Search Results (25)

In this paper, we investigate the existence of multiple solutions for a Kirchhoff-type equation with Dirichlet boundary conditions defined on locally finite graphs. Our study extends some previous results on nonlinear Laplacian equations to the more complex Kirchhoff equation which incorporates a nonlocal term. By employing an abstract three critical points theorem that is based on Morse theory, we provide sufficient conditions that guarantee the existence of at least three distinct solutions, including two strictly positive solutions. We also present an example to verify our results. Full article

This entry gives an overview of the main data structures and approaches used for a two-dimensional representation of the terrain surface using a digital elevation model (DEM). A DEM represents the elevation of the earth surface from a set of points. It is used for terrain analysis, visualisation and interpretation. DEMs are most commonly defined as a grid where an elevation is assigned to each grid cell. Due to its simplicity, the square grid structure is the most common DEM structure. However, it is less adaptive and shows limitations for more complex processing and reasoning. Hence, the triangulated irregular network is a more adaptive structure and explicitly stores the relationships between the points. Other topological structures (contour graphs, contour trees) have been developed to study terrain morphology. Topological relationships are captured in another structure, the surface network (SN), composed of critical points (peaks, pits, saddles) and critical lines (thalweg, ridge lines). The SN can be computed using either a TIN or a grid. The Morse Theory provides a mathematical approach to studying the topology of surfaces, which is applied to the SN. It has been used for terrain simplification, multi-resolution modelling, terrain segmentation and landform identification. The extended surface network (ESN) extends the classical SN by integrating both the surface and the drainage networks. The ESN can itself be extended for the cognitive representation of the terrain based on saliences (typical points, lines and regions) and skeleton lines (linking critical points), while capturing the context of the appearance of landforms using topo-contexts. Full article

Connected autonomous vehicles (CAVs) face constraints from multiple traffic elements, such as the vehicle, road, and environmental factors. Accurately quantifying the vehicle’s operational status and driving risk level in complex traffic scenarios is crucial for enhancing the efficiency and safety of connected autonomous driving. To continuously and dynamically quantify the driving risks faced by CAVs in the road environment—arising from the front, rear, and lateral directions—this study focused s on the self-driving particle characteristics that enable CAVs to perceive their surrounding environment and make driving decisions. The vehicle-to-vehicle interaction behavior was analogized to the inter-molecular interaction relationship, and a molecular Morse potential model was applied, coupled with the vehicle dynamics theory. This approach considers the safety margin and the specificity of driving styles. A multi-layer decoder–encoder long short-term memory (LSTM) network was employed to predict vehicle trajectories and establish a risk quantification model for vehicle-to-vehicle interaction behavior. Using SUMO software (win64-1.11.0), three typical driving behavior scenarios—car-following, lane-changing, and yielding—were modeled. A comparative analysis was conducted between the risk field quantification method and existing risk quantification indicators such as post-encroachment time (PET), deceleration rate to avoid crash (DRAC), modified time to collision (MTTC), and safety potential fields (SPFs). The evaluation results demonstrate that the risk field quantification method has the advantage of continuously quantifying risk, addressing the limitations of traditional risk indicators, which may yield discontinuous results when conflict points disappear. Furthermore, when the half-life parameter is reasonably set, the method exhibits more stable evaluation performance. This research provides a theoretical basis for the dynamic equilibrium control of driving risks in connected autonomous vehicle fleets within mixed-traffic environments, offering insights and references for collision avoidance design. Full article

This study presents the description of the parameterization of sound speed distribution in the Sea of Japan in the presence of a synoptic eddy. An analytical representation of the background sound speed profile (SSP) on its periphery is proposed. The perturbation of sound speed directly associated with the presence of an eddy is investigated. The proposed parameterization of the background SSP leads to a Sturm–Liouville problem for normal mode computation, which is equivalent to the eigenvalue problem for the Schrödinger equation with the Morse potential. This equivalence leads to simple analytical formulae for normal modes and their respective horizontal wavenumbers. It is shown that in the presence of an eddy causing moderate variations in sound speed, the standard perturbation theory for acoustic modes can be applied to describe the variability in horizontal wavenumbers across the area in which the eddy is localized. The proposed parameterization can be applied to the sound propagation modeling in the Sea of Japan. Full article

