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This paper deals with two new subclasses of holomorphic and bi-univalent functions in the open unit disk defined by generalized bivariate Fibonacci polynomials. In this paper the coefficient bounds are estimated for | a 2 | and | a 3 | which a 2 and a 3 are the Taylor–Maclaurin coefficients of the functions belonging to these new subclasses. Then, the Fekete–Szegö problem is handled for the functions in these subclasses. Also, several remarks are presented. The results of this paper generalize certain earlier results in the literature. [ABSTRACT FROM AUTHOR]

Let R_k(t) := |P_k(e^{it})|^2 and S_k(t) := |Q_k(e^{it})|^2, where P_k and Q_k are the usual Rudin-Shapiro polynomials of degree n-1 with n=2^k. In a recent paper we combined close to sharp upper bounds for the modulus of the autocorrelation coefficients of the Rudin-Shapiro polynomials with a deep theorem of Littlewood to prove that there is an absolute constant A>0 such that the equations R_k(t) = (1+\eta)n and S_k(t) = (1+\eta)n have at least An^{0.5394282} distinct solutions in [0,2\pi) whenever \eta is real, |\eta | < 2^{-8}, and k is sufficiently large. In this paper we show that the equations R_k(t)=(1+\eta)n and S_k(t)=(1+\eta)n have at least (1/2-|\eta |-\varepsilon)n/2 distinct solutions in [0,2\pi) for every \eta \in (-1/2,1/2), \varepsilon > 0, and sufficiently large be k \geq k_{\eta,\varepsilon }. [ABSTRACT FROM AUTHOR]

This study aims to present the modified SVM polynomial method in order to evaluate insurance data. The research methodology discusses classical and modified SVM polynomial methods by R programming, and uses performance profiles to create the most preferable methods. It offers a new algorithm called an accurate evaluating algorithm as the way to construct the modified SVM polynomial method. The classical SVM polynomial method is also represented as the main idea in finding the modified polynomial SVM method. Model Performance Evaluation (MPE), Receiver Operating Characteristics (ROCs) Curve, Area Under Curve (AUC), partial AUC (pAUC), smoothing, confidence intervals, and thresholds are further named an accurate evaluating algorithm, employed to build the modified SVM polynomial method. The research paper also presents the best performance profiles based on the computing time and the number of iterations of both classical and modified SVM polynomial methods. Performance profiles show numerical comparisons based on both methods involving insurance data also displayed in this paper. It can be concluded that applying an accurate evaluating algorithm on the modified SVM polynomial method will improve the data accuracy up to 86% via computing time and iterations compared to the classical SVM polynomial method, which is only 79%. This accurate evaluating algorithm can be applied to various large-sized data by utilizing R programming with changing any suitable kernels for that data. This vital discovery will offer solutions for faster and more accurate data analysis that can benefit researchers, the private sector, or governments struggling with data. [ABSTRACT FROM AUTHOR]

China aims to reduce carbon dioxide emissions and achieve peak carbon and carbon neutrality goals. Natural gas, as a high-quality fossil fuel energy, is an important transition resource for China in the process of carbon reduction, so it is necessary to predict China's natural gas demand. In this paper, a novel natural gas demand combination forecasting model is constructed to accurately predict the future natural gas demand. The Lasso model and the polynomial model are used to build a combinatorial model, which overcomes the shortcomings of traditional models, which have low data dimensions and poor prediction abilities. In the modeling process, the cross-validation method is used to adjust the modeling parameters. By comparing the performance of the combinatorial forecasting model, the single forecasting model and other commonly used forecasting models, the results show that the error (2.99%) of the combinatorial forecasting model is the smallest, which verifies the high accuracy and good stability advantages of the combinatorial forecasting model. Finally, the paper analyzes the relevant data from 1999 to 2022 and predicts China's natural gas demand in the next 10 years. The results show that the annual growth rate of China's natural gas demand in the next 10 years will reach 13.33%, at 8.3 × 1011 m3 in 2033, which proves that China urgently needs to rapidly develop the gas supply capacity of gas supply enterprises. This study integrates the impact of multiple factors on the natural gas demand, predicts China's natural gas demand from 2023 to 2033, and provides decision-making support for China's energy structure adjustment and natural gas import trade. [ABSTRACT FROM AUTHOR]

