Search Results (13)

For the inner product space, we have Appolonius’ identity. From this identity, Park and Th. M. Rassias induced and investigated the quadratic functional equation of the Apollonius type. And Park and Th. M. Rassias first introduced an Apollonius-type additive functional equation. In this work, we investigate an Apollonius-type additive functional equation in 2-normed spaces. We first investigate the stability of an Apollonius-type additive functional equation in 2-Banach spaces by using Hyers’ direct method. Then, we consider the instability of an Apollonius-type additive functional equation in 2-Banach spaces. Full article

This contribution compares the final sections of Acts of the Apostles and Philostratus’ Life of Apollonius of Tyana. Through this comparison, I aim to show that these two writings resemble one another in their attention to travel as a literary theme. Both Acts and Life employ this theme to communicate their message and, in their narrative endings, set up their implied readers as travelers who are meant to continue the journeys of the protagonists in these writings. At the same time, Acts and Life differ in how exactly they envision their readers to continue the journeys of their protagonists. I will argue that these similarities and differences can be explained by the shared social and intellectual climate that Acts and Life inhabit: both writings result from discourses on travel and self that were rife among intellectuals in the Roman Empire in the first three centuries of our era, irrespective of their ethnic, legal, or cultural affiliations. Full article

An important action-planning problem is considered for participants of the pursuit–evasion game with multiple pursuers and a high-speed evader. The objects of study are mobile robotic systems and specifically small unmanned aerial vehicles (UAVs). The problem is complicated by the presence of significant wind loads that affect the trajectory and motion strategies of the players. It is assumed that UAVs have limited computing resources, which involves the use of computationally fast and real-time heuristic approaches. A novel and rapidly developing intelligent–geometric theory is applied to address the discussed problem. To accurately calculate the points of the participant’s rapprochement, we use a geometric approach based on the construction of circles or spheres of Apollonius. Intelligent control methods are applied to synthesize complex motion strategies of participants. A method for quickly predicting the evader’s trajectory is proposed based on a two-layer neural network containing a new activation function of the “s-parabola” type. We consider a special backpropagation training scheme for the model under study. A simulation scheme has been developed and tested, which includes mathematical models of dynamic objects and wind loads. The conducted simulations on pursuit–evasion games in close to real conditions showed the prospects and expediency of the presented approach. Full article

The multi-UAV task assignment problem in large-scale group-to-group interception scenarios presents challenges in terms of large computational complexity and the lack of accurate evaluation models. This paper proposes an effective evaluation model and hierarchical task assignment framework to address these challenges. The evaluation model incorporates the dynamics constraints specific to fixed-wing UAVs and improves the Apollonius circle model to accurately describe the cooperative interception effectiveness of multiple UAVs. By evaluating the interception effectiveness during the interception process, the assignment scheme of the multiple UAVs could be given based on the model. To optimize the configuration of UAVs and targets, a hierarchical framework based on the network flow algorithm is employed. This framework utilizes a clustering method based on feature similarity and interception advantage to decompose the large-scale task assignment problem into smaller, complete submodels. Following the assignment, Dubins curves are planned to the optimal interception points, ensuring the effectiveness of the interception task. Simulation results demonstrate the feasibility and effectiveness of the proposed scheme. With the increase in the model scale, the proposed scheme has a greater descending rate of runtime. In a large-scale scenario involving 200 UAVs and 100 targets, the runtime is reduced by $84.86\%$. Full article

This essay provides an introduction to the question of the contribution of the ancient sophists to aesthetics in Western art. It commences by examining the persistent analogies to visual arts in negative and positive discussions of sophistry, both philosophical and rhetorical, and proceeds to examine sophistic rhetoric in Gorgias, Aristides, Lucian, Philostratus and Byzantine ekphrasis, culminating with Philostratus’ discussions of mimesis and phantasia in Apollonius of Tyana. The discussions of the relation of being and nonbeing in Gorgias’ On Nonbeing and in Plato’s Sophist form the ontological core of sophistic claims about imaginative invention and the sophistic advancement of voluntary illusion (apatē) as a means to poetic “justice” or “truth”. Such claims should be considered in the light of the epistemological and ontological skepticism propounded by Gorgias. Although the opprobrium attached to sophistry obscures its later influence, we can nevertheless discern a sophistic aesthetic tradition focused on the reflective reception of artworks that re-emerges in the Renaissance. In the last section, I adumbrate the lines of study for examining a sophistic Renaissance in the visual arts, with attention to antiquarianism as an area where the significance of the beholder’s imaginative projection suggests the endurance—or revitalization—of sophistic aesthetics. Full article

Multi-agent differential games usually include tracking policies and escaping policies. To obtain the proper policies in unknown environments, agents can learn through reinforcement learning. This typically requires a large amount of interaction with the environment, which is time-consuming and inefficient. However, if one can obtain an estimated model based on some prior knowledge, the control policy can be obtained based on suboptimal knowledge. Although there exists an error between the estimated model and the environment, the suboptimal guided policy will avoid unnecessary exploration; thus, the learning process can be significantly accelerated. Facing the problem of tracking policy optimization for multiple pursuers, this study proposed a new form of fuzzy actor–critic learning algorithm based on suboptimal knowledge (SK-FACL). In the SK-FACL, the information about the environment that can be obtained is abstracted as an estimated model, and the suboptimal guided policy is calculated based on the Apollonius circle. The guided policy is combined with the fuzzy actor–critic learning algorithm, improving the learning efficiency. Considering the ground game of two pursuers and one evader, the experimental results verified the advantages of the SK-FACL in reducing tracking error, adapting model error and adapting to sudden changes made by the evader compared with pure knowledge control and the pure fuzzy actor–critic learning algorithm. Full article

A self-organizing cooperation strategy for multiple unmanned surface vessels (USVs) to pursue intelligent evaders in the case of a dynamic obstacle vessel is proposed. Firstly, a self-organizing cooperative hunting strategy is proposed to form an Apollonius circle. According to the escape strategies of evaders under different encirclement states, the pursuers are divided into pursuit group and ambush group. The pursuit group drives the evaders into the ambush area and completes the encirclement together with the ambush group. In order to better deal with the dynamic obstacle ships encountered in the pursuit process in the dynamic ocean environment, the artificial potential field-based collision avoidance method for inter-USV and dynamic collision avoidance strategy for encountering obstacle ships based on International Regulations for Preventing Collisions at Sea (COLREGs) are proposed. The simulation results show that the algorithm can make the pursuers complete the encirclement of the evaders and has good obstacle avoidance performance and flexibility in the environment with dynamic obstacle ships. Full article

This article proposes a possible path devoted to upper secondary school and early university students, as well as training teachers, with the aim to build a conscious approach to the learning/teaching of the conics, which uses, for an educational purpose, the close relationship between conics as loci of points of a plane and conics as sections of a cone. In this path, we will refer to some elements taken from the history of mathematics relating to a particular conic: the parabola. These elements could help students to discover and realize the transition from a parabola considered as a curve in a plane to the same parabola considered on a cone of which it is a section, as well as the inverse passage, and to grasp the profound link between two presentations of the same geometric object. Both steps will be carried out through constructions made with the use of the GeoGebra dynamic geometry software. In addition, it will be highlighted how the construction of conics by points has allowed the creation of lenses and mirrors, which represents a practical application of geometry very relevant to physics and astronomy. Such a practical approach could help students to overcome the difficulty in understanding conics by making the argument less abstract. Moreover, this path could build up an environment in which teachers and students could explore some semiotic registers and their changes, through which Mathematics expresses itself. In the final part, an educational experiment of the path that was proposed will be shown to the students of the Master’s degree course of “mathematics education” at the Department of Mathematics and Computer Science of the University of Calabria. The results of this experiment are described in detail and seem to confirm that the twofold view of the parabola as a section of a cone and as locus of points of a plane helps the students in understanding its meaning in both theoretical and applicative fields. Full article

This study aims to document the extent to which the use of digital technology enhances and extends high school teachers’ problem-solving strategies when framing their teaching scenarios. The participants systematically relied on online developments such as Wikipedia to contextualize problem statements or to review involved concepts. Likewise, they activated GeoGebra’s affordances to construct and explore dynamic models of tasks. The Apollonius problem is used to illustrate and discuss how the participants contextualized the task and relied on technology affordances to construct and explore problems’ dynamic models. As a result, they exhibited and extended the domain of several problem-solving strategies including the use of simpler cases, dragging orderly objects, measuring objects attributes, and finding loci of some objects that shaped their approached to reasoning and solve problems. Full article

In 1639, the treatise Prodromi catoptricorum et dioptricorum sive Conicorum operis ad abdita radii reflexi et refracti mysteria praevij et facem praeferentis. Libri quatuor priores by Claude Mydorge was printed in Paris. This volume, which followed the printing of his first two books in 1631, has resonance especially in the writings of those who, after him, addressed the conics. This fact raises the question of who Mydorge was and what his knowledge of the “doctrine” of the conics was, what is the most appropriate cultural context in which to properly read this writing, and finally, what is the place of its content in the development of thought placed between the Veteres and the Recentiores. In this paper, I attempt to elaborate an answer to these different questions, with the aim of emphasizing how the author reads and interprets the first books of Apollonius’ Conics. Neither the treatise, nor the figure of Mydorge, have received much attention in the current literature, although he was estimated as a savant in Paris and he was believed by Descartes to be one of the greatest mathematicians of his time. Full article

Wi-Fi fingerprinting has been widely used for indoor localization because of its good cost-effectiveness. However, it suffers from relatively low localization accuracy and robustness owing to the signal fluctuations. Virtual Access Points (VAP) can effectively reduce the impact of signal fluctuation problem in Wi-Fi fingerprinting. Current techniques normally use the Log-Normal Shadowing Model to estimate the virtual location of the access point. This would lead to inaccurate location estimation due to the signal attenuation factor in the model, which is difficult to be determined. To overcome this challenge, in this study, we propose a novel approach to calculating the virtual location of the access points by using the Apollonius Circle theory, specifically the distance ratio, which can eliminate the attenuation parameter term in the original model. This is based on the assumption that neighboring locations share the same attenuation parameter corresponding to the signal attenuation caused by obstacles. We evaluated the proposed method in a laboratory building with three different kinds of scenes and 1194 test points in total. The experimental results show that the proposed approach can improve the accuracy and robustness of the Wi-Fi fingerprinting techniques and achieve state-of-art performance. Full article

In this work, we investigate the generalized Hyers-Ulam stability of the Apollonius type additive functional equation in modular spaces with or without ${\Delta }_2$ -conditions. We study the same problem in fuzzy Banach spaces and $\beta$ -homogeneous Banach spaces. We show the hyperstability of the functional equation associated with the Jordan triple product in fuzzy Banach algebras. The obtained results can be applied to differential and integral equations with kernels of non-power types. Full article

We investigate two different textures of smectic A liquid crystals. These textures are particularly symmetric when they are observed at crossed polars optical microscopy. For both textures, a model has been made in order to examine the link between the defective macroscopic texture and the microscopic disposition of the layers. We present in particular in the case of some hexagonal tiling of circles (similar to the Apollonius tiling) some numeric simulation in order to visualize the smectic layers. We discuss of the nature of the smectic layers, which permit to assure their continuity from one focal conic domain to another adjacent one. Full article