Inversion Formula Research Articles

Harmonic analysis associated with the linear canonical Fourier–Jacobi transform

The main objective of this paper is to develop a new harmonic analysis related to a Fourier–Jacobi type operator Δ α , β m of the real line. We define and study the linear canonical Fourier–Jacobi transform F α , β m . We study some important properties, inversion formula, Plancherel theorem, Paley–Wiener theorem and Riemann–Lebsgue lemma. An application in solving a generalized heat equation is given.

Inversion formulas for the attenuated conical Radon transform: Plane and cylinder case

Since the invention of Compton camera imaging, the conical Radon transform, which maps a given function defined on 3-dimensional Euclidean space to its surface integrals over cones, has been studied intensively. The problem of recovering such a function from its unrestricted conical Radon transform is overdetermined, since the set of all cones depends on the three dimensions of the vertex, the two dimensions of the central axis, and the one-dimensional opening angle. Therefore, various types of restricted conical Radon transforms have also been studied. However, most of these studies have neglected the attenuation of the medium. This article presents the study of attenuated conical Radon transforms with vertices on a plane (referred to as the plane case) or the cylinder (referred to as the cylinder case) in a 3-dimensional space. In all cases, the function is integrated over conical surfaces with an additional weight that decreases with the distance to the vertex of the cones. The main results provide explicit inversion formulas for the attenuated conical Radon transform in the plane and in the cylinder case.

A Comparative View of Becker, Lomnitz, and Lambert Linear Viscoelastic Models

We compare the classical viscoelastic models due to Becker and Lomnitz with respect to a recent viscoelastic model based on the Lambert W function. We take advantage of this comparison to derive new analytical expressions for the relaxation spectrum in the Becker and Lomnitz models, as well as novel integral representations for the retardation and relaxation spectra in the Lambert model. In order to derive these analytical expressions, we have used the analytical properties of the exponential integral and the Lambert W function, as well as the Titchmarsh’s inversion formula of the Stieltjes transform. In addition, we prove some interesting inequalities by comparing the different models considered, as well as the non-negativity of the retardation and relaxation spectral functions. This means that the complete monotonicity of the rate of creep and the relaxation functions is satisfied, as required by the classical theory of linear viscoelasticity.

Dispersion relation from Lorentzian inversion in 1d CFT

Starting from the Lorentzian inversion formula, we derive a dispersion relation which computes a four-point function in 1d CFTs as an integral over its double discontinuity. The crossing symmetric kernel of the integral is given explicitly for the case of identical operators with integer or half-integer scaling dimension. This derivation complements the one that uses analytic functionals. We use the dispersion relation to evaluate holographic correlators defined on the half-BPS Wilson line of planar N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills, reproducing results up to fourth order in an expansion at large t’Hooft coupling.

Octonion quadratic-phase Fourier transform: inequalities, uncertainty principles, and examples

In this article, we define the octonion quadratic-phase Fourier transform (OQPFT) and derive its inversion formula, including its fundamental properties such as linearity, parity, modulation, and shifting. We also establish its relationship with the quaternion quadratic-phase Fourier transform (QQPFT). Further, we derive the Parseval formula and the Riemann–Lebesgue lemma using this transform. Furthermore, we formulate two important inequalities (sharp Pitt’s and sharp Hausdorff–Young’s inequalities) and three main uncertainty principles (logarithmic, Donoho–Stark’s, and Heisenberg’s uncertainty principles) for the OQPFT. To complete our investigation, we construct three elementary examples of signal theory with graphical interpretations to illustrate the use of OQPFT and discuss their particular cases.

