Triangle Function Research Articles

Joint grey time windows and two-dimensional loading design for modelling and solving multi-objective vehicle routing problems

This study defines grey time windows as periods during which customers expect deliveries at a specific time, with an allowance for a certain amount of advance or delay. For model solving, the grey time window is first whitened using the whitening triangle function based on different satisfaction thresholds and corresponding delivery costs. A tightness function is then designed for two-dimensional loading, which measures the effectiveness of different key points and loading postures. Finally, the Iori dataset was utilised to verify the effectiveness of the proposed solution algorithm. To the best of our knowledge, this is the first time the joint design concept of grey time windows and the whitening method have been introduced. Additionally, a grey opportunity-constrained programming model was constructed, with the model transformation and solution methods provided. Simulation results demonstrate that the proposed two-dimensional loading algorithm is both simple and efficient, and that the constructed grey chance-constrained programming model is solvable and has practical application value. Ultimately, this study provides insights for logistics companies on optimising delivery plans under various time window concepts.

On generalizations of some fixed point theorems in semimetric spaces with triangle functions

In the present study, we prove generalizations of Banach, Kannan, Chatterjea, Ćirić-Reich-Rus fixed point theorems, as well as of the fixed point theorem for mapping contracting perimeters of triangles. We consider corresponding mappings in semimetric spaces with triangle functions introduced by Bessenyei and Páles. Such an approach allows us to derive corollaries for various types of semimetric spaces such as metric spaces, ultrametric spaces, and b-metric spaces. The significance of these generalized theorems extends across multiple disciplines, such as optimization, mathematical modeling, and computer science. They may serve to establish stability conditions, demonstrate the existence of optimal solutions, and improve algorithm design.

Wilson networks in AdS and global conformal blocks

We develop the relation between gravitational Wilson line networks, defined as a particular product of Wilson line operators averaged over the cap states, and conformal correlators in the context of the AdS2/CFT1 correspondence. The n-point sl(2,R) comb channel global conformal block in CFT1 is explicitly calculated by means of the extrapolate dictionary relation from the gravitational Wilson line network with n boundary endpoints stretched in AdS2. Remarkably, the Wilson line calculation directly yields the conformal block in a particularly simple form which up to the leg factor is given by the comb function of cross-ratios. It is also found that the comb channel structure constants are expressed in terms of factorials and triangle functions of conformal weights which form determines fusion rules for a given 3-valent vertex. We obtain analytic expressions for the Wilson line matrix elements in AdS2 which are building blocks of the Wilson line networks. We analyze general cap states and specify those which lead to asymptotic values of the Wilson line networks interpreted as boundary correlators of CFT1 primary operators. The cases of (in)finite-dimensional sl(2,R) modules carried by Wilson lines are treated on equal footing that boils down to consideration of singular submodules and their contributions to the Wilson line matrix elements.

Switched-Capacitor-Based Hybrid Clamped Converter for Wide Power Factor Applications

The clamped multilevel converter generally adopts the zero-sequence voltage injection method to realize the voltage balancing of the DC link capacitors. However, the relationship between the reference modulation voltage and the duty ratio of phase neutral point current is often a triangle or trapezoidal function. When the power factor is low, the required optimal zero-sequence voltage will be out of bounds, which will lead the capacitor voltage ripples to become larger or even divergent. Therefore, a novel switched-capacitor-based hybrid clamped(SCB-HC) converter is proposed in this article. The new redundant switching states are obtained by introducing the boost characteristics of the switched-capacitor(SC) circuit. And a hybrid phase-shifted pulse width modulation(PS-PWM) is developed to extend its functional relationship. This ensures the peak current that flow out of the neutral points is fewer, then the capacitor voltage drift is reduced significantly. Moreover, the proposed hybrid PS-PWM method not only ensures that SC circuit switches series and parallel patter once during a carrier period, but also achieves the voltage balancing of SC and DC link midpoint capacitor naturally under ideal conditions. On top of that, the capacitor voltage balancing method is designed to ensure independent and decoupled control of all capacitor voltages. Simulations and experiments show that the proposed topology, modulation method and voltage control method are practical and feasible for wide power factor applications.

