Synchronous Mode Research Articles

QUAN ĐIỄM CỦA GIẢNG VIÊN VỀ VIỆC THỰC HIỆN CHƯƠNG TRÌNH TIẾNG ANH TRỰC TUYẾN ĐỒNG BỘ TẠI TRƯỜNG ĐẠI HỌC MỞ TP. HỔ CHÍ MINH

This study investigates lecturers' opinions on the non-major English program level A, which is currently being implemented in a synchronous online mode at Ho Chi Minh City Open University. A survey was conducted with the participation of lecturers who teach online for the program. Quantitative data were collected through questionnaires with the participation of all 23 lecturers, and qualitative data were obtained through interviews with 10 lecturers. The findings show that the lecturers believe that the program has prepared students quite well for the course and that both lecturers and students have the capacity and conditions to participate in this type of program. However, they still face various difficulties when teaching Speaking and Pronunciation skills as well as classroom management. These difficulties are due to the nature of synchronous online teaching, poor internet connection from students, technical issues, and most notably, some students’ attitudes that are not suitable for online learning. The lecturers also contributed some opinions on the regulations of the university on students’ behavior and exams for non-major English classes in synchronous online mode. Based on the above results, we propose some solutions to improve the effectiveness of the program.

The impact of the virtual classroom on the development of oral communicative competence in qualifying secondary school learners: A case study of the provincial direction of national education in Agadir

The period, mandated by health decision-makers in Morocco, was characterised by a certain improvisation in terms of the approaches used to enssure some form of pedagogical continuity. WhatsApp groups emerged as one of the methods adopted in an imposed manner by most teachers who were content with the digital tools available. In this context, our article aims to analyze the technical, didactic and pedagogical functioning of distance sessions programmed in synchronous mode, in order to determine whether virtual classes simulated real ones. It also seeks to study their effect on the development of oral communicative competence in Moroccan high school students. For this, we employed a mixed methodology: online questionnaire and semi-directive online interview process. The results indicated that no oral production was carried out, and hence there is a barrier to achieve an effective level of communicative competence during virtual sessions.

Effects of Process Parameters on Microstructure Evolution of Pure Aluminum Target under Clock 135° Collaborative Hot Rolling: A Simulation Study

Based on the results of current research and experiments, clock 135° hot rolling has been widely considered to be the preferred method in target rolling. However, it has been found that the process parameters are of vital importance during the rolling. Therefore, this work focuses mainly on the investigation of the effects of the roll speed ratio and offset distance on the microstructure evolution of a pure aluminum (Al) target under clock 135° collaborative hot rolling. Taking the ultra-thick pure Al metal circular billet as a research object, firstly, the evolutionary behavior of the effective strain (ES) and grain refinement for a rolled piece under the clock counterclockwise 135° synchronous and asynchronous (the speed ratios between rollers are set to be 1:1.05, 1:1.1, 1:1.2, respectively) rolling modes has been comparatively studied based on numerical simulation by DEFORM-3D software. It has been shown that a large ES value of 5.835 mm/mm is obtained in the lower surface layer by counterclockwise 135° asynchronous rolling with a roll–speed ratio of 1:1.2. Meanwhile, the average grain size below 80 µm accounts for ~61.8% of the total grains. These results demonstrate that the clock counterclockwise 135° asynchronous rolling method with a roll speed ratio of 1:1.2 should be an ideal strategy in obtaining finer grains. Unfortunately, nevertheless, the maximum degree of the bad plate shape is generated after forging by clock asynchronous rolling with a speed ratio of 1:1.2. As a consequence, clock counterclockwise 135° snake rolling with a 30 mm offset distance was proposed on the basis of a rolling speed ratio of 1:1.2, which perfectly corrected the warping plate shape caused by clock asynchronous rolling. What is more important, the minimum damage value of 1.60 was achieved accordingly. Meanwhile, the ES value increased in the core of the plate for the clock counterclockwise 135° snake rolling with all of the four offset distances compared to clock synchronous rolling. This study should be significantly conducive to guidance on setting process parameters in the industrial production of hot rolling metal or alloy targets.

Is the Term "Food Consistency" Used Consistently in Consumer Science? An Exploratory Study of Consumer Association and Conceptualization.

