Triangular Maps Research Articles

Triangular Fuzzy Cognitive Maps Method for Modelling Interrelation between Causal and Trigger Factors Towards Students' Mathematics Problem-Solving Ability

The purpose of this study is to model the interrelation between causal and trigger factors towards students' mathematics problem-solving ability by using the triangular fuzzy cognitive maps (TrFCM) method. Selection weaknesses and limitations in the method of relational analysis cause the interrelation and influence between variables not to be visualised and do not reveal the characteristics of the actual interaction. As a result, this study demonstrates TrFCM as a more effective way of analysing the relationship between variables based on the complexity that happens in analysing causal factors and triggers for students' problem-solving abilities in mathematics. The results of the influential relations map (IRM) demonstrate that emotion and metacognition are the triggers for problem-solving ability. While executive function is the main cause of success in completing mathematics problems, it is also influenced by additional factors such as motivation, attention, and working memory. These causal and triggering factors also mobilise parts of students' cognitive and behavioural performance to improve the process of solving mathematics problems. Based on the outcomes of this study, computational intelligence methods like fuzzy systems give useful procedures for analysing data from expert surveys. The TrFCM method offers a more accurate relational analysis procedure in modelling interrelation between human factors.

Density estimation via measure transport: Outlook for applications in the biological sciences

AbstractOne among several advantages of measure transport methods is that they allow or a unified framework for processing and analysis of data distributed according to a wide class of probability measures. Within this context, we present results from computational studies aimed at assessing the potential of measure transport techniques, specifically, the use of triangular transport maps, as part of a workflow intended to support research in the biological sciences. Scenarios characterized by the availability of limited amount of sample data, which are common in domains such as radiation biology, are of particular interest. We find that when estimating a distribution density function given limited amount of sample data, adaptive transport maps are advantageous. In particular, statistics gathered from computing series of adaptive transport maps, trained on a series of randomly chosen subsets of the set of available data samples, leads to uncovering information hidden in the data. As a result, in the radiation biology application considered here, this approach provides a tool for generating hypotheses about gene relationships and their dynamics under radiation exposure.

On stable and quasi-chaotic regimes in a one-dimensional unimodal mapping obtained by modeling the dynamics of a biological population

The paper considers the properties of the difference equation describing the dynamics of the animal population, obtained earlier in the framework of studies of tundra communities. We have considered a special case in which the model is represented by a one-parameter difference equation that defines a one-dimensional unimodal mapping of a segment into itself, similar to the well-known triangular (tent) mapping, supplemented by a region with a constant value. A change in the mapping parameter generates a bifurcation scenario, in which stability zones arise, characterized by orbits of a constant period, interspersed with zones with more complicated, “quasi-chaotic” regimes. Based on the properties of the n-iterated triangular mapping, a necessary and sufficient condition for the localization of cyclic orbits in the considered type of unimodal mappings is formulated, which makes it possible to identify stability regions for any given period n. On the basis of this condition an algorithm for detecting stability zones is proposed. The main subject of the study is the fractal properties of the set, which is the complement of the obtained set of stability regions to the entire domain of mapping definition. The dynamics of the DH (n) value is obtained, the limit of which at n → ∞ is equal to the fractal dimension dH . It is shown that in the studied range of n (2 ≤ n ≤ 22), the DH (n) < 0.9, which suggest that dH < 1. If so, then according to the definition of a fractal set, its topological dimension is dT = 0, which means that the complement of the set of stability regions consists of isolated points.

Minimality and distributional chaos in triangular maps

The result of this paper contributes to the classification of triangular maps of the square with zero topological entropy stated by A. N. Sharkovsky in the 1980s. The problem was if a triangular map of the square such that its any ω-limit set contains unique minimal set can be distributionally chaotic. So far such result was disproved only for the class of triangular maps non-decreasing on fibres [L. Paganoni, J. Smítal, Strange distributionally chaotic triangular maps, Chaos Solitons Fractals 26(2) (2005), pp. 581–589]. In this paper, we solve the problem in negative for all triangular maps of the square, correcting the original result from Balibrea and Smítal [Strong distributional chaos and minimal sets, Topology appl. 156 (2009), pp. 1673–1678].

