Algebraic Features Research Articles

COMPLETE CLASSIFICATION OF QUADRATIC IRRATIONALS WITH PERIOD TWO

This article presents a comprehensive investigation into the classification of quadratic irrationals with period two in their continued fraction representations. Building upon foundational results in Number Theory, particularly in the context of continued fractions and Pell's equation, the study reveals intricate relationships between quadratic irrationals and their periodic structures. The main object of study is √N and properties of its continued fractions. While it is well-known that continued fractions of √N is periodic with periodic part being palindrome, the distribution of the lengths of the periodic parts are far from being complete. Our main goal will be to focus on the period two case and provide a complete characterization. The research's proved theorems clarify the conditions under which the period length is exactly two and give an insight into the underlying algebraic features. Additionally, it delves deeper by offering numerical analysis and illustrations demonstrating the distribution of period lengths among quadratic irrationals. This research opens up new paths for future studies on quadratic irrationals and how they're shown as continued fractions.

AN ENCRYPTION ALGORITHM EMPLOYING GRAPHS

With the advancement of technology, maintaining secrecy is a crucial concern that requires a variety of skills. A scientific method for protecting communication against unauthenticated access is cryptography. In cryptography, there are several encryption techniques for data security. It has been suggested that new nonstandard encryption techniques are needed to shield communication from conventional threats. This work presents a method that uses graphs together with some algebraic features to provide some new encryption techniques for safe message transfer. The transmission of secret communications will be safer because of the suggested encryption techniques.

Neutrosophic Fuzzy Tribonacci ℐ-Lacunary Statistical Convergent Sequence Spaces

Abstract The aim of this paper is to introduce and investigate some neutrosophic fuzzy tribonacci ℐ-lacunary statistical convergent sequence spaces by utilizing the domain of regular tribonacci matrix A = (ajk ). Morever, we also put forward various algebraic and topological features of these convergent sequence spaces and establish several interesting inclusion relations.

Investigating dispersive optical solitons with the generalized stochastic perturbed Schrödinger–Hirota equation incorporating power-law nonlinearity and multiplicative white noise

This article aims to introduce the generalized stochastic perturbed Schrödinger–Hirota equation, which incorporates multiplicative white noise in the Itô sense. The study focuses on investigating the stochastic optical soliton solutions related to the governing model to understand the characteristics and properties of these soliton solutions. We can achieve this by utilizing the enhanced Kudryashov method and the enhanced direct algebraic approach. The study stands out for its innovative approach to examining the effects of multiplicative white noise on stochastic optical soliton solutions through the lens of the generalized stochastic perturbed Schrödinger–Hirota equation. Utilizing the enhanced Kudryashov method alongside the novel enhanced direct algebraic method, this research effectively identifies and delineates a range of soliton solution types, such as bright, dark, singular, and straddled solitons. This thorough investigation not only underscores the efficacy and efficiency of these methods but also sheds light on the impact of noise on the structural and dynamic properties of soliton solutions, supported by illustrative visualizations. A notable finding is that all solutions derived using the enhanced direct algebraic method feature a nonzero coefficient for the highest-order term of the auxiliary equation. The diversity of solutions obtained through these approaches includes bright, dark, singular, and straddled solitons. Additionally, the study reveals Weierstrass and Jacobi elliptic doubly periodic solutions, which can transition into various soliton types under specific parameter conditions, offering a broader understanding of soliton behavior and properties.

On the Construction of the Properties of t - Norms in the Q-Fuzzy T-Sub algebra and Ideals in the BP-Algebra

In the present piece, we propound t- norms of the QFTSA ὦȴȵŦὦȴὦȵ in BP-Algebra and also t- norms of the QFTI ὦὦȴὦȴɉŦŦὦȴȵɉὦȵὦȵ , ȴȵɉĜ of the BP-Algebra and we assay some of their features. Likewise, we define Cartesian products of QFTSAs and QFTI of BP algebras. These, as well as other algebraic features, are bandied in depth. Objective: This article generally aimed at investigating t- norms concepts. It can also be applied to the algebraic properties of Q-Fuzzy T-Ideals, Q-Fuzzy T-Sub algebra and also defined for Cartesian products of QFTSAs and QFTI of BP algebras.Methods: t -norms concepts can apply for the algebraic properties of Q-Fuzzy T-Ideals, and T-Sub algebra related to the theorems, lemma, and examples we find the proof.Findings: In the present study, we propound ȶ- norms of the QFTSA ὦȴȵŦὦȴὦȵ in BP-Algebra and also ȶ- norms of the QFTI ὦὦȴὦȴɉŦŦὦȴȵɉὦȵὦȵ , ȴȵɉĜ of the BP-Algebra and we assay some of their features. Likewise, we define Cartesian products of QFTSAs and QFTI of BP algebras. These, as well as other algebraic features, are bandied in depth. Novelty: This study has defined new concepts of t-norms that apply to Q-fuzzy T-Ideals and Q fuzzy T-sub-algebra in BP algebras. Keywords: Fuzzy Set (FS), Q-Fuzzy Sets (QFS), Q-Fuzzy Subset (QFSb), Q-Fuzzy T-Ideal (QFTI), Q-Fuzzy T-Subalgebra (QFTSA)