This paper is concerned with an important matrix condition in compressed sensing known as the restricted isometry property (RIP). We demonstrate that testing whether a matrix satisfies RIP is \ssr NP-hard. As a consequence of our result, it is impossible to efficiently test for RIP provided \ssr P\ne\ssr NP. [ABSTRACT FROM PUBLISHER]

In this paper, we develop a novel method to construct Goethals–Seidel (GS) sequences with special structures. In the existing methods, utilizing Turyn sequences is an effective and convenient approach; however, this method cannot cover all GS sequences. Motivated by this, we are devoted to designing some sequences that can potentially construct all GS sequences. Firstly, it is proven that a quad of ± 1 polynomials can be considered a linear combination of eight polynomials with coefficients uniquely belonging to { 0 , ± 1 } . Based on this fact, we change the construction of a quad of Goethals–Seidel sequences to find eight sequences consisting of 0 and ± 1 . One more motivation is to obtain these sequences more efficiently. To this end, we make use of the k-block, of which some properties of (anti) symmetry are discussed. After this, we can then look for the sequences with the help of computers since the symmetry properties facilitate reducing the search range. Moreover, we find that one of the eight blocks, which we utilize to construct GS sequences directly, can also be combined with Williamson sequences to generate GS sequences with more order. Several examples are provided to verify the theoretical results. The main contribution of this work is in building a bridge linking the GS sequences and eight polynomials, and the paper also provides a novel insight through which to consider the existence of GS sequences. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to discuss a generalization of the bubble transform to differential forms. The bubble transform was discussed in Falk and Winther (Found Comput Math 16(1):297–328, 2016) for scalar valued functions, or zero-forms, and represents a new tool for the understanding of finite element spaces of arbitrary polynomial degree. The present paper contains a similar study for differential forms. From a simplicial mesh T of the domain Ω , we build a map which decomposes piecewise smooth k-forms into a sum of local bubbles supported on appropriate macroelements. The key properties of the decomposition are that it commutes with the exterior derivative and preserves the piecewise polynomial structure of the standard finite element spaces of k-forms. Furthermore, the transform is bounded in L 2 and also on the appropriate subspace consisting of k-forms with exterior derivatives in L 2 . [ABSTRACT FROM AUTHOR]

It is well known that in C [ X ] , the product of two polynomials is equal to the product of their greatest common divisor and their least common multiple. In a recent paper, we proved a similar relation between the ranks of matrix polynomials. More precisely, the sum of the ranks of two matrix polynomials is equal to the sum of the rank of the greatest common divisor of the polynomials applied to the respective matrix and the rank of the least common multiple of the polynomials applied to the respective matrix. In this paper, we present three new proofs for this result. In addition to these, we present two more applications. [ABSTRACT FROM AUTHOR]

The aim of this paper is to extend the domain of the Grace's theorem, Schur-Szegö composition theorem and Cohn-Egerváry theorem from the set of complex numbers to the set of bicomplex numbers. [ABSTRACT FROM AUTHOR]

In conventional finite element simulations, foil windings with thin foils and with a large number of turns require many mesh elements. This renders models quickly computationally infeasible. This paper uses a homogenized foil winding model and approximates the voltage distribution in the foil winding domain by globally supported polynomials. This way, the small-scale structure in the foil winding domain does not have to be resolved by the finite element mesh. The method is validated successfully for a stand-alone foil winding example and for a pot inductor example. Moreover, a transformer equipped with a foil winding at its primary side is simulated using a field-circuit coupled model. [ABSTRACT FROM AUTHOR]

