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This paper shows a 2-step procedure for obtaining a variable-bandwidth recursive digital filter whose structure contains cascaded second-order (2nd-order) sections. Such a cascade-form structure is insensitive to the round-off noises that come from filter-coefficient quantizations in hardware implementations. This paper also shows how to utilize a 2-step procedure to get a variable-bandwidth recursive filter that is absolutely stable. The first step (Step-1) of the 2-step procedure designs a series of constant-bandwidth filters for approximating a series of evenly discretized variable specifications, and the second step (Step-2) fits the coefficient values obtained from Step-1 by employing individual polynomials. To ensure the stability of the resultant constant-bandwidth filters in Step-1, coefficient transformations are first executed on the 2nd-order transfer function's denominator-coefficients, and then each coefficient of both numerator and transformed denominator is found as an individual polynomial. Once all the polynomials are obtained, the polynomials corresponding to the transformed denominator are further converted to composite functions for ensuring the stability. Hence, the 2-step procedure not only produces a cascade-form variable-bandwidth filter that has low quantization errors, but also ensures the stability. A lowpass example is included for verifying the achieved stability and showing the high approximation accuracy. [ABSTRACT FROM AUTHOR]

The walk matrix of an n-vertex graph G with adjacency matrix A, denoted by $ W(G) $ W (G) , is $ [e,Ae,\ldots,A^{n-1}e] $ [ e , Ae , ... , A n − 1 e ] , where e is the all-ones vector. Let $ G\circ P_m $ G ∘ P m be the rooted product of G and a rooted path $ P_m $ P m (taking an endvertex as the root), i.e. $ G\circ P_m $ G ∘ P m is a graph obtained from G and n copies of $ P_m $ P m by identifying each vertex of G with an endvertex of a copy of $ P_m $ P m . Mao et al. [A new method for constructing graphs determined by their generalized spectrum. Linear Algebra Appl. 2015;477:112–127.] and Mao and Wang [Generalized spectral characterization of rooted product graphs. Linear Multilinear Algebra. 2022. DOI:10.1080/03081087.2022.2098226.] proved that, for m = 2 and $ m\in \{3,4\} $ m ∈ { 3 , 4 } , respectively \[ \det W(G\circ P_m)=\pm a_0^{\lfloor\frac{m}{2}\rfloor}(\det W(G))^m, \] det W (G ∘ P m) = ± a 0 ⌊ m 2 ⌋ (det W (G)) m , where $ a_0 $ a 0 is the constant term of the characteristic polynomial of G. Furthermore, in the same paper, Mao and Wang conjectured that the formula holds for any $ m\ge 2 $ m ≥ 2. In this paper, we verify this conjecture using the technique of Chebyshev polynomials. [ABSTRACT FROM AUTHOR]

This paper deals with the analysis of the orthogonality of a monic polynomial sequencedefined as a linear combination of a sequence of monic orthogonal polynomialswithwherefor. Moreover, we obtain the relation between the corresponding linear functionals as well as an explicit expression for the sequence of monic orthogonal polynomials. We obtain the connection between the Jacobi matrices associated withand, respectively, by using an LU factorization. Some special cases of the above type relation are analysed. [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]

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]

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]

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]

This paper investigates the asymptotic stability of neutral stochastic pantograph systems with Markovian switching (NSPSMS). The stochastic LaSalle theorem has been set up to locate the attractive sets for NSPSMS without linear growth condition. Subsequently, combining the stochastic LaSalle theorem and uniformly continuous theory, almost surely asymptotic stability and pth moment asymptotic stability are analysed. Furthermore, we also present one new criterion on the pth moment polynomial stability and almost surely polynomial stability for NSPSMS, where the coefficients of the delay term are time invariable. Finally, three examples are provided to exhibit the efficiency of the proposed results. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in a special class of integer matrices, namely the power-product matrix, defined with two positive integers n and d. Each matrix element is computed by a power-product of two weak compositions of d into n parts. The power-product matrix has several interesting applications such as the power-sum representation of polynomials and the difference-of-convex-sums-of-squares decomposition of polynomials. We investigate some properties of this matrix including: nonsingularity, sparsity and determinant. Based on techniques in enumerative combinatorics, we prove that the power-product matrix is nonsingular and the number of nonzero entries can be computed exactly. This matrix shows sparse structure which is a good feature in numerical computation of its inverse required in some applications. Special attention is devoted to the computation of the determinant for n = 2 whose explicit formulation is obtained. [ABSTRACT FROM AUTHOR]