The post-pandemic context has changed the modes for collecting data in sensory and consumer science. The objectives of this research were to analyze consumers' associations of food consistency and to study two virtual modes of the Free Word Association test (FWA). This test was administered to 209 consumers (180 women, 29 men, 18-45 years old) asynchronously (i.e. self-administered) and synchronously (i.e. face to face interviews). The Cognitive Salience Index (CSI) was calculated, and the structure of the social representation was analyzed. Correspondence analysis showed that food consistency was a mixture of concepts related to structure, hardness and several aspects of auditory (e.g., Crunchy, Crispy), tactile (e.g. Smooth, Spreadable) and oral texture (e.g., Creamy, Gummy). Slightly consistent food was associated with something soft, liquid or semisolid, and very consistent food to something hard and resistant. Consistent food was more related to "very" than to "slightly consistent." The CSI depended on the stimulus presented (p < 0.05). Regarding the social representation structure, the central core had the highest CSI for all stimuli (CSI ≥ 0.13, p < 0.05). Consumers defined "very consistent, consistent and slightly consistent food" by naming more foods in the synchronous mode than in the asynchronous one. In the asynchronous mode, consumers took more time to complete the test. The virtual FWA test (asynchronous or synchronous) showed some differences in the associations of term consistency, due to the lack of spontaneity in the first minute. It is important to adjust the methodologies to standardize the times in both modes.

Collective oscillations in a three-dimensional spin model with non-reciprocal interactions

Abstract We study the onset of collective oscillations at low temperature in a three-dimensional spin model with non-reciprocal short-range interactions. Performing numerical simulations of the model, the presence of a continuous phase transition to global oscillations is confirmed by a finite-size scaling analysis, yielding values of the exponents β and ν compatible with both the three-dimensional XY and Ising equilibrium universality classes. By systematically varying the interaction range, we show that collective oscillations in this spin model actually result from two successive phase transitions: a mean-field phase transition over finite-size neighborhoods, which leads to the emergence of local noisy oscillators, and a synchronization transition of local noisy oscillators, which generates coherent macroscopic oscillations. Using a Fokker–Planck equation under a local mean-field approximation, we derive from the spin dynamics coupled Langevin equations for the complex amplitudes describing noisy oscillations on a mesoscopic scale. The phase diagram of these coupled equations is qualitatively obtained from a fully-connected (mean-field) approximation. This analytical approach allows us to clearly disentangle the onset of local and global oscillations, and to identify the two main control parameters, expressed as combinations of the microscopic parameters of the spin dynamics, that control the phase diagram of the model.

Strong coupling yields abrupt synchronization transitions in coupled oscillators

Coupled oscillator networks often display transitions between qualitatively different phase-locked solutions—such as synchrony and rotating wave solutions—following perturbation or parameter variation. In the limit of weak coupling, these transitions can be understood in terms of commonly studied phase approximations. As the coupling strength increases, however, predicting the location and criticality of transition, whether continuous or discontinuous, from the phase dynamics may depend on the order of the phase approximation—or a phase description of the network dynamics that neglects amplitudes may become impossible altogether. Here we analyze synchronization transitions and their criticality systematically for varying coupling strength in theory and experiments with coupled electrochemical oscillators. First, we analyze bifurcations analysis of synchrony and splay states in an abstract phase model and discuss conditions under which synchronization transitions with different criticalities are possible. In particular, we show that such conditions can be understood by considering the relative contributions of higher harmonics to the phase dynamics. Second, we illustrate that transitions with different criticality indeed occur in experimental systems. Third, we highlight that the amplitude dynamics observed in the experiments can be captured in a numerical bifurcation analysis of delay-coupled oscillators. Our results showcase that reduced order phase models may miss important features that one would expect in the dynamics of the full system. Published by the American Physical Society 2024

A novel technology for lithium extraction through low-temperature synergistic roasting of α-spodumene with lepidolite