This is the second paper in a series of two. The goal of the series is to give a polynomial-time algorithm for the 4-coloring problem and the 4-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results, this completes the classification of the complexity of the 4-coloring problem for graphs with a connected forbidden induced subgraph. In this paper we give a polynomial time-algorithm that starts with a 4-precoloring of a graph with no induced six-vertex path and outputs a polynomial-sized collection of so-called excellent precolorings. Excellent precolorings are easier to handle than general ones, and, in addition, in order to determine whether the initial precoloring can be extended to the whole graph, it is enough to answer the same question for each of the excellent precolorings in the collection. The first paper in the series deals with excellent precolorings, thus providing a complete solution to the problem. [ABSTRACT FROM AUTHOR]

This paper presents an investigation of the uniqueness problem of derivatives of a meromorphic function and its difference polynomial in view of a partially sharing. As a consequence of the main result, we improve the recent result of W. J. Chen and Z. G. Huang with the weaker hypotheses and also supplement several results in particular cases. [ABSTRACT FROM AUTHOR]

An R-moduleM is called δ-weakly Hopfian if any δ-small surjective endomorphism of M is an automorphism. In this paper we explore various properties of δ-weakly Hopfian modules, shedding light on their distinct characteristics. Additionally, we examine the δ-weakly Hopficity of modules over polynomial and truncated polynomial rings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some new generalizations of classical descent and inversion statistics on signed permutations that arise from the work of Sack and Úlfarsson [18], and called k-width descents and k-width inversions of type A ([8]). Using the aforementioned new statistics, we derive new generalizations of Eulerian polynomials of type A, B and D. We establish also the 7-positivity of the Eulerian "width-k" polynomials. Referring to Petersen's paper [16], we give a combinatorial interpretation of finite sequences associated with these new polynomials using quasi-symmetric functions and a partition P. [ABSTRACT FROM AUTHOR]

This paper presents the visualization process of finding the roots of a complex polynomial - which is called polynomiography - by the Caputo fractional derivative. In this work, we substitute the variable-order Caputo fractional derivative for classic derivative in Newton's iterative method. To investigate the proposed root-finding method, we apply it for two polynomials p(z) = z5 - 1 and p(z) = -2z4 + z3 + z2 - 2z - 1 on the complex plane and compute the MNI and CAI parameters. Presented examples show that through the expressed process, we can obtain very interesting fractal patterns. The obtained patterns show that the proposed method has potential artistic application. [ABSTRACT FROM AUTHOR]

The semigeneric threshold arrangement is the hyperplane arrangement defined by xi + xj = ai, where 1 ≤ i < j ≤ n and ai's are generic elements. In this paper, we obtain a necessary and sufficient condition for subarrangements of the semigeneric threshold arrangement to be central from the perspective of simple graphs. Combining it with Whitney's theorem, we provide a formula for the characteristic polynomials of the semigeneric threshold arrangement and its subarrangements. [ABSTRACT FROM AUTHOR]

In this paper, we first prove a relation between the critical points of the skew-circulant matrix and the eigenvalues of its principal matrix. Furthermore, we reprove the inequality about the zeros of a polynomial and its critical points by using the properties of skew-circulant matrix, which is to show that we can not only find the skew-circulant matrix, but also give more structure matrices to prove this inequality. [ABSTRACT FROM AUTHOR]

This paper introduces a new modification to an iterative method for solving Cauchy problems (IVPs) based on an inverse polynomial technique. The proposed method is proven to be consistent, stable, and convergent. We also demonstrate the consistency property of the One-stage scheme and the Two-stage scheme, as well as the stability property of the proposed method. To validate the accuracy of our approach, we conduct several numerical experiments and present the results graphically. Our findings show that the proposed method outperforms existing approaches in terms of accuracy and efficiency. Additionally, we discuss the implications of our results for future research in this field. Overall, this paper provides a valuable contribution to the numerical solution of IVPs and lays the groundwork for further exploration in this area. [ABSTRACT FROM AUTHOR]

