Computer Science and Mathematics

Abstract: Standard numerical invariants can agree even when the integral geometry assembling singular strata differs. We make this hidden structure computable for double-octic Calabi–Yau threefolds. Motivated by Hodge atoms, we associate to an admissible eight-plane arrangement A a local Hodge atom inventory \(HA_{\mathrm{loc}}(A)\), recording singular-stratum multiplicities, and a source assembly complex \(HA_R^{(2)}(A)=[R\langle\mathcal L(A)\rangle\xrightarrow{d_A}R\langle\mathcal P(A)\rangle]\), recording how double lines meet selected multiple points. The notation HA marks their HodgeCY role; neither object is yet a mixed-Hodge-theoretic Hodge atom spectrum. Exact computations show that 239,240,241 share one local inventory but determine two rational and three integral assembly types. In particular, 239 and 240 are isomorphic over QQ but not over ZZ, proving that rational assembly does not determine its integral refinement. More strikingly, 84 and 84a have identical local inventories and Hodge data \((h^{1,2},h^{1,1},e)=(0,40,80)\), yet distinct integral and automorphism-resolved assembly types. Thus source assembly detects coefficient- and symmetry-sensitive structure invisible to local counts and Hodge numbers. For explicit quartic perturbations of 84 and 84a, we also certify zero-dimensional saturated Jacobian schemes of degree 112.

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Abstract: We develop a symmetric-polynomial framework for the study of one-parameter families of Poncelet triangles. For triangles inscribed in the unit circle and circumscribed about a central conic, we show that the elementary symmetric polynomials of the vertices depend linearly on a single complex parameter. It follows that every symmetric rational function of the vertices is a rational function of this parameter, providing a unified algebraic approach to the investigation of geometric invariants. As applications, we recover several known invariance results and establish new ones involving angles, orthic triangles, tangential triangles, polar circles, and other classical constructions. We also obtain explicit formulas and loci for several associated geometric objects, leading to new invariance phenomena within Poncelet families.

Abstract: Africa remains the most under-addressed continent on the planet, with an estimated 60–70% of streets and dwellings lacking formal, standardised addresses. This deficit has profound consequences for governance, emergency response, e-commerce, logistics, and the daily lives of over 1.4 billion people. We present a hybrid graph-theoretic and machine-learning framework for automated address generation trained on high-quality open data from three European cities (Stuttgart, Paris, and Bern) and progressively adapted to seven African cities (Kigali, Dakar, and Kampala as training cities; Dar es Salaam, Nairobi, Kinshasa, and Lagos as unseen test cities). The framework models street networks as primal and dual graphs, applies the Intersection Continuity Negotiation algorithm for stroke detection, and deploys four supervised classifiers for connectivity error detection, road-type classification, building-to-street assignment, and address quality scoring. Building footprints are fused from three complementary sources (OpenStreetMap, Google Open Buildings, Microsoft Building Footprints) and augmented with SRTM elevation data and the World Settlement Footprint 3D (WSF 3D) dataset. Across ten study cities, the pipeline produces 3.2 million merged building footprints, 467,000 street edges, and 880,464 quality-scored addresses, and a five-version iterative improvement cycle reduces the EU–Africa address quality gap by 54.2%. To validate African address outputs independently, we construct a cross-validation database of 1,000 verifiable addresses of known organisations across the seven African cities; cross-validation achieves a geometric match rate of 79.6% within 100 m and 83.5% within 150 m. The system is implemented as an open-source QGIS plugin and ArcGIS Toolbox.

Abstract: Many results in quadrilateral geometry are traditionally stated for convex quadrilaterals. In this paper, we show that several of these results remain valid for self-intersecting quadrilaterals. In particular, we prove that, for any vertex, the maltitudes (midpoint altitudes) corresponding to two adjacent sides and the associated diagonal are concurrent; we call this point the malticenter of the vertex. This result holds uniformly for both simple and self-intersecting quadrilaterals. We further relate malticenters to classical centers (centroid, circumcenter, and orthocenter) via homotheties. Finally, we prove a conjecture of [1], previously observed for convex quadrilaterals, showing that \[ \operatorname{Area}(ABCD)=\operatorname{Area}(H_AH_BH_CH_D) \] holds for all quadrilaterals, where \(H_A,H_B,H_C,H_D\) are the orthocenters of the triangles formed by the vertices of $ABCD$.

