To analyse the localized behaviour of a signal, the more general ‘windowed Boas transform’ is introduced in this article. Various properties of the windowed Boas transform are discussed, including convolution and correlation. An interplay between windowed Boas and windowed Hilbert transform is given. Also, the inversion formula for the windowed Boas transform is provided. Furthermore, we establish the relation between the windowed Boas transform and the Fourier transform. Lastly, the spectral energy and the energy difference of both the transforms is derived.

Microwave permittivity of limestone and quicklime at high temperatures

We analyze shock thermodynamics of the source-free ten-moment Gaussian closure in one-dimensional variation, focusing on fast shocks. The longitudinal subsystem (ρ,u,Pxx) is algebraically identical to the Euler equations with γ=3, while the transverse degrees of freedom enter through a Schur complement that is advected in smooth flow and transmitted unchanged through fast shocks. For transversely isotropic shear-free states, this structure yields a closed entropy–anisotropy theory: the post-shock anisotropy decomposes exactly into a reversible (isentropic) component and an irreversible enhancement, ΔA(r,Δs;R1)=9R1r2 (e2Δs−1)(R1r2e2Δs+2)(R1r2+2), where r=ρ2/ρ1 is the compression ratio, Δs is the Gaussian entropy jump, and R1=Pxx,1/Pyy,1 is the upstream pressure ratio. The weak-shock conversion factor is κ(R1)=18R1/(R1+2)2, and the strong-shock saturation is ΔA→9/(4R1+2). For the full three-dimensional pressure tensor, we prove that fast shocks transmit the normalized shear vector p/Pxx and the normalized Schur complement C=ρ−1(Π−ppT/Pxx) unchanged, implying detP2=β(r) r2 detP1 and Δs=12 ln (β/r3) independent of shear. Inversion formulas allow Δs to be inferred directly from second moments, providing analytical benchmarks for numerical methods and diagnostics for kinetic simulation data. Numerical verification confirms all predictions.

Truck platoons improve freight efficiency, but their high-speed safety under varying road conditions remains unclear. This study investigates the maximum safe speed of platoons by considering tire force saturation and longitudinal‒lateral coupling. A hierarchical cooperative control framework is proposed, including cooperative control, control allocation, and action execution. A five-degree-of-freedom platoon model with lane-keeping is developed, and a distributed coupled sliding mode controller computes the desired tire forces. Feasible forces are obtained through constrained optimisation within tire limits, while steering angles and torques are derived from an inverse magic formula model. Finite-time convergence and string stability are theoretically guaranteed. Matlab/Simulink–TruckSim co-simulations show the proposed scheme improves platoon safety compared with decoupled methods. Furthermore, the maximum safe speed is quantified under different adhesion and curvature conditions. The results provide insights into the safe operating envelope of platoons and support their integration into intelligent transport systems.

This paper presents a rigorous comparative review of five pivotal integral transforms: Laplace, Sumudu, Elzaki, Aboodh, and the recently introduced RAHMOH transform. We establish a unified theoretical framework to analyze kernel structures, derive the complex inversion formula for the RAHMOH operator, and formally prove its mathematical equivalence to the Laplace transform. Unlike previous surveys, we derive analytical solutions for integer-order ODEs and PDEs, as well as fractional differential equations (FDEs) using all methods, confirming that while they are mathematically isomorphic, they differ significantly in algebraic pathways. Specifically, the analysis identifies the RAHMOH transform as a generalized ''bridge'' operator, encapsulating the scaling properties of Sumudu and the decay properties of Laplace through its dual-variable kernel. Furthermore, numerical simulations via MATLAB validate the consistency of the RAHMOH transform, demonstrating its dimensional stability and accuracy in modeling both dissipative and fractional systems.

ABSTRACT The special affine Fourier transform, a time‐shifted and frequency‐modulated variant of the linear canonical transform, has emerged as a powerful tool in signal processing and optics. In this paper, we investigate the inversion formula of the windowed special affine Fourier transform via Riemann sums. We show that for specific window functions, the Riemann sums are well‐defined on and converge to the target function (or signal) as the sampling step size tends to zero. Utilizing the Poisson summation formula, we further prove the validity of the inversion formula in for all . Additionally, numerical experiments with representative window functions are presented at the end of this paper to verify the effectiveness of the proposed inversion formula.