Finite commutative semifields of odd characteristic correspond to Dembowski-Ostrom polynomials. A new proof of this fact is the main result of this paper. The paper also discusses strong isotopy of commutative semifields and shows that the limit on the number of strong isotopy classes can be obtained from a general theorem on commutative loops. By this theorem for each commutative loop Q the commutative isotopes form at most | N μ : (N μ) 2 N λ | classes with respect to the strong isotopy, where N μ and N λ are the middle and the left nucleus of Q. Loops Q1 and Q2 are said to be strongly isotopic if there exists an isotopism Q 1 → Q 2 of the form (α , α , γ) . [ABSTRACT FROM AUTHOR]

In this paper, left Atkinson linear relations defined in a Banach space are considered. The behaviour of such notion in polynomials is studied and a spectral mapping theorem for the corresponding spectrum is deduced. Let $ \boldsymbol {\mathcal {R}} $ R denote any of the following classes: left (right) Atkinson, Fredholm, left (right) Weyl, Weyl, left (right) Browder, Browder linear relations. For a closed and everywhere defined linear relation $ \mathcal {T}, $ T , we prove that under suitable conditions, $ \mathcal {T} = \mathcal {T}_{\mathcal {M}} \oplus \mathcal {T}_{\mathcal {N}}, $ T = T M ⊕ T N , where $ \mathcal {T}_{\mathcal {M}}\in \boldsymbol {\mathcal {R}} $ T M ∈ R and $ \mathcal {T}_{\mathcal {N}} $ T N is a bounded nilpotent operator if and only if $ \mathcal {T} $ T admits a Kato-decomposition and 0 is not an interior point of the $ \boldsymbol {\mathcal {R}} $ R spectrum of $ \mathcal {T}, $ T , $ \sigma _{\boldsymbol {\mathcal {R}}}(\mathcal {T}). $ σ R (T). As an application, some connections results between $ \sigma _{\boldsymbol {\mathcal {R}}}(\mathcal {T}) $ σ R (T) and $ \sigma _{\boldsymbol {\mathcal {DK}}}(\mathcal {T}) $ σ DK (T) are established, where $ \sigma _{\boldsymbol {\mathcal {DK}}}(\mathcal {T}) $ σ DK (T) denotes the Kato-decomposition spectrum of $ \mathcal {T}. $ T. [ABSTRACT FROM AUTHOR]

In a previous paper, we introduced the Collatz polynomials $ P_N\left (z \right) $ P N (z) , whose coefficients are the terms of the Collatz sequence of the positive integer N. Our work in this paper expands on our previous results, using the Eneström-Kakeya Theorem to tighten our old bounds of the roots of $ P_N\left (z \right) $ P N (z) and giving precise conditions under which these new bounds are sharp. In particular, we confirm an experimental result that zeros on the circle $ \ensuremath {\{z\in \ensuremath {\mathbb {C}}:\left | z \right | = 2\}} $ { z ∈ C : | z | = 2 } are rare: the set of N such that $ P_N\left (z \right) $ P N (z) has a root of modulus 2 is sparse in the natural numbers. We close with some questions for further study. [ABSTRACT FROM AUTHOR]

