Abstract We establish new expansion formulas connecting parabolic and Bessel waves in both directions, covering both ideal nondiffracting beams and their Gaussian-apodized (Helmholtz–Gauss) counterparts. Beyond deriving new closed-form expressions for the expansion coefficients in terms of hypergeometric functions, we introduce a simple unified recursive scheme that generates all coefficients, bypassing the numerical evaluation of hypergeometric functions and significantly reducing computational complexity. The recurrence relations not only improve numerical evaluation of the coefficients but also reveal an underlying algebraic structure common to all coefficient families, enabling more transparent derivations and straightforward inversion of the expansion formulas. Additionally, we show that the expansion coefficients admit an equivalent formulation in terms of orthonormal functions whose polynomial component is proportional to continuous Hahn polynomials. For parabolic–Gauss beams, we further derive the normalization constant in terms of a compact and rapidly convergent series built from the same orthonormal functions. Numerical validations show that the recursive approach enables high-fidelity reconstruction of parabolic beams from the more accessible Bessel basis.

A considerable amount of research has been devoted to the study of strong commutativity-preserving (SCP) maps, and numerous results have been established in the context of various algebraic structures, particularly prime and semiprime rings. However, despite this significant progress, the concept of SCP maps has not yet been explored in connection with multiplicative (generalized)-derivations. This observation reveals a clear gap in the existing literature. Motivated by this, the present work aims to initiate and develop the study of strong commutativity-preserving properties for multiplicative (generalized)-derivations. The primary objective of this paper is to investigate the behavior of a multiplicative (generalized)-derivation [Formula: see text] associated with the map [Formula: see text] on a semiprime ring [Formula: see text] under certain commutativity-related identities within a non-zero ideal [Formula: see text] of [Formula: see text]. In particular, we focus on the following conditions : (1) [Formula: see text] for all [Formula: see text]; (2) [Formula: see text] for all [Formula: see text]; (3) [Formula: see text] for all [Formula: see text].

In recent years, the study of generalized fuzzy structures in algebraic systems has attracted considerable attention due to their ability to represent uncertainty and bipolar information. In this paper, we introduce the notion of cubic bipolar ideals in the framework of Sheffer stroke Hilbert algebras. This concept integrates the descriptive capability of cubic sets with the dual representation of bipolar information, providing a broader perspective for investigating algebraic structures associated with the Sheffer stroke operation. We establish the definition of cubic bipolar ideals and investigate several of their fundamental properties. In particular, the structural behavior of these ideals is examined within Sheffer stroke Hilbert algebras. Furthermore, the preservation of cubic bipolar ideals under algebraic homomorphisms is analyzed through the study of images and preimages. The Cartesian product of cubic bipolar ideals is also discussed, and conditions ensuring the stability of the resulting structures are obtained. The results presented here contribute to the development of fuzzy algebraic theory and extend existing approaches to Sheffer stroke-based algebraic systems.

In this paper, all biderivations of the gap-p Virasoro algebra 𝔤 are presented. As applications, the forms of linear commuting maps and the commutative post-Lie algebra structures on the gap-p Virasoro algebra are characterized.

In this short review paper, a detailed analysis of six two-dimensional quantum superintegrable systems in flat space is presented. It includes the Smorodinsky–Winternitz potentials I–II (the Holt potential), the Fokas–Lagerstrom model, the 3-body Calogero and Wolfes (equivalently, [Formula: see text] rational, or [Formula: see text]) models and the Tremblay–Turbiner–Winternitz (TTW) system with integer index k. It is shown that all of them are exactly-solvable, thus, confirming the Montreal conjecture (2001); they admit algebraic forms for the Hamiltonian and both integrals (all three can be written as differential operators with polynomial coefficients without a constant term), they have polynomial eigenfunctions with the invariants of the discrete symmetry group of invariance taken as variables, they have hidden (Lie) algebraic structure [Formula: see text] with various k and they possess a (finite order) polynomial algebras of integrals. Each model is characterized by infinitely-many finite-dimensional invariant subspaces, which form the infinite flag. Each subspace coincides with the finite-dimensional representation space of the algebra [Formula: see text] for a certain k. In all presented cases, the algebra of integrals is a 4-generated [Formula: see text] infinite-dimensional algebra of ordered monomials of degrees 2–5, which is a subalgebra of the universal enveloping algebra of the hidden algebra.

