Computer Science and Mathematics

Abstract: We construct Recursive Interval Geometry (RIG) as a strictly inductive algebra over \( R=\mathbb{Z}[\Omega^{-1}] \), generated by the binary structural projections \( \textrm{pos} \) and \( \textrm{len} \). On the resulting filtered configuration space \( \mathcal I \) we introduce an \( \Omega \)-weighted inner product and its induced norm \( \|\cdot\|_\Omega \), yielding an anisotropic pre-Hilbert geometry with an explicit orthogonal decomposition across recursive scales and an ultrametric-like, depth-dominant metric profile. We establish the associated Cauchy--Schwarz inequality and develop Gevrey-type structural estimates that quantify the decay of higher recursive layers. We further describe measure-theoretic and topological properties of the arithmetic substrate underlying the recursion. Finally, we define a canonical recursive tensor product driven by the principal component of the first argument, and prove its basic algebraic and metric properties, including depth additivity and strict \( \Omega \)-norm factorization. These results provide a compact, parameter-minimal mathematical foundation: beyond the discrete resolution parameter \( \Omega \), no additional free parameters are introduced, making the framework suitable for subsequent physical constructions.

Abstract: Let $\gamma_{1},\gamma_{2},\cdots,\gamma_{n}\in (0,1]$ be the orders and $\psi_{1},\psi_{2},\cdots,\psi_{n}$ be the time-dependent functions that are associated with incommensurate $\widehat{\psi}$-Caputo real-order systems. Is it possible to extend the Lyapunov stability theory for such advanced systems? In this paper, we state and prove new Lyapunov stability theorems that provide efficient, sufficient criteria for the stability of equilibria in such systems when one discovers adequate Lyapunov functions. We also address new conjectures that introduce new perspectives on stability tests with space-time partial derivatives of Lyapunov functions for such systems.

Abstract: We construct radial dual lattice graphs for the Eisenstein, Hurwitz, and E₈ lattices using admissible hyperspherical inversion. The inversion induces exact bijections between outer zone vertices and rational inner zone representatives, and it gives transported-edge graph isomorphisms once the radial-dual edge relation is defined. We verify the norm relation, involution, shell compression, and finite-shell adjacency identities using exact arithmetic. Composing the radial inversion with the Moxness E₈-to-H₄ folding matrix H₄fold gives a candidate golden linear-radial dual compressor Υᵣ for Cycle Clock Theory (CCT) workflows; on the E₈ root shell, the top 4×8 projection block Π of H₄_fold maps the 240 roots into two 120-point layers (the regular 600-cell H₄ and its golden-ratio scaled copy H₄Φ, with radius ratio Φ, equivalently squared-norm ratio Φ²). For larger shells, this compressor is validated on finite domains and proposed as a proof target for full cycle-clock enumeration. The framework aligns naturally with the classical divisor-sum expression c₈(n) = 240 σ₃(n) for E₈ shell multiplicities and makes it computationally useful for folded inner zone enumeration: our benchmark reports 83,000–260,000× speedups over exact-arithmetic Jacobi theta polynomial expansion at moderate shell indices. Realized via the Cayley integers in 8D, the construction aligns conceptually with octonion-based models in quasicrystalline quantum gravity while remaining strictly algebraic and geometric, operating on rational extensions without approximations, floating-point drift, continuous relaxations, or information loss. The construction offers a practical exact-arithmetic method for shelling and scaling calculations, while full global injectivity of Υᵣ on L⁸, 8D-to-4D graph-isomorphism preservation across arbitrary shells, and end-to-end CCT simulation integration remain proof obligations for future work. The entire construction is formulated inside the real Clifford algebra Cl(8) as the enveloping associative algebra: A₂, D₄, and E₈ are realized as the grade-1 root sub-systems of Cl(2), Cl(4), and Cl(8) respectively, and the classical Eisenstein, Hurwitz, and Cayley integer rings are recovered as the even subalgebras Cl⁺(d) acting on these generators. Under this enveloping algebra we adopt the uniform packing-radius root-length convention ⟨α,α⟩ = 2 (Euclidean length √2) across all three lattices, in agreement with the Bourbaki/Conway–Sloane/Viazovska normalization and with the maximally dense sphere-packing radius in each dimension; the canonical admissible inversion radius is therefore r = √2 uniformly, and the hyperspherical inversion ιᵣ is Clifford-equivariant under Pin(d) ⊂ Cl(d).

