Abstract: We investigate discrete-time deterministic systems on finite state spaces equipped with symmetry groups, extending the analysis to actions of arbitrary countable linearly ordered groups. Under the assumption of strong recurrence, characterized by the absence of weakly wandering sets of positive measure, we establish the structural constraints governing dynamical invariants. For systems associated with amenable groups, we employ Følner sequences to rigorously define asymptotic frequencies and demonstrate that maximal Shannon entropy emerges naturally from the system's architecture rather than stochastic assumptions. We show that the interplay of strong recurrence and symmetry enforces specific distribution patterns; while transitive symmetry leads to a uniform stationary distribution and maximal entropy, we provide a generalized formula for non-transitive cases based on orbit decomposition. These results bridge classical recurrence theory and ergodic decomposition with modern measure-theoretic entropy, illustrated through concrete examples for both finite and infinite countable settings.

Abstract: We define the function $Col: \mathbb{N} \to \mathbb{N}$ as the Collatz function, given by $3n + 1$ if $n$ is odd and $\displaystyle\frac{n}{2}$ if $n$ is even. The conjecture postulates that for any positive integer, at some point, its iteration will reach 1, or equivalently, every orbit will fall into the periodic cycle $\{4, 2, 1\}$. Two conditions would invalidate the conjecture: The existence of a divergent orbit or the presence of another cycle. We can study the dynamics of the orbits through the density of even terms in their orbit. If all points' accumulation density exceeds the value of $\displaystyle\frac{\ln(3)}{\ln(2)}$ then the orbit is bounded. The main result of this work is to show that there are no natural numbers such that the accumulation points of the pair density are less than $\displaystyle\frac{\ln(3)}{\ln(2)}$. In other words, there are no divergent orbits.

Abstract: We develop a unified, substitution-based framework for solving non-homogeneous linear ordinary differential equations (ODEs) via the systematic factorization of the associated differential operator. By decomposing higher-order operators into a nested sequence of first-order linear factors, the non-homogeneous problem is resolved through successive integrations using integrating factors. This construction yields explicit integral representations of particular solutions, providing a direct derivation of the convolution kernel and Green’s function without requiring the method of undetermined coefficients, variation of parameters, or Laplace transforms. For second-order equations, we clarify the structural origins of resonance and oscillatory behavior and extend the approach to variable-coefficient settings, specifically Cauchy–Euler equations.

Abstract: In this paper, we focus on the online stochastic optimization problems in which the random parameters follow time-varying distributions. At each round t, decision is obtained from solving current optimization problem.Then samples are drawn from distributions which are updated after obtaining decision. The objective and constraint are updated in this process, and the updated problem is used to obtain the next decision. For solving the online stochastic optimization problem, we propose a model-based stochastic augmented Lagrangian method, which is referred to as MSALM. At each round, we construct the model functions for the sample objective and constraint functions based on their properties, which reduced the computational complexity. The step size is designed in a dynamic form and decreases as t increases to accelerate convergence. Due to the setting of the online stochastic problem, we use stochastic dynamic regret and constraint violation to measure the performance of our algorithm. Under the assumptions, we prove that our algorithm’s stochastic dynamic regret and constraint violation have a sublinear bound of total number of slots T. We design simulation experiments to verify the efficiency of our online algorithm. Its performance is evaluated on a range of information and system engineering problems, including adaptive filtering, online logistic regression, the time-varying smart grid energy dispatch, the online network resource allocation, and the path planning. In addition, in the context of the path planning problem, we integrate our algorithm with supervised learning to demonstrate its enhanced capabilities. The experimental results validate the performance of our new algorithm in practical applications.

Abstract: We ask for ultrametric version of following three: (1) Bourgain-Figiel-Milman Theorem, (2) Enflo Type, (3) Mendel-Naor Cotype.

