Search Results (142)

Arnold tongues are wedge-shaped regions in parameter space associated with mode locking and synchronization phenomena in nonlinear dynamical systems. The Caputo fractional standard map extends the classical standard map by incorporating long-memory effects through fractional derivatives and is known to generate Arnold tongue structures as the fractionality parameter approaches unity. In this paper, we investigate the fractional standard map applied to matrix-valued state variables, with particular emphasis on systems governed by high-dimensional nilpotent matrices. We show that the interplay between fractional memory and nilpotent algebra produces hierarchical families of Arnold tongues associated with divergent dynamics. This phenomenon is not observed in either the classical standard map or the non-fractional standard map of nilpotent matrices alone. For idempotent matrices, the fractional standard map retains the same level of dynamical complexity as its scalar counterpart. For nilpotent matrices, higher-order terms induce coupling between the map coefficients, giving rise to substantially richer dynamical behavior. This combination of fractional memory and nilpotent algebra provides a systematic framework for studying higher-dimensional nonlinear dynamics beyond the scalar setting. To support numerical investigations, an efficient computational scheme for the auxiliary parameters is derived and calibrated using the H-rank algorithm, which provides a concise measure of algebraic complexity in sequences generated by dynamical systems. Numerical simulations reveal hierarchical structures of Arnold tongues of divergence together with characteristic divergence rates of the auxiliary parameters. The hierarchical level of a given auxiliary parameter is identified as a key quantity determining the algebraic complexity of the transient dynamics, with potential implications for information encoding in applications exploiting transient dynamical processes. Full article

Let $\mathcal{A}=\{p+Jq:p,q\in \mathbb{C},J^2=-1\}$ be the commutative bicomplex-type algebra in which J commutes with the scalar imaginary unit. A Cauchy–Riemann-type operator $\overline{D}$ is studied on domains in ${\mathbb{C}}^2$. In the active coordinates $\xi =z_1-i{z}_2$ and $\eta =z_1+i{z}_2$, the equation $\overline{D}f=0$ is diagonal in the idempotent basis: the $e_{+}$-component is holomorphic in $\xi$ with $\eta$ as the parameter, while the $e_{-}$-component is holomorphic in $\eta$ with $\xi$ as the parameter. The expression $e_{+}F\left(\xi \right)+e_{-}G\left(\eta \right)$ is the parameter-independent subcase. From this decomposition, one obtains a slice characterization, a criterion for separatedness, a comparison with ordinary holomorphic functions of two complex variables, active-variable Cauchy formulas and estimates, local series with parameter-dependent coefficients, reflection symmetry, and Hardy and Bergman kernel lifts on the separated Hilbert spaces. Full article

Selecting the right energy storage system (ESS) for grid integration is a high-stakes decision involving conflicting technical, economic, environmental, and risk criteria under deep uncertainty. The existing fuzzy multi-criteria decision-making (MCDM) methods either fail to capture neutral or abstaining expert judgments or treat evaluation criteria as independent, which is an unrealistic assumption in complex engineering decisions. To address both limitations simultaneously, this study develops four new aggregation operators by extending the Bonferroni mean (BM) into the q-rung picture fuzzy sets (q-RPFSs) framework: the q-RPFBM-based, q-RPFWBM-based, q-RPFGBM-based, and q-RPFWGBM-based operators. Unlike the existing q-RPFS operator families (Dombi, Frank, Fermatean, Yager, Maclaurin), which aggregate criteria independently, BM-based operators explicitly model pairwise interactions among criteria with a structurally distinct aggregation logic that is especially critical when criteria such as cost, risk, reliability, and environmental impact are mutually correlated. The theoretical validity of the operators is confirmed through proofs of idempotency, monotonicity, and boundedness. Applied to a comprehensive ESS selection problem for Türkiye (covering nine alternatives across nineteen sub-criteria and five main criteria, including an explicit risk dimension), the framework consistently identifies pumped hydro storage as the optimal choice. Sensitivity analyses under varying q, s, and t parameters, as well as perturbed criterion weights, confirm the robustness of this ranking. The proposed framework offers energy planners and decision-makers a principled and transparent tool for evaluating ESS under high uncertainty and criterion interdependence. Full article

We introduce and study several new classes of rings defined by idempotent conditions. A ring R is called i-semicommutative if $(1-ab)aRb(1-ab)=0$ whenever $ab$ is a nonzero idempotent. This property lies strictly between semicommutativity and i-reversibility (where $ab$ nonzero idempotent forces $ba$ idempotent). We also define i-reduced rings (if $a^2$ is a nonzero idempotent then $a^3=a$) and i-domains (if $ab$ is a nonzero idempotent then $a\in \mathfrak{I}\left(b\right)$ or $b\in \mathfrak{I}\left(a\right)$). Basic properties are established, including closure under subrings, behaviour of corners, and connections with classical ring concepts. We characterize these properties for triangular matrix rings and for full matrix rings over commutative rings. Several examples illustrate the independence of the new notions from abelian, reversible and semicommutative rings. Open questions are posed concerning direct finiteness and matrix rings over noncommutative rings. Full article