Abstract: Using generalized Chapple--Euler relation, we prove that a triangle of unit circumradius may be circumscribed about a central conic with foci \( a_1,a_2\in\mathbb C \) if and only if the major axis of the conic has length \( |1-\overline{a_1}a_2| \). This characterization applies uniformly to both types of central conics.Our main result shows that Marden's theorem yields a parametrization of every such Poncelet family. The elementary symmetric polynomials of the vertices of a circumscribed triangle admit explicit formulas in terms of the foci \( a_1,a_2 \) and a unimodular parameter \( \lambda \in \mathbb C \). As a consequence, the classical degree-3 Blaschke-product equation arises directly from Marden's theorem and remains valid without requiring the foci to lie inside the unit disk.The parametrization provides a unified framework for deriving geometric invariants of families of circumscribed triangles. Using Marden's theorem, we provide a short proof of a classical theorem which states that the circumcircle of a triangle formed by three tangents to a parabola passes through the focus. Finally, we establish a higher-degree symmetric parametrization for partial fractions with finite numbers of poles, leading to generalized Möbius-product equations. The associated residues satisfy a partition-of-unity identity extending the classical Marden weights, thereby placing Marden's theorem within a broader residue-theoretic framework and suggesting possible higher-degree analogues in Poncelet geometry.

Abstract: We prove that the total area of the power circles associated with an odd p-gon inscribed in and circumscribed about a pair of homothetic ellipses remains invariant throughout the Poncelet family. Our proof is based on a simple affine averaging principle, which also yields several related quadratic invariants, including explicit formulas for the sums of the squared distances from the center to the vertices, to the side midpoints, as well as for the sum of the squared side lengths. As an application, we show that a convex Poncelet pentagon and the corresponding star Poncelet pentagon, both circumscribed about the same inellipse, have equal total power-circle area. These results unify several metric invariants of odd Poncelet polygons within a common affine-geometric framework.

Abstract: We study the adjunction-defect calculus underlying MacPherson--Vilonen gluing. For an open--closed decomposition \(X=U\sqcup Z\), recollement gives adjoint triples \(j_!\dashv j^*\dashv j_*\) and \(i^*\dashv i_*\dashv i^!\). We package this data as a MacPherson--Vilonen adjunction package and compute the twist--cotwist defects of its four constituent adjunctions. The nontrivial open-adjunction defects are \(T_{D^b_c(X)}(j_!\dashv j^*)\simeq i_*i^*(-)[-1]\) and \(T_{D^b_c(X)}(j^*\dashv j_*)\simeq i_*i^!(-)[1]\). By contrast, full-faithfulness in genuine recollement forces several open and closed defects to vanish, so imposing sphericality on all four raw recollement adjunctions is degenerate. Thus compatible sphericalization is formulated only conditionally, requiring shared-object data and transportability of compatibility morphisms. The main unconditional result identifies the boundary residual \(\operatorname{BRes}_{Z}(M):=\operatorname{Cone}(j_!j^*M\to j_*j^*M)\) as an extension \(i_*i^*M\to \operatorname{BRes}_{Z}(M)\to i_*i^!M[1]\to\). Iterating over closed-stratum filtrations yields residual towers, with filtered--graded, duality, motivic, and schober-theoretic outlooks.

Abstract: We develop a sharp local-to-global transition calculus for monotone profiles whose centered holomorphic extension maps a horizontal strip into the unit disk. Schwarz-Pick contraction gives the pointwise bounds |G′| ≤ [π/(4a)](1 − G²) and 0 ≤ F′ ≤ [π/(2a)]F(1 − F), where G = 2F − 1. Integration yields optimal two-point, anchored-envelope, and threshold-duration inequalities. Equality at one real point, equality for one distinct pair, or contact with an anchored envelope at one later point forces the complete logistic profile. A speed-defect identity measures accumulated transition delay, while quantile-spacing, variance, and area bounds show that every admissible transition is at least as dispersed as the logistic extremizer. We separate the disk-contractivity hypothesis from spectral growth: compact spectral support supplies an entire function of exponential type, while strip-to-disk contractivity is the geometric condition that produces the nonlinear speed limit. The resulting framework is a portable analytic theorem; a domain realization supplies its physical clock and dynamics.