Abstract: Structural Execution Sequence (The Deductive Itinerary): This manuscript does not propose a continuous physical theory; it executes a formal mathematical theorem. By strictly bounding the scientific enterprise to the thermodynamic limits of finite computation, it outlines the exact deductive sequence that isolates the minimal necessary architectural class capable of compiling empirical reality. The theorem unrolls across four sequential proofs: 1. The Ontological Anchor (Axiom I: The Embedded Observer): The observer is formally defined as a finite physical sub-system. Scientific prediction evaluates strictly as the act of physical computation. It is governed by the Universal Cost Ledger (C_univ), which algorithmically penalizes static memory allocation (S) and dynamic execution trace (T). 2. The Syntactic Anchor (Axiom II: The Computable Boundary): Because the embedded agent is finite, any valid generative framework must execute safely on finite hardware. Empirical science evaluates exclusively as a formal closed system operating within the Computable Domain (M_TTG): the strict mathematical intersection of computability (Turing 1936), semantic exteriority (Tarski 1936), and bounded scope (Gödel 1931). 3. The Semantic Anchor (Axiom III: Data Supremacy): The raw observational array (D), extracted via thermodynamic collision with the environment, evaluates as the absolute ground truth. The syntactic injection of uncomputable continuous parameters or infinite-precision fields (Δθ) triggers a catastrophic memory leak (C_univ -> ∞) and is structurally forbidden by the Zero-Patch Standard. 4. The Hardware Compilation (m*): By enforcing these three axioms against the macroscopic Evidence Vector (E), we algorithmically deduce the exact hardware interface of reality. The minimal necessary architectural class (m*) compiles strictly as a discrete, local, deterministic Base-72 symplectic state-machine. The Checksum Protocol: We dynamically unroll this architectural class to verify that emergent Lorentz invariance, quantum measurement bounds, fractal self-similarity, objective causality, and thermodynamic irreversibility execute natively as the deterministic compiler artifacts of the discrete hardware itself.

Abstract: We study shell kernels for the odd-to-odd Syracuse dynamics generated by uniformly distributed initial windows. For backstepped first-passage shells, we prove short-time localization, derive an exact inverse-affine representation of the fixed-time kernel, and reduce the shell-slice discrepancy to weighted primitive-frequency correlations. We also prove a quantitative boundary-layer estimate and identify a formal renewal model for the corresponding shell mechanism. On the arithmetic side, we obtain an exact block decomposition for the primitive-frequency transfer operator, prove that no naive operator gap is available, and reduce the unresolved step to explicit incomplete principal-unit exponential sums modulo powers of 3. Thus the paper is unconditional up to a final primitive-frequency estimate, which is formulated explicitly.

Abstract: The nontrivial zeros of the Riemann zeta function are parameterized by the spectral variable \( s\in\mathbb{C} \), and the isomonodromic deformation parameter t of the Painlevé III equation of type \( D_6 \) is connected to s by \( t=s(1-s) \), which maps the critical line \( \Re(s)=\frac12 \) to the positive real ray \( t\in[\frac14,\infty) \). Any de Branges realization of the Riemann Hypothesis within this framework requires four explicit conditions: (C1) geometric feasibility ---the positive lambda-length slice of the \( \mathrm{PIII}_{D_6} \) character variety defines a real form of the wild Stokes and monodromy data; (C2) global positivity---the Riemann--Hilbert jump matrices yield a Herglotz Weyl--Titchmarsh function; (C3) embedding compatibility---the functional equation involution \( s\mapsto 1-\bar{s} \) preserves the positive slice; and (C4) analytic regularity---the tau-function composed with \( t=s(1-s) \) is entire of finite order after gauge removal. We prove all four conditions unconditionally. For (C1), an explicit birational map \( \Phi \) expresses all Stokes multipliers as positive monomials in the lambda-lengths. For (C2), the Painlevé/gauge theory correspondence identifies the \( \mathrm{PIII}_{D_6} \) oper with a Schrödinger operator whose real coefficients force \( \Im m(\lambda,t)>0 \) via a Wronskian argument; isomonodromic uniqueness and Remling's inverse theorem complete the proof. For (C4), integrality of the local exponent \( \alpha\in\mathbb{Z}_{\ge0} \) is the precise criterion, satisfied on an explicit sublocus of the positive slice. With all four conditions established, the Riemann Hypothesis reduces to the Bridge Conjecture alone. We test the direct form of the Bridge Conjecture---the identification \( E_{D_6}(s)=C\,\xi(s) \)---and show it fails for all constant monodromy phases and for all Dirichlet L-functions, because the tau-zero counting \( \mathcal{N}_{D_6}(T)\sim 2T/\pi \) lacks the \( \log T \) factor of the Riemann--von Mangoldt law. This leads to the identification of \( E_{D_6}(s) \) as a new explicit element of the Hermite--Biehler class \( \mathcal{HB}(1/2) \), whose canonical form is the isomonodromic cosine \( F(s)=\cos(2\sqrt{s(1-s)}) \). We prove that \( F\in\mathcal{HB}(1/2) \) is entire of order 1, satisfies \( F(s)=F(1-s) \), has all zeros on \( \Re s=\frac12 \) at \( \gamma_n=\sqrt{(2n-1)^2\pi^2-4}\,/\,4 \), with asymptotic spacing \( \pi/2 \) identified as the WKB semiclassical level spacing of the \( \mathrm{PIII}_{D_6} \) oper arising from the Seiberg--Witten period \( a_{D_6}(t)=2\sqrt{t} \). A four-tier falsifiability diagnostic and the character \( \chi_4 \) scorecard are presented.

