In this work, we present the generalized framework of preinvex mapping namely the \(n\)-fractional polynomial \(s\)-like \(\mathtt{m}\)-preinvex mapping. Several fundamental propositions are suggested to clarify the structural features of this newly proposed concept, followed by a detailed study of its rich algebraic properties. Additionally, we obtain a new lemma using a \(k\)-fractional integral operator that is essential to the analysis that follows. We derive refinements of Ostrowski type inequalities in the context of fractional calculus based on the lemma. The finding presented herein offer meaningful refinements and novel insights, significantly extending and enriching the existing literature on mathematical inequalities and fractional convex analysis.

A delay differential model is developed for an infectious disease in which latent infection, early infectiousness, reporting, unreported symptomatic infection, and removal are represented separately. The incidence term is nonnegative, unreported cases add to rather than subtract from infectious pressure, and a latent accounting state preserves population balance across the fixed delay. Continuous nonnegative histories are specified on the complete delay interval. Positivity, boundedness, the population identity, and the invasion reproduction number are established. A functional-correction procedure is formulated as a coupled mean closure followed by Picard iteration; this avoids treating compartment averages as independent quantities and provides a computable local error criterion. Reproducible synthetic experiments use a population of \(10^{6}\), a four-day latency, and a baseline reproduction number of \(2.20\). Without a transmission change, incidence peaks on day \(100.50\) at \(1881.58\) infections per \(100{,}000\) persons per day, with \(84.35\%\) cumulative infection by day 180. A \(60\%\) transmission reduction beginning on day 35 lowers the day-42 effective reproduction number to \(0.876\), the peak to \(25.93\) per \(100{,}000\) per day, and cumulative infection to \(1.68\%\). Holding the reproduction number fixed while increasing the delay from zero to eight days shifts the peak from day \(51.95\) to day \(140.08\) and reduces its height from \(3334.63\) to \(1332.50\) per \(100{,}000\) per day. On a one-day correction interval, the maximum scaled error decreases from \(1.52\times10^{-1}\) at first order to \(7.92\times10^{-3}\) at second order and \(1.99\times10^{-6}\) at fifth order. These experiments are numerical benchmarks rather than forecasts; disease-specific prediction requires calibrated data and identifiability analysis.

Set-valued operations are useful when one input state admits several chemically distinct outcomes, but an ordinary powerset does not distinguish alternative outcomes from co-products and does not preserve stoichiometric multiplicity. This paper develops a type-consistent framework in which atomic species generate a free commutative multiset monoid, elementary hyperoperations return nonempty sets of product multisets, and \((m,n)\)-superhyperoperations act between explicitly defined iterated nonempty powersets. Domain-specific feasibility predicates impose balance, energetic, kinetic, and selection constraints without changing the common algebraic type. Two general results are established: elementary balance is preserved along every finite admissible path, and the \((0,1)\) construction reduces exactly to the underlying hyperoperation. The framework is specialized to biochemical, geochemical, photochemical, electrochemical, radiochemical, neurochemical, and medicinal-chemical systems. Machine-checkable stoichiometric residuals are reported for representative channels. The quantitative checks include the Daniell-cell value \(E^\circ_{\mathrm{cell}}=1.103\,\mathrm{V}\), the photon energy \(hc/(365\,\mathrm{nm})=3.3968\,\mathrm{eV}\), and evaluated \(^{212}\mathrm{Bi}\) branching fractions of \(64.06\%\) and \(35.94\%\), which sum to \(100.00\%\). These checks validate the internal consistency of the formal examples; they are not presented as new laboratory measurements.

Acute viral conjunctivitis can generate rapidly growing outbreaks in schools and other close-contact settings, but intervention comparisons are meaningful only when the underlying transmission model is dimensionally consistent and its reproduction numbers are reproducible. We formulate a closed-population susceptible–exposed–infectious–recovered (SEIR) model with two distinct controls: a dimensionless reduction in effective contact, \(u_1\), and treatment-supported case management, \(u_2\), which increases removal from infectious contact at a maximum rate \(\tau\). The basic reproduction number is \(R_0=\beta/\gamma\), the controlled invasion number is \(R_c=\beta(1-u_1)/(\gamma+\tau u_2)\), and the state-dependent effective reproduction number is \(R_e(t)=R_cS(t)/N\). These expressions satisfy the required identity \(R_c(0,0)=R_0\) and yield an explicit control threshold. The baseline scenario is calibrated to the published estimate \(R_0=2.64\) for acute haemorrhagic conjunctivitis, with a 1.5-day latent period and a 5-day infectious period. Contact reduction alone crosses the invasion threshold at \(u_1=0.621\), whereas the assumed case-management range does not do so alone. A combined scenario \((u_1,u_2)=(0.50,0.80)\) gives \(R_c=0.660\) and limits new infections to \(0.289\%\) of initially susceptible individuals in a 10,000-person illustrative cohort, compared with \(90.938\%\) without control. A 10,000-draw uncertainty analysis gives a \(93.9\%\) probability that the combined scenario maintains \(R_c<1\). Independent numerical solvers agree to within \(4.4\times10^{-10}\) persons, and population is conserved to within \(7.7\times10^{-11}\) persons. The results support combined intervention thresholds as transparent scenario analyses; they do not constitute a clinical ranking of treatments or a population-specific forecast without local calibration.