Circuit fault detection model using multiclass support vector machine

ABSTRACT Fault detection in a complex circuit is a tedious process, and it requires specialised manpower to detect and localise the faults. Manual detection is quite time consuming and might be wrong some times. Identification of faults automatically by analysing the circuit using transforms and machine learning algorithm is presented in this research work. A hardware model and a software model are developed to generate the test and train samples, and they are used in simulation analysis to detect the faults. A simple adiabatic adder using metal-oxide-semiconductor field-effect transistor is used in the hardware module, and multiple techniques like adaptive median filtering, Hilbert transform, geometric algebraic scale-invariant feature transform and multiclass support vector machine are used in the simulation model to detect the faults in the circuit. All the stages of simulation analysis results are presented to validate the performance of the proposed model. Normal and faulty conditions are accurately detected by the proposed model with maximum detection accuracy, which reduces the human efforts in designing and developing a circuit.

Lefschetz duality for local cohomology

Since the 1974 paper by Peskine and Szpiro, liaison theory via complete intersections, and more generally via Gorenstein varieties, has become a standard tool kit in commutative algebra and algebraic geometry, allowing to compare algebraic features of linked varieties. In this paper we develop a liaison theory via quasi-Gorenstein varieties, a much broader class than Gorenstein varieties. As applications, we derive a connectedness property of quasi-Gorenstein subspace arrangements generalizing previous results by Benedetti and the first author, and we deduce the classical topological Lefschetz duality via the Stanley-Reisner correspondence.

DataDTA: a multi-feature and dual-interaction aggregation framework for drug-target binding affinity prediction.

Accurate prediction of drug-target binding affinity (DTA) is crucial for drug discovery. The increase in the publication of large-scale DTA datasets enables the development of various computational methods for DTA prediction. Numerous deep learning-based methods have been proposed to predict affinities, some of which only utilize original sequence information or complex structures, but the effective combination of various information and protein-binding pockets have not been fully mined. Therefore, a new method that integrates available key information is urgently needed to predict DTA and accelerate the drug discovery process. In this study, we propose a novel deep learning-based predictor termed DataDTA to estimate the affinities of drug-target pairs. DataDTA utilizes descriptors of predicted pockets and sequences of proteins, as well as low-dimensional molecular features and SMILES strings of compounds as inputs. Specifically, the pockets were predicted from the three-dimensional structure of proteins and their descriptors were extracted as the partial input features for DTA prediction. The molecular representation of compounds based on algebraic graph features was collected to supplement the input information of targets. Furthermore, to ensure effective learning of multiscale interaction features, a dual-interaction aggregation neural network strategy was developed. DataDTA was compared with state-of-the-art methods on different datasets, and the results showed that DataDTA is a reliable prediction tool for affinities estimation. Specifically, the CI of DataDTA is 0.806 and the R value is 0.814 on the test dataset, which is higher than other methods. The codes of DataDTA are available at https://github.com/YanZhu06/DataDTA. Supplementary data are available at Bioinformatics online.

Two new families of finitely generated simple groups of homeomorphisms of the real line

The goal of this article is to exhibit two new families of finitely generated simple groups of homeomorphisms of R. These families are strikingly different from existing families owing to the nature of their actions on R, and exhibit surprising algebraic and dynamical features. The first construction provides the first examples of finitely generated simple groups of homeomorphisms of R that also admit minimal actions by homeomorphisms on the torus. The second construction provides the first examples of finitely generated simple groups of homeomorphisms of R which also admit a minimal action by homeomorphisms on the circle. This also provides new examples of finitely generated simple groups that admit nontrivial homogeneous quasimorphisms (and therefore have infinite commutator width), also being the first such left orderable examples.

