Computer Science and Mathematics

Abstract: We introduce and investigate Einstein bitopological gyrogroups, namely Einstein gyrogroups endowed with two compatible topologies. The construction is motivated by the Einstein velocity addition law from special relativity and by the gyrogroup theory developed by Ungar. The additional topological structure permits one to distinguish two possibly different notions of continuity, convergence, compactness, and local regularity on the same Einstein gyrogroup. We first recall the Einstein addition law on the open unit ball and the associated gyrogroup structure. We then introduce bitopological Einstein gyrogroups and define pairwise continuous gyrogroup operations, pairwise continuous gyrations, and bitopological Einstein gyrohomomorphisms. Several basic properties are established, including pairwise continuity of translations, inversion, and gyrations. We further study pairwise Hausdorffness, pairwise compactness, separation properties, and the relationship between the two topologies. Examples are given in which the two topologies coincide and in which they are genuinely distinct. Finally, we discuss Einstein bitopological gyrovector spaces and indicate connections with hyperbolic geometry, Lorentz transformations, and future developments in differential and geometric structures on bitopological gyrogroups.

Abstract: We introduce and investigate a smooth version of a bitopological gyrogroup, which we call a bitopological Lie gyrogroup. The construction combines the algebraic structure of gyrogroups with two compatible smooth topologies on the same underlying manifold. The first topology describes one smooth structure, while the second provides an independent but compatible smooth structure. We formulate the notions of bitopological gyrogroup, Lie gyrogroup, bitopological Lie gyrogroup, smooth gyrohomomorphism, smooth gyration, and bitopological Lie subgyrogroup. Several structural properties are established. In particular, we study left translations, gyrations, tangent spaces, the two induced differential structures, compatibility of the two tangent bundles, and the behavior of smooth gyrohomomorphisms. We prove that the classical Lie group situation is recovered when all gyrations are trivial. We also discuss the Einstein and M”obius gyrogroups as fundamental examples and explain how their standard smooth topology gives rise to natural bitopological variants. The paper provides a differential-geometric framework for studying nonassociative smooth group-like structures with two interacting topologies. This framework is intended as a starting point for the development of Lie theory, invariant differential geometry, and representation theory for bitopological gyrogroups.

Abstract: Partially Negative Dimensional Product manifolds, abbreviated PNDP-manifolds, are a recently proposed class of geometric objects constructed from Einstein warped products in which the fiber is allowed to possess a virtual, or derived, negative dimension. The original construction was introduced by Pigazzini, Özel, Linker and Jafari [1]. The purpose of this paper is not to develop the most general theory of PNDP-manifolds, but to explain the basic ideas in an accessible way. We begin with ordinary manifolds, products, warped products and Einstein manifolds. We then explain why the phrase “negative dimension” should be interpreted carefully: it is not the dimension of an ordinary smooth manifold in the classical sense. Rather, it is a virtual dimension arising naturally in derived-geometric or obstruction-theoretic descriptions. We introduce the basic PNDP construction, derive its dimension formula, discuss the important zero-dimensional or “point-like” case, and give elementary examples. Finally, we explain how PNDP geometry is related to hidden dimensions, desuspension, Einstein geometry and possible applications. The paper is intended as a pedagogical introduction for readers who know elementary differential geometry but have never encountered PNDP geometry.

Abstract: We introduce and investigate the notion of a strongly star S-closed topological space. This notion is obtained by combining the closed-cover formulation of strongly S-closed spaces with the finite-star technique used in star covering properties. We establish several elementary properties of strongly star S-closed spaces, investigate their relationship with strongly S-closed spaces and star-compact spaces, and study hereditary properties, continuous images, closed subspaces, products, and separation conditions. We also give examples illustrating that strongly star S-closedness is genuinely weaker than strongly S-closedness. Finally, we formulate several questions concerning the relationship between strongly star S-closedness and other star covering properties.

