Abstract: Fractional-order models provide an effective framework for studying epidemiological processes with memory effects. In this study, a nonlinear fractional-order SEIHR model for pneumonia transmission is proposed using the Caputo derivative. The model is analyzed within the framework of nonlinear functional analysis, where the system is represented by a nonlinear operator on a suitable Banach space. Fundamental qualitative properties, including positivity and boundedness of solutions, are rigorously established. Disease-free and endemic equilibrium points are derived, and the basic reproduction number is obtained via the next-generation operator approach. Local and global stability of equilibrium are investigated using fractional-order spectral conditions and Lyapunov functions. Numerical simulations based on the fractional Adams–Bashforth–Moulton method support the theoretical results and illustrate the influence of memory effects on pneumonia transmission dynamics.

Abstract: Linear systems with nonnegative or positive coefficients play a central role in the analysis of dynamical processes where admissible states are required to be positive. This paper studies a linear system governed by a positive matrix and interprets it as a spec-tral problem motivated by a subsidy allocation model. The analysis is carried out within the framework of the Perron–Frobenius theory and relies on classical results of linear algebra, in particular Perron’s theorem and Wielandt’s lemma. Using purely theoretical methods, we show that a fair allocation is characterized by a positive eigenvector asso-ciated with the spectral radius of the underlying matrix. The positivity and primitivity of the matrix guarantee the existence and uniqueness of this eigenvector up to scaling, while the convergence of matrix powers ensures the stability of the resulting allocation independently of initial conditions. These results demonstrate that fairness and stability arise as intrinsic consequences of the spectral structure of positive matrices. The paper provides a rigorous mathematical interpretation of equilibrium and stability in linear dynamical systems and illustrates the relevance of positive matrix theory in the study of structured linear models.

Abstract: Representational alignment, defined as correspondence between distinct representations of the same underlying structure, is usually evaluated using coordinate-level similarity in high-dimensional spaces, together with correlation-based measures, subspace alignment techniques, probing performance and mutual predictability. However, these approaches do not specify a baseline for the level of agreement induced solely by dimensional compression, shared statistical structure or symmetry. We develop a methodological framework for assessing representational alignment using the Borsuk-Ulam theorem as a formal constraint. Representations are modeled as continuous maps from a state space endowed with a minimal symmetry into lower-dimensional descriptive spaces. In this setting, the Borsuk-Ulam theorem provides a lower bound on the identification of symmetry-paired states that must arise under dimensional compression. Building on this bound, we define representational alignment in terms of shared induced equivalence relations rather than coordinate-level similarity. Alignment is quantified by testing whether distinct models collapse the same symmetry-related states beyond what is guaranteed by topological necessity alone. The resulting metrics are architecture-independent, symmetry-explicit and compatible with probe-based comparisons, enabling controlled null models and scale-dependent analyses. Our framework supports testable hypotheses concerning how alignment varies with representation dimension, compression strength and symmetry structure, and applies to both synthetic and learned representations without requiring access to internal model parameters. By grounding alignment assessment in a well-defined topological constraint, this approach enables principled comparison of representations while remaining neutral with respect to the semantic or ontological interpretation of learned features.

Abstract: This paper establishes a unified mathematical framework independent of strong regularity constraints on initial data and external forces, and rigorously proves the existence, uniqueness, and stability of global smooth solutions for the 3D incompressible Navier-Stokes equations. The framework covers three classes of initial data: $H^{s}$-bounded, purely $L^{2}$-bounded, and locally weakly singular, with external forces restricted only to $L^{2}([0, \infty) ; L^{2}(\mathbb{R}^{3}))$. The core innovation lies in the trinity framework of compactly supported mollifier regularization, uniform double limit energy estimates, and Galerkin iteration, which seamlessly adapts to both weakly regular practical scenarios and highly regular ideal scenarios without structural reconstruction. Key conclusions include: (1) Local weak singularities of initial data vanish instantaneously for $t>0$, and solutions are globally smooth in $C^{\infty}((0, \infty) ; H^{\infty}(\mathbb{R}^{3}))$; (2) High-regularity initial data and external forces yield solutions with arbitrary-order smoothness at $t=0$ and for all subsequent time, excluding finite-time blow-up; (3) Turbulent "apparent singularities" are interpreted as spatiotemporal high-frequency oscillations of smooth solutions, without relying on physical assumptions. This work fills the gap in weakly regular well-posedness theory and provides rigorous mathematical support for ideal scenario analysis.