Coupled vibration characteristics of marine shafting with complex change of inclination under ice loads

With the intensifying exploitation of the resources and the waterway of the Arctic region in worldwide, the variable inclination propulsion system are gradually used in the ice-sailing ship, its design and strength study are of great engineering significance and application value. A variable inclination propulsion shafting with universal joint is introduced and its mechanical model is established in this paper. By combine torsional stress and inclination angle calculation, a decoupling algorithm based on Newmark is designed, and then the analysis of ice loads response in both frequency domain and time domain are completed. The results show that due to the complex triangle function relation among the various components of the shafting, there is an interaction between the propeller and the transmission shafting. In particular, when considering the dynamic change of inclination, nonlinear bending-torsional coupling vibration may occur and further lead to the enhancement effect of torsional vibration. Among the three working conditions defined by DNV, the bending-torsional coupling effect vary with rotation speed and inclination angle, and is mainly concentrated in high frequency bands. In the ice loads condition 2, with the largest time domain excitation amplitude and the smallest high order component, it’s vibration response is the strongest, and the change law of dangerous phase angle is similar to the bending moment. With the increase of initial inclination angle, the amplitude of low frequency vibration decreases, yet the amplitude of high frequency vibration increase. This trend is more obvious with the increase of rotation speed, and more high frequency resonance points appear in the condition 1 and condition 3.

Using Fourier series to obtain cross periodic wall response factors

Wall periodic response factors are a very usual calculation method of transient heat transfer through building envelope elements (walls, roofs …) in steady periodic conditions, used in popular heat load calculation procedures as ASHRAE’s RTS method [Spitler, Jeffrey D., Daniel E. Fisher, and Curtis O. Pedersen. 1997. “The Radiant Time Series Cooling Load Calculation Procedure.” ASHRAE Transactions 103 (2): 503–515]. This response factors, time sampled heat flux responses of a multi-layer wall to a 24h-periodic unit triangle function, can be obtained by means of multiple methods: Laplace’s method, state space method, frequency domain methods, etc. These methods are numerical since there is no analytical way of obtaining these response factors. The aim of this work is, taking advantage of the periodic nature of excitations, use Fourier series to represent boundary conditions, and this way find an easier and less computationally demanding procedure to calculate these response factors. Additionally, the convergence of these Fourier series will be analysed to determine the minimum set of frequencies needed to ensure a fixed admissible error for wall periodic response factors.

Multiplicative brownian motion stabilizes traveling wave solutions and dynamical behavior analysis of the stochastic Davey–Stewartson equations

This article investigates the traveling wave solutions of the stochastic Davey–Stewartson equations based on the theory of the complete discriminant system of quadratic polynomials. Firstly, the stochastic Davey–Stewartson equations are transformed into the nonlinear ordinary differential equations through traveling wave transformation. By using Maple software, some two-dimensional and three-dimensional phase portraits are drawn. Secondly, according to the classification of the roots of quadratic polynomial equation, new traveling wave solutions of the stochastic Davey–Stewartson equations are obtained by using the complete discriminant system, which include rational function solutions, triangle function solutions and the Jacobi elliptic function solutions. Finally, three-dimensional-surface plots and two-dimensional-shape plots of some obtained solutions are drawn by using Maple software.

New optical soliton solutions to the coupled fractional Lakshmanan–Porsezian–Daniel equations with Kerr law of nonlinearity

This article studies the optical soliton solutions of the coupled fractional Lakshmanan–Porsezian–Daniel equations with kerr’s law nonlinearity based on the complete discriminant system of cubic polynomials. By means of traveling wave transformation, the fractional Lakshmanan–Porsezian–Daniel equations are simplified into two ordinary differential equations. Then, according to the classification of the roots of cubic polynomials, the optical soliton solutions of the fractional model with Kerr’s law nonlinearity are traversed. Then new traveling wave solutions are obtained, including the rational solutions, triangle function solutions, implicit function solutions and Jacobian elliptic function solutions, and the corresponding solutions are shown in two-dimensional and three-dimensional figures. Compared with previous related research, the exact optical soliton solutions of the LPD equation system for nonlinear Kerr’s law presented in this study are more completely and clearly classified, and researchers are easier to obtain these solutions.

A note on generalized four-point inequality

In 2017, M. Bessenyei and Z. Páles [1] introduced a definition of a triangle function that generates a concept of a generalized triangle inequality in semimetric spaces. Inspired by this concept, we discuss already known inequalities in metric spaces that relate six distances determined by four points and introduce a definition of the generalized four-point inequality in semimetric spaces. Conditions under which quasisymmetric mappings and quasi-Möbius mappings between semimetric spaces preserve such an inequality are obtained.

