Computer Science and Mathematics

Abstract: Join every point of a circle to its image under an arithmetic map, and a curve appears that no single chord draws: an envelope. For the doubling and tripling maps these are the cardioid and the nephroid; more generally, \( x\mapsto kx \bmod 1 \) produces an epicycloid with k-1 cusps, which is similar to the main hyperbolic component of the Multibrot set of \( z^k+c \). In particular, the cardioid of the times table is, up to the factor -4/3, the main cardioid of the Mandelbrot set. We call such curves arithmetic envelopes and use them as a geometric toolbox. Labelling the odd integers by dyadic rationals identifies them with the Prüfer 2-group; in these coordinates multiplication by 3 is a piecewise linear circle homeomorphism, conjugate to the rotation by log₂3. Our main result determines the region left free by its chords, observed by Henkel and called the convexoid: it is convex and bounded by a segment of a diameter and by arcs of two epicycloids with cusps at 1/5 and -1/7. The pedal curve of the same family is heart-shaped and lies on the algebraic curves \( x=\rho\,T_5(\rho) \) and \( x=\rho\,T_7(\rho) \) with \( \rho^2=x^2+y^2 \). Lifted into space, the chords form ruled surfaces whose apparent contour is the envelope; on the number helix \( s=\log_2x \), multiplication by 3 is a screw motion, and the chords wind around a helix of radius \( |\cos(\pi\log_23)| \). In the Klein model all chords are hyperbolic geodesics, so the convexoid is hyperbolically convex. We close with connections to open problems, from Furstenberg's \( \times2 \) \( \times3 \) conjecture to the Mandelbrot set, and with applications.

Abstract: We prove Petty's conjecture for all finite-dimensional real \(L_p\) subspaces with \(2\le p\le4\). Every \(d\)-dimensional real linear subspace of \(L_p\), over any measure space, contains \(d+1\) equidistant points. More strongly, every equilateral subset extends to exactly \(d+1\) points, preserving its original points and common distance. Every equilateral set is affinely independent, so every maximal equilateral set has cardinality \(d+1\). This extends the \(2\le p\le4\) case of Kusner's conjecture from coordinate spaces to arbitrary subspaces and complements the constructions of linear excess for every fixed \(p>4\). The proof combines conditional positivity of the powered distance kernel, a circumradius estimate, and Brouwer's fixed-point theorem. The main theorems are formalised in Lean.

Abstract: We study extension-sensitive Hecke transport of conifold mixed-Hodge modules through elementary \(GL_2\) modifications of Higgs bundles. For the determinantal threefold node, an invariant-line incidence family realizes two small resolutions and their \(A_1\) Springer intersection. Smooth Higgs evaluation and a codimension-two Gysin calculation preserve the source Yoneda class, while first-jet vanishing theorems establish nonempty admissible loci in explicit genus and determinant ranges. On a rank-one modified-Higgs target open \(U\), a canonical kernel-branch inverse proves nonvanishing of both ordinary and exceptional unit-kernel pushforwards. On a smaller saturated open $U'$, the unframed inverse correspondence is finite flat of degree two. Under specified equivariant mixed-Hodge and descent hypotheses, the surviving morphism defines a nonzero degree-two derived class, giving a nontrivial Postnikov attachment between adjacent perverse cohomology objects. We further identify the Springer intersection with a bounded affine-Grassmannian convolution slice and establish morphism-level compatibility under explicit specialization hypotheses.

Abstract: Starting from the representation of De Moivre’s matrix formula combined with the Kronecker product, we derive a 4 × 4 matrix M_θ that encodes algebraic properties with topological implications in four dimensions. This matrix describes trans- formations associated with symmetry and antisymmetry, encompassing distortion, dilation/contraction, and combined shear in a four-dimensional framework. We de- compose M_θ into its symmetric (S) and antisymmetric (A_anti ) components, a result that characterizes key features of the Alpha Group, including the formation of a tan- gent plane at infinity and its corresponding topological and geometric consequences. To investigate the system’s dynamics, we perform Monte Carlo simulations for varying values of the parameter θ, revealing attractor behavior and geometric transitions between Euclidean and hyperbolic regimes in projected three-dimensional trajectories.

