We present a unified, set–theoretic framework that extends molecular graphs to hypergraphs and superhypergraphs via iterated power sets. We define Molecular Graphs, Molecular HyperGraphs, and Molecular SuperHyperGraphs, and develop four complements over them: Weighted, Rough, Neural, and Multipolar frameworks. We prove concise inclusion results—most notably, that the Weighted Molecular SuperHyperGraph strictly contains both Weighted Molecular HyperGraphs and (unweighted) Molecular SuperHyperGraphs—while preserving alternating–path distances under canonical embeddings. Compact examples (e.g., methane, ethanol, acetic acid) illustrate how atoms, bonds, functional groups, and higher–order motifs appear as vertices, hyperedges, and superedges under rank constraints. We also provide implementation–agnostic message–passing rules for variable–arity interactions, enabling property prediction and hierarchical analysis in chemistry and chemical biology. This paper is devoted to theoretical analysis, and it is hoped that quantitative studies by domain experts will be developed in future work.

For the first time, the chemical composition of essential oils obtained from the leaves and flowers of Mediterranean rosemary (Salvia rosmarinus) collected in forested areas surrounding Jerusalem has been investigated. Gas chromatographic analysis revealed that the essential oils are dominated by structurally unusual terpenoids containing a cyclobutane ring. A comprehensive survey of the literature further demonstrates that S. rosmarinus is a prolific natural source of cyclobutane-containing terpenes, with approximately 40 such compounds reported to date. These rare terpenoid frameworks are of particular interest due to their uncommon ring strain and distinct biosynthetic origin. In addition, available data on the biological activities of the major cyclobutane-containing terpenes are summarized, highlighting their antimicrobial, anti-inflammatory, antioxidant, and cytotoxic properties. Collectively, these findings underscore rosemary as a unique producer of cyclobutane terpenoids and emphasize the pharmacological potential of this rare class of natural products.

Let R be a ring with identity. The nil-clean graph of R is a graph, denoted by G_NC(R), whose vertex-set is the set R, and where two distinct vertices x and y are adjacent if and only if x + y is a nil-clean element of R. An element r ∈ R is called a nil-clean element if it can be decomposed a sum of an idempotent and a nilpotent element of R. Let G be a finite undirected graph. An automorphism φ of G is a permutation on the vertex-set V(G) such that the graph preserves adjacency, that is, φ(v₁) is adjacent to φ(v₂) if and only if v₁ is adjacent to v₂. The set of all automorphisms of G together with the composition operation of permutations forms the automorphism group of G. In this paper, we firstly compute the order of the automorphism groups of nil-clean graphs for the ring ℤ_n. And then we determine the structure of the automorphism groups of G_NC(ℤ_n) for n = p^k, pq, 2^kp^l, where p, q are distinct primes and k, l are positive integers.

A subfamily of Dyck words called tight Dyck words is seen to correspond, via a “castling” procedure, to the vertex set of an ordered tree T. From T, a “blowing” operation recreates the whole family ol Dyck words. The vertices of T can be elementarily updated all along T. This simplifies an edge-supplementary arc-factorization view of Hamilton cycles of odd and middle-levels graphs found by T. Mütze et al. This takes into account that the Dyck words represent: (a) the cyclic and dihedral vertex classes of odd and middle-levels graphs, respectively, and (b) the cycles of their 2-factors, as found by T. Mütze et al.

Summability methods for trigonometric Fourier series play a fundamental role in approximation theory and signal processing. Among them, Fejér means provide a classical regularization tool ensuring uniform convergence for continuous functions. In this paper, we investigate an operator constructed as the arithmetic mean of the first $n$ Fejér means. This approach leads to an additional averaging procedure and naturally strengthens the smoothing effect compared to a single Fejér mean of the same order. The operator is studied both in the time and frequency domains. In the time domain, it is represented as a convolution operator with a positive summability kernel. Its normalization and structural properties are established, including preservation of constants and removability of the singularity at the origin. In the frequency domain, the operator is described via its Fourier multipliers, obtained as averages of the corresponding Fejér multipliers. Their monotonic decay with respect to the harmonic index is analyzed, which provides insight into the enhanced attenuation of high-frequency components. A discrete (interpolation-type) analogue defined on a uniform grid is also introduced and interpreted as a quadrature approximation of the continuous convolution representation. Explicit representations of the operator and its kernel are derived. The smoothing character of the method is justified theoretically and confirmed numerically for periodic signals with additive noise. The experiments demonstrate improved suppression of high harmonics compared to classical Fejér summation of the same order. The proposed operator can be regarded as a strengthened low-pass Fourier multiplier method and may be effectively applied to smoothing and filtering of one-dimensional periodic signals.