Inversion Formulas by the Use of Bessel Beams of Integer and Fractional Orders

Inversion Formulas by the Use of Bessel Beams of Integer and Fractional Orders

Determination of body length regression formulas for the golden grey mullet (Chelon auratus, Risso 1810) based on otoliths found in the feces of the Caspian Seal (Pusa caspica Gmelin, 1788)

The Caspian Seal, the sole mammal endemic to the Caspian Sea, is listed in the IUCN Red List as a species facing the threat of extinction, a status similarly recognized and/or assigned in all countries in the Caspian region.The study of seal nutrition using caprological methods is of great interest for understanding animal adaptation to deteriorating habitat conditions and assessing the adequacy of their food base in the sea. This method involves analyzing the contents of feces from ichthyophages to detect undigested fish otoliths.The results of studying the seal's diet during periods of haulout on breeding grounds showed that the frequency of occurrence of Chelon auratus otoliths ranged from 2-3% to 27.7% of all fish otoliths detected in feces.This study aims to assess changes in Chelon auratus otoliths as they pass through seals' gastrointestinal tracts and estimate formulas for recovering the linear dimensions of fish consumed by these seals.The research indicates that otolith growth slows with fish growth but does not cease entirely. Allometry is evident in the different ratios of otolith length and width to fish length in various size groups and the change in otolith shape as fish grow.To recover the body length of fish consumed by seals based on otoliths found in their feces, applying inverse calculation formulas obtained when studying the relationship between otolith growth and fish body growth using collection material obtained directly from fish is possible.Further research should focus on refining the recovery formula for the length of large fish by dividing the length into several segments, each described by separate formulas describing the relationship between fish body length and otolith length and width growth.

Investigation of solution behavior of Differential Equations by Sumudu methods of random complex Partial Differential Equations

In this study, solutions of random complex partial differential equations were found using the two-dimensional Sumudu transformation method(STM). The initial conditions of a deterministic equation or the non-homogeneous part of the equation are transformed into random variables to obtain a random complex partial differential equation. With the help of the properties of two-dimensional Sumudu and inverse Sumudu transformation, an approximate analytical solution of a complex partial differential equation with random constant coefficients was obtained by selecting a random variable with an initial condition of Normal and Gamma distribution. The probability characteristics of the resulting solutions, such as expected value and variance, were obtained and graphically shown with the help of the Maple package program.

Reproducing formulas for L2-multiplier operators related to the k-Hankel transform

ABSTRACT The aims of this paper are to study the L 2 -multiplier operators, T m , σ , associated with k-Hankel transform. We derive three Calderón's reproducing formulas. Initially, by an admissibility condition we get the inversion formula for T m , σ ( φ ) . Furthermore, we achieve quadratic convergence as our second formula. Additionally, by utilizing the extremal function, we obtain the best approximation formula.

Proof of a conjectured Möbius inversion formula for Grothendieck polynomials

Proof of a conjectured Möbius inversion formula for Grothendieck polynomials

Some approximate inversion formulas for the Whittaker multiplier operators

Some approximate inversion formulas for the Whittaker multiplier operators

A microlocal and visual comparison of 2D Kirchhoff migration formulas in seismic imaging*

Abstract The term Kirchhoff migration refers to a collection of approximate linearized inversion formulas for solving the inverse problem of seismic tomography which entails reconstructing the Earth’s subsurface from reflected wave fields. A number of such formulas exists, the first dating from the 1950 s. As far as we know, these formulas have not yet been mathematically compared with respect to their imaging properties. This shortcoming is to be alleviated by the present work: we systematically discuss the advantages and disadvantages of the formulas in 2D from a microlocal point of view. To this end we consider the corresponding imaging operators in an unified framework as pseudodifferential or Fourier integral operators. Numerical examples illustrate the theoretical insights and allow a visual comparison of the different formulas.

