Abstract: Buckled Rings, also known as Pressurized Elastic Circles, were first studied by Maurice Lévy , and then by Halphen and Greenhill . These planar curves can be described as critical points for a variational problem, namely the integral of a quadratic polynomial in the geodesic curvature of a curve. Thus they are a generalization of elastic curves, and they are solitary wave solutions to a flow in the (three-dimensional) filament hierarchy. An example of such a curve is the Kiepert Trefoil, which has three leaves meeting at a central singular point. Such a variational problem can be considered for curves in other surfaces. In particular, a recent paper of I. Castro et al found many examples of such curves in the unit sphere. In this article, which is in part an extension of that paper, we consider a new family of such curves, having a discrete dihedral symmetry about a central singular point. That is, these are spherical analogues of the Kiepert curve. We determine such curves explicitly using the notion of a Killing field, which is a vector field along a curve which is the restriction of an isometry of the sphere. The curvature k of each such curve is given explicitly by an elliptic function. If the curve is centered at the south pole of the sphere and has minimum value ρ, then k−ρ is linear in the height above the pole.

Abstract: In this article, we explored and investigated a novel class of fuzzy sets, called k-fuzzy γI-open (k-FγI-open) sets in fuzzy ideal topological spaces (FITSs) based on Sostak՚s sense. The class of k-FγI-open sets is contained in the class of k-fuzzy strong β-I-open (k-FSβI-open) sets and contains all k-fuzzy pre-I-open (k-FPI-open) sets and k-fuzzy semi-I-open (k-FSI-open) sets. We also introduced and studied the interior and closure operators with respect to the classes of k-FγI-open sets and k-FγI-closed sets. However, we defined and discussed novel types of fuzzy I-separation axioms using k-FγI-closed sets, called k-FγI-regular spaces and k-FγI-normal spaces. Thereafter, we displayed and studied the notion of fuzzy γI-continuity (FγI-continuity) using k-FγI-open sets. Furthermore, we presented and characterized the notions of fuzzy weak γI-continuity (FWγI-continuity) and fuzzy almost γI-continuity (FAγI-continuity), which are weaker forms of FγI-continuity. Finally, we introduced and investigated some new fuzzy γI-mappings via k-FγI-open sets and k-FγI-closed sets, called FγI-open mappings, FγI-closed mappings, FγI-irresolute mappings, FγI-irresolute open mappings, and FγI-irresolute closed mappings.

Abstract:

In the present article, we define and investigate the notion of r-fuzzy γ-open (r-F-γ-open) sets as a generalized novel class of fuzzy open (F-open) sets on fuzzy topological spaces (F T Ss) in the sense of Šostak. This class is contained in the class of r-F-β-open sets and contains all r-F-pre-open and r-F-semi-open sets. However, we introduce the interior and closure operators with respect to the classes of r-F-γ-open and r-F-γ-closed sets, and study some of their properties. After that, we define and discuss the notions of F-γ-continuous (respectively (resp. for short) F-γ-irresolute) functions between F T Ss (M, ℑ) and (N, F). Also, we display and investigate the notions of F-almost (resp. F-weakly) γ-continuous functions, which are weaker forms of F-γ-continuous functions. We also showed that F-γ-continuity ⇒ F-almost γ-continuity ⇒ F-weakly γ-continuity, but the converse may not be true. Next, we present and characterize new F-functions via r-F-γ-open and r-F-γ-closed sets, called F-γ-open (resp. F-γ-irresolute open, F-γ-closed, F-γ-irresolute closed, and F-γ-irresolute homeomorphism) functions. The relationships between these classes of functions were discussed with the help of some examples. We also introduce some new types of F-separation axioms, called r-F-γ-regular (resp. r-F-γ-normal) spaces via r-F-γ-closed sets, and study some properties of them. Lastly, we explore and discuss some new types of F-compactness, called r-F-almost (resp. r-F-nearly) γ-compact sets using r-F-γ-open sets.