Abstract: This paper introduces almost semi-Moscow topological groups, where the group operations are almost semi-continuous, and the space is almost semi-Moscow. Basic properties of almost semi-Moscow spaces and almost semi-Moscow topological groups are established. Characterizations via semi-\( \theta \)-open sets are obtained, and product stability is proved. A stronger subclass, the almost semi-Moscow almost topological groups, is introduced and studied. For this subclass, fundamental properties are established, closure identities are derived, and results on regularly open subgroups, quotient groups, and products are proved. Under extremally disconnectedness, it forms an almost Moscow topological group, yielding open Boolean subgroups and strong PT-group structure.

Abstract: We propose a geometric framework for studying Calabi–Yau type conditions on partially negative dimensional product manifolds (PNDP-manifolds). The construction is motivated by the PNDP framework introduced by Pigazzini, Özel, Linker and Jafari, in which the fiber is represented by a smooth manifold together with an obstruction bundle and consequently carries a virtual dimension that may be negative. The main idea of the present paper is to replace the ordinary tangent bundle of a PNDP-manifold by a formal virtual tangent object. For a PNDP model with base B and derived fiber represented by a pair (M, E), we consider TvirP = TB ⊕ TM − E. This leads naturally to a virtual first Chern class cvir 1 (P) = c1(TB) + c1(TM) − c1(E). We define a virtual Calabi–Yau PNDP condition by the vanishing of this class and formulate a stronger condition involving a compatible Kähler structure and a virtual Ricci-flat equation. Several examples and structural consequences are given. The paper is deliberately formulated as a proposed extension: the virtual topological conditions are precise in K-theoretic language, whereas the analytic theory of virtual Ricci-flat metrics is presented as a research program rather than as an established theorem.

Abstract: We introduce a framework for studying Lie algebroids on manifolds equipped with two topologies. The underlying smooth manifold is assumed to carry two topologies compatible with the same smooth structure, while the Lie algebroid data may be considered simultane- ously with respect to these topologies. We dene bitopological Lie algebroids as pairs of Lie algebroid structures over a common smooth base and investigate their anchors, brackets, morphisms, and cohomology. We dis- tinguish the case of two independent Lie algebroid structures from the stronger situation in which the two structures are identied by a vector bundle isomorphism. Examples involv- ing tangent Lie algebroids, action Lie algebroids, and cotangent Lie algebroids of Poisson manifolds are discussed. We also formulate compatibility conditions for pairs of Lie algebroid structures and establish elementary results concerning their sections and cohomology. The framework pro- vides a basis for further investigations of bitopological Lie groupoids, Lie bialgebroids, and geometric structures dened by two interacting dierential systems.

Abstract: We introduce and study a framework for Lie groupoids equipped with two smooth man- ifold structures on their spaces of objects and arrows. The associated topologies are allowed to dier, and the groupoid structure maps are required to satisfy appropriate smoothness conditions with respect to both structures. We formulate the basic definitions of bitopological manifolds, bitopological Lie groupoids, and morphisms between them. We investigate the relationship between the two topologies, the behavior of units and inverses, and the construction of the Lie algebroid associated with a bitopological Lie groupoid. Several examples are presented, including ordinary Lie groupoids, pair groupoids, and action groupoids. We also discuss restrictions, pullbacks, and compatibility conditions for the two induced Lie algebroid structures. The purpose of this paper is to provide a common framework for studying Lie groupoid structures in which two distinct topological or smooth structures play simultaneous roles.

Abstract: We introduce and investigate a differential-topological framework for manifolds equipped with two interacting topologies. A bitopological manifold is defined as a set endowed with two manifold topologies together with compatible smooth structures. The two topologies are allowed to be different, and consequently one obtains two notions of convergence, closure, interior, compactness, continuity, and smoothness. We formulate the notions of pairwise continuity, pairwise smoothness, bitopological charts, bitopological atlases, pairwise Hausdorffness, pairwise compactness, and pairwise connectedness. We establish basic properties of these structures and describe conditions under which the two topologies coincide. Several examples are given, including the diagonal case of an ordinary smooth manifold, manifolds arising from two diffeomorphic but distinct topologies, product constructions, and bitopological manifolds associated with ordered manifolds. The framework is motivated by the classical theory of bitopological spaces initiated by Kelly and developed subsequently through pairwise separation, quasi-uniformity, covering properties, and related constructions. The purpose of the present paper is to transfer these ideas to a differential-geometric setting and to provide a convenient language for future studies of geometry on spaces carrying two compatible topological structures.

