Computer Science and Mathematics

Abstract: This paper aims to formulate well-defined Rota-Baxter operators on Hopf π-algebras. To this end, we construct convolution algebras over π-graded structures, and further investigate the properties of antipodes and homomorphisms of Hopf π-algebras. Finally, we present the definition of Rota-Baxter Hopf π-algebras and construct valid examples.

Abstract: We prove that finitary strongly taut paracompact locally small spaces are mild (all their subspaces are strict). We apply this result to show that all paracompact locally definable spaces over o-minimal structures are mild.

Abstract: This paper investigates the precise upper bound of the third-order Hankel determinant for the inverse functions within the function classes $\mathcal S^{*}_{e}$, $\mathcal C_{e}$, $\mathcal S^{*}_{\mathcal L}$, and $\mathcal C_{\mathcal L}$. Additionally, it establishes the upper bound for the second-order Hankel determinant of the logarithmic coefficient in function class $\mathcal S^{*}_{\mathcal L}$, as well as the third-order Hankel determinant for function class $\mathcal C_{car}$.

Abstract: By means of the weight functions, the idea of introduced parameters and the well known reverse Hardy’s integral inequality, an extended reverse Hardy-type inequality is obtained and then a new reverse half-discrete Hilbert-type inequality with the general homogeneous kernel, as well as multiple lower limit function and remainder sum is given. The equivalent statements of the best value related to several parameters are considered. As applications, the equivalent forms and some particular examples are provided.

Abstract: We study an Eulerian pantograph equation with proportional delay by means of a logarithmic change of variables, which transforms the problem into a constant-shift equation on the half-line. The main object is the maximal left-transport realization on L^p(0,∞), with derivative domain W^1,p(0,∞) and no boundary condition at the endpoint. For the characteristic function m(μ)=vμ+B+Cexp(Lμ), we prove that the spectrum is the closure of the image of the open left half-plane under m: the open image gives the point spectrum, the boundary image gives the approximate spectrum, and the closure gives the full spectrum. The associated semigroup has a sharp norm identity with exponent ReB+|C|, yielding the exact stability threshold ReB+|C|<0. We compare this half-line result with the full-line Fourier multiplier realization and with the Mellin-root description of pantograph modes. Hermite functions are used only as a computational representation tool for histories and translations; they do not replace the half-line spectral theorem. The same logarithmic translation structure also suggests a natural connection with Fourier Neural Operator parameterizations of finite-history input-output maps, although the spectral results proved here are independent of any neural approximation scheme.

Abstract: We study a coefficient-space normalization for univariate polynomials: given P(x) = aₙxⁿ + ... + a₁x + a₀, shift to Q(y) = P(y+φ) at a critical point φ satisfying P′(φ) = 0. This eliminates the linear coefficient of Q and, for the companion matrix construction, typically reduces the condition number κ substantially. The method generalizes a classical quadratic identity (the vertex of a parabola sits at −b/2a) to arbitrary degree, and we call it the Vertex Shift Method (VSM). We establish that the shifted companion matrix is similar to the spectrally translated original companion matrix, a direct-conditioning selection rule for choosing among the n−1 available critical points, a dynamic extension (Curved Shift) for time-varying polynomial families together with its breakdown condition and a hybrid tracking algorithm, and a lower bound exposing a structural limitation of similarity-based conditioning transformations in general, including but not limited to diagonal balancing. Re-audited benchmarks on four polynomial families give conditioning gains from 1× (correctly, no gain, on an already-balanced Chebyshev polynomial) to 3519× (Wilkinson W₁₀); five additional legacy instances are reported separately because their exact original data were not preserved, with one of them (Wilkinson W₁₅) independently re-audited under the current method. We report failure modes and the regimes in which balancing outperforms VSM as openly as the regimes in which it does not, and give two validated applications: a singularity-avoiding companion representation for singular Markov generator polynomials, and Kerr black hole geodesic conditioning, whose coefficient vector is reconstructed exactly from explicit physical parameters using the primary-source radial-potential formulas of Compère and Druart and Teo.

