Interpolation Formula Research Articles

Lipschitz closed injective hull ideals and Lipschitz interpolative ideals

In this paper we present two constructions: the Lipschitz closed injective hull ideals and the Lipschitz interpolative ideals of a Lipschitz operator ideal. These procedures aim to construct new Lipschitz operator ideals between pointed metric spaces and Banach spaces. The new ideals are characterized by specific criteria that determine whether a Lipschitz operator belongs to them, using summability properties and interpolation formulas.

Determination of the mean duration of recent infection and false recency rate for the HIV triplex multiplex bead assay.

We developed the HIV Triplex multiplex bead assay to identify and serotype HIV infection with high sensitivity and specificity; and distinguish recent from long-term HIV-1 infections. It can facilitate accurate incidence estimation, while reducing the number of tests and blood collected, which is highly desirable for use in future studies and surveys. Using previously collected, treatment-naive longitudinal seroconversion HIV-1 positive panels and specimens from individuals infected for >12 months, we determined the assay's mean duration of recent infection (MDRI) and false-recency rate (FRR) respectively, at various mean fluorescent intensity (MFI) cutoffs. We tested seroconversion specimens (N = 814) from 142 individuals infected with HIV-1 subtypes B, C, or AE, and 1341 cross-sectional specimens from individuals infected >12 months. The MFI cutoffs of 1000 to 2000 were evaluated for recency classification, including an MFI of 1250 corresponding to the limiting antigen avidity enzyme immunoassay (LAg-EIA) cutoff of 1.5 normalized optical density for MDRI and FRR. We used four statistical methods: Methods 1 and 2 used the empirically balanced observation time approach. Method 2 MFI values were raised to power = 1.33, based on a repeated measures model to linearize the relationship between MFI and time points, whereas Method 1 was not linearized. Methods 3 and 4 employed quadratic and linear interpolations for each seroconversion panel. FRR was calculated by dividing the number of specimens misclassified as recent by the total number of specimens tested. MDRI values ranged from 135-146 days at MFI = 1000 to 229-279 days at MFI = 2000 by the 4 methods. FRR varied from 0.15%-1.27% with increasing MFI cutoff. At MFI = 1250, the average MDRI of 4 methods was 169 days and ranged from 159-183 with overlapping 95% CIs and FRR = 0.52%. The HIV Triplex assay demonstrates a longer dynamic range compared to current HIV recency assays with a low FRR for cutoffs examined. With a longer dynamic range and low FRR, the MDRI for recent infection can be extended as appropriate to detect more recent infections, increasing the value of incidence assays benefiting public health surveillance and future surveys.

3D Controlled‐source electromagnetic modelling in anisotropic media using secondary potentials and a cascadic multigrid solver

AbstractQuantitative interpretation of the data from controlled‐source electromagnetic methods, whether via forward modelling or inversion, requires solving a considerable number of forward problems, and multigrid methods are often employed to accelerate the solving process. In this study, a new extrapolation cascadic multigrid method is employed to solve the large sparse complex linear system arising from the finite element approximation of Maxwell's equations using secondary potentials. The equations using secondary potentials are discretized by the classic nodal finite element method on nonuniform rectilinear grids. The resulting linear systems are solved by the extrapolation cascadic multigrid method with a new prolongation operator and preconditioned Stabilized bi‐conjugate gradient method smoother. High‐order interpolation and global extrapolation formulas are utilized to construct the multigrid prolongation operator. The extrapolation cascadic multigrid method with the new prolongation operator is easier to implement and more flexible in application than the original one. Finally, several synthetic examples including layered models, models with anisotropic anomalous bodies or layers, are used to validate the accuracy and efficiency of the proposed method. Numerical results show that the extrapolation cascadic multigrid method improves the efficiency of 3D controlled‐source electromagnetic forward modelling a lot, compared with traditional iterative solvers and some state‐of‐the‐art methods or software (e.g., preconditioned flexible generalized minimal residual method, emg3d) in the considered models and grid settings. The efficiency benefit is more evident as the number of unknowns increases, and the proposed method is more efficient at low frequencies. The extrapolation cascadic multigrid method can also be used to solve systems of equations arising from related applications, such as induction logging, airborne electromagnetic, etc.