Abstract: Quantum mechanics elevates non-commutativity from a mathematical curiosity to a physical axiom: observables are promoted to operators, and Poisson brackets are replaced by commutators. This ``quantization procedure'' has remained a postulate for a century, with no explanation of \textit{why} nature demands non-commuting observables. We argue that the answer has been hiding in plain sight since 1843: the quaternion algebra \(\HH\), whose defining relation \(\mathbf{i}\mathbf{j} = -\mathbf{j}\mathbf{i} = \mathbf{k}\) makes it the unique non-commutative normed division algebra over \(\R\).We develop this argument in three tiers of increasing rigor. First, we prove that the 44-vector \(\mathbb{Z}_3\)-triality vacuum lattice---a zero-parameter discrete structure established in prior work---is \textit{exactly} the orbit closure of the quaternion basis under \(\mathbb{Z}_3 \subset \SU(2)\). The three closure operations are unified as conjugations by a single \(q_T \in \SU(2)\) (Proposition~3.1), and a group-theoretic proof via the \(A_2\) root lattice decomposition of \(\R^3 = V_0 \oplus V_1\) yields $|L_{44}| = 44$ (Theorem~3.1). Second, we derive the canonical commutation relation $[\hat{x},\hat{p}] = i\hbar$ from quaternion algebra through the constructive chain \(\HH \to \HH_\C \cong M(2,\C) \to \mathcal{F}_B(\C^2) \to\) Heisenberg algebra, with \(\hbar = \sqrt{3}\,S_0\) where \(\sqrt{3}\) is the geometric quantum number of the \(\mathbb{Z}_3\) rotation angle and \(S_0\) the fundamental action scale of the lattice. Third, we construct the full Hilbert space framework via the GNS representation of \(\HH\) as a C$^*$-algebra (Appendix~A).The Pauli algebra, Dirac algebra, and spin-statistics connection admit progressively weaker claims: the Pauli algebra is identically the quaternion algebra (theorem); the Dirac algebra inherits quaternion structure through \(\mathrm{Cl}(3,0) \cong \HH \oplus \HH\) (structural); the spin-statistics connection is a geometric interpretation suggested by the framework. What was discovered as a mathematical structure in 1843, applied as a computational technique in 1925, and axiomatized as a postulate ever since, is revealed as the geometric foundation of quantum non-commutativity: \textit{quaternion multiplication is the mathematical origin of why observables fail to commute.

Abstract: This article investigates generalized trans-Sasakian manifolds and their rich geometric structure. We present a method for constructing two additional almost contact metric structures from a given generalized trans-Sasakian manifold. A complete classification of these structures is obtained through the study of their integrability and normality properties. We further examine these structures when they are realized as real hypersurfaces of complex space forms. Several existence and non-existence results are established.