The increasing relevance of nanofluids in biomedical heat and mass transfer applications has driven the need for more biologically realistic models that can accurately represent the micro-scale dynamics of blood flow. Motivated by this need, this research introduces a novel bio-convective formulation by coupling the effects of nanoparticle-enhanced conductivity with microorganism-induced convection, providing a comprehensive theoretical framework for bioconvective transport of non-Newtonian bio-nanofluids over a nonlinear stretching surface. The model is intended as an idealized representation of shear-driven transport mechanisms relevant to microfluidic and bio-inspired thermal systems. The governing boundary-layer equations for momentum, energy, and microorganism concentration are transformed via similarity variables and solved numerically using the Runge–Kutta–Fehlberg (RKF45) method with the shooting technique. The results reveal that the inclusion of motile microorganisms significantly modifies the flow structure, reducing their accumulation near the surface with increasing nanoparticle concentration, radiation, and magnetic effects. Comparisons between gold and silver-based nanofluids reveal that gold-based suspensions maintain higher thermal energy levels, accompanied by increased viscous resistance and diminished microorganism transport. Parametric analyses indicate that higher nanoparticle concentrations and magnetic field strength lead to reduced velocity and microorganism density, while enhancing the fluid’s temperature due to augmented viscous and Joule heating. Furthermore, increasing the nonlinear stretching parameter and Prandtl number improves convective cooling but restricts microorganism transport. While biomedical applications are discussed for motivation, the present configuration does not represent a patient-specific arterial geometry.

In this paper, we shall prove an existence of solution for the constrained evolution variational inequality problem of finding \(\xi\in \mathcal{K}\subseteq Y= L^{p}(0, T; W_{0}^{1,p}(\Omega))\) (where \(\mathcal{K}\) is a nonempty, closed, convex and symmetric subset of \(Y\)) with \(p\geq 2\), \(T>0\), \(\xi^{\prime}\in Y^*\) and \(\xi(0)=\xi(T)\), such that \[\langle \mathcal{A}\xi-\Delta_{p}\xi, v-\xi\rangle + \langle Q\xi, v-\xi\rangle +\Phi(v)-\Phi(\xi) \geq \langle f^*, v-\xi\rangle,\tag{*}\] for all \(v\in \mathcal{K}\), where \(f^*\in Y^*\), \[\begin{align}\langle Q\xi, w\rangle =&\sum\limits_{i=1}^{N}{\int_{\Gamma}{q_{i}(x, t, \xi(x,t), \nabla \xi(x,t))\frac{\partial w(x,t)}{\partial x_i}dxdt}}\\&+\int_{\Gamma}{q_{0}(x,t, \xi(x,t), \nabla \xi(x,t)) w(x,t)dxdt},~w\in Y,~\xi\in Y,\end{align}\tag{**}\] \(\Gamma =[0, T]\times\Omega\), \(\Omega\) is a nonempty, bounded and open subset of \(\mathbf{R}^{N}\) with \(N\in Z_{+}\), \(\Delta_{p}\) is the \(p\)-Laplacian operator, \(q_i: \overline{\Omega}\times [0, T]\times \mathbf{R}\times \mathbf{R}^{N}\to \mathbf{R}\) satisfies mild conditions, \(\mathcal{A}\xi =\xi^{\prime}\) (where \(\xi^{\prime}\) is the derivative of \(\xi\) in the sense of distributions) and \(\Phi: Y\supseteq D(\Phi)\to \mathbf{R}\cup\{\infty\}\) is a proper, convex and lower-semi-continuous function. In order to address problems like (\(\ast\)), we shall establish new maximal monotonicity results for the sum of two maximal monotone operators \(\mathcal{N}\) and \(\mathcal{M}\) defined from reflexive Banach space into its dual, provided that \(\text{int}(D(\psi_{\mathcal{M}})- D(\psi_{\mathcal{N}}))\neq\emptyset\), where \(\psi_{N}\) and \(\psi_{M}\) are corresponding convex functions introduced by Simons. The question of maximal monotonicity of two maximal monotone operators is one of the outstanding problems in monotone operator theory. The significant contribution of Rockafellar gave a foundation in the study of nonlinear problems. In this paper, we give new maximality results, which present generalizations of the existing criteria, and provide a positive solution for Simons problem . With the help of these results, existence of solution for (\(\ast\)) is proved possibly allowing \(D(\mathcal{A}) \cap \text{int}{\mathcal{K}}=\emptyset\).

