Abstract Symplectic integrators with long-term preservation of integrals of motion are introduced for the guiding-center motion of plasma particles in toroidal magnetic fields of general topology. An efficient transformation to canonical coordinates from cylindrical and flux-like coordinates is discussed and applied using one component of the magnetic vector potential as a spatial coordinate. This choice is efficient in both, theoretical and numerical developments and marks a generalization of magnetic flux coordinates. The transformation enables the application of conventional symplectic integration schemes formulated in canonical coordinates, as well as variational integrators on the guiding-center system, without requiring magnetic flux coordinates. Symplectic properties and superior efficiency of the implicit midpoint scheme compared to conventional non-symplectic methods are demonstrated on perturbed tokamak fields with magnetic islands and stochastic regions. The presented results are benchmarked against symplectic integration in canonicalized flux coordinates and Boozer coordinates on energetic particle losses in 3D magnetic configurations.

Hybrid High-Order Approximations of Div-Curl Systems on Domains with General Topology

Machine failures can generally be divided into soft failures caused by the normal usage of the equipment itself and hard failures caused by external impacts. Soft failures and hard failures may also affect each other, meaning that external impacts can accelerate the degradation of the equipment's performance and make it more prone to failures as the number of impacts increases. We introduce a new maintenance and reliability model which considers the dependencies between hard and soft failure modes for individual machine types, allowing hard failure thresholds to be dynamically adjusted. This model, based on a manufacturing system of general topology, not only incorporates degradation-shock dependence, but also includes a reliability constraint. The model is set up such that once the degradation level of any machine exceeds a predetermined threshold, maintenance and repair actions, both imperfect and perfect, will be executed. In order to determine the optimal thresholds for maintenance and upkeep strategy, we develop a simulation optimisation algorithm, known as the Jaya-Imperialist Competitive Algorithm (JICA). This method is based on the Imperialist Competitive Algorithm (ICA) and combines the Jaya Algorithm with an adaptive penalty function. Empirical analysis has confirmed that, compared to a set of benchmark algorithms, the algorithm proposed in this study can obtain a superior maintenance and repair strategy while demonstrating greater efficiency. We also perform sensitivity analysis on several important parameters, which can facilitate a deeper understanding of the optimisation model and support optimal decision-making.

Spiking Neural Networks (SNNs) exhibit their optimal information-processing capability at the edge of chaos, but tuning them to this critical regime in reservoir-computing architectures usually relies on costly trial-and-error or plasticity-driven adaptation. This work presents an analytical framework for configuring in the critical regime a SNN-based reservoir with a highly general topology. Specifically, we derive and solve a mean-field equation that governs the evolution of the average membrane potential in leaky integrate-and-fire neurons, and provide an approximation for the critical point. This framework reduces the need for an extensive online fine-tuning, offering a streamlined path to near-optimal network performance from the outset. Through extensive numerical experiments, we validate the theoretical predictions by analyzing the network’s spiking dynamics and quantifying its computational capacity using the information-based Lempel-Ziv-Welch complexity near criticality. Finally, we explore self-organized quasi-criticality by implementing a local homeostatic learning rule for synaptic weights, demonstrating that the network’s dynamics remain close to the theoretical critical point. Beyond AI, our approach and findings also have significant implications for computational neuroscience, providing a principled framework for quantitatively understanding how (neuro)biological networks exploit criticality for efficient information processing. The paper is accompanied by Python code, enabling the reproducibility of the findings.

The article examines the problem of using the ideas of set theory and general topology in philosophy on the example of book by Japanese philosopher Tanabe Hajime (1885–1962) The Development of Historicism in Mathematics. It is known that topological models have been successfully applied in physics in quantum field theory. Tanabe Hajime, along with Pavel Florensky and Vladimir Ern, was one of the first to raise the question of the applicability of topology in the social and humanitarian sciences, generating with its help a model of history. Tanabe uses the tools of topology, such as the concepts of a set, its density, neighborhood, con­tinuum, section, etc., to describe and explain the mechanisms underlying social reality not as a spatial phenomenon, but as a historical time continuum. The article focuses on such components of Tanabe’s philosophy as the concept of the conti­nuity of time and history, the features of understanding the present and the relation­ship between time modes, criticism of the unification of time and space, and the con­cept of historicism. The result that the author of the article came to is the clarification of the role and place of mathematical terms and theories in the philosophical system of Tanabe Hajime, who was not limited to the use of mathematical analogies, but also actually raised the question of mathematical modeling of history.

