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.
We study a topology on the space of continuous functions between metric spaces, constructed from an admissible ideal on the natural numbers and a fixed dense sampling of the domain. Two functions are declared close whenever their pointwise deviations at the sampling points are negligible with respect to the ideal, and the resulting topology is defined via a canonical uniform structure. We show that under a compatibility condition linking the sampling sequence to the ideal, this topology is Hausdorff and, in fact, coincides with the classical topology of uniform convergence; without this compatibility condition, the topology can be strictly coarser than the uniform topology and can fail to be Hausdorff. We characterise ordinary and ideal convergence of function sequences with respect to this topology, showing that pointwise, uniform, and topological ideal convergence collapse to a two-level hierarchy once compatibility is assumed. We also show that the resulting function space is pseudometrisable in general and metrisable under compatibility, and consequently is automatically Fréchet–Urysohn and strictly Fréchet–Urysohn whenever compatibility holds, independently of any covering property of the domain.
Let \(G\) be a finite group. The product order relatively prime graph (PORP graph) \(\Gamma_{porp}(G)\) is the graph with vertex set \(V(\Gamma_{porp}(G))=G\) and two distinct vertices \(a\) and \(b\) are adjacent if \((o(a),o(ab))=1\) or \((o(b),o(ab))=1\). In this paper, we investigate the product order relatively prime graph associated with the dihedral group \(D_n\). We study the graph for two important classes of dihedral groups, namely \(D_{p^\alpha}\), where \(p\) is a prime and \(\alpha\ge1\), and \(D_{pq}\), where \(p\) and \(q\) are distinct odd primes. We determine the graph structure and establish several structural properties, including the degree sequence, total number of edges, independence number, clique number and structural decomposition. Furthermore, we investigate graph-theoretic properties such as planarity, bipartiteness, Eulericity, Hamiltonicity, and characterize the conditions under which the graph is complete.
A fractional-order predator–prey model (FOPPM) based on the Caputo derivative of order \(\alpha\) (\(0 < \alpha \le 1\)) is developed to examine the joint influence of a strong Allee effect in the prey and a nonlinear anti-predator feedback acting on the predator. The formulation couples cubic prey growth and bilinear predation with a saturating feedback loss, thereby representing threshold-dependent recovery, trophic conversion, and density-dependent behavioural regulation. The system admits three boundary equilibria, while coexistence equilibria are determined by a corrected cubic algebraic equation. Its discriminant classifies one- and three-real-root regimes, but ecological feasibility further requires \(A<X^*<K\), \(Y^*>0\), and \(kmX^*-d>0\). Nullcline geometry shows that at most two roots can be biologically feasible; the actual number is therefore established by direct root filtering for each parameter set. Local stability is characterised through the Jacobian \(\mathcal{J}(E^*)\) and the classical eigenvalue test for integer-order dynamics, while the fractional-order model requires satisfaction of Matignon’s criterion \(|\arg(\lambda_i)|>\frac{\alpha\pi}{2}, \; i=1,2.\) The coexistence equilibrium may switch between stability and instability under variations in carrying capacity \(K\), Allee threshold \(A\), interaction rate \(m\), or feedback intensity \(\varepsilon\). Because the Jacobian eigenvalues do not depend on \(\alpha\), fractional memory does not rotate them; instead, decreasing \(\alpha\) widens the admissible Matignon sector and changes the rate and shape of transient relaxation. A weighted population estimate establishes positivity, uniform boundedness, and global existence. Since the extinction equilibrium is always locally stable under a strong Allee effect, coexistence cannot be globally attractive on the entire positive quadrant. A Volterra-type logarithmic functional is therefore formulated as a basin-restricted Mittag–Leffler certificate on positively invariant sets separated from the coordinate axes. The theoretical results are tested with a fractional Adams–Bashforth–Moulton predictor–corrector (ABM–PC) scheme. A grid-refinement study gives observed orders \(1.91\), \(1.90\), and \(1.96\) for \(\alpha=0.9\), consistent with the expected order \(\min\{2,1+\alpha\}\). An independent \(\alpha=1\) comparison with a high-accuracy Runge–Kutta solution gives maximum componentwise errors below \(3.5\times10^{-4}\). Parameter sweeps across \((K,A,m,\varepsilon,\alpha)\) then quantify changes in feasibility, stability, and transient persistence. For \(\alpha=1\), a trace-zero crossing with positive determinant and transversality is the classical Hopf candidate. For \(0<\alpha<1\), crossing the Matignon boundary is instead a fractional stability-sector transition; it is not, by itself, a Hopf bifurcation. Moreover, autonomous Caputo systems of non-integer order do not possess exact nonconstant periodic solutions, so the computed oscillations are interpreted as memory-dependent transients rather than limit cycles. Overall, the FOPPM provides a coherent framework for studying memory, Allee thresholds, and behavioural feedback within one model. Corrected equilibrium algebra, explicit feasibility filtering, fractional stability theory, reproducible convergence checks, and regenerated simulations together distinguish true asymptotic conclusions from finite-time memory effects.
We introduce Lipschitz classes \(\mbox{Lip}^M(\eta,2)\) and \(\mbox{Lip}^M(\eta,\infty)\) of functions associated with the canonical Sturm–Liouville operator \(\mathscr{L}^M\); and we prove a new versions of Titchmarsh and Boas-type theorems for the canonical Sturm–Liouville transform \(\mathscr{F}^M\). An application to the canonical Sturm–Liouville multipliers is given. Titchmarsh and Boas-type theorems for the canonical Fourier–Bessel transform and the canonical Fourier–Jacobi transform are special cases of this work.
We study convergence of sequences of nonempty closed sets in a metric space through the power-type Musielak–Orlicz modular \(\Phi_n(t)=t^{p_n}\). We introduce Wijsman \((p_n)\)-statistical convergence, Wijsman \((p_n)\)-strong Cesàro convergence, and a lacunary block-mean statistical version associated with a lacunary sequence \(\theta=(k_r)\). In the bounded-exponent case \(1\le p_n\le p_+<\infty\), we prove that, for Wijsman bounded sequences, these three modes of convergence are equivalent whenever the lacunary moderate growth condition holds. In the power-type setting this condition is equivalent to \(h_r/k_r\to 0\). We also give comparison results and examples showing why boundedness of the exponent sequence is essential. For random closed sets, we prove a pathwise Cesàro transfer result: at a fixed base point, almost sure modular strong Cesàro control, together with almost sure boundedness of the distance evaluations and bounded exponents, yields almost sure convergence of the Cesàro averages of the distance functionals to the Wijsman limit.
This study investigates a class of number-theoretic problems related to representations of large integers as combinations of prime numbers and quartic terms, extending ideas originating from the Estermann problem. The authors focus on a refined ternary decomposition involving almost equal components, employing analytic methods from additive number theory. Central to the analysis are estimates of trigonometric sums of Weyl type and their applications to bounding error terms in asymptotic formulas. By developing sharper bounds for short exponential sums, the paper establishes a new asymptotic formula for the fourth-degree analogue under constrained conditions. To bridge theory and computation, we further introduce a numerical framework based on Gegenbauer polynomial expansions, which efficiently approximates the associated singular series and exponential integrals with spectral accuracy. The proposed algorithm leverages discrete orthogonality and Gaussian quadrature to achieve high-precision validation of the asymptotic predictions. The obtained results improve previously known bounds, provide rigorous numerical verification, and contribute to a deeper understanding of additive structures involving primes and polynomial terms. These findings have potential implications for related problems in analytic number theory and the development of spectral methods for arithmetic functions.
