Computer Science and Mathematics

Abstract: We separate four levels associated with the Riemann Hypothesis and strengthen the selected \(\Xi\)-certificate layer by an explicit theta-kernel atlas theorem. The resulting framework is a theory of RH-specific analytic and number-theoretic reasoning as exact selected sign-fibre reasoning. The theta kernel, theta-atlas certificates, winding certificates, Li-negative certificates, Lagarias-negative certificates, zero-count certificates, and half-plane-cover certificates are coordinate-bearing RH-specific channels. We make precise their absorption into one enriched selected RH criterion package. The number theory does not escape selected-logical nullity; it supplies the exact sign fibres to which nullity applies.First, at the raw formula level, the Dirichlet evaluator interprets \(\zeta(s)\) only by the Dirichlet series on \(\Rea(s)>1\). Under strict atomic strong Kleene semantics, the usual strip formula \[ \RHzero: \quad \forall s\bigl((0<\Rea(s)<1\wedge\zeta(s)=0)\Rightarrow\Rea(s)=1/2\bigr) \] has value \(\Gap\). Moreover, arbitrary strip-total completions realize arbitrary zero sets in the critical strip. Therefore no bivalent RH sign is invariant across such completions. Analytic continuation is treated as the selector forming the classical proposition \(\RH\).Second, after analytic selection, we work with \[ \Xi(z)=\xi(1/2+\ii z), \qquad \mathcal U=\{x+\ii y:x>0,\ 0<y<1/2\}, \] so that \[ \RH\iff Z(\Xi;\mathcal U)=\varnothing. \] The main analytic/certification contribution in the present framework is the conversion of the classical theta-kernel representation \[ \Xi(z)=4\int_0^\infty \Phi_+(u)\cos(zu)\,\dd u \] into a finite rational atlas-certificate system with explicit majorants and quantitative cell-count bounds. Here \[ \Phi_+(u) = \sum_{n=1}^{\infty} \left( 2\pi^2 n^4 e^{9u/2} - 3\pi n^2 e^{5u/2} \right)e^{-\pi n^2e^{2u}}, \] and we derive explicit majorants \[ \sup_{|z|\le R}|\Xi^{(j)}(z)| \le \mathfrak M_j(R), \] where \[ \mathfrak M_j(R) = 4\left( 2\pi^2G_4I_j(R+9/2) + 3\pi G_2I_j(R+5/2) \right), \] \[ G_m=\sum_{n=1}^{\infty}n^m e^{-\pi(n^2-1)}, \qquad I_j(\alpha)=\int_0^\infty u^j e^{\alpha u}e^{-\pi e^{2u}}\,\dd u. \] For rational radii \(R\), and more generally for any supplied rational upper radius \(R^+\ge R_0\), all ingredients admit finite rational upper certificates. If the analytic radius is \(R_0\), then the certified estimate is \[ \sup_{|z|\le R_0}|\Xi^{(j)}(z)| \le \mathfrak M_j(R^+). \]For \[ K_N= \left[\frac1N,N\right] \times \left[\frac1N,\frac12-\frac1N\right]\subset\mathcal U \qquad(N\ge5), \] we define a finite decidable rational certificate predicate \[ \ThetaAtlas(N,c) \] certifying zero-freeness of \(\Xi\) on \(K_N\). It is sound and existentially complete on zero-free windows; from supplied rational lower-bound data it is effectively constructible. It is quantitative: if \[ \delta_N:=\min_{z\in K_N}|\Xi(z)|>0 \] and \[ M_N\ge \sup_{|z|\le N+2}|\Xi'(z)| \] is certified, then an accepted atlas exists with explicitly bounded cell count \[ \left( 1+ \left\lceil \frac{2\sqrt2\,M_N^+}{\Delta_N} \left(N-\frac1N\right) \right\rceil \right) \left( 1+ \left\lceil \frac{2\sqrt2\,M_N^+}{\Delta_N} \left(\frac12-\frac2N\right) \right\rceil \right), \] where \[ M_N^+=\max(1,M_N), \qquad \Delta_N=\min(1,\delta_N). \] In finite rational certificate records, \(\sqrt2\) is replaced by any certified rational upper bound \(q_2>0\) satisfying \(q_2^2>2\). Thus \[ \RH\iff \forall N\ge5\,\exists c\,\ThetaAtlas(N,c). \] This is the coordinate-bearing positive \(\Xi\)-channel used later by the selected package.Third, exact winding certificates provide the finite negative channel \[ \neg\RH\iff\exists w\,\BadXi(w), \] and the theta-atlas channel provides the all-stage positive channel \[ \RH\iff\forall N\ge5\,\exists c\,\ThetaAtlas(N,c). \] The quantitative boundary-mesh bound for winding certificates includes an explicit certified tube-radius term, because a finite boundary mesh must be fine enough not only for variation control of \(\Xi\), but also for the enclosing domain disks to remain inside the certified compact guard. We also compare the \(\Xi\)-negative channel with classical RH criteria: Li-negative certificates and Lagarias-negative certificates are exact finite negative channels for \(\neg\RH\), and the corresponding witness fibres are pairwise computably replay-equivalent.Fourth, these number-theoretic criteria are absorbed into one enriched selected RH criterion package. The finite negative fibres include: \[ \exists w\,\BXi(w) \iff \exists v\,\BLi(v) \iff \exists v\,\BLag(v) \iff \neg\RH. \] The all-stage positive fibres include: \[ \forall n\,\exists c\,\CTheta(n,c) \iff \forall n\,\exists c\,\CZero(n,c) \iff \forall n\,\exists c\,\CHP(n,c) \iff \RH. \] Thus Li, Lagarias, theta-atlas, winding, zero-count, and