Abstract: Utilizing the concept of rough upper approximation neighborhood systems in simple graphs, this paper introduces a novel class of topologies on vertex sets. We delve into identifying the specific graphs that induce either the indiscrete or discrete topology, unraveling essential topological properties within this classification. Exploring further, we delve into the continuity and isomorphism of graph mappings. Subsequently, we apply these findings to enhance radar chart graphical methods through the analysis of corresponding graph structures.

Abstract:

Alternative proof is given for an earlier presented result that if a link in 3-space bounds a proper oriented surface (without closed component) in the upper half 4-space, then the link bounds a proper oriented ribbon surface in the upper half 4-space which is a renewal embedding of the original surface.

Abstract: In this paper, we prove that the complex four dimensional compact holomorphic symplectic manifold we found earlier is not formal. This gives another strong consequence that it is not a topological Kahler manifold. We also conjecture that this is true for the higher dimensional ones.

Abstract: Urban planning is a critical discipline that influences the development and sustainability of cities. In this paper, we describe a holistic methodology for automatically extracting, mapping, and visualizing building footprints in Karachi using Python on Google Colab. This method uses cloud-based technologies to automate the extraction of data from open-source satellite imagery, process this into meaningful spatial formats, and visualize building footprints with minimal human intervention. The methodology is for GIS developers, urban planners, and other stakeholders in urban management, giving a scalable and efficient solution to the complex task of urban spatial analysis. The GoogleBuildingMiner tool, within the mapminer allows easy access to building footprint data by querying the Google Maps API and other relevant sources. The successful extraction of spatial data over 1 km areas compatible with any GIS software demonstrates the wide applicability of the method for extracting the building footprint on a global scale. The results show how much Python-driven automation might increase the accuracy of the data extracted, reduce the time involved in manual digitization, and standardize the output formats into an acceptable format for subsequent analyses and decision-making purposes.

Abstract:

Topological genomics offers a novel framework for understanding how genomic structures influence neural circuit differentiation, integrating principles of differential topology and statistical mechanics homology. This approach examines how genomic topological invariants, such as persistent loops and Betti numbers, correspond to functional transformations in neural circuits. By employing persistent homology to analyze genomic interaction maps and differential topology to model neural circuit formation, we identify key transitions that govern differentiation. Statistical mechanics complements this framework by modeling energy landscapes and phase transitions that reflect the emergent properties of neural architectures. Together, these interdisciplinary methods elucidate the role of topological features in genetic regulation and neural circuit specialization, with implications for neurodevelopment, pathology, and artificial intelligence.

Abstract: It is well known that the four ancient Greek triangle centers and others have homogeneous barycentric coordinates that are polynomials in the sidelengths a; b; c of a triangle ABC. For example, the circumcenter is represented by the polynomial a(b2 + c2 - a2). It is not so well known that triangle centers having barycentric coordinates such as tanA : tanB : tanC are also representable by polynomials, in this case, by p(a; b; c) : p(b; c; a) : p(c; a; b), where p(a; b; c) = a(a2 + b2 - c2)(a2 + c2 - b2). This paper presents and discusses polynomial representations of triangle centers that have barycentric coordinates of the form f(a; b; c) : f(b; c; a) : f(c; a; b), where f depends on one or more of the functions in the set fcos; sin; tan; sec; csc; cotg. The topics discussed include innite trigonometric orthopoints, the n-Euler line, and symbolic substitution.

Abstract: The Bloch sphere is a powerful geometrical representation of qubit states, fundamental to understanding quantum information theory. This paper explores the stereographic projection of the Bloch sphere, focusing on its utility in addressing decoherence and quantum error correction, two major challenges in quantum computing. By mapping the complex quantum states onto a two-dimensional plane, the stereographic projection simplifies the visualization and manipulation of qubit transformations. We investigate how this projection aids in identifying and correcting errors that arise from decoherence---such as phase flips and bit flips---while maintaining the integrity of quantum states. Additionally, we investigate the influence of decoherence on qubits through this projection, demonstrating how error correction protocols can be optimized by mapping errors onto the complex plane. The analysis highlights the potential of stereographic projection in developing more robust strategies for mitigating decoherence, thereby advancing the overall performance of quantum systems.