Abstract: In this paper, we first introduce the notion of a Nijenhuis operator on Leibniz triple systems, which can generate a trivial deformation. Then we use Nijenhuis operators to define product structures on a Leibniz triple system. There exists a product structure on a Leibniz triple system if and only if the Leibniz triple system is the direct sum of two subalgebras. There are some special product structures, each of which corresponds to a special decomposition of a Leibniz triple system. Parallelly, we study a complex structure on a Leibniz triple system. Finally, we add a compatibility condition between a product structure and a complex structure to introduce the notion of a complex product structure on a Leibniz triple system.

Abstract: This paper presents an operator-theoretic proof of the Riemann Hypothesis. The proof is organized so as not to identify the zeros of the completed zeta function with eigenvalues at the outset. Instead, three independent pieces of data are built: an analytic operator setting on a weighted Hilbert space, a coefficient-space arithmetic trace that evaluates the Euler-product prime-power contribution, and singular-boundary data constructed inside the analytic Hilbert-space framework. These data are then placed in a common Hilbert space $X = \mathcal{K}_R \oplus J_{\mathrm{arith}}\mathcal{H}_{\mathrm{arith}} \oplus \operatorname{Ran}\Pi_{\mathrm{res}}$, where the prime-power term is evaluated exactly on the arithmetic summand and the residual part is removed by passing to the canonical representative modulo $\operatorname{Ran}\Pi_{\mathrm{res}}$. The remaining effective $\mathcal{K}_R$-projected component is thus represented as the $\Pi_R$-projection onto the singular-boundary subspace $\mathcal{K}_R$. From this residual-free $\mathcal{K}_R$-component, a boundary-distribution comparison map is constructed. The functional equation for $\xi$ induces a boundary reflection $\Theta_R$, which descends to a bounded self-adjoint involution $\mathcal{S}_R$ on $\mathcal{K}_R$. The resulting signed boundary-distribution comparison kernel is realized, by Schatten-class smoothing estimates, as a self-adjoint Hilbert--Schmidt operator $K=K^* \in \mathfrak{S}_2$. This construction uses the functional equation, the boundary-distribution framework, and the orthogonal projection structure; it does not assume the location of the zeros of $\xi$, nor any positivity, Herglotz, or spectral localization statement equivalent to the Riemann Hypothesis. The operator $K$ defines the regularized Fredholm determinant $F_K(s) = e^{a_K+b_K(s-1/2)} \det_2(I+i(s-1/2)K)$, where the constants $a_K, b_K$ fix only the value and first logarithmic derivative at $s=1/2$. The comparison with the completed zeta function is carried out through a central Cauchy--Laplace regularization. The central comparison topology is fixed independently of the pairings $\mu_L$ and $\mu_\xi$. The finite-window counterterm is defined algebraically from central and endpoint finite jets before either pairing is evaluated, so the regularization does not encode the desired equality. Finite-window central cutoffs converge to the central kernel in this topology, and the two central pairings extend continuously to it. The finite-window residual-free equality therefore passes to the central limit and gives equality of the central logarithmic derivatives of $F_K$ and $\xi$. Together with the central normalization, this yields a local analytic equality, and the identity theorem gives $F_K(s) \equiv \xi(s)$ on the whole complex plane. Finally, since $K$ is self-adjoint, every zero of $F_K$ arises from a nonzero eigenvalue $\lambda_j$ of $K$ and is therefore of the form $s=1/2+i/\lambda_j$ for $\lambda_j \in \mathbb{R}\setminus\{0\}$. The global identity $F_K \equiv \xi$ therefore places every nontrivial zero of $\xi$, and hence of $\zeta$, on the critical line.

Abstract: The Gauss-Markov Theorem is central to linear statistical inference, assuring that Ordinary Least Squares (OLS) is the Best Linear Unbiased Estimator (BLUE) under the classical assumptions. But textbook proofs typically involve two assumptions - column full rank of the design matrix and spherical error covariance - that are often violated in practice. In this paper, we provide a single unified proof encompassing both rank deficiency and non-spherical errors using linear algebra: orthogonal projections, the Moore-Penrose pseudoinverse, and the Loewner positive semi-definite matrix order. We sequentially address both of these conditions, culminating in a unified theorem allowing for rank deficient design matrices and arbitrary positive definite error covariances. Our proof takes the form of three lemmas: (1) OLS fitted values are the unique orthogonal projection onto the column space of the design matrix; (2) any alternative linear unbiased estimator is dominated by OLS in the positive semi-definite order; and (3) Generalized Least Squares is BLUE under non-spherical errors by way of a Cholesky decomposition argument. To bridge matrix theory and practical biostatistics, we apply this framework to baseline clinical data from the WHELD dementia study, demonstrating how the pseudoinverse resolves exact multicollinearity and why robust estimators are necessary for non-spherical intra-class correlations. This approach serves as a formal reference for researchers and an illustrative pedagogic tool.