In this paper, a simple and efficient no-reference image quality assessment technique based on polynomial coefficient and reduced Gerschgorin circle is presented. The dominant features and dynamics of the test images are captured using the polynomial coefficient. Moreover, the image focus measure is modeled using the concept of reduced Gerschgorin's higher circle bound. The presented image focus measure is tested on different synthetic images datasets such as LIVE, TID2008, CSIQ, Cornel and IVC. Further, the proposed approach is also examined on real-time images captured by the camera in different conditions. It is observed that the proposed approach demonstrates important properties of focus measures such as unimodality and contrast invariance. Moreover, the presented technique is robust under the different varying noisy conditions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a one-dimensional singularly perturbed problem of the convection–diffusion type. The problem is solved numerically by the local discontinuous Galerkin (LDG) method on a Bakhvalov-type mesh. Here we propose new numerical fluxes and new penalty parameters in the LDG method and prove the supercloseness of the LDG method in an energy norm. Besides, a variant of the energy norm is proposed. It is proved that the method is convergent uniformly in the perturbation parameter with an improved order of k + 1 in the new norm (k is the degree of the piecewise polynomial used in the LDG method). [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of symmetric orthogonal polynomials, which generalizes the q-Dunkl-classical polynomials and also provides new characterizations of symmetric q-semiclassical orthogonal polynomials. Both the notion of a class and a criterion for determining the class will be given. As an application, several examples of q-Dunkl-semiclassical forms of the class $ s \leqq 2 $ s ≦ 2 are highlighted. [ABSTRACT FROM AUTHOR]

In this paper we use an approximate analytical method for numerical solution of one dimensional moving boundary problem. We consider the oxygen diffusion problem where the oxygen is allowed to diffuse into a sick cell which absorbs and immobilizes oxygen at a constant rate. Our main problem consists in tracking the moving boundary that represents the oxygen penetration depth inside the sick cell. We can find an accurate solution which is obtained by a polynomial of third and fourth degree and we show some mistakes in the paper published by Seval Çatal in (App.Math.Comput 145:361–369, 2003). [ABSTRACT FROM AUTHOR]

This paper contains a biographical sketch of the late Japanese number theorist Jitsuro Nagura (1901–68) based on material provided by his eldest son Riichi Nagura, and a synopsis of his two surviving mathematical papers, the first a well-known article on primes in short intervals, and the second an unpublished draft of a proof relating to complex polynomials. [ABSTRACT FROM AUTHOR]

The fabrication of artificial heterostructures is mainly based on substitution systems. We present simple ways to construct double-sided versions of the Fibonacci, Prouhet-Thue-Morse, paperfolding, period doubling and Golay-Rudin-Shapiro sequences. We also construct a generic instance of the two-dimensional Prouhet-Thue-Morse structure and explore its symbolic complexity. The complexity turns out to be polynomial and hence, the entropy goes to zero. [ABSTRACT FROM AUTHOR]

The paper presents a solution to the problem of constructing the set of causal discrete controllers for strictly causal multidimensional discrete objects based on the application of determinant polynomial equations (DPE). It is shown that the properties of the solutions of the corresponding DPE depend on the type of the mathematical description of the controlled object. If the object is described by a backward polynomial model in terms of the B-shift operator ζ, it is shown that the corresponding DPE generates only causal solutions. This allows us to construct a parameterisation of the set of transfer matrices of causal stabilising controllers in a form that corresponds to the multidimensional Youla-Kučera parameterisation. For the case where the object is given as a forward polynomial model in the F-shift operator z, special techniques are developed to extract the set of all stabilising controllers from the subset of causal solutions. [ABSTRACT FROM AUTHOR]

The generalized L'vov–Kaplansky conjecture states that for any finite-dimensional simple algebra A the image of a multilinear polynomial on A is a vector space. In this paper, we prove it for the algebra of octonions $ \mathbb {O} $ O over a field F satisfying certain specified conditions (in particular, we prove it for quadratically closed fields, and for the field $ \mathbb R $ R ). In fact, letting V be the space of pure octonions in $ \mathbb {O} $ O , we prove that the image set must be either $ \{0\} $ { 0 } , F, V or $ \mathbb {O} $ O . We discuss possible evaluations of semihomogeneous polynomials on $ \mathbb {O} $ O and of arbitrary polynomials on the corresponding Malcev algebra. [ABSTRACT FROM AUTHOR]

