For all positive non-square integer multipliers, there are infinitely many triangular numbers that are multiples of other triangular numbers. With a simple change of variables, one obtains a Pell equation, whose odd solutions provide the indices of the many infinitely triangular numbers multiple of other triangular numbers. General algebraic expressions of fundamental solutions of Pell equations are found for the multiplier expressed in function of the closest integer square. Finally, recurrent relations yielding the triangular numbers and their multiples and indices are calculated for non-square multipliers.
We introduce a class of mixed general bivariational inequalities in a real Hilbert space and show that several known models, including mixed variational inequalities, bivariational inequalities, general variational inequalities, and complementarity problems, arise as special cases. An auxiliary-principle framework is then used to derive predictor–corrector type iterative schemes. A basic descent estimate is established under a g-partially relaxed monotonicity assumption, and a convergence theorem is obtained under natural continuity and uniqueness hypotheses. The presentation has been streamlined to make the algorithmic steps explicit, and a scalar example is included to illustrate the resolvent formulation.
Let \(\eta\) be a fixed positive integer. Let \(S\) be a subset of \(\mathbb{Z}\), \(\star:S\times S\to \mathbb{Z}\) be a binary function, and \(\zeta_{\eta}:\{\xi\in \mathbb{Z}:\gcd(\xi,\eta)=1\}\to \{0,1\}\) be a function. For a simple connected graph \(G\) of order \(n\), a bijective function \(f:V(G)\to S\) (where \(|S|=n\)) is called an arithmetic cordial labeling modulo \(\eta\) under the arithmetic structure \(\langle S,\zeta_\eta,\star\rangle\) if the induced function \(f_\eta^*:E(G)\to \{0,1\}\), defined by \(f_\eta^*(ab)=1\) whenever \(\gcd(f(a)\star f(b),\eta)= 1\) and \(\zeta_\eta(f(a)\star f(b))=1\); otherwise, \(f_\eta^*(ab)=0\), satisfies the condition \(|e_{f_\eta^*}(0)-e_{f_\eta^*}(1)|\leq 1\), where \(e_{f_\eta^*}(i)\) is the number of edges with label \(i\) (\(i=0,1\)). In this paper, we explore the arithmetic cordial labeling of some graphs under conditions imposed on the function \(\zeta_\eta\). The graphs included are star graphs, ladder graphs, alternate cycle snake graphs, join graphs, corona graphs, and tensor product graphs.
This article proposes a new extension of fixed-point theorems in the context of b-fuzzy metric spaces based on Geraghty-type inequalities. We present the concept of pair upper \((F,h)\)-class functions, which play a key role in establishing existence and uniqueness of fixed points for contractive situations. These outcomes extend and generalize some eminent fixed-point theorems in fuzzy and b-metric spaces. We support our results by proper example.
Classical orthogonal polynomials of the Askey-Wilson scheme have many different properties, e.g. they satisfy differential and recurrence equations and they have hypergeometric representations, Rodrigues formulas, generating functions, moment representations, etc. In this paper we concentrate on finding multiple hypergeometric representations for the polynomial sequences belonging to the classical continuous and classical discrete classes that are defined on a linear lattice. Currently such a database is not available. Using computer algebra, especially Zeilberger’s algorithm, it is possible to prove such identities and therefore the paper is accompanied by a Maple worksheet containing derivations or proofs of all given identities, most of which are new.
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.