Fast three-dimensional simulation based on data constraints and methods to improve the realism of knitted sweaters

The purpose of this research is to use the methods of data constraints for parametric design and three-dimensional simulation of knitted sweaters to achieve the purpose of enhancing the simulation effect. First, based on the basic plates of knitted sweaters as a reference, the parameters of the curve parts of the plates were constrained, and the constraint relationships were applied to build the parametric design models. After entering the user parameters, the changed plates can be obtained. On this basis, the transformation of flat plates to three-dimensional surfaces was carried out. The triangular grid mapping relationships between two-dimensional plates and three-dimensional surfaces were constructed, and the physical properties were further specified for the mesh particles. When the surface texture parameters of the two-dimensional plane are designed, the three-dimensional surface will change accordingly. In addition, this research put forward the stitching and matching methods of the seams of knitted sweaters. The joint size was equally divided, and the number of coils was converted into the number of triangular mesh. Point-to-point mesh was used to realize the accurate counterpoint fusion of coils. The feasibility of parametric and virtual design methods was verified by taking an example, which provides ideas for the computer-aided design of knitted sweaters and promotes the improvement of production efficiency without wasting raw materials.

Concentric triangular mapping with iterative substitution method for PAPR reduction in OCDM system without SI

Abstract OCDM, a new MCM scheme, has been proposed in the literature. Like widely used OFDM, OCDM also has a drawback of high PAPR. PAPR reduction schemes like SLM and PTS require information about phase factors i.e. SI used at the transmitting end for PAPR reduction for recovery of original data signal at the receiver. Transmission and recovery of SI is always a concern. In this paper a new CTM-IS method has been proposed for PAPR reduction of OCDM signal without SI. Main objective of the proposed scheme is to improve the error performance of SI free SLM and PTS based schemes without any degradation in PAPR reduction. PAPR performance and SER performance, over AWGN as well as multipath Rayleigh fading channels, of OCDM signal with proposed CTM-IS scheme is evaluated. Analytical expression for SER of the proposed CTM-IS scheme over AWGN channel is also derived. The computational complexity of the proposed CTM-IS scheme is also presented. Performance of the proposed CTM-IS scheme is compared with the existing MSM combined with SLM (MSM-SLM), CCM combined with SLM (CCM-SLM) and OM combined with SLM (OM-SLM) schemes. The proposed CTM-IS scheme has demonstrated to offer a theoretical power gain of about 3.01 dB, 2.55 dB, and 14.23 dB over MSM-SLM, CCM-SLM and OM-SLM schemes, respectively. Proposed CTM-IS scheme provides better SER performance with respect to MSM-SLM, CCM-SLM and OM-SLM without any loss in the PAPR performance. Moreover, computational complexity of CTM-IS scheme is less than that of MSM-SLM and CCM-SLM schemes.

Scalable Bayesian Transport Maps for High-Dimensional Non-Gaussian Spatial Fields

A multivariate distribution can be described by a triangular transport map from the target distribution to a simple reference distribution. We propose Bayesian nonparametric inference on the transport map by modeling its components using Gaussian processes. This enables regularization and uncertainty quantification of the map estimation, while resulting in a closed-form and invertible posterior map. We then focus on inferring the distribution of a nonstationary spatial field from a small number of replicates. We develop specific transport-map priors that are highly flexible and are motivated by the behavior of a large class of stochastic processes. Our approach is scalable to high-dimensional distributions due to data-dependent sparsity and parallel computations. We also discuss extensions, including Dirichlet process mixtures for flexible marginals. We present numerical results to demonstrate the accuracy, scalability, and usefulness of our methods, including statistical emulation of non-Gaussian climate-model output. Supplementary materials for this article are available online.

Complex chaotic attractor via fractal process with parabolic map and triangular map

Complex chaotic attractor via fractal process with parabolic map and triangular map

Modular fuzzy control for nonlinear multipoint based core model of TMI PWR under variable fuel cycle conditions

This paper provides design of modular fuzzy controller based on reactor power for multipoint kinetic model of TMI PWR, under transients and variable operational fuel cycle conditions. The validated four-point kinetic model is developed considering six groups of delayed neutron precursors, reactivity feedbacks and diffusion among selected nodes. Fuel burn-up effects and power variations are modeled using the variable thermal coefficients. The ability of adaptive multi-module fuzzy controller for reactor power is evaluated and compared at start and end of burn-up cycle. The power dependant nonlinearity of reactor core and time dependant variations of reactivity characteristics are handled with modular approach of TS fuzzy control. The performance monitoring and control improvements are achieved by online parametric identification of process model. The four modules are distributed on the power base line using the triangular mapping. Design of fuzzy controller is based on the optimized gain requirement at different power levels, error and error change. Control output is mapped as biased linear function of selected input variables using TS type fuzzy logic inference. Safety analysis is performed with Lyapunov function criteria for multipoint model. Transient analysis is performed for power tracking and accident handling ability of proposed control design and compared with the PID controller. The power variation among the adjacent nodes is reduced up to 5% under load following transients with the proposed model. Modular fuzzy approach found proficient in handling the changing operational conditions and variable dynamics of the system.

