Birkhoff Interpolation Research Articles

A global progressive image secret sharing scheme under multi-group joint management.

Diverging from traditional secret sharing schemes, group secret sharing schemes enable the recovery of secret information through collaborative efforts among groups. Existing schemes seldom consider the issue of the secrecy level of image information between different groups. Therefore, we propose a global progressive image secret sharing scheme under multi-group joint management. For inter-group relations, multiple groups with different priority levels are constructed using the approach of bit-polar decomposition. In this arrangement, higher-level groups obtain clearer secret image information. For intra-group relations, a participant-weighted secret sharing scheme is constructed based on Chinese Remainder Theorem and Birkhoff interpolation, in which the participants' secret sub-shares are reusable. During the recovery process, the sub-images can be recovered within the intragroup with the corresponding level. Groups collaborate through lightweight overlay operations to obtain different layers of secret images, achieving a global progressive effect. Analysis results show that the scheme is both secure and practical for group secret sharing.

Real-Time Virtual Sensing for Dynamic Vibration of Flexible Structure via Fiber Bragg Grating Sensors

Real-time monitoring of the dynamic vibrations of beam-type structures indicates important implications in aerospace engineering applications, in which the additional sensing information about vibration parameters, including displacements, rotational angles, angular rates, as well as accelerations, facilitates the improvement of the system’s safety. This article proposed a virtual sensing method with the usage of fiber Bragg grating (FBG) strain sensors to directly estimate the vibration parameters from the measured strains. The algebraic relationships of displacement–strain and rotation–strain on the sensor’s installing points as well as the boundary points are obtained based on the Birkhoff interpolation method, and meanwhile, the explicit expressions on angular rates and accelerations are derived. Furthermore, a tracking differentiator is used to recover the strain signals and their first- and second-order derivatives from the noisy measurements in order to real-time estimate the angular rates and accelerations of the vibrating beam. Differing from the classical modal-based method, complex modal decomposition is not required in the proposed method. The comparison of numerical simulation results reveals that the proposed method has advantages over the modal-based method in accuracy and efficiency. Furthermore, the laboratory experiments indicate that the proposed method is effective for real-time estimating vibration parameters from the noisy measured strains.

Optimal Birkhoff Interpolation and Birkhoff Numbers in Some Function Spaces

Optimal Birkhoff Interpolation and Birkhoff Numbers in Some Function Spaces

A new efficient hierarchical multi-secret sharing scheme based on linear homogeneous recurrence relations

Hierarchical secret sharing is an important key management technique since it is specially customized for hierarchical organizations with different departments allocated with different privileges, such as the government agencies or companies. Hierarchical access structures have been widely adopted in secret sharing schemes, where efficiency is the primary consideration for various applications. How to design an efficient hierarchical secret sharing scheme is an important issue. A famous hierarchical secret sharing (HSS) scheme was proposed by Tassa based on Birkhoff interpolation. Later, based on the same method, many other HSS schemes were proposed. However, these schemes all depend on Polya’s condition, which is a necessary condition, not a sufficient condition. It cannot guarantee that Tassa’s HSS scheme always exists. Furthermore, this condition needs to check the non-singularity of many matrices.We propose a hierarchical multi-secret sharing scheme based on the linear homogeneous recurrence (LHR) relations and the one-way function. In our scheme, we select m linearly independent homogeneous recurrence relations. The participants in the highly-ranked subsets γ1,γ2,⋯,γj-1 join in the jth subset to construct the jth LHR relation. In addition, the proposed hierarchical multi-secret sharing scheme just requires one share for each participant. Besides, our scheme is both perfect and ideal. Furthermore, our scheme avoids many checks of the non-singularity of many matrices in the presented hierarchical secret sharing schemes. Although we need to publish more public values, our scheme reduces the computational complexity of the hierarchical secret sharing schemes from exponential time to polynomial time, i.e., O(nkm-1logn), which is relatively more efficient than schemes in the literature.

Hermite–Birkhoff Interpolation Polynomial of Minimum Norm in Hilbert Space

Hermite–Birkhoff Interpolation Polynomial of Minimum Norm in Hilbert Space

Solving a Birkhoff Interpolation Problem for (0,1,5) data

Abstract: The following research work deals with a special type of Birkhoff interpolation problem in which we have 3 set of data prescribed in the unit interval I = [0, 1]. Data values are the function value, first derivatives and fifth derivatives prescribed at nodes of the unit interval. We obtained a unique spline interpolating the given data along with the convergence problem.

