In this paper, we present new sharp forms of Iyengar-type integral inequalities for differentiable and twice-differentiable functions. Using exact integral identities and the Cauchy-Schwarz inequality, we derive optimal \(L^2\)-bounds for the deviation between the midpoint value and the integral mean of a function. These results refine and extend several classical Hermite-Hadamard and Iyengar inequalities.

This study introduces the Secant–Weibull Autoregressive Conditional Duration (SW–ACD) model and its extension with exogenous calendar effects (SW–ACD–X) . The primary innovation is the integration of the Secant-Weibull distribution as the innovation law, which allows the framework to capture non-monotonic intensity shapes, such as unimodal and bathtub patterns, that are typically inaccessible to standard monotonic models. A significant methodological contribution is the SW–ACD–X model, which endogenously incorporates intraday seasonality into the conditional mean equation. This joint estimation strategy provides an integrated alternative to traditional two-step pre-filtering by simultaneously capturing the interaction between deterministic diurnal patterns and stochastic duration clustering. The numerical properties of the proposed models are assessed through Monte Carlo simulations, which demonstrate asymptotic consistency while highlighting inherent identification challenges in small-sample regimes. Model estimation is implemented using a dual approach: Frequentist Maximum Likelihood and Bayesian Hamiltonian Monte Carlo (HMC) via the No-U-Turn Sampler (NUTS) in RStan. Empirical application to high-frequency IBM transaction data shows that the SW–ACD–X exhibits promising fit advantages over established benchmarks, including the W–ACD–X, LW–ACD–X, G-ACD-X , and Lomax–ACD–X models. Comprehensive model selection based on AIC, BIC, WAIC, and LOOIC confirms that the proposed model is a robust tool for analyzing market microstructure, liquidity dynamics, and the complex patterns of high-frequency durations.

In this paper, we introduce and study the class of modified \((p, h)\)-convex stochastic processes, which unifies and extends several existing notions of convexity in the stochastic setting. We establish fundamental arithmetic properties of this class and derive Hermite–Hadamard-type inequalities using classical and fractional Katugampola-type operators. We also investigated Ostrowski-type and Jensen-type inequalities in a simple and unified framework. Our results generalize and unify many existing results in the literature, providing a comprehensive framework for the study of convex stochastic processes.

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.