Abstract: We study finite orbit-sum termination for the two-sided exponential iteration generated by \( f(x)=2^x-1 \) and its inverse on the unit interval. For \( w\in(0,1] \), put \( u_k(w)=f^k(w) \) for \( k\in\mathbb Z \). A binary digit sequence \( a=(a_k)_{k\in\mathbb Z} \) with $a_0=1$ is normalized by \( \sum_{k\in\mathbb Z}a_k u_k(w)=1. \) Thus the expansion scale is generated by the point being expanded, rather than by an external base or partition, and finite termination is governed by finite orbit-sum equations. We prove existence and uniqueness of normalization roots for admissible digit sequences, construct the associated greedy code, and characterize finite termination by finite orbit-sum hitting equations compatible with the greedy order. The finite terminal set is countable and has Lebesgue measure zero. On the arithmetic side, the first positive and first negative boundary roots are transcendental, and the first positive second-order boundary root is irrational. Assuming Schanuel's conjecture, we exclude all non-trivial rational finite terminal points and prove transcendence for all purely positive finite roots, for all two-term boundary roots paired by the mirror identity, and for all roots with one first negative layer and arbitrary finite positive support.

Abstract: The mathematization of science is undergoing a structural transformation driven by the rise of computation and data-intensive methods. While classical mathematization relied on explicitly defined laws and formal structures, contemporary scientific practice increasingly encounters mathematical objects that arise as outcomes of dynamical and algorithmic processes. This paper introduces the notion of {\emph computationally emergent structures} to describe entities generated and stabilized through the interaction of parameterized models, optimization dynamics, and data. We develop a minimal formal framework in which such structures are characterized as asymptotic outcomes of learning dynamics and show that, in over parameterized regimes, they are selected by implicit variational principles not specified a priori. This framework provides a unified account of implicit regularization, kernel regimes, and stability phenomena in modern learning systems. These results show that contemporary learning systems operate according to implicit variational principles in which geometry, dynamics, and data jointly determine effective mathematical structure. They thereby identify a shift from representation to dynamical emergence, extending the scope of mathematization toward a theory of structure formation grounded in computation.

Abstract: Partial metrics have shown to be useful dissimilarity measures when incomplete information, partial states, or inherent uncertainty is involved. The main characteristic of this kind of distances is allowing non-zero self-distances. This distinctive property makes them particularly appropriate for applications to computer science, artificial intelligence, pattern recognition and bioinformatics. Nevertheless, in these fields it is often more relevant to quantify the amount of shared information between objects rather than their dissimilarity. In this context, similarity metrics have proven to be a valuable tool. The literature has suggested the existence of a duality relationship between partial metrics and similarity metrics. In this paper we investigate such a relationship. Specifically, we focus on identifying the properties of functions that induce a similarity metric from a partial metric in the sense of O’Neill. We provide a characterization of these functions, showing that they coincide with the class of strictly decreasing and convex functions on the set of non-negative real numbers. We also show that these functions preserve the topology and the partial order, that is, the partial order and topology generated by the induced similarity metric and by the original partial metric are the same. Besides, we characterize the class of functions capable of generating an O’Neill partial metric from a similarity metric showing that such a class is formed by strictly decreasing and concave functions on the set of real numbers. In this case we also show that the partial orders and the topologies generated by the induced partial metric and by the original similarity metric coincide. The results are supported and clarified by appropriate examples.

Abstract: The aim of this paper is to show that Cantor's absolute infinite Ω can consistently be proven to exist by introducing a consistent class theory NBG_Ω that is sufficiently strong. Gödel's first incompleteness theorem asserts that an axiomatic theory cannot be simultaneously (1) arithmetical, (2) axiomatizable, (3) consistent, (4) complete, and (5) countable. When axiomatizability is generalized to non-countable theories, a self-sufficient theory can be defined as possessing the first four of these five properties. An Ω-expanded theory is a countable theory that is extended with Ω-many omni-independent axioms, where Ω is axiomatized as a class cardinality so large that it cannot be proven to exist as a set cardinality in any Ω-consistent first-order set theory. ZFC_Ω, an Ω-expansion of ZFC, is subsequently shown to be self-sufficient. NBG_Ω, which extends ZFC_Ω to the class level, can then consistently prove the existence of Ω.

