Computer Science and Mathematics

Abstract: For a smooth projective complex variety X, we introduce typed Hodge sectors, single-sector algebraicity targets, and expanded targets in which several typed mechanisms admit to a common receiving structure while retaining their separate hypotheses and Chow-level certification. We prove that every certified span \( V^p(\mathcal T) \) lies in \( \operatorname{cl}_X(CH^p(X)_{\mathbf Q}) \), and develop direct-sum aggregation, fiber-product comparison, and certificate-preserving receiver widening. We distinguish structural widening from certification-strict widening, show that shared reception need not enlarge the certified span, and characterize strict gain by a shared-gain quotient \( \Delta_{\mathrm{sh}}^p \). Under maximal rational certification this quotient vanishes, revealing the construction-relative nature of rational gain. We also introduce integral constructional defect quotients \( D_{\tau,\mathcal T}^p(X)\ \)and exhibit a receiver widening that strictly reduces integral defect while leaving the rationalized span unchanged. Coverage remains a separate condition throughout.

Abstract: Let \(X\) be a Tychonoff space and let \(bX\) be a Hausdorff compactification of \(X\). We study the boundary-approach density \(d_b^{\partial}(X)\), an embedding-sensitive cardinal measuring how small a single subset of \(X\) can be while its closure reaches the whole remainder. We examine its behavior under products and as the compactification varies. For a non-empty family \((X_i)_{i\in I}\) of non-empty Tychonoff spaces with Hausdorff compactifications \(b_iX_i\), put \(P=\prod_{i\in I}X_i\) and \(K=\prod_{i\in I}b_iX_i\). We obtain an exact trichotomy. If all factors are compact, the value is \(0\). If exactly one factor is non-compact, the value is the maximum of the boundary-approach density of that factor and the density of the remaining compact product. If at least two factors are non-compact, the value is exactly \(\text{dens}(P)\), independently of the chosen factor compactifications. For a non-compact locally compact Tychonoff space \(X\), the one-point and Stone–Čech compactifications attain the minimum and maximum of the boundary-approach spectrum. If \(X\) also admits a clopen decomposition \(X=\coprod_{\xi<\tau}G_\xi\), where \(\tau\) is infinite and each \(G_\xi\) is non-empty and non-\(\omega\)-bounded, then every infinite cardinal \(\lambda\leq\tau\) is realized by a compactification \(b_\lambda X\), and the compactifications may be chosen to form a chain in the compactification order. When \(\tau=\text{dens}(X)\), the spectrum is the full interval \(\{\lambda:\omega\leq\lambda\leq\text{dens}(X)\}\). In particular, every non-compact locally compact metrizable space has this full spectrum. We also obtain an exact closed-core formula, an intrinsic Stone–Čech characterization, the identity \(d_\beta^{\partial}(X)=\text{dens}(X)\) for non-compact metrizable \(X\), and monotonicity of \(d_\beta^{\partial}\) under continuous maps with dense image.

Abstract: We classify all compact homogeneous smooth pseudoconvex complex Finsler manifolds. We apply a result of Tits on compact complex homogeneous space, or of H. C. Wang and Hano-Kobayashi on the classification of compact complex homogeneous manifolds with a compact reductive Lie group. In particular, They are homogeneous complex torus bundles over rational projective homogeneous manifolds. One can simply construct an invariant complex Finsler structure by the given isotropic subgroup whichis a subgroup of U(n) at any given point, and then transfer it to the whole manifold with the group action.