This paper investigates the coupled thermoelastic interactions within an \(n\)-dimensional rectangular parallelepiped domain under time-dependent boundary conditions, formulating a hyperbolic system based on the Cattaneo-Vernotte principle to account for finite-speed thermal wave propagation. The mixed boundary value problem, incorporating non-homogeneous Dirichlet conditions and Cauchy initial data for displacement and temperature fields, is solved analytically via the Generalized Fourier Principle, yielding a unified solution expressed as an \(n\)-dimensional eigenfunction expansion. To validate the analytical findings and address complex configurations, a Fibonacci Collocation Spectral Method (FCSM) evaluated at Chebyshev–Gauss–Lobatto nodes is developed. Rigorous error analysis in \(L^2\) and \(L^\infty\) norms confirms spectral convergence under appropriate regularity assumptions. Numerical experiments in one, two, and three dimensions demonstrate exponential error decay from \(\mathcal{O}(10^{-3})\) to \(\mathcal{O}(10^{-14})\) with moderate polynomial degrees, establishing a robust theoretical and computational framework for analyzing wave-like thermoelastic behavior in high-precision engineering and advanced materials applications.

We study the Diophantine problem of determining for which positive integers \(M\) the sum of \(M\) consecutive squares beginning at \(a^{2}\) can itself be a square, namely \[\sum\limits_{i=0}^{M-1}(a+i)^{2}=s^{2}.\] Using the necessary conditions established by Beeckmans, we derive sharper congruence restrictions on the parameter \(M\). In particular, we prove that no solution exists when \(M\equiv5,6,7,8\) or \(10\left(\text{mod}\,12\right)\). For the remaining congruence classes \(M\equiv0,1,2,4,9\) or \(11\left(\text{mod}\,12\right)\), we obtain refined necessary conditions, namely \(M\equiv0\) or \(24\left(\text{mod}\,72\right)\); \(M\equiv1,2\) or \(16\left(\text{mod}\,24\right)\); \(M\equiv9\) or \(33\left(\text{mod}\,72\right)\); or \(M\equiv11\left(\text{mod}\,12\right)\), together with the corresponding congruence restrictions on \(a\) and \(s\). These classes should be interpreted only as necessary compatibility conditions; they do not, on their own, establish the existence of solutions. The remaining residue class \(M\equiv3\left(\text{mod}\,12\right)\) is examined separately by means of a recursive residue-class sieve that yields computational evidence against solvability, although no complete symbolic exclusion is claimed. Finally, when \(M\) is itself a square and a solution exists, we show that necessarily \(M\equiv1\left(\text{mod}\,24\right)\) and \(\left(M-1\right)/24\) is a generalized pentagonal number.

We establish a localized Bochner-type rigidity theorem for harmonic maps between Riemannian manifolds. Let f : (M, g) → (M̂, ĝ) be a harmonic map from a compact manifold. Instead of assuming global nonpositivity of the sectional curvature of the target manifold, we impose a curvature bound localized along the image f(M), expressed in terms of the maximal sectional curvature encountered along this image. We prove that if the minimal Ricci curvature of the domain dominates this image–dependent curvature bound through a quantitative curvature pinching inequality involving the maximal energy density of f, then the map must be constant. In the critical case of equality, we obtain a homothetic classification: the differential of f is parallel and the image f(M) is totally geodesic. Thus, the theorem replaces global curvature sign assumptions by an image–dependent curvature domination principle and provides a localized analogue of classical Yano–Ishihara–type rigidity results.