In this paper, we formulate the determinantal polynomials of weighted shift matrices with palindromic weights (a,br,ar²,. . . ,br2n-3,ar2n-2,c,ar2n-2,br2n-3, . . . ,ar²,br,a), (a,br,ar², . . . ,ar2n-2,br2n-1,c,br2n-1,ar2n-2, . . . ,ar²,br,a), (a,br,ar², . . . ,br2n-3,ar2n-2,c,c,ar2n-2,br2n-3, . . . ,ar²,br,a) and (a,br,ar², . . . ,ar2n-2,br2n-1,c,c,br2n-1,ar2n-2, . . . ,ar²,br,a). Also, we obtain an explicit expression of the numerical radius for each of the weighted shift matrices using these determinantal polynomials. The purpose of this paper is to generalize the results given in [12] and [4]. [ABSTRACT FROM AUTHOR]

This paper surveys the theory of functional identities and its applications. No prior knowledge of the theory is required to follow the paper. [ABSTRACT FROM AUTHOR]

In this paper, we firstly introduce the Gaussian Leonardo polynomial sequences {GLen(X)}⃵n=0 and we obtain Binet's formula, generating function of this sequence. Moreover, we define the matrix GL(X) in the form of 3 x 3. Finally, we study on the coding and decoding applications of the Leonardo number by using the Leonardo matrix. [ABSTRACT FROM AUTHOR]

In the present paper, we consider a special class of third-order polynomial evolutionary equations. These equations, via Lie theory admit the same one-parameter point transformations which leave the equations invariant. Reductions with these invariant functions lead to highly nonlinear third-order ordinary differential equations. We use a power series to establish interesting solutions to the reduced equations, whereby recurrence relations occur and convergence of the series may be tested. Finally, the conserved vectors of the class are constructed. [ABSTRACT FROM AUTHOR]

In this paper, we present a well-organized strategy to estimate the fractional advectiondiffusion equations, which is an important class of equations that arises in many application fields. Thus, Lagrange square interpolation is applied in the discretization of the fractional temporal derivative, and the weighted and shifted Legendre polynomials as operators are exploited to discretize the spatial fractional derivatives of the space-fractional term in multi-term time fractional advection-diffusion model. The privilege of the numerical method is the orthogonality of Legendre polynomials and its operational matrices which reduces time computation and increases speed. A second-order implicit technique is given, and its stability and convergence are investigated. Finally, we propose three numerical examples to check the validity and numerical results to illustrate the precision and efficiency of the new approach. [ABSTRACT FROM AUTHOR]

Many works have been done in recent years as to explorations for degenerate versions of some special polynomials and numbers, which began with the pioneering work of Carlitz on the degenerate Bernoulli and degenerate Euler polynomials. This paper is focused on the study of some properties, recurrence relations and identities related to the degenerate r-Bell polynomials, the two variable degenerate Fubini polynomials and the degenerate r-Stirling numbers of the second kind. Especially, we express the power series ∑∞n=0∑nk=0(k+r)p,λxnn! in terms of the degenerate r-Bell polynomials, of the degenerate r-Stirling numbers of the second kind and of the degenerate Fubini polynomials. [ABSTRACT FROM AUTHOR]

It is remarkable that, in recent years, intensive studies have been done for degenerate versions of many special polynomials and numbers and have yielded many interesting results. The aim of this paper is to study the generalized degenerate (r, s)-Stirling numbers of the second and their natural extensions to polynomials, namely the generalized degenerate (r, s)-Bell polynomials, arising from certain 'degenerate boson normal ordering.'Wederive some properties, explicit expressions and generating functions for those numbers and polynomials. The generalized degenerate (r, s)-Stirling numbers of the second and the degenerate boson normal ordering are respectively degenerate versions of the generalized (r, s)-Stirling numbers of the second and the boson normal ordering studied earlier by Blasiak-Person-Solomon. [ABSTRACT FROM AUTHOR]

