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IN THIS ISSUE

Research Papers

ASME J. Risk Uncertainty Part B. 2018;4(4):041001-041001-7. doi:10.1115/1.4039243.

As a common type system, multistate weighted k-out-of-n system is of great importance in reliability engineering. The components are usually treated as independent from each other. It is usually not that case in real life and the components are dependent. On the other hand, the performance of the components degrades over time, leading to the change of the components' weight at the same time. As a result, the present paper provides a method to evaluate the dynamic reliability of multistate weighted k-out-of-n: G system with s-dependent components. The degradation of the components follows a Markov process and the components are nonrepairable. Copula function is used to model the s-dependence of the components. The LZ-transform for a discrete-state continuous-time Markov process is combined, and the explicit expression for the survival function and the mean time to failure (MTTF) of the system is obtained. A small electricity generating system is studied based on our method in the illustration, and detailed comparison result is made for dependent case and independent case. Dynamic reliability with varied levels of electricity generation conforming to the actual situation for this generating system is also calculated.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2018;4(4):041002-041002-12. doi:10.1115/1.4039149.
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Bayesian networks (BNs) are being studied in recent years for system diagnosis, reliability analysis, and design of complex engineered systems. In several practical applications, BNs need to be learned from available data before being used for design or other purposes. Current BN learning algorithms are mainly developed for networks with only discrete variables. Engineering design problems often consist of both discrete and continuous variables. This paper develops a framework to handle continuous variables in BN learning by integrating learning algorithms of discrete BNs with Gaussian mixture models (GMMs). We first make the topology learning more robust by optimizing the number of Gaussian components in the univariate GMMs currently available in the literature. Based on the BN topology learning, a new multivariate Gaussian mixture (MGM) strategy is developed to improve the accuracy of conditional probability learning in the BN. A method is proposed to address this difficulty of MGM modeling with data of mixed discrete and continuous variables by mapping the data for discrete variables into data for a standard normal variable. The proposed framework is capable of learning BNs without discretizing the continuous variables or making assumptions about their conditional probability densities (CPDs). The applications of the learned BN to uncertainty quantification and model calibration are also investigated. The results of a mathematical example and an engineering application example demonstrate the effectiveness of the proposed framework.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2018;4(4):041003-041003-8. doi:10.1115/1.4039941.

The gear door lock system (GDLS) is a hydraulic and mechatronic system with high degree of complexity and uncertainty, making the performance assessment of the system especially intractable. We develop copula models to estimate the reliability of GDLS with dependent failure modes. Based on the working principle of the GDLS, kinematic and dynamic model with imperfect joints is built in which Latin hypercube sampling (LHS) and kernel smoothing density are utilized to obtain the marginal failure probabilities. Then, copula models are utilized to describe the dependence between the two function failure modes. Furthermore, to be more accurate, mixed copula models are developed. The squared Euclidean distance is adopted to estimate the parameters of the above reliability models. Finally, the Monte Carlo simulation is conducted to evaluate different reliability models.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2018;4(4):041004-041004-7. doi:10.1115/1.4039464.

A series of pedestrian sideswipe impacts were computationally reconstructed; a fast-walking pedestrian was collided laterally with the side of a moving vehicle at 25 km/h or 40 km/h, which resulted in rotating the pedestrian's body axially. Potential severity of traumatic brain injury (TBI) was assessed using linear and rotational acceleration pulses applied to the head and by measuring intracranial brain tissue deformation. We found that TBI risk due to secondary head strike with the ground can be much greater than that due to primary head strike with the vehicle. Further, an “effective” head mass, meff, was computed based upon the impulse and vertical velocity change involved in the secondary head strike, which mostly exceeded the mass of the adult head-form impactor (4.5 kg) commonly used for a current regulatory impact test for pedestrian safety assessment. Our results demonstrated that a sport utility vehicle (SUV) is more aggressive than a sedan due to the differences in frontal shape. Additionally, it was highlighted that a striking vehicle velocity should be lower than 25 km/h at the moment of impact to exclude the potential risk of sustaining TBI, which would be mitigated by actively controlling meff, because meff is closely associated with a rotational acceleration pulse applied to the head involved in the final event of ground contact.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2018;4(4):041005-041005-7. doi:10.1115/1.4039465.