Abstract: In this paper, we investigate the growth behavior of the m−th derivatives (m≥1) of arbitrary algebraic polynomials in bounded and unbounded domains whose boundaries are piecewise quasismooth curve and possess interior and exterior zero angles of power type, within weighted Lebesgue spaces. We derive estimates for the growth of these derivatives in terms of the behavior of the weight function, the geometric properties of the boundary, and the tangency exponents of the boundary arcs at their junction points. The obtained estimates are established both in the closure of the bounded domain and at exterior points, while the pointwise estimates in the unbounded domain additionally reflect the location of the point within the domain.

Abstract: This paper proves the Riemann Hypothesis by constructing, rather than postulating, a self-adjoint Hilbert--Schmidt determinant model for the completed zeta function. The proof is organized around a single principle: the zeros of the completed zeta function are not used as spectral data. Instead, the classical explicit formula is first converted into a finite-window comparison problem in an orthogonal Hilbert-space decomposition. Sections 2--5 build the three ingredients needed for this conversion: the analytic compact-resolvent framework, the coefficient-space arithmetic trace for the prime-power contribution, and the singular-boundary component. These are placed in the ambient decomposition \(X=\mathcal K_R\oplus J_{\mathrm{arith}}\mathcal H_{\mathrm{arith}}\oplus\operatorname{Ran}\Pi_{\mathrm{res}}\). Passing to the canonical comparison representative modulo \(\operatorname{Ran}\Pi_{\mathrm{res}}\) leaves an effective \(\mathcal K_R\)-component, while the arithmetic summand accounts for the Euler-product term. The finite-part structure used in this comparison is separated by type. The contour-coordinate ledger and the seam/Gram/LCI transport are fixed from the finite-window contour convention, the logarithmic representative, and finite readout coordinates. The scalar probe used for the classical comparison is fixed separately by the geometric Guinand--Weil contour normal form, which uses the classical boundary normal form of the completed zeta logarithmic derivative, namely the gamma-factor, Euler-product, and functional-equation normal forms, but takes no zero divisor as input; it is not chosen from prescribed zero-side values, from the determinant side, from a \(\mathcal K_R\)-side comparison equality, or from RH. The operator-side functional is defined from the \(\mathcal K_R\)-projection of the canonical comparison representative, whereas the classical explicit-formula ledger is introduced separately and identified with the completed zeta logarithmic derivative only at the final target-identification stage. Section 6 closes the proof. The centered Mellin seam involution \(w\mapsto -w\) descends to a self-adjoint involution on \(\mathcal K_R\). Its signed boundary-distribution kernel is realized, by a Sobolev-reference Schatten-four sandwich estimate, as a self-adjoint Hilbert--Schmidt operator \(K=K^*\in\mathfrak S_2\). This gives the intrinsic determinant factor \(F_K^0(s)=\det\nolimits_2(I+i(s-\tfrac12)K)\) and the comparison function \(F_K(s)=e^{a_{\mathrm{EF}}+b_{\mathrm{EF}}(s-\frac12)}F_K^0(s)\), where \(a_{\mathrm{EF}}\) and \(b_{\mathrm{EF}}\) are central constants of the explicit-formula ledger; they are not supplied from \(\xi\) at the construction stage. The finite-window comparison quotient is then passed to the central Cauchy--Laplace family. On the \(K\)-side, the finite-part realized functional is identified with the determinant trace through finite-window scalar coefficients, cyclic tensor contractions, finite-rank compression, and the Hilbert--Schmidt limit. On the classical side, the explicit-formula ledger is a bookkeeping total, so that after the Archimedean and arithmetic contributions are removed its zero-side residual is identified, by a finite-window Guinand--Weil residue theorem, with the central logarithmic derivative of the completed zeta function; the zeros of \(\xi\) enter only as the residue side of Cauchy's theorem. These two independently obtained transform identities give \(\frac{d}{dw}\log F_K(\frac12+w)=\frac{d}{dw}\log \xi(\frac12+w)\) near \(w=0\). The central scalar target identification gives local analytic equality, and the identity theorem yields \(F_K(s)\equiv\xi(s)\). Since \(K\) is self-adjoint, every zero of \(F_K\) coming from a nonzero eigenvalue \(\lambda_j\in\mathbb R\setminus\{0\}\) has the form \(s=\frac12+\frac{i}{\lambda_j}\). The identity \(F_K=\xi\) therefore places every nontrivial zero of \(\xi\), and hence of \(\zeta\), on the critical line.