Common fixed point theorem for pair of quasi triangular α-orbital admissible mappings in complete metric space with application

In this paper, we establish a common fixed point theorem for pair of quasi triangular α-orbital admissible with an interpolative (φ, ψ)- Banach-Kannan-Chatterjea type Z-contraction mappings with respect to simulation function in complete metric space. An illustrative example is furnished to validate our main result. Our result extends the result of M. S. Khan et al. [15]. As an application, we provide the existence of a solution for a nonlinear Fredholm integral equations.

Extended Fuzzy Rule Suram for Coffee Drying System

Extended fuzzy rule Suram is an algorithm developed to control a drying system using diesel as an energy source by modifying the value of the fuzzy membership function {0.5,1]. For a coffee drying room control system with solar energy, the bleak rule is based on fuzzy logic with variables of weather, air condition and wind speed. This algorithm is applied to the coffee drying process. The state variable membership function is represented in error values ​​and changes in error with a typical triangular and trapezoidal map for weather variables, air conditions, while wind speed is expressed in terms of wind speed. The results of the analysis of experiments with 16 fuzzy rules to control system output according to weather conditions obtained optimization of the use of solar energy by minimizing the use of electrical energy by heating. This algorithm also adjusts the coffee drying schedule by controlling the chamber, namely through temperature and humidity control. The results of the application of this algorithm show that the efficiency of electrical energy reaches 40,86%.

Remarks on the Belitskii–Lyubich and discrete Markus–Yamabe conjectures

Abstract In this paper we prove the Belitskii–Lyubich conjecture for triangular and gradient mappings. These results resemble those obtained for the discrete Markus–Yamabe conjecture. However, the proofs are quite different, which sheds new light on the subject. We complete the picture by showing that the general versions of the two conjectures can be turned into theorems at little cost, simply by relaxing their spectral condition.

Fast quantum image encryption scheme based on multilayer short memory fractional order Lotka-Volterra system and dual-scale triangular map

The Caputo fractional order Lotka-Volterra system is time-consuming in practical applications, since its starting point is fixed. To tackle this problem, a short memory fractional order Lotka-Volterra system (SMFrLVS) is proposed, where the chaotic attractor of the short memory fractional order Lotka-Volterra system is achieved by the predictor-corrector method. Then, a multilayer fractional order Lotka-Volterra system with short memory (MSMFrLVS) is introduced, whose chaotic behaviors are explored via Poincare sections and frequency power spectra. A quantum image encryption algorithm is proposed by combining MSMFrLVS with quantum dual-scale triangular map. A quantum circuit of the dual-scale triangular map is designed with ADDER-MOD2n. At the permutation stage, the plaintext image is transformed into quantum form with the generalized quantum image representation model. The resulting quantum image is divided into sub-blocks and scrambled by the quantum dual-scale triangular map. Subsequently, the intra and the inter permutation operations on bit-planes are realized by sorting pseudo-random sequence and by quantum Gray code, respectively. At the diffusion stage, the initial values of the MSMFrLVS are generated with a plaintext correlation mechanism. The ciphertext image can be acquired by carrying out three-level diffusion operations. It is demonstrated that the proposed quantum image encryption algorithm performs better than some typical image encryption algorithm in terms of security, robustness, computational complexity and encryption speed.

Petrie Paths in Triangular Normalizer Maps

This study is devoted to investigate the Petrie paths in the normalizer maps and regular triangular maps corresponding to the subgroups $\Gamma_0(N)$ of the modular group $\Gamma$. We show that each regular triangular map admits a closed Petrie path. Thus, for each regular map, we find the Petrie length of the corresponding map.

