Polynomial Operators Research Articles

A Reformulated Krein Matrix for Star-Even Polynomial Operators with Applications

In its original formulation the Krein matrix was used to locate the spectrum of first-order star-even polynomial operators where both operator coefficients are nonsingular. Such operators naturally arise when considering first-order-in-time Hamiltonian PDEs. Herein the matrix is reformulated to allow for operator coefficients with nontrivial kernel. Moreover, it is extended to allow for the study of the spectral problem associated with quadratic star-even operators, which arise when considering the spectral problem associated with second-order-in-time Hamiltonian PDEs. In conjunction with the Hamiltonian-Krein index (HKI) the Krein matrix is used to study two problems: conditions leading to Hamiltonian-Hopf bifurcations for small spatially periodic waves, and the location and Krein signature of small eigenvalues associated with, e.g., $n$-pulse problems. For the first case we consider in detail a first-order-in-time fifth-order KdV-like equation. In the latter case we use a combination of Lin's method, the HKI, and the Krein matrix to study the spectrum associated with $n$-pulses for a second-order-in-time Hamiltonian system which is used to model the dynamics of a suspension bridge.

Spectral Analysis of Operator Polynomials and Second-Order Differential Operators

Изучение спектральных свойств операторных полиномов сводится к изучению соответствующих спектральных свойств операторов, заданных операторной матрицей. Результаты применяются к исследованию дифференциальных операторов второго порядка с помощью сопоставления им соответствующих дифференциальных операторов первого порядка и использования их свойств, связанных с обратимостью. Библиография: 16 названий.

Investigation of the effects of upward and downward continuation operations on regional trends in parts of Southern Niger Delta of Nigeria using gravity data

Analytical operations of upward and downward continuations were applied to structural trends obtained from airborne gravity data in parts of Southern Niger Delta of Nigeria with varying depths of 500 to 1,000 m to observe the effects of the continuations on the structural trends. The data were first gridded and the regional effects separated from the residuals using a second-order polynomial operation. The continuation operations show that at a depth of 500 m, both upward and downward continuations did not affect the observed trends much as the difference between the original trends and the ones with continuation operations were almost the same. However, with continuation depths of 1,000 to 5, 000 m, the upward continuations enhanced the sharpness of the structural trends with increasing depths while the downward continuations suppressed and distorted the structural trends with increasing depths. The results confirm the instability in downward continuation operations with increasing depths for long-wavelength regional trends since the operations increases the noise levels exponentially and hence distorts the sharpness of the trends. The enhancement of the regional trends by the upward continuation shows it as a stable operation for long-wavelength regional trends for proper visualization.

Multi-secret Image Sharing Scheme based on coding method

Image Secret sharing scheme is an effective way to share a secret message by distributing the shares to different owners which has become a relatively independent research field. The classical Shamir’s scheme is based on Lagrange interpolation polynomial, and the utility of the scheme is greatly compromised by the complex polynomial operations cost. Visual cryptography is a special image sharing scheme, which uses the human visual system to recover the secret information. Multi-secret shared visual cryptography is an important research part of visual cryptography. It can share several secret images, and have important significance for perfecting the theoretical system of visual cryptography and expanding the application fields of visual cryptography. In this paper, based on the research of the internal relations between the coding theory and the secret sharing, a Multi-secret Image Sharing scheme is proposed, the generation of the shares is realized by the binary Cauchy encoding. It satisfies the general characters of the secret image sharing schemes, and the algorithm complexity is low, which is easy for implementation.

A certain subclass of bi-univalent analytic functions introduced by means of the q-analogue of Noor integral operator and Horadam polynomials

A certain subclass of bi-univalent analytic functions introduced by means of the q-analogue of Noor integral operator and Horadam polynomials

Commutative algebraic monoid structures on affine spaces

We study commutative associative polynomial operations [Formula: see text] with unit on the affine space [Formula: see text] over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric varieties.

The effect of the serious game Tempoly on learning arithmetic polynomial operations

Research on learning through serious games has been gaining importance in recent years. We develop the serious game Tempoly, concerning the learning of the four arithmetic operations on polynomials. A quasi-experimental study was carried out with 95 students of 8th grade, to evaluate the reactions of the students to the game, the effects of the use of the game in the students’ learning, and the use of the game through time. According to data collected through a questionnaire, we may conclude that students liked to play the game, considered that the game helped them learning the polynomial operations and that what they learned in class was useful to play the game. It was concluded that students who played the game simultaneously with the classes improved their results in a pre/post-test more than the students who did not play the game and through a Mann-Whitney test, the results showed statistical differences. The study revealed that the students that had more time to play the game did play more out of the class.

DIVIDED POWER ALGEBRAS OVER AN OPERAD

AbstractThe purpose of this paper is to give a characterisation of divided power algebras over a reduced operad. Such a characterisation is given in terms of polynomial operations, following the classical example of divided power algebras. We describe these polynomial operations in two different ways: one way uses invariant elements under the action of the symmetric group and the other coinvariant elements. Our results are then applied to the case of level algebras, which are (non-associative) commutative algebras satisfying the exchange law.