Abstract: The Collatz conjecture is a long-standing open problem in number theory characterized by simple local rules and unexpectedly complex global behavior. Despite extensive numerical verification and diverse analytical approaches, no classical proof of convergence is currently known. In this work, we present a structural reinterpretation of the Collatz map within Viscous Time Theory (VTT), a heuristic framework for dissipative information flow in discrete dynamical systems. Rather than addressing the conjecture through additive or probabilistic reasoning, we model Collatz iterations as trajectories evolving within an informational field governed by admissibility, curvature, and dissipation constraints. We introduce quantitative measures of informational curvature (ΔC), admissibility (Φα), and dissipation, and apply them to large-scale validated Collatz trajectories. Independent computational validation confirms that Φα remains strictly subcritical and that the mean informational curvature is negative across all tested initial conditions. These results indicate that local expansion events do not accumulate into global divergence, but are structurally confined within a stable informational basin. We classify the Collatz dynamics as a dissipative system operating in a non-additive informational regime, denoted Metamorphosis 4, in which classical conservation-based reasoning ceases to be applicable while identity persistence remains intact. This approach does not constitute a proof in the axiomatic sense, but provides a validated structural explanation for the absence of divergent trajectories. The proposed framework offers a new perspective on Collatz-type problems and suggests broader applicability to discrete dynamical systems exhibiting apparent paradoxical behavior.

Abstract: This study introduces a novel neural network-based symbolic computation algorithm (NNSCA) for obtaining exact solutions to the (3+1)dimension Jimbo-Miwa equation. By integrating neural networks with symbolic computation, NNSCA addresses the limitations of conventional approaches, enabling the derivation and visualization of exact solutions. The neural network architecture is meticulously designed, and the partial differential equation is transformed into algebraic constraints via Maple, establishing a closed-loop solution framework. NNSCA offers a generalized paradigm for investigating high-dimensional nonlinear partial differential equations, highlighting its substantial application prospects.

Abstract: In this paper, we extend the Singh-Chatterjea type contractive mappings from metric spaces to b-metric spaces. Additionally, we introduce extra terms into the mappings to extend and generalize several fixed point results within the framework of b-metric spaces.

Abstract: We define a concrete ``truncated quasiconformal energy'' $E_\xi(T)$ associated to the Riemann $\xi$--function on a height window $|t|<T$. The definition is geometric: one selects a canonical family of disjoint level corridors, anchored to representative off--critical zeros (one per level, if any exist), and considers the least possible quasiconformal dilatation needed to move those symmetric puncture pairs toward the critical line subject to a corridor--control constraint. We then prove sharp extremal--length lower bounds of the form \[ E_\xi(T)\ \ge\ \log\!\left(\frac{\Mod(\Gamma^{\mathrm{src}}_\xi(T))}{\Mod(\Gamma^{\mathrm{tgt}}_\xi(T))}\right), \qquad d_\xi(T):=\tfrac12 E_\xi(T)\ \ge\ \tfrac12\log\!\left(\frac{\Mod(\Gamma^{\mathrm{src}}_\xi(T))}{\Mod(\Gamma^{\mathrm{tgt}}_\xi(T))}\right), \] and we compute the moduli explicitly in terms of corridor widths in a uniform level decomposition. These inequalities are unconditional consequences of extremal length and do not prove the Riemann Hypothesis. Their role is to produce a mathematically precise ``energy ladder'' $T\mapsto E_\xi(T)$: each finite window yields a finite-stage energy optimization problem, while any divergence $E_\xi(T)\to\infty$ as $T\to\infty$ is an infinite-energy obstruction to a global bounded-distortion axis-landing deformation in the chosen corridor-controlled class.

Abstract: This article is devoted to constructing of fractional powers of operators and their matrix approximations. A key feature of this study is the use of a spectral approach that remains applicable even when the base operator does not generate a semigroup. Our main results include the convergence rate of matrix approximation, derived from resolvent estimates, and a practical algorithm for constructing matrix approximations. The theory is supported by examples.