Abstract Tubular Structures (TS) play a central role in the human body due to their wide range of applications. Many native structures consist of TS with different properties. These structures fulfill a variety of biological and mechanical functions. In some cases these structures need to be replaced. SMES (melt electrospinning) offers a process to manufacture artificial TS. The employed 3D printer (X350pro, German RepRap), an SMES process, is modified by connecting a negative highvoltage source to a collector on the print bed while the nozzle is grounded. For the manufacture of TS the collector is a rotating metal tube. A Scanning electron microscope (SEM) (TM-1000, Hitachi) is used to obtain information on the general fiber topology and, in particular, the fiber bonding. The fiber diameters are determined with a fiber analysis program (MAVIfiber2d, Fraunhofer Institute) using SEM images of sputtered (Coater SCD 040, Balzers Union Limited) samples. The topography change of TS by varying the extrusion multiplier and the rotational speed is examined macroscopically. In order to evaluate the process stability of the build-up, a print series is examined. For the optimized standard print, reliably closed TS can be produced with an average fiber diameter in the range of 30.43 to 35.28 μm. The fiber topology can be modified by changing parameters like speed and extrusion multiplier. The developed manufacturing process for tubular nonwoven structures thus offers a versatile manufacturing process that can be investigated in more detail for specific TS in medical applications.

Abstract We develop a finite element method for an elliptic Maxwell boundary value problem on polyhedral domains in ℝ 3 {\mathbb{R}^{3}} with a general topology. Our method is based on a Hodge decomposition approach that leads to standard scalar elliptic problems and elliptic saddle point problems for vector potentials that have previously been investigated in the study of fluid flow problems. We carry out an error analysis that does not involve assumed regularity of the solution and present corroborating numerical results.

We study the emergent behaviors of the Lohe model on the unit hyperbolic sphere H d \mathbb {H}^d under more general interconnection topologies. In previous literature, the Lohe model on H d \mathbb {H}^d was studied under the complete graph. Using the LaSalle invariance principle, we show that the Lohe model on H d \mathbb {H}^d converges to a synchronized state for directed graphs containing spanning trees if all oscillator frequencies are equal. Moreover, we show that exponential convergence is achieved by analyzing the error dynamics of the Lohe model on H d \mathbb {H}^d . We also provide several numerical simulations and compare them with theoretical results .

A space [Formula: see text] is called [Formula: see text]-normal (respectively, [Formula: see text]-normal) if there exist a normal space [Formula: see text] and a bijection [Formula: see text] such that [Formula: see text] is a homeomorphism for any cellular-compact (resp. [Formula: see text]-bounded) subset [Formula: see text] of [Formula: see text]. We investigate the relationship between [Formula: see text]-normal space and [Formula: see text]-normal space and also study the basic topological properties of [Formula: see text]-normal and [Formula: see text]-normal space. In the course of this study, we prove the submetrizable spaces are [Formula: see text]-normal which answers affirmatively, a question raised earlier, and yields a very simple proof of the fact [Formula: see text]-space and Niemytzki plane are [Formula: see text]-normal, proved quite painstakingly in Applied General Topology, Vol 21, No. 1 (2020). A large class of spaces that are [Formula: see text]-normal is given here.

Abstract We study quantum gravity corrections to the no-boundary wavefunction describing a universe with spatial topology S 1 × S 2. It has been suggested that quantum effects become increasingly important when the size of the circle is large relative to the sphere. In this paper, we confirm this claim by an explicit four-dimensional one-loop calculation of the gravitational path integral preparing such a state. In the process, we clarify some aspects of the gravitational path integral on complex spacetimes. These quantum corrections play a crucial role in ensuring that the norm of the wavefunction is naturally expressed in terms of a path integral over S 2 × S 2 at the classical level. We extend some of the analysis to more general spatial topologies, as well as to the inclusion of fermions.