Complete Planar Impedance Conservation and PML Boundary Value Solutions

This paper shows that a straightforward way to evaluate planar impedance conservation boundary values of arbitrary boundary operators and orientations is to directly copy the construction from the well-understood 1-D Robin impedance-matching solution and then let the boundary rotate freely. Copying from this classical solution, hence, requires no use of split field and any of its stretched coordinate reformulations. In fact, it actually gives the complete general superposition solution, which also explains that the existing PML considers only a rational boundary operator and solves only a coordinately aligned example. The construction thus embraces both general order polynomial and nonpolynomial boundary operators and noncoordinately aligned boundary solutions to be put on top of the PML orthogonal terminations. A constructive source-field energy coupling relation is also introduced to simplify the impedance conservation analysis to read as the triple product identity. Evident by the various forms of Maxwell equations, we elaborate in detail that boundary value equations can also be written in various equivalent and endless redundant forms.

Inverse Problems for a Class of Degenerate Evolution Equations with Riemann - Liouville Derivative

Abstract Unique solvability and well-posedness issues are studied for linear inverse problems with a constant unknown parameter to fractional order differential equations with Riemann – Liouvlle derivative in Banach spaces. Firstly, well-posedness criteria for the inverse problem with the Cauchy type initial conditions to the differential equation in a Banach space that solved with respect to the fractional derivative is obtained. This result is applied to search of sufficient conditions for the unique solution existence of the inverse problem for equation with linear degenerate operator at the Riemann – Liouville fractional derivative. It is shown that the presence of the matching conditions for the data of the problem excludes the possibility of the well-posedness consideration for the degenerate inverse problem with the Cauchy type condition. But for the inverse problem with the Showalter – Sidorov type conditions it is found the criteria of the well-posedness. Abstract results are used to the search of conditions of the unique solvability for an inverse problem to a class of partial differential equations of time-fractional order with polynomials of elliptic differential operators with respect to the spatial variables.

On weighted uniform boundedness and convergence of the Bernstein–Chlodovsky operators

A recent article of the first author presented a constructive proof for the Weierstrass approximation theorem for a wide class of weighted spaces of continuous functions defined on [ 0 , ∞ ) , using Chlodovsky's generalization of the Bernstein polynomial operators. Here, we investigate necessary and sufficient conditions related to the uniform boundedness of a sequence of such operators and to their convergence properties when the weight is a classical Freud weight of the form w ( x ) = e − x α , α ≥ 1 .

An Efficient Solution Towards Secure Homomorphic Symmetric Encryption Algorithms

In this paper, we consider Homomorphic Encryption (HE) to process over encrypted data in order to achieve user privacy. We present a framework solution to provide a high level of security for the symmetric HE algorithms. The proposed solution introduces a dynamic structure and dynamic diffusion’s primitives that enhance existing symmetric HE algorithms and overcome their weaknesses. We apply this solution to a well known symmetric homomorphic approach, the PORE (Polynomial Operation for Randomization and Encryption) approach. The security analysis of the proposed solution shows that it ensures a high level of security without performance degradation. It is also evaluated against different attacks. This leads to secure and efficient HE Algorithms for practical implementations.

Efficient and Lossless Polynomial-Based Secret Image Sharing for Color Images

In today's era of developed network, it is particularly important to ensure the security of information. Secret image sharing is a kind of secret information protection technology with loss tolerance by sharing a secret image to a number of shadow images. Most of the researches on secret image sharing focus on grayscale images, but the images we use in our daily life are mostly in color, so the research on the secret image sharing for color image is very meaningful and has great application value. The traditional color image sharing scheme is to share and recover R, G, and B color planes separately, which cause that the time of sharing and recovering one color image is at least triple that of sharing and recovering one grayscale image with the same size as the color image. In this paper, we propose an efficient and lossless secret image sharing scheme for color images. We concatenate the R, G, and B values of every color pixel into a long integer and share it by the polynomial-based secret image sharing, thus the sharing efficiency of color images is greatly improved by reducing the number of sharing operations to 1/3 of the traditional scheme. And we achieve lossless recovery by taking GF(16777259) as the finite field of polynomial operations. 16777259 is the smallest prime number greater than 2 24 - 1, so we intend to screen the random numbers to drop the invalid share values. Theoretical analyses and experiments are used to prove the effectiveness of the proposed scheme.