α-Spodumene and lepidolite have garnered significant attention because of the booming market of lithium (Li) salts. However, α-spodumene requires a high-temperature phase transition step at ≥1050 °C, and lepidolite needs a large amount of sulfate to extract Li. The economic and environmental benefits of these processes need to be improved. Herein, an innovative scheme, low-temperature synergistic roasting of α-spodumene with lepidolite, was proposed to efficiently extract Li. The conditions for phase transition roasting, acid baking, and water leaching were sequentially optimized. The mechanisms underlying the synchronous phase transition of α-spodumene and lepidolite were studied using various characterization methods. Results indicated that lepidolite lowers the transition temperature of spodumene from the α-phase to the β-phase by 150 °C. During the synergistic roasting, fluorine escapes from the lepidolite lattice and attacks SiO, AlO, and LiO bonds within the α-spodumene lattice. Under optimal conditions, 97.60 % of the Li was extracted, while the leaching ratios of Na, K, Al, and Si were only 50.96 %, 10.96 %, 5.75 %, and 0.30 %, respectively. Current scheme is a promising low-carbon technology that synchronously achieves the low-temperature phase transition of α-spodumene and additive-free Li extraction from lepidolite, offering superior economics, environmental benefits, and practicality compared with the existing methods.

Neuron configuration enhances the synchronization dynamics in ring networks with heterogeneous firing patterns

Neuron synchronization is fundamental for brain dynamics. While several efforts have been made to understand neuron coupling between tonic and bursting neurons, the position of neurons in a neural network and its relationship with synchronization is not fully understood. This work investigates the impact of neuronal heterogeneity on firing pattern transitions and synchronization in networks comprising tonic and bursting neurons. Using the Huber-Braun model, we explore two configurations: one with neurons grouped closely and another with maximal separation. Our findings reveal that while increased coupling strength generally promotes synchronization, the specific mix of neuron types and their spatial configuration crucially modulate both synchronization and firing pattern transitions, leading to phenomena such as cluster synchronization and global phase synchrony. The results indicate that the configuration with maximal separation and with mostly bursting neurons synchronizes more quickly. Firing pattern transitions are also configuration-dependent. For example, in the network with half tonic and half bursting, the configuration grouped closely undergoes diverse complex dynamics transition in the synchronized state, while the other configuration synchronize with chaotic bursting dynamics. Behaviors like cluster synchronization are observed in this network. The study underscores the significance of considering neuronal heterogeneity in understanding network dynamics.

Eco-friendly second derivative synchronous fluorescence spectroscopic method for simultaneous determination of rosuvastatin and ezetimibe in pharmaceutical capsules

The fixed dose combination of Rosuvastatin and ezetimibe has recently received approval from the FDA for the treatment of elevated levels of low-density lipoprotein cholesterol in adults. Herein, an eco-friendly and highly sensitive spectrofluorimetric method was developed and validated for simultaneous determination of rosuvastatin and ezetimibe in commercial capsules. The developed method involved synchronous fluorescence spectroscopy combined with second derivative spectroscopy to resolve the overlapping fluorescence spectra of rosuvastatin and ezetimibe. The studied drugs were measured in synchronous mode at Δλ of 40 and their recorded synchronous fluorescence spectra were derivatized into second-order spectra, enabling the selective quantification of rosuvastatin and ezetimibe at 370 nm and 312 nm, respectively. Optimization studies regarding to the influence of buffer pH, incorporation of surfactant, choice of diluting solvent, and synchronous Δλ were carried out. The method was validated using the validation characteristics listed in ICH Q2(R1). The calibration curves displayed satisfactory linear relationships across the calibration range of 0.1–2 µg/mL for rosuvastatin and 0.05–3 µg/mL for ezetimibe. The methodology demonstrated robustness to minor modifications in the procedural parameters and selectivity in quantifying the studied drugs in synthetic mixed solutions and commercial capsules without interference. Furthermore, the level of environmental friendliness and sustainability of the suggested spectrofluorimetric approach was assessed in relation to two previously documented methodologies utilizing the AGREE metric. The findings indicated that the suggested method demonstrated a notably superior level of sustainability in comparison to the documented methods.

Single-Plane Versus Synchronous Biplane Ultrasound Imaging During Vascular Access Procedures: Which Is Better and How Can We Improve?