We employ well-known concepts from statistical physics, quantum field theories, and general topology to study magnetic reconnection and topology change and their connection in incompressible flows in the context of an effective field theory without appealing to magnetic field lines. We consider the dynamical system corresponding to wave packets moving with Alfvén velocity x[over ̇](t):=V_{A}(x,t) whose trajectories x(t) define pathlines, which naturally provides a mathematical way to estimate the rate of magnetic topology change. A considerable simplification is attained, in fact, by directly employing well-known concepts from hydrodynamic turbulence without appealing to the complicated notion of magnetic field lines moving through plasma, which may prove even more useful in the relativistic regime. Continuity conditions for magnetic field allow rapid but continuous divergence of pathlines, shown to imply reconnection, but not discontinuous divergence, which would change topology. Thus, topology can change only due to time-reversal symmetry breaking, e.g., by dissipative effects. In laminar and even chaotic flows, the separation of pathlines at all times remains proportional to their initial separation, argued to correspond to slow reconnection, and topology changes by dissipation with a rate proportional to resistivity. In turbulence, pathlines diverge superlinearly with time independent of their initial separation, i.e., fast reconnection, and magnetic topology changes by turbulent dissipation with a rate independent of small-scale plasma effects. The crucial role of turbulence in enhancing topology change and reconnection rates originates from its ability to break time-reversal invariance and make the flow superchaotic. In fact, due to the loss of Lipschitz continuity of the magnetic field in turbulence, pathlines separate superlinearly even if their initial separation tends to vanish, unlike deterministic chaos. This superchaotic behavior is an example of spontaneous stochasticity in statistical physics, sometimes called the real butterfly effect in chaos theory to distinguish it from the butterfly effect, in which trajectories can diverge exponentially only if initial separation remains finite. If 3D reconnection is defined as magnetic topology change, it can be fast only in turbulence where both reconnection and topology change are driven by spontaneous stochasticity, independent of any plasma effects. Our results strongly support the Lazarian-Vishniac theory of turbulent reconnection.

With a growing number of quantum networks in operation, there is a pressing need for performance analysis of quantum switching technologies. A quantum switch establishes, distributes, and maintains entanglements across a network. In contrast to a classical switching fabric, a quantum switch is a two-sided queueing network. The switch generates Link-Level Entanglements (LLEs) across links that connect users to the switch, which are then fused to process the network's entanglement requests. First, we characterize the capacity region that is defined as the set of entanglement request rates for which there exists a scheduling policy stabilizing the system. We then show that a sequence of Max-Weight policies that we propose achieve throughput optimality in the asymptotic sense. Our proof techniques analyse a two-time scale separation phenomenon at the fluid scale for a general switch topology. This allows us to demonstrate that the optimal fluid dynamics are given by a scheduling algorithm that solves a certain average reward Markov Decision Process.

In this paper, we present the concept of GN-topological space by introducing a topology defined on a subfamily of the family consisting of all convergent nets in any general topological space. Inspired by the method concept, which is a type of function whose domain is a set of convergent sequences and whose range is a set of real numbers, we define a different type of function, which we call the N-method, whose range is a general topological space and whose domain is the GN-topological space defined on this topological space. After stating the elements that led us to conduct this research and the sources of inspiration for us, we examine the properties of some types of continuities on the concept of the N-method, whose basis is the concept of an h-open set, and reveal the relations between them. In the following section, we similarly introduce the nano GN-topological space and nano N-method concepts and examine the behaviors of the continuity types, which are based on the concept of the nano h-open set and nano N-method. In the conclusion, we outline the impact we expect our studies to have on the scientific world.

Abstract Computing free energy is a fundamental problem in statistical physics. Recently, two distinct
methods have been developed and have demonstrated remarkable success: the tensor-network-
based contraction method and the neural-network-based variational method. Tensor networks are
accurate, but their application is often limited to low-dimensional systems due to the high compu-
tational complexity in high-dimensional systems. The neural network method applies to systems
with general topology. However, as a variational method, it is not as accurate as tensor networks.
In this work, we propose an integrated approach, tensor-network-based variational autoregressive
networks (TNVAN), that leverages the strengths of both tensor networks and neural networks: com-
bining the variational autoregressive neural network’s ability to compute an upper bound on free
energy and perform unbiased sampling from the variational distribution with the tensor network’s
power to accurately compute the partition function for small sub-systems, resulting in a robust
method for precisely estimating free energy. To evaluate the proposed approach, we conducted
numerical experiments on spin glass systems with various topologies, including two-dimensional
lattices, fully connected graphs, and random graphs. Our numerical results demonstrate the supe-
rior accuracy of our method compared to existing approaches. In particular, it effectively handles
systems with long-range interactions and leverages GPU efficiency without requiring singular value
decomposition, indicating great potential in tackling statistical mechanics problems and simulating
high-dimensional complex systems through both tensor networks and neural networks.