Let G be a graph, and let $ A(G) $ A (G) be the adjacency matrix of G. The computation of permanent of $ A(G) $ A (G) is #p-complete. Computing permanent of $ A(G) $ A (G) is of great interest in quantum chemistry, statistical physics, among other disciplines. In this paper, we characterize the ordering of permanents of adjacency matrices of all graphs obtained from regular complete bipartite graph $ K_{p, p} $ K p , p by deleting six edges. As an application, we show that all graphs with a perfect matching obtained from $ K_{p, p} $ K p , p with six edges deleted are determined by their permanental polynomials. [ABSTRACT FROM AUTHOR]

Connection between the interlacing of the zeros and the orthogonality of a given sequence of polynomials is done by K. Driver. In this paper, we attempt to extend this result to some particular cases of d-orthogonal polynomials. In fact, first, we characterize the 2-orthogonality of a given sequence $ \{ P_{n} \}_{n\geq 0} $ { P n } n ≥ 0 , with the existence of a certain ratio $ c_n $ c n expressed by means of the zeros of $ P_n $ P n . Then, for the $ (d+1) $ (d + 1) -fold symmetric polynomials, $ \{ P_{n} \}_{n\geq 0} $ { P n } n ≥ 0 , such that $ P_n $ P n has $ q_n $ q n distinct positive real zeros, $ n=(d+1)q_n+j, j=0,\ldots,d $ n = (d + 1) q n + j , j = 0 , ... , d , we study the connection between the interlacing of these zeros, the d-orthogonality and the positivity of the ratio $ c_n $ c n . Finally, we give necessary and sufficient conditions on the zeros of a given sequence $ \{ P_{n} \}_{n\geq 0} $ { P n } n ≥ 0 , that will assure that this sequence satisfies a particular $ (d+1) $ (d + 1) -order recurrence relation. Many examples to illustrate the obtained results are given. [ABSTRACT FROM AUTHOR]

The fractional q-calculus has attracted the interest of a large number of academics over the last four decades or so, due mainly to a wide range of applications that cover natural sciences to social sciences. Many fractional q-calculus operators, particularly those involving various q-special functions, have been deeply studied and widely applied. In this paper, we aim to establish certain image formulas of Saigo fractional q-integral operators involving the product of generalized q-hypergeometric series and a general class of q-polynomials that are primarily expressed in terms of generalized q-hypergeometric series in a systematic manner. We demonstrate their use by studying q-Konhouser biorthogonal polynomials and q-Jacobi polynomials. Additionally, some fascinating special cases of our main findings are taken into consideration, and pertinent connections between some of the findings presented here and those from earlier studies are also made. [ABSTRACT FROM AUTHOR]

In this paper, an efficient Machine Learning-based Legendre wavelet method (ML-LWM) is utilised to estimate the parameters in ship damping models. Damping is critical for the roll motion response of a ship in waves. To the best of our knowledge, there is no Legendre Wavelet Method (LWM) has been reported to estimate the damping and restoring moments in ship dynamics models. LWM is applied to estimate roll angle, damping coefficients and restoring moments. Some numerical examples are given to demonstrate the validity and applicability of the proposed ML-LWM. The ML-LWM results are compared with the results obtained by the Homotopy Perturbation Method (HPM). Also, the proposed results are validated with the experimental data. Satisfactory agreement with experimental and HPM is noticed. The efficiency of the proposed wavelet method is confirmed by CPU runtime. [ABSTRACT FROM AUTHOR]

The development of certain aspects of hybrid special polynomials after incorporating monomiality principle, operational rules, and other properties and their aspects is obvious and indisputable. The study presented in this paper follows this line of research. By using the monomiality principle, new outcomes are produced, thus in line with prior facts, our aim is to introduce the degenerate 2D bivariate Appell polynomials Rmj (u, v; h). Further, we obtain some wellknown main properties and explicit forms satisfied by these polynomials. Also, generating relations corresponding to the degenerate 2D bivariate Bernoulli, Euler, and Genocchi polynomials are established, they will further be helpful in establishing the corresponding results for these polynomials. [ABSTRACT FROM AUTHOR]