Abstract: Cohn and Rajagopal conjectured that no six-dimensional kissing configuration with Szöllősi's configuration Q₅ or with their configuration R₅ as a cross section can contain 72 or more points. We prove that such a configuration has at most 70 points, and we exhibit configurations with 64 points. The argument combines three ingredients: an elementary bound on the heights of points above a cross section in terms of its covering constant; a positive-kernel bound on each side of the section, together with its equality case; and explicit polar and deep-hole certificates. The upper bound 70 and a manual polar certificate also appear in a manuscript of Iuzhakov and Vetrov; we incorporate their certificate with attribution, so the proof of this bound and the 64-point constructions can be checked without exhaustive computation. The same method shows that every 72-point kissing configuration in R⁶ with a D₅ or L₅ cross section is isometric to one of the four classical configurations, and that a seven-dimensional kissing configuration with an E₆ cross section has at most 126 points, with exactly two configurations attaining 126 up to isometry. We also prove explicit neighborhood bounds of 40, 72 and 126 in dimensions five, six and seven; for E₆ the radius is \( \sqrt{113/448}-1/2>1/450 \).

Abstract: Revised fuzzy metrics express separation through continuous triangular conorms and induce metrizable topologies with open and closed balls. We formulate fuzzy-metric roundness in the revised language and prove that the closure of every open revised fuzzy ball is contained in the corresponding closed ball. A two-point counterexample shows that equality can fail, correcting an overly general ball-closure assertion in the literature. We characterize nonroundness by attainment of a positive minimum by a radial map on an open set not containing its centre, and show that the standard revised fuzzy metric induced by a metric is round exactly when the original metric is round. We obtain factorwise characterizations of roundness for finite maximum products and of both roundness and completeness for countable weighted maximum products. We also prove that every nontrivial revised fuzzy metric admits a topologically equivalent nonround truncation. After establishing the separation-based Cantor characterization of completeness, we derive local fixed-point principles from decreasing families of invariant round balls with vanishing revised fuzzy diameter. Unlike algebraic contraction theorems, these principles require no prescribed global contractive inequality. A nonlinear example shows that the method applies to a self-map that is not a Banach contraction on the relevant invariant domain.

Abstract: Let \((X,\mathcal{R})\) be a relator space and let \(\tau_{\mathcal{R}}\) denote the family of relator-open subsets of \(X\). We investigate the family \(hO(X,\mathcal{R})\) of \(h\)-open subsets defined by the condition \[ A \subseteq \operatorname{int}_{\mathcal{R}}(A \cup V) \] for every nonempty proper relator-open set \(V\). The purpose of this paper is to formulate structural conditions on \(\mathcal{R}\) under which \(hO(X,\mathcal{R})\) is a topology, to characterize the equality \[ hO(X,\mathcal{R}) = \tau_{\mathcal{R}}, \] and to study when the \(h\)-open family is strictly finer than the relator topology. We establish several sufficient conditions in terms of the relator interior operator and its interaction with finite intersections. We also introduce \(h\)-separation axioms \(h\)-\(T_0\), \(h\)-\(T_1\), \(h\)-\(T_2\), \(h\)-\(R_0\), and \(h\)-\(R_1\) directly for relator spaces. The associated topology \((X, hO(X,\mathcal{R}))\) provides a natural framework for these axioms. The results connect the theory of relators with the classical theory of generalized topologies and with the recent development of \(h\)-open sets.