New fractional pseudospectral methods with accurate convergence rates for fractional differential equations

The main purpose of this paper is to introduce generalized fractional pseudospectral integration and differentiation matrices using a family of fractional interpolants, called fractional Lagrange interpolants. We develop novel approaches to the numerical solution of fractional differential equations with a singular behavior at an end-point. To achieve this goal, we present efficient and stable methods based on three-term recurrence relations, generalized barycentric representations, and Jacobi-Gauss quadrature rules to evaluate the corresponding matrices. In a special case, we prove the equivalence of the proposed fractional pseudospectral methods using a suitable fractional Birkhoff interpolation problem. In fact, the fractional integration matrix yields the stable inverse of the fractional differentiation matrix, and the resulting system is well-conditioned. We develop efficient implementation procedures for providing optimal error estimates with accurate convergence rates for the interpolation operators and the proposed schemes in the $L^{2}$-norm. Some numerical results are given to illustrate the accuracy and performance of the algorithms and the convergence rates.

Fair hierarchical secret sharing scheme based on smart contract

Secret sharing has a number of practical applications in network-based scenarios, such as key transfer protocols, attribute-based encryption and secure multiparty computation. However, existing secret sharing schemes cannot efficiently achieve fairness. They either rely on a trusted third party or require multiple rounds of communication. In this paper, we propose the first decentralized and fair hierarchical threshold secret sharing (HTSS) scheme using blockchain. In the scheme, secret shares are distributed to different levels of parties, and any authorized subset of parties can obtain the secret. We leverage a smart contract to force all participants to commit to the secret shares; otherwise, the committer either reveals his secret share within a certain time frame or pays a fine. Thus, unlike previous HTSS schemes, the participants can reconstruct the secret fairly using Birkhoff interpolation without a trusted party and complete the computation in one round. We formally prove that our scheme is secure. We evaluate the performance of the scheme by implementing our scheme on Ethereum’s official test network. Our experiments show that our scheme can run reasonably fast and is practical.

Simultaneous approximation of Birkhoff interpolation and the associated sharp inequalities

This paper gives a kind of sharp simultaneous approximation error estimation of Birkhoff interpolation [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] is the Birkhoff interpolation based on [Formula: see text] pairs of numbers [Formula: see text] with its P[Formula: see text]lya interpolation matrix to be regular. First, based on the integral remainder formula of Birkhoff interpolation, we refer the computation of [Formula: see text] to the norm of an integral operator. Second, we refer the values of [Formula: see text] and [Formula: see text] to two explicit integral expressions and the value of [Formula: see text] to the computation of the maximum eigenvalue of a Hilbert–Schmidt operator. At the same time, we give the corresponding sharp Wirtinger inequality [Formula: see text] and sharp Picone inequality [Formula: see text].

TOMIC FUNCTIONS AND LACUNARY INTERPOLATION SERIES IN BOUNDARY VALUE PROBLEMS FOR PARTIAL DERIVATIVES EQUATIONS AND IMAGE PROCESSING

In the paper we consider and solve the problem of construction of the so called tomic functions – the systems of infinitely differentiable functions which while retaining many important properties of the shifts of atomic function up(x) such as locality and representation of algebraic polynomials and being based on the atomic functions nevertheless have nonuniform character and therefore allow to take into account the inhomogeneous and changing character of the data encountered in real world problems in particular in boundary value problems for partial differential equations with variable coefficients and complex geometry of domains in which these boundary value problems must be solved. The same class of tomic functions can be applied to processing,denoising and sparse storage of signals and images by lacunary interpolation. The lacunary or Birkhoff interpolation of functions in which the function is being restored by the values of derivatives of orderin points in which values of function and derivatives of order k<r are unknown is of great importance in many real world problems such as remote sensing. The lacunary interpolation methods using the tomic functions possesss important advantages over currently widely applied lacunary spline interpolation in view of infinite smoothness of tomic functions.The tomic functions can also be applied to connect (to stitch) atomic expansions with different steps on different intervals preserving smoothness and optimal approximation properties. The equations for of construction oftomic functions tofuj(x) –analogues of the basic functions of the generalized atomic Taylor expansions are obtained – which are needed for lacunary (Birkhoff) interpolation. For the applications in variational and collocation methods for solving bondary value problems for partial derivative and integral equations the tomic functions ftupr,j(x) are obtained that are analogues of B-splines and atomic functions fupn(x). Using similar methods, the tomic functions based on other atomic functions such as Ξn(x) can be obtained.