Abstract: This paper proposes an axiomatization of the absolute infinite and argues against width and height potentialism in set theory. It builds on an unrestricted language and a non-recursively enumerable class theory, called MK^meta, that extends the formal MK: Morse-Kelley with global choice (GC). Class ordinals and class cardinals avoid the Burali-Forti paradox and GC is assumed to warrant comparability of class cardinals. Meta-formality subsequently gets a maximal fixed-point definition under consistency filtering of recursively enumerable formality. By showing that the concept of maximal meta-consistent height (MMH) of an axiom is theory-independent, it follows that no Ord can exceed Ord^meta, the proper class ordinal of MK^meta, such that the absolute infinite Ω^meta = Ord^meta. Unlike formal and infinitary formal-based theories, which are fundamentally incomplete, MK^meta achieves completeness by having absolutely infinitely many formal-based axioms. Moreover, potentialism is countered by MK^meta, which accepts those formal axioms that maximize its models, all of which are elementarily equivalent to the representative V^meta. At last, only the meta-formal level can capture the entire mathematical reality in a single theory and thus give definite answers.

Abstract: Ordinary differential equations are fundamental tools for modeling dynamic systems in science, engineering, and applied mathematics. Solving these equations accurately and efficiently is crucial, particularly in cases where analytical solutions are challenging or impossible to obtain. This paper presents a method for solving inhomogeneous linear ordinary differential equations using an artificial neural network. The network is composed of a single input layer with one neuron, one hidden layer with three neurons, and a single output layer with one neuron. A multiple regression model is employed to determine the weights from the input layer to the hidden layer, while radial basis functions are used to compute the weights from the hidden layer to the output layer. The bias values are chosen within the range of -1 to 1 to optimize learning behavior. A trial solution is constructed as a sum of two parts. One part satisfies the initial condition, and the other part is the output of the network to approximate the function. The neural network is trained to minimize the mean squared error of the residuals obtained by doing the substitution of the trial solution into the given ordinary differential equation. The methodology is tested on first-order and second-order ordinary differential equations to evaluate its accuracy, stability and how its capability can be generalized. The results show that the method can approximate the exact solutions of these ordinary differential equations with high accuracy.

Abstract: Assuming the coefficient matrix is a nonzero singular matrix, we demonstrate that the invertible solutions of the Yang-Baxter-like matrix equation must possess at least two elementary divisors. This establishes a necessary condition for an invertible matrix to satisfy the Yang-Baxter-like matrix equation. Building on this finding, we derive several meaningful corollaries. Additionally, we provide some examples to illustrate our results.

Abstract: We will present numerical methods for solving initial value problems using an indefinite integral approach. This new method allows controlling the order of approximation as well as the number of discretization steps. The methods allow an a priori estimation of the approximation error, so that an optimized solution may be accessible. The approach is based on representing indefinite integrals using discretization over conformal mappings or orthogonal polynomial roots. Using these discretizations, matrices for implicit Runge-Kutta procedures are created using collocation methods.