Abstract: We prove a smooth algebraic transfer formula for intersection Hirzebruch classes of pure-dimensional complex algebraic varieties. If \(\pi\colon X\to Y\) is a smooth algebraic morphism of pure fiber dimension between pure-dimensional complex algebraic varieties, then \(\Ty(T_{X/Y})\cap \pi_{\BM}^*\IT_{y*}(Y)=\IT_{y*}(X)\), and specializing at \(y=1\) gives the \(L\)-class transfer formula \(L^*(T_{X/Y})\cap \pi_{\BM}^*\IT_{1*}(Y)=\IT_{1*}(X)\). The formula completes a basic pair of functorial operations for intersection Hirzebruch classes: it is the smooth-morphism counterpart to Banagl's algebraic Gysin formula for \(\IT_{1*}\) under upwardly normally nonsingular embeddings, and it establishes the smooth algebraic transfer formula anticipated in Banagl's work on \(L\)-homology bundle transfer for Witt spaces. Both base and total space may be singular; only the morphism is required to be smooth, so the formula computes the intersection Hirzebruch class of a smooth family directly from the class of its base together with the relative tangent correction. Beyond the class-level statement, we isolate the precise structural input governing its object-level refinement: we show that whether the canonical comparison \(\pi^*\IC_Y^H[r]\simeq\IC_X^H\) holds as an isomorphism of mixed Hodge modules reduces to a single compatibility, namely that shifted smooth pullback commutes with the \(j_!/j_*\) boundary package of the intersection Hodge module, equivalently with the associated Beilinson--MacPherson--Vilonen gluing data. This identifies exactly what must be supplied to upgrade Banagl's class-level identity in \(K_0(\MHM)\) to an object-level isomorphism. We also establish the smooth pullback formula for the full motivic Hirzebruch transformation \(\MHT_{y*}\), prove compatibility with composition of smooth morphisms, recover the vector bundle and zero-section cases, and combine the smooth theorem with Banagl's algebraic Gysin theorem to obtain a restricted lci-type composition principle. The results are illustrated through examples, including projective bundles over Schubert varieties.

Abstract: Let X be a complex algebraic variety with a two-stratum decomposition X = U ⊔ S, where j : U ,→ X is open and i : S ,→ X is closed. MacPherson and Vilonen associate this configuration with a zig-zag category Z(X, S) and a functor μ : P(X) → Z(X, S) from the category of perverse sheaves on X, P(X). The construction packages a perverse sheaf Q^∗ by its open restriction j^∗ Q^∗, the two closed terms H^−d(i^! Q^∗) and H^−d(i^∗ Q^∗), and the exact sequence H^−d−1(i^∗ Rj∗ j∗ Q∗) → H^−d(i^! Q^∗) → H^−d(i∗ Q∗) →H^−d(i^∗ Rj^∗ j^∗ Q^∗). In this note we construct the corresponding zig-zag category Z_Nori(X, S) and zig-zag functor μ_Nori : PNM(X) → Z_Nori(X, S) in the setting of perverse Nori motives. The construction uses the same object, morphism, and exact-sequence data, with the terms interpreted in the perverse Nori motivic formalism. We prove that realization sends Z_Nori(X, S) to the classical zig-zag category and that the square relating μ_Nori to the classical MacPherson–Vilonen functor μ commutes up to natural isomorphism. The purpose of the paper is to make the zig-zag construction available as a reusable tool in perverse Nori motivic categories.