A kind of sharp Wirtinger inequality

In this study, we give a kind of sharp Wirtinger inequality \t\t\t∥f∥p≤Cr,p,q∥f(r)∥qfor all 1≤p,q≤∞,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\Vert f \\Vert _{p}\\le C_{r,p,q} \\bigl\\Vert f^{(r)} \\bigr\\Vert _{q} \\quad \\text{for all } 1\\le p,q\\le \\infty , $$\\end{document} where f is defined on [0,1] and satisfies f^{(k_{1})}(0)=f^{(k _{2})}(0)=cdots =f^{(k_{s})}(0)=f^{(m_{s+1})}(1)=cdots =f^{(m_{r})}(1)=0 with 0le k_{1}< k_{2}<cdots <k_{s}le r-1 and 0le m_{s+1}< m_{s+2}< cdots <m_{r}le r-1. First, based on the Birkhoff interpolation, we refer the computation of C_{r,p,q} to the norm of an integral-type operator. Second, we refer the values of C_{r,1,1} and C_{r,infty ,infty } to explicit integral expressions and the value of C_{r,2,2} to the computation of the maximal eigenvalue of a Hilbert–Schmidt operator. Finally, we give three examples to show our method.

The Role of Hilbert–Schmidt SVD basis in Hermite–Birkhoff interpolation in fractional sense

The Hermite–Birkhoff (HB) interpolation is an extension of polynomial interpolation that appears when observation gives operational information. Additional capabilities in HB interpolation by considering fractional operators are created, as it occur in many applied systems. Due to the nice properties in kernel-based approximation, we intend to apply it to solve the HB interpolation in the fractional sense. We show the standard basis in kernel-based approximation is often insufficient for computing a stable solution in fractional HB interpolation. Because of the inherent ill-condition of kernel-based methods, we investigate the fractional HB interpolation using alternate Hilbert–Schmidt SVD (HS–SVD) bases, since it provides a linear transformation which can be applied analytically, and therefore, is able to remove a significant portion of the ill-conditioning. Also, the convergence and stability of the fractional HB interpolation using HS–SVD method are discussed. Numerical results show that in solving fractional HB interpolation, although the standard basis for many positive definite kernels is ill-conditioned in the flat limit, the HS–SVD basis solves the existing problem.

A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Differential Equations

In this paper, we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial differential equations (PDEs). Different from many other approaches, the nonstandard singular basis functions are defined from some generalised Birkhoff interpolation problems through explicit inversion of some prototypical fractional initial value problem (FIVP) with a smooth source term. As such, the singularity of the new basis can be tailored to that of the singular solutions to a class of time-fractional PDEs, leading to spectrally accurate approximation. It also provides the acceptable solution to more general singular problems.

Implicit surface reconstruction based on generalized radial basis functions interpolant with distinct constraints

• We propose a generalized radial basis function interpolant with difference constraints. • Various types of constraints are derived to satisfy the interpolation conditions . • An adaptive iterative algorithm is used for approximating gradient and tangent constraints. • The numerical results reveal the improved performance of our method. In this work, we present a new algorithm for reconstruction of implicit surfaces from a set of cloud points with normals (Hermite data), based on the generalized radial basis functions interpolant with various types of constraints. Our key contribution is a novel construction of difference constraints in the interpolant to satisfy certain linear functional data, named as the domain constraints, the difference constraints of the gradient and the difference constraints of the tangent. To avoid ambiguous constraints, we apply an iterative algorithm to determine the adaptive distance in the difference constraints. The extended interpolant can not only provide an exact interpolation to the function values, but also approximate the function's derivatives by constructing a difference operator along the gradient or tangent direction. We interpolate the sampling points using our method by constructing a signed distance field in the geometry domain. Compared to the Hermite–Birkhoff interpolation with RBFs, the interpolant we use has faster solution efficiency. The experimental results of several data sets show the reliability and performance of our method in a variety of practical scenarios.