Biquaternion ambiguity function and associated uncertainty principles

A mathematical tool used to analyze signals in high-dimensional areas is the biquaternion ambiguity function. While typical ambiguity functions deal with time delay and Doppler frequency, biquaternion ambiguity functions extend the concept to rotations and scaling factors. Biquaternion ambiguity functions offer a more thorough representation of signal ambiguity in these high-dimensional spaces, which can help develop better signal processing algorithms for a range of uses. In this paper, we introduce the notion of biquaternion ambiguity function which is formed from conventional ambiguity function by invoking biquaternion algebra. We first establish the various properties, namely dilation, translation, shift, nonlinearity, boundedness, reconstruction formula, Moyal’s formula, and inversion formula. An interesting relationship between the biquaternion Fourier transform and the biquaternion ambiguity function is also established. Furthermore, we establish the various versions of uncertainty inequalities namely, Pitt’s inequality, sharp Hausdorff–Young inequality, sharp Local uncertainty principle and logarithmic uncertainty principle for biquaternion ambiguity function. Some potential applications are also presented at the end.

The Generalized Mehler–Fock Transform over Lebesgue Spaces

This paper focuses on establishing boundedness properties and Parseval–Goldstein-type relations for the generalized Mehler–Fock transform initially introduced by B. L. J. Braaksma and B. M. Meulenbeld (Compositio Math., 18(3):235–287, 1967). Also, we derive an inversion formula for this transform over Lebesgue spaces.

The performance of Tiangong-2 InIRA in gravity anomaly recovery from fused conventional nadir-observing altimeter data

Satellite altimetry has emerged as the primary data source for deriving marine gravity, owing to advancements in satellite altimetry technologies. The Tiangong-2 interferometric imaging radar altimeter (TG2 InIRA) is a new generation wide-swath altimeter (WSA) launched by China on 15 September 2016. TG2 InIRA employs a short-baseline interferometry measurement method at a small incidence angle, thereby enabling the precise measurement of wide-swath sea surface heights (SSHs). The present study demonstrates the efficacy of utilizing data from the TG2 InIRA in conjunction with data from the CryoSat-2 (CS2) satellite to recover marine gravity anomalies. In the Northwest Pacific region (15°N ∼ 25°N, 140°E ∼ 150°E), the TG2 InIRA data were filtered to a 2 km × 2 km grid and any anomalous values were excluded using a mean sea surface model. Deflections of the vertical (DOVs) were calculated using the remove-restore technique, and the inverse Vening-Meinesz formula was applied to invert the gravity anomaly. A power spectral density analysis was conducted to assess variations in gravity anomalies at ship locations in the frequency domain. The results, based on the SIO V31.1 (DOV) model, indicate that the gridded vertical deflections derived from CS2 and TG2 InIRA data achieve more balanced accuracy in the east–west and north–south directions. The integration of TG2 InIRA and CS2 data yielded gravity results exhibiting a root mean square (RMS) of 0.519 mGal lower than those derived from TG2 InIRA data alone, and 0.034 mGal lower than those derived from CS2 data. These findings were derived from shipborne gravity data. Furthermore, the inversion of gravity anomalies from TG2 InIRA demonstrated an enhanced capability in the reduction of high-frequency gravity noise.

Efficacy of intermittent pneumatic compression on breast cancer-related upper limb lymphedema: a systematic review and meta-analysis in clinical studies.

Complete decongestive therapy (CDT) and intermittent pneumatic compression (IPC) are the most common combination of treatments in breast cancer-related upper limb lymphedema. The effects of IPC as an addition to CDT are inconsistent in different studies. This meta-analysis aimed to explore whether IPC could bring additional benefits to CDT. Literatures were retrieved from databases with full-text publications ranging from January 1995 to March 2024. Fixed-effect models were applied to subsequent analysis if no heterogeneity was detected by using the Inverse formula. Publication bias was assessed using the Begg's test and Eagger's test. Twelve studies were finally included for further analysis. Results showed that additional application of IPC to CDT could further improve lymphedema within 4 weeks after the treatment period [standard mean difference (SMD) =-0.2 mL, 95% confidence interval (CI): -0.33 to -0.07 mL]. However, this additional benefit was weakened within about 9.4±2.6 weeks' follow-up duration after ceasing physical therapy (SMD =-0.15 mL, 95% CI: -0.33 to 0.04 mL). Periodically continuous treatment should be suggested to maintain the effect of CDT + IPC to promote lymph drainage and lymphedema improvement. Nonetheless, the treatment involved in the studies ranged from 4 to 12 weeks, therefore potential bias might exist.