half-plane-cover criteria are not external routes around selected-logical nullity. Once made exact, they become sign fibres of the same enriched selected RH package. A finite negative certificate is literal occupation of the negative fibre and is therefore exactly \(\neg\RH\). A complete positive certificate stream is literal occupation of the positive fibre and is therefore exactly \(\RH\). Before such occupation is supplied, the mere availability, exactness, and replay equivalence of the criteria remain selected-logically null.Fifth, arithmetizing the finite negative verifier gives a literal \(\Pi_1\) sentence \[ \RHarith\equiv \forall w\,\forall u\,\neg\BadXiCheckArith(w,u) \] satisfying \[ \mathbb N\models\RHarith\iff\RH. \] The arithmetic formula is obtained from complete low-level Turing/register-machine accepting computation histories; no high-level halting assertion is hidden in the matrix. The final version fixes a concrete bounded tableau-access convention, for instance a Gödel-\(\beta\)-style bounded access relation, and every subroutine not among the fixed bounded access primitives is displayed by its own finite subtrace. Thus the matrix is bounded after definitional trace elimination.For every \(T\supseteq\mathsf Q\), \[ \neg\RH\Rightarrow T\vdash\neg\RHarith, \] so \[ T\nvdash\neg\RHarith\Rightarrow\RH. \] If \(T\) is \(\Sigma_1\)-sound, then \[ T\vdash\neg\RHarith\iff\neg\RH. \]The final layer is universal selected-logical nullity, together with its enriched criterion version. At the level of arbitrary resolvers, the semantic nullity theorem has the following unconditional metatheoretic form: for every consistent theory \(T\supseteq\mathsf Q\), \[ \SoundT(\Resolver)\wedge\InvSel(\Resolver) \Rightarrow \Resolver\equiv\varnothing. \] Here \(\InvSel\) is full replacement invariance: the resolver is unchanged under arbitrary replacement of one exact selected package by another. This condition is not asserted for arbitrary coordinate-bearing RH-specific resolvers. Rather, it is the semantic target of the pure selected-logical regime. The consistency hypothesis on \(T\) is essential for proof-status labels: if \(T\) were inconsistent, then \(\mathbf{Prov}_T\) and \(\mathbf{Ref}_T\) would be common sound labels for every package under the label semantics used in this paper.The meta theory then discharges the invariance premise for pure selected-parametric resolvers. In the stipulated pure resolver language the package variable is opaque: one may use only selected bridge forms that are valid for all exact selected packages, and one may not inspect concrete fibres, package codes, truth or proof-status oracles, or coordinate-bearing analytic data. The selected parametricity theorem proves, by induction on the generation of pure resolver terms, \[ \PureSel(\Resolver)\Rightarrow\InvSel(\Resolver). \] Consequently, for every consistent \(T\supseteq\mathsf Q\), \[ \SoundT(\Resolver)\wedge\PureSel(\Resolver) \Rightarrow \Resolver\equiv\varnothing. \] Thus the invariance premise is a semantic hypothesis for arbitrary resolvers, but it is a theorem for the intended pure selected-logical regime.This gives a nullity-matching structure: \[ \BivRes_{\EDir}(\RHzero)=\varnothing, \] \[ \CompInvRes_{\mathscr C_{\Strip}}(\RHzero)=\varnothing, \] and, for every consistent \(T\supseteq\mathsf Q\), \[ \bigcap_{\mathcal P\in\Pkg}\Act_T(\mathcal P)=\varnothing, \] \[ \bigcap_{\mathcal P^\ast\in\PkgStar}\Act_T(\mathcal P^\ast)=\varnothing. \] The first is raw Dirichlet-clause bivalence nullity; the second is completion-invariant bivalence nullity; the third is ordinary exact-package selected-label nullity; and the fourth is enriched exact-package selected-label nullity. The canonical resolver induced by the full invariant selected-logical closure \(\Gamselmax\) is invariant by construction, so for every consistent \(T\supseteq\mathsf Q\), \[ \LogRes_T^{\Unif}(\Gamselmax;\RHsharp)=\varnothing. \] For the enriched selected RH criterion package \(\PRHstar\), \[ \SoundT(\Resolver^\ast)\wedge\PureSel^\ast(\Resolver^\ast) \Rightarrow \Resolver^\ast(\PRHstar)=\varnothing \] for every consistent \(T\supseteq\mathsf Q\).Consequently, the theta-atlas theorem and the universal nullity theorem are complementary. The theta, winding, Li, Lagarias, zero-count, and half-plane-cover criteria supply coordinate-bearing RH-specific fibres. The universal nullity theorem says that the unoccupied enriched fibre architecture, when viewed in the pure replacement-invariant selected-logical regime, carries no invariant truth/proof-status label. Exact number-theoretic criteria do not bypass nullity; they become its fibres. Thus the paper is a theory of RH-specific analytic reasoning with an effect discipline: non-null RH-specific conclusions occur by fibre occupation or by leaving the pure invariant regime.