Abstract: We study finite-node threefold degenerations with ordinary double points through nearby and vanishing cycles, intersection spaces, and Hodge atoms at Calabi--Yau conifold degenerations. Under the Banagl--Budur--Maxim specialization hypotheses, trivial local Milnor monodromy gives \(\mathcal{IS}_{X_0}^H\simeq\Psi\). The nearby/intersection-space carrier has weight filtration with \(\operatorname{Gr}_2^W\Psi\simeq K\), \(\operatorname{Gr}_3^W\Psi\simeq IC_{X_0}^H\), and \(\operatorname{Gr}_4^W\Psi\simeq\Phi\), while global relations give \(\operatorname{im}N=V_{\mathrm{glob}}\), \(N^2=0\), and \(\ker N=V_{\mathrm{glob}}^\perp\). In the Calabi--Yau setting, this filtration gives a Hodge-atom realization: the rigid Hodge atom is carried by the pure sector \(\operatorname{Gr}_3^W\), the degeneration-side flexible Hodge atom by \(\operatorname{Gr}_4^W\), and the full nearby realization contains the monodromy-dual flexible sector \(\operatorname{Gr}_2^W\). Equivalently, up to the fixed Tate normalization, the Hodge-atom carrier is \(\Psi/W_2\Psi\). For the classical $125$-node quintic, the relation rank is $24$ and the limiting middle cohomology has weight dimensions 204=101+2+101.

Abstract: Let H(A) be the Hattori space associated with A ⊆ R, and put B = R \ A. Hernández-Hernández, Ramírez-Chávez and Rojas-Hernández asked ([9], Problem 7.8) for characterizations of when H(A)³, all finite powers H(A)ⁿ, or H(A)^ω are Lindelöf. Using the square theorem and the perfect set property dichotomy, we show that if B has the perfect set property, then the following conditions are equivalent: B is countable, H(A) is second countable, H(A)² is Lindelöf, H(A)³ is Lindelöf, H(A)ⁿ is Lindelöf for every integer n ≥ 2, and H(A)^ω is Lindelöf. This applies in particular when B is analytic or Borel. We also show that the perfect set property assumption cannot be removed for higher powers. In ZFC there is a set A ⊆ R such that B has cardinality c and contains no subspace homeomorphic to 2^ω in the Euclidean topology, while H(A)² is Lindelöf and H(A)ⁿ is not Lindelöf for every n ≥ 4. In particular, H(A)^ω is not Lindelöf. The corresponding implication for the cube remains open.

Abstract: We give a partial answer to a problem of Hernández-Hernández, Ramírez-Chávez and Rojas-Hernández concerning the Lindelöf property of function spaces over one-point extensions of countable free sums. For a free filter \(\mathcal{F}\) on \(\omega\), let \(L_{\mathcal{F}}\) denote the space of real sequences that converge to zero along \(\mathcal{F}\). A coding lemma identifies the descriptive complexity of this auxiliary space by showing that \(L_{\mathcal{F}}\) is \(K\)-analytic if and only if \(\mathcal{F}\) is analytic as a subspace of \(2^\omega\). If \(\mathcal{F}\) is analytic and \((X_n)_{n<\omega}\) is a sequence of non-empty compact spaces, then\[ C_p\left(\bigoplus_{n<\omega} X_n\right)\text{ is Lindelöf} \quad\Longrightarrow\quad C_p(X_{\mathcal{F}})\text{ is Lindelöf}. \]Moreover, if \(C_p(X_{\mathcal{F}})\) is Lindelöf, then \(C_p\left(\bigoplus_{n\in\omega\setminus A} X_n\right)\) is Lindelöf for every \(A\in\mathcal{F}\). Consequently, for constant compact fibres and analytic free filters, \(C_p(X_{\mathcal{F}}(K))\) is Lindelöf and only if \(C_p(K)^\omega\) is Lindelöf. If the free sum has a countable network, no descriptive-set-theoretic restriction on the filter is needed.