Abstract: We introduce and investigate a relator-theoretic version of h-open sets. A relator space is a pair X(R) = (X,R), where R is a nonempty family of binary relations on X. Relator spaces provide a common framework for uniform spaces, ordered structures, generalized topological spaces and several relational constructions. We define the relator interior and use it to introduce h-open subsets relative to a relator. We establish basic permanence properties, characterize the associated h-interior, and give sufficient conditions under which the family of all relator h-open sets forms a topology. We further show that the construction contains the classical theory of h-open sets as a special case. Several examples demonstrate the distinction between ordinary relator openness and h-openness. Finally, we discuss monotone relator maps and the behavior of h-open sets under inverse images. The results suggest that the h-construction is naturally viewed as an operation on relational structures rather than only on ordinary topological spaces.

Abstract: We study corrected characteristic transport along composite algebraic-geometric constructions. For an admissible geometric operation \(F\), a characteristic transformation \(\Phi\), and its Riemann--Roch-corrected operator \(F_{\Phi,*}^{\mathrm{RR}}\), the discrepancy \(\Delta_\Phi(F;\xi):=\Phi(F_!\xi)-F_{\Phi,*}^{\mathrm{RR}}(\Phi(\xi))\) defines a characteristic transport defect. Starting from established one-stage Riemann--Roch, Verdier specialization, Gysin, and motivic characteristic-class formulas, we develop composition and support-propagation laws for defects along finite geometric routes. In the singular setting, Sch{\"u}rmann's Verdier--Riemann--Roch defect propagates through later proper, smooth, and compatible Gysin stages; for a nodal plane cubic, the vanishing-cycle contribution is computed explicitly as \(-\mathbf 1_{\{p\}}\mapsto[p]\mapsto[\mathrm{pt}]\). We organize stagewise data into comparative transport profiles and give support-graded criteria for distinguishing routes whose actions agree on a prescribed cohomological sector. For transfer--return correspondences, an explicit Jacobian family arising from finite covers of curves of genus at least two yields residuals in \(\ker(\rho_{L_A})\setminus\ker(\operatorname{cl})\). We also establish corrected smooth--Gysin compatibility in Cartesian squares. The resulting formalism is diagnostic and factorization-sensitive rather than algebraicity-producing.

Abstract: We introduce and investigate a separation property that we call an R1-Urysohn property. A topological space (X, τ ) is called R1-Urysohn if, whenever x, y ∈ X satisfy cl x ̸= cl y, there exist disjoint open sets U, V ∈ τ such that cl x ⊆ clU and cl y ⊆ cl V. The definition is formulated in terms of closures of singleton sets, and therefore naturally interacts with the classical R0 and R1 separation axioms. We establish basic relationships with R0, R1, Hausdorff, regular, and Urysohn spaces. We prove that every Urysohn space is R1-Urysohn and that every R1 space is R1-Urysohn. Several examples are given to show that the new condition does not automatically imply the classical R1 or Hausdorff properties. We also investigate the behavior of the property under subspaces, products, continuous maps, and quotient constructions.

Abstract: Let H(A) be the Hattori space associated with A ⊆ R, and put B = R \ A. Kulesza announced at the 2018 Joint Mathematics Meetings that Lindelöfness of H(A)²ⁿ forces Lindelöfness of H(A)²ⁿ⁺¹ for every positive integer n. The meeting abstract states this implication without a proof. This paper proves the case n = 1 in ZFC. H(A)² is Lindelöf ⇒ H(A)³ is Lindelöf. The proof passes from an uncountable closed discrete subset of the cube to a perfect set with injective coordinate projections and uses the one-sided local bases to obtain three cocountable Pareto-maximality sets with countable triple intersection. The same one-sided geometry gives a pointwise half-space criterion for affine discreteness and, when H(A)² is Lindelöf, a four-coordinate obstruction for uncountable closed discrete monotone traces. Known results on mixed Sorgenfrey–Euclidean products lead to hereditary-Lindelöf criteria for finite powers, a countable-power characterization, and a finite-power Lindelöf reduction. For every n ∈ N, the power H(A)ⁿ is hereditarily Lindelöf if and only if the Sorgenfrey power Bⁿ is Lindelöf, and H(A)^ω is hereditarily Lindelöf if and only if every finite Sorgenfrey power of B is Lindelöf. A transfinite construction gives A ⊆ R for which H(A)² and H(A)³ are Lindelöf, whereas H(A)⁴ contains a closed discrete affine line of cardinality \( \mathfrak c. \) For this ZFC example, B² is not Lindelöf and the second and third Hattori powers are not hereditarily Lindelöf.