Abstract: For starlike, convex and bounded turning functions linked with nephroid function, the sharp upper bounds of the third-order Hankel determinant, the third-order Hankel determinant of inverse functions and the second-order Hankel determinant of logarithmic coefficients are computed.

Abstract: By using recently established refinements of Jessen and converse Jessen-type inequalities, together with improved characterizations of strongly convex functions, we derive new Jensen–Mercer type functional inequalities. The obtained results extend and sharpen several known inequalities for convex functions. As applications, we establish new integral inequalities which improve some recently published estimates. In addition, we obtain inequalities involving generalized logarithmic means. Finally, we derive corresponding probabilistic versions of the main results, which lead to refinements of several known results from the literature.

Abstract: We study positive–negative guarded systems of language equations over a fixed finite alphabet. The ambient space is the complete ultrametric space of all formal languages equipped with a length-based distance, where two languages are close whenever they agree on all words up to a sufficiently large length. The systems considered here contain both positive recursive dependencies and negative dependencies expressed through language complements. To handle this mixed structure, we introduce a suitable product order on pairs of languages and prove that the associated system operator has the weak monotone property. We show that complement is an isometry for the length-based ultrametric and establish a signed wrapping estimate for guarded positive and negative language terms. These estimates lead to an ordered contraction principle for comparable pairs. As a consequence, the canonical lower and upper Picard iterations converge to the same limit, which is the unique fixed pair of the system. We also derive an explicit convergence rate and a finite-depth certification result: after a prescribed number of iterations, the approximants agree with the fixed-point semantics on all words below a given length. Additional symmetry assumptions are shown to force the unique fixed pair to be diagonal, reducing the system to a single language equation. Finally, we discuss an application to trace-based policies for tool-using AI agents. In this interpretation, finite executions of an agent are represented as words over an alphabet of observable tool-events, and the two components of the fixed point provide a stable semantics for policy-defined admissible and risky trace classes. The resulting framework gives a mathematically certified method for finite-depth analysis of recursive trace-based policies based on ultrametric fixed-point techniques.

