Algebraic Properties Research Articles

A New Class of Operators: LM-operators

This paper introduces a new class of operators called LM- operators, extending the concepts of L-weakly and M-weakly compact operators within the framework of Banach lattices. While L-weakly and M-weakly compact operators have established properties, LM-compact operators retain some of these characteristics but also diverge in certain respects. The study explores the relationships between LM-compact operators and some existing classes, and examines their algebraic properties and duality aspects.

GENERALIZED n-POLYNOMIAL P-FUNCTIONS WITH SOME RELATED INEQUALITIES AND THEIR APPLICATIONS

Abstract. In this paper, we introduce the notion of generalized n-polynomial P-function. We explore some algebraic properties of this function class. Additionally, we establish a new trapezium type inequality for this generalized class of functions and derive several refinements of the trapezium type inequality for functions whose first derivative in absolute value at a certain power is generalized n-polynomial P-function. Finally, we conclude our paper by exploring some applications of the results we have obtained in the context of special means. Our novel findings generalize previously known results in the literature.

HG-CONVEX FUNCTIONS

In this paper, the concept of Hg-convex function is given for the first time in the literature. Some inequalities of Hadamard’s type for Hg-convex functions are given. Some algebraic properties of Hg-convex functions and special cases are discussed. In addition, we establish some new integral inequalities for Hg-convex functions by using an integral identity.

ON THE MODULAR SEQUENCE SPACES GENERATED BY THE CESÀRO MEAN

In this paper, the seminormed Ces\`aro difference sequence space \( \ell(\mathcal{F}_j, q, g, r, \mu, \Delta_{({s})}^{t}, \mathcal{C})\) is defined by using the generalized Orlicz function. Some algebraic and topological properties of the space \(\ell(\mathcal{F}_j, q, g, r, \mu, \Delta_{({s})}^{t}, \mathcal{C}) \) are investigated. Various inclusion relations for this sequence space are also studied.

RESTRICTED AND EXTENDED THETA OPERATIONS OF SOFT SETS: NEW RESTRICTED AND EXTENDED SOFT SET OPERATIONS

Since its introduction by Molodtsov in 1999, soft set theory has gained widespread recognition as a method for addressing uncertainty-related issues and modeling uncertainty. It has been used to solve several theoretical and practical issues. Since its introduction, the central idea of the theory-soft set operations-has captured the attention of scholars. Numerous limited and expanded businesses have been identified, and their attributes have been scrutinized thus far. We present a detailed analysis of the fundamental algebraic properties of our proposed restricted theta and extended theta operations, which are unique restricted and extended soft set operations. We also investigate these operations’ distributions over various kinds of soft set operations. We demonstrate that, when coupled with other types of soft set operations, the extended theta operation forms numerous significant algebraic structures, such as semirings in the collection of soft sets over the universe, by taking into account the algebraic properties of the extended theta operation and its distribution rules. This theoretical subject is very important from both a theoretical and practical perspective since soft sets' operations form the foundation for numerous applications, including cryptology and decision-making procedures.

Steenrod operator of the dihedral homology of A_∞-algebras

In this study, we examine the interaction between Steenrod operations and dihedral homology within the framework of A-infinity algebras, aiming to understand how these operations, initial in algebraic topology, contribute to homological invariants and influence dihedral homology structures. We begin by exploring the initial properties of A-infinity algebras as generalizations of associative algebras, focusing on their volume to support Steenrod operations. From there, we delve into the effects of these operations within dihedral homology, uncovering their role in revealing deeper algebraic structures and improving our understanding of homology theories. We also analyze the connections between Steenrod operations and projective varieties over finite fields, emphasizing their actions in derived categories and their significance in the context of α-characteristic fields. By defining Steenrod operators within dihedral homology, we explain the complex relationships between these algebraic structures and operations derived from homology theory. Through specific examples and theoretical models, we demonstrate how these interactions advance our understanding of homological invariants and provide valuable tools and perspectives for the broader fields of algebraic topology and homological algebra.

ON AN OVER-RING C(X)∆ OF C(X)

Our aim in this paper is to introduce a ring of functions defined on a topological space X having a special property. By C(X)∆ we denote the set of all realvalued functions defined on the topological space X, the discontinuity set of elements of which are members of ∆ ⊆ P(X), where ∆ satisfies the following properties: (i) for each x ∈ X, {x} ∈ ∆, (ii) for A, B ∈ P(X) with A ⊆ B, B ∈ ∆ implies that A ∈ ∆ and (iii) for A, B ∈ ∆, A ∪ B ∈ ∆. This C(X)∆ is an over-ring of C(X), moreover, C(X) ⊆ C(X)F ⊆ C(X)∆ ⊆ RX. The ring C(X)∆ is also almost regular. We study the ∆-completely separated sets and C∆-embedded subsets of X. Complete characterizations of fixed maximal ideals are then done and algebraic properties of C(X)∆ have been studied. In [6], the authors have introduced FP-spaces, for which the ring C(X)F is regular. Here we have generalized the notion of FP-spaces in the context of C(X)∆, so that the ring in question becomes regular. As a result, ∆Pspaces have been introduced, it has been proved that every P-space is a ∆P-space and examples are given in support of the fact that there exist ∆P-spaces which are not P-spaces.