Highlights Biplane ultrasound imaging reduces need for probe manipulation during procedure. Providers note clinical benefit of biplane ultrasound imaging for vascular access. Biplane disadvantages include probe size, imaging quality, size of the screen. Thorough didactic and practical education is essential for biplane success. Abstract Background: Use of ultrasound guidance for vascular access procedures is commonplace in inpatient and outpatient care settings. Standard ultrasound probes offer the operator a single-plane view, necessitating rotation of probe to attain dual complimentary views. This mechanical probe rotation increases technical difficulty of ultrasound use. The purpose of this study is to evaluate the rate of cannulation success and efficiency of a synchronous biplane ultrasound mode in ultrasound-guided arterial line placement as compared with a standard single-view ultrasound mode in the operating room setting. Methods: Patients scheduled for elective surgery in which a radial arterial catheter would be used for hemodynamic monitoring were approached for consent to this study. Patients were randomized to either undergo placement with single-plane view versus synchronous biplane view; outcomes were recorded. Providers were provided preprocedural ultrasound education as well as the option of a short hands-on experience; their level of experience was noted. Results: Placement time of a peripheral arterial catheter was longer and required more attempts to be successful using synchronous biplane imaging as compared with single-plane imaging across providers of all skill/experience levels. Subjectively, providers noted the benefit of synchronous biplane imaging in vascular access; however, disadvantages including probe size, quality of imaging, and size of the screen were noted. Discussion: Thorough education regarding the use and functionality of biplane synchronous imaging in vascular access is essential. Additional guided-practice time with experienced operators could also be helpful to overcome the challenges observed in this study.

What Mistakes Can Be Made When Performing the Electrical Cardioversion Procedure?-Analysis of Emergency Medical Team Performance during the Championships in Emergency Medicine.

Medical personnel carrying out electrical cardioversion (EC) procedures must remember to have the R-wave sync mode switched on, use the correct energy and maintain personal safety. The defibrillators used by medical response teams most often switch out of cardioversion mode once a shock is delivered. Therefore, this mode must be switched on again before subsequent shocks are delivered. The main aim of the study was to assess the ability of emergency medical teams participating in emergency medicine championships to perform EC. The research was a retrospective observational study and was based on an analysis of the evaluation sheets from two tasks simulating the management of a patient with unstable tachycardia conducted during the International Winter Emergency Medicine Championships. Three-person teams consisting of paramedics and representing the Polish emergency services were included in the study. The team representing the championship organiser and the few foreign teams participating in the competition were excluded from the study. The decision to conduct EC was taken by 36 teams (83.72%) in 2015 and 27 teams (87.10%) in 2019. In both editions of the championships, during consecutive shocks, the percentage of actions performed correctly decreased significantly-switching on synchronisation mode in 2015 (94.4%, 83.33%, 72.22%) and in 2019 (100%, 88.89%, 81.48%); correct energies in 2015 (91.67%, 80.56%, 77.78%) and in 2019 (92.59%, 85.19%, 81.48%); shocks in a safe manner in 2015 (94.44%, 94.44%, 91.67%) and in 2019 (100%, 96.30%, 96.30%). Teams participating in the assessed tasks in a significant majority of cases correctly qualified the patient for EC, and correctly carried out the actions required for this procedure. It is of particular note that with every subsequent shock, the percentage of shocks carried out without the sync mode increased significantly.

Resonance-induced synchronization in coupled phase oscillators with bimodal frequency distribution and periodic coupling.

Synchronization behaviors in globally coupled phase oscillators with symmetric bimodal frequency distribution and periodic coupling are studied. It is found that by a proper setting of the frequency of the periodic coupling, the synchronization propensity of the oscillators can be markedly improved. Specifically, we show that when the frequency of the periodic coupling matches the distance of the central frequencies in the distribution, the critical coupling characterizing the onset of synchronization can be substantially decreased. The mechanism behind the phenomenon of periodic-coupling-enhanced synchronization is analyzed by the methods of Ott-Antonsen ansatz and synchronization transition tree, and it is revealed that the synchronization enhancement is attributed to the resonance between the synchronization clusters and the periodic coupling.

Analysis of the dynamical behavior of discrete memristor-coupled scale-free neural networks

The synchronization of neural networks is crucial for neural information processing and represents a key feature of various functional brain diseases. Memristors are ideal electronic components for mimicking biological synapses, among which discrete memristors have the advantage of fast computing speed and are often used in memristor-based neural networks. For these reasons, this paper proposes a novel discrete memristor-coupled Scale-Free neural network (DMSNN). Phase diagrams and time series of membrane potential are employed to analyze the firing pattern coexistence of individual neurons in the network. Furthermore, Spatiotemporal patterns, heat maps of the Spearman correlation coefficient matrix and the values of neuron membrane potential at a particular time point are adopted to declare the spatio-temporal dynamics of the complex neural network, encompassing asynchronization, chimeric state, synchronization and synchronization transition. The study also identifies the phenomenon of topology-induced coexistence and elucidates the underlying reasons for the emergence of chimeric states in the DMSNN as the coupling strength increases. Finally, a hardware implementation platform is constructed using a highly integrated SSD202 processor to validate the accuracy of the DMSNN. The results are consistent with the numerical simulations.