Abstract: It is posited that the two balanced ternary systems, (-1, 0, 1) and (-i, 0, i), are positioned on the line of real numbers and on the axis of complex numbers, respectively. In the event that the system is reduced to a single entity, the digits of the resulting system will be as follows: { -1, -i, 0, i, 1} The set (-1, -i, 0, i, 1) is transformed into a base five system. In this article and the following ones, I will outline the aforementioned relationship and its considerable potential for implementation in the domains of computer technology and a novel programming language. In addition to laying the groundwork for the trivalent system, which was clearly and brilliantly developed by Jan Łukasiewicz, we can expand beyond the {third middle defined by Aristotle in Chapter 9 of his treatise "De Interpretatione", which was written in opposition to the Stoics' determinism. This perspective enables us to extend the law of middles to the fourth, fifth, sixth, and so on, while adhering to the principles of polyvalent systems. This generates a proliferating field of probabilities where we can establish a chain of closely related probabilities, link by link, where each one is equally likely to be true or false. This allows us to approach or separate from the local truth or lie. I understand that the concepts of truth and falsehood, as developed by mathematical logic in a bivalent system, refer to a particular truth or lie. Thus, absolute truth is universal and impossible to know. However, it is not necessary to know absolute truth because what affects us in our daily lives is local truth or local falsehood. Then, it is appropriate to discern between a local truth universally accepted and a falsehood that can also be accepted as true, as well as the distinction between a true truth and a falsehood that could also be a true lie. In this article, we will analyze up to the third dimension (3D) which is composed by the following structures: i. A polyvalent system of "fifth truth degree", where the fifth middle is introduced. ii. A balanced system of base seven, in which seven coordinated points are introduced. iii. This balanced system operates within the Ternary Balanced system. iv. The Ternary base number defines the lowest and highest limits. v. Every volumetric body is founded on its complex plane, but empty space, between the volumetric bodies is a volume of its respective dimension. vi. Every mathematical operation can be developed directly as (ST110i0)(1T0S1) or (ST110i0)/(1T0S1) without requiring the complex polynomial form. A polyvalent system allows us to construct volumes of bodies, then surface of volumetric bodies, then volumes of volumetric bodies, then surface of volumes of volumetric bodies, and so forth. I briefly glance beyond the seventh base to the eleventh, thirteenth, and fifteenth bases.

Abstract: Goldbach’s conjecture, one of the oldest and most resilient problems in number theory, has traditionally been approached through additive and combinatorial methods. Despite extensive numerical verification and partial results, a structural explanation for its apparent universality remains elusive. In this work, we propose a reinterpretation of Goldbach’s conjecture within the framework of Viscous Time Theory (VTT), introducing an informational–geometric perspective in which prime numbers are treated as stable coherence attractors in an informational field. Within this framework, the pairing of two primes summing to an even integer is no longer viewed as a purely combinatorial coincidence, but as a coherence-driven event governed by informational balance and minimal decoherence pathways. We introduce measurable informational parameters, notably ΔC (coherence variation) and ΔI (informational imbalance), and show how they provide a natural ordering principle for prime pairing phenomena. The conjecture is thus reframed as a manifestation of structural stability in an informational field, rather than as a purely arithmetic property. While no classical proof is claimed, this approach offers a unifying conceptual model that accounts for the persistence of Goldbach-type pairings and connects number theory with broader informational and geometric principles. The results suggest that Goldbach’s conjecture may be interpreted as a specific instance of a more general coherence pairing mechanism in discrete informational systems. The proposed framework is further supported by large-scale numerical validation up to even integers, revealing smooth scaling behavior, bounded curvature, and stable coherence-field signatures consistent with the theoretical model.

Abstract: Multivariable optimization is an essential mathematical exercise in daily engineering design and troubleshooting. To this end, simplex multivariable optimization method is a powerful optimization approach that has served many engineering disciplines well over the years. One such simplex algorithm is the Nelder Mead algorithm. But, the reflection step of the Nelder Mead algorithm may increase the number of iterations needed to arrive at the optimal point, as it reflects the starting point to the opposite side of the function. This work proposes an automated two-stage adjustable ratio simplex optimization method that first search within and around the optimization surface for a good starting point, followed by a narrow and more refined search for the optimal point. For both stages of the new simplex algorithm, only contraction and extension steps are used, and this helps to remove possible oscillatory effects common to other simplex algorithms as the iterations progress. Demonstrative use of the new simplex algorithm on optimizing the coefficients of a quadratic function reveals good accuracy and speed as compared to the Nelder Mead algorithm which uses significantly more iterations. Future testing should be conducted with other optimization functions as well as objective functions. Interested readers are invited to explore and expand on the work reported herein.