Abstract Over the past decades, transformations between different classes of eigenvalue problems have played a central role in the development of numerical methods for eigenvalue computations. One of the most well-known and successful examples of this is the companion linearization for polynomial eigenvalue problems. In this paper, we construct a transformation that equivalently reframes a specific type of eigenvalue problem with eigenvector nonlinearities (NEPv) into an eigenvalue problem with eigenvalue nonlinearities (NEP). The NEPv class considered consists of nonlinearities expressed as sums of products of matrices and scalar functions, where the scalar functions depend nonlinearly on the eigenvector. Our transformation defines scalar eigenvalue nonlinearities through a polynomial system, resulting in NEP nonlinearities of algebraic type. We propose methods to solve the polynomial system, one of which involves a multiparameter eigenvalue problem (MEP). We adapt well-established NEP solvers to this setting, with the most effective strategy being a combination of deflation and a locally quadratically convergent iterative method. The efficiency and properties of the approach are illustrated by solving a problem related to a modification of a Gross-Pitaevskii equation (GPE). The simulations are reproducible and publicly available.

This paper presents a systematic algebraic construction of noncyclic generalizations of BCH and Srivastava codes over Galois rings [Formula: see text] The proposed codes are defined via parity-check matrices whose entries are carefully chosen from the Galois ring, leading to determinants of the Alternant type. This structure, when combined with careful selection of ring elements to ensure key determinants are units, allows us to derive a rigorous lower bound on the minimum distance, providing a theoretically guaranteed error-correcting capability. We explicitly construct these codes and compute their core parameters. A comparative analysis with classical constructions over finite fields shows that for the same length [Formula: see text] and designed distance d, the ring-based construction achieves the same dimension k but with symbols drawn from a larger alphabet of size [Formula: see text] This yields a codebook of size [Formula: see text] representing an increase in information density (bits per codeword) compared to the field-based codebook of size [Formula: see text] The increased information rate comes at the cost of greater algebraic complexity in implementation, while the guaranteed minimum distance remains unchanged. This work establishes a foundational framework for applying advanced algebraic structures in noncyclic coding theory, with implications for modern communication systems requiring robust error control.

In this paper, we present the spinor structure associated with the Hyperbolic Clifford algebra of a real n-dimensional vector space V, which is denoted by ClHV. Unlike the standard Clifford algebra, the Hyperbolic Clifford algebra Cl(HV) simultaneously accommodates both multiforms and multivectors in a single algebraic structure, making it the natural framework—known as the “mother algebra”—for the study of superfields in theoretical physics and for generalizing the Clifford bundle formalism to hyperbolic structures arising in gravitational theories. The orthogonal groups and orthogonal transformations associated to the hyperbolic space HV are presented. The Clifford–Lipschitz group and the Pin and Spin groups associated with ClHV are defined. Then, the frame bundle and spinor structure associated to Hyperbolic Clifford algebra is derived.

This paper presents a unified mathematical and computational framework for analyzing melodic structures in North Indian classical music through a novel concept termed the raga-restricted operation. This operation provides an algebraic mechanism to model permissible note transitions while enforcing the grammatical constraints of a raga. The proposed framework integrates algebraic structures, graph-theoretic representations, and stochastic modeling using Markov chains to achieve a coherent and computationally tractable description of melodic behavior. The methodology is applied to two representative ragas, Raga Yaman and Raga Bhupali, where melodic sequences are generated under raga constraints and analyzed using directed graphs, transition probability matrices, and stationary distributions. The results demonstrate that the framework preserves the intrinsic structure of ragas while enabling quantitative analysis of transition dynamics, probabilistic dependencies, and long-term behavior. From a computational perspective, the model supports systematic sequence generation, prediction of note transitions, and structural analysis of melodic patterns. Thus, the proposed approach establishes a rigorous connection between music theory and mathematical modeling, contributing to the advancement of computational musicology and probabilistic analysis of raga-based systems.

This paper proposes a recursive method for constructing intra-resolvable balanced incomplete block designs (BIBDs). The approach exploits the algebraic and geometric structure of finite projective geometries over Galois fields to generate resolvable designs with improved efficiency in terms of the number of blocks and treatment replications. The recursive procedure produces symmetric and uniform designs that are particularly suitable for high-dimensional settings. By systematically nesting resolvable blocks, we derive a new class of balanced n-ary designs that are both economical and scalable. These designs hold significant value for the statistical community, offering broad applicability in resource-constrained experimental environments such as precision agriculture, high-throughput drug screening, and computer-based simulation studies. We provide theoretical foundations through explicit constructions and comparative evaluations, demonstrating the advantages of our method over classical approaches.