Abstract:

Let $\mathcal{M}(\text{Spin}(8,\mathbb{C}))$ be the moduli space of $\text{Spin}(8,\mathbb{C})$-Higgs bundles over a compact Riemann surface $X$ of genus $g\geq 2$. It admits a system, called Hitchin integrable system, induced by the Hitchin map, whose fibers are Prym varieties. Also, the triality automorphism of $\text{Spin}(8,\mathbb{C})$ acts on $\mathcal{M}(\text{Spin}(8,\mathbb{C}))$ and those Higgs bundles that admit a reduction of structure group to $G_2$ are fixed points of this action. This defines a map of moduli spaces of Higgs bundles $\mathcal{M}(G_2)\rightarrow\mathcal{M}(\text{Spin}(8,\mathbb{C}))$. In this work, the action of the triality automorphism is extended to an action on the Hitchin integrable system associated to $\mathcal{M}(\text{Spin}(8,\mathbb{C}))$. In particular, it is checked that the map $\mathcal{M}(G_2)\rightarrow\mathcal{M}(\text{Spin}(8,\mathbb{C}))$ restricts to a map at the level of the Prym varieties induced by the Hitchin map. Necessary and sufficient conditions are also provided for the Prym varieties associated with the moduli spaces of $G_2$ and $\text{Spin}(8,\mathbb{C})$-Higgs bundles to be disconnected. Finally, some consequences are drawn from the above results in relation to the geometry of the Prym varieties involved.

Abstract: Our main result is that chaos in dimension $n+1$ is a one-dimensional geometrical object embedded in a geometrical object of dimension $n$ which corresponds to a $n$ dimensional object which is either singular or non-singular. Our main result is then that this chaos occurs in the first case as either on an isolated or non-isolated singularity. In the first case this chaos is either boundary chaos or spherical chaos which is what happens also in the non-singular case. In the case of an isolated singular geometry one has chaos which can either be boundary, spherical or tubular chaos. We furthermore prove that the prime numbers display quantum behaviour.

Abstract: The amount of solar energy produced is directly proportional to the surface area of solar panels exposed to sun rays. To increase the productivity of solar panels, a number of techniques are proposed by curving the surface or protruding the panel surface to maximise the surface area, e.g. Protruding panel surfaces with shapes such as pyramidal, cone, spherical cap, sinusoidal wavy curve, etc. These techniques are analysed to determine whether the panel surface area increases. The geometrical design analysis showed that, depending on the chosen surface curving method, the panel can achieve a remarkable increase in surface area.The system advisor model was used to assess a sample of solar cells with standard measurements of 6 by 6 inches that were organised in a zigzag pattern with different angles of inclination (10°, 15°, 20°, 25°, 30°, and 35°). Overall, the maximum power pump and annual AC energy performance increase by 1.4%, 4.2%, 6.8%, 10.8%, 14.9%, and 21.6%, respectively, over a one-year period of productivity. However, by applying the concept of solar panel efficiency degradation due to tilting solar panels at an angle by Mamun et al. (2022), for each 5° increase in tilt angle beyond the ideal 10°, the efficiency decreases by approximately 0.76%. In conclusion, the overall efficiency is greater than that of standard flat solar panels, provided the tilt angle is within a reasonable range.

Abstract: This paper explores the properties of projective vector fields on semi-Riemannian manifolds. The main result establishes that if a projective vector field η on such a manifold is also a conformal vector field with potential function ψ, then η must either be homothetic, or the vector field ζ, which is dual to dψ, is a light-like vector field. Additionally, it is shown that a complete Riemannian manifold admits a projective vector field that is also conformal and non-Killing if and only if it is locally Euclidean. The paper also presents other results related to the characterization of Killing and parallel vector fields using the Ricci curvature and the Hessian of the function given by the inner product of the vector field.