Dedekind-type DC sums and their properties are defined in terms of Euler functions. Ma et al. recently introduced poly-Dedekind-type DC sums and demonstrated that they satisfy a reciprocity relation. In this paper, we introduce the degenerate poly-Euler polynomials and numbers, and we also consider the reciprocity relations of degenerate poly-Dedekind-type DC sums. Equivalently, several properties and identities of degenerate poly-Euler functions are obtained. [ABSTRACT FROM AUTHOR]

Recently, Kim-Kim introduced the truncated degenerate Bell polynomials and numbers. In this paper, weintroduce the truncated Lah-Bell polynomials and numbers. We obtain some identities, recurrence relations and properties. Furthermore, we also introduce the truncated modified Lah-Bell polynomials and numbers, truncated degenerate Lah-Bell polynomials and numbers and we derive some properties and identities involving with those polynomials and numbers. [ABSTRACT FROM AUTHOR]

In this paper, we provide a generating function for mix type Apostol-Genocchi polynomials of order η associated with Bell polynomials. We also derive certain important identities of Apostol Genocchi polynomials of order η based on Bell polynomials, such as the correlation formula, the implicit summation formula, the derivative formula, some correlation with Stirling numbers and their special instances. Moreover, we discover some symmetric identities and their related known results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the input–output finite-time (IO-FT) stabilisation problem for a class of continuous-time periodic piecewise polynomial systems (PPPSs) with immeasurable states and external disturbances via state estimation-based robust reliable controller. Firstly, to reflect the actuality, a parameter uncertainty that exhibits the random nature and the actuator faults with nonlinear character are considered in the addressed PPPSs and the control scheme, respectively. In detail, the randomness of uncertain parameters is portrayed by the stochastic variable and it is presumed to pursue the Bernoulli distributions. Secondly, to estimate the immeasurable states of PPPSs, a periodic piecewise polynomial observer is designed depending on the output of PPPSs. The main intent of this article is to devise a state estimation-based robust reliable controller to ascertain the IO-FT stabilisation of the PPPSs. Moreover, by bridging the Lyapunov stability theory, linear matrix inequality technique and IO-FT stability theory, the required IO-FT stabilisation conditions in the frame of linear matrix inequality are procured for the system under consideration. Eventually, simulation results of the addressed PPPSs are shown in line with the proposed analytical findings, revealing the competence, inherent capability and utility of the devised control protocol. [ABSTRACT FROM AUTHOR]

Let p 1 < p 2 < ⋯ < p n be positive numbers, then for any integer m the Loewner matrix associated with the function x m is given by L m = (p i m − p j m) / (p i − p j) i , j = 1 n. A question was left open in a paper [Bhatia R, Friedland S, Jain T. Inertia of loewner matrices. Indiana Univ Math J. 2016;65:1251–1261] by Bhatia, Friedland and Jain to find formulas for the determinants of the matrices L m in the case n = 3. We aim to answer this question firmly in terms of the Schur polynomials in this paper. [ABSTRACT FROM AUTHOR]

In his paper, 'Random walks and orthogonal polynomials: some challenges', F. A. Grunbaum gave the polynomials Q n (x) orthogonal with respect to the weight 4 p q − x 2 1 − x 2 on (− 4 p q , 4 p q) explicitly as Q n (x) = (q p) n 2 ((2 − 2 p) T n (x 2 p q ) + (2 p − 1) U n (x 2 p q )) , where T n and U n are, respectively, the Chebyshev polynomials of the first and second types. In this paper, similarly, we introduce the polynomials R n (x) defined by R n (x) = (q p) n 2 ((2 − 2 p) V n (x 2 p q ) + (2 p − 1) W n (x 2 p q )) , where V n and W n are, respectively, the Chebyshev polynomials of the third and fourth types, then we give the three term recurrence relation of the polynomials R n . Second, we give the kernel K n Q (x , 4 p q) where K n Q (x , y) is the Christoffel–Darboux formula for the polynomials Q n . Finally we give the integral of Q n function of T n and we show how we deduce that Q n (2 p q tan ⁡ (π 4 x)) is orthogonal with respect to the weight ϕ (x) = π p q 1 + tan 2 ⁡ (π 4 x) 1 − 4 p q tan 2 ⁡ (π 4 x) 1 − tan 2 ⁡ (π 4 x) on (− 1 , 1). [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of computing the nearest linearly structured polynomial matrix with prescribed distinct eigenvalues. The problem deals with the computation of the minimal structured perturbation to the coefficient matrices so that the perturbed polynomial matrix has the prescribed eigenvalues and is the nearest to the given polynomial matrix. In recent years, a series of papers have been published on the perturbation of polynomial matrices with prescribed eigenvalues; however, a linear structure-preserving result does not exist so far to the best of our knowledge. In this paper, we have proposed an optimization based approach where we have reformulated the problem as a constrained optimization problem. Towards the end, a few numerical case studies are presented which demonstrate the efficiency and usefulness of our proposed method. [ABSTRACT FROM AUTHOR]

