Triangular Distribution Research Articles

Centrifuge and analytical modeling of counterweight retaining walls under translation mode

Counterweight retaining walls (CRWs) are a variant of gravity-retaining walls that feature a pressure relief platform (RP) on the backfill side to reduce total earth pressure, offering a cost-effective option compared to traditional forms. However, the intricate interactions between CRWs and backfill, coupled with the lack of clear design guidelines, have restricted their use. This study examines the behavior of CRWs in translation mode using three different centrifuge model tests, each with varying fill heights. Observations show a slightly concave slip surface through the wall heel and a nearly planar slip surface extending from the edge of the RP to the back of the wall in an active limit state. Backfill movement can be classified into translation, sliding, and stable zones. Earth pressure distribution on the upper wall follows a trapezoidal pattern, whereas the pressure on the lower wall exhibits a triangular distribution. Earth pressure on the RP is approximately linear, peaking near the RP’s edge. Introducing a friction angle mobilization factor for the potential first slip surface improves accuracy in calculating earth pressure. The tests were replicated to validate the analytical model for earth pressure calculations, using the adaptive finite-element limit analysis method.

Propagation properties of partially coherent vector beam with multiple off-axis vortex phases

In this paper, we investigate the propagation properties of radially polarized rectangular-symmetric cosine-Gaussian Schell-model (RCGSM) beam with multiple off-axis vortex phases by using Fourier transform and convolution method based on the unified theory of coherence and polarization. The results indicate that the radially polarized RCGSM beam has self-splitting properties and can be split into four identical lobes due to its unique spatial coherence structure. Furthermore, the modulation of multiple off-axis vortex phases can be acted on each lobe. For high coherence, the spot arrays with triangular or square hollow light intensity distribution can be generated in the focal plane by modulating the number of off-axis vortices (<i>N</i><sub>0</sub>) and the beam order, and the corresponding state of polarization on each lobe presents an inverted triangular or oblique square elliptic distribution. However, for small coherence, the modulation effect of multiple off-axis vortex phases disappears and the light intensity of each lobe degenerates into a quasi-Gaussian distribution, whereas its state of polarization keeps invariant, which is independent of the beam order and coherence length. In addition, the beam still has a certain self-healing ability for one of the off-axis vortex phases partially blocked by an obstacle, but it will be destroyed for completely blocking, resulting in a notch on each lobe.

Salah-Mohammad triangular distribution Curve is an Alternative to the Laffer curve

Salah-Mohammad triangular distribution Curve is an Alternative to the Laffer curve

Разработка технологических схем и обоснование параметров крепления восточного борта карьера «Железный» тросово-канатными анкерами на основе формирования и анализа больших данных

Productivity of a mining enterprise is limited by the possibility of its transportation subsystem, which efficiency improvement is an urgent scientific and practical task. Studying the processes of formation and transformation of material flows allows us to clarify the design methodology of continuous haulage systems. The normative methodology of belt conveyors selection is based on treating the material flows as normally distributed random variables, with the point value, i.e. the irregularity coefficient, being used as the main indicator. There are doubts in methodological justification of such an approach. The normal law of distribution in the classical variant adequately describes a random variable changing within infinite limits, and real mine material flows via the conveyor lines are random variables with unilateral constraints. The majority of tasks on finding the design ranges of material flows can be solved using experimentally established distribution functions (or probability density functions) of random quantities of material flows. Based on the publications in recent years, as well as on our own experimental studies, it is proposed to describe material flows by piecewise linear probability density functions, in particular the triangular ones, and to summarize material flows on the basis of general theoretical provisions of classical probability theory. The paper solves a particular problem of finding an analytical solution of the sum of two random material flows defined by the triangular distribution laws and compares the results of numerical integration of the probability densities of the material flows.

