Search Results (101)

We introduce a synchronization procedure for clocks based on the Einstein–Landauer framework. Clocks are modeled as discrete, macroscopic devices operating at a thermal equilibrium temperature T. Synchronization is achieved by transmitting photons from one clock to another; the absorption of a photon by a clock reduces the uncertainty in its timekeeping. The minimum energy required for this reduction in uncertainty is determined by the Landauer bound. We distinguish between the time-bearing and non-time-bearing degrees of freedom of the clocks. A reduction in uncertainty under synchronization in the time-bearing degrees of freedom necessarily leads to heat dissipation in the non-time-bearing ones. The minimum energy dissipation in these non-time-bearing degrees of freedom is likewise given by the Landauer limit. The same is true for mechanical synchronization of clocks. We also consider lattices of clocks and analyze synchronization using a Ramsey graph approach. Notably, clocks operating at the same temperature may be synchronized using photons of different frequencies. Each clock is categorized as either synchronized or non-synchronized, resulting in a bi-colored complete graph of clocks. By Ramsey’s theorem, such a graph inevitably contains a triad (or loop) of clocks that are either all synchronized or all non-synchronized. The extension of the Ramsey approach to infinite lattices of clocks is reported. Full article

We investigate the behavior of solutions of second-order elliptic Dirichlet problems for a convex domain by using a “gliding hump” technique and prove that there are no contour limits at a specified point of the boundary of the domain. Then we consider two-dimensional domains which have a reentrant (i.e., nonconvex) corner at a point P of the boundary of the domain. Assuming certain comparison functions exist, we prove that for any solution of an appropriate Dirichlet problem on the domain whose graph has finite area, there are infinitely many curves of finite length in the domain ending at P along which the solution has a limit at P. We then prove that such behavior occurs for quasilinear operations with positive genre. Full article

The Ramsey approach is applied to analyses of the kinematics of systems built of non-relativistic, motile point masses/particles. This approach is based on colored graph theory. Point masses/particles serve as the vertices of the graph. The time dependence of the distance between the particles determines the coloring of the links. The vertices/particles are connected with orange links when particles move away from each other or remain at the same distance. The vertices/particles are linked with violet edges when particles converge. The sign of the time derivative of the distance between the particles dictates the color of the edge. Thus, a complete, bi-colored Ramsey temporal graph emerges. The suggested coloring procedure is not transitive. The coloring of the links is time-dependent. The proposed coloring procedure is frame-independent and insensitive to Galilean transformations. At least one monochromatic triangle will inevitably appear in the graph emerging from the motion of six particles due to the fact that the Ramsey number $R\left(\mathrm{3,3}\right)=6.$ This approach is extended to the analysis of systems containing an infinite number of moving point masses. An infinite monochromatic (violet or orange) clique will necessarily appear in the graph. Applications of the introduced approach are discussed. The suggested Ramsey approach may be useful for the analysis of turbulence seen within the Lagrangian paradigm. Full article

A new family of graph operations based on multipartite graph with an arbitrary number of parts is defined and their applications are explored in this paper. The complete spectra of graphs derived from multipartite graphs are determined. Because the adjacency matrix of the multipartite graph is symmetric, we can use it to generate an unlimited number of special symmetric graphs. Methods for generating countless new families of integral graphs using these multipartite graph operations have been presented. By applying these multipartite graph operations, we can construct infinitely many orderenergetic graphs from orderenergetic or non-orderenergetic graphs. Additionally, infinite pairs of equienergetic and non-cospectral graphs can be generated through these new operations. Moreover, this kind of graph operation can also be used to construct other special graphs related to eigenvalues and energy. Full article

This paper addresses the challenge of achieving fast and accurate transient stability analysis and emergency control in power systems, which are crucial for reliable grid operation under disturbances. To this end, we propose a spatio-temporal graph deep learning approach leveraging Diffusion Convolutional Gated Recurrent Units (DCGRUs) for transient stability assessment and coherent generator group prediction. Unlike traditional methods, our approach explicitly represents transient responses as spatio-temporal graph data, capturing both topological and dynamic dependencies. The DCGRU model effectively extracts these features, and the predicted coherent generator groups are incorporated into the single-machine infinite-bus equivalence method to design an emergency generator tripping scheme. Simulation analysis results on both benchmark and real-world power grids validate the proposed method’s feasibility and effectiveness in enhancing transient stability analysis and emergency control. Full article