Chow forms and complete intersections in the projective space

We prove that every Hodge cohomology class of bidegree (q,q) on a projective manifold X can be recovered from its image by the Chow transformation restricted to a suitable irreducible algebraic component of the space Cq−1(X) of effective algebraic cycles in X of dimension q−1. An application to the problem of the approximation by algebraic cycles is given. In the case of the cohomology class of an effective algebraic cycle, this injectivity at the cohomological level is a consequence of the inversion formula for the Chow transform of a conormal. When X=PN, the inversion formula for the conormal is extended to the case of the conormal of any closed positive (q,q)-current. An inversion formula for the Radon transform, defined on the Grassmannian, of smooth functions is involved and is also used to obtain a characterization of Chow forms of complete intersections in the projective space, expressed by means of the Capelli differential operators.

Coupled 2<i>D</i>-Fractional Wavelet Transform

We introduce a new fractional wavelet transform using the convolution for the coupled fractional Fourier transform. We derive that the fractional wavelet transform satisfies all the expected properties such as Parseval identity, inversion formula, convolution theorem. We also characterize the range of the fractional wavelet transform on . Finally, we establish the uncertainty principle for the fractional wavelet transform.

On the multi‐parameters identification of concrete dams: A novel stochastic inverse approach

AbstractThis paper introduces a novel stochastic inverse method that utilizes perturbation theory and advanced intelligence techniques to solve the multi‐parameter identification problem of concrete dams using displacement field monitoring data. The proposed method considers the uncertainties associated with the dam displacement monitoring data, which are comprised of two distinct sources: the first is related to stochastic mechanical properties of the dam, and the second is due to observation errors. The displacements at different measuring points generated by dam mechanical properties exhibit spatial correlation, while the observation errors at different points can be considered statistically random. In this context, the inversion formulas are derived for unknown stochastic parameters of the dam by combining perturbation equations and Taylor expansion methods. An improved meta‐heuristic optimization method is employed to identify the mean of stochastic parameters, while mathematical and statistical methods are used to determine the variance of stochastic parameters. The feasibility of the proposed method is verified through numerical examples of a typical dam section under different conditions. Additionally, the paper discusses and demonstrates the applicability of this method in a practical dam project. Results indicate that this method can effectively capture the uncertainty of dam's mechanical properties and separates them from observation errors.

Decomposition of the Unitary Representation of $SL_{2}(\mathbb{R})$ on the Upper Half Plane into Irreducible Components

The main purpose of the paper is to find the inversion formula for the covariant transform. This formula is equivalent to the decomposition of the unitary representation of $SL_{2}(\mathbb{R})$ on the upper half plane into irreducible components. We consider an eigenvalue $1+s^{2}$ of the Casimir operator: $$d\rho_{k}(C)=-4v^{2}\left(\partial_{u}^{2}+\partial_{v}^{2}\right),\quad \text{where}\, k=0.$$ To find the inversion formula, first we study the representation of $SL_{2}(\mathbb{R})$, $\rho_{k}$ and $\rho_{\tau}$, induced from the complex characters of $K$ and $N$ respectively. Then, we find the induced covariant transform $\mathcal{W}_{\varphi_{0}}^{\rho_{k}}$ with $N$-eigenvector to obtain a transform in the space $\FSpace{L}{2}(SL_{2}(\mathbb{R}/N)$. Thereafter, we compute the contravariant transform with $K$-eigenvector $$\mathcal{M}_{\phi_{0}}^{\rho_{\tau}}:\FSpace{L}{2}(SL_{2}(\mathbb{R})/N)\rightarrow \FSpace{L}{2}(SL_{2}(\mathbb{R})/K).$$