Уточнение методов расчета грузопотоков конвейерных транспортных систем угольных шахт

Productivity of a mining enterprise is limited by the possibility of its transportation subsystem, which efficiency improvement is an urgent scientific and practical task. Studying the processes of formation and transformation of material flows allows us to clarify the design methodology of continuous haulage systems. The normative methodology of belt conveyors selection is based on treating the material flows as normally distributed random variables, with the point value, i.e. the irregularity coefficient, being used as the main indicator. There are doubts in methodological justification of such an approach. The normal law of distribution in the classical variant adequately describes a random variable changing within infinite limits, and real mine material flows via the conveyor lines are random variables with unilateral constraints. The majority of tasks on finding the design ranges of material flows can be solved using experimentally established distribution functions (or probability density functions) of random quantities of material flows. Based on the publications in recent years, as well as on our own experimental studies, it is proposed to describe material flows by piecewise linear probability density functions, in particular the triangular ones, and to summarize material flows on the basis of general theoretical provisions of classical probability theory. The paper solves a particular problem of finding an analytical solution of the sum of two random material flows defined by the triangular distribution laws and compares the results of numerical integration of the probability densities of the material flows.

Frequency-tunable quartic soliton in nonlinear negative-index metamaterials

An exact quartic soliton solution along with its constraint conditions in metamaterials (MMs) is presented on the basis of the nonlinear Schrödinger equation (NLSE) with second-, third- and fourth-order diffraction. Through analyzing the parameter relations based on the Drude model, it is demonstrated that the quartic soliton can only exist in the negative-index region (NIR) of self-focusing nonlinear MMs. With insight into the underlying parameter relations of quartic soliton, it is found that by selecting suitable frequency of incident wave, the quartic soliton in negative-index MMs (NIMs) exhibits frequency tunability or frequency stability. Particularly, the quartic solitons with different frequencies can propagate in an identical velocity, which provides a theoretical basis for realizing frequency division multiplexing (FDM) technology in nonlinear MMs. Different from pure-quartic soliton (PQS) in ordinary materials exhibiting oscillatory decaying tails in spatial domain and a flat top in frequency domain on logarithmic scale, the quartic soliton reported here possesses triangular distributions in both spatial and frequency domains. The presented results contribute to a fundamental theoretical support for future application of quartic soliton and are beneficial for exploring new solitary waves in MMs.

Geometric Programming Problems with Triangular and Trapezoidal Twofold Uncertainty Distributions

Geometric Programming Problems with Triangular and Trapezoidal Twofold Uncertainty Distributions

Statistical Estimation Of Value Range

This paper presents a statistical model of Value Range based on Triangular Distribution which adapts the concept of Control Limits in the Control Chart Theory. It is the purpose/aim of the paper to contribute on the methodological issue of Uncertainty which takes the probabilistic nature of uncertainty into account. 
 Using market comparable data from two cases of commercial land valuation in East Jakarta and Tangerang County, the analytical results of the model suggests that Coefficient of Variation is a key aspect to consider whether the Value Range should be constructed at ± 1s or 2s. The sensitivity analysis indicates that the model works well with less data and data set with narrower variation is generally more favourable.
 The practical value of the model is its potential use as a method for valuers to self-examine their valuation output. The model may as well be used as a method to complement fairness analysis, especially when there is a request for external review or even an appeal.

Geometry of deviation measures for triangular distributions

Triangular distributions are widely used in many applications with limited sample data, business simulations, and project management. As with other distributions, a standard way to measure deviations is to compute the standard deviation. However, the standard deviation is sensitive to outliers. In this paper, we consider and compare other deviation metrics, namely the mean absolute deviation from the mean, the median, and the quantile-based deviation. We show the simple geometric interpretations for these deviation measures and how to construct them using a compass and a straightedge. The explicit formula of mean absolute deviation from the median for triangular distribution is derived in this paper for the first time. It has a simple geometric interpretation. It is the least volatile and is always better than the standard or mean absolute deviation from the mean. Although greater than the quantile deviation, it is easier to compute with limited sample data. We present a new procedure to estimate the parameters of this distribution in terms of this deviation. This procedure is computationally simple and may be superior to other methods when dealing with limited sample data, as is often the case with triangle distributions.