Abstract: Discrete global grid systems (DGGSs) organize geographic data as indexed cells whose neighborhoods support spatial queries. We study exact halo construction on a triangular DGGS and its use in buffer coverage. Rather than store refined connectivity, the architecture reconstructs adjacency using compact integer triangular indices and an Atlas of Connectivity Maps. A graphics processing unit (GPU) resident hash frontier admits each new cell once. Probe reduction and reusable output staging reduce repeated work and delivery costs. Measurements through level 28 compare serial and twenty thread central processing unit (CPU) baselines using flat hash tables, on an Intel Core i7 12700K and NVIDIA RTX 4070. The largest halo contains 10.8 million cells and reaches page locked (pinned) host memory in 13.3 ms; its returned container copy is measured separately. Small queries favor CPU execution. Independent mesh references validate Pentakis adjacency, while the performance runs check cumulative output agreement. A stopping argument connects complete rings to conservative distance coverage, conditional on valid source and distance bounds. A separate World Geodetic System 1984 boundary buffer study includes source covering and distance filtering; CPU execution is faster in all seven cases. Numerical tests do not certify floating point distance bounds.

Abstract: Our earlier work generated quality-scored addresses for ten cities from street graphs and supervised classifiers. Here we restate that pipeline mathematically and apply it, completely, to one commune: Gombe, the administrative centre of Kinshasa (Democratic Republic of the Congo; km2, buildings from OpenStreetMap and Google Open Buildings, addresses already in use). The street network is treated as an embedded planar graph whose oriented segments (darts, half-edges) carry the two sides of every street, and whose faces are the urban blocks. In this language we (i) prove that the published street-continuity score never joins straight continuations and give a corrected, name-aware matching; (ii) define an invertible oriented-segment address (S,σ,s,d) (street, side, position, offset) from which the human-readable address is rendered; (iii) formalise anchored house numbering (Theil–Sen fit, parity projection, order preservation and infill without renumbering) and block (îlot) addressing on faces; (iv) introduce a conditional-logit model over the candidate street sides of each building, with an optional Markov random field on the building adjacency graph; and (v) derive and simulate how input errors propagate through the graph. On Gombe, every building receives an address; anchored numbers are within ±10 of the number in use for % of test addresses (published model: %). The Euler relation holds exactly on the dart structure (|V|−|E|+|F|=2C=), and the nearest street lies on the building’s own face for % of buildings. The street-side choice model agrees with the street of existing addresses for % of them in leave-one-out validation, against % for the V8 rules and % for the nearest street; the Markov prior does not improve on it. Monte Carlo error experiments show that local errors stay local except through two global quantities, the betweenness ranks of provisional street codes and the commune numbering rate, which carry most of the propagated damage (% of the addresses changed by a missing segment are re-ranked codes). Both should therefore be frozen once addresses are issued.

Abstract: The purpose of this paper is to introduce and investigate a basic ideal-theoretic extension of the notion of a Moscow space. A classical Moscow space is a topological space in which the closure of every open set can be represented as a union of G_δ-subsets of the ambient space. We introduce the notion of an I-G_δ set by requiring a set to differ from a G_δ set by an element of a prescribed ideal I. This leads to the definition of an ideal Moscow space. We establish elementary relationships between Moscow spaces and ideal Moscow spaces, including monotonicity with respect to the ideal, invariance under homeomorphisms, and sufficient conditions based on countable pseudocharacter. Examples illustrate the definition and its dependence on the ideal. We also discuss heredity, products, and mappings, distinguishing proved statements from questions that require additional hypotheses. The proposed framework provides a starting point for a systematic study of Moscow-type properties in ideal topological spaces.

Abstract: The hyperstructural model of redox reactions organises the oxidation states of an element into an H_v-group whose hyperoperation returns the set of thermodynamically accessible products. We recast this model in the language of general topology. To every hyperoperation ⊕ on a nite set H we attach a topology τ_⊕, the reaction topology, generated by the family of hyperproducts {x ⊕ y} as a subbasis. We prove that τ_⊕ is an Alexandrov topology whose minimal neighbourhoods are N(x) = ∩ {x ⊕ y : x ∈ x ⊕ y}; that its specialisation preorder admits the chemical reading y accompanies x in every reaction product containing x''; that τ_⊕ is T₀ exactly when no two species are reactionindistinguishable, and discrete exactly when every species is separated from all others by some reaction; and that the connected components of (H, τ_⊕) are the components of the comparability graph of that preorder. We then show that the hyperoperation is continuous for the Vietoris hyperspace topology whenever it is constant on minimal neighbourhoods, so that every H_v-group with discrete reaction topology is a topological H_v-group, and that the quotient by the fundamental relation β^∗ is discrete if and only if the β^∗-classes are open. The constructions are worked out on the redox data for tin, iron, copper and manganese: the hypergraphs induced by the iron H_v-group yield, respectively, a disconnected space with components {Fe³⁺} and {Fe, Fe²⁺}, a connected T₀ space, a connected non-T₀ space with two Kolmogorov classes, and an indiscrete space, while the copper hypergraph yields a connected, non-discrete T₀ space.