Bernstein-Riemann interpolation formula for arbitrary continuous functions on a segment

Bernstein-Riemann interpolation formula for arbitrary continuous functions on a segment

Influence of specularity factor on heat transport in nanoribbons of different sizes

In this paper, the phonon heat transfer in two-dimensional conductors with different types of phonon reflection from the boundaries is examined. Heat conductors with arbitrary ratios of width W to length L are studied assuming that the mean free path between phonon-phonon collisions is infinite. The integral equations for the angular distribution functions of the incident and reflected phonons at a specific point on the conductor boundary were proposed. To solve these integral equations a new iterative method is proposed. The proposed iterative approach formally corresponds to taking into account subsequent collisions of phonons with the edges of the conductor. It ensures the convergence of the desired solution for any W/L ratio and for any value of the specularity coefficient p. Interpolation formulae are found to describe with sufficient accuracy the solution of the system of integral equations in the entire region of the specularity coefficient p from 0 to 1, and the W/L ratios ranging from 10 to 0.01. These formulae allow the construction of the isolines of the thermal conductance coefficient values, from which it is possible to determine the necessary values of W/L and parameter p to get the desired value of the thermal conductance.

Derivative Sampling of Periodic and Nonperiodic Band-Limited Signals — Fourier Spectrum and Interpolation Formula

Derivative Sampling of Periodic and Nonperiodic Band-Limited Signals — Fourier Spectrum and Interpolation Formula

Geodesic metrics for RBF approximation of some physical quantities measured on sphere

The radial basis function (RBF) approximation is a rapidly developing field of mathematics. In the paper, we are concerned with the measurement of scalar physical quantities at nodes on sphere in the 3D Euclidean space and the spherical RBF interpolation of the data acquired. We employ a multiquadric as the radial basis function and the corresponding trend is a polynomial of degree 2 considered in Cartesian coordinates. Attention is paid to geodesic metrics that define the distance of two points on a sphere. The choice of a particular geodesic metric function is an important part of the construction of interpolation formula.We show the existence of an interpolation formula of the type considered. The approximation formulas of this type can be useful in the interpretation of measurements of various physical quantities. We present an example concerned with the sampling of anisotropy of magnetic susceptibility having extensive applications in geosciences and demonstrate the advantages and drawbacks of the formulas chosen, in particular the strong dependence of interpolation results on condition number of the matrix of the system considered and on round-off errors in general.

Analytical solution of two non‐identical edge cracks in an infinite strip under anti‐plane shear wave

AbstractThis article presents an extensive analytical solution addressing the interaction between two non‐identical edge cracks in an infinite orthotropic strip under anti‐plane shear waves. Most studies assume identical cracks or single edge crack in a strip, but this research breaks new ground by considering cracks of different sizes. By incorporating mixed‐type boundary conditions, the study derives dual integral equations. These equations are then transformed into a singular integral equation of Cauchy type with the aid of a trial solution and contour integration technique. The singular integral equation is further converted into a system of integral equations, which are solved numerically utilizing Jacobi polynomials. The obtained solutions are utilized to derive expressions for the stress intensity factor (SIF) and crack opening displacement (COD) at the crack tip using Krenk's interpolation formulae. The derived results are presented graphically and compared against existing solutions for single edge crack and symmetric edge cracks in static scenario.

Comment to the article “Mean flow induced by longitudinal libration of a fluid-filled rotating container bounded by two conical surfaces”

Developing the ideas from the author’s previous publication in the European Journal of Mechanics / B Fluids (https://doi.org/10.1016/j.euromechflu.2024.02.008), an interpolation formula is proposed for the angular velocity of mean retrograde flow in a fluid-filled oblate spheroid with arbitrary eccentricity, in the limit of very low longitudinal libration frequencies.

Quantum Cryptography for Enhanced Data Security: A Comparative Survey of Data Encryption and Decryption of Error Correction

Data information security is a crucial concern that ought to be managed to help protect vital data. Cryptography is one of the conventional approaches for securing data and is generally considered a fundamental data security component that provides privacy, integrity, confidentiality, and authentication. In this paper, a comparative survey of Rivest-Shamir-Adleman (RSA) data security algorithm is proposed in order to compare their security strength. In doing so, we integrate traditional RSA and Gaussian Interpolation formulas to test their security level. The integration raised the security strength of RSA to the fifth degree,while the Gaussian First Forward Interpolation was used to encrypt the ASCII values of the message after which the traditional RSA was used to encrypt and decrypt the message in the second and third levels. Comparative data security analysis was performed using four different algorithms; RSA, SRNN, 2-Key pair algorithms, and the results showed that when the data size was small, the encryption and decryption times were lower for the proposed algorithm but higher when the data size was big. Thus the two public keys used and some mathematical relations made the eavesdropper not to get much knowledge about the key and therefore, unable to decrypt the message.