In this paper, we demonstrate that when saddle point reduction is applicable, there is a clear relationship between the Morse index and the critical groups before and after the reduction. As an application of this result, we use saddle point reduction along with the critical point theorem to show the existence of periodic solutions in second-order Hamiltonian systems. Full article

Geographic information systems have undergone rapid growth for decades. Topology has provided valuable modeling tools in the development of this field. Formal verification ofthe model of topological spatial relations can provide a reliable guarantee for the correctness of geographic information systems. We present a proof of the topological spatial relations model that has been formally verified in the Coq proof assistant. After an introduction to the formalization of the axiomatic set theory of Morse–Kelley, the formal description of the elementary concepts and properties of general topology is developed. The topological spatial relations between two sets are described by using the concept of the intersection value. Finally, we formally proved the topological spatial relations between two sets which are restricted to the regularly closed and the planar spatial regions. All the proof details are strictly completed in Coq, which shows that the correctness of the theoretical model for geographic information systems can be checked by a computer. This paper provides a novel method to verify the correctness of the topological spatial relations model. This work can also contribute to the creation and validation of various geological models and software. Full article

In this paper, we consider the existence and multiplicity of nontrivial solutions for discrete elliptic Dirichlet problems ${\Delta }_1^{2}u(i-1,j)+{\Delta }_2^{2}u(i,j-1)=-f((i,j),u(i,j)),(i,j)\in \Omega ,u(i,0)=u(i,T_2+1)=0i\in \mathbb{Z}(1,T_1),u(0,j)=u(T_1+1,j)=0j\in \mathbb{Z}(1,T_2),$ which have a symmetric structure. When the nonlinearity $f(·,u)$ is resonant at both zero and infinity, we construct a variational functional on a suitable function space and turn the problem of finding nontrivial solutions of discrete elliptic Dirichlet problems to seeking nontrivial critical points of the corresponding functional. We establish a series of results based on the existence of one, two or five nontrivial solutions under reasonable assumptions. Our results depend on the Morse theory and local linking. Full article

The LiDAR point cloud has been widely used in scenarios of automatic driving, object recognition, structure reconstruction, etc., while it remains a challenging problem in line structure extraction, due to the noise and accuracy, especially in data acquired by consumer electronic devices. To address the issue, a line structure extraction method based on the persistence of tensor feature is proposed, and subsequently applied to the data acquired by an iPhone-based LiDAR sensor. The tensor of each point is encoded, voted, and aggregated by its neighborhood, and further decomposed into different geometric features in each dimension. Then, the line feature in the point cloud is represented and computed using the persistence of the tensor feature. Finally, the line structure is extracted based on the persistent homology according to the discrete Morse theory. With the LiDAR point cloud collected by the iPhone 12 Pro MAX, experiments are conducted, line structures are extracted from two different datasets, and results perform well in comparison with other related results. Full article

Computation of circuit complexity has gained much attention in the theoretical physics community in recent times, to gain insights into the chaotic features and random fluctuations of fields in the quantum regime. Recent studies of circuit complexity take inspiration from Nielsen’s geometric approach, which is based on the idea of optimal quantum control in which a cost function is introduced for the various possible path to determine the optimum circuit. In this paper, we study the relationship between the circuit complexity and Morse theory within the framework of algebraic topology, which will then help us study circuit complexity in supersymmetric quantum field theory describing both simple and inverted harmonic oscillators up to higher orders of quantum corrections. We will restrict ourselves to $\mathcal{N}=1$ supersymmetry with one fermionic generator $Q_{\alpha }$. The expression of circuit complexity in quantum regime would then be given by the Hessian of the Morse function in supersymmetric quantum field theory. We also provide technical proof of the well known universal connecting relation between quantum chaos and circuit complexity of the supersymmetric quantum field theories, using the general description of Morse theory. Full article

In this paper, an attempt to unify two important lines of thought in applied optimization is proposed. We wish to integrate the well-known (dynamic) theory of Pontryagin optimal control with the Pareto optimization (of the static type), involving the maximization/minimization of a non-trivial number of functions or functionals, Pontryagin optimal control offers the definitive theoretical device for the dynamic realization of the objectives to be optimized. The Pareto theory is undoubtedly less known in mathematical literature, even if it was studied in topological and variational details (Morse theory) by Stephen Smale. This reunification, obviously partial, presents new conceptual problems; therefore, a basic review is necessary and desirable. After this review, we define and unify the two theories. Finally, we propose a Pontryagin extension of a recent multiobjective optimization application to the evolution of trees and the related anatomy of the xylems. This work is intended as the first contribution to a series to be developed by the authors on this subject. Full article

We address the basic question in discrete Morse theory of combining discrete gradient fields that are partially defined on subsets of the given complex. This is a well-posed question when the discrete gradient field V is generated using a fixed algorithm which has a local nature. One example is ProcessLowerStars, a widely used algorithm for computing persistent homology associated to a grey-scale image in 2D or 3D. While the algorithm for V may be inherently local, being computed within stars of vertices and so embarrassingly parallelizable, in practical use, it is natural to want to distribute the computation over patches $P_i$, apply the chosen algorithm to compute the fields $V_i$ associated to each patch, and then assemble the ambient field V from these. Simply merging the fields from the patches, even when that makes sense, gives a wrong answer. We develop both very general merging procedures and leaner versions designed for specific, easy-to-arrange covering patterns. Full article

This paper presents the results of calculating the thermodynamic properties of aluminum, copper, and their binary alloys under isothermal and shock compression. The calculations were performed by a theoretical equation of state based on perturbation theory. The pair Morse potential was used to describe the intermolecular interaction in metals. The calculation results are in good agreement with the experimental data and the results of molecular dynamics modeling performed in this work using the LAMMPS software package. Furthermore, it is shown that the equation of state based on the perturbation theory with the corresponding potential of intermolecular interaction can be used to calculate the thermodynamic properties of gaseous (fluid) systems and pure metals and their binary alloys. Full article

In this paper we present a survey of Fermat metrics and their applications to stationary spacetimes. A Fermat principle for light rays is stated in this class of spacetimes and we present a variational theory for the light rays and a description of the multiple image effect. Some results on variational methods, as Ljusternik-Schnirelmann and Morse Theory are recalled, to give a description of the variational methods used. Other applications of the Fermat metrics concern the global hyperbolicity and the geodesic connectedeness and a characterization of the Sagnac effect in a stationary spacetime. Finally some possible applications to other class of spacetimes are considered. Full article

We are interested in evaluating the state of drivers to determine whether they are attentive to the road or not by using motion sensor data collected from car driving experiments. That is, our goal is to design a predictive model that can estimate the state of drivers given the data collected from motion sensors. For that purpose, we leverage recent developments in topological data analysis (TDA) to analyze and transform the data coming from sensor time series and build a machine learning model based on the topological features extracted with the TDA. We provide some experiments showing that our model proves to be accurate in the identification of the state of the user, predicting whether they are relaxed or tense. Full article

The article develops our understanding of social capital by analyzing social capital as an organizational phenomenon. The analysis is based on qualitative data consisting of interviews and documents obtained from six different kindergartens in Norway. Kindergartens are used as a “prism” through which we can understand how social capital is formed—and the mechanisms that shape the development of various forms of networks within welfare organizations. More specifically we look at drop-in kindergartens. The specific purpose of these kindergartens is to provide open and inclusive arenas that promote integration and community. We find that the kindergartens vary in the degree to which they succeed in building bridging forms of networks and communities. Using concepts from organizational theory and Wenger’s (1998) theory of communities of practice, we find that formal organizational factors such as ownership, organizational goals, profiling, location, and educational content impact the formation of bridging forms of social capital. The composition of the user groups and the user groups’ motivation for participating most clearly affect the conditions for community formation. The composition of the user groups is the result of a number of organizational factors and organizational mechanisms. Kindergartens that have a heterogeneous user group, and a user group with a community orientation (Morse 2006), are more successful at creating bridging types of social networks. Full article