This work develops a reduced-order model for the transverse deformation of a skeletal-muscle fascicle during an imposed contraction cycle. The muscle is represented as a tensioned fiber coupled to a Kelvin–Voigt foundation, so that elastic storage and rate-dependent dissipation are included explicitly. A through-thickness displacement ansatz is introduced without the coordinate singularity present in earlier formulations, and compatible fixed-end and quiescent initial conditions are imposed. Spatial variation of fiber tension is treated by a bounded perturbation expansion. Projection onto the fixed-end eigenfunctions yields a hierarchy of damped modal equations whose solutions are written with the correct mode-dependent Green function and time convolution. The second-order approximation is evaluated for a reproducible dimensionless test problem and compared with an independent conservative finite-difference solution advanced by the average-acceleration Newmark method. For a tension-heterogeneity amplitude of \(0.20\), the global relative root-mean-square difference is \(0.441\%\), the maximum time-resolved relative \(L^2\) difference after the initial loading interval is \(3.16\%\), and the maximum midpoint displacement difference is \(8.04\times10^{-4}\). Grid refinement from 201 to 401 spatial points changes the final finite-difference solution by \(6.49\times10^{-6}\) in relative \(L^2\) norm. The formulation is intended as a transparent benchmark and rapid reduced-order approximation; it does not replace three-dimensional active-muscle models or constitute clinical validation.

Colombian numbers, also known as self-numbers, are natural numbers that do not have any generator under the digit-addition process \(\phi(n)=n+s(n)\), where \(s(n)\) denotes the sum of digits of \(n\). Here, a generator of \(n\) is any integer \(x\) such that \(\phi(x)=n\). In this paper we study both structural and computational aspects of these numbers, including Kaprekar’s test for their identification and the concept of junction numbers. We present computational counts of numbers with zero, one, and two generators in intervals of the form \([10^k+1,10^{k+1}]\), revealing highly regular digit patterns. Furthermore, we analyze the sequences \(X\) and \(Y\), defined by selecting the largest and smallest generators respectively, and compare their lengths across the same intervals. The results show a remarkable stability in the relative frequencies of the three possible outcomes (\(X\) shorter, \(Y\) shorter, or equal length) suggesting the existence of asymptotic regularities. Finally, we prove that every non-Colombian number can be obtained by successive applications of \(\phi\) starting from a Colombian number.

An integrable version of SLIR (former SEIR) epidemiological model is introduced with the time dependence of the force of infection. The explicit formulas for special solutions are obtained for this model. From these formulas by classical analysis methods one can obtain the formulas for important metrics of spread of disease: maximum number of infectious people and their corresponding peak times. The usefulness of explicit formulas is illustrated by application to the spread of Covid-19. It is shown in the model example that the time dependence of the force of infection produces the bimodal dynamics of infectious people. Multimodal behavior of the spread of disease can be visualized in graphs of time behavior of numbers of Covid-19 infectious people (see https://www.worldometers.info/coronavirus). These multiple waves are explained biologically by different variants of Covid-19 virus. In previous SLIR models multimodal distribution of the number of infectious people appeared only in numerical simulations or under the assumption that there is an external periodic action.

We study binary colorings of the rectangular grid \(G_{m,n}\) through the evolution of their white and black monochromatic clusters. Extending Mansour’s finite-automaton framework, we encode column-by-column configurations by frontier states whose full active-cluster partition is necessarily noncrossing. To refine this planar restriction, we associate with each frontier partition a block-adjacency graph \(Q(\Pi)\), whose vertices are the frontier blocks and whose edges join blocks containing consecutive frontier positions. We prove that \(Q(\Pi)\) is always connected and that the block coloring induced by every reachable state is a proper two-coloring of \(Q(\Pi)\); hence \(Q(\Pi)\) is bipartite. For \(m=4\), the bipartite criterion is complete: exactly \(13\) of the \(14\) noncrossing partitions satisfy it, each admitting precisely two complementary proper block colorings. Explicit witnesses realize all \(26\) resulting nonempty states, so the complete reachable automaton has \(27\) states after adjoining the initial state. Using an exact labeled transition certificate, we derive a reduced rational generating function for the total cluster count \(S_{4,n}\), an explicit closed formula, a minimal linear recurrence of order five, and the asymptotic cluster density under the uniform coloring measure. Exact computations for \(1\leq m\leq8\) support the conjecture that reachability is characterized by the bipartiteness of the block-adjacency graph.

Fractional ordinary differential equations modeling diffusion-like processes are important tools for describing systems where transport is unusual or depends on past behavior, such as in porous media, biological systems, or viscoelastic materials. Since exact solutions are rarely available, we rely on numerical methods to solve them. In this work, we introduce a spectral collocation method that uses Delannoy polynomials to approximate solutions of these equations involving the Caputo derivative. By building operational matrices for the Caputo derivative, the method directly constructs the algebraic system, which is then solved efficiently using Newton’s method. Tests show that the method is highly accurate, converges quickly, and is computationally efficient. These results suggest that Delannoy polynomials are a strong alternative to classical polynomial bases for spectral solutions of fractional differential problems.