Valuation of transactive energy (TE) systems should be supported by a structured and systematic approach to uncertainty identification, assessment, and treatment in the interest of risk-informed decision making. The proposed approach, a variation of fault tree analysis, is anticipated to support valuation analysts in analyzing conventional and transactive system scenarios. This approach allows for expanding the entire tree up to the level of minute details or collapsing them to a level sufficient enough to get an overview of the problem. Quantification scheme for the described approach lends itself for valuation. The method complements value exchange analysis, simulation, and field demonstration studies. The practicality of the proposed approach is demonstrated through uncertainty assessment of the smart grid interoperability panel peak heat day scenario.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2018;4(4):041006-041006-17. doi:10.1115/1.4039558.

This paper examines the variability of predicted responses when multiple stress–strain curves (reflecting variability from replicate material tests) are propagated through a finite element model of a ductile steel can being slowly crushed. Over 140 response quantities of interest (QOIs) (including displacements, stresses, strains, and calculated measures of material damage) are tracked in the simulations. Each response quantity's behavior varies according to the particular stress–strain curves used for the materials in the model. We desire to estimate or bound response variation when only a few stress–strain curve samples are available from material testing. Propagation of just a few samples will usually result in significantly underestimated response uncertainty relative to propagation of a much larger population that adequately samples the presiding random-function source. A simple classical statistical method, tolerance intervals (TIs), is tested for effectively treating sparse stress–strain curve data. The method is found to perform well on the highly nonlinear input-to-output response mappings and non-normal response distributions in the can crush problem. The results and discussion in this paper support a proposition that the method will apply similarly well for other sparsely sampled random variable or function data, whether from experiments or models. The simple TI method is also demonstrated to be very economical.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2018;4(4):041007-041007-8. doi:10.1115/1.4039467.

The development of robust and adaptable methods of grasping force optimization (GFO) is an important consideration for robotic devices, especially those which are designed to interact naturally with a variety of objects. Along with considerations for the computational efficiency of such methods, it is also important to ensure that a GFO approach chooses forces which can produce a stable grasp even in the presence of uncertainty. This paper examines the robustness of three methods of GFO in the presence of variability in the contact locations and in the coefficients of friction between the hand and the object. A Monte Carlo simulation is used to determine the resulting probability of failure and sensitivity levels when variability is introduced. Two numerical examples representing two common grasps performed by the human hand are used to demonstrate the performance of the optimization methods. Additionally, the method which yields the best overall performance is also tested to determine its consistency when force is applied to the object's center of mass in different directions. The results show that both the nonlinear and linear matrix inequality (LMIs) methods of GFO produce acceptable results, whereas the linear method produces unacceptably high probabilities of failure. Further, the nonlinear method continues to produce acceptable results even when the direction of the applied force is changed. Based on these results, the nonlinear method of GFO is considered to be robust in the presence of variability in the contact locations and coefficients of friction.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2018;4(4):041008-041008-13. doi:10.1115/1.4039784.

Optimal sizing of peak loads has proven to be an important factor affecting the overall energy consumption of heating ventilation and air-conditioning (HVAC) systems. Uncertainty quantification of peak loads enables optimal configuration of the system by opting for a suitable size factor. However, the representation of uncertainty in HVAC sizing has been limited to probabilistic analysis and scenario-based cases, which may limit and bias the results. This study provides a framework for uncertainty representation in building energy modeling, due to both random factors and imprecise knowledge. The framework is shown by a numerical case study of sizing cooling loads, in which uncertain climatic data are represented by probability distributions and human-driven activities are described by possibility distributions. Cooling loads obtained from the hybrid probabilistic–possibilistic propagation of uncertainty are compared to those obtained by pure probabilistic and pure possibilistic approaches. Results indicate that a pure possibilistic representation may not provide detailed information on the peak cooling loads, whereas a pure probabilistic approach may underestimate the effect of uncertain human behavior. The proposed hybrid representation and propagation of uncertainty in this paper can overcome these issues by proper handling of both random and limited data.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2018;4(4):041009-041009-9. doi:10.1115/1.4039471.