The performance of fiber-reinforced composites depends on the structure and continuity of high-performance fiber. However, fiber discontinuities due to insufficient carrier capacity during the braiding process occur frequently, which is a major obstacle to the high quality of preforms. This work proposes a general topology optimization method of carrier capacity to reduce fiber discontinuity due to yarn change. The movement path of the carrier was calculated. Collision detection of the carrier was performed by an interfering judgment between facets. The genetic algorithm was used to find the optimal structural parameter values of the carrier. To verify the effectiveness of the proposed method, braiding experiments were conducted using optimized carriers. The results show that the optimal carrier capacity of a three-layer spherical braiding machine is 24.3% higher than the carrier currently used. The frequency of yarn changes is significantly reduced, and fiber continuity is effectively ensured. The optimization method can provide reference values of carrier size for different types of braiding equipment, avoiding repeated trial-and-error costs. It can realize high-efficiency production and has great application value.

Our main aim is to define the relation between the fuzzy set and topological spaces. However the fuzzy sets hypothesis goes in a developing phase, now a days it is accepted spacious and precious applications. We also develop the generalised structure which includes several fields such as point set topology, algebraic topology, and differential topology. Georg Cantor, David Hilbert, Felix Hausdorff, Maurice Frechet and Henri Poincare studied its basic properties based on Euclidean space and some geometrical structure. For that purpose, we have tried to offer more fundamental description of these two topics. In the fuzzy topological spaces section. We derive definition and also proved of theorems. The existing consequences in this analysis signify that large numbers of the fundamental concepts from general topology might be expanded enthusiastically to fuzzy topological spaces.

This paper introduces a novel method for the topology optimization of multi-material structures. The core innovation lies in minimizing an appropriate norm of the transfer matrix that links external loads to system outputs. Several formulations of structural compliance are considered within a framework that accommodates both single and multiple inputs and outputs. The multimaterial interpolation scheme follows the approach proposed by Yi and collaborators, which is adapted to the general topology optimization framework proposed by Venini and collaborators. While the proposed method is inherently capable of addressing the dynamic response of multimaterial structures, this study focuses exclusively on static topology optimization. The extension to dynamic topology optimization is deferred to a fortcoming study, which requires a multimaterial interpolation of inertial properties that is currently in an advanced development stage. Finally, a series of numerical case studies illustrate the key features of the proposed approach.

We address the problem of approximating a set-valued function F, where F:[a,b]→K(Rd) given its samples {F(a+ih)}i=0N, with h=(b−a)/N. We revisit an existing method that approximates set-valued functions by interpolating signed-distance functions. This method provides a high-order approximation for general topologies but loses accuracy near points where F undergoes topological changes. To address this, we introduce new techniques that enhance efficiency and maintain high-order accuracy across [a,b]. Building on the foundation of previous publication, we introduce new techniques to improve the method’s efficiency and extend its high-order approximation accuracy throughout the entire interval [a,b]. Particular focus is placed on identifying and analyzing the behavior of F near topological transition points. To address this, two algorithms are introduced. The first algorithm employs signed-distance quasi-interpolation, incorporating specialized adjustments to effectively handle singularities at points of topological change. The second algorithm leverages an implicit function representation of Graph(F), offering an alternative and robust approach to its approximation. These enhancements improve accuracy and stability in handling set-valued functions with changing topologies.

Abstract Supply and manufacturing networks in the chemical industry involve diverse processing steps across different locations, rendering their operation vulnerable to disruptions from unplanned events. Optimal responses should consider factors such as product allocation, delayed shipments, and price renegotiation, among other factors. In such context, we propose a multiperiod mixed‐integer linear programming model that integrates production, scheduling, shipping, and order management to minimize the financial impact of such disruptions. The model accommodates arbitrary supply chain topologies and incorporates various disruption scenarios, offering adaptability to real‐world complexities. A case study from the chemical industry demonstrates the scalability of the model under finer time discretization and explores the influence of disruption types and order management costs on optimal schedules. This approach provides a tractable, adaptable framework for developing responsive operational plans in supply chain and manufacturing networks under uncertainty.

The article presents a variant of design and implementation of the discipline “General Topology” within the framework of the optional module “Elements of Modern Mathematics” of the master’s educational programme of higher education in the direction of training 44.04.01 Pedagogical Education, “Mathematical Education” specialisation. The authors propose an original approach to selecting the content for practical classes, consisting in selection of a set of research problems previously solved within graduation works. The article describes the methods of working with these problems in class, consisting in sequential construction of ways to solve them with subsequent comparison with existing solutions. The method of designing and implementing the content allows students to master complex material within the framework of research technology at a level accessible to them.