Abstract: We give a constructive high-dimensional escape sequence for the equation (∆ − x ·∇)u = u associated with the symmetric Ornstein–Uhlenbeck operator in Gaussian space. Let (a_i)_i≥1 be a positive square summable sequence and let B_n = {x ∈ Rⁿ : ∑ⁿ_i=1 a_i² x_i² < 1}. We construct functions u_n that are continuous on Rⁿ, smooth on both sides of ∂B_n, solve the positive spectral equation away from ∂B_n, and have finite Gaussian H¹ energy. The construction uses a single real harmonic polynomial, Re(x1 +ix2)^m_n with m_n = ⌊n^1/8⌋, multiplied by the finite-energy Tricomi branch of the separated radial Ornstein–Uhlenbeck equation and then extended into B_n by the weighted Dirichlet principle. The exterior energy has a lower bound of order (2π)^n/2n^−1/2(2m_n/e)^m_n, whereas the interior minimizing energy is bounded by (2π)^n/2n^CC_a^m_n. Hence the ratio of total Gaussian H¹ energy to the energy inside B_n tends to infinity. The proof is written with all non-standard notation defined explicitly, and two examples, including an ℓ¹-small sequence with ∑_i a_i < 1, are included as checks of the hypotheses.

Abstract: In this article, a class of rough generalized Marcinkiewicz operators is considered. Under the condition that the singular kernel belongs to the space Lq(Bs−1), the boundedness of these operators are confirmed from the space of homogeneous Triebel–Lizorkin functions to the Lp(Rd+1) space. Moreover, appropriate Lp bounds are obtained which allow us to utilize Yano’s extrapolation procedure to prove the boundedness of the aforementioned operators under weaker assumptions on the kernels. In this work, several known past results are generalized, extended, and improved.

Abstract: This paper introduces the radius of integrability, a quantitative invariant that transforms the qualitative ϵ-δ formulation of Riemann integration into a measurable property of function spaces. For a Riemann integrable function and a prescribed accuracy ϵ, the radius identifies the largest partition mesh δ that guarantees every tagged Riemann sum approximates the integral within the specified error. The framework is developed for both compact domain intervals, via pointwise and uniform radii, and unbounded intervals, through the tail integrability radius which quantifies the necessary truncation window for improper integrals. Key theoretical results include the establishment of a bottleneck identity relating local and global mesh requirements and a structural theorem showing that for C1 integrands, the radius is asymptotically governed by the inverse of the function’s total variation. Furthermore, this work completes a hierarchical program of regularity radii—encompassing convergence, continuity, and differentiability—by revealing a dimensional progression of geometric anchors. We demonstrate that while continuity is anchored at a point and differentiability at a line, integrability is anchored at a two-dimensional region. The theory is illustrated through explicit computations for several classical functions, including the normal density, the stretched exponential, and the Thomae function, providing a new quantitative lens for classifying integrable functions based on their partition sensitivity and tail decay regimes.

Abstract: This paper introduces a novel class of convex functions associated with a strip domain and establishes the upper bounds for the coefficients of initial terms, as well as second and third-order Hankel determinants. It provides exact upper bounds for the third-order Hankel determinants of both the inverse of starlike functions and convex functions, along with the upper bounds for the second-order Hankel determinants of the logarithmic coefficients related to these functions.

Abstract: The Google Quantum AI whitepaper [1] established that the secp256k1 elliptic curve discrete logarithm problem (ECDLP-256) the signature primitive securing the majority of public blockchain networks can be broken with fewer than 500,000 physical qubits in approximately nine minutes on a fast-clock cryptographically relevant quantum computer (CRQC), a roughly 20-fold reduction from prior estimates. That analysis addresses cryptocurrency nodes and does not model the structurally distinct risk profile of IoT/WSN nodes participating in blockchain networks. This paper closes that gap. We demonstrate that gateway-mediated transaction submission, combined with the duty-cycle wakeup latency endemic to IEEE 802.15.4-class sensor nodes, can extend the on-spend attack window beyond Bitcoin’s ten-minute block time making constrained WSN nodes more vulnerable to on-spend attacks than standard cryptocurrency clients for certain deployment configurations. We introduce a formal HNDL Risk Score (HRS) that jointly quantifies harvest-now/decrypt-later exposure, on-spend vulnerability, and migration feasibility as a function of node hardware class and deployment lifespan. We benchmark all four NIST-standardised post-quantum cryptography algorithms (ML-KEM-512, ML-DSA-44, FN-DSA/FALCON-512, SLH-DSA-128s) on representative WSN hardware (Raspberry Pi Pico RP2040 and ESP32) and identify FN-DSA/FALCON-512 as the superior candidate for WSN-blockchain transaction authentication, offering 666-byte signatures at 71.6 ms mean signing time on Cortex-M0+ hardware. A concrete, tiered migration roadmap is proposed for healthcare WSN, industrial IIoT, and smart city sensor deployments.

Abstract: This research explores the existence of solutions for a class of random fractional differential equations characterized by bounded delay, specifically within the context of Fréchet spaces. By integrating the properties of noncompactness measures with a generalized Darbo fixed point approach, we establish existence results for the associated Darboux problem. To illustrate the practical utility of these analytical results, a representative example is provided.

Abstract: This paper investigates weighted composition-differentiation operators acting between Bers- type spaces defined on generalized Hua domains of the first kind. By establishing a key norm inequality for functions in these spaces, we derive necessary and sufficient conditions for the boundedness and compactness of such operators.

Abstract: This paper introduces a quantitative refinement of the classical concept of differentiability within the space of real functions. Shifting the focus from the qualitative existence of the derivative to a scale-sensitive framework, we define two new invariants of the radii of differentiability: the radius of pointwise differentiability and the radius of uniform differentiability. These radii quantify the maximal horizontal scale over which the first-order Taylor approximation remains valid for a prescribed error tolerance ε. The theoretical development establishes a robust set of structural properties, including scaling laws, monotonicity, and behavior under function composition and sums. We provide a rigorous characterization of these invariants, demonstrating that the property of having an infinite radius of differentiability is uniquely characteristic of affine functions. A significant portion of the study is dedicated to the “bottleneck identity,” which reconciles local and global regularity by expressing the uniform radius as the infimum of its pointwise counterparts. Furthermore, we explore the interplay between these differentiability radii and the radius of continuity, utilizing the Fundamental Theorem of Calculus to prove that the process of integration yields a strict improvement in the local regularity profile. Finally, the utility of the proposed framework is demonstrated through explicit computations for several classes of elementary functions, including polynomials and trigonometric maps.

Abstract: Our result is in line with the Beurling and Alcantara-Bode equivalent formulations of the Riemann Hypothesis (RH). Also, it intends as a numerical method to supply the lack of the methods in literature for investigation of the injectivity of linear bounded operators on separable Hilbert spaces. The criteria exploit the operator approximation positivity properties on finite dimension subspaces having their union a dense set covering a wide range of linear, bounded operators. For operators that are not positive definite, taking their associated Hermitian, it consists of: a Hermitian Hilbert-Schmidt operator whose family of finite rank approximations built on a dense set having the positivity parameters inferior bounded, has a null space containing only 0, i.e. containing no not null elements. We obtained the injectivity for the Alcantara-Bode integral operator connected to Riemann Zeta function, that is in fact the equivalent formulation of the RH.

Abstract: The selection of locations for logistics distribution centers poses a significant challenge in logistics network planning. Traditional methods often demonstrate limited accuracy in solutions and a tendency to become trapped in local optima when addressing large-scale, multi-constraint location models. To address these shortcomings, this study introduces a firefly algorithm enhanced by genetic mutation strategies (GVFA) to optimize the location of distribution centers. Within the framework of the standard firefly algorithm, we incorporate an adaptive step-size decay mechanism and a mutation operator. The movement step size adjusts dynamically based on iteration counts, while a mutation probability of 5\% is implemented to maintain population diversity, effectively reducing the risk of premature convergence. A specialized boundary-handling strategy ensures that the search process remains within the feasible solution space, guiding the population toward the global optimum. Experiments were conducted using latitude-longitude coordinates and logistics demand data from 159 Cainiao Post stations in Hengyang City, resulting in the construction of a location model aimed at minimizing total costs. The findings confirm the efficiency and stability of our method in optimizing distribution center locations, thereby providing a novel intelligent optimization approach for the siting of logistics distribution centers.

Abstract: In 1829, Cauchy derived an upper bound for every root of a complex polynomial using the maximum of the absolute values of the coefficients. In 1931, Montel derived an upper bound using the sum of the absolute values of the coefficients. We derive noncommutative versions of the Cauchy and Montel bounds.

Abstract: Let $\mathbb{K}$ be a non-Archimedean valued field. Let \begin{align*} p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in \mathbb{K}[z], \quad a_n \neq 0. \end{align*} If $\lambda \in \mathbb{K}$ satisfies $p(\lambda)=0$, then we show that \begin{align*} |\lambda|\leq \min \left\{1, \frac{1}{|a_n|^\frac{1}{n}}\left(\max_{0\leq j \leq n-1}|a_j|\right)^\frac{1}{n}\right \} \end{align*} or \begin{align*} 1\leq |\lambda|\leq \frac{1}{|a_n|}\max_{0\leq j \leq n-1}|a_j|. \end{align*} This is the non-Archimedean version of the Cauchy upper bound for every root of a complex polynomial derived by Cauchy in 1829. Our bound is different from the non-Archimedean bound obtained by Nica and Sprague [Am. Math. Mon., 2023].

Abstract: We study a class of wrapping operators acting on the space of formal languages over a fixed finite alphabet. The underlying space is equipped with a length-based ultrametric, in which two languages are close whenever they coincide on all sufficiently short words. We prove that every wrapping operator generated by a finite family of guards with positive total guard length is a contraction. As a consequence, Banach’s contraction principle yields existence and uniqueness of a fixed point for the corresponding recursive language equation, together with convergence of the Picard iteration from an arbitrary initial language. We also obtain an explicit quantitative estimate for the rate of convergence. This makes it possible to determine how many iterations are sufficient to recover the fixed point correctly on all words up to a prescribed length. Several examples illustrate the theory, including operators with different guard lengths and a case showing that convergence in the length-based ultrametric does not coincide with set-theoretic convergence. An application to recursive structures and document validation is also presented, including recursive data formats, abstract syntax trees, and a restricted fragment of JSON schemas. The results provide a formal foundation for validation together with explicit bounds for correctness on inputs of bounded length.

Abstract: The standard ε–δ definition of continuity is inherently quantitative, yet the precise dependence of the admissible radius δ on the accuracy ε and the base point x₀ is rarely treated as an independent mathematical object. In this paper, we introduce the radius of continuity through two variants: the radius of pointwise continuity and the radius of uniform continuity, defined as explicit numerical invariants that capture the maximal symmetric neighborhood on which a real-valued function maintains a prescribed tolerance. We establish the fundamental structural properties of these radii, including their behavior under algebraic operations such as sums, products, and compositions, and demonstrate their inverse relationship to the classical modulus of continuity. Furthermore, we prove that the finiteness pattern of these radii characterizes constant versus non-constant functions. To illustrate the utility of this framework, we derive closed-form expressions for the pointwise radius of quadratic polynomials and the uniform radius of the normal probability density function. These examples highlight how the radius of continuity encodes geometric and probabilistic features, such as local curvature and global scale parameters. Ultimately, this perspective bridges the gap between real analysis and quantitative methods in metric geometry, offering a concrete measure of the stability of a function's continuity.

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

Abstract: Massera and Schaffer [Ann. Math. (2), 1958] established a breakthrough upper bound for the Clarkson angle between two nonzero vectors in a normed linear space. Maligranda [Am. Math. Mon., 2006] improved Massera-Schaffer upper bound for the Clarkson angle. Pecaric and Rajic [Math. Inequal. Appl., 2007] extended Maligranda's inequality to finitely many nonzero vectors. We derive a finite field version of Massera-Schaffer-Maligranda-Pecaric-Rajic inequality.