Some applications of group-theoretic Rips constructions to the classification of von Neumann algebras

In this paper we study various von Neumann algebraic rigidity aspects for the property (T) groups that arise via the Rips construction developed by Belegradek and Osin in geometric group theory \cite{BO06}. Specifically, developing a new interplay between Popa's deformation/rigidity theory \cite{Po07} and geometric group theory methods we show that several algebraic features of these groups are completely recognizable from the von Neumann algebraic structure. In particular, we obtain new infinite families of pairwise non-isomorphic property (T) group factors thereby providing positive evidence towards Connes' Rigidity Conjecture. In addition, we use the Rips construction to build examples of property (T) II$_1$ factors which posses maximal von Neumann subalgebras without property (T) which answers a question raised in an earlier version of \cite{JS19} by Y. Jiang and A. Skalski.

Observability of Discrete-Time Two-Time-Scale Multi-Agent Systems with Heterogeneous Features under Leader-Based Architecture

This paper investigates the observability of discrete-time two-time-scale multi-agent systems with heterogeneous features under leader–follower architecture. First, a singular perturbation difference model for the discussed system is established based on consensus agreement. Second, to eliminate the numerical ill-posed problem that may arise from the singularly perturbed small parameter that distinguishes different time scales in the observability analysis, the order of the system model is reduced using the boundary layer theory of the singular perturbation system to obtain a slow-time-scale subsystem and a fast-time-scale subsystem. Then, based on the matrix theory, some algebraic and graphical features that guarantee the observability of the system are obtained. Finally, the validity of the theoretical results is verified by a numerical example.

Embedding integrable superspin chain in string theory

Using results on topological line defects of 4D Chern-Simons theory and the algebra/cycle homology correspondence in complex surfaces S with ADE singularities, we study the graded properties of the sl(m|n) chain and its embedding in string theory. Because of the Z2-grading of sl(m|n), we show that the (m+n)!/m!n! varieties of superspin chains with underlying super geometries have different cycle homologies. We investigate the algebraic and homological features of these integrable quantum chains and give a link between graded 2-cycles and genus-g Rieman surfaces Σg. Moreover, using homology language, we yield the brane realisation of the sl(m|n) chain in type IIA string and its uplift to M-theory. Other aspects like graded complex surfaces with sl(m|n) singularity as well as super magnons are also described.

A Detailed Study of Mathematical Rings in q-Rung Orthopair Fuzzy Framework

Symmetry-related problems can be addressed by means of group theory, and ring theory can be seen as an extension of additive group theory. Ring theory, a significant topic in abstract algebra, is currently active in a diverse range of study domains across the disciplines of mathematics, theoretical physics and coding theory. The study of ideals is vital to the theory of rings in a wide range of ways. The uncertainties present in the information are addressed well by the q-rung orthopair fuzzy set (q-ROFS). Considering the significance of ring theory and the q-ROFS, this article defines q-rung orthopair fuzzy ideals (q-ROFIs) in conventional rings and investigates its various algebraic features. We introduce the notion of q-rung orthopair fuzzy cosets (q-ROFCs) of a q-ROFI and demonstrate that, under certain binary operations, the collection of all q-ROFCs of a q-ROFI forms a ring. In addition, we provide a q-rung orthopair analog of the fundamental theorem of ring homomorphism. Furthermore, we present the notion of q-rung orthopair fuzzy semi-prime ideals (q-ROFSPIs) and provide a comprehensive explanation of their many algebraic properties. Finally, regular rings were characterized using q-ROFIs.

A signature-based algorithm for computing the nondegenerate locus of a polynomial system

Polynomial system solving arises in many application areas to model non-linear geometric properties. In such settings, polynomial systems may come with degeneration which the end-user wants to exclude from the solution set. The nondegenerate locus of a polynomial system is the set of points where the codimension of the solution set matches the number of equations.Computing the nondegenerate locus is classically done through ideal-theoretic operations in commutative algebra such as saturation ideals or equidimensional decompositions to extract the component of maximal codimension.By exploiting the algebraic features of signature-based Gröbner basis algorithms we design an algorithm which computes a Gröbner basis of the equations describing the closure of the nondegenerate locus of a polynomial system, without computing first a Gröbner basis for the whole polynomial system.

Robust Spectrum Sensing Based on Phase Difference Distribution

In the era of 5G and beyond, massive wireless connectivity will be further intensified, leading to more severe spectrum shortages. Therefore, as a promising way to solve the spectrum shortage, spectrum sensing will play a more vital role. However, traditional sensing schemes are not adequate since they are sensitive to noise uncertainty and frequency mismatch. To address the problem, we propose two sensing schemes robust to noise uncertainty and frequency mismatch by leveraging the phase difference (PD) distribution. First, we derive the approximate PD distribution under Rayleigh fading channels. It consists of cosine components, which is hardly affected by noise uncertainty and frequency mismatch and differs significantly from the PD distribution of noise. Afterwards, leveraging the geometric and algebraic features of PD distribution, we propose two sensing schemes, including PD distribution shape recognition detection (SRD) and PD distribution cosine feature recognition detection (CRD), to detect spectrum holes with known CF. We further improve them for blind sensing scenarios with unknown frequency and propose the PD-based frequency estimation scheme. Simulation results show that when the signal-to-noise ratio (SNR) is lower than 0 dB, the robustness of the proposed schemes is much higher than benchmarks.