Inverse topological index problems involve determining whether a graph exists with a given integer as its topological index. One such index, the Mostar index, $Mo\left(G\right)$, is defined as $Mo\left(G\right)={\sum }_{uv\in E\left(G\right)}|n_u\left(e\right|G)-n_v\left(e\right|G)|,$ where $n_u\left(e\right|G)$ and $n_v\left(e\right|G)$ represent the number of vertices closer to vertex u than v and closer to v than u, respectively, for an edge $e=uv$. The inverse Mostar index problem has gained significant attention recently. In their work, Alizadeh et al. [Solving the Mostar index inverse problem, J. Math. Chem. 62 (5) (2024) 1079–1093] proposed the following open problem: “Which nonnegative integers can be realized as Mostar indices of c-cyclic graphs, for a given positive integer c?”. Subsequently, one of the present authors [On the inverse Mostar index problem for molecular graphs, Trans. Comb. 14 (1) (2024) 65–77] conjectured that, except for finitely many positive integers, all other positive integers can be realized as the Mostar index of a c-cyclic graph, where $c\ge 3$. In this paper, we address the inverse Mostar index problem for c-cyclic graphs. Specifically, we construct infinitely many families of symmetric c-cyclic structures, thereby demonstrating a solution to the inverse Mostar index problem using an infinite family of such symmetric structures. By providing a comprehensive proof of the conjecture, we fully resolve this longstanding open problem. Full article

For an undirected graph G, a zero-sum flow is an assignment of nonzero integer weights to the edges such that each vertex has a zero-sum, namely the sum of all incident edge weights with each vertex is zero. This concept is an undirected analog of nowhere-zero flows for directed graphs. We study a more general one, namely constant-sum $\mathbb{A}$-flows, which gives edge weights using nonzero elements of an additive Abelian group $\mathbb{A}$ and requires each vertex to have a constant-sum instead. In particular, we focus on two special cases: $\mathbb{A}={\mathbb{Z}}_k$, the finite cyclic group of integer congruence modulo k, and $\mathbb{A}=\mathbb{Z}$, the infinite cyclic group of integers. The constant sum under a constant-sum $\mathbb{A}$-flow is called an index of G for short, and the set of all possible constant sums (indices) of G is called the constant sum spectrum. It is denoted by $I_k\left(G\right)$ and $I\left(G\right)$ for $\mathbb{A}={\mathbb{Z}}_k$ and $\mathbb{A}=\mathbb{Z}$, respectively. The zero-sum flows and constant-sum group flows for regular graphs regarding cases $\mathbb{Z}$ and ${\mathbb{Z}}_k$ have been studied extensively in the literature over the years. In this article, we study the constant sum spectrum of nearly regular graphs such as wheel graphs $W_n$ and fan graphs $F_n$ in particular. We completely determine the constant-sum spectrum of fan graphs and wheel graphs concerning ${\mathbb{Z}}_k$ and $\mathbb{Z}$, respectively. Some open problems will be mentioned in the concluding remarks. Full article

Graphical games describe strategic interactions among a specified network of players. The threshold protocol game is a graphical game that models the adoption of a lesser-used product in a population when individuals benefit by using the same product. The threshold protocol game has historically been considered using infinite, simple graphs. In general, however, players might value some relationships more than others or may have different levels of influence in the graph. These traits are described by weights on graph edges or vertices, respectively. Relative comparisons on arbitrarily weighted graphs have been studied for a variety of graphical games. Alternatively, graph labelings are functions that assign values to the edges and vertices of graphs based on a particular set of rules. This work demonstrates that the outcome of the threshold protocol game can be characterized on a magic square-generalization labeled graph. There are a variety of graph labelings that generalize the concept of magic squares. In each, the labels on similar sets of graph elements sum to a constant. The constant sums of magic square-generalization labelings mean that each player experiences a constant level of influence without needing to specify the value of players relative to one another. The game outcome is compared across different types and features of labelings. Full article

The threshold protocol game is a graphical game that models the adoption of an idea or product through a population. There are two states players may take in the game, and the goal of the game is to motivate the state that begins in the minority to spread to every player. Here, the threshold protocol game is defined, and existence results are studied on infinite graphs. Many generalizations are proposed and applied. This work explores the impact of graph topology on the outcome of the threshold protocol game and consequently considers finite graphs. By exploiting the well-known topologies of complete and complete bipartite graphs, the outcome of the threshold protocol game can be fully characterized on these graphs. These characterizations are ideal, as they are given in terms of the game parameters. More generally, initial conditions in terms of game parameters that cause the preferred game outcome to occur are identified. It is shown that the necessary conditions differ between non-bipartite and bipartite graphs because non-bipartite graphs contain odd cycles while bipartite graphs do not. These results motivate the primary result of this work, which is an exhaustive list of achievable game outcomes on bipartite graphs. While possible outcomes are identified, it is noted that a complete characterization of when game outcomes occur is not possible on general bipartite graphs. Full article

An even ring system G is a simple 2-connected plane graph with all interior vertices of degree 3, all exterior vertices of either degree 2 or 3, and all finite faces of an even length. G is angularly connected if all of the peripheral segments of G have odd lengths. In this paper, we show that every angularly connected even ring system G, which does not contain any triple of altogether-adjacent peripheral faces, has a perfect matching. This was achieved by finding an appropriate edge coloring of G, derived from the proof of the existence of a proper face 3-coloring of the graph. Additionally, an infinite family of graphs that are face 3-colorable has been identified. When interpreted in the context of the inner dual of G, this leads to the introduction of 3-colorable graphs containing cycles of lengths 4 and 6, which is a supplementation of some already known results. Finally, we have investigated the concept of the Clar structure and Clar set within the aforementioned family of graphs. We found that a Clar set of an angularly connected even ring system cannot in general be obtained by minimizing the cardinality of the set A. This result is in contrast to the previously known case for the subfamily of benzenoid systems, which admit a face 3-coloring. Our results open up avenues for further research into the properties of Clar and Fries sets of angularly connected even ring systems. Full article

In some location scenarios where the location information of nodes cannot be mastered in advance, the anchor-free location technology is particularly important. In order to reduce the complicated calculation and eliminate the accumulated error in the traditional anchor-free location algorithm, a new anchor-free location algorithm based on transition coordinates is proposed in this paper. This algorithm is different from the traditional methods such as minimum cost function or inverse matrix. Instead, N initial coordinates are randomly generated as the starting position of the transition coordinates, and the position increment between the transition coordinates and the real coordinates of the node is constantly modified. After K iterations, the convergent position coordinates are finally infinitely close to the real position coordinates of N nodes, and the computational complexity is less than most existing algorithms. As follows, the factors that affect the performance of the algorithm are investigated in the simulation experiment, including the topology structure, positioning accuracy and the total number of nodes, etc. The results show great advantages compared with the traditional anchor-free positioning algorithm. When the topology structure of the initial coordinates changes from a square to a random graph, the number of iterations increases by 15.79%. When the positioning accuracy increased from 1% to 1‰, the number of iterations increased by 36.84%. When the number of nodes N is reduced from 9 to 4, the number of iterations is reduced by 63.16%. In addition, the algorithm can also be extended to the field of moving coordinates or three-dimensional spatial positioning, which has broad application prospects. Full article