Abstract: Let \( \theta \) be an ordinal and let \( (N_\alpha)_{\alpha\leq\theta} \) be a continuous increasing tower of closed normal subgroups of a Hausdorff topological group \( G \), with \( N_0=\{e\} \). Write \( \rho G \) for the Raikov completion of \( G \), and put \( K_\alpha=\overline{N_\alpha}^{\rho G} \) and \( H_\alpha=GK_\alpha \). The successive differences and the corresponding limit differences of the groups \( H_\alpha \) mark the first stages at which points of the Raikov completion become old quotient points. The set of stages with nonempty strata can be prescribed arbitrarily even under pseudocompactness. For every nonzero ordinal \( \theta \) and every set \( S\subseteq(0,\theta] \), there is a Hausdorff pseudocompact Boolean group with a continuous tower of closed normal subgroups such that every member of the tower is pseudocompact and the nonempty strata occur exactly at the stages in \( S \). The terminal quotient can simultaneously be chosen isomorphic to \( \mathbb Z/2\mathbb Z \), and for every \( \delta<\theta \) the successive factor \( N_{\delta+1}/N_\delta \) is compact exactly when \( \delta+1\notin S \). Thus a tower may have a single nonempty stratum at a limit stage even when every successive factor is compact. The realization theorem uses an exact description of the fibers of quotient extensions. For a closed normal subgroup \( N \) of \( G \), let \( \widehat q:\rho G\to\rho(G/N) \) extend the quotient homomorphism and write \( K=\overline N^{\rho G} \). The fiber in \( \rho G\setminus G \) over an embedded quotient point \( gN \) is \( g(K\setminus N) \), homeomorphic to the Raikov remainder \( \rho N\setminus N \), while for a point\( y\in\widehat q(\rho G)\setminus(G/N) \) the whole \( \widehat q \)-fiber lies in \( \rho G\setminus G \) and is a copy of \( \rho N \). For precompact kernels this gives a canonical decomposition, and nested quotients give the two-stage identity underlying the transfinite filtration. For compact kernels, pseudocompactness of the source and quotient remainders is equivalent. For noncompact precompact kernels, the source remainder maps onto the entire quotient completion, and countable Abelian examples exhibit both pseudocompact and non-pseudocompact quotient remainders.

Abstract: Let \(\mathcal{I}\) be an admissible ideal on \(\mathbb{N}\), and let \(\operatorname{Fin}\) denote the ideal of finite subsets of \(\mathbb{N}\). Zhou--Liu--Liu--Lin asked whether every regular space with a \(\sigma\)-hereditarily closure-preserving \(\mathcal{I}\)-\( sn \)-network is \(\mathcal{I}\)-\( sn \)-metrizable. The central condition is local HCP-finiteness, denoted \(\operatorname{HCF}(\mathcal{I})\). The standard \(\mathfrak{p}^{-}\) property implies this condition. If \(\mathcal{K}\) has \(\mathfrak{p}^{-}\) and \(\mathcal{J}\) has local HCP-finiteness, then \(\mathcal{K}\otimes\mathcal{J}\) has local HCP-finiteness. For varying inner ideals, the heterogeneous sum has local HCP-finiteness if the outer ideal has \(\mathfrak{p}^{-}\) and, outside a \(\mathcal{K}\)-small set of rows, local HCP-finiteness and cross-row Kat\v{e}tov absorption hold. At Kat\v{e}tov's first limit stage, the heterogeneous theorem proves \(\operatorname{HCF}(\operatorname{Fin}^{\omega})\), while constant-inner lifting gives \(\operatorname{HCF}(\operatorname{Fin}^{\omega+r})\) for \(0<r<\omega\). The question therefore has an affirmative answer for each of these ideals, although none has \(\mathfrak{p}^{-}\). For an increasing sequence of admissible ideals on a fixed countable set whose union is a proper ideal, local HCP-finiteness passes to the union if cofinally many stages are \(K\)-uniform and have local HCP-finiteness. Two rank-\(\omega\) limit ideals satisfy the hypotheses of the increasing-union theorem. For the tree-derived hierarchy of Pelayo G\'omez, the theorem answers the same question affirmatively for the \(\boldsymbol{\Sigma}^{0}_{\omega}\) ideal \(\mathcal{H}_{<\omega}\). The inductive limit \(\operatorname{Fin}_{\omega}\), whose finite approximating stages are isomorphic to support lifts of finite Fubini powers, satisfies the same metrization conclusion. Both limit ideals have separation rank \(\omega\) and fail \(\mathfrak{p}^{-}\). Thus \(\mathfrak{p}^{-}\) is sufficient but not necessary for the metrization conclusion.