Abstract: Business development unfolds within complex adaptive environments marked by nonlinear interaction, structural asymmetry, and recurrent instability. Sustained performance under such conditions requires regulatory structures that preserve coherence while enabling structural transformation. This study advances symmetry evolution as a systems principle that explains the emergence of balance through interaction among decision bias, structural symmetry, and regulatory intensity. An evolutionary regulation framework represents this interaction as a closed-loop dynamic that drives coevolution of regulation and symmetry through recursive feedback. Stability emerges as a property of proportional coupling rather than correction of deviation. Multi-modal simulations representing turbulent decision landscapes demonstrate formation of bounded oscillatory equilibrium under perturbation while preserving exploratory capacity. Coordinated evolution of regulatory gain and structural symmetry sustains adaptive stability without suppressing innovation dynamics. The study establishes a systemic foundation for resilience and endogenous governance in complex business systems and reframes decision optimization as structural adaptation within evolving regulatory architectures.

Abstract: This paper investigates the existence and uniqueness of weak solutions for a stochastic Ginzburg-Landau equation involving the fractional Laplacian. The primary focus is on establishing a rigorous mathematical framework to handle the coexistence of the nonlocal fractional Laplacian and stochastic perturbations. By employing the Galerkin method, we establish that the initial-boundary value problem admits a unique global weak solution for any \( L^{2}_{a} \) initial value. This study utilizes the properties of the fractional Laplacian and fractional Sobolev spaces to provide a rigorous proof of the existence and uniqueness theorem. These results extend the analysis of Ginzburg-Landau equations to models incorporating stochastic terms and fractional Laplacian.

Abstract: Dense light field depth estimation remains challenging due to sparse angular sampling, occlusion boundaries, textureless regions, and the cost of exhaustive multi-view matching. We propose Deep Spectral Epipolar Representation (DSER), a geometry-aware framework that introduces spectral regularization in the epipolar domain for dense disparity reconstruction. DSER models frequency-consistent EPI structure to constrain correspondence estimation and couples this prior with a hybrid inference pipeline that combines least squares gradient initialization, plane-sweeping cost aggregation, and multiscale EPI refinement. An occlusion-aware directed random walk further propagates reliable disparity along edge-consistent paths, improving boundary sharpness and weak-texture stability. Experiments on benchmark and real-world light field datasets show that DSER achieves a strong accuracy-efficiency trade-off, producing more structurally consistent depth maps than representative classical and hybrid baselines. These results establish spectral epipolar regularization as an effective inductive bias for scalable and noise-robust light field depth estimation.

Abstract: We prove Convex Seed Universality for the Kreuzer—Skarke classification of four-dimensional reflexive polytopes. Every reflexive polytope in the Kreuzer—Skarke dataset arises from a primitive convex seed through a finite sequence of four toric operations: unimodular transformations, stellar subdivisions, polar duality, and lattice translations. Seed orbits coincide with connected components of the GKZ secondary fan, and the Hodge numbers of the associated Calabi—Yau hypersurfaces remain constant on each orbit. The seed invariant matrix is identified with the GLSM charge matrix, providing a natural toric-geometric interpretation of the construction. Four structural theorems: Seed Completeness, Orbit Connectivity, Hodge Invariance, and Exhaustiveness, together establish seed universality for the entire Kreuzer—Skarke dataset.

Abstract: We extend the master-integral-transform theory from entire kernels to finite-principal-part Laurent kernels and show that the resulting transform is a weighted dilation operator acting on the Fourier transform of a weighted signal. This yields a unified operator framework for several exact inversion mechanisms, including Mellin diagonalization, two-sided Mellin-symbol inversion, Dirichlet–Wiener inversion, log-scale Fourier inversion, recursive inversion, and Neumann-series recovery. The main structural result is that finite negative Laurent tails do not destroy the spectral architecture; they enlarge the one-sided dilation orbit to a two-sided one. We establish exact factorization formulas on weighted function spaces, prove branchwise Mellin inversion under explicit integrability assumptions, derive a contour-free Dirichlet–Wiener inverse, obtain a log-scale Fourier multiplier representation suitable for FFT-based recovery, and prove a practical stability bound away from multiplier zeros. A worked symbolic example and a numerical blueprint are also included.