Accurate correction of radial lens distortion is a fundamental requirement in computer vision and photogrammetry, as geometric inaccuracies directly affect 3D reconstruction, mapping, and geospatial measurements, particularly in high-precision imaging systems. In this study, we propose a fully analytical, non-iterative method for truncated inverse modeling of radial lens distortion, applicable to general radial distortion polynomials that contain constant terms. Unlike classical truncated Lagrange series reversion, which relies on recursive expansion and combinatorial series construction, the proposed formulation determines inverse distortion coefficients directly through a system of constrained algebraic inverse polynomials. This enables deterministic computation of inverse parameters without iterative refinement, numerical root finding, or combinatorial complexity. The method was evaluated using ultra-wide-angle smartphone camera imagery exhibiting severe barrel distortion modeled by an eighth-degree depressed radial distortion polynomial. Its performance was compared with a commonly used iterative inverse modeling approach. The analytical formulation demonstrated improved numerical stability and substantially reduced reprojection errors when correcting highly nonlinear distortion profiles, achieving sub-pixel accuracy in image rectification. In contrast, the iterative approach exhibited instability and significantly larger reprojection errors under identical conditions. These results demonstrate that the proposed framework provides a general, robust, and repeatable solution for inverse radial distortion modeling, particularly for high-order polynomial models. The method offers clear practical advantages for camera calibration pipelines in photogrammetry, remote sensing, robotics, and other applications requiring high-fidelity imaging.

Consideration of the statistical stability of mathematical models of measurement objects, which significantly affects the accuracy of the statistical (probabilistic) estimates used, is considered. This problem is relevant because there is no effective criterion for distinguishing deterministic and random sequences. Accounting is based on a two-level probability distribution density in the form of a conversion formula. The numerator of the formula characterizes the errors of the systematic component of the hypothetical probability distribution as the observed deviations from it of the statistical distribution of measurement data, and the denominator is the unobservable components of the random component. The convergence of a number of repeated measurements is a necessary condition for the correctness of the statistical (probabilistic) estimates obtained. The formulation of such a problem is possible due to the fact that convolution in the form of an inversion formula is considered as the distribution of the sum of two random terms, when the second term characterizes the statistical stability of the first term. The direct solution of the problem by functional transformations of probability distributions leads to very cumbersome results. Within the framework of the interpolation concept, A. N. Kolmogorov's axiomatic approach is adopted, in which probability is represented by a positive real random variable characterized by a probability distribution function, which is a second-order distribution. It is established that the indicator of statistical stability of a mathematical model, the probability of agreement with the data of joint measurements, or kappa, is a reproducibility criterion, summarizes the statistics of the criteria of agreement of A. N. Kolmogorov and N. V. Smirnov – the distances between the distribution functions, and contains the distance according to the variation of V. Feller. The growth trend in the statistics of the probability of agreement with an increase in the sample size for probability distributions directly characterizes the degree of statistical stability of measurement data and the reliability of statistical inference logic in measuring problems of identifying probability distributions, and also complements confi dence probability as a characteristic of the quality of mathematical models of measurement objects.

We propose a novel generalized wavelet transform termed the offset linear canonical wavelet transform (OLCWT) that combines the multiresolution capabilities of the classical wavelet transform with the time-frequency localization power of the offset linear canonical transform (OLCT). The OLCWT inherits the desirable properties of both transforms, offering a new perspective in the analysis of non-stationary signals with localized structures in the offset linear canonical domain. A key feature of our construction is the use of a simplified yet effective convolution structure associated with the OLCT, which reduces computational complexity and enhances analytical tractability. We rigorously establish the fundamental properties of OLCWT including the inner product relation, admissibility condition, and inversion formula. The composition of two OLCWTs is studied, and discrete variants (semi-discrete and fully discrete) are also developed. Examples and graphical interpretations are provided to illustrate the theoretical findings.