In this study, a generalized and advanced form of the near soft set theory (NST) framework is proposed for information aggregation (IA) processes. The primary motivation of the study is to address the lack of similarity-based uncertainty modeling in the literature by integrating the parametric structure of soft sets with the similarity-oriented structure of nearness approximation spaces. Within this framework, the AND-product and OR-product operations are introduced as the main methodological tools, and their algebraic structures are analyzed in detail. It is mathematically demonstrated that these operations satisfy fundamental properties such as idempotency, absorption, distributivity, and De Morgan identities. The principal original contribution of the study is the development of a novel Uni–Int-based decision-making mechanism that enables the systematic distinction between strong and acceptable alternatives. In addition, the boundary frequency indicator (br), which quantitatively evaluates the reliability of objects under perceptual uncertainty and is introduced for the first time in the literature, is proposed. The applicability of the proposed model is demonstrated through a real-estate selection problem, and a sensitivity analysis is conducted to reveal the determining effect of the nearness parameter r on decision granularity. The obtained findings indicate that the proposed NST framework provides a more flexible, more discriminative, and structurally robust decision-support model than classical approaches, particularly for similarity-based IA problems.

Abstract While General Fractional Calculus has successfully expanded the scope of memory operators beyond power-laws, standard formulations remain predominantly restricted to the half-line via Riemann-Liouville or Caputo definitions. This constraint artificially truncates the system's history, limiting the thermodynamic consistency required for modeling processes on unbounded domains. To overcome these barriers, we construct the Weighted Weyl-Sonine Framework, a generalized formalism that extends non-local theory to the entire real line without history truncation.Unlike recent algebraic approaches based on conjugation for finite intervals, we develop a rigorous harmonic analysis framework. Our central contribution is the Generalized Spectral Mapping Theorem, which establishes the Weighted Fourier Transform as a unitary diagonalization map for these operators. This result allows us to rigorously classify and solve distinct physical regimes under a single algebraic structure. We explicitly derive exact solutions for diffusive relaxation (governed by Complete Bernstein Functions), inertial wave propagation (exhibiting oscillatory dynamics), and retarded aging (via distributed order), proving that our framework unifies the description of anomalous transport and wave mechanics in complex, time-deformed media.

Abstract This paper addresses the problem of achieving quadratic convergence in quaternion phase retrieval. To this end, we propose a novel augmented quaternion Gauss–Newton (AQGN) framework for recovering quaternion signals from magnitude-only measurements. The method is developed based on the generalized Hamilton–real (GHR) calculus, which rigorously preserves the intrinsic quaternion algebraic structure while enabling efficient second-order optimization without component-wise decomposition. The main contributions are threefold. \textit{i)} We derive the AQGN update rule within the GHR calculus framework. \textit{ii)} We prove that the resampled AQGN achieves quadratic convergence under spectral initialization with high probability, provided $\mathcal{O}(n \log n)$ quaternion Gaussian measurements. \textit{iii)} We further show that the algorithm attains an $\epsilon$-accurate solution within $\mathcal{O}(\log_2 \log_2 \epsilon^{-1})$ iterations, demonstrating a substantial reduction in iteration complexity compared with existing quaternion-based optimization methods. Extensive numerical experiments on both synthetic datasets and real-world images demonstrate that the proposed method generally outperforms state-of-the-art approaches in terms of recovery accuracy, convergence speed, and computational efficiency. Overall, the proposed framework establishes a provably efficient second-order method for hypercomplex phase retrieval, providing both theoretical guarantees and practical scalability for high-dimensional quaternion signal recovery.

Insertion-deletion (insdel for short) codes have received extensive attention due to their ability to correct synchronization errors. It is usually a very challenging problem to determine the insdel distances of linear codes. In this paper, a good lower bound on the insdel distance of a linear code with a certain algebraic structure is provided and it indeed gives an affirmative answer to an open problem proposed by Hao Chen (IEEE Transactions on Information Theory, 68(8): 5126–5132, 2022). Applying this lower bound to binary first-order Reed-Muller codes and binary simplex codes, we obtain that they possess linear subcodes that can correct arbitrary insdel errors while maintaining Hamming distances robustness. The application of this bound on Reed-Muller codes completely solves an open problem left by Lara Dolecek and Venkat Anantharam (IEEE Transactions on Information Theory, 53(4): 1430–1443, 2007). In order to enhance the code rate, we further perform puncturing on these subcodes and determine the Hamming distances of the punctured subcodes while ensuring that their insdel distances remain unchanged.