Dihedral beams

Abstract In this work, a group theory-based formulation that introduces new classes of dihedral-symmetric beams is presented. Our framework leverages the algebraic properties of the dihedral group of rotations and reflections to transform input beams into closed-form families of dihedral-invariant wavefields, which will be referred to as dihedral beams. Each transformation is associated with a specific dihedral group in such a way that each family of dihedral beams exhibits the symmetries of its corresponding group. Our approach is inspired by one of the outcomes of this work: elegant Hermite–Gauss beams can be described as a dihedral interference pattern of elegant traveling waves, a new set of solutions to the paraxial equation also developed in this paper. Particularly, when taking elegant traveling waves as input beams, they transform into elegant dihedral beams possessing quasi-crystalline properties and including features like phase singularities, self-healing, and pseudo-nondiffracting propagation, as well as containing elegant Hermite and Laguerre–Gauss beams as special cases. Our approach can be applied to arbitrary scalar and vector input beams and constitutes a general group-theory formulation that can be extended beyond the dihedral group.

Multi-party key exchange scheme based on supertropical semiring

In an era where digital communication and data sharing are everywhere, ensuring the security of sensitive information is important. Cryptographic protocols are essential for establishing secure communication over potentially insecure networks. This paper introduces a novel three-party key exchange protocol based on a newly constructed supertropical semiring. By leveraging the unique algebraic properties of supertropical semirings, our protocol aims to provide enhanced security against known attacks on tropical based cryptographic schemes. We demonstrate the security and efficiency of our protocol through detailed analysis and examples, showing its robustness in protecting confidential information.

H-infinity consensus control of fractional-order multi-agent systems

This paper considers the H-infinity consensus problem of fractional-order multi-agent systems (FOMASs) with external disturbances ( 0 < α < 1 ). Using matrix spectral decomposition and the algebra properties of the Kronecker product, the H-infinity consensus problem in FOMASs is converted to an H-infinity problem for a group of fractional-order systems (FOSs). Based on the bounded real lemma of H-infinity stability for FOS in term of linear matrix inequalities (LMIs), a necessary and sufficient condition for the existence of an H-infinity controller achieving consensus in FOMASs is derived. A multi-step procedure for constructing controller is further presented. In the simulation part of the paper, the viability of the proposed method is demonstrated by a simulation example. A comparative analysis of the performance of controllers designed based on integer-order models and fractional-order models is subsequently conducted to elucidate the distinct advantages of method in this paper.

Set-Theoretical Solutions for the Yang–Baxter Equation in GE-Algebras: Applications to Quantum Spin Systems

This manuscript presents set-theoretical solutions to the Yang–Baxter equation within the framework of GE-algebras by constructing mappings that satisfy the braid condition and exploring the algebraic properties of GE-algebras. Detailed proofs and the use of left and right translation operators are provided to analyze these algebraic interactions, while an algorithm is introduced to automate the verification process, facilitating broader applications in quantum mechanics and mathematical physics. Additionally, the Yang–Baxter equation is applied to spin transformations in quantum mechanical spin-12 systems, with transformations like rotations and reflections modeled using GE-algebras. A Cayley table is used to represent the algebraic structure of these transformations, and the proposed algorithm ensures that these solutions are consistent with the Yang–Baxter equation, offering new insights into the role of GE-algebras in quantum spin systems.

On Linear Operators in Hilbert Spaces and Their Applications in OFDM Wireless Networks

This paper explores the application of Hilbert topological spaces and linear operator algebra in the modelling and analysis of OFDM signals and wireless channels, where the channel is considered as a linear time-invariant (LTI) system. The wireless channel, when subjected to an input OFDM signal, can be described as a mapping from an input Hilbert space to an output Hilbert space, with the system response governed by linear operator theory. By employing the mathematical framework of Hilbert spaces, we formalise the representation of OFDM signals, which are interpreted as elements of an infinite-dimensional vector space endowed with an inner product. The LTI wireless channel is characterised by using bounded linear operators on these spaces, allowing for the decomposition of complex channel behaviour into a series of linear transformations. The channel’s impulse response is treated as a kernel operator, facilitating a functional analysis approach to understanding the signal transmission process. This representation enables a more profound understanding of channel effects, such as fading and interference, through the eigenfunction expansion of the operator, leading to a spectral characterization of the channel. The algebraic properties of linear operators are leveraged to develop optimal solutions for mitigating channel distortion effects.