In this work, we establish the relationships between weak e-reversibility and weak e-semicommutativity, introducing weak e-reflexive rings as a natural generalization of reflexive and e-reflexive rings. We demonstrate that, under specific conditions, weak e-reflexivity and weak e-reversibility coincide in Baer rings, and we provide characterizations via corner subrings and semicentral idempotents. Furthermore, we introduce e-nilpotent reflexive rings and examine their structural stability and connections within the class of generalized reflexive rings. A comprehensive analysis is provided regarding the behavior of these properties under polynomial and Dorroh extensions, as well as within matrix rings and quotient structures. Full article

The growing complexity of modern decision-making environments, characterized by multi-dimensional data, uncertainty, and dynamic behavior, demands advanced mathematical frameworks for effective information aggregation. Although fractional fuzzy tensor (FFT) models provide a powerful tool for representing such complex systems by integrating fuzzy logic, tensor structures, and fractional dynamics, the lack of suitable aggregation mechanisms significantly limits their practical applicability. To address this challenge, this paper proposes a novel family of Bonferroni mean-based aggregation operators within the fractional fuzzy tensor environment. The proposed framework extends the classical Bonferroni mean to multi-dimensional fractional fuzzy settings, enabling the effective modeling of interrelationships among criteria while preserving the structural and dynamic properties of FFTs. Specifically, four aggregation operators—namely, the fractional fuzzy tensor Bonferroni mean (FFT-BM), weighted Bonferroni mean (FFT-WBM), ordered Bonferroni mean (FFT-OBM), and hybrid Bonferroni mean (FFT-HBM)—are systematically developed. A comprehensive theoretical analysis is conducted to investigate fundamental properties such as idempotency, monotonicity, boundedness, commutativity, and stability, thereby establishing the mathematical consistency and reliability of the proposed operators. Furthermore, a structured multi-criteria decision-making (MCDM) algorithm is formulated, incorporating tensor construction, aggregation, evaluation, and sensitivity analysis phases to handle complex uncertain information effectively. To demonstrate the practical applicability of the proposed framework, a real-world case study related to disaster management decision-making is presented. The results are further validated through quantitative comparative analysis with classical and recent aggregation operators, revealing improved discrimination power, robustness, and ranking consistency. Additionally, sensitivity analysis confirms the stability of the proposed approach under varying parameters. The findings indicate that the proposed Bonferroni mean-based aggregation framework significantly enhances the capability of FFT models in handling high-dimensional, uncertain, and dynamic decision-making problems. This study not only strengthens the theoretical foundation of aggregation in tensor-based fuzzy environments but also provides a flexible and reliable decision-support tool for complex real-world applications. Full article

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 ($b_r$), 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. Full article

We undertake a thorough examination of George Boole’s claim that, as he discovered by means of his algebra, the law of idempotence is “more fundamental” than the law of non-contradiction (The Laws of Thought, Chapter III, Proposition IV). There is a paucity of sources investigating this subject (with a notable exception being (Béziau 2018)). We query Boole’s claim; we examine if and how we can make sense of it; we identify the notable Aristotelian precedent of philosophical reflections on relative fundamentality of logical principles; and we inquire as to what philosophical view of logic is consistent with Boole’s way of thinking about logical principles. Boole’s thinking is apparently burdened by a metaphysically laden view of logic. We argue in detail that it is a radically different way of thinking about logic—a formalist view that regards logic as manipulation of symbolic resources, congenial to logical positivism—which allows us to make some tentative sense of claims about relative fundamentality of logical laws, insofar as we can define such a notion in a meaningful way. However, on the other hand, entanglements in metaphysically laden phantasmagorias fail to support (or perhaps even fail to make sense of) Boole’s claim. In order to substantiate the metalogical and philosophical–logical claims, we advance and construct formal derivations within different Boolean languages with a view to showing how idempotence is primary in some formal systems, but it is derivable (from non-contradiction) in other systems. Hence, Boole’s claim, as we can make sense of it (as relative derivability), is language-dependent, and we argue that this is consistent with a certain philosophical view of what logic is. Full article

