A double Roman dominating function (DRDF) f on a given graph G is a mapping from V ( G ) to {0, 1, 2, 3} in such a way that a vertex u for which f ( u ) = 0 has at least a neighbor labeled 3 or two neighbors both labeled 2 and a vertex u for which f ( u ) = 1 has at least a neighbor labeled 2 or 3. The weight of a DRDF f is the value w ( f ) = ∑ u ∈ V ( G ) f ( u ). The minimum weight of a DRDF on a graph G is called the double Roman domination number γ dR ( G ) of G . A Sunlet graph S n is constructed from an n -cycle by attaching one pendant vertex to each vertex of the cycle. This paper investigates the double Roman domination of Sunlet graphs S n for integer n ≥3. Using three lemmas, we derive the exact formula γ dR ( S n )=⌈7 n /3⌉. We further analyze whether Sunlet graphs belong to the family of double Roman graphs. By comparing the domination number γ ( S n )= n with the known inequality 2 γ ( G )≤ γ dR ( G )≤3 γ ( G ) for double Roman graphs, we prove that S n fails to satisfy the required inequality, which implies no Sunlet graph is a double Roman graph. The obtained result gives an explicit expression for the double Roman domination number of this class of unicyclic graphs and enriches the research achievements of double Roman domination theory for unicyclic graphs.
This paper introduces and investigates a novel generalization of operator similarity, termed (α,β)--almost similarity, which extends the concept of almost similar operators by incorporating two real parameters. We establish fundamental properties of this new equivalence relation, demonstrating that it forms an equivalence class on the space of bounded linear operators on a Hilbert space. Key results include the invariance of spectrum, point spectrum, and approximate point spectrum under this relation. The study also defines the class of (α,β)-𝔗 operators, an expansion of the classical 𝔗-operator concept, and explores its relationship with (α,β)--almost similarity. Furthermore, we analyze the connections between similarity, unitary equivalence, and (α,β)--almost similarity, providing conditions under which these relations coincide, particularly for self-adjoint and projection operators. The results contribute to the broader understanding of operator equivalence relations and their spectral implications.
Cantilever balconies are widely used in modern buildings due to their architectural flexibility and efficient use of space. However, their structural performance is highly sensitive to environmental loading because their bending resistance is concentrated at the fixed support. In real service conditions, these structures are simultaneously subjected to static loads, wind-induced aerodynamic forces, seasonal thermal effects, and seismic ground acceleration. Most conventional analyses treat these effects independently, which may underestimate cumulative deflection and lead to inaccurate serviceability predictions. This study develops a unified mathematical model to quantify the total tip deflection of a cantilever balcony subjected to combined static, wind, thermal, and seismic loading. The formulation is based on Euler–Bernoulli beam theory and linear elasticity assumptions. Closed-form analytical expressions are derived for each loading component and integrated using the principle of superposition to obtain a compact total deflection equation. Numerical simulations are performed for aluminum, steel, reinforced concrete, and carbon fiber composites under representative environmental conditions. Results show that thermal effects become dominant in high-temperature environments for materials with large coefficients of thermal expansion, while seismic effects become significant in regions with high peak ground acceleration. Among the materials considered, carbon fiber composites consistently exhibit the smallest total deflection due to their high stiffness and low thermal sensitivity, while reinforced concrete shows the largest deformation due to its lower elastic modulus. The proposed model provides a mathematically consistent framework for evaluating cantilever balcony performance under multi-hazard environmental loading and offers a useful decision-support tool for preliminary structural design and material selection.
This paper shows that every integer and odd integer are a sum of a prime and two squares. We solved this by transforming the problem of the number of such representations into finding the number of integer solutions to the corresponding Diophantine equation. To find integer solutions to nonlinear or higher-order Diophantine equations is very challenging, whereas counting the number of integer solutions for the linear Diophantine equations is comparatively easier. The methods involve the combinatorial approximation method to obtain the number of integer solutions of the Diophantine equation with the countable solution sets. First, the number of integer solutions to the linear Diophantine equations with integer sets or sequences as solution sets can be determined by combinatorial methods. Second, the approximate number of integer solutions of the linear Diophantine equations with integer sets or sequences as solution sets is obtained using the approximate method. Finally, for cases where the solution sets and the number of solution sets of the variables in the linear Diophantine equations are either the same or different, we propose the approximate combinatorial method to compute the approximate number of integer solutions of the linear Diophantine equations whose solution sets are integer subsets or sequences of integer subsets. The results are consistent with the existing conclusions. The purpose of this paper is to verify the adaptability and correctness of the approximate combinatorial methods for solving the number of integer solutions of the general Diophantine equations with countable solution sets.
We prove that rank-zero elliptic curves over generate positive-rank elliptic curves through the unit circle, via quadratic twisted models . This construction demonstrates rank evolution from zero to infinite rational points, complementing high-rank families. All rank-zero status and positive-rank emergence rigorously verified computationally. SMC(2020): 11G05, 14H52, 11Y50.