An improved pulsed neutron method for gas reservoirs identification in the complex buried hill formation of the Bohai Sea

We give conditions for a locally finite join-semilattice $P$ to have the property that for any functions $f:P\to \mathbb{C}$ and $g:P\to \mathbb{C}$ not identically zero and linked by the Möbius inversion formula, the support of at least one of $f$ and $g$ is infinite. This generalises and gives an entirely poset-theoretic proof of a result of Pollack. Various examples and nonexamples are discussed.

This paper introduces a collocation algorithm for numerically solving the third-order Gilson–Pickering equation (GPE) and the classical Rosenau–Hyman equation (RHE). We employ newly developed shifted Pell polynomials as basis functions. Novel formulas for these polynomials are devised and utilized in constructing the proposed algorithm. Specifically, we establish a new power form and its inversion formula, along with an explicit formula for derivatives of the shifted Pell polynomials, from which the operational matrices of derivatives (OMDs) are derived. These matrices facilitate the conversion of nonlinear dispersive models into systems of algebraic equations, efficiently solved using Newton’s iterative technique. The error analysis of the shifted Pell expansion is discussed in depth. Several numerical examples, including the RHE, its fourth-order variant, and the Fornberg–Whitham equation, are provided to demonstrate the method’s performance and accuracy. Comparative results are also reported.

We consider special upwinding Petrov-Galerkin discretizations for convection-diffusion problems. For the one dimensional case with a standard continuous linear element as the trial space and a special exponential bubble test space, we prove that the Green function associated to the continuous solution can generate the test space. In this case, we find a formula for the exact inverse of the discretization matrix, that is used for establishing new error estimates for other bubble upwinding Petrov-Galerkin discretizations. We introduce a quadratic bubble upwinding method with a special scaling parameter that provides optimal approximation order for the solution in the discrete infinity norm while avoiding exponential test functions. Provided the linear interpolant has standard approximation properties, we prove optimal approximation estimates in standard L 2 and H 1 norms. The quadratic bubble method is extended to a two dimensional convection diffusion problem. The proposed discretization produces optimal L 2 and H 1 convergence orders on subdomains that avoid the boundary layers. The tensor idea of using an efficient upwinding Petrov-Galerkin discretization along each stream line direction in combination with a standard discretizations for the orthogonal direction(s) can lead to efficient new discretization methods for multidimensional convection dominated models.

In this paper, we construct and analyse Bessel and Flett potentials associated with the heat and Poisson semigroups in the framework of the ( k , 1 ) -generalized Fourier transform. We establish fundamental properties of these potentials and derive an explicit inversion formula for the Flett potential using a wavelet-like transform. Furthermore, we introduce a β-semigroup B k ( β , t ) , which enables the formulation of an inversion formula for the Riesz potential. As a unifying extension, we define and investigate bi-parametric potentials J k ( α , β ) , which generalize both the Bessel potential and the Flett potential. In addition, we define the associated function spaces.

In this article, we present new theoretical findings on specific polynomials that generalize the concept of telephone numbers, namely, Telephone polynomials (TelPs). Several new formulas are developed, including expressions for higher-order derivatives, repeated integrals, and moment formulas of TelPs. Moreover, we derive explicit connections between the derivatives of TelPs and the two classes of symmetric and non-symmetric polynomials, producing many formulas between these polynomials and several celebrated polynomials such as Hermite, Laguerre, Jacobi, Fibonacci, Lucas, Bernoulli, and Euler polynomials. The inverse formulas are also obtained, expressing the derivatives of well-known polynomial families in terms of TelPs. Furthermore, some novel linearization formulas (LFs) with some classes of polynomials are established. Finally, some new definite and indefinite integrals of TelPs are established using some of the developed relations.

Due to the high frequency of fifth-generation (5G) signals, which leads to an extremely large computational scale when traditional algorithms solve the channel path loss, it is necessary to seek a fast solution method for the 5G channel path loss in substations in order to achieve a fast adaptation of the channel to the signal receiver and to ensure the reception quality of the 5G signals. Aiming at the problem that traditional algorithms suffer from extremely high computational complexity when dealing with the dyadic reflection–diffraction coefficients, a method based on singular value decomposition is proposed to reduce the dimensionality of the channel matrix for solution. Firstly, the ray tube model is used to divide the channel, and the incident angle information within the channel matrix is chunked through the nodes to discard the duplicates and those that contribute very little to the channel path loss. Then, matrix dimensionality reduction is achieved by the singular value decomposition algorithm, and the dimensionality-reduced channel matrix is substituted into the sum-vector inverse wrap-around coefficient solution formula to achieve the fast solution of 5G channel path loss. Finally, a comparison of the computational results of the proposed algorithm with those of the traditional algorithm is carried out by taking the 5G base station antenna of AAU5270E as an example and using the computational results of the experimental measurement data as a benchmark. The results show that the accuracy loss of the method proposed in the paper is only 1.31%, the compression of the data is 84.89, and the order of magnitude of the computation is 105 lower than that of the traditional algorithm. Future research could further integrate real-time channel data to achieve dynamic adaptive optimization, while extending this dimensionality reduction framework to high-dimensional complex channel modeling such as the sixth generation (6G), thereby promoting the continuous development of the algorithm in terms of real-time performance and generalization capability.

In this article, we derive a unified inversion formula for the k-dimensional totally geodesic Radon transform on simply connected spaces of constant curvature, namely R n , H n , and S n . The unification of the formulae is motivated from the observation that the geometries of constant curvature spaces are interlinked. To obtain the unified inversion formula, we use wavelet-like transforms.

Abstract Following J. Radon’s 1917 celebrated paper, F. John has tried to solve Radon’s problem on the set of all circles of fixed radius a in R 2 . But he rapidly realized that the task was really challenging. Much later returning to this problem in R 3 , he succeeded, using a special solution of a Cauchy problem for the wave equation, to establish an inversion formula under the form of an infinite series of ‘partial’ function reconstructions, each one of them expressible in terms of spherical means of the Radon data on spheres of increasing radius ( 2 m + 1 ) a with m = 0 , 1 , 2 , … . The aim of this paper is to provide a proof of the validity of F. John’s result by reproducing his remarkable inversion formula via a calculation of the formal inversion kernel of this transform based on the polar expansion of 1 / sin ⁡ z . We then show that a Fourier approach to the previous Cauchy problem for the R 3 wave equation is the logical framework for finding F. John’s special wave function as well as for establishing the corresponding inversion scheme. As no solution to this problem in R 2 pre-exists, we move to compute the formal inversion kernel of this transform with the pole expansion of 1 / J 0 ( z ) , where J 0 ( z ) is a Bessel function to obtain a new similar series type inversion formula. Interestingly this result emerges naturally from the Fourier approach of the Darboux equation in R 2 , instead of the wave equation.

ABSTRACT To achieve an efficient time‐frequency representation of higher‐dimensional signals, we introduce the notion of the Clifford‐valued bendlet transform, utilizing the geometric and algebraic properties intrinsic to bendlets and Clifford algebras. The bendlet system, a second‐order shearlet with bent elements, enables the precise characterization of the curvature of discontinuities. Clifford algebras, meanwhile, convert geometric objects into fundamental computational elements and define universal operators applicable to all types of geometric elements. In this work, we rigorously investigate the essential properties of the proposed transform using operator theory and Clifford‐valued Fourier transforms. We present a detailed analysis of the convergence of inversion formulae and the boundedness of the associated localization operators. Furthermore, we thoroughly explore certain classes of uncertainty principles for the Clifford‐valued bendlet transform. Our findings establish a robust theoretical foundation for the practical application of this transform in multidimensional signal processing.

Abstract This work studies two Compton cameras with linear and circular detectors and includes the effect of arbitrary attenuation on the model. To this end, we introduce two operators that consist of attenuated integrals over V-lines with a swinging symmetry axis and a fixed opening angle. The effect of attenuation is modeled by negative exponential weights that depend on general attenuation maps. We establish some useful equivalences between these V-line integrals and the attenuated Radon transform on straight lines. With the attenuation map known, these equivalences enable analytic reconstruction. We propose an approach built upon Natterer’s inversion formula. We perform numerical simulations to illustrate the efficiency of the proposed reconstruction framework.