Tensile Force Distribution And Development Within Geogrid-Reinforced Retaining Wall

Abstract Geogrid-reinforced earth retaining walls used to improve soil quality, and provide additional shear strength in the soil mass through the tensile strength in the reinforcement layers. A numerical model was developed by finite element code PLAXIS2D, of a segmental facing geogrid-reinforced retaining wall. This research has been carried out to investigate the effect of loading increments, loading increments width, loading increments location, facing inclination angle, geogrid inclination angle, and geogrid-soil friction factor, on the behaviour of a geogrid-reinforced soil retaining wall. The results show that the failure plane occurred in the reinforced zone at the mid-height, this observation contradicted the triangular distribution with depth assumed in conception methodologies for reinforced soil retaining wall. The distribution of peak tensile strength with depth was bilinear at high loading increments and became trapezoidal at low loading ones. Furthermore, it was found that the behaviour of a geogrid-reinforced soil retaining wall is independent of loading increments width beyond 0.5H. It also seems that the loading increments location can change the shape and the position of the peak tensile strength. It also seems that the geogrid inclination angle has a major effect on the lateral facing displacements and safety factor.

A Model of Cumulative Risk of Disease Progression Among Patients with Indolent Systemic Mastocytosis

Introduction Indolent systemic mastocytosis (ISM), a clonal mast cell disease driven by the KIT D816V mutation in ~95% of cases, can cause lifelong debilitating cutaneous, gastrointestinal, and neurocognitive symptoms with some cases progressing to advanced systemic mastocytosis (AdvSM). While several studies have quantified variable rates of disease progression from ISM to AdvSM based on the absolute frequency of these events, these studies were limited by short median follow-up time considering the long clinical course of this condition sometimes in decades, thereby leading to underestimation of the cumulate rates of progression from ISM to AdvSM. We conducted a Markov analysis using population level data in the US to estimate the lifetime risk of progression for ISM patients. Methods We modeled the total course of SM as a discrete-time Markov chain, enabling us to estimate cumulative rates of disease progression based on fixed-interval empirical observations from the literature and, where there is no literature available, expert opinion. The Markov model assumes that patients enter the model as at-risk ISM, not-at-risk ISM, or de novo AdvSM, based on the assumed relative incidence of each subtype. Not-at-risk patients will then move to death after a period of time, with an annualized risk of death from disease. At-risk patients with ISM will remain in that state, progress to AdvSM (which may be tyrosine kinase inhibitor treated or not) or die. Finally, those who progress to AdvSM will either remain in their state or die. See Figure1 for details, where each model parameter is reflected as a line between states and modeled based on an annualized rate of state transition. See Figure 2 for a full list of model parameters. Two hundred trials were performed, in which the model parameters were statistically sampled from a range of likely values, based on a triangular distribution between minimum, mode, and maximum. For each trial, the disease course for 5000 patients with SM was then simulated from initial SM diagnosis until death. Results Based on the rates of disease incidence, progression, treatment, and survival documented within various studies, we estimate the cumulative risk (± standard deviation) of disease progression from ISM to AdvSM to be 18.0% ± 3.1%, based on a probabilistic sampling of the underlying assumptions. Conclusions This study suggests that the cumulative rate of disease progression among patients with ISM may be higher than previously recognized. In our earlier work, we showed a higher rate of disease progression in ISM patients with high symptom burden (Mukherjee S, et al. ASH 2022). These data highlight the importance of close monitoring for signs of progression and warrant further research to identify subsets of ISM patients who may have a higher underlying risk of progression.