We introduce the generalized degenerate Euler-Genocchi polynomials as a degenerate version of the Euler-Genocchi polynomials. In addition, we introduce their higher-order version, namely the generalized degenerate Euler-Genocchi polynomials of order a, as a degenerate version of the generalized Euler-Genocchi polynomials of order a. The aim of this paper is to study certain properties and identities involving those polynomials, the generalized falling factorials, the degenerate Euler polynomials of order a, the degenerate Stirling numbers of the second kind, and the 'alternating degenerate power sum of integers'. [ABSTRACT FROM AUTHOR]

The recent trends in collecting huge datasets have posed a great challenge that is brought by the high dimensionality and aggravated by the presence of irrelevant dimensions. Machine learning models for regression is recognized as a convenient way of improving the estimation for empirical models. Popular machine learning models is support vector regression (SVR). However, the usage of principal component analysis (PCA) as a variable reduction method along with SVR is suggested. The principal component analysis helps in building a predictive model that is simple as it contains the smallest number of variables and efficient. In this paper, we investigate the competence of SVR with PCA to explore its performance for a more accurate estimation. Simulation study and Renal Failure (RF) data of SVR optimized by four different kernel functions; linear, polynomial, radial basis, and sigmoid functions using R software, version (Rx64 3.2.5) to compare the behavior of e-SVR and v-SVR models for different sample sizes ranges from small, moderate to large such as: 50, 100, and 150. The performance criteria are root mean squared error (RMSE) and coefficient of determination R2 showed the superiority of e-SVR over v-SVR. Furthermore, the implementation of SVR after employing PCA improves the results. Also, the simulation results showed that the best performing kernel function is the linear kernel. For real data the results showed that the best kernels are linear and radial basis function. It is also clear that, with e-SVR and v-SVR, the RMSE values for almost kernel functions decreased with increasing sample size. Therefore, the performance of e-SVR improved after applying PCA. In addition sample size n = 50 gave good results for linear and radial kernel. [ABSTRACT FROM AUTHOR]

In the current paper we provide a proof of NP-completeness for the Cell Formation Problem (CFP) with the fractional grouping efficacy objective function. First the CFP with a linear objective function is considered. Following the ideas of Pinheiro et al. (2016) we show that it is equivalent to the Bicluster Graph Editing Problem (BGEP), which is known to be NP-complete due to the reduction from the 3-Exact 3-Cover Problem – 3E3CP (Amit, 2004). Then we suggest a polynomial reduction of the CFP with the linear objective to the CFP with the grouping efficacy objective. It proves the NP-completeness of this fractional CFP formulation. Along with the NP-status our paper presents important connections of the CFP with the BGEP and 3E3CP. Such connections could be used for "transferring" of known theoretical properties, efficient algorithms, polynomial cases, and other features of well-studied graph editing and exact covering problems to the CFP. [ABSTRACT FROM AUTHOR]

Erdös–Lax inequality relates the sup norm of the derivative of a polynomial along the unit circle to that of the polynomial itself (on the unit circle). This paper aims to extend the classical Erdös–Lax inequality to the polar derivative of a polynomial by using the extreme coefficients of the given polynomial. The obtained results not only enrich the realm of Erdös–Lax-type inequalities but also offer a promising avenue for diverse applications where these inequalities play a pivotal role. To illustrate the practical significance of our results, we present a numerical example. It vividly demonstrates that our bounds are considerably sharper than the existing ones in the extensive literature on this captivating subject. [ABSTRACT FROM AUTHOR]

A dynamic visualization of Julia and Mandelbrot fractals involves creating animated representations of these fractals that change over time or in response to user interaction which allows users to gain deeper insights into the intricate structures and properties of these fractals. This paper explored the dynamic visualization of fractals within Julia and Mandelbrot sets, focusing on a generalized rational type complex polynomial of the form Sc(z) = azn + b/zm + c, where a; b; c ∈ C with |a| > 1 and n;m ∈ N with n > 1. By applying viscosity approximation-type iteration processes extended with s-convexity, we unveiled the intricate dynamics inherent in these fractals. Novel escape criteria was derived to facilitate the generation of Julia and Mandelbrot sets via the proposed iteration process. We also presented graphical illustrations of Mandelbrot and Julia fractals, highlighting the change in the structure of the generated sets with respect to the variations in parameters. [ABSTRACT FROM AUTHOR]