Integrating Google Classroom and Project-Based Learning for Enhanced Synchronous and Asynchronous Education

Some of them are present long before but other have been influenced by the modern challenges those are present have also radically changed the world why educational is organized and delivered, as numerous institutions have been forced from in-person classroom keeper instructing to a range of online modes. A critical tool for online learning has been Google Classroom, which combines synchronous learning (real-time) and asynchronous learning (self-paced). This work explores implementation, benefits, challenges, and effectiveness of Google Classroom fused with Project-Based Learning (PBL) for enhancing learning opportunities especially in English as Second Language (ESL) education. Results suggest that the hybrid mode allows for a balance between being less strict and enabling engagement, motivation, and achievement in those learning with other hybrid modes and those strictly synchronous or asynchronous modes.

A new fixed-time sliding mode control scheme for synchronization of chaotic systems

Chaotic synchronization is crucial in the field of secure communication, and fixed-time synchronization has realistic application prospects and demands. Aiming at the traditional sliding mode synchronization control method with chattering problem, based on the Lyapunov stability theory and incorporating the continuous time fixed time stability theorem, this paper proposed a new fixed-time sliding mode control scheme for synchronization of chaotic systems. The traditional finite-time sliding mode synchronization is compared with the proposed fixed-time sliding mode synchronization scheme and the results are discussed. The efficacy of the controller is validated using MATLAB simulations, which eliminates the chatter problem in the traditional sliding mode synchronization scheme and has the advantage of short synchronization time. In addition, the parameters of the controller can be set flexibly, which is an advantage of the fixed-time synchronous control scheme.

A new approach to instantaneous power based control of DFIG using active disturbance rejection control

This paper presents the design and control of a wind energy converter system that uses a doubly fed induction generator. It introduces a model that incorporates instantaneous active and reactive power as state variables to represent both the stator and rotor sides of the doubly fed induction generator. Subsequently, a linear active disturbance rejection-based control regulator is proposed for the control loops. The parameters of the linear active disturbance rejection controllers (ADRC) are adjusted by two methods: the single parameter tuning approach and the genetic algorithm (GA) method. The performance of the suggested controller is ultimately compared with a tuned proportional-integral (PI) control regulator in the MATLAB environment. In order to examine the performance of the system, various test scenarios are taken into account. These scenarios include different percentages of load change in both sub-synchronous and super-synchronous modes of operation, faults with different fault periods are considered in both modes of operation. The conventional PI controller-based approach requires 35 control cycles to reach steady state under load change conditions and 50 control cycles under symmetrical fault conditions, regardless of whether the system is operating in sub synchronous or super synchronous mode. The single parameter-based tuned ADRC approach only requires 15 control cycles for load change and 20 control cycles for symmetrical fault scenarios. In contrast, for both the load change and symmetrical fault scenarios, the ADRC with GA-based parameter tuning reaches steady state in just 5 control cycles. Furthermore, a case involving wind speed variation is presented to demonstrate the maximum power point tracking capability of the proposed control structure. Similarly, for the robustness test of the control regulator, variations in resistance and inductance are introduced to observe their impact on the response of the control regulators. Finally, a real-time hardware in loop setup is presented to validate the control strategy during the load change in the sub-synchronous mode of operation of the doubly fed induction generator.

Formation of Vibration Fields for a Mechatronic Platform Driven by Dual Asynchronous Motors

This paper investigates the formation of vibration fields in a mechatronic setup driven by dual induction motors, relying on the controlled synchronization of unbalanced rotors. The proposed algorithm enables precise control over rotor speeds and phase shifts. Experimental results from a multi-resonance vibration laboratory setup demonstrate this approach’s ability to form the vibration fields. The ability to control these fields is crucial for applications such as vibratory transportation and the mixing of bulk materials. The results obtained can ensure a diverse picture of the complex trajectories of motion for various points of the platform, primarily in the screens, making various useful effects for vibration technologies. Additionally, the practical value of this research is that in the case of double synchronous mode the ordinate of the lower point of the trajectory is lower than in the case of single synchronous mode, which improves the efficiency of unloading and prevents congestion formation. The experimental data highlight the practical advantages and potential improvements in efficiency and reliability offered by this method.