On the algebraic property of locally convex topological algebras

On the algebraic property of locally convex topological algebras

On Stanley–Reisner Rings with Linear Resolutions

For a graph G, Bayer–Denker–Milutinović–Rowlands–Sundaram–Xue study in [1] a new graph complex Δkt(G), namely the simplicial complex with facets that are complements to independent sets of size k in G. They are interested in topological properties such as shellability, vertex decomposability, homotopy type, and homology of these complexes. In this paper we study more algebraic properties, such as Cohen-Macaulayness, Betti numbers, and linear resolutions of the Stanley-Reisner ring of these complexes and their Alexander duals.

Symmetric group gauge theories and simple gauge/string dualities

Abstract We study two-dimensional topological gauge theories with gauge group equal to the symmetric group Sn and their string theory duals. The simplest such theory is the topological quantum field theory of principal Sn fiber bundles. Its correlators are equal to Hurwitz numbers. The operator products in the gauge theory for each finite value of n are coded in one partial permutation algebra. We propose a generalization of the partial permutation algebra to the symmetric orbifold topological quantum field theory of any seed theory and show that the theory factorizes into marked partial permutation combinatorics and seed Frobenius algebra properties. Moreover, we exploit the established correspondence between Hurwitz theory and the stationary sector of Gromov–Witten theory on the sphere to prove an exact gauge/string duality. The relevant field theory is a grand canonical version of Hurwitz theory and its two-point functions are obtained by summing over all values of the instanton degree of the maps covering the sphere. We stress that one must look for a multiplicative basis on the boundary to match the bulk operator algebra of single string insertions. The relevant boundary observables are completed cycles.

Weak exactness and amalgamated free product of von Neumann algebras

Weak exactness and amalgamated free product of von Neumann algebras

Insight into soft binary piecewise lambda operation: a new operation for soft sets

AbstractThe notion of soft set operations is one of the key concept for soft set theory as the theory has been progressing, both theoretically and practically, based on this notion. As proposing new soft set operations, deriving their algebraic properties, and studying the algebraic structure of soft sets from the perspective of soft set operations offer a comprehensive understanding of their applications as well as the appreciation of how soft set algebra can be applied to classical and nonclassical logic, in this study, a new soft set operation, called the “soft binary piecewise lambda operation" is proposed. Since one of the the main objective of abstract algebra is to analyze the properties of the operations defined on a set to classify the algebraic structures, the operation’s full properties and its distributions over other soft set operations are investigated to reveal which algebraic structures the operation forms individually, and together with other soft set operations in the collection of soft sets over the universe. It is showed that the operation forms a noncommutative semigroup and a right-left system, besides semi-rings and near-semi-rings together with certain types of soft set operations under certain conditions in the collection of soft sets over the universe. Since such in-depth analyses advance our knowledge of the applications of soft sets over a range of field, this novel operation may serve as an inspiration to create new perspectives for addressing issues related to parametric data, soft set-based cryptography, or decision-making techniques in practical settings, business, and technology.

Semi-Overlap Functions on Complete Lattices, Semi-Θ-Ξ Functions, and Inflationary MTL Algebras

As new kinds of aggregation functions, overlap functions and semi overlap functions are widely applied to information fusion, approximation reasoning, data classification, decision science, etc. This paper extends the semi-overlap function on [0, 1] to the function on complete lattices and investigates the residual implication derived from it; then it explores the construction of a semi-overlap function on complete lattices and some fundamental properties. Especially, this paper introduces a more generalized concept of the ‘semi-Θ-Ξ function’, which innovatively unifies the semi-overlap function and semi-grouping function. Additionally, it provides methods for constructing and characterizing the semi-Θ-Ξ function. Furthermore, this paper characterizes the semi-overlap function on complete lattices and the semi-Θ-Ξ function on [0, 1] from an algebraic point of view and proves that the algebraic structures corresponding to the inflationary semi-overlap function, the inflationary semi-Θ-Ξ function, and residual implications derived by each of them are inflationary MTL algebras. This paper further discusses the properties of inflationary MTL algebra and its relationship with non-associative MTL algebra, and it explores the connections between some related algebraic structures.

Some algebraic and analytical properties of a class of two-place functions

Some algebraic and analytical properties of a class of two-place functions

Mixed skew cyclic codes over rings

This paper explores different types of skew cyclic codes by generating special subclasses with additional desirable properties. Specifically, we are interested in skew cyclic codes over mixed rings. We study some algebraic and structural properties of these codes and their constructions. We study skew cyclic codes over the mixed alphabet ring $\mathbb{F}_q(\mathbb{F}_q+v\mathbb{F}_q)$ under a mixed automorphism $(\theta,\tilde{\theta})$ and we give the structure of these codes for an arbitrary length via the non-commutative ring $\mathbb{F}_q[x,\theta](\mathbb{F}_q+v\mathbb{F}_q)[x,\tilde{\theta}]$. A condition for the existence of linear complementary dual (LCD) codes (which play an important role in practical applications such as armoring implementations against side-channel attacks and fault injection attacks) are explored specifically for skew cyclic codes.