Abstract: This paper deepens the understanding of the geometric relationships between a regular plane curve and its stereographic lift onto the unit sphere S2. Central to this framework is the Stereo graphically Arc-Length (SAL) parametrization, a non-linear differential constraint that synchronizes coordinate time identically with the canonical spherical arc length t. Within this domain, we construct three automated dimensional-reduction architectures to classify locally 3D space curves based on the continuous evolution of the parameter t: a tangent indicatrix relaxation through a single flat projection channel, a coordinate projection relaxation utilizing a coupled dual-screen setup linked by a shared coordinate bridge, and an invariant phase-space representation inside the standalone γ-plane mapping fundamental 3D curvature and torsion fields. These local classifications rest on our local classification of plane curves, via the characterization of plane curves with polynomial curvature. Related to this classification scheme is the Conformal Binormal Invariant, an out-of-plane functional evaluated within flat coordinate channels. The framework is illustrated by a detailed example: the conical spiral.

Abstract: Games on knot and link diagrams provide a combinatorial setting in which topological invariants can be used to certify terminal outcomes and to guide winning strategies. Separately, bitopological game theory studies situations in which two topologies are relevant to the same underlying set or function space. This paper formulates a bitopological extension of finite knot-diagram games. A bitopological knot game consists of a finite diagram-state space, two topologies, player-dependent admissibility neighborhoods subordinate to those topologies, a turn rule, a terminal payoff, and a knot or link invariant used to evaluate terminal states. We prove a strategic quotient theorem: when a quotient map preserves terminal payoffs and the sets of successor classes available to each player, the quotient game has exactly the same minimax value and winning regions. We then give a two-topology version and explain how invariant-valued terminal payoffs can reduce a diagram game to a finite invariant game. A finite example shows that changing one player's admissibility system can change the value even when the terminal invariant is unchanged. Applications to the Knotting--Unknotting Game and the KnotLink Game are discussed as extensions, not as claims that the published winning-strategy results for those games have been reproved or superseded. The main contribution is a precise framework and a theorem that can be applied once the admissibility systems and quotient hypotheses are verified for a concrete family of knot or link shadows.

Abstract: We develop a groupoid-based framework for gerbes on orbifolds equipped with two compatible topologies. The smooth topology determines the underlying orbifold, whereas a second topology records additional topological structure on the levels of its presenting groupoid. We define paired banded gerbes, formulate their Dixmier–Douady invariants, and establish classification and Morita-invariance results under explicit descent and coefficient hypotheses. In particular, we prove that decorated pairs of circle-banded gerbes are classified by a product of two groupoid-cohomology groups. We then formulate a relative classification for compatible pairs using a homotopy fibre product of gerbe-classification groupoids. We prove that both constructions are invariant under Morita equivalences that preserve the two topological sites and their coefficient systems. A finite-quotient example demonstrates why orbifold isotropy cannot in general be discarded in favor of the coarse orbit space. The example also provides a test that any proposed bitopological Dixmier–Douady theory must satisfy. The standard orbifold-gerbe theory is distinguished throughout from the additional paired structure introduced here. The classification statements are conditional on explicitly stated descent hypotheses; no automatic equivalence between arbitrary pairs of topologies is assumed.

Abstract: We introduce a framework for topological games on neutrosophic bitopological spaces. The underlying structure consists of a set equipped with two neutrosophic topologies, allowing the two players to interact with distinct notions of openness. After fixing explicit conventions for neutrosophic unions, intersections, interiors, and threshold covering, we define one-selection and finite-selection games, cross-interior covering conditions, strategies, and winning plays. We establish general implications between winning strategies and selection principles, prove a threshold monotonicity result, and investigate the behavior of the games under equality of the two topologies. Examples illustrate the difference between the two topologies and the role of the cross-interior condition. We also prove that the proposed framework reduces to the corresponding classical bitopological game in the crisp-valued case. The definitions and results are formulated so that their hypotheses and conclusions can be checked directly. The paper distinguishes the classical background of bitopological spaces and selection games from the particular neutrosophic game framework developed here.