Abstract: A pure geometric construction is presented that recovers a substantial portion of classical mathematics from a single primitive figure: the non-proper Archimedean conical helix. Starting from three geometric axioms that govern continuous coiling, radial growth under maximal cohesive spread, and indecomposable flux conservation, the basic objects of Euclidean geometry are reconstructed—points, lines, planes, congruence, circles, triangles, parallels, and ruler-and-compass constructions—strictly as geometric shadows of the helix. From the same structure, obtaining the fundamental constants π, the golden ratio, the Fibonacci sequence, the trigonometric ratios, the imaginary unit, and the prime numbers (as irreducible cycle lengths). Geometric arithmetic operations, the successor function, and an induction principle are then introduced, followed by a complete pure-geometric development of the calculus (limits, derivatives, integrals, series, and vector calculus), real and complex analysis, measure theory, and the core structures of differential geometry (tangent bundle, Riemannian metric from flux, covariant derivative, curvature as holonomy, geodesics, and the Laplace–Beltrami operator). The construction concludes with geometric sets, geometric functions, and a pure geometric form of wave–particle duality embodied by the helix itself. Throughout, no external arithmetic, analytic, or set-theoretic primitives are assumed; every notion is derived from the helix and its intrinsic geometric operations.

Abstract: We study the topology associated with the family of h-open sets and use it to organize several separation properties in a unified way. After recalling the classical closure, kernel and separation axioms, we distinguish the original topology τ from the associated topology τ_h = hO(X) and introduce the h-specialization preorder and the h-kernel. The axiom h-R₀ is characterized as the natural symmetry property of this preorder and by the equality of singleton h-closures and h-kernels. We prove the relations h-T₁ ⇔ h-T₀ + h-R₀ and h-T₂ ⇔ h-T₀ + h-R₁, and characterize h-R₁ by singleton h-θ-closures. We then develop the theory of h-difference sets and the axioms h-D₀, h-D₁ and h-D₂, proving h-D₀ ⇔ h-T₀ and h-D₁ ⇔ h-D₂. A corrected characterization in terms of h-neat points and several preservation results under h-irresolute mappings are obtained. Finally, singleton h-derived sets and h-semisimplicity are related to strong h-regularity.

Abstract: Let (X,τ) be a topological space and let τ_h(X) denote the topology formed by the h-open subsets of X. We develop h-density, h-separability and countable h-dense homogeneity systematically through the associated space (X, τ_h(X)). We prove that h-density and h-separability are precisely ordinary density and separability in the associated topology, and that h homeomorphisms are exactly homeomorphisms between associated spaces. Consequently, (X,τ) is h-countable dense homogeneous (h-CDH) if and only if (X, τ_h(X)) is countable dense homogeneous (CDH). A classical theorem on CDH spaces then yields that every h-CDH space is h-T₁. Hence its h-specialization preorder is equality, hCl({x}) = hKer({x}) = {x} for every point, and the pointwise closure–kernel defect measured by the h-RT condition vanishes identically. We give explicit examples showing that or dinary CDH and h-CDH are incomparable in general, while they coincide in Hausdorff spaces. We formulate a product-transfer principle for arbitrary families under compatibility of associated and product topologies and show, using a metrizable counterexample, that compatibility alone does not make h-CDH productive. Finally, we relate the theory to h-normality, including an infinite-product transfer criterion, and formulate several questions concerning normality, product compatibility and converses to the h-RT consequence.

Abstract: We introduce and investigate forms of regularity and normality defined by means of h-open sets. Using the associated topology τ_h = hO(X), we interpret h-regularity as a relative regularity property in which points and closed sets of the original topology are separated by τ_h-open sets, whereas strong h-regularity is precisely the ordinary regularity of (X, τ_h). We establish characterizations of h-regularity, study its preservation under suitable mappings, compare the new notions with their classical counterparts, and examine related separation properties.