Abstract: We study the finite integral information retained by rational Hodge-cycle constructions. For a smooth projective complex variety \(X\), the standard integral Hodge defect group in codimension \(p\) is \(Z^{2p}(X)=H^{2p}(X,\mathbb Z(p))_{\mathrm{Hdg}}/ \operatorname{im}(\operatorname{cl}_{X,\mathbb Z})\). If \(L_{\mathrm{Hdg}}\subseteq H^{2p}(X,\mathbb Z(p))_{\mathrm{Hdg}}\) is a finitely generated typed Hodge sector, its actual algebraic part is \(L_{\mathrm{alg}}^{\mathrm{act}}=L_{\mathrm{Hdg}}\cap \operatorname{im}(\operatorname{cl}_{X,\mathbb Z})\). A specified rational Hodge-cycle construction \(\mathcal C\) supplies, when sound, a subgroup \(L_{\mathrm{alg}}^{\mathcal C}\subseteq L_{\mathrm{alg}}^{\mathrm{act}}\). Thus one obtains an actual sector defect \(D_{\mathrm{act}}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/ L_{\mathrm{alg}}^{\mathrm{act}}\) and a constructional defect \(D_{\mathcal C}(L_{\mathrm{Hdg}})=L_{\mathrm{Hdg}}/ L_{\mathrm{alg}}^{\mathcal C}\). We prove that rational coverage, \(L_{\mathrm{alg}}^{\mathcal C}\otimes\mathbb Q= L_{\mathrm{Hdg}}\otimes\mathbb Q\), is equivalent to the finiteness of \(D_{\mathcal C}(L_{\mathrm{Hdg}})\). Under this hypothesis the rational residual defect vanishes and the remaining defect is the finite saturation quotient. If the proof supplies an integer \(N\) with \(N L_{\mathrm{Hdg}}\subseteq L_{\mathrm{alg}}^{\mathcal C}\), then \(N\) annihilates the constructional defect. We compare constructional annihilators with external annihilators of actual sector defects. In particular, when a diagonal-decomposition or torsion-order argument supplies an annihilator \(\operatorname{Tor}(X)\) for the actual sector defect, and \(\mathcal C\) supplies a constructional annihilator \(N\), we obtain \(\exp\operatorname{im}(D_{\mathcal C}(L_{\mathrm{Hdg}})\to D_{\mathrm{act}}(L_{\mathrm{Hdg}}))\mid \gcd(N,\operatorname{Tor}(X))\). This controls the actual obstruction detected by the construction, but not the whole constructional defect. We exhibit this distinction on \(X=\mathbb P^1\times\mathbb P^1\): the degree-two map \(f=g\times\operatorname{id}\), with \(g([x:y])=[x^2:y^2]\), has \(f^*(\mathbb ZH_1\oplus\mathbb ZH_2)=\mathbb Z(2H_1)\oplus\mathbb ZH_2\), giving \(D_{\mathcal C}\cong\mathbb Z/2\mathbb Z\) while \(D_{\mathrm{act}}=0\). Thus constructional defects can be nonzero even when no actual integral Hodge obstruction remains. The resulting defect profiles fall into three basic types: multiplier-supported, saturation-supported, and realization-loss. Finite trace and norm maps, group averaging, rational algebraic projectors, correspondence images, isogenies, Prym and Weil-type comparisons, specialization terms, regulator kernels, and Bockstein finite-coefficient sources are treated as instances of this trichotomy. The paper does not prove new cases of the rational Hodge conjecture; it gives a sectorwise formalism for extracting the finite integral content of rational algebraicity arguments.

Abstract: This paper introduces almost Moscow topological groups, a novel class of topological algebraic structures that unifies the almost continuity with the G_δ-regularity. The study establishes that every almost Moscow space is δ-extremally disconnected, with strict converse failure. The framework yields sharp characterizations such as an almost Moscow topological group is extremally disconnected iff locally countably S-closed and km-perfect, further a locally rc-Lindelöf mildly Hausdorff induces δ-extremally disconnected. Additionally, semi-regularizations of locally countably rc-paracompact groups are ω-extremally disconnected. Consequently, Hausdorff semi-regular first countable locally countably rc-paracompact members are discrete. In the extremally disconnected setting, every such group contains an open Boolean subgroup. Moreover, locallycountably S-closed km-perfect almost Moscow topological groups are extremally disconnected topological groups that are strong PT-groups. Results on δ-closures, rc-Lindelöf equivalences, and homomorphic permanence establish a robust foundation for algebraic operations under weakened continuity, with applications to completion theory, quotient structures, and generalized topological dynamics.

Abstract: We study the functional analytic structure of weighted pullback spaces D^w_ϕ defined on the open unit interval (0, 1). Rather than prescribing the underlying coordinate transport map directly, we generate it implicitly through the automatic compactification of the nonlinear parametric relation R^p + qR = e^v, where p ∈ (0, 1] regulates left-endpoint algebraic compression and q ≥ 0 governs right-endpoint domain redistribution. We prove that D^w_ϕ is isometrically isomorphic to a classical weighted Hilbert space L²((0, 1), ρ(x) dx), explicitly constructing the two-sided inverse operator to establish surjectivity. Using an orthogonal spatial decomposition based on disjoint supports on the transport line, we isolate independent boundary integration thresholds. We derive precise directional variations of the localized energy densities with respect to continuous parameter deformations, clarifying the analytical balance between coordinate transport and weight redistribution components. Finally, we establish a global admissibility criterion for dual power-law singular profiles, verify the density of algebraic polynomials, and characterize the action of the parametric variation fields.