Fast Mesh Refinement in Pseudospectral Optimal Control

Mesh refinement in pseudospectral (PS) optimal control is embarrassingly easy: simply increase the order of the Lagrange interpolating polynomial, and the mathematics of convergence automates the distribution of the grid points. Unfortunately, as increases, the condition number of the resulting linear algebra increases as ; hence, spectral efficiency and accuracy are lost in practice. In this paper, Birkhoff interpolation concepts are advanced over an arbitrary grid to generate well-conditioned PS optimal control discretizations. It is shown that the condition number increases only as in general, but it is independent of for the special case of one of the boundary points being fixed. Hence, spectral accuracy and efficiency are maintained as increases. The effectiveness of the resulting fast mesh refinement strategy is demonstrated by using polynomials of over one-thousandth order to solve a low-thrust long-duration orbit transfer problem.

Hierarchical Threshold Secret Image Sharing Scheme Based on Birkhoff Interpolation and Matrix Projection

This paper focuses on how to protect confidential image based on hierarchical threshold secret sharingscheme, against fake shadow attacks, collusion attacks and shadow information leakage problem.Inspired by existing research, we propose a novel hierarchical threshold secret sharing scheme basedon Birkhoff interpolation and matrix projection, hierarchical secret distribution mathematical processesand hierarchical threshold reconstruction mathematical processes are proposed in detail inthis paper, by designing random matrix generation, polynomial multiple derivatives, and Birkhoffinterpolation method in Galois field. Simulations and analysis validate the proposed scheme cantolerate fake shadow attacks and collusion attacks, and has the ability to avoid information leakage.Experiments also prove that shadow secret embedding capacity of secret cover image is bigger thanthe existing schemes.

GENERALIZED INTERPOLATION HERMITE – BIRKHOFF POLYNOMIALS FOR ARBITRARY-ORDER PARTIAL DIFFERENTIAL OPERATORS

This article is devoted to the problem of construction and research of the generalized Hermite – Birkhoff interpolation formulas for arbitrary-order partial differential operators given in the space of continuously differentiable functions of many variables. The construction of operator interpolation polynomials is based both on interpolation polynomials for scalar functions with respect to an arbitrary Chebyshev system, and on the generalized Hermite – Birkhoff interpolation formulas obtained earlier by the authors for general operators in functional spaces. The presented operator formulas contain the Stieltjes integrals and the Gateaux differentials of an interpolated operator. An explicit representation of the error of operator interpolation was obtained. Some special cases of the generalized Hermite – Birkhoff formulas for partial differential operators are considered. The obtained results can be used in theoretical research as the basis for constructing approximate methods for solution of some nonlinear operator-differential equations found in mathematical physics.

English

English

A Class of Birkhoff Type Interpolation and Applications

In this paper a class of Birkhoff type interpolation problem on arbitrary knots is studied. This class of Birkhoff interpolation is a generalization of a Birkhoff problem, which is included in one of the articles in the referenced list. Because the behavior of first and second order derivatives are more important than the higher order, therefore, for a basic function, a better approximation will be obtained than higher order derivatives. The low error value calculated for some functions using this class represents the relative importance of this class of Birkhoff interpolation problem. One of the uses of this class from the Birkhoff polynomials is to review and propose a quadrature formula for this class. One of the ways to reduce the error for interpolation polynomials of this class of Birkhoff interpolation problem is to select the Chebyshev polynomial zeros as nodal points, which can be used to solve numerical differential equations.

Image reconstruction from scattered Radon data by weighted positive definite kernel functions

We propose a novel kernel-based method for image reconstruction from scattered Radon data. To this end, we employ generalized Hermite–Birkhoff interpolation by positive definite kernel functions. For radial kernels, however, a straightforward application of the generalized Hermite–Birkhoff interpolation method fails to work, as we prove in this paper. To obtain a well-posed reconstruction scheme for scattered Radon data, we introduce a new class of weighted positive definite kernels, which are symmetric but not radially symmetric. By our construction, the resulting weighted kernels are combinations of radial positive definite kernels and positive weight functions. This yields very flexible image reconstruction methods, which work for arbitrary distributions of Radon lines. We develop suitable representations for the weighted basis functions and the symmetric positive definite kernel matrices that are resulting from the proposed reconstruction scheme. For the relevant special case, where Gaussian radial kernels are combined with Gaussian weights, explicit formulae for the weighted Gaussian basis functions and the kernel matrices are given. Supporting numerical examples are finally presented.