Abstract: We formulate a general framework for finite, perfect-information, two-player games whose positions are knot or link diagrams and whose terminal payoffs depend on the represented topological objects. The central question is when a knot or link invariant can determine a game's outcome, bound its strategic value, or distinguish positions that are strategically inequivalent. We separate classical knot-theoretic input from game-theoretic conclusions and prove general sufficient conditions based on terminal payoff factorization, invariant-compatible move relations, reachable invariant sets, and minimax recursion. We also give a finite-state quotient theorem and a counterexample showing why a terminal invariant alone generally cannot determine a nonterminal position's value. The results are abstract framework theorems. The paper does not claim that the framework, or the specific conclusions below, exhausts the existing literature on knot games.

Abstract: How many equilateral points can ℓₚⁿ contain beyond n + 1? For every fixed real p > 4, we construct equilateral sets for which the number of points beyond n + 1 grows at least linearly with n. For every fixed dimension, we prove that the maximum is n + 1 at all sufficiently large finite p, with an explicit threshold. The least dimension admitting any fixed number of extra points is p¹⁺ᵒ⁽¹⁾ as p → ∞. The constructions and finite bounds are formalised in Lean. Thus p = 4 marks the boundary: n + 1 is exact in every dimension for 2 ≤ p ≤ 4, while at every fixed finite p > 4 the excess grows at least linearly with dimension.

Abstract: We investigate whether the specialization and extension structures of singular Hodge atoms can be organized compatibly with Hecke correspondences and their composition, as a step toward potential connections with geometric Langlands. Inspired by Gaitsgory’s nearby-cycle approach to affine-Hecke convolution, we develop explicit mixed-Hodge-module comparisons for GL₂ and GL₃ Hecke collisions, using finite-dimensional incidence geometry and Grothendieck–Springer resolutions. Our principal result reveals that derived composition and coefficient-level extensions retain fundamentally different information: in a transverse compound-A₂ collision, the composition-defect triangle splits while all three underlying coefficient extensions remain nonsplit. A complementary GL₂ calculation shows that isomorphic nearby cycles and trivial unipotent monodromy can conceal distinct specialization morphisms and nontrivial extensions. These results establish concrete, morphism-sensitive models for studying singular Hodge-theoretic information under geometric composition. They motivate a systematic progression toward higher-rank Hecke geometry, global Springer-orbit gluing, and prospective Hecke–Hitchin and geometric-Langlands realizations.

Abstract: We develop a topological framework for group-like algebraic systems whose multiplication is genuinely multi-valued. The central object is a hyperoperation ◦ : G × G −→ P∗(G), where P∗(G) denotes the family of all nonempty subsets of G. We focus on the compactvalued case ◦ : G × G −→ K(G), where K(G) is the hyperspace of all nonempty compact subsets of G, and impose continuity with respect to the full Vietoris topology. This provides a natural refinement of the usual upper hyperspace approach to topological hyperstructures. We formulate the notion of a Vietoris multi-valued topological group using the axioms of a polygroup, rather than using weaker identity and inverse conditions. This choice guarantees a strong algebraic connection with ordinary groups and permits the construction of fundamental quotients. We establish basic properties of continuous multi-valued translations, inversion, division, compact products, closed graphs, products, subgroups, and continuous homomorphisms. A central role is played by complete parts. We introduce Vietoris-complete multi-valued topological groups, prove that complete open neighborhoods generate natural left and right neighborhood systems, and construct canonical left and right uniform structures. We then investigate the fundamental relation and prove that, under a natural complete-part hypothesis, the fundamental quotient carries a canonical topological group structure. We also establish a singleton-collapse theorem showing that the proposed theory contains ordinary topological groups as a rigid special case. Several examples are developed, including finite hypergroups, quotient constructions, compact orbit hyperstructures, and finite-valued operations. The paper concludes with a comparison with Marty hypergroups, polygroups, and locally compact convolution hypergroups, together with open problems concerning completion, invariant measures, Lie-type structures, bitopological extensions, and multi-valued groupoids.