This paper presents a new approach for the unified Chebyshev polynomials (UCPs). It is first necessary to introduce the three basic formulas of these polynomials, namely analytic form, moments, and inversion formulas, which will later be utilized to derive further formulas of the UCPs. We will prove the basic formula that shows that these polynomials can be expressed as a combination of three consecutive terms of Chebyshev polynomials (CPs) of the second kind. New derivatives and connection formulas between two different classes of the UCPs are established. Some other expressions of the derivatives of UCPs are given in terms of other orthogonal and non-orthogonal polynomials. The UCPs are also the basis for additional derivative expressions of well-known polynomials. A new linearization formula (LF) of the UCPs that generalizes some well-known formulas is given in a simplified form where no hypergeometric forms are present. Other product formulas of the UCPs with various polynomials are also given. As an application to some of the derived formulas, some definite and weighted definite integrals are computed in closed forms. [ABSTRACT FROM AUTHOR]

In this paper, the vector extension operation is proposed to replace the de Boor‐Cox formula for a fast algorithm to B‐spline basis functions. This B‐spline basis function based on vector extending operation is implemented in the class and shape transformation (CST) parameterization method in place of the traditional Bézier polynomials to enhance the local ability of control and accuracy to represent an airfoil shape. To calculate the k‐degree B‐spline function's nonzero values, the algorithm can improve the computing efficiency by 2k+1 times. [ABSTRACT FROM AUTHOR]

The Boolean map χ n : F 2 n → F 2 n , x ↦ y defined by y i = x i + (x i + 1 + 1) x i + 2 (where i ∈ Z / n Z ) is used in various permutations that are part of cryptographic schemes, e.g., Keccak-f (the SHA-3-permutation), ASCON (the winner of the NIST Lightweight competition), Xoodoo, Rasta and Subterranean (2.0). In this paper, we study various algebraic properties of this map. We consider χ n (through vectorial isomorphism) as a univariate polynomial. We show that it is a power function if and only if n = 1 , 3 . We furthermore compute bounds on the sparsity and degree of these univariate polynomials, and the number of different univariate representations. Secondly, we compute the number of monomials of given degree in the inverse of χ n (if it exists). This number coincides with binomial coefficients. Lastly, we consider χ n as a polynomial map, to study whether the same rule ( y i = x i + (x i + 1 + 1) x i + 2 ) gives a bijection on field extensions of F 2 . We show that this is not the case for extensions whose degree is divisible by two or three. Based on these results, we conjecture that this rule does not give a bijection on any extension field of F 2 . [ABSTRACT FROM AUTHOR]

Ascon, a family of algorithms that supports authenticated encryption and hashing, has been selected as the new standard for lightweight cryptography in the NIST Lightweight Cryptography Project. Ascon's permutation and authenticated encryption have been actively analyzed, but there are relatively few analyses on the hashing. In this paper, we concentrate on preimage attacks on Ascon-Xof. We focus on linearizing the polynomials leaked by the hash value to find its inverse. In an attack on 2-round Ascon-Xof, we carefully construct the set of guess bits using a greedy algorithm in the context of guess-and-determine. This allows us to attack Ascon-Xof more efficiently than the method in Dobraunig et al., and we fully implement our attack to demonstrate its effectiveness. We also provide the number of guess bits required to linearize one output bit after 3- and 4-round Ascon's permutation, respectively. In particular, for the first time, we connect the result for 3-round Ascon to a preimage attack on Ascon-Xof with a 64-bit output. Our attacks primarily focus on analyzing weakened versions of Ascon-Xof, where the weakening involves setting all the IV values to 0 and omitting the round constants. Although our attacks do not compromise the security of the full Ascon-Xof, they provide new insights into their security. [ABSTRACT FROM AUTHOR]