Abstract: Recently, Abbas introduced the notion of an h-open set as a class of generalized open sets in a topological space. The subsequent corrigendum and addendum of Sharma, Saproo, Billawria and Digra clarified the theory and established that the family of all h-open sets is itself a topology, denoted by τ_h, without any T_1/2 assumption. Motivated by this topological interpretation, we introduce and study the separation axioms h-D₁ and h-D₂ by means of h-difference sets. We prove that these two axioms coincide, relate them to the classical D_i axioms in the associated topology (X, τ_h), examine their interaction with h-T_i and h-symmetric spaces, and obtain preservation results under h-irresolute and h-continuous mappings. Finite and infinite examples are included to distinguish the original topology from the associated topology of h-open sets. We also correct the overly strong assertion that h-symmetry alone implies h-T₁: the additional h-T₀ hypothesis is essential. In addition, we place the h-R₀, h-R_H and h-R_D conditions in the associated-topology framework, establish a compatible subspace theorem, and clarify the interaction between h-compactness and h-T₂. Three vector diagrams summarize the structural relationships among the separation axioms considered.

Abstract: Deciding whether a vertex of a cubical complex lies on a locally well-composed surface reduces to a finite question: examine the occupancy pattern of the eight voxels incident to the vertex and decide whether the resulting active incidence structure is connected and 2-regular. We give a symbolic account of this decision. Writing \(n_E(S)\) and \(n_F(S)\) for the numbers of active edges and faces determined by elementary local activity rules, we show that every active face has degree exactly two and every active edge has degree two or four, yielding the identity \(n_F(S)=n_E(S)+k(S)\) for a single defect count \(k(S)\), and hence a surface criterion equivalent to \(\mathrm{comps}(S)=1\wedge n_E(S)=n_F(S)\). We further prove that the active incidence structure is invariant under complementation as a literal identity, not merely up to isomorphism, and combine this duality with a finite case analysis to classify explicitly which pairs \((|S|,d_1(S))\) of cardinality and Hamming-adjacency count are realized by a surface state, yielding an exact threshold classification as a proved corollary. The classification is established by a complete finite case analysis up to cube symmetries for \(|S|\le4\) and by duality for \(|S|\in\{5,6,7\}\).

Abstract: Let \(\pi:\mathcal X\to\Delta\) be a projective one-parameter degeneration of complex threefolds whose general fiber is smooth and whose central fiber has finitely many ordinary double points, and suppose that a finite group \(G\) acts algebraically and fiberwise on \(\mathcal X\). We construct an equivariant refinement of the corrected rigid--flexible conifold degeneration package. The variation-cone mixed-Hodge-module carrier admits a canonical \(G\)-linearization, and its node-supported flexible quotient carries the rational representation \(V_{\mathrm{van}}\cong\bigoplus_{[p]\in\Sigma/G}\operatorname{Ind}_{G_p}^{G}V_p\), where \(G_p\) is the stabilizer of \(p\) and \(V_p\) is its one-dimensional local vanishing representation. Frobenius reciprocity determines the irreducible symmetry multiplicities, yielding a canonical equivariant limiting rigid--flexible profile whose forgetful image is the ordinary conifold profile. We also construct an explicit projective \(S_4\)-equivariant smoothing of a four-nodal cubic threefold for which \(G_p\simeq S_3\) and \(V_p\simeq\operatorname{sgn}_{S_3}\), giving \(V_{\mathrm{van}}\cong\operatorname{sgn}_{S_4}\oplus(V_{\mathrm{std}}\otimes\operatorname{sgn}_{S_4})\). Thus the node \(G\)-set alone does not determine the equivariant vanishing representation. Finally, we prove that \(G\) acts by automorphisms of the limiting mixed Hodge structure and that its rational \(G\)-isotypic components are sub-mixed-Hodge structures preserved by nilpotent monodromy.