Optimizing Low-Complexity Analog Mappings for Low-Power Sensors With Energy Scheduling Capabilities

Power consumption is a major challenge for a massive deployment of wireless sensors in Internet of Things (IoT) networks. This article studies the use of analog joint source-channel coding (AJSCC) mappings in low-power sensing schemes. In particular, we propose a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">triangular</i> mapping geometry as a low-complexity dimension reduction mapping. The proposed triangular mapping is employed for analog compression of multiple sensor readings into one signal and, thus, limits the need for power-hungry analog-to-digital conversion and processing at the sensing nodes. A comprehensive performance analysis of the proposed triangular mapping in terms of the mean squared error (MSE) performance is provided analytically and verified numerically. The problem of mapping adaptation to different source distributions is also studied. Moreover, the proposed triangular mapping is adopted in an energy scheduling problem in which the sensing nodes schedule their use of the received powers at different time instants and adjust the mapping parameters accordingly with the goal of minimizing the sum distortion at the receiver. We present a fast low-complexity algorithm for optimal energy scheduling and verify its performance in comparison with commercial convex optimization solvers. It is shown that the proposed mapping provides a very good MSE performance compared to the AJSCC benchmarks despite having a much lower complexity circuit implementation.

Common Fixed-Points Technique for the Existence of a Solution to Fractional Integro-Differential Equations via Orthogonal Branciari Metric Spaces

The idea of symmetry is a built-in feature of the metric function. In this paper, we investigate the existence and uniqueness of a fixed point of certain contraction via orthogonal triangular α-orbital admissible mapping in the context of orthogonal complete Branciari metric spaces endowed with a transitive binary relation. Our results generalize and extend some pioneering results in the literature. Furthermore, the existence criteria of the solutions to fractional integro-differential equations are established to demonstrate the applicability of our results.

Commensuration effects on skyrmion Hall angle and drag for manipulation of skyrmions on two-dimensional periodic substrates

We examine the dynamics of an individually driven skyrmion moving through a background lattice of skyrmions coupled to a 2D periodic substrate as we vary the ratio of the number of skyrmions to the number of pinning sites across commensurate and incommensurate conditions. As the skyrmion density increases, the skyrmion Hall angle is nonmonotonic, dropping to low or zero values in commensurate states and rising to an enhanced value in incommensurate states. Under commensuration, the driven skyrmion is channeled by a symmetry direction of the pinning array and exhibits an increased velocity. At fillings for which the skyrmion Hall angle is zero, the velocity has a narrow band noise signature, while for incommensurate fillings, the skyrmion motion is disordered and the velocity noise is broad band. Under commensurate conditions, multi-step depinning transitions appear and the skyrmion Hall angle is zero at low drives but becomes finite at higher drives, while at incommensurate fillings there is only a single depinning transition. As the gyrotropic component of the skyrmion dynamics, called the Magnus force, increases, peaks in the velocity that appear in commensurate regimes cross over to dips, and new types of directional locking effects can arise in which the skyrmion travels along other symmetry directions of the background lattice. At large Magnus forces, and particularly at commensurate fillings, the driven skyrmion can experience a velocity boost in which the skyrmion moves faster than the applied drive due to the alignment of the Magnus-induced velocity with the driving direction. In some cases, an increase of the Magnus force can produce regimes of enhanced pinning when the skyrmion is forced to move along a nonsymmetry direction of the periodic pinning array. This is in contrast to systems with random pinning, where increasing the Magnus force generally reduces the pinning effect. We demonstrate these dynamics for both square and triangular substrates and map out the different regimes as a function of filling fraction, pinning force, and the strength of the Magnus force in a series of dynamic phase diagrams.

Design of a reflective LED automotive headlamp lighting system based on a free-form surface.

In this paper, a design scheme of an automobile low-beam light illumination system with a reflector free-form surface is presented. Per the energy grid mapping theory and edge ray theory, an oblique triangular energy mapping scheme was proposed, and the free-form reflector was obtained by geometric iterative calculation. The Monte Carlo ray tracing method was used to simulate the system. The simulation results met the regulatory requirements of ECE R112, and the light efficiency could reach 81%. The length, width, and height of the reflector were 55.61mm×22.02mm×23.70mm. The reflector achieves a special light type through precise light distribution, does not need other shielding objects, and has the advantage of small size.

On \({\mathcal{F}}\)-Contractions for Weak α-Admissible Mappings in Metric-Like Spaces

In the paper, we consider some fixed point results of F-contractions for triangular α-admissible and triangular weak α-admissible mappings in metric-like spaces. The results on F-contraction type mappings in the context of metric-like spaces are generalized, improved, unified, and enriched. We prove the main result but using only the property (F1) of the strictly increasing mapping F:0,+∞→−∞,+∞. Our approach gives a proper generalization of several results given in current literature.

Elliptic Problems in Curved Domains Using Cubature Points Based Triangular Spectral Elements and Isoparametric Mappings

Elliptic Problems in Curved Domains Using Cubature Points Based Triangular Spectral Elements and Isoparametric Mappings