We investigate a quantum walk on a ring represented by a directed triangle graph with complex edge weights and monitored at a constant rate until the quantum walker is detected. To this end, the first hitting time statistics are recorded using unitary dynamics interspersed stroboscopically by measurements, which are implemented on IBM quantum computers with a midcircuit readout option. Unlike classical hitting times, the statistical aspect of the problem depends on the way we construct the measured path, an effect that we quantify experimentally. First, we experimentally verify the theoretical prediction that the mean return time to a target state is quantized, with abrupt discontinuities found for specific sampling times and other control parameters, which has a well-known topological interpretation. Second, depending on the initial state, system parameters, and measurement protocol, the detection probability can be less than one or even zero, which is related to dark-state physics. Both return-time quantization and the appearance of the dark states are related to degeneracies in the eigenvalues of the unitary time evolution operator. We conclude that, for the IBM quantum computer under study, the first hitting times of monitored quantum walks are resilient to noise. However, a finite number of measurements leads to broadening effects, which modify the topological quantization and chiral effects of the asymptotic theory with an infinite number of measurements. Full article

Most electrical energy generation systems are based on synchronous generators; as a result, their analysis always provides interesting findings, especially if an approach different to those traditionally studied is used. Therefore, an approach involving the modeling and simulation of a synchronous generator connected to an infinite bus through a transmission line in a bond graph is proposed. The behavior of the synchronous generator is analyzed in four case studies of the transmission line: (1) a symmetrical transmission line, where the resistance and inductance of the three phases $(a,b,c)$ are equal, which determine resistances and inductances in coordinates $(d,q,0)$ as individual decoupled elements; (2) a symmetrical transmission line for the resistances and for non-symmetrical inductances in coordinates $(a,b,c)$ that result in resistances that are individual decoupled elements and in a field of inductances in coordinates $(d,q,0)$; (3) a non-symmetrical transmission line for resistances and for symmetrical inductances in coordinates $(a,b,c)$ that produce a field of resistances and inductances as individual elements decoupled in coordinates $(d,q,0)$; and (4) a non-symmetrical transmission line for resistances and inductances in coordinates $(a,b,c)$ that determine resistances and inductance fields in coordinates $(d,q,0)$. A junction structure based on a bond graph model that allows for obtaining the mathematical model of this electrical system is proposed. Due to the characteristics of a bond graph, model reduction can be carried out directly and easily. Therefore, reduced bond graph models for the four transmission line case studies are proposed, where the transmission line is seen as if it were inside the synchronous generator. In order to demonstrate that the models obtained are correct, simulation results using the 20-Sim software are shown. The simulation results determine that for a symmetrical transmission line, currents in the generator in the d and q axes are −25.87 A and 0.1168 A, while in the case of a non-symmetrical transmission line, these currents are −26.14 A and 0.0211 A, showing that for these current magnitudes, the generator is little affected due to the parameters of the generator and the line. However, for a high degree of non-symmetry of the resistances in phases a, b and c, it causes the generator to reach an unstable condition, which is shown in the last simulation of the paper. Full article

An infinite homogeneous tree is a special type of graph that has a completely symmetrical structure in all directions. For an infinite homogeneous tree $T=(\mathcal{V},\mathcal{E})$ with the natural distance d defined on graphs and a weighted measure $\mu$ of exponential growth, the authors introduce the variable Lebesgue space $L^{p(·)}\left(\mu \right)$ over $(\mathcal{V},d,\mu )$ and investigate it under the global Hölder continuity condition for $p(·)$. As an application, the strong and weak boundedness of the maximal operator relevant to admissible trapezoids on $L^{p(·)}\left(\mu \right)$ is obtained, and an unbounded example is presented. Full article

This work is devoted to the interaction of nucleotides. The goal of this study is to learn or try to learn how the interaction between nucleotides with exposure to a liquid takes place. Will the interacting forces of the nucleotides be sufficient to approach the incision? A numerical imitation of the interaction is conducted using the discrete element method and a Gears predictor–corrector as part of the integrated scheme. In this work, the results reflect the dynamics of nucleotides: velocity, displacement, and force graphs are presented with and without the effect of the liquid. During changes caused by the influence of a liquid, the nucleotide interaction transforms and passes three stages: a full stop, one similar to viscous damping, and one similar to non-dissipative behaviors. The main contribution of this work is a better understanding of the behavior of infinitely small objects that would be difficult to observe in vivo. The changing influence of a liquid can transform into certain effects. As a result, a model is provided, which can be based on the results of well-known physical experiments (DNA unzipping) for modeling nucleotide interactions. Full article