A note on generalized four-point inequality

In 2017, M. Bessenyei and Z. Páles [1] introduced a definition of a triangle function that generates a concept of a generalized triangle inequality in semimetric spaces. Inspired by this concept, we discuss already known inequalities in metric spaces that relate six distances determined by four points and introduce a definition of the generalized four-point inequality in semimetric spaces. Conditions under which quasisymmetric mappings and quasi-Möbius mappings between semimetric spaces preserve such an inequality are obtained.

On the definition of probabilistic metric spaces by means of fuzzy measures

Metric spaces are defined in terms of a space and a metric, or distance. Probabilistic metric spaces are a useful extension of metric spaces where the distance is a distribution instead of a number. In this way, we can take into account uncertainty. Then, the triangle inequality is replaced by a condition based on triangle functions on the distributions. In this paper we introduce F-spaces. This is a new type of probabilistic metric spaces which is based on fuzzy measures (also known as non-additive measures and capacities). We prove some properties that describe which families of fuzzy measures are compatible with which type of triangle functions. Then, we show how we can use Sugeno, Choquet integrals, and, in general, any other fuzzy integral as a tool for building these spaces. We show how these results can be used to compute distances between functions. We illustrate the example comparing three types of means when applied to a set of databases. The example uses Sugeno λ-measures to illustrate the theoretical results presented in the paper.

Triangular bright solitons in nonlinear optics and Bose-Einstein condensates.

We demonstrate what we believe to be novel triangular bright solitons that can be supported by the nonlinear Schrödinger equation with inhomogeneous Kerr-like nonlinearity and external harmonic potential, which can be realized in nonlinear optics and Bose-Einstein condensates. The profiles of these solitons are quite different from the common Gaussian or sech envelope beams, as their tops and bottoms are similar to the triangle and inverted triangle functions, respectively. The self-defocusing nonlinearity gives rise to the triangle-up solitons, while the self-focusing nonlinearity supports the triangle-down solitons. Here, we restrict our attention only to the lowest-order fundamental triangular solitons. All such solitons are stable, which is demonstrated by the linear stability analysis and also clarified by direct numerical simulations. In addition, the modulated propagation of both types of triangular solitons, with the modulated parameter being the strength of nonlinearity, is also presented. We find that such propagation is strongly affected by the form of the modulation of the nonlinearity. For example, the sudden change of the modulated parameter causes instabilities in the solitons, whereas the gradual variation generates stable solitons. Also, a periodic variation of the parameter causes the regular oscillation of solitons, with the same period. Interestingly, the triangle-up and triangle-down solitons can change into each other, when the parameter changes the sign.

Explicit soliton solutions of (3+1)-dimensional nonlinear Schrödinger equation with time variable coefficients

Based on the extended zero curvature equation, we proposed a (3+1)-dimensional nonlinear Schrödinger equation with time variable coefficients. This model has important application in the dispersion and nonlinearity managed fiber laser. In this paper, taking advantage of four different assumptions, soliton solutions and triangle function solutions of (3+1)-dimensional nonlinear Schrödinger equation with time variable coefficients are derived. Both amplitude and wave speed are found to be related to time-dependent coefficients.

On the categories of probabilistic approach groups: Actions

Starting with the category of probabilistic approach groups, we show that the category of approach groups can be embedded into the category of probabilistic approach groups as a bicoreflective subcategory; further, considering a category of probabilistic topological convergence groups, we show that the category of probabilistic topological convergence groups is isomorphic to the category of probabilistic approach groups under so-called triangle function ?: ?+ ? ?+ ?? ?+, where ?+ is the set of all distance distribution functions that plays a central role for probabilistic metric spaces. Moreover, if we allow this triangle function ? to be sup-continuous, then we can show that the category of probabilistic metric groups can be embedded into the category of probabilistic approach groups as a coreflective subcategory. Furthermore, we demonstrate that every T1 probabilistic topological convergence group satisfying so-called (PM) axiom is probabilistic metrizable. Finally, among others, introducing a category of probabilistic approach transformation groups, we show that the category of probabilistic topological convergence transformation groups is isomorphic to the category of probabilistic approach transformation groups; this solves an open problem that proposed in one of our earlier papers. Moreover, we prove that the category of probabilistic metric transformation groups is isomorphic to the category of probabilistic metric probabilistic convergence transformation groups.