Abstract: This paper investigates the existence of pseudosolutions for a class of fractional retarded dynamic equations on time scales in Banach spaces endowed with the weak topology. The proposed model combines fractional dynamics, explicit delay effects, and hybrid continuous–discrete temporal structures within a unified analytical framework. The analysis is performed by means of the ∆-Henstock–Kurzweil–Pettis integral, allowing significantly weaker regularity assumptions than those required by classical integration theories. The existence result is established using the De Blasi measure of weak noncompactness together with Kubiaczyk’s fixed point theorem for weakly sequentially continuous operators. The obtained theorem extends several existing results on fractional differential equations and dynamic equations on time scales by incorporating explicit delays and generalized integration into a common framework. The obtained results provide a unified analytical framework for studying hereditary systems evolving on hybrid time domains.

Abstract: This paper investigates the constrained 2-maxian problem defined on block graphs, a special class of composite graphs containing trees and cacti as subclasses. The problem requires placing two facilities on a block graph such that the Euclidean/topological distance between the two facilities is bounded by a given upper limit, while maximizing the total weighted maximum distance from all client vertices to the nearest facility. We first analyze the structural properties of block graphs and the vertex optimality of the constrained 2-maxian solution, proving that at least one facility in any optimal solution must be located at a vertex of the block graph. Based on this key property, we design an efficient polynomial-time algorithm for the constrained 2-maxian problem on block graphs. We further discuss the differences and connections between constrained and unconstrained 2-maxian problems, and compare the proposed method with existing algorithms for trees, cycles and cactus graphs. Numerical analysis verifies the correctness and efficiency of the algorithm.

Abstract: We develop a theta-kernel framework for a positivity problem associated with the Riemann $\Xi$-function, working with \[ D(z)=\int_0^\infty \Phi(u)\cos(zu)\,du=\frac12\Xi(z). \] After the change of variables \[ a=\frac{u+v}{2},\qquad b=\frac{u-v}{2}, \] the growth derivative of \(|D(x+iy)|^2\) is transformed into a phase-aligned paired-tail problem for the two-variable kernel \[ M(a,b)=\Phi(a+b)\Phi(a-b). \] We establish an exact residual identity after cancellation of all longitudinal boundary modes and derive a uniform transverse integration-by-parts hierarchy implying asymptotic suppression of the higher transverse remainder. A narrow curvature-transition layer is identified via a transition functional \(F_\lambda\), and the associated Riccati transition curve is shown to possess a smooth geometry with certified negative signed curvature. Post-crest block positivity is proved analytically for sufficiently large frequencies and certified numerically on a compact domain. The envelope function \[ Q_y(a)=2\int_0^a M(a,b)\cosh(2yb)\,db \] is shown numerically to possess a unique crest for every \(y>0\). Finally, we formulate a conditional bridge from complete block positivity to positivity of the full theta-kernel integral and hence to the growth criterion for the Riemann hypothesis. Several key questions remain open, including analytic crest uniqueness and pre-crest block positivity. Thus the paper provides a rigorous analytic and geometric framework surrounding theta-kernel positivity for the Riemann $\Xi$-function without claiming a proof of the Riemann hypothesis.