Abstract: We develop a topological approach to homotopy theory in which ordinary single-valued maps are replaced by suitable multi-valued functions. The main purpose is to formulate a coherent homotopy theory for set-valued mappings by using compact-valued upper semicontinuous maps, Vietoris maps, admissible mappings, selectors, and hyperspaces. We introduce multi-valued homotopies and study their elementary properties, including homotopy invariance, composition, products, retractions, and induced maps on homology. A central point of the construction is that a multi-valued map \( F:X\multimap Y \) can be represented, under suitable hypotheses, by a diagram \( X\longleftarrow \Gamma\longrightarrow Y \) whose left-hand map is a Vietoris map. This permits homological information to be transported from the multi-valued setting to the ordinary category of topological spaces. We establish several basic homotopy results and formulate a multi-valued version of the homotopy category. We also discuss the relationship with classical homotopy theory, hyperspace topology, selection theory, and Lefschetz-type fixed point theory. The paper is intended as a foundational framework for studying homotopy-theoretic questions in settings where a point may have several possible images.

Abstract: We introduce a framework for studying homotopy theory through neutrosophic multi-valued functions. The basic idea is to combine the set-valued nature of multivalued mappings with the three-component description of truth, indeterminacy, and falsity characteristic of neutrosophic set theory. A neutrosophic multi-valued function is represented by a family of nonempty subsets together with three membership profiles describing truth, indeterminacy, and falsity. We define neutrosophic continuity, neutrosophic homotopy, neutrosophic homotopy equivalence, neutrosophic selectors, and admissible neutrosophic multi-valued functions. We establish elementary properties of neutrosophic homotopy, including reflexivity, symmetry, transitivity, compatibility with composition, and product constructions. We further introduce a Vietoris-type representation and use it to formulate induced maps on homology. The resulting framework contains ordinary homotopy theory and ordinary multivalued homotopy theory as special cases. It also provides a natural setting for describing topological phenomena in which several possible images coexist with different degrees of truth, indeterminacy, and falsity. The paper concludes with a proposed neutrosophic homotopy category and several problems concerning fundamental groups, higher homotopy groups, fixed point theory, and categorical extensions.

Abstract: We construct a mathematical framework for a proposed Timeless Yang–Mills Operator inspired by the timeless-space and variable quantum-vacuum-density program of Davide Fiscaletti and Amrit S. Šorli, together with the mass-gap proposal of Šorli and Čelan based on minimal vacuum excitation and the Planck–Einstein relation. The fundamental object is a time-independent gauge configuration space, while the numerical ordering of physical changes is not taken as a coordinate of the configuration space. We define a gauge-invariant vacuum functional on the quotient A/G, construct a formal factorized positive operator having this functional as its vacuum state, and formulate a gauge-reduced excitation operator for compact simple groups. Explicit SU(2) and SU(3) examples are developed using the corresponding Lie-algebra invariants. A proposed nonperturbative coercivity inequality is then introduced. We prove that this inequality implies a strictly positive spectral gap and establish stability under bounded gauge-invariant perturbations. The main result is a dedicated conditional theorem: if the proposed operator satisfies a precise list of self-adjointness, gauge-invariance, locality, reflection positivity, continuum-limit, nontriviality, and uniform coercivity assumptions, then Osterwalder–Schrader reconstruction yields a nontrivial quantum Yang–Mills theory with a Clay-type mass gap. We emphasize that the theorem is conditional and does not claim a solution of the Clay Millennium Problem. The hard unresolved step is to establish the required assumptions for the genuine four-dimensional continuum Yang–Mills theory.

Abstract: We answer six questions concerning ideal topological groups posed by Jafari and Rajesh in Questions and Answers in General Topology 44 (2026), 17–19. The questions concern minimality, sequential com-pleteness, precompactness, τ-precompactness, separately I-continuous multiplication, and jointly ideal-continuous multiplication. The first two questions require care because the generalized notion of I-continuity is weaker than ordinary continuity and therefore classical compactness arguments cannot simply be transferred without additional hypotheses. We formulate sufficient ideal-compactness conditions under which the desired conclusions follow. For Questions 3–6 we give explicit construc-tions. In particular, compact groups equipped with the ideal of finite subsets provide τ-precompact examples for every infinite cardinal τ. Furthermore, the finite group Z₃ with the non-discrete topology τ = ∅, 1, 2, Z₃ and ideal I = ∅, 0 provides a genuinely non-classical example: it is an ideal topological group although it is not an ordinary topological group. Consequently it supplies explicit examples for both separately ideal-continuous semitopological groups and jointly ideal-continuous paratopological groups.