Abstract:

We establish new Lyapunov stability theory for ψ-Caputo fractional-order systems by strengthening Lyapunov functions under reasonable guiding wings of Class-K_∞ functions and their fractional derivative inequalities. The new generalized ψ-Gronwall inequalities and conceptual definitions of stability that are linked with the ψ-Mittag-Leffler function were introduced. Our main results are Lyapunov stability theorems whenever one finds a potential Lyapunov function that has upper and lower bounds and obeys typical Lyapunov fractional differential inequalities along imagined real trajectories of such systems. This theory works with some typical worked-out dynamic models, in which the stability dynamics are discussed.

Abstract: In this article we study semi-Markov decision processes (SMDPs) where the pay-off criterion is limiting ratio average, generally known as undiscounted pay-off. Here we consider the action space of the decision maker to be possibly countably infinite. However, we do not put any restriction on the reward function. We prove the existence of a near-optimal or ϵ-optimal strategy of the decision maker which turns out to be a deterministic semi-stationary. An efficient algorithm is discussed to compute a near-optimal pure semi-stationary strategy for such SMDP model. Also under some standard ergodicity conditions, we propose an optimality equation of these SMDP models.

Abstract: Suppose \( A=I-PQ^T, \) where both \( P \)and \( Q \) are \( n\times 3 \) complex matrices and det\( (Q^T P)\neq 0 \). By analyzing the Jordan canonical form of the coefficient matrix \( A, \) we find all the anti-commuting solutions \( X, \) which satisfy \( AX=-XA, \) of the Yang-Baxter-like matrix equation(YBME) \( AXA=XAX \).

Abstract: We report the use of our group’s hierarchical Bayesian implementation of the Multi-dimensional Nominal Categories Model followed by standard factor rotations of the principal dimensions to obtain 29 curated sparse dimensions from a set of 203,564 (104,998 pre and 98,566 post) administrations of a multiple-choice concept test in mechanics. We emphasize our careful attention to issues common to fitting such multi-parameter models to large data sets: a novel set of filters to remove administrations from non-conscientious testees, use of Bayesian methods to avoid overfitting, selecting the best transformations to find easily identifiable sparse dimensions, and verification and pruning of these using bootstrap samples. We demonstrate that most dimensions are invariant across different demographically different samples of students as well as between pre-instruction vs post-instruction samples. Most sparse dimensions correspond to well-known misconceptions in mechanics.

Abstract: For a Hutchinson iterated function system (IFS), a Banach contraction on a complete metric space, or a finite-metric dynamical system, a natural question is: at which resolution σ does the contraction's geometric structure (fractal attractor, basin of attraction, periodic part) become optimally visible? We answer this by introducing a scale-selection principle: define the observation scale σ_c := argmax_σ χ(σ), where χ(σ) = |dO(σ)/d log σ| is the susceptibility of a resolution-dependent observable, and prove that σ_c exists under explicit boundary-regularity hypotheses.The framework's main quantitative results are three theorems specialised to Banach contractions and IFS: (i) a geometric scaling identity σ_c = qL for affine Banach contractions with operator norm q and basin scale L, applying directly to Hutchinson IFS with σ_c ∼ q · diam(K⋆); (ii) a discrete Banach theorem on finite metric structures under uniform Lipschitz Lip_d(f) = q < 1, giving an exact collapse-time N⋆ = ⌈log(∆/d_min) / log(1/q)⌉; (iii) a spectral concentration theorem placing σ_c at the inverse log-spectral-gap of the transfer operator at fixed positive noise. A stability lemma for canonical normalisation under smooth windowing and a parametric Banach correspondence observation complete the technical core.The framework is stated explicitly in the non-expansive Lipschitz regime Lip_d(f) ≤ 1 on the metric side, and at fixed positive noise ε ∈ (0, 1) on the spectral side. A four-type classification of operations by injectivity structure organises the broader landscape; cross-domain empirical evidence anchored on a peer-reviewed NISQ-hardware measurement of σ_c is summarised. The middle-thirds Cantor set IFS appears as the principal worked example.

Abstract: The article analyzes the linear-quadratic optimal stabilization problem in the so-called "critical case", namely, the situation is considered when the spectrum of the system matrix contains purely imaginary eigenvalues or when the standard conditions of positive definiteness of the weight matrices of the quality functional are violated. Methods for regularizing critical problems by perturbing the system matrices and the functional are investigated, and algorithms for decomposing multidimensional problems into a set of one-dimensional canonical systems are proposed. The results are of practical importance for constructing optimal synthesis in various engineering and economic systems, in particular, the results can be used for stabilizing unmanned aerial vehicles, robotic complexes and intelligent power grids.

Abstract: Binary Waring decomposition seeks to express a homogeneous binary form as a minimal sum of powers of linear forms. In the binary setting, Sylvester’s theorem gives a classical algebraic route for rank determination and parameter recovery through structured Hankel/catalecticant matrices. Although this procedure is exact and interpretable in ideal arithmetic, practical rank identification may become unstable when the input coefficients are contaminated by noise or when the underlying roots are close to degenerate configurations. This paper develops a data-driven rank inference framework coupled with certified Sylvester reconstruction for robust binary Waring decomposition. The proposed method first converts the coefficient sequence into a Hankel-aware graph that captures recurrence-induced dependencies among polynomial coefficients. A graph neural network is then used to infer plausible rank candidates from this structured representation. Instead of accepting a single prediction directly, the framework performs explicit Sylvester reconstruction and algebraic residual verification for candidate ranks. To further improve decision reliability, a lightweight meta-verification module integrates reconstruction residuals, model confidence scores, and stability-related indicators to select the most credible rank. Experiments on large-scale synthetic binary forms demonstrate that the proposed approach improves rank identification accuracy and verified reconstruction success under low-to-moderate noise, while maintaining the transparency and auditability of classical symbolic–numeric computation. These results suggest that data-driven rank inference can serve as an effective front-end for algebraically certified reconstruction, especially in numerically ambiguous regimes where fixed threshold-based Sylvester implementations are fragile.

Abstract: For a fixed integer \( n\ge2 \), let \( \Delta_{n-1} \) denote the standard probability simplex in $\( \mathbb R^n \)$. We introduce and analyze the self-power escort transformation \( T_n:\Delta_{n-1}\to\Delta_{n-1} \) defined by \( \begin{equation*}T_n(p)_i=\frac{p_i^{p_i}}{\sum_{j=1}^n p_j^{p_j}},\qquad i=1,\ldots,n,\end{equation*} \) with the continuous boundary convention \( 0^0=1 \). Unlike the usual power escort transformation, whose exponent is an external parameter, the exponent here is the coordinate itself; equivalently, each coordinate is reweighted by \( \exp(p_i\log p_i) \). The paper proves that this elementary self-feedback rule has a rigid global dynamics. Every boundary point is immediately activated, every orbit enters an explicitly described compact core after one step, the uniform distribution is the unique fixed point, and the Hilbert log-diameter contracts exponentially after core entrance. Consequently every orbit converges exponentially to the uniform distribution, and no nontrivial periodic orbit exists. We then compute the complete linear spectrum and second-order normal form at the uniform state, obtaining a sign transition between the binary and higher-dimensional systems. For \( n\ge4 \), the post-core Hilbert contraction constant is sharpened to a two-variable min--max problem involving only the smallest and largest coordinates. We also classify the entire one-step uniformization fiber \( T_n^{-1}(\nu_n) \): besides the vertices and the uniform point, each dimension \( n\ge3 \) has exactly one nonuniform interior orbit type, up to permutation. Finally, in the binary case we prove global contraction in logit coordinates and an arithmetic escape phenomenon: algebraic irrational inputs become transcendental after one step, while rational nonuniform inputs become transcendental after two steps.

Abstract: We present a rigorous and comprehensive development of the Vertex Shift Method(VSM), a derivative-based polynomial normalization technique extending Loh’s quadraticmidpoint insight [1] to arbitrary-degree polynomials and time-dependent dynamicalsystems. Given a polynomial P(x) of degree n ≥2, VSM identifies a critical point φsatisfying P′(φ) = 0 and applies the shift x= y+ φ, producing a normalized form Q(y)with no linear term.We prove: (i) the linear-term elimination theorem—the shifted polynomial has zerolinear coefficient by construction; (ii) the spectral translation theorem µi = λi−φ, all rootsshift rigidly by exactly φ; (iii) companion-matrix similarity via S(φ) = exp(φD); (iv) theCurved Shift Theorem for time-varying polynomials via the Implicit Function Theorem;and (v) explicit conditioning reduction bounds. A central contribution is the identificationof VSM as a structure-aware preconditioning operator in coefficient space:eliminating a1 reduces coefficient imbalance—the principal driver of ill-conditioning incompanion matrices.Numerical experiments on eight polynomial families demonstrate conditioning improve-ments of 30×to 2.3 ×105×, 15–57% QR iteration reductions, and 3–20×root-accuracyimprovements. All results are fully reproducible via the companion Jupyter notebook.

Abstract: In this paper, we are concerned with the existence of crossing periodic solutions for a class of second order discontinuous undamped Duffing equations. By applying the Poincaré-Bohl theorem, we obtain several existence results of 2π crossing periodic solutions.