CSR strategy composition and leaf traits for herbaceous plants in garden design

Low community stability and sustainability, accompanied by high maintenance cost was an important ecological problem in planted herbaceous communities in many cities. Investigation showed that the overwhelmingly majority of garden designers ignored the subtle interactions between plants and plants in community itself, as well as the fitness of plants to certain environment. Fortunately, CSR strategy methodology based on leaf traits could be a useful tool for reflecting plant functional groups, thus we can have a deeper understanding of the characteristics of various herbaceous plants. In this study, we used 318 common herbaceous plants in Shaanxi as experimental material to investigate leaf area (LA), leaf fresh weight (LFW) and leaf dry weight (LFW) of them. Their CSR strategies were determined using the 'StrateFy' analysis tool and were presented in a triangular distribution. The results showed that S-selection was responsible for the vast majority of herbaceous plants in garden design, C- selection took second place, and R- selection was the least. Sub-strategy (CS, S/CS, CR, CS/CSR, S/CSR and so on) also played a vital role in garden plants, which together accounted for 90.88% from all plant strategies. In terms of the representative families, the strategies of Labiatae and Asteraceae distributed widely, and the trade-off appeared in C-S axes of Gramineae and Liliaceae. Among three leaf traits, LA was of great importance to C- strategy, LDMC to S- strategy, and SLA was important to R-strategy. CSR strategy composition indicates the power of artificial selection, and reasonable collocationof different CSR strategies can create a better community. Thus, we propose that future investigations on CSR strategies in different regions all over the world should provide a reliable database for garden herbaceous species selection.

Impact of Alcohol Consumption on Multiple Sclerosis Using Model-based Standardization and Misclassification Adjustment Via Probabilistic Bias Analysis.

The etiology of multiple sclerosis (MS) is still not well-demonstrated, and assessment of some risk factors like alcohol consumption has problems like confounding and measurement bias. To determine the causal effect of alcohol consumption on MS after adjusting for alcohol consumption misclassification bias and confounders. In a population-based incident case-control study, 547 patients with MS and 1057 healthy people were recruited. A minimally sufficient adjustment set of confounders was derived using the causal directed acyclic graph. The probabilistic bias analysis method (PBAM) using beta, logit-logistic, and triangular probability distributions for sensitivity/specificity to adjust for misclassification bias in self-reporting alcohol consumption and model-based standardization (MBS) to estimate the causal effect of alcohol consumption were used. Population attributable fraction (PAF) estimates with 95% Monte Carlo sensitivity analysis (MCSA) intervals were calculated using PBAM and MBS analysis. Bootstrap was used to deal with random errors. The adjusted risk ratio (95% MCSA interval) from the probabilistic bias analysis and MBS between alcohol consumption and MS using the three distribution was in the range of 1.93 (1.07 to 4.07) to 2.02 (1.15 to 4.69). The risk difference (RD) in all three scenarios was 0.0001 (0.0000 to 0.0005) and PAF was in the range of 0.15 (0.010 to 0.50) to 0.17 (0.001 to 0.47). After adjusting for measurement bias, confounding, and random error alcohol consumption had a positive causal effect on the incidence of MS.

Lean performance evaluation of dairy cow farming processes with mathematical model development

Management sciences, especially controlling, are undergoing extensive changes. Various mathematical-statistical and IT solutions help organizations develop their systems and extract relevant information from data structures, thereby supporting effective decision-making and organizational operations. Agriculture is a special sector, but management functions apply here too, and modern tools and methods support the implementation of efficient operations. Dairy farming is a particularly important sector in agriculture, where controlling systems are developed in response to special requirements. The controlling activity of the sector is significantly influenced by the development of digitalization, which has resulted in the monitoring of processes and organizational performance evaluation becoming more extensive and detailed. Lean management is also an excellent method for increasing the efficiency of processes and organizational performance in the agricultural sector, which can be used to optimize processes, identify unnecessary activities and ensure appropriate task control. In this way, controlling systems must take into account the importance of lean management and contribute to the performance evaluation of organizational lean. In our study, we develop a lean performance evaluation model through an extended case study, which is suitable for evaluating the lean performance of the examined dairy farm. We used a fuzzy triangular function to evaluate the performance of enterprises, which represents the value of the given parameter with a triangular distribution. With dynamically growing and usable data, the model is able to provide continuous feedback on organizational lean operation and possible intervention points.

