel Peninsula, Poland

The coast is a dynamic zone where constantly occurring hydrodynamic and morphodynamic processes affect the shape of the shore. The paper presents a method based on spatial and spectral analysis of changes in the coastline position based on data obtained from aerial photographs interpretation and Fourier analysis, on the example of the Hel Peninsula. The Hel Peninsula is one of the most interesting accumulation forms of the Polish Baltic coast, where dynamic changes of the seashore cause the occurrence of time-varying sections of accumulation-abrasion of the coastline. For the purpose of detecting the coastline changes, historical aerial photographs from the years 1947, 1957, 1963, 1991 were used. It was assumed that the over-40-year research period, which includes the obtained series of aerial photographs, would allow for a sufficient study of the long-term shoreline changes, which allow for distinguishing the length of characteristic coastline undulations. The quasi-wave signal of the shoreline changes obtained from the aerial photographs interpretation, after using Fourier analysis, enabled an effective and precise identification of the coastline undulation. The spatial analyses, showed that the Hel Peninsula is clearly divided into a part subjected to accumulation processes and an abraded one. Furthermore, the dynamics of coastline changes was determined, showing that the abrasive processes were intensifying. Moreover, spectral Fourier analysis allowed for the precise identification of coastline undulations with dominant lengths. The obtained results of spatial and spectral analysis indicate that abrasive-accumulation sections with a length of about 0.3–4.5 km dominate on the Hel Peninsula shoreline.

On the injectivity of the shifted Funk–Radon transform and related harmonic analysis

AbstractNecessary and sufficient conditions are obtained for injectivity of the shifted Funk–Radon transform associated with k-dimensional totally geodesic submanifolds of the unit sphere Sn in ℝn+1. This result generalizes the well known statement for the spherical means on Sn and is formulated in terms of zeros of Jacobi polynomials. The relevant harmonic analysis is developed, including a new concept of induced Stiefel (or Grassmannian) harmonics, the Funk–Hecke type theorems, addition formula, and multipliers. Some perspectives and conjectures are discussed.

Dynamic Harmonic Distortion Analysis and Mitigation Strategies for DC Third Rail Systems

Harmonic distortions in DC third rail systems can cause overheating of electric motors and transformers. Single-tuned filters and shunt active harmonic filters (SAHFs) are often used to mitigate the harmonic distortions. However, there is a lack of studies on the effects of train dynamics on harmonic distortions. This paper aims to investigate the influence of dynamic train behaviors for a DC third rail system and provide recommendations for the design of single-tuned filters and SAHFs to mitigate the harmonic distortions. The traction power supply system of a DC third rail system in Malaysia is modeled using ETAP-eTraX and MATLAB-Simulink software for the investigation. The ETAP-eTraX software is used to accurately compute the dynamic behavior of the train, while MATLAB-Simulink allows for the assessment of the impact of train behavior on the rail track, as well as the harmonic effect of the railway power network on the train. The findings showed that the SAHF exhibits strong adaptability and superior filtering performance compared to the single-tuned filter in addressing dynamic harmonic distortion in traction power supply systems. This study emphasizes the significance of incorporating harmonic mitigation devices, particularly for managing dynamic harmonic distortion based on actual train consumption patterns.

Dual spaces vs. Haar measures of polynomial hypergroups

Many symmetric orthogonal polynomials (Pn(x))n∈N0 induce a hypergroup structure on N0. The Haar measure is the counting measure weighted with h(n)≔1/∫RPn2(x)dμ(x)≥1, where μ denotes the orthogonalization measure. We observed that many naturally occurring examples satisfy the remarkable property h(n)≥2(n∈N). We give sufficient criteria and particularly show that h(n)≥2(n∈N) if the (Hermitian) dual space N0̂ equals the full interval [−1,1], which is fulfilled by an abundance of examples. We also study the role of nonnegative linearization of products (and of the resulting harmonic and functional analysis). Moreover, we construct two example types with h(1)<2. To our knowledge, these are the first such examples. The first type is based on Karlin–McGregor polynomials, and N0̂ consists of two intervals and can be chosen “maximal” in some sense; h is of quadratic growth. The second type relies on hypergroups of strong compact type; h grows exponentially, and N0̂ is discrete.