A novel uncertainty quantification routine in the genre of adaptive sparse grid stochastic collocation (SC) has been proposed in this study to investigate the propagation of parametric uncertainties in a stall flutter aeroelastic system. In a hypercube stochastic domain, presence of strong nonlinearities can give way to steep solution gradients that can adversely affect the convergence of nonadaptive sparse grid collocation schemes. A new adaptive scheme is proposed here that allows for accelerated convergence by clustering more discretization points in regimes characterized by steep fronts, using hat-like basis functions with nonequidistant nodes. The proposed technique has been applied on a nonlinear stall flutter aeroelastic system to quantify the propagation of multiparametric uncertainty from both structural and aerodynamic parameters. Their relative importance on the stochastic response is presented through a sensitivity analysis.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2018;4(4):041010-041010-9. doi:10.1115/1.4039357.

Go-karts are a common amusement park feature enjoyed by people of all ages. While intended for racing, contact between go-karts does occur. To investigate and quantify the accelerations and forces which result from contact, 44 low-speed impacts were conducted between a stationary (target) and a moving (bullet) go-kart. The occupant of the bullet go-kart was one of two human volunteers. The occupant of the target go-kart was a Hybrid III 50th percentile male anthropomorphic test device (ATD). Impact configurations consisted of rear-end impacts, frontal impacts, side impacts, and oblique impacts. Results demonstrated high repeatability for the vehicle performance and occupant response. Go-kart accelerations and speed changes increased with increased impact speed. Impact duration and restitution generally decreased with increased impact speed. All ATD acceleration, force, and moment values increased with increased impact speed. Common injury metrics such as the head injury criterion (HIC), $Nij$, and $Nkm$ were calculated and were found to be below injury thresholds. Occupant response was also compared to published activities of daily living data.

Commentary by Dr. Valentin Fuster
Select Articles from Part A: Civil Engineering

Technical Papers

ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering. 2018;4(2):. doi:10.1061/AJRUA6.0000965.
Abstract

Abstract  Stochastic soil modeling aims to provide reasonable mean, variance, and spatial correlation of soil properties with quantified uncertainty. Because of difficulties in integrating limited and imperfect prior knowledge (i.e., epistemic uncertainty) with observed site-specific information from tests (i.e., aleatoric uncertainty), a reasonably accurate estimate of the spatial correlation is significantly challenging. Possible reasons include (1) only sparse data being available (i.e., one-dimensional observations are collected at selected locations); and (2) from a physical point of view, the formation process of soil layers is considerably complex. This paper develops a Gaussian Markov random field (GMRF)-based modeling framework to describe the spatial correlation of soil properties conditional on observed electric cone penetration test (CPT) soundings at multiple locations. The model parameters are estimated using a novel stochastic partial differential equation (SPDE) approach and a fast Bayesian algorithm using the integrated nested Laplace approximation (INLA). An existing software library is used to implement the SPDE approach and Bayesian estimation. A real-world example using 185 CPT soundings from Alameda County, California is provided to demonstrate the developed method and examine its performance. The analyzed results from the proposed model framework are compared with the widely accepted covariance-based kriging method. The results indicate that the new approach generally outperforms the kriging method in predicting the long-range variability. In addition, a better understanding of the fine-scale variability along the depth is achieved by investigating one-dimensional residual processes at multiple locations.

Topics:
Modeling , Soil
ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering. 2018;4(2):. doi:10.1061/AJRUA6.0000950.
Abstract

Abstract  Metamodeling techniques have been widely used as substitutes for high-fidelity and time-consuming models in various engineering applications. