Existing studies mostly infer the existence of intrinsic spatial mass from physical intuition through a qualitative analysis of wave–particle duality (e.g., empirical application of the zero equation) but lack a proof process that meets mathematical rigor. The core contribution of this paper lies in constructing a strict “axiom–lemma–theorem” derivation system based on the discrete group Z₂ (describing the binary reciprocity of “particle nature–spatial nature”) and the complex linear space M (representing mass states), combined with the kernel space theory of linear functionals and the positivity of Hermitian inner products.

Intrinsic spatial mass is a core concept in the mass mathematics system that connects abstract mathematical structures to the properties of physical spacetime. A rigorous proof of its existence is crucial for ensuring the theoretical consistency of this field. Using discrete group representation theory, complex linear space theory, and linear functional analysis as tools, this paper constructs a theoretical framework comprising four core axioms (Discrete Symmetry Group Axiom, Complex Mass Space Axiom, Zero equation Constraint Axiom, and Physical Observability Axiom). Through a three-level derivation chain—non-emptiness of the kernel of a linear functional–non-vanishing of intrinsic spatial mass–consistency of discrete group action—the existence of intrinsic spatial mass is strictly proven. This study shows that intrinsic spatial mass has three cases in the complex mass space M≅C2: m_is = im_rp; m_rs = −im_ip; and m_is = -m_ip (where m_p is the particle mass, m_s is spatial mass, r is the real number, and i is the imaginary number). It satisfies both physical observability (its squared norm is a non-negative real number) and discrete symmetry consistency (closure under Z₂ group action). This proof provides core support for the axiomatic construction of mass mathematics and offers a new perspective on the mathematical interpretation of wave–particle duality in quantum mechanics.

Zero-divisor graphs provide a powerful bridge between commutative algebra and graph theory by encoding annihilation relationships among zero divisors of a finite ring. While parameters such as connectivity, chromatic number, and classical domination have been extensively explored, Roman domination and double Roman domination in zero-divisor graphs remain largely uncharacterized. These parameters model optimal resource allocation under different levels of protection and redundancy. This study aims to compute and analyze the Roman domination number and the double Roman domination number for key families of finite commutative rings and to establish structural relationships between algebraic properties of rings and optimal domination behavior.

Methods:
For each selected ring family, zero-divisor graphs are constructed from annihilation relations. Roman and double Roman domination are formulated as constrained optimization problems, supported by algebraic–graph theoretic lemmas, degree-based bounds, and annihilator structure analysis. Exact values are obtained using integer linear programming (ILP), while large graphs are studied using heuristic algorithms informed by neighborhood structures and annihilator classes.

Results:
Preliminary analyses reveal that vertices with maximal annihilator degree play a central role in optimal Roman and double Roman labelings. Structural features such as star-like patterns, complete subgraphs, and decompositions from the Chinese Remainder Theorem significantly reduce domination cost. Across ring families, distinct algebraic characteristics—particularly annihilator chains and idempotent behavior—strongly influence domination numbers.

Conclusion:
This work establishes new theoretical bounds, exact values, and structural characterizations of Roman and double Roman domination in zero-divisor graphs. These findings also support an application framework interpreting domination assignments as minimum-cost emergency-station placement strategies, demonstrating the practical relevance of the studied parameters.

This paper focuses on the investigation of commutativity properties of the factor ring ℜ/P, where P is assumed to be a prime ideal of an arbitrary ring ℜ. The main objective is to determine sufficient algebraic conditions under which the quotient structure ℜ/P exhibits commutative behavior. To achieve this goal, the study employs the concept of generalized P-derivations ℧ and ⨿, which are constructed in association with the P-derivations χ and ∝, respectively. These generalized mappings are required to satisfy specific functional identities that create intrinsic links between the ring ℜ and its prime ideal P.

By analyzing these identities, we establish several results that reveal how the interaction between generalized P-derivations and the underlying ring structure influences the commutativity of ℜ/P. The approach adopted in this work allows for a unified treatment of various derivation-like operators and highlights their effectiveness in deriving commutativity criteria in the presence of prime ideals. Furthermore, a number of related consequences and supplementary observations are discussed to place the main results within a broader algebraic context.

In order to emphasize the necessity of assuming the primeness of P, illustrative examples are presented showing that the obtained conclusions may no longer hold if this assumption is weakened or removed. These examples demonstrate that the primeness condition plays a crucial role in guaranteeing the validity of the established identities and the resulting commutativity of the factor ring. Overall, the results contribute to the ongoing study of derivations and generalized derivations in ring theory and provide further insight into their applications to factor rings determined by prime ideals.

The interplay between algebraic structures and graph theory has emerged as a dynamic and fruitful area of research in modern mathematics. By establishing connections between graphs and groups, mathematicians have developed innovative approaches to visualize and analyze the intrinsic properties of groups using graph-theoretic tools. These connections enhance our understanding of group theory and provide new perspectives on how graph theory can be effectively applied within algebraic frameworks.