Ovoids of the parabolic quadric Q(6, q) of PG (6 , q) have been largely studied in the last 40 years. They can only occur if q is an odd prime power and there are two known families of ovoids of Q(6, q), the Thas-Kantor ovoids and the Ree-Tits ovoids, both for q a power of 3. It is well known that to any ovoid of Q(6, q) two polynomials f 1 (X , Y , Z) , f 2 (X , Y , Z) can be associated. In this paper we classify ovoids of Q(6, q) with max { deg (f 1) , deg (f 2) } < (1 6.3 q) 3 13 - 1 . [ABSTRACT FROM AUTHOR]

The planar slope number psn (G) of a planar graph G is the minimum number of edge slopes in a planar straight-line drawing of G. It is known that psn (G) ∈ O (c Δ) for every planar graph G of maximum degree Δ . This upper bound has been improved to O (Δ 5) if G has treewidth three, and to O (Δ) if G has treewidth two. In this paper we prove psn (G) ≤ max { 4 , Δ } when G is a Halin graph, and thus has treewidth three. Furthermore, we present the first polynomial upper bound on the planar slope number for a family of graphs having treewidth four. Namely we show that O (Δ 2) slopes suffice for nested pseudotrees. [ABSTRACT FROM AUTHOR]

In this paper, an observer-based tracking control problem is considered for a polynomial system with parameter uncertainties and external disturbances. Noting that some states are difficult to measure in practice, a polynomial state observer is employed to obtain the unknown states. Then a robust observer-based tracking controller is designed to force the states of the polynomial uncertain system to follow the ones of the reference model and satisfy $ H_\infty $ H ∞ performance. By using a full-state-dependent homogeneous polynomial function, the solvability condition is derived for the observer-based tracking control strategy, which reduces conservatism. Sum of squares technique is utilised to calculate the corresponding observer and controller, effectively dealing with the calculation problem in polynomial systems. Simulation examples are given to illustrate the effectiveness of the presented control strategy. [ABSTRACT FROM AUTHOR]

In this paper, we give a complete description of the generalized hypergeometric functions, introduced by Faraut and Kor´anyi on the Cartan domain. We establish some Gauss type contiguous relations between these functions on the two Cartan domains of type I2 and type IV4 analogous to the classical relations in the one variable case. [ABSTRACT FROM AUTHOR]

This paper presents a new method for regional stabilization of discrete-time nonlinear systems by static output feedback based on polynomial Lyapunov functions. The considered class of nonlinear systems subject to time-varying parameters can be described by difference-algebraic representation and an equivalent polytopic model is obtained, making it possible to apply Lyapunov theory and linear matrix inequality-based tool. In this sense, a convex optimization problem in terms of LMI is provided to guarantee the closed-loop system's robust stabilization and to enlarge the estimated domain of attraction (DoA). The proposed control design is a one-step approach that can be applied to systems with nonlinear output matrices. It requires no iterative algorithms or congruence transformation, and auxiliary decision variables are introduced only aiming at less conservative results. Besides that, the use of polynomial Lyapunov functions allows us to obtain asymmetric and non-convex estimated DoA, reducing the conservativeness. Numerical examples are provided to illustrate the effectiveness and advantages of the proposed approach. [ABSTRACT FROM AUTHOR]

In the current distributed environment, reversible data hiding in encrypted domain (RDH-ED) cannot grant corresponding privileges according to users' identity classes. To address this issue, this paper proposes a hierarchical authorization structure embedding scheme based on secret image sharing (SIS) and users' hierarchical identities. In the first embedding, the polynomial coefficient redundancy generated in the encryption process of the SIS is utilized by the image owner. For the second, the participants are categorized into two parts. One is core users with adaptive difference reservation embedding, and the other is ordinary users with pixel bit replacement embedding. At the time of reconstruction, more than one core user must provide pixel differences, which grants more privileges to core users. The experimental results demonstrate that the average embedding rate (ER) of the test images is 4.3333 bits per pixel (bpp) in the (3, 4) threshold scheme. Additionally, the reconstructed image achieves a PSNR of +∞ and an SSIM of 1. Compared to existing high-performance RDH-ED schemes based on secret sharing, the proposed scheme with a larger ER maintains strong security and reversibility. Moreover, it is also suitable for multiple embeddings involving multilevel participant identities. In conclusion, the results underscore the efficacy of our technique in achieving both security and performance objectives within a complex distributed setting. [ABSTRACT FROM AUTHOR]