Abstract: One of the defining challenges of our time is the management of urban traffic. As cities grow denser and more complex, the limitations of traditional fixed-time signal systems become increasingly difficult to ignore. This paper explores how Reinforcement Learning (RL) can offer a smarter and more adaptive alternative for traffic signal control (TSC), showing its potential to optimize traffic flow and reduce congestion. To investigate this potential, we relied on computer simulations as a training and evaluation environment. The results show that RL-based controllers have good adaptive abilities and work better than traditional traffic signal strategies. However, there are still critical challenges, such as scalability to large networks, non-stationarity of the environment, and enforcement of safety constraints that are key barriers to real-world deployment. These findings highlight the significant promise of RL for future traffic management, but also stress the need for further research on robustness, reliability and generalizability of these systems.

Abstract: The axiom of maximal cardinality (AMC) asserts that there exists a class cardinal Ω that is greater than any set cardinal in any theory T that can be consistently extended to a class theory proving the existence of Ω. Three theorems are then proven to show that (1) AMC is not vacuously true, (2) AMC is not self-referentially inconsistent, and (3) Ω is a maximal cardinality. The maximality of this Ω implies it is Cantor's absolute infinite. While Gödel's incompleteness theorems presuppose the existence of a theory that interprets the notion 'countably recursive theory', AMC's theory interprets the notion 'theory'.

Abstract: This study addresses the protracted global crisis in mathematical education and investigates fundamentally new, high-technology pathways to overcome it, moving systematically from empirical diagnosis to technological renaissance. The first part of the work provides a rigorous cross-national analysis of cognitive decline and the degradation of mathematical pedagogy in traditionally dominant scientific nations, specifically the United States, France and Russia. Measures to stimulate youth engagement with mathematics through immersion into the historical context of mathematical concepts and operations, combined with the author’s empirical experiments in testing students using historical-mathematical problems, are described. The second half of the study (Parts II–IV) elevates the discourse to the contemporary technological landscape of 2026, demonstrating how Reasoning AI—an advanced class of artificial intelligence—can be strategically deployed as an autonomous digital mentor. We illustrate how these advanced models can automate the longitudinal diagnostic screening of underlying cognitive processes, pinpoint structural misconceptions, and act as a “digital Socrates” that forces students back to rigorous logical principles rather than providing ready-made answers. Furthermore, the paper shows how Reasoning AI decimates routine programming barriers, enabling students to channel their mathematical intuition directly into the empirical exploration of non-linear dynamic systems and fractal topologies. Finally, the study warns against the anthropocentric risks of “AI sycophancy” and the mechanical patterns of standardized testing, arguing for a radical paradigm shift that restores mathematics as a living language for investigating the real world.

Abstract: The Yang-Baxter-like matrix equation has wide applications in various fields and holds significant theoretical research value. Suppose the coefficient matrix is an n by n diagonalizable complex matrix which has three different nonzero eigenvalues. We get some distinctive properties of solutions of the Yang-Baxter-like matrix equation.

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 ϒ_r for Cycle Clock Theory (CCT) workflows; on the E₈ root shell, the top 4×8 projection block Π of H₄fold 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. We show that the inversion exchanging the two folded layers is the geometric-mean inversion at radius r² = 4/Φ, that one global rescaling brings it to the canonical admissible radius r = √2, and that it is the self-aligned member of a geometric-mean inversion family whose other two rungs swap the G₂ and F₄ shell pairs. 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 ϒ_r 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 ι_r is Clifford-equivariant under Pin(d) ⊂ Cl(d).

Abstract: We study support-determined constraints for eigenvectors of nonnegative symmetric tensors whose support may contain repeated indices. Such tensors are naturally encoded by uniform multi-hypergraphs, where each multiedge is represented by an exponent vector \( \alpha\in\mathbb N_0^n\ \) with \( |\alpha|=k\ \). Replacing the ordinary vertex-edge incidence matrix by the exponent-incidence matrix, we show that every nonzero-eigenvalue H-eigenvector satisfies linear incidence constraints in the transformed coordinates \( y_i=x_i^k\ \). These constraints are invariant under positive scalar edge weights and reduce to the usual support-incidence constraints for ordinary squarefree hypergraphs. We also describe the positive branch of the resulting constraint variety and prove a positive-weight realization criterion: a positive vector \( x\ \) can be realized as an eigenvector of some positive edge-weighting of a fixed multi-hypergraph if and only if \( x^{[k]}\ \) lies in the positive cone generated by the exponent-incidence columns. Thus the exponent-incidence constraint variety gives a linear algebraic relaxation, while the positive exponent-incidence cone gives the exact positive-weight realization region.