Abstract: In the theory of mixed-type equations, there are many works in bounded domains with smooth boundaries bounded by a normal curve for first and second-kind mixed-type equations. In this paper, for a second-kind mixed-type equation in an unbounded domain whose elliptic part is a horizontal half-strip, a Bitsadze-Samarskii type problem is investigated. The uniqueness of the solution is proved using the extremum principle, and the existence of the solution is proved by the Green’s function method and the integral equations method. When constructing the Green’s function, the properties of Bessel functions of the second kind with imaginary argument and the properties of the Gauss hypergeometric function are widely used. Visualization of the solution to the Bitsadze-Samarskii type problem is performed, confirming its correctness from both mathematical and physical points of view.

Abstract: Calculating finite sums and products of trigonometric functions is an important and fascinating problem, a straightforward method is using infinite series or infinite product. In this paper, we calculate four finite sums and a finite product of trigonometric functions using this method, which contributes to a deeper understanding of this problem.

Abstract: Golden section is a proven, robust and high utility single variable optimization method of use in many engineering disciplines. In its typical implementation, a fixed ratio (also known as Golden ratio) is used to partition the left-right bracket in such a way that the two internal points are strategically placed to close-in on the minimum point. Practical usage of this fixed ratio method shows good performance in progressive movement of x2 internal point to the minimum value. However, the fixed ratio method could not capture or follow the curve of rapidly changing function well. Herein, an improved Golden section method with dynamic ratio for partitioning the bracket is formulated and tested on a demonstrative case to optimize a quadratic function. Due to lack of computational power on a GPU budget laptop, more tests could not be conducted on this dynamic ratio Golden section method. Initial results obtained showed that the dynamic ratio does place the x2 internal point closer to the right bracket in regions of rapidly changing value. Overall, the dynamic ratio Golden section method reached convergence to the minimum point in 3 iterations compared to 12 iterations for the conventional fixed ratio Golden section method in a demonstrative case.

Abstract: This study explores the FitzHugh-Nagumo model, a mathematical system used to simulate the activity of neurons. We apply the Adomian Decomposition Method (ADM) to generate approximate solutions using polynomial series. While these formula-based approximations are highly accurate for capturing short-term changes, the analysis reveals a critical limitation: they eventually fail over longer timeframes. This is because the neuron model is designed to produce stable, repeating cycles (oscillations), whereas polynomial approximations naturally grow to infinity rather than looping back. Consequently, this analytical method cannot accurately reproduce the neuron's long-term, rhythmic behavior. To accurately capture the long-term dynamics and spiking behavior of the neuron, numerical integration approaches are the necessary and most reliable option.

Abstract: The divisibility of integer shifts by multiplicative functions has long been a central topic in analytic number theory, originating from Lehmer’s study of φ(n) | n − 1 and later refinements concerning φ(n) | n + a. While extensive progress has been made for the Euler totient function, analogous questions for the Carmichael function λ(n) have remained largely inaccessible to existing techniques, particularly for fixed positive shifts a > 1. This paper investigates the divisibility relation λ(n) | n + a, establishing that for every fixed integer a ≥ 2 the set of such integers n has natural density zero unconditionally, and is finite under standard equidistribution assumptions on primes in square-modulus progressions, such as the Elliott–Halberstam conjecture. The argument combines structural and analytic methods. A valuation obstruction shows that for squarefree integers with sufficiently many prime factors, divisibility fails whenever a prime power divides λ(n) to higher order than it divides n + a. Analytic input from the Bombieri–Vinogradov theorem controls the distribution of small prime divisors of p_i − 1, while stronger equidistribution hypotheses are used to force the appearance of squared prime factors and trigger the obstruction. The only remaining configuration is an exceptional schema in which all prime divisors p_i of n satisfy p_i ≡ 1 (mod M ) with M | (a + 1) and p_i − 1 squarefree, a case shown to yield at most one admissible prime for each modulus. Together these results provide a complete structural classification of the divisibility condition λ(n) | n + a across integer shifts, highlighting the finiteness of all positive shifts a ≥ 2, the contrasting behavior of the case a = 1, and the existence of infinite families for negative shifts. The methods extend naturally to other multiplicative functions, suggesting a broader framework for shifted divisibility phenomena. Extensive computational verification for all n ≤ 2 × 106 finds no counterexamples, providing strong numerical support for the theoretical conclusions.