We equip a topological space $(X,\tau )$ with a function $\mathfrak{a}:X\to \tau$ satisfying the single axiom $x\in \mathfrak{a}\left(x\right)$. The resulting triple $(X,\tau ,\mathfrak{a})$, which we call an aura topological space, provides a point-to-open-set assignment that differs from all existing auxiliary structures in topology—ideals, filters, grills, primals, and the various non-classical frameworks based on fuzzy, soft, or neutrosophic sets. The aura-closure operator ${\mathrm{cl}}_{\mathfrak{a}}\left(A\right)=\{x\in X:\mathfrak{a}\left(x\right)\cap A\ne ⌀\}$ is shown to be an additive Čech closure operator; it satisfies extensivity, monotonicity, and finite additivity, but idempotency fails in general. Iterating ${\mathrm{cl}}_{\mathfrak{a}}$ transfinitely yields a Kuratowski closure whose topology ${\tau }_{\mathfrak{a}}^{\infty }$ satisfies ${\tau }_{\mathfrak{a}}^{\infty }={\tau }_{\mathfrak{a}}\subseteq \tau$, where ${\tau }_{\mathfrak{a}}$ is the collection of all $\mathfrak{a}$-open sets. We introduce $\mathfrak{a}$-semi-open, $\mathfrak{a}$-pre-open, $\mathfrak{a}$-$\alpha$-open, and $\mathfrak{a}$-$\beta$-open sets, determine the complete hierarchy among these classes and their classical counterparts, and separate all non-coinciding classes by counterexamples on finite spaces as well as on the real line. The notions of $\mathfrak{a}$-convergence of sequences and the corresponding continuity notions and their decompositions are studied. Separation axioms $\mathfrak{a}$-$T_i$$(i = 0, 1, 2)$ are introduced, and it is proved that $\mathfrak{a}$-$T_1$ and $\mathfrak{a}$-$T_2$ are equivalent. A detailed comparison with ideals, filters, grills, and primals highlights the distinctive features of the aura framework. Full article

Motivated by the fact that the Fibonacci sequence is the simplest nontrivial second-order recurrence with a rational generating function, we develop a Fibonacci-weighted Hardy theory for bicomplex holomorphic functions. Starting from the coefficient norm ${\sum }_{n\ge 0}{\left|a_n\right|}^2/F_{n+1}$, we obtain a bicomplex Hilbert module whose reproducing kernel is governed by ${(1-t-t^2)}^{-1}$ and whose maximal disk of holomorphy is determined sharply by the nearest kernel singularity, giving the radius ${\rho }_F={\phi }^{-1/2}$ (the square-root inverse of the golden ratio $\phi$). The arithmetic recurrence makes several objects fully explicit: we derive closed formulas for the kernels through the idempotent decomposition of $\mathbb{BC}$, compute exact norms of the shift powers and a golden-ratio spectral radius, and package the local theory into a sheaf of Fibonacci-holomorphic germs that are compatible with the bicomplex idempotent splitting. We also treat $(p,q)$-Fibonacci weights, obtaining a one-parameter family of rational kernels ${(1-pt-q{t}^2)}^{-1}$ and corresponding operator bounds. In addition to providing a concrete bicomplex model within weighted Hardy theory, the resulting explicit kernels furnish benchmark examples for kernel-based interpolation and for the operator theory of unilateral weighted shifts. Full article

This paper establishes a graph-theoretic framework for idealization semigroups arising from Bruck–Reilly extensions. Building on a recent study by Wazzan and Ozalan, we introduce five graph families—${\Gamma }_E$, ${\Gamma }_0$, ${\Gamma }_{\mathrm{Cay}}$, ${\Gamma }_{\mathcal{K}}$, and $\Gamma \left(G_k\right)$—each encoding a distinct algebraic facet of $S{B}_i(\bowtie )B$. We prove explicit correspondences linking combinatorial invariants to algebraic structure: diameter captures generating efficiency and semilattice height; girth signals short relations; chromatic number bounds idempotent cardinalities and $\mathcal{D}$-class counts; clique number measures maximal commuting subsets; and Laplacian spectra encode ideal size and Schützenberger groups. Our central result demonstrates that Green’s relations are combinatorially recoverable from graph pairs. For commutative $S{B}_i(\bowtie )B$, $({\Gamma }_E,{\Gamma }_{\mathcal{K}})$ uniquely determines $\mathcal{J}$-order, $\mathcal{D}$-classes, and $\mathcal{H}$-classes via neighborhood inclusions, bipartite components, and automorphism orbits, yielding the first algorithmic reconstruction of ideal-theoretic structure from graph data. The framework is implemented in SageMath as a reproducible open-source toolkit validated on concrete examples. This work synthesizes algebraic graph theory, semigroup theory, and computational mathematics into a unified algebraic-combinatorial dictionary, providing both new analytical tools and a methodological template for studying algebraic constructions via graph invariants. Full article

