Narrow Results

The primary objective of this study is to analyze the Hyers–Ulam stability of fractional derivatives for the Black–Scholes model, involving two underlying asset systems and utilizing Caputo fractional derivatives. We employ a fixed-point approach to examine the existence and uniqueness of solutions and to investigate the Hyers–Ulam stability of the given problem. Additionally, we analyze the graphical behavior of the obtained results, demonstrating that the analytical method is highly efficient and delivers precise results for determining approximate numerical solutions. The findings highlight the significant role of fractional approaches in studying nonlinear systems of scientific and physical importance. Furthermore, the graphical analysis, considering various fractional orders and parameter values, unveils new insights and intriguing phenomena associated with the Black–Scholes model.

Objectives: To model cortical connections in order to characterize their oscillatory behavior and role in the generation of spontaneous electroencephalogram (EEG). Methods: We studied averaged responses to single pulse electrical stimulation (SPES) from the non-epileptogenic hemisphere of five patients assessed with intracranial EEG who became seizure free after contralateral temporal lobectomy. Second-order control system equations were modified to characterize the systems generating a given response. SPES responses were modeled as responses to a unit step input. EEG power spectrum was calculated on the 20s preceding SPES. Results: 121 channels showed responses to 32 stimulation sites. A single system could model the response in 41.3% and two systems were required in 58.7%. Peaks in the frequency response of the models tended to occur within the frequency range of most activity on the spontaneous EEG. Discrepancies were noted between activity predicted by models and activity recorded in the spontaneous EEG. These discrepancies could be explained by the existence of alpha rhythm or interictal epileptiform discharges. Conclusions: Cortical interactions shown by SPES can be described as control systems which can predict cortical oscillatory behavior. The method is unique as it describes connectivity as well as dynamic interactions.

We consider nonlinear transport equations with non-local velocity describing the time-evolution of a measure. Such equations often appear when considering the mean-field limit of finite-dimensional systems modeling collective dynamics. We address the problem of controlling these equations by means of a time-varying bounded control action localized on a time-varying control subset of small Lebesgue measure. We first define dissipativity for nonlinear transport equations in terms of Lie derivatives of a Lyapunov function depending on the measure. Then, assuming that the uncontrolled system is dissipative, we provide an explicit construction of a control law steering the system to an invariant sublevel of the Lyapunov function. The control function and the control domain are designed in terms of the Lie derivatives of the Lyapunov function. In this sense the construction can be seen as an infinite-dimensional analogue of the well-known Jurdjevic–Quinn procedure. Moreover, the control law presents sparsity properties in the sense that the support of the control is small. Finally, we show that our result applies to a large class of kinetic equations modeling multi-agent dynamics.

Differential inclusions arising in population biology are analysed and a numerical approach to solve the corresponding initial value problems is proposed. These differential inclusions originate from optimal myopic strategies and from projecting differential equations onto a viability set. We show that such differential inclusions may piecewise be considered either as ordinary differential equations or as differential-algebraic equations with Index 2. The numerical method is based on this observation. Simulations for one population of predators feeding on two populations of prey and for a system of two competing populations whose growth is constrained by a limited resource are given.