Dynamic analysis of new two‐dimensional fractional‐order discrete chaotic map and its application in cryptosystem

A new fractional difference equation two‐dimensional triangle function combination discrete chaotic map (2D‐TFCDM) based on Caputo derivative is proposed. Using the bifurcation diagram, the maximum Lyapunov exponent, and the phase diagram, the numerical solutions of the fractional difference equations are obtained, and the chaotic behavior is observed numerically. After encrypting the key with elliptic curve cryptosystem, the fractional map is developed as an encryption algorithm and applied to color image encryption. Finally, the proposed encryption system is systematically analyzed from five main aspects, and the results show that the proposed encryption system has a good encryption effect.

New Optical Solitons in Bragg Grating Fibers for the Nonlinear Coupled ( 2 + 1 )-Dimensional Kundu-Mukherjee-Naskar System via Complete Discrimination System Method

In this paper, the coupled Kundu-Mukherjee-Naskar (KMN) model in Bragg grating fibers is considered to retrieve some new optical soliton solutions in ( 2 + 1 ) dimensions. Plenty of new exact solutions, including rational function solutions and triangle function solutions, in addition to the Jacobian elliptic function solutions, are obtained by using a complete discrimination system method. The 3D-surface plots, 2D-shape plots, and corresponding 2D contour plots of some obtained solutions are drawn, which provides a visualized structures and propagation of solitons. The novel results are new and show the effectiveness of the proposed method.

Fuzzy logic based elbow strength prediction for lateral epicondylitis

We proposed and implemented a system based on fuzzy logic Toolbox. We designed such a system to measure the strength of the upper limb elbow of a tennis player. We have tried to predict the injuries and movements for the sports player, For that we used the Triangle and Trapezoidal membership function for better prediction of the elbow strength. We have selected Two input parameters Elbow Flexion Angle and Torque in the system development using fuzzy logic toolbox in Matlab. We have considered these two inputs and based on that we have developed membership functions. After we get the elbow strength levels we will be able to predict the level of injuries. By measuring the elbow strength, comments can be made on the chances of player getting elbow related injuries.

On quasisymmetric mappings in semimetric spaces

The class of quasisymmetric mappings on the real axis was first introduced by Beurling and Ahlfors in 1956. In 1980 Tukia and Väisälä considered these mappings between general metric spaces. In our paper we generalize the concept of a quasisymmetric mapping to the case of general semimetric spaces and study some properties of these mappings. In particular, conditions under which quasisymmetric mappings preserve triangle functions, Ptolemy's inequality and the relation "to lie between" are found. Considering quasisymmetric mappings between semimetric spaces with different triangle functions we give a new estimate for the ratio of diameters of two subsets, which are images of two bounded subsets. This result generalizes the well-known Tukia-Väisälä inequality. Moreover, we study connections between quasisymmetric mappings and weak similarities which form a special class of mappings between semimetric spaces.

New Exact Solutions to the Lakshmanan–Porsezian–Daniel Equation with Kerr Law of Nonlinearity

In this study, some new exact travelling wave solutions to the Lakshmanan–Porsezian–Daniel (LPD) equation with Kerr law of nonlinearity are retrieved by the complete discrimination system for the polynomial method. Under the travelling wave transformation, the LPD equation is reduced to an ordinary differential equation. The new exact travelling wave solutions including rational solutions, triangle function solutions, solitary wave solutions, and Jacobian elliptic function solutions are obtained and graphically illustrated using three-dimensional and two-dimensional graphs. Comparing with the previous results for LPD equation, some of new solutions in this work such as elliptical solution are not studied, which shows the complete discrimination system method is efficient.

Application of fuzzy finite difference scheme for the non-homogeneous fuzzy heat equation

A numerical framework based on fuzzy finite difference is presented for approximating fuzzy triangular solutions of fuzzy partial differential equations by considering the type of $$[gH-p]-$$ differentiability. The fuzzy triangle functions are expanded using full fuzzy Taylor expansion to develop a new fuzzy finite difference method. By considering the type of gH-differentiability, we approximate the fuzzy derivatives with a new fuzzy finite difference. In particular, we propose using this method to solve non-homogeneous fuzzy heat equation with triangular initial-boundary conditions. We examine the truncation error and the convergence conditions of the proposed method. Several numerical examples are presented to demonstrate the performance of the methods. The final results demonstrate the efficiency and the ability of the new fuzzy finite difference method to produce triangular fuzzy numerical results which are more consistent with existing reality.