Abstract: We introduce a framework for studying groups and groupoids equipped with two Polish topologies on their underlying sets. The principal objective is to combine the structural properties of topological groups and topological groupoids with the descriptive-set-theoretic features of Polish spaces. We dene Polish bitopological groups, Borel-compatible Polish bitopological groups, and Polish bitopological groupoids. We establish elementary permanence properties, formulate criteria for closed subgroups and products, and investigate the interaction between the two induced Borel structures. For groupoids, we examine the Polish structure of the space of composable arrows and the continuity of the structural maps with respect to each topology. Examples and counterexamples clarify the distinction between having two Polish topolo- gies and having two topologies that generate the same Borel sets. We conclude with several questions concerning descriptive set theory, orbit equivalence relations, and the extension of the framework to Polish groupoid actions.

Abstract: Hyperbolic geometry explains many structural features of real-world networks, yet existing hyperbolic generative network models with community organization such as the nonuniform popularity-similarity optimization (nPSO) are restricted to unipartite graphs. Here we introduce a bipartite extension of the nPSO model called nBPSO, motivated by the prevalence of community organization in real-world bipartite networks and the emerging evidence of hyperbolic hierarchical connectivity in trained sparse neural networks. nBPSO provides explicit control over link density, degree distribution, average clustering, and community organization. We engineer the model to generate consecutive and concatenated bipartite layers, enabling initialization of dynamic sparse training multilayer perceptron neural networks with tunable hyperbolic organization for each layer. We find that task-agnostic (unaligned with the task) hyperbolic random network initialization degrades the performance of sparse neural networks compared with uniform random initialization. This degradation arises because the hyperbolic random initialization imposes an arbitrary hierarchical prior into the sparse topology. Dynamic sparse training may then need to optimize the weights and reorganize this arbitrary task-agnostic hierarchy into a task-driven one. By contrast, a uniform random initialization imposes at start less structural bias and may therefore be easier to reconfigure during training toward the hierarchical topology suited to the task. This work motivates the need for data-informed hyperbolic initialization methods which generate sparse topologies with a hierarchical prior closer to the task, facilitating the further learning of a task-oriented hierarchical organization during dynamic sparse training. In addition, nBPSO represents the first hyperbolic bipartite network generative model that allows the mesoscale community structure to be explicitly controlled through model parameters, making it a useful tool for broader complex systems applications where community organization is central to accurate and interpretable modeling.

Abstract: We introduce a framework for studying BatalinVilkovisky (BV) algebras associated with bitopological spaces. A bitopological space is a triple (X, τ₁, τ₂) consisting of a set X endowed with two topologies. We dene a bitopological BV-algebra as a pair of BV- algebra structures connected by specied comparison maps, and we formulate compatibility conditions for the products, BV operators, and induced Gerstenhaber brackets. We establish elementary structural results concerning morphisms, diagonal structures, and the recovery of ordinary BV-algebras. We then discuss applications to free loop spaces and string topology. In particular, we explain how two topologies on a manifold may lead to two loop-space homology theories and identify additional conditions needed to compare their string topology operations. The framework separates formal algebraic consequences from geometric existence questions.

Abstract: We introduce and investigate a class of bitopological spaces whose two component topologies are Polish. Such spaces provide a natural meeting point between the theory of bitopological spaces and descriptive set theory. We distinguish several levels of Polishness: componentwise Polishness, common Polishness, and pairwise Polishness. Basic separation, countability, Baire-category, product, subspace, and mapping properties are established. We also investigate the interaction between the two Borel structures and show that, under natural compatibility assumptions, the identity maps between the two topological structures are Borel isomorphisms. Particular attention is given to quasi-metrizable bitopological spaces, since a quasi-metric naturally determines two opposite topologies. Finally, examples are presented showing that the two component topologies may be Polish while their common join need not be Polish without additional hypotheses.