Abstract: The sum-of-divisors function σ(n) has been studied since antiquity, most often in connection with perfect and abundant numbers, yet its behavior under additive divisibility constraints has not been systematically classified. The paper considers the problem of determining, for a fixed integer a, the positive integers n for which σ(n) | n + a. It is shown that for every fixed integer a ≥ 2, only finitely many positive integers n satisfy this relation. The proof reduces the divisibility condition to a size dichotomy: either n < a, yielding only finitely many possibilities, or σ(n) = n + a, which is equivalent to a fixed-value equation for the sum of proper divisors. It is then shown that this equation admits only finitely many solutions for each fixed a. Special cases are described explicitly. When a = 1, the relation σ(n) | n + 1 holds only for n = 1 and for prime n. When a = 0, the condition reduces to σ(n) = 2n, recovering the classical perfect numbers. For a < 0, the inequality σ(n) > n for all n > 1 excludes all but trivial cases. These results complete the classification of shifted divisibility for σ(n) and close the sequence initiated by analogous investigations of φ(n) and λ(n), identifying σ as the terminal case in which multiplicative divisibility collapses to finiteness.

Abstract: We develop Real Differentiation: a fibre- and stack-sensitive language for infinitesimal change whose basic objects are labelled (“energy-tagged”) fields and operators. The formalism is designed to enforce a no branch mixing principle: differentiation, integration, and PDE operators act fibrewise on each labelled branch, while still permitting monodromy-aware transport when labels are organised as a covering space or stack of analytic continuations. The core construction is a jet-theoretic endofunctor D, defined via the first infinitesimal neighbourhood of the diagonal. In the constant-label case \( \mathbb{E}\mathcal{x} \) , we show that the labelled topos (Sh(\( \mathcal{B} \))/\( \mathcal{X} \) )/\( \mathbb{E}\mathcal{x} \) decomposes canonically as a product of copies of Sh(\( \mathcal{B} \))/\( \mathcal{X} \) , so that labelled objects are literally families indexed by energy values. This makes “no branch mixing” a theorem: all constructions in the labelled topos act componentwise. In the stacky/local-system case, labels are organised by an étale map (or stack) \( \mathcal{E} \) → \( \mathcal{X} \) ; labelled objects are sheaves on the total space and carry monodromy by descent. We further define directional derivatives by contraction of the universal derivation with vector fields; we prove Leibniz and chain rules, including a smooth/analytic chain rule under explicit functional-calculus hypotheses. Higher jets are related to differential operators, and an infinite-jet comonad governs the differential-operator calculus. Finally, we introduce controlled energy mixing via correspondences of label objects, providing a principled way to model coupled branch dynamics.

Abstract: In this paper we investigate the weighted $L^p$ boundedness of generalized Marcinkiewicz integrals $\mathcal{M}^{(\varepsilon)}_{\mathbf{K}}$ over multiple symmetric domains. Under the conditions $\mathbf{K}\in L^{q}( \mathbb{B}^{{{m}}-1}\times \mathbb{B}% ^{{{n}}-1})$, $q>1$, we stablish suitable weighted $L^p$ bounds for the integrals $\mathcal{M}^{(\varepsilon)}_{\mathbf{K}}$. These bounds are combined with an extrapolation argument of Yano so we obtain the weighted $L^p$ boundedness of $\mathcal{M}^{(\varepsilon)}_{\mathbf{K}}$ from the Triebel-Lizorkin space $\overset{.}{F}_{p}^{0,\varepsilon}(\omega_1,\omega_2)$ to the space $L^p(\omega_1,\omega_2)$ under the weak conditions $\mathbf{K}$ lie in the space $ B_q^{(0,\frac{2}{\varepsilon}-1)}(\mathbb{B}^{m-1}\times\mathbb{B}% ^{n-1})$ or in the space $L(\log L)^{2/\varepsilon}(\mathbb{B}^{m-1}\times\mathbb{B}% ^{n-1})$. Our findings are essential improvements and extension of several known findings in the literature.