Abstract: Defect data attached to a morphism admit two orthogonal refinements. The first is factorization sensitive: a factorization \( A\xrightarrow{u}B\xrightarrow{v}A^\dagger \) with vu = R, gives a residual triple (Cone(u), Cone(v), Cone(R)) constrained by the octahedral axiom. The second is coefficient-sensitive: an excess object is the cone of a comparison morphism between integral or finite-coefficient residual data and its rational or mixed-Hodge realization. We organize these two directions into a residual–excess matrix. The main residual witness is a finite ordinary-double-point conifold degeneration. Saito divisor gluing produces a node-supported Hodge interface \( W^H_\Sigma=\bigoplus_{p_k\in\Sigma}i_{k*}\mathbb Q^H_{\{p_k\}}(-1) \) factoring finite-node monodromy through \( \psi^H_{\pi,1}\to W^H_\Sigma\to\psi^H_{\pi,1}(-1) \). The variation cone yields the corrected extension \( 0\to IC^H_{X_0}\to P^H_{\mathrm{var},\Sigma}\to W^H_\Sigma\to0 \), whose class decomposes into nodewise Ext residual classes. Ordinary double points calibrate the zero-excess regime because their Milnor fibers are integrally torsion-free. Diaz’s Enriques-product Bockstein mechanism calibrates the nonzero-excess regime through integral torsion killed by rationalization. The resulting framework compares finite-node Saito gluing with integral Hodge obstruction channels without conflating rational residual data with coefficient-change defects.

Abstract: Almost paracontact almost paracomplex manifolds equipped with a pair of structure-compatible metrics, one Riemannian and the other pseudo-Riemannian, are considered. The difference tensor between the two corresponding Levi-Civita connections is studied within the framework of the known Manev–Staikova classification of these manifolds. The obtained characterizations of the basic classes in terms of the difference tensor are illustrated by means of Lie groups of the lowest dimension.

Abstract: We initiate the study of solitons for the Gauss curvature flow on surfaces immersed in three-dimensional Lorentzian Walker manifolds. A soliton is a surface whose shape is preserved along the flow, evolving purely by the ambient isometries. Working with both the extrinsic and intrinsic Gauss curvature, we compute the fundamental forms of the relevant families of invariant surfaces and reduce the soliton condition to an ordinary differential equation on the generating curve. Our main contributions are threefold. First, we prove a rigidity theorem: a surface invariant under a one-parameter group of isometries is a soliton for that same group if and only if it is flat. Second, in the strictly Walker case we show that the only solitons with respect to the canonical parallel null field are the coordinate planes. Third, for the remaining Killing fields we classify the solitons via a phase-plane analysis of the associated ODE systems and describe the qualitative geometry of the solution surfaces. These results provide a Lorentzian analogue of the classification recently carried out by Belli and López in the Riemannian solvable Lie group Sol.

Abstract: In the historical development of various fields of mathematics, significant advances have occurred in areas such as algebra, abstract algebra, group theory, and numerous other mathematical and scientific domains. Contributions from mathematicians such as Dio- phantus, Goldbach, Euler, Girolamo Cardano, Johannes Kepler, Poncelet, Henri Poincaré, George Cantor, Felix Klein, David Hilbert, and Hermann Weyl have been fundamental, particularly in the pursuit of increasingly complex and deeper structures within geometry and topology. In this work, the division operation in the Alpha group is defined by analogy with the Kronecker tensor product. The representation of quaternion theory, based on De Moivre’s theorem, is employed for the construction of the matrices. The Alpha Group di- vision operation is then applied to analyze the various tensor metrics resulting from plane rotations over the interval from 0 to 2π radians. Since the general transformation kernel of the 4 × 4 matrix is defined within the Alpha group, it is possible to observe the variabil- ity associated with the tangent and cotangent functions that constitute the transformation matrix. The Alpha group, defined through a generalized division operation, thus provides a geometric and topological representation of infinity via the kernel transformation of the 4 × 4 matrix. Ultimately, this work seeks to connect the ideas developed by Poncelet and Cantor regarding the formation of imaginary elements in infinite projections with the con- cept of different types of infinity, as interpreted through the application of group theory.