Exploration of the rapid growth of high-molecular-weight oligomers in the later stage of diisocyanate self-polymerization

Diisocyanate trimers are important products in the polyurethane industry. Large amounts of high-molecular-weight oligomers, which change the physical and chemical properties of the final products, are generated in the later stages of the synthesis process. It is important to understand the self-polymerization of high-molecular-weight oligomers. The relationship between this phenomenon and the PDI monomer concentration was first experimentally demonstrated. Then the self-polymerization of pentamethylene diisocyanate (PDI) was investigated via Dmol3 calculations using the linear synchronous transition (LST) and quadratic synchronous transition (QST) methods. Calculations revealed a new synthesis pathway with a reaction energy barrier of 31.548 kcal/mol, leading to the conversion of pentamers into decamers. In the initial stage of the polymerization reaction, the PDI monomer concentration was high, and the reaction followed a step-growth polymerization. However, as the PDI monomer concentration decreased and the PDI pentamer concentration increased in the later stage of the reaction, a new synthesis pathway emerged, leading to the direct synthesis of high-molecular-weight oligomers from the available oligomers. This study further demonstrates the self-polymerization of diisocyanate and provides a comprehensive understanding of its synthesis process. By inhibiting this new pathway can reduce the content of high-molecular-weight oligomers, increase yields, and reduce the product viscosity, all of which are beneficial for chemical process optimization.

Optimized estimation of marine deflection of the vertical from multibeam laser altimeter data of ICESat-2

SUMMARY Satellite altimetry data, with its increasing density and quality, has become the primary source for marine deflection of the vertical (DOV) and gravity anomaly modelling. Limited by orbital inclinations, the precision of the meridian component of the gridded deflection of the vertical (GDOV) calculated by traditional altimetry satellites is significantly better than that of the prime vertical component, and the excessive precision difference between these two components restricts the inversion precision of marine gravity anomaly model. The study of cross-track deflection of the vertical (CTDOV) is enabled by the multibeam synchronous observation mode of the new laser altimetry satellite, Ice, Cloud and Land Elevation Satellite-2 (ICESat-2). Based on the remove-restore method, residual geoid gradients are first calculated in this paper using three approaches: along-track (A-T), cross-track (C-T) and an integration of along-track and cross-track. Vertical deflections are then computed on a 1′ × 1′ grid using the least squares collocation (LSC) method, and the precision is verified against the SIO V32.1_DOV model. An optimized combination is proposed to address the issue of precision differences between the meridian and prime vertical components, and to enhance the precision of DOV inversion. A new DOV combination is formed by combining the meridian component from along-track deflection of the vertical (ATDOV) with the prime vertical component from cross-track deflection of the vertical (CTDOV) based on the remove-restore method. The Philippine Sea (0°–35°N, 120°–150°E) is selected as the test area to verify the feasibility of the optimized combination. The results indicate that the optimized combination of the meridian and prime vertical components achieved test precision of 2.63 and 3.33 μrad, respectively, when compared against the SIO V32.1_DOV model. The precision gap between the components is effectively narrowed by this approach, which maintains the precision of the meridian component and enhances that of the prime vertical component, thereby achieving optimal inversion precision for gravity anomalies.

Synchronization transitions in adaptive Kuramoto-Sakaguchi oscillators with higher-order interactions.

Coupled oscillators models help us in understanding the origin of synchronization phenomenon prevalent in both natural and artificial systems. Here, we study the coupled Kuramoto oscillator model having phase lag and adaptation in higher-order interactions. We find that the type of transition to synchronization changes from the first-order to second-order through tiered synchronization depending on the adaptation parameters. Phase lag enables this transition at a lower exponent of the adaptation parameters. Moreover, an interplay between the adaptation and phase lag parameters eliminates tiered synchronization, facilitating a direct transition from the first to second-order. In the thermodynamic limit, the Ott-Antonsen approach accurately describes all stationary and (un)stable states, with analytical results matching those obtained from numerical simulations for finite system sizes.