The origins of this interdisciplinary relationship can be traced back to the pioneering work of Arthur Cayley in 1878, who introduced what are now famously known as Cayley graphs. These graphs serve as visual representations of groups, where each vertex corresponds to a group element, and edges reflect the relationships defined by a generating set. Cayley’s groundbreaking idea laid the foundation for a vast and ongoing exploration of the rich interactions between group theory and graph theory, which continues to inspire contemporary research across various mathematical domains.

In this study, we contribute to this line of research by investigating different notions of commutativity degrees in finite groups and exploring their algebraic properties. Building on these findings, we construct new classes of graphs associated with finite groups and examine their structural characteristics. We further illustrate our results by considering several examples of special types of groups, offering fresh insights into the mutual influence of algebraic and graph-theoretic perspectives.

This paper presents formal proof of the Emergent Continuum Hypothesis (ECH), a principle positing that the mathematical continuum is not a fundamental, axiomatic entity but a macroscopic phenomenon emerging from a discrete underlying reality. We demonstrate that a specific, non-trivial continuum is the necessary and unique limit of a system built from the set of prime numbers. The proof is constructed in four parts. First, we define a sequence of finite, directed metric spaces derived from the primes. The metric is determined by a novel, asymmetric weight function where the interaction between any two primes is mediated by the entire system; this interaction strength is based on the p-adic norms of the gap between the primes, evaluated against all primes in the system. Second, we prove that this sequence of spaces is a Cauchy sequence in the measured Gromov–Hausdorff metric, and therefore, converges to a complete, path-connected geodesic space, which we identify as the Emergent Continuum. Third, we prove that this convergence is critically dependent on the deep arithmetic nature of the rules, showing that simpler, non-arithmetic rules fail to produce a stable, non-trivial limit. Finally, we prove that the canonical Laplacian operator on this Emergent Continuum possesses a spectrum with eigenvalue spacing statistics that necessarily follow the Gaussian Unitary Ensemble (GUE). This is shown to be a direct consequence of the intrinsic asymmetry in our rules of assembly, which breaks time-reversal symmetry and induces the quantum chaotic behavior observed in number theory. This work establishes a rigorous mathematical bridge between discrete arithmetic and continuous analysis, offering a new paradigm for foundational questions in mathematics.

This work presents a novel geometric approach to control theory on curved surfaces by establishing a precise connection between isomonodromic deformations of logarithmic connections on principal bundles and control systems on Riemann surfaces. The key insight is that optimal trajectories in control systems can be understood as geodesics with respect to a metric determined by monodromy data. The framework begins by characterizing control systems in terms of logarithmic connections on principal bundles over Riemann surfaces, where the residues at singular points encode the control structure. We establish that controllability is equivalent to the residues generating the full Lie algebra of the structure group under commutator brackets, providing a geometric criterion that can be verified algebraically. The main result demonstrates that optimal control trajectories minimizing a quadratic cost functional correspond precisely to isomonodromic deformations, where the monodromy representation remains constant under parameter variation. This correspondence extends naturally to systems with non-holonomic constraints, where the eigenvalue structure of the residue matrices determines the growth vector of the constraint distribution. The symmetry algebra of the constrained system is shown to be isomorphic to the Lie algebra generated by the residues. We illustrate the theory through a detailed computational example involving principal bundles with special unitary structure group over a hyperbolic surface of genus two.

This paper establishes several results on fuzzy β-continuous and M-fuzzy β-continuous mappings between fuzzy topological spaces. Equivalent characterizations of fuzzy β-continuous maps are obtained. In particular, it is proved that a mapping is fuzzy β-continuous if and only if the inverse image of every fuzzy closed set is fuzzy β-closed. Additional characterizations are derived using β-closure operators, where necessary and sufficient conditions of the form βcl(f⁻¹(V)) ≤ f⁻¹(βcl(V)) are established for arbitrary fuzzy sets. The central contribution of this work concerns the composition of fuzzy β-continuous mappings. It is shown that the composition of two fuzzy β-continuous mappings need not be fuzzy β-continuous in general; however, sufficient conditions are provided under which the composition becomes fuzzy β-continuous, particularly when one of the mappings satisfies stronger continuity conditions such as fuzzy pre-continuity or fuzzy semi-continuity. Several propositions formalize these results, and explicit examples illustrate that the corresponding converse statements fail. This theorem clarifies the structural behavior of fuzzy β-continuity under composition and provides useful criteria for analyzing the stability of such mappings in fuzzy topological spaces. The relationships between fuzzy β-continuous mappings and other classes of fuzzy mappings are also examined. It is proved that every fuzzy continuous, fuzzy pre-continuous, and fuzzy semi-continuous mapping is fuzzy β-continuous, while the reverse implications do not hold in general. The notion of M-fuzzy β-continuous mappings is studied, and it is shown that every M-fuzzy β-continuous map is fuzzy β-continuous, although the converse fails. Finally, for a bijective mapping between fuzzy topological spaces, it is proved that if βint(A) ≤ f⁻¹(int(f(A))) for any fuzzy set A, then f(βcl(A)) ≤ cl(f(A)).