This paper presents exact dual semi-definite programs (SDPs) for robust SOS-convex polynomial optimization problems with affinely adjustable variables in the sense that the optimal values of the robust problem and its associated dual SDP are equal with the solution attainment of the dual problem. This class of robust convex optimization problems includes the corresponding quadratically constrained convex quadratic optimization problems and separable convex polynomial optimization problems, and it employs a general bounded spectrahedron uncertainty set that covers the most commonly used uncertainty sets of numerically solvable robust optimization models, such as boxes, balls and ellipsoids. As special cases, it also demonstrates that explicit exact dual SDP and second-order cone programming (SOCP) in terms of original data hold for the robust two-stage convex quadratic programs with quadratic constraints and the robust two-stage separable convex quadratic programs under an ellipsoidal uncertainty set, respectively. Finally, the paper illustrates the results via numerical implementations of the developed SDP duality scheme on adjustable robust lot-sizing problems with nonlinear costs under demand uncertainty. [ABSTRACT FROM AUTHOR]

This paper focuses on the quasi-orthogonality between sequences of d -orthogonal polynomials and their characterization in terms of linear combinations as well as the left product of vector of linear forms by matrix polynomial. Therefore, structure relations for Hahn-classical d -orthogonal polynomials are provided. [ABSTRACT FROM AUTHOR]

The tensor product non-weight Virasoro modules M (V , λ 0) ⊗ ⨂ i = 1 m Ω (λ i , α i) are studied in this paper, where Ω (λ i , α i) and M (V , λ 0) (with λ 0 , λ i , α i ∈ C ∗ and V being an irreducible module over a certain finite dimensional Lie algebra related to the Virasoro algebra) are irreducible Virasoro modules, respectively, defined in Liu and Zhao [Generalized polynomial modules over the Virasoro algebra. Proc Amer Math Soc. 2016;144:5103–5112] and Lü and Zhao [Irreducible Virasoro modules from irreducible Weyl modules. J Algebra. 2014;414:271–287]. The necessary and sufficient condition for M (V , λ 0) ⊗ ⨂ i = 1 m Ω (λ i , α i) to be irreducible is obtained. Then we determine the necessary and sufficient condition for two such irreducible modules to be isomorphic. At last, we show that the irreducible L -modules M (V , λ 0) ⊗ ⨂ i = 1 m Ω (λ i , α i) are new for m>1. [ABSTRACT FROM AUTHOR]

This paper provides the exact, complete, and possibly non connected robust D -stability domain for linear uncertain dynamical systems, which polynomially depend upon a given number of real parameters. Derived in terms of a pencil of multiple degrees pertaining to the perturbed system's dynamics and to the characteristic function of the considered quadratic matrix inequality (QMI) region, the bounds of these robust D -stability domains, in the parameter space, turn out to be delimited by the finite, real and unduplicated generalised eigenvalues of this pencil, indicating when the system's eigenvalues cross the boundaries of the considered QMI region. The paper also shows that these generalised eigenvalues coincide with those of an augmented pencil of degree just one, thus rendering the computation of these generalised eigenvalues, in general, a standard and relatively simple operation. [ABSTRACT FROM AUTHOR]