Abstract: The stability of equilibrium of general nonautonomous fractional-order systems remains a long-standing challenging problem; going beyond fractional derivatives of adequate Lyapunov functions seems crucial. We put forward new conjectures by using Lyapunov functions and introduce the fractional Lyapunov linearization method to both autonomous and non-autonomous fractional-order systems. We prove some new Lyapunov theorems that develop sufficient conditions for local asymptotic stability in nonlinear systems. In light of conjectures, we establish two new stability theorems by means of Krasovskii’s method, which enables the construction of a Lyapunov function for autonomous fractional order systems. As applications, we consider three examples and demonstrate our results to examine their stability.

Abstract: Ordered pairs of normalized real numbers form a closed arithmetic system once a normalization function is fixed. Their algebraic and matrix-theoretic extensions are governed by a spectral two-channel structure. The normalization function is not specialized: it is only assumed to be continuous, strictly increasing from \( \mathbb{R} \) onto \( (0,1) \), and symmetric in the sense that \( \varphi(x)+\varphi(-x)=1. \) For an OPNs element \( \alpha=(\mu_{\alpha},\nu_{\alpha}) \), we define the spectral-coordinate map \( S_{\varphi}(\alpha)=(p_{\alpha},q_{\alpha}) \), where \( p_{\alpha} = -\varphi^{-1}(\mu_{\alpha}) - \varphi^{-1}(\nu_{\alpha}), \qquad q_{\alpha} = \varphi^{-1}(\mu_{\alpha}) - \varphi^{-1}(\nu_{\alpha}) \). The scalar theory is reorganized as a two-channel real algebra with two primitive idempotents \( e_p \) and \( e_q \). Intrinsically, this algebra is a reduced semisimple Artinian real algebra with exactly two primitive central idempotents. The main result of this paper is an idempotent Smith theory for polynomial matrices over OPNs. Although the polynomial coefficient ring \( \mathcal{O}_{\varphi}[t] \) has zero divisors, every OPNs polynomial matrix admits a Smith normal form whose invariant factors have the two-channel form \( d_i(t)=f_i(t)e_p+g_i(t)e_q, \qquad f_i(t),g_i(t)\in\mathbb{R}[t]. \) The divisibility chain, determinantal ideals, finite-presentation classification over \( \mathcal{O}_{\varphi}[t] \), and uniqueness of these invariant factors are controlled channel by channel, while the resulting invariant factors remain intrinsic OPNs objects. As the constant-matrix shadow of this Smith theory, every rectangular OPNs matrix is equivalent, under invertible OPNs row and column transformations, to a rectangular diagonal matrix whose only possible diagonal entries are \( 1_{\mathcal{O}}, \qquad e_p, \qquad e_q, \qquad 0_{\mathcal{O}}. \) The resulting matrix theory gives canonical descriptions of kernels, images, cokernels, linear systems, determinant theory, the Cayley--Hamilton theorem, minimal polynomial theory, rational canonical classification, regular spectral rectangles, similarity classification, and symmetric matrix inertia.It is also shown that the four-pivot geometry is irreducible: the rank pair controls the stable block-sum monoid, the factorization preorder, orbit dimensions, orbit closures, and the associated determinantal ideal chain. In the polynomial theory, projective rank pairs yield explicit nonfree projective modules over \( \mathcal{O}_{\varphi}[t] \).

Abstract: We derive a Riesz-Frechet representation for bounded linear functionals defined on the padic Hilbert spaces introduced by Kalisch [Ann. of Math. (2), 1947]. We also notice the surprising difference between the Archimedean case and the non-Archimedean case (exact non-Archimedean version of Riesz-Frechet representation fails).