Digital signature schemes are a cornerstone of modern public key cryptography, providing essential security services such as authentication, data integrity, and non-repudiation in open communication networks. Most classical digital signature constructions are based on commutative algebraic structures and rely on a single hard mathematical problem. In recent years, non-commutative algebraic frameworks have attracted increasing attention due to their potential to offer enhanced security and resistance to emerging cryptanalytic techniques. In this paper, we propose a novel digital signature scheme based on elliptic curves defined over a finite non-commutative ring. We first introduce a non-commutative ring $R$ constructed from an elliptic curve over the finite ring $ \mathbb{F}_{q}[\varepsilon]$ where $\varepsilon^4=\varepsilon^3$ and $(char(\mathbb{F}_q)\neq 2,3)$ \cite{1}. The proposed construction combines two well-known computationally hard problems: the elliptic curve discrete logarithm problem and the conjugacy problem in non-commutative rings. The proposed digital signature algorithm is designed by exploiting the interaction between these two problems, resulting in a hybrid cryptographic scheme that strengthens security compared to classical approaches based on a single hardness assumption. The scheme ensures the fundamental security properties required for digital signatures, including authentication, message integrity, and non-repudiation. Furthermore, we present a comprehensive security analysis of the proposed scheme and evaluate its resistance against four common types of cryptographic attacks. The results demonstrate that the use of non-commutative algebraic structures provides a promising direction for the design of secure and efficient digital signature schemes.

Aggregation functions play a fundamental role in mathematics, particularly in the combination of multiple measurements into a single quantity. They also allow the construction of a single mathematical structure of the same type from a family of given structures. Early work in this area dates back to Boršík and Doboš, who studied the aggregation of metrics. Their results motivated further research on the aggregation of more general structures, including quasi-pseudometrics, fuzzy quasi-pseudometrics, and indistinguishability operators.

All these structures induce a topology on their underlying sets, which raises a natural question: does the aggregation process preserve the associated topological properties? In particular, one may ask whether the topology generated by an aggregated (quasi-)(pseudo)metric coincides with the product topology of the individual spaces. For metric aggregation functions on products, this problem was solved by Boršík and Doboš through the introduction and characterization of strongly metric aggregation functions.

In this work, we extend their approach to the broader setting of quasi-pseudometric aggregation functions on products. We introduce the notion of strongly (quasi-)(pseudo)metric aggregation functions on products and provide a complete characterization of this class. A function is said to be strongly aggregating if, for every family of (quasi-)(pseudo)metric spaces, the topology induced by the aggregated (quasi-)(pseudo)metric agrees with the corresponding product topology.

As a key illustration of our results, we prove that a (quasi-)metric aggregation function on products is strongly aggregating if and only if it is continuous at the zero vector. This condition yields a concise and transparent topological characterization.

In conclusion, our results generalize the classical theory of strongly metric aggregation functions to the quasi-pseudometric framework and establish explicit relationships between the different classes of strongly quasi-pseudometric aggregation functions on products.

Lie-Rinehart algebras provide an algebraic abstraction of Lie algebroids, encoding simultaneously a Lie algebra structure and a module structure over a commutative algebra, together with a compatible anchor map acting by derivations. Since their introduction by Herz, these structures have played a central role in the algebraic formulation of differential geometry, Poisson geometry, and deformation theory, serving as the natural algebraic setting for derivations, differential forms, and cohomology on commutative algebras. An analogous construction for a general Lie algebroid has been introduced by Iglesias and Marrero under the name of a generalized Lie algebroid. In parallel, Jacobi algebras generalize Poisson algebras by allowing the Leibniz rule to hold up to a derivation, capturing algebraically the geometry of contact and locally conformal symplectic manifolds. While Poisson algebras correspond to Lie algebroids via the cotangent bundle construction, Jacobi algebras require a more flexible framework that incorporates both a Lie bracket and a distinguished derivation.

In our talk, we describe generalized Lie-Rinehart algebras as an extension of Lie-Rinehart algebras and give some examples. These structures extend classical Lie-Rinehart algebras by allowing the anchor to take values in first-order differential operators. We characterize generalized symplectic manifolds and contact manifolds in terms of generalized symplectic Lie-Rinehart algebras and we show the existence of Jacobi structure induced by a generalized symplectic Lie-Rinehart algebra. We introduce the notion of symplectomorphisms of generalized Lie-Rinehart algebras and describe them particularly in the case of contact manifolds, generalized symplectic manifolds and in the case of non-degenerate Jacobi structures