S. Frisch, showed that the integer-valued polynomials on upper triangular matrix ring Int T n (K) (T n (D)) : = { f ∈ T n (K) [ x ] | f (T n (D)) ⊆ T n (D) } is a ring, where D is an integral domain with field of fractions K. Let R 1 ⊆ R 2 be commutative rings with identity. In this paper, we study the set Int T n (R 2) (Ω , T n (R 1)) : = { f ∈ T n (R 2) [ x ] | f (Ω) ⊆ T n (R 1) } for some subsets Ω ⊆ T n (R 1) . We generalize Frisch's result and show that Int T n (R 2) (T n (R 1)) : = Int T n (R 2) (T n (R 1) , T n (R 1)) is a ring. We state a lower bound for the Krull dimension of the integer-valued polynomials on upper triangular matrix rings. Finally, we state the concept of Skolem closure of an ideal of the integer-valued polynomials on upper triangular matrix rings and as a consequence, we obtain a classification of maximal ideals of the integer-valued polynomials on upper triangular matrix rings. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we use differential operator to present new identities and provide alternative proofs for certain established identities related to geometric polynomials. Additionally, by using the differential operator associated with geometric polynomials, Rolle’s theorem and a theorem of Wang and Yeh, we present some results on the real rootedness of polynomials. [ABSTRACT FROM AUTHOR]

AbstractIn this paper we study certain combinatorial attributes of Ind-schemes of polynomial automorphisms in positive characteristic. In particular, we prove that over an algebraically closed field K of positive characteristic ≠2 every automorphism of the group of origin-preserving automorphisms of the polynomial algebra K[x1,…,xn] (n≥3), which fixes every diagonal matrix, preserves, up to composition with a linear inner automorphism, every tame automorphism. [ABSTRACT FROM AUTHOR]

In the present paper we shall obtain the following result: Let n ≥ 2 and m ≥ 1 be integers. Let p (x 1 , ... , x m) be a polynomial with zero constant term in non-commutative variables over an algebraically closed field K. Suppose that 1 ≤ r ≤ n − 1 , where r is the order of p. Then p (T n (K)) is a dense subset of T n (K) (r − 1) (with respect to the Zariski topology). This is a solution of a problem presented by Gargate and de Mello and a supplement to a result obtained by Panja and Prasad in 2023. [ABSTRACT FROM AUTHOR]

In this paper, we provide a local method to find compositional inverses of all PPs, some new PPs and their compositional inverses are given. [ABSTRACT FROM AUTHOR]

AbstractInspection of raw materials, semi-finished products or final products is an important part of a quality assurance system. Acceptance sampling is a procedure used to decide either to accept or reject suppliers’ submitted lots in accordance with a specific standard. In this manuscript, designing acceptance sampling plans for lots with polynomial profile as quality characteristic is considered. To use previous lot information on deciding for accept or reject the available lot, exponentially weighted moving average (EWMA) statistic based on process yield index (Spk) of polynomial profiles with two-sided specification limits is applied. For this purpose, three acceptance sampling plans of repetitive group sampling (RGS), resubmitted and also combined RGS-resubmitted plans are developed in this paper. The Parameters of the proposed plans are determined using non-linear optimization model. Performance of the proposed acceptance sampling plans are compared using numerical methods and Monte-Carlo simulations. Simulation results emphasize that the proposed combined sampling plan mostly requires less average sampling number (ASN) and consequently leads to decreasing inspection time and costs. The application of the proposed combined plan is also demonstrated using a real case. [ABSTRACT FROM AUTHOR]

Polynomials with an exponential equivalent to the extent of a characteristic partition and coefficients proportional to the multiplicity/occurrence of the accompanying partition are termed as counting polynomials. These polynomials are a well-known method for describing a chemical graph's molecular invariants as polynomials. It is possible to deduce several key topological invariants from polynomials by either directly taking their value at a certain point or by calculating derivatives or integrals of the polynomial. A topological invariants is a real number correlated with a network that predicts the physico-chemical properties. Chemical modeling, drug design, and structural activity relations use invariants. This paper aims to find the counting polynomials such as Sadhana, omega, PI and theta polynomial of certain graphene nanostructures. In furthermore, topological invariants of specific graphene nanostructures that are related to these counting polynomials are investigated. [ABSTRACT FROM AUTHOR]