Two‐dimensional adaptive Whittaker–Shannon Sinc‐based zooming

AbstractIn this work, we introduce a novel image zooming methodology that transitions from a nonadaptive Sin‐based approach to an adaptive Sinc‐based zooming technique. The two techniques base their theoretical foundation on the Whittaker–Shannon interpolation formula and the Nyquist theorem. The evolution into adaptive Sinc‐based zoom is accomplished through the use of two novel concepts: (1) the pixel‐local scaled k‐space and (2) the k‐space filtering sigmoidal function. The pixel‐local scaled k‐space is the standardized and scaled k‐space magnitude of the image to zoom. The k‐space filtering sigmoidal function scales the pixel‐local scaled k‐space values into the numerical interval [0, 1]. Using these two novel concepts, the Whittaker–Shannon interpolation formula is elaborated and used to zoom images. Zooming is determined by the shape of the Sinc functions in the Whittaker–Shannon interpolation formula, which, in turn, depends on the combined effect of the pixel‐local scaled k‐space, the sampling rate, and the k‐space filtering sigmoidal function. The primary outcome of this research demonstrates that the Whittaker–Shannon interpolation formula can achieve successful zooms for values of the sampling rate significantly greater than the bandwidth. Conversely, when the sampling rate is much greater than the bandwidth, the nonadaptive technique fails to perform the zoom correctly. The conclusion is that the k‐space filtering sigmoidal function is identified as the crucial parameter in the adaptive Sinc‐based zoom technique. The implications of this research extend to Sinc‐based image zooming applications.

Bernstein–Riemann Interpolation Formula for Arbitrary Continuous Functions on an Interval

Bernstein–Riemann Interpolation Formula for Arbitrary Continuous Functions on an Interval

A SBFEM formula for the mixed-order hexahedron interpolation based on serendipity elements

The hexahedron elements have been widely used in theoretical research and engineering applications because of their simple formulation and fine analytical performance. In this paper, a flexible mixed-order hexahedron interpolation is proposed based on the SBFEM theory, which is summarized as follows: (1) The interpolation functions are constructed for boundary surfaces by introducing the “Serendipity element”, which allows for a combination of linear and quadratic interpolation; (2) Two approaches for order conversion in hexahedron elements have been applied and investigated; (3) The accuracy and applicability of the proposed method are verified using several classical examples and engineering practice. In this way, the proposed method is capable of handling abundant patterns of different order combinations, thus the analysis performance of hexahedron elements has been improved. The precision of the proposed method is demonstrated by comparing it with the theoretical solution and the classical isoparametric FEM. By comparison, the computational efficiency can be improved by approximately 40–55 % with satisfactory accuracy. In general, the proposed method offers a new alternative channel for simulating particular problems, such as bending and stress concentration.

An efficient numerical method based on ultraspherical polynomials for linear weakly singular fractional Volterra integro‐differential equations

The linear weakly singular fractional Volterra integro‐differential equations involving the Caputo derivative have solutions whose derivatives are unbounded at the left endpoint of the integration interval. In this paper, we use suitable transformations to prevail on this nonsmooth behavior. We used the product integration method based on the new fractional basis function to solve these equations, which led to the production of the new interpolation formula and weights for the method. We investigate the convergence of the presented method. The proposed scheme is employed to solve some numerical examples to test its efficiency and accuracy.

The discrete analogue of high-order differential operator and its application to finding coefficients of optimal quadrature formulas

The discrete analog of the differential operator plays a significant role in constructing interpolation, quadrature, and cubature formulas. In this work, we consider a discrete analog Dm(hβ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$D_{m}(h\\beta )$\\end{document} of the differential operator d2mdx2m+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\frac{d^{2m}}{dx^{2m}}+1$\\end{document} designed specifically for even natural numbers m. The operator’s effectiveness in constructing an optimal quadrature formula in the L2(2,0)(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L_{2}^{(2,0)}(0,1)$\\end{document} space is demonstrated. The errors of the optimal quadrature formula in the W2(2,1)(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$W_{2}^{(2,1)}(0,1)$\\end{document} space and in the L2(2,0)(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L_{2}^{(2,0)}(0,1)$\\end{document} space are compared numerically. The numerical results indicate that the optimal quadrature formula constructed in this work has a smaller error than the one constructed in the W2(2,1)(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$W_{2}^{(2,1)}(0,1)$\\end{document} space.

Simple and General Unitarity Conserving Numerical Real-Time Propagators of the Time-Dependent Schrödinger Equation Based on Magnus Expansion.