A NEW SOFT SET OPERATION: COMPLEMENTARY SOFT BINARY PIECEWISE UNION (∪) OPERATION

Molodtsov’s soft set theory has been used in various disciplines both theoretically and practically. It is an effective mathematical tool for dealing with uncertainty. Since its debut, numerous types of soft set operations have been presented and applied. In this study, we analyze the fundamental algebraic features of a novel soft set operation which we call complementary soft binary piecewise union operation. Additionally, by examining the distribution of complementary soft binary piecewise union operation over all other soft set operations’ type, it aims to contribute to the literature on soft sets by revealing relationships between this new soft set operation and other types of soft set operations. Moreover, we prove that the set of all the soft sets with a fixed parameter set together with the complementary soft binary piecewise union operation and the soft binary piecewise intersection operation is a zero-symmetric near-semiring.

Rough I_((λ,μ) )-Statistical Convergence of Double Sequences in Gradual Normed Linear Spaces

The aim of this paper is to we examine the notion of gradually rough I_((λ,μ) )-statistical convergence of double sequences in gradual normed linear spaces (GNLS). In addition, we define the concept of gradually rough I_((λ,μ) )-statistical limit set of double sequences and obtain some algebraic and topological features of this set. Theorems are proved in the light of GNLS theory approach. Results are obtained via different perspective and new examples are established to justify the counterparts and indicate existence of introduced notions. We produce significant results that present several fundamental properties of this notion. The results established in this research work supplies an exhaustive foundation in GNLS and make a significant contribution in the theoretical development of GNLS in literature. The original aspect of this study is the first wholly up-to-date and thorough examination of the features and implementations of new introduced notions in GNLS.

Elaborating the word problem for free idempotent-generated semigroups over the full transformation monoid

With each semigroup one can associate a partial algebra, called the biordered set, which captures important algebraic and geometric features of the structure of idempotents of that semigroup. For a biordered set $$\mathscr {E}$$ , one can construct the free idempotent-generated semigroup over $$\mathscr {E}$$ , $$\mathsf {IG}(\mathscr {E})$$ , which is the free-est semigroup (in a definite categorical sense) whose biorder of idempotents is isomorphic to $$\mathscr {E}$$ . Studies of these intriguing objects have been recently focusing on their particular aspects, such as maximal subgroups, the word problem, etc. In 2012, Gray and Ruškuc pointed out that a more detailed investigation into the structure of the free idempotent-generated semigroup over the biorder of $$\mathscr {T}_n$$ , the full transformation monoid over an n-element set, might be worth pursuing. In 2019, together with Gould and Yang, the present author showed that the word problem for $$\mathsf {IG}(\mathscr {E}_{\mathscr {T}_n})$$ is algorithmically soluble. In a recent work by the author, it was showed that, for a wide class of biorders $$\mathscr {E}$$ , the algorithmic solution of the word problem revolves around the so-called vertex groups, which arise as certain subgroups of direct products of pairs of maximal subgroups of $$\mathsf {IG}(\mathscr {E})$$ . In this paper we determine these vertex groups for the case when $$\mathscr {E}$$ is the biorder of idempotents of $$\mathscr {T}_n$$ .

Fibonacci Ideal Convergence on Intuitionistic Fuzzy Normed Linear Spaces

The main goal of this article is to present the notion of Fibonacci I-convergence of sequences on intuitionistic fuzzy normed linear space. To accomplish this goal, we mainly investigate some fundamental properties of the newly introduced notion. Then, we examine the Fibonacci I-Cauchy sequences and Fibonacci I completeness in the construction of an intuitionistic fuzzy normed linear space. Some intuitionistic fuzzy Fibonacci ideal convergent spaces have been established. Further, we prove on some algebraic and topological features of these convergent sequence spaces.

Orlicz-lacunary convergent triple sequences and ideal convergence

In the present paper we introduce and study Orlicz lacunary convergent triple sequences over n-normed spaces. We make an effort to present the notion of $g_{3}$-ideal convergence in triple sequence spaces. We examine some topological and algebraic features of new formed sequence spaces. Some inclusion relations are obtained in this paper. Finally, we investigate ideal convergence in these spaces.