Abstract: We introduce and study the separation axiom h-R_T , designed to provide a pointwise measure of the discrepancy between the h-closure and the h-kernel of a singleton in the associated topology τ_h. Using the specialization preorder of (X, τ_h), we obtain structural characterizations of h-R₀, h-T₀, and h-R_T . We prove that every h-R_T space satisfies a conditional singleton-derivedset property and an h-R_H closure-intersection condition, while the conjunction of h-T₀ and h-R_T implies the global h-R_D condition. Explicit finite examples separate h-R_T from h-R₀, h-T₀, and h-R_D. We establish preservation results for compatible subspaces and finite products, characterize locally h-indiscrete spaces through the associated topology, and introduce the weakly h-R₀ axiom inspired by Di Maio’s weakly R₀ condition. For weakly h-R₀ spaces, we obtain a kernel characterization, a preservation theorem for injective always h-closed maps, and a product theorem under an explicit compatibility condition on the associated topologies.

Abstract: An involution of an exceptional geometry of type E₇ is called regular if its fix structure, viewed as simplicial complex, is a building. Involutions which do not act trivial on the underlying field correspond to Galois descent and were treated a long time ago by Jacques Tits. In the present paper, we classify the regular involutions of geometries of type E₇ that act trivial on the underlying (arbitrary) field, which we assume not to have characteristic 2. As a result, we discover new subgeometries of the exceptional geometry of type E₇.

Abstract: In this paper, we introduce the notion of strong contractions. These contractions resemble the strong triangle inequality in the definition of strong b-metric spaces [5]. We obtain some fixed point theorems for these new contractions in the setting of metric spaces with an illustrative example. Finally, we apply our result to the Fredholm integral equation.

Abstract: Local topological singularities on digital or voxel grids motivated well-composedness conditions excluding them; P-well-composedness lifts this regularity into an intrinsic, order-theoretic setting on posets. We study whether P-well-composedness is inherited by the strict neighborhood of a face in a finite embedded cubical complex. Let X = F(K) be the face poset of a finite nonempty set of grid n-cubes, with ambient rank n ≤ 3, and let N_h = θ′_X (h) carry the induced order. We prove N_h is P-well-composed whenever X is, provided h is a vertex or has no strict coface in the border ΔX, via a coherence argument for vertices and a finite, computer-assisted enumeration otherwise. The remaining rank-three case, an edge with a strict coface in the border, is posed as a No-Shrinkage Conjecture, with computational evidence reported separately.

Abstract: The closest point on a circular helix to a query point---a primitive of track fitting, computer-aided design, and robotics---reduces exactly to Kepler's equation \( u+e\sin u=M \), whose eccentricity \( e=a\rho/b^{2} \) can be arbitrarily large. We determine the complete solution structure of this problem. The cylinder law states that for \( rho < = b^{2}/a \) the squared distance is convex with a unique minimizer. The count law gives, for \( e>1 \), the exact number of stationary points \( N=2⌈{(M+c)/2\pi⌉ }-2⌊{(M-c)/2\pi}⌋-3 \) with \( c=\arccos(-1/e)+\sqrt{e^{2}-1} \); pairs of stationary points are born and annihilated on explicit spiral surfaces, which we identify with the focal surfaces of the helix, whose cuspidal edge is the evolute helix \( \rho=b^{2}/a \). The count grows like \( 2e/\pi \), so the often-quoted bound of three roots fails beyond \( e^{*}\approx4.6033 \). The principal-branch theorem shows that the global minimizer is always the unique root in the branch \( k_{0}=\mathrm{round}(M/2\pi) \), with the exact exception of two symmetric minimizers when \( M\equiv\pi\ (\mathrm{mod}\ 2\pi) \); the set of ties is the cut locus, terminating on the evolute helix. The minimizer is provably well-conditioned, \( |du^{*}/dM|< 1/(e-1) \), and conditioning degenerates only as \( O(\sqrt\delta) \) on the focal surfaces and as \( O(\delta^{1/3}) \) on the evolute. A hybrid bisection--Newton solver, a two-helix extension, and an exhaustive numerical verification complete the paper.