Abstract: This paper explores a novel approach to establishing the topological equivalence of manifolds embedded in ℝ

Abstract: In this paper we investigate the differential topological properties of a subclass of singular space--subcarteisan spaces. First, we get further result on partition of unity for differential spaces. Second, we establish the tubular neighborhood theorem for subcartesian spaces with constant structural dimension. Third, we generalize the concept of Morse functions on smooth manifolds to differential spaces. For subcartesian space with constant structural dimension, we provide examples of Morse functions; With the assumption that the subcartesian space can be embedded as a bounded subset of an Euclidean space, we prove that smooth bounded functions on this space can be approximated by Morse functions; We study the infinitesimal stability of Morse functions on subcartesian spaces. Classical results on Morse functions on smooth manifolds can be treated directly as corollaries of our results here.

Abstract: This study presents two interconnected mathematical models: one describing the stability and bifurcation dynamics of ecological biomes, and another simulating the progressive deformation and structural failure of a toroidal object. Both models explore topological changes in complex systems, bridging concepts from ecology, materials science, and mathematical physics. The first part introduces a dynamic model of ecological biomes, incorporating seasonal oscillations and external pressures. We define equations for topology updates, seasonal effects, and pressure impacts, culminating in a bifurcation condition that represents a critical shift in the biome's equilibrium state. This model provides insights into the resilience and tipping points of ecological systems under varying environmental stresses.The second part extends these concepts to a three-dimensional toroidal structure. We present parametric equations describing the torus geometry and develop a time-dependent deformation model that includes both stochastic and deterministic elements. A novel twisting deformation component enhances the realism of the structural changes. We define a critical threshold for maximum displacement, beyond which the torus undergoes structural failure, analogous to the bifurcation point in the biome model. Both models are implemented computationally, providing visual representations of the systems' evolution over time. The biome model illustrates the path to and consequences of bifurcation, while the torus simulation depicts the stages of deformation leading to structural break. This work contributes to the understanding of topological changes and critical transitions in both ecological and physical systems. By presenting these models in parallel, we highlight the universal nature of stability, bifurcation, and failure across different domains. The methodologies presented here have potential applications in ecology, materials science, structural engineering, and theoretical physics, offering a framework for studying the limits of system integrity under stress.

Abstract: In this paper the one-to-one correspondences and equivalences between ideals, primals, filters and grills are introduced. It is shown the local functions and the topological spaces induced by them are the same. From this point of view, the topological properties with respect to one topology can be derived from topological properties valid in the corresponding topology.

Abstract: This paper explores the geometric impossibility of angle trisection within Euclidean geometry, with a particular focus on the 90° angle. It presents a rigorous proof that highlights the fundamental limitations of various approaches to angle trisection, highlighting their failure to achieve universality. The paper critiques modern proofs that allow for the trisection of certain angles while dismissing others, arguing that such methods are inherently flawed due to their lack of geometric consistency. The paper introduces a new perspective through a new property named “inconsistent property”, a concept derived from geometric operations involving proportional magnitudes. This property reveals contradictions within the framework of Euclidean geometry, paralleling the logical challenges faced in proving the impossibility of trisecting an arbitrary angle. By drawing comparisons between traditional and contemporary proofs, the paper demonstrates that inconsistencies arise not only from specific cases but also from broader geometric principles. The findings reaffirm the robustness of Euclidean geometry in addressing the trisection problem and challenge the validity of alternative proofs that do not adhere to universal geometric principles. This paper contributes to a clearer understanding of the limitations of angle trisection methods and encourages further investigation into the technical implications and broader impact of the angle impossibility proof.

Abstract: The grand antiprism A is an outlier among the uniform 4-polytopes, since it is not obtainable from Wythoff’s construction. Its symmetry group G(A) has been incorrectly described as [[10,2+,10]] or even as an `ionic diminished Coxeter group’. In fact, G(A) is another group of order 400, namely the group ±[D10×D10]·2, in the notation of Conway and Smith.