We study double-circulant codes over a class of semi-local rings arising from the idempotent construction $R={\mathbb{Z}}_{p^2}+u{\mathbb{Z}}_{p^2}$, where $u^2=u$, and p is an odd prime. Although both algebraic settings considered admit this presentation, they correspond to two distinct rings depending on whether the additional relation $pu=0$ is imposed or not. These two configurations induce different ideal lattices and symmetry properties, which play a decisive role in the structure and enumeration of codes. Exploiting the Chinese Remainder Theorem, we describe self-dual and linear complementary dual (LCD) double-circulant codes in a unified, componentwise manner. Exact enumeration formulas are derived by reducing the corresponding duality constraints to norm equations over finite fields and unramified Galois extensions of ${\mathbb{Z}}_{p^2}$. We further construct explicit ${\mathbb{F}}_p$-linear Gray maps from $R^{2n}$ to ${\mathbb{F}}_p^{6n}$ in the degenerate case $pu=0$ and to ${\mathbb{F}}_p^{8n}$ in the standard case $pu\ne 0$, and show that these maps preserve self-duality and the LCD property. Assuming a standard primitive-root hypothesis on the code length, as predicted by Artin’s primitive root conjecture, we establish asymptotic existence bounds for the Gray images of both LCD and self-dual double-circulant codes via a probabilistic argument. The degenerate case $pu=0$ yields a shorter Gray expansion and a stronger self-dual entropy threshold, while the case $pu\ne 0$ leads to a larger self-dual ensemble with distinct asymptotic characteristics. Full article

We study the Moore–Penrose inverse of idempotent operators on Hilbert $C^{*}$-modules. First, we extend the computation of the Moore–Penrose inverse of an idempotent operator and its difference from the range projection to this setting. This leads to an explicit formula for the Moore–Penrose inverse of the sum of an idempotent and its adjoint. Furthermore, we establish a decomposition of an idempotent operator into a product of two commuting idempotents and clarify the relationship between their Moore–Penrose inverses and that of the original operator. We also analyze spectral properties and operator norms, obtaining sharp norm bounds. Full article

This article presents an ontology-driven Digital Twin (DT) for hotel front-desk operations that fuses two real-time data streams: (i) physiological and activity signals from wrist-worn wearables assigned to staff, and (ii) 3D people-positioning and occupancy events captured by reception-area cameras using a proprietary implementation of Optical Camera Communication (OCC). Building on a previously proposed front-desk ontology, the semantic model is extended with positional events, zone semantics, and wearable-derived workload indices to estimate queue state, staff workload, and service demand in real time. A vendor-agnostic, property-based REST API specifies the DT interface in terms of observable properties, including authentication and authorization, idempotent ingestion, timestamp conventions, version negotiation, integrity protection for signed webhooks, rate limiting and backoff, pagination and filtering, and privacy-preserving identifiers, enabling any compliant backend to implement the specification. The proposed layered architecture connects ingestion, spatial reasoning, and decision services to dashboards and key performance indicators (KPIs). This article details the positioning pipeline (calibration, normalized 3D coordinates, zone mapping, and confidence handling), the wearable workload pipeline, and an evaluation protocol covering localization error, zone classification, queue-length estimation, and workload accuracy. The results indicate that a spatially aware, ontology-based DT can support more balanced staff allocation and improved guest experience while remaining technology-agnostic and privacy-conscious. Full article

In this paper, for any real number λ, we transform the complete max-plus semiring ${\mathbb{R}}_{\infty }$ into a commutative, complete, additively idempotent semiring ${\mathbb{R}}_{\infty }^{\lambda }$, called the lower λ-truncation of ${\mathbb{R}}_{\infty }$. It is obtained by removing from ${\mathbb{R}}_{\infty }$ all real numbers smaller than λ, inheriting the addition operation, shifting the original products by −λ, and appropriately modifying the residuum operation. The purpose of lower truncations is to transfer the iterative procedures for computing the greatest presimulations and prebisimulations between max-plus automata, in cases where they cannot be completed in a finite number of iterations over ${\mathbb{R}}_{\infty }$, to ${\mathbb{R}}_{\infty }^{\lambda }$, where they could terminate in a finite number of iterations. For instance, we prove that this necessarily happens when working with max-plus automata with integer weights. We also show how presimulations and prebisimulations computed over ${\mathbb{R}}_{\infty }^{\lambda }$ can be transformed into presimulations and prebisimulations between the original automata over ${\mathbb{R}}_{\infty }$. Although they do not play a significant role from the standpoint of computing presimulations and prebisimulations, for theoretical reasons we also introduce two types of upper truncations of the complete max-plus semiring ${\mathbb{R}}_{\infty }$. Full article