Let R be a (semi-) prime ring with extended centroid C, let f(X1, ... , Xk) be a multilinear polynomial over C in k noncommutative indeterminates which is not central-valued on R and let g be a generalized derivation of R. In this paper, we completely characterize the form of g and the structure of R such that (g(f(x1, ... , xk))m − γf(x1, ... , xk)n)s = 0 for all x1, ... , xk ∈ R, where γ ∈ C and m, n, s are fixed positive integers. Our results naturally improve and generalize the theorems obtained by Huang and Davvaz in [Generalized derivations of rings and Banach algebras, Comm. Algebra (2013); 43, 1188–1194] and the theorems recently obtained by De Filippis et al. in [Generalized derivations with nilpotent, power-central and invertible values in prime and semiprime rings, Comm. Algebra (2019); 47, 3025–3039]. Moreover, we describe a revised version of the theorem obtained by Huang in [On generalized derivations of prime and semiprime rings, Taiwanese J. Math. (2012); 16, 771–776.] [ABSTRACT FROM AUTHOR]

In this paper, the planar polynomial of a graph is introduced and some of its properties are discussed. The planar polynomial of the graph G is real-rooted if and only if G is planar. This polynomial is not EE-invariant. Some useful results about the roots of planar polynomial of complete graphs are presented. Moreover, all the graphs whose planar polynomial is of degree five are characterized. [ABSTRACT FROM AUTHOR]

Let I (Γ) be the homogeneous ideal of a finite set Γ in P n of d points in linearly general position. In 1972, Saint-Donat proved that if d ≤ 2 n then I (Γ) is generated by completely decomposable quadratic polynomials. Later, R. Treger generalized this result by showing that I (Γ) is generated by forms of degree ≤ m if d ≤ mn for some m ≥ 2 . This paper aims to generalize Saint-Donat's work in a different direction by proving that the degree m piece of I (Γ) can be generated by completely decomposable forms of degree m if and only if d ≤ mn . Communicated by Daniel Erman [ABSTRACT FROM AUTHOR]

Let L = ∑ i , j a i j ∂ x i ∂ x j + ∑ i b i ∂ x i + c be an elliptic operator with smooth enough coefficients, u a solution to the equation Lu = 0 in Ω ⊂ R n , ω ⊂ Ω an open set. As is well known, if u | ω = 0 then u = 0 everywhere in Ω. Let P be a polynomial of variables τ 1 , ... , τ m , functions u 1 , ... , u m the solutions to Lu = 0; let p (x) = P (u 1 (x) , ... , u m (x)). If the coefficients of the operator are (real) analytic then u 1 , ... , u m , p are analytic and p | ω = 0 implies p ≡ 0 in Ω. Is the same true for smooth but not analytic coefficients? The question also concerns the polynomials of harmonic quaternion fields. In general, the answer turns out to be negative: the paper provides the relevant counterexamples. [ABSTRACT FROM AUTHOR]

Let k be an arbitrary field, Λ be a k-algebra and V be a Λ -module. When it exists, the universal deformation ring R (Λ , V) of V is a k-algebra whose local homomorphisms to R parametrize the lifts of V up to R ⊗ k Λ , where R is any complete, local commutative Noetherian k-algebra with residue field k. Symmetric special biserial algebras, which coincide with Brauer graph algebras, can be viewed as generalizing the blocks of finite type p-modular group algebras. Bleher and Wackwitz classified the universal deformation rings for all modules for symmetric special biserial algebras with finite representation type. In this paper, we begin to address the tame case. Specifically, let Λ be any 1-domestic, symmetric special biserial algebra. By viewing Λ as generalized Brauer tree algebras and making use of a derived equivalence, we classify the universal deformation rings for those Λ -modules V with stable endomorphism ring isomorphic to k. The latter is a natural condition, since it guarantees the existence of the universal deformation ring R (Λ , V) . [ABSTRACT FROM AUTHOR]