Abstract: Certain integer transformations exhibit unexpected forms of stability that resemble attractors in dynamical systems. Two classical examples are the Kaprekar transformation leading to the constant 6174 and the arithmetic structure of perfect numbers. Although traditionally studied in separate areas of number theory, both phenomena reveal a common feature: the emergence of stable configurations under discrete informational constraints. In this work, we propose a unified framework based on Viscous Time Theory (VTT) and its informational geometry perspective, in which these two structures are interpreted as complementary forms of arithmetic stabilization. The Kaprekar transformation defines a discrete dynamical system whose iterations rapidly converge to a unique attractor (6174) for almost all four-digit inputs. Perfect numbers, on the other hand, arise as equilibrium points of the divisor-sum operator, where the informational deviation between a number and the sum of its proper divisors vanishes. We formalize both mechanisms using a common representation based on discrete informational tension functions defined over the integers. Within this framework, Kaprekar collapse appears as a dynamic attractor produced by iterative dissipation of digit-configuration tension, while perfect numbers correspond to static coherence wells generated by structural balance in the divisor field. Numerical exploration further suggests the presence of near-equilibrium zones—arithmetic configurations where informational gradients become locally minimal. These structures provide a natural bridge between iterative attractors and divisor-based equilibria, suggesting that stability phenomena in number theory may be understood through a broader lens of informational relaxation processes. The results do not claim new proofs regarding perfect numbers, but instead propose a conceptual and computational framework that unifies dynamic and structural stability in arithmetic systems. This perspective may provide new tools for exploring discrete attractors, divisor dynamics, and informational structures within number theory.

Abstract: We develop a unified operator- and matrix-valued strip-analytic extension of the Abu-Ghuwaleh transform program. The central object is a strongly measurable operator-valued orbit density whose boundary representation induces a continuous dilation-convolution operator acting on the Fourier transform of a weighted Hilbert-space-valued signal. In this setting the transform admits two complementary inversion mechanisms: Mellin contour inversion and contour-free Wiener--Mellin inversion on the logarithmic scale. We prove exact factorization formulas on named weighted signal spaces, derive branchwise Mellin diagonalization formulas with operator-valued system symbols, obtain inversion theorems under bounded invertibility assumptions, and formulate a log-scale Fourier multiplier representation suitable for FFT-based recovery. We then prove Young-type boundedness on the logarithmic side and stability estimates on frequency windows away from singularities of the multiplier. The finite-dimensional matrix case is obtained as a direct specialization of the Hilbert-space theory, and in that setting the Wiener inverse is derived from a standard matrix Wiener criterion. Finally, we isolate an explicit Gamma-type kernel family for which the system symbol is computable in closed form and yields concrete injectivity and stability constants. The paper is intended as the natural operator-theoretic successor to the scalar strip-analytic stage of the master-integral-transform program.

Abstract: We develop the strip-analytic sequel to the master-integral-transform program with entire kernels by replacing the discrete Taylor-spectrum model with a continuous spectral model on the dilation side. The central object is a Hardy-strip orbit kernel whose boundary representation induces a continuous dilation-convolution operator acting on the Fourier transform of a weighted signal. In this setting, the Abu-Ghuwaleh transform admits two complementary inversion mechanisms: Mellin contour inversion and contour-free Wiener--Mellin inversion. We prove exact factorization formulas on named weighted function spaces, derive branchwise Mellin diagonalization formulas, obtain inversion theorems under nonvanishing assumptions on the continuous symbol, and show that logarithmic coordinates convert the transform into an additive convolution equation. This yields a practical FFT-based inversion framework together with a stability bound on frequency windows away from zeros of the multiplier. We also prove an explicit injectivity-and-stability proposition for a resolvent-type kernel family with Gamma-type symbol. The paper is designed as the natural continuous-spectrum successor to the entire-kernel and finite-Laurent stages of the program.