Operator interpolation formulas of Hermitе type with arbitrary multiplicity nodes based on identity transformations of function

The problem of construction and research of Hermite interpolation formulas with nodes of arbitrary multiplicity for operators given in functional spaces of one and two variables is considered. The construction of operator interpolation polynomials is based both on interpolation polynomials for scalar functions with respect to an arbitrary Chebyshev system and on identity transformations of functions. The reduced operator formulas contain the Stieltjes integrals and the Gateaux differentials of an interpolated operator and are invariant for a special class of operator polynomials of appropriate degree. For some of the obtained operator polynomials, an explicit representation of the interpolation error is found. Particular cases of Hermite formulas based both on the integral transformations of Hankel, Abel, Fourier and on the Fourier sine (cosine) transform are considered. The application of separate interpolation formulas is illustrated by examples. The presented results can be used in theoretical research as the basis for construction of approximate methods for solving integral, differential and other types of nonlinear operator equations.

On modeling and complete solutions to general fixpoint problems in multi-scale systems with applications

This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e., the general fixed point problem is first reformulated as a nonconvex optimization problem, its well-posedness is discussed based on the objectivity principle in continuum physics; then the canonical duality theory is applied for solving this challenging problem to obtain not only all fixed points, but also their stability properties. Applications are illustrated by problems governed by nonconvex polynomial, exponential, and logarithmic operators. This paper shows that within the framework of the canonical duality theory, there is no difference between the fixed point problems and nonconvex analysis/optimization in multidisciplinary studies.

Metric approximation of set-valued functions of bounded variation

In this paper we approximate univariate set-valued functions (SVFs) of bounded variation with range consisting of general (not necessarily convex) compact sets. The approximation operators adapted to SVFs are local operators such as the symmetric Schoenberg spline operator, the Bernstein polynomial operator and the Steklov function. All operators are adapted by using metric linear combinations. Error bounds, obtained in the averaged Hausdorff metric, provide rates of approximation similar to those for real-valued functions of bounded variation.

Polynomial jump-diffusions on the unit simplex

Polynomial jump-diffusions constitute a class of tractable stochastic models with wide applicability in areas such as mathematical finance and population genetics. We provide a full parameterization of polynomial jump-diffusions on the unit simplex under natural structural hypotheses on the jumps. As a stepping stone, we characterize well-posedness of the martingale problem for polynomial operators on general compact state spaces.

Piecewise Polynomial Algorithms for the Analysis of Processes in Inhomogeneous Media

The authors propose, theoretically substantiate, and programmatically implement high-precision numerical-analytical algorithms for approximation of problems solutions in inhomogeneous media on the basis of linear polynomial operators.

Microvibration isolation by adaptive feedforward control with asymmetric hysteresis compensation

Abstract This paper proposes an improved adaptive filtered-x normalized least-mean-square (FxNLMS) algorithm to achieve active microvibration isolation. Physically, microvibration disturbance is attenuated by a custom-developed magnetostrictive-actuated device. Whereas, magnetostrictive materials characterize hysteresis. Particularly, when the amplitude of the input current is large, the hysteresis loop describes strong asymmetry and nonlinearity. To avoid this phenomenon, in some studies, the actuators are limited to move over a small range, and thus the potential range of the isolated microvibration is restrictive. Then Jiles-Atherton model and Bouc-Wen model are utilized to compensate hysteresis, however these models are non-analytical with certain approximation. In order to compensate hysteresis more accurately, Preisach model, a popular operator-based model, is presented. However, due to the double integrals in this model, the hysteresis identification and inversion process is complicated. Therefore, we reduce hysteresis based on a modified Prandtl-Ishlinskii (PI) model which is more simple but effective. This model utilizes the polynomial operators to describe hysteresis asymmetry and nonlinearity. The inverse hysteresis is then applied to compensate the magnetostrictive-induced hysteresis and to alleviate microvibration combining with the conventional FxNLMS algorithm. An experimental setup is fabricated and some experimental investigations are conducted. Experimental results show that the improved FxNLMS algorithm enhances the performance of microvibration isolation effectively—in the presence of single-tone (4 Hz at amplitude of 250 μm) and double tone (2 Hz mixed with 6 Hz at amplitude of 250 μm) excited microvibration, the isolation ratios of the improved FxNLMS controller are 20.79 dB and 15.32 dB respectively, while those of the conventional FxNLMS controller are 13.83 dB and 10.65 dB respectively.

Cosmological coherent state expectation values in loop quantum gravity I. Isotropic kinematics

This is the first paper of a series dedicated to loop quantum gravity (LQG) coherent states and cosmology. The concept is based on the effective dynamics program of Loop Quantum Cosmology, where the classical dynamics generated by the expectation value of the Hamiltonian on semiclassical states is found to be in agreement with the quantum evolution of such states. We ask the question of whether this expectation value agrees with the one obtained in the full theory. The answer is in the negative, Dapor and Liegener (2017 arXiv:1706.09833). This series of papers is dedicated to detailing the computations that lead to that surprising result. In the current paper, we construct the family of coherent states in LQG which represent flat (k = 0) Robertson–Walker spacetimes, and present the tools needed to compute expectation values of polynomial operators in holonomy and flux on such states. These tools will be applied to the LQG Hamiltonian operator (in Thiemann regularization) in the second paper of the series. The third paper will present an extension to cosmologies and a comparison with alternative regularizations of the Hamiltonian.