Comparison of the clinical efficacy of three cannulated screws with parallel distribution and inverted triangular distribution in the treatment of femoral neck fractures in the elderly.

The aim of the present study was to investigate the clinical efficacy of three cannulated screws with parallel distribution in comparison with the 'gold standard' of inverting three cannulated screws with triangular distribution, in the treatment of femoral neck fractures in the elderly. A total of 106 elderly patients with femoral neck fractures treated at the Department of Orthopedics of the First Affiliated Hospital of Nanjing Medical University (Jiangsu Provincial Hospital) from October 2018 to March 2020 were retrospectively analyzed and divided into groups A and B. Group A consisted of 51 patients with a mean age of 73.3±6.8 years; and group B consisted of 55 patients with a mean age of 74.5±7.3 years. Group A was treated with closed reduction + inverted triangular distribution of three cannulated screws, while group B was treated with closed reduction + parallel distribution of three cannulated screws, and the general surgical conditions, including fracture reduction quality, fracture healing, incidence of avascular necrosis of the femoral head, and functional recovery scale (FRS) score of hip joint function were assessed in both groups. All patients were followed up for an average of 14.8 months. Compared with group A, group B had significant advantages in operation time, number of times intraoperative fluoroscopy was performed, number of intraoperative guide wire adjustments, and proportion of postoperative referrals to the intensive care unit (P<0.05). There were no significant differences in fracture reduction quality, fracture healing, incidence of avascular necrosis of the femoral head, and FRS score of hip joint function between groups A and B (P>0.05). For elderly patients with femoral neck fractures, distributing three cannulated screws in parallel after closed reduction achieved similar clinical efficacy to 'gold standard' inverted triangular distribution, and had obvious advantages in operation time, with significantly reduced surgical difficulty. This procedure is therefore deemed worthy of promotion and clinical application in the primary hospitals of China.

Implementation of a triangular probabilistic distribution for optimal parametrization of the SEIR model recovery rates with delay.

Dynamical analysis of epidemiological models received significant attention after the global disaster of the Covid-19 pandemic. Although much of the attention has focused on choosing the appropriate model to describe and interpret the epidemiological data and on predicting the spread of the disease, working on uncovering the hidden dynamics by analyzing the predictive data and studying the algebraic properties of the models such as the invariant spaces, the center manifolds, and the Lyapunov functions in the general parameter space still have the potential to make a significant contribution to our understanding of the dynamics of spread. In this paper, in order to determine the recovery rate based on characteristic model parameters, the triangular probabilistic distribution is implemented in an epidemic delay differential equation with delays. By defining generic coefficients derived from field surveys and population characteristics, we propose a way to reduce the number of model parameters that need to be obtained during model fitting in order to find an effective way to handle the cultural and physiological diversity of societies and isolate the effect of counter measures for the epidemics.

A Millimeter-Wave Polarization-Reconfigurable Double-Folded Antenna Array

This paper introduces a double-folded, polarization reconfigurable, dual antenna array for millimeter-wave (mm-wave) communications. The proposed structure reconfigures its polarization by supporting either left- or right-handed circularly polarized radiation with a maximum realized gain that reaches up to 15.8 dBic for each state. Reconfiguration is achieved through the activation of a single-pole-double-throw (SPDT) switch. A dedicated folded feeding network with a triangular power distribution is designed to achieve the required in-phase power distribution between the ten radiating elements that compose each antenna array. The folded feeding network eliminates the need for a stacked design with integrated vias, which reduces undesired interferences and inductive parasitic effects. The fabricated prototype validates the operation of the proposed reconfigurable structure where a measured axial ratio that is less than 3 dB is achieved between 28.8 GHz and 31.2 GHz with a measured half-power beamwidth of 13° along the direction of maximum radiation.