A novel quantum-safe key encapsulation algorithm, called Multivariate Polynomial Public Key (MPPK), was recently proposed by Kuang, Perepechaenko, and Barbeau. Security of the MPPK key encapsulation mechanism does not rely on the prime factorization or discrete logarithm problems. It builds upon the NP-completeness of the modular Diophantine equation problem, for which there are no known efficient classical or quantum algorithms. Hence, it is resistant to known quantum computing attacks. The private key of MPPK comprises a pair of multivariate polynomials. In a companion paper, we analyzed the performance of MPPK when these polynomials are quadratic. The analysis highlighted the MPPK high decapsulation time. We found that, while maintaining the security strength, the polynomials can be linear. Considerable performance gains are obtained for the decapsulation process. In this article, we benchmark the linear case and compare the results with the previous quadratic case. [ABSTRACT FROM AUTHOR]

The paper proposes the method for constructing Chebyshev approximation of multivariable functions with a relative error using a power expression. It involves building an intermediate Chebyshev approximation of values rooted to the corresponding degree of the approximated function by a polynomial with a relative error. Parameters for the polynomial approximation are computed as the boundary mean-power approximation through an iterative scheme using the least squares method with a variable weight function. The authors provide test examples, confirming the fast convergence of the method for constructing the Chebyshev approximation for the functions of one, two, and three variables using power expression. [ABSTRACT FROM AUTHOR]

Herein, we present an algorithm for handling fractional delay differential equations (FDDEs). Chebyshev polynomials (CPs) class of the seventh kind is a subclass of the generalized Gegenbauer (ultraspherical) polynomials. The members of this class make up the basis functions in this paper. Our suggested numerical algorithm is derived using new theoretical findings about these polynomials and their shifted counterparts. We will use the Tau method to convert the FDDE with the governing conditions into a linear algebraic system, which can then be solved numerically using a suitable procedure. We will give a detailed discussion of the convergence and error analysis of the shifted Chebyshev expansion. Lastly, some numerical examples are provided to verify the precision and applicability of the proposed strategy. [ABSTRACT FROM AUTHOR]

Due to the high approximation power and simplicity of computation of smooth radial basis functions (RBFs), in recent decades they have received much attention for function approximation. These RBFs contain a shape parameter that regulates their approximation power and stability but its optimal selection is challenging. To avoid this difficulty, this paper follows a novel and computationally efficient strategy to propose a space of radial polynomials with even degree that well approximates flat RBFs. The proposed space, Hn, is the shifted radial polynomials of degree 2n. By obtaining the dimension of Hn and introducing a basis set, it is shown that Hn is considerably smaller than P2n while the distances from RBFs to both Hn and P2n are nearly equal. For computation, by introducing new basis functions, two computationally efficient approaches are proposed. Finally, the presented theoretical studies are verified by the numerical results. [ABSTRACT FROM AUTHOR]

The present paper deals with the Durrmeyer construction of operators based on a class of orthogonal polynomials called Apostol-Genocchi polynomials. For the proposed operators, we first establish a global approximation result followed by its convergence estimate in terms of usual, r-th and weighted modulus of continuity. We further study the asymptotic type results such as the Voronovskaya theorem and quantitative Voronovskaya theorem. Moreover, we estimate the rate of pointwise convergence of the proposed operators for functions of bounded variation defined on the interval (0, ∞). Finally, the results are validated through graphical representations and an absolute error table. [ABSTRACT FROM AUTHOR]