Complex systems, as interwoven miscellaneous interacting entities that emerge and evolve through self-organization in a myriad of spiraling contexts, exhibit subtleties on global scale besides steering the way to understand complexity which has been under evolutionary processes with unfolding cumulative nature wherein order is viewed as the unifying framework. Indicating the striking feature of non-separability in components, a complex system cannot be understood in terms of the individual isolated constituents’ properties per se, it can rather be comprehended as a way to multilevel approach systems behavior with systems whose emergent behavior and pattern transcend the characteristics of ubiquitous units composing the system itself. This observation specifies a change of scientific paradigm, presenting that a reductionist perspective does not by any means imply a constructionist view; and in that vein, complex systems science, associated with multiscale problems, is regarded as ascendancy of emergence over reductionism and level of mechanistic insight evolving into complex system. While evolvability being related to the species and humans owing their existence to their ancestors’ capability with regards to adapting, emerging and evolving besides the relation between complexity of models, designs, visualization and optimality, a horizon that can take into account the subtleties making their own means of solutions applicable is to be entailed by complexity. Such views attach their germane importance to the future science of complexity which may probably be best regarded as a minimal history congruent with observable variations, namely the most parallelizable or symmetric process which can turn random inputs into regular outputs. Interestingly enough, chaos and nonlinear systems come into this picture as cousins of complexity which with tons of its components are involved in a hectic interaction with one another in a nonlinear fashion amongst the other related systems and fields. Relation, in mathematics, is a way of connecting two or more things, which is to say numbers, sets or other mathematical objects, and it is a relation that describes the way the things are interrelated to facilitate making sense of complex mathematical systems. Accordingly, mathematical modeling and scientific computing are proven principal tools toward the solution of problems arising in complex systems’ exploration with sound, stimulating and innovative aspects attributed to data science as a tailored-made discipline to enable making sense out of voluminous (-big) data. Regarding the computation of the complexity of any mathematical model, conducting the analyses over the run time is related to the sort of data determined and employed along with the methods. This enables the possibility of examining the data applied in the study, which is dependent on the capacity of the computer at work. Besides these, varying capacities of the computers have impact on the results; nevertheless, the application of the method on the code step by step must be taken into consideration. In this sense, the definition of complexity evaluated over different data lends a broader applicability range with more realism and convenience since the process is dependent on concrete mathematical foundations. All of these indicate that the methods need to be investigated based on their mathematical foundation together with the methods. In that way, it can become foreseeable what level of complexity will emerge for any data desired to be employed. With relation to fractals, fractal theory and analysis are geared toward assessing the fractal characteristics of data, several methods being at stake to assign fractal dimensions to the datasets, and within that perspective, fractal analysis provides expansion of knowledge regarding the functions and structures of complex systems while acting as a potential means to evaluate the novel areas of research and to capture the roughness of objects, their nonlinearity, randomness, and so on. The idea of fractional-order integration and differentiation as well as the inverse relationship between them lends fractional calculus applications in various fields spanning across science, medicine and engineering, amongst the others. The approach of fractional calculus, within mathematics-informed frameworks employed to enable reliable comprehension into complex processes which encompass an array of temporal and spatial scales notably provides the novel applicable models through fractional-order calculus to optimization methods. Computational science and modeling, notwithstanding, are oriented toward the simulation and investigation of complex systems through the use of computers by making use of domains ranging from mathematics to physics as well as computer science. A computational model consisting of numerous variables that characterize the system under consideration allows the performing of many simulated experiments via computerized means. Furthermore, Artificial Intelligence (AI) techniques whether combined or not with fractal, fractional analysis as well as mathematical models have enabled various applications including the prediction of mechanisms ranging extensively from living organisms to other interactions across incredible spectra besides providing solutions to real-world complex problems both on local and global scale. While enabling model accuracy maximization, AI can also ensure the minimization of functions such as computational burden. Relatedly, level of complexity, often employed in computer science for decision-making and problem-solving processes, aims to evaluate the difficulty of algorithms, and by so doing, it helps to determine the number of required resources and time for task completion. Computational (-algorithmic) complexity, referring to the measure of the amount of computing resources (memory and storage) which a specific algorithm consumes when it is run, essentially signifies the complexity of an algorithm, yielding an approximate sense of the volume of computing resources and seeking to prove the input data with different values and sizes. Computational complexity, with search algorithms and solution landscapes, eventually points toward reductions vis à vis universality to explore varying degrees of problems with different ranges of predictability. Taken together, this line of sophisticated and computer-assisted proof approach can fulfill the requirements of accuracy, interpretability, predictability and reliance on mathematical sciences with the assistance of AI and machine learning being at the plinth of and at the intersection with different domains among many other related points in line with the concurrent technical analyses, computing processes, computational foundations and mathematical modeling. Consequently, as distinctive from the other ones, our special issue series provides a novel direction for stimulating, refreshing and innovative interdisciplinary, multidisciplinary and transdisciplinary understanding and research in model-based, data-driven modes to be able to obtain feasible accurate solutions, designed simulations, optimization processes, among many more. Hence, we address the theoretical reflections on how all these processes are modeled, merging all together the advanced methods, mathematical analyses, computational technologies, quantum means elaborating and exhibiting the implications of applicable approaches in real-world systems and other related domains.

Many robotic control architectures perform a continuous cycle of sensing, reasoning and acting, where that reasoning can be carried out in a reactive or deliberative form. Reactive methods are fast and provide the robot with high interaction and response capabilities. Deliberative reasoning is particularly suitable in robotic systems because it employs some form of forward projection (reasoning in depth about goals, pre-conditions, resources and timing constraints) and provides the robot reasonable responses in situations unforeseen by the designer. However, this reasoning, typically conducted using Artificial Intelligence techniques like Automated Planning (AP), is not effective for controlling autonomous agents which operate in complex and dynamic environments. Deliberative planning, although feasible in stable situations, takes too long in unexpected or changing situations which require re-planning. Therefore, planning cannot be done on-line in many complex robotic problems, where quick responses are frequently required. In this paper, we propose an alternative approach based on case-based policy learning which integrates deliberative reasoning through AP and reactive response time through reactive planning policies. The method is based on learning planning knowledge from actual experiences to obtain a case-based policy. The contribution of this paper is two fold. First, it is shown that the learned case-based policy produces reasonable and timely responses in complex environments. Second, it is also shown how one case-based policy that solves a particular problem can be reused to solve a similar but more complex problem in a transfer learning scope.

The mass separator of super heavy elements also known as MASHA setup from the automation systems point of view is described. The setup uses ISOL (Isotope Separation On-Line) method and represents the modular online mass spectrometer where masses of new elements from helium up to mass A=450 are measured directly in the course of their synthesis with accelerated heavy-ion beams. During the last years the control, the data acquisition and the beam diagnostics systems were modified. The modernization was in changing the platforms to the modern one from the CAMAC to the NI PXI and from the SmartBOX to the Wago PLC and the NI CompactRIO. The new systems were integrated and tested on the beam experiments with heavy ions at the U-400M cyclotron. The future experimental program of the MASHA setup for study of the physical and chemical properties of superheavy elements at the SHE factory is discussed.

With the mutual promotion and development of computing technology, communication technology and control technology, building control systems by multiple machines has become a trend. The reliability of communication mechanism in these systems is the basic guarantee of system effectiveness. In this paper, a communication mechanism suitable for centralized and distributed control systems is proposed, which attempts to provide an effective framework for the design of communication subsystems in the system. The mechanism includes the logical structure of the communication network, RPC of Client-Server used in a control system, protocol and its basic functions. The basic function of the protocol is concretization and leaves a large expansion space, so this mechanism can be directly applied to the engineering design of medium and small control system, and has good flexibility and scalability.

Using singularity theory tools we study the time averaged problem for control systems and profit densities on the circle. We show that the optimal strategy always can be selected within periodic cyclic motions and stationary strategies. When the problem depends additionally on a one dimensional parameter, we analyze generic transitions between these two types of strategies under change of the parameter and classify all generic singularities of the averaged profit as a function of the parameter.