Generalised triangular distributions for ordinal deep learning: Novel proposal and optimisation

Deep learning techniques for ordinal classification have recently gained significant attention. Predicting an ordinal variable, that is, a variable that demonstrates a natural relationship between categories, is of relevance for a number of real-world problems in various fields of knowledge. For example, a medical diagnosis can occur at different stages of the disease. Applying standard classifiers to ordered labels can lead to errors in distant categories, when errors in an ordinal problem ideally tend to be produced in adjacent classes because of their similarity. To address this issue, we propose a soft labelling approach based on generalised triangular distributions, which are asymmetric and different for each class. The parameters of these distributions are determined using a metaheuristic and are specifically adapted to the given problem. Moreover, this approach enables the model to avoid errors in distant classes (e.g. classifying a patient with a severe disease as healthy). A comprehensive comparison was performed using eight datasets and five performance metrics. The main advantage of the proposed soft-labelling approach is that it adapts the distributions to each problem, resulting in greater flexibility and better performance. The results and statistical analysis show that the proposed methodology significantly outperforms all other methods.

Life Cycle Carbon Dioxide Emissions and Sensitivity Analysis of Elevators

With the intensification of climate warming, the carbon dioxide emissions from high-energy-consuming elevators have attracted increasing societal attention. The assessment of carbon dioxide emissions, particularly the boundaries and strategies of carbon dioxide emissions accounting, lacks systematic research. However, an efficient evaluation of elevator carbon dioxide emissions is beneficial for improving elevator energy utilization. A carbon dioxide emissions accounting method and inventory analysis of a life cycle for an elevator is proposed to measure the carbon dioxide emissions from production to disposal. In addition, a new assessment indicator, namely, annual carbon dioxide emissions per ton·kilometer, is proposed to evaluate the carbon dioxide emissions for different types of elevators. The lifetime carbon dioxide emissions of the elevator and its sensitivity to influencing factors were assessed. The results indicate that the carbon dioxide emissions in the four stages of manufacturing, installation, operation and maintenance, and demolition and scraping contributed 41.31%, 0.92%, 57.32% and 0.44%, respectively. The annual carbon dioxide emissions of the elevator were about 27.18 kgCO2/t·km. The four primary factors affecting CO2 emissions were electricity consumption, printed circuit boards, low-alloy steel and chrome steel in descending order. Their probability distribution characteristics all obeyed triangular or uniform distributions. The median of their 95% confidence intervals was about 73,800. Their coefficients of variation were all below 2.1%. The effective strategies for energy conservation and carbon reduction were suggested by the life cycle impactor assessment. They also provide guidance for sustainable elevators.

The arithmetic of triangular Z-numbers with reduced calculation complexity using an extension of triangular distribution

Information that people rely on is often uncertain and partially reliable. Zadeh introduced the concept of Z-numbers as a more adequate formal construct for describing uncertain and partially reliable information. Most existing applications of Z-numbers involve discrete ones due to the high complexity of calculating continuous ones. However, the continuous form is the most common form of information in the real world. Simplifying continuous Z-number calculations is significant for practical applications. There are two reasons for the complexity of continuous Z-number calculations: the use of normal distributions and the inconsistency between the meaning and definition of Z-numbers. In this paper, we extend the triangular distribution as the hidden probability density function of triangular Z-numbers. We add a new parameter to the triangular distribution to influence its convexity and concavity, and then expand the value's domain of the probability measure. Finally, we implement the basic operations of triangular Z-numbers based on the extended triangular distribution. The suggested method is illustrated with numerical examples, and we compare its computational complexity and the entropy (uncertainty) of the resulting Z-number to the traditional method. The comparison shows that our method has lower computational complexity, higher precision and lower uncertainty in the results.