In this paper it is justified in full length that there is a genuine need for the giant subject ‘ALGEBRA’ to have a new but unique Algebraic Structure. Consequently, a new algebraic structure “Region” is introduced, and its properties are studied. This paper introduces a new algebraic structure called Region, providing a unified framework for interactions between internal multiplication and scalar multiplication commonly used in algebra. The novelty lies in combining field structure, vector space structure, and compatibility in a single framework. Without the algebraic structure “Region” the subject ‘ALGEBRA’ can not validate many elementary algebraic computations being frequently practiced by the mathematicians, researchers and students in their daily works in the last centuries; unbelievable and surprising, but it is true.

Error-Correcting Code (ECC) is one of the solutions that have been widely developed to overcome errors, with the aim of detecting and correcting errors that occur. Among the various types of ECC, Goppa code have proven to be effective in maintaining data integrity. This study focuses on examining how the algebraic structure of the Goppa code is utilized for ECC, thereby enabling an understanding of how elements in the field can be transformed into binary vectors and polynomials to construct the parity-check matrix and the generator matrix. The algebraic structure is then analyzed to determine whether it can be optimized to detect and correct errors in the encoding and decoding processes. Subsequently, a simple program will be developed using the SageMath programming language, followed by simulations. The simulation results are expected to optimize the error-correction capacity, thereby demonstrating an improvement in the error-correction capability of the Goppa code during the decoding process.

We study the algebraic structure of DNA codes constructed over the ring R=F_2 [u,v,w] ⟨u^2=v^2,uv=0,w^2=w⟩, which is a commutative local Frobenius non-chain ring. We define a gray map over R and generate DNA codes using the images of the gray map. We define reversible DNA codes and reversible complement DNA codes over the ring.

In this paper, we investigate the computational complexity of the knapsack problem and subset sum problem for the following tropical algebraic structures. We consider the semigroup of square matrices of size k × k with non-negative entries over the max-plus algebra and the semigroup of square matrices of size k × k with positive entries over the max-times algebra. We prove that the knapsack problem and the subset sum problem for these structures are 𝖭𝖯-complete. We demonstrate that there are pseudo-polynomial algorithms to solve these problems. Also, we show that for the latter semigroup, there are polynomial generic algorithms to solve the knapsack problem and the subset sum problem.

Let $\mathfrakB$ be a graded commutative ring with unity, and let $\mathfrakW$ be a graded unital $\mathfrakB$-module. This study introduces and develops the concept of graded strongly $J_gr^Soc$-2-absorbing submodules, a natural extension of graded $J_gr$-2-absorbing submodules within the framework of graded module theory. The motivation for this generalization stems from the need to better capture the interplay between graded algebraic structures and the behaviors of certain radicals and socles under graded operations. A properly graded submodule $N$ of $\mathfrakW$ is defined as a graded strongly $J_gr^Soc$-2-absorbing submodule if, for all $b,uin h( \mathfrakB )$ and $cin h( \mathfrakW )$, the containment $bucin N$ implies that at least one of the following conditions holds: $bcin N + ( J_gr( \mathfrakW ) \cap Soc^gr( \mathfrakW ) )$, $ucin N + ( J_gr( \mathfrakW ) \cap Soc^gr( \mathfrakW ) )$, or $buin ( N + ( J_gr( \mathfrakW ) \cap Soc^gr( \mathfrakW ) ):_\mathfrakB\mathfrakW )$. Several fundamental properties of these submodules are established, along with characterizations that distinguish them from related graded structures. Moreover, the investigation reveals meaningful connections between these submodules and the graded socle and graded Jacobson radical of the module, offering new insights into their algebraic significance.

Mekler's construction is a powerful technique for building purely algebraic structures from combinatorial ones.Its power lies in the fact that it allows various model-theoretic tameness properties of the combinatorial structure to transfer to the algebraic one.In this paper, we push this ideology much further, describing a broad class of properties that transfer through Mekler's construction.This technique subsumes many well-known results and opens avenues for many more.As a straightforward application of our methods, we obtain transfer principles for stably embedded pairs of Mekler groups and construct strictly NFOP k pure groups for all k >2 .We also answer a question of Chernikov and Hempel on transfer of burden.