Magnus expansion (ME) provides a general way to expand the real-time propagator of a time-dependent Hamiltonian within the exponential such that the unitarity is satisfied at any order. We use this property and explicit integration of Lagrange interpolation formulas for the time-dependent Hamiltonian within each time interval and derive approximations that preserve unitarity for the differential time evolution operators of general time-dependent Hamiltonians. The resulting second-order approximation is the same as using the average of Hamiltonians for two end points of time. We identify three fourth-order approximations involving commutators of Hamiltonians at different times and also derive a sixth-order expression. A test of these approximations along with other available expressions for a two-state time-dependent Hamiltonian with sinusoidal time dependences provides information on the relative performance of these approximations and suggests that the derived expressions can serve as useful numerical tools for time evolution in time-resolved spectroscopy, quantum control, quantum sensing, real-time ab initio quantum dynamics, and open system quantum dynamics.

An Efficient Procedure to Compensate for the Errors Due to the Probe Mispositioning in a Cylindrical Near-Field Facility.

This paper deals with the compensation of the probe mispositioning errors occurring in a cylindrical near-field (NF) facility due to the imprecise control of the linear and azimuthal positioners allowing the cylindrical scanning and/or to their limited resolution and to defects in the rails guiding the linear motion. As a result, 3-D errors in the positioning of the probe at any sampling point, as prescribed by the adopted non-redundant representation, affect the accuracy of the NF measurements. An efficient procedure is here proposed to properly compensate for these errors. It involves two steps. The former allows one to correct the mispositioning errors due to the deviation of each actual sampling point from the nominal measurement cylinder. The latter makes use of an iterative technique to restore the NF samples at any sampling point fixed by the used non-redundant representation from the ones obtained at the previous step and affected by 2-D mispositioning errors. Once these steps have been fruitfully applied, the so-compensated NF samples are effectively interpolated through a 2-D optimal sampling interpolation (OSI) formula to accurately reconstruct the input data required to perform the traditional cylindrical near-to-far-field transformation. The OSI representation is here developed by considering an elongated antenna under test as enclosed either in a prolate spheroid or in a cylinder terminated by two half spheres (rounded cylinder) in order to make the representation effectively non-redundant. Numerical test results, which thoroughly prove the efficacy of the devised procedure in correcting even severe 3-D mispositioning errors, are reported.

New discretization of [formula omitted]-Caputo fractional derivative and applications

In the present paper, two approximations to evaluate the ψ-Caputo fractional derivative are developed using the linear and the quadratic polynomial interpolations. We present a study of the pointwise error for each approximation and illustrate some particular cases that correspond to approximations of the well known fractional derivatives, such as: Caputo, Katugampola and Hadamard fractional derivatives. In order to elucidate the investigated results, we present some examples for each approximation. For concreteness, we show some applications where we solve initial value problems and problems involving fractional sub-diffusion equations. Finally, some concluding remark are presented.

CONCEPT OF DESHA IN AYURVEDIC CLASSICAL TEXTS: AN EFFORTS TO GEOGRAPHICAL MAPPING USING MODERN TECHNIQUES AND DATA

Objectives: This study aims to map each census district of India using the concept and classification of Desha available in classical Ayurvedic text using modern techniques and tools. Method: Rainfall, as well as Potential Evapotranspiration data obtained from WRIS (IMD Grid), was mapped using GIS software, and the district wise average was calculated for the value of the Raster layer obtained by IWD Interpolation and Moisture Index with the formulae of Thornthwaite and Mather and simplified by Venkat Raman & Krishnan was calculated. Every district was assigned to a particular class based upon the analogy of Aridity they had using a classification proposed by Reddy & Reddy in 1973. Result & Conclusions: The overall average Moisture Index for India is -3.43, which placed the country into the class of Anupa Sadharan Desha. At the same time, it has been observed that approximately 59% of the total geographic areas fall under the class of Jangal and Jangal Sadharan Desha. The western coastal area, north-eastern states, Suderban and Kaveri Delta, as well as the Himalayan Biosphere region, form part of Anupa Desha, while Rajasthan to Kutchh falls under the category of Jangal Desha.

Virial coefficients of the uniform electron gas from path-integral Monte Carlo simulations.

The properties of plasmas in the low-density limit are described by virial expansions. Analytical expressions are known from Green's function approaches only for the first three virial coefficients. Accurate path-integral Monte Carlo (PIMC) simulations have recently been performed for the uniform electron gas, allowing the virial expansions to be analyzed and interpolation formulas to be derived. The exact expression for the second virial coefficient is used to test the accuracy of the PIMC simulations and the range of validity of the interpolation formula of Groth etal. [Phys. Rev. Lett. 119, 135001 (2017)0031-900710.1103/PhysRevLett.119.135001], and we discuss the fourth virial coefficient, which is not exactly known yet. Combining PIMC simulations with benchmarks from exact virial expansion results would allow us to obtain more accurate representations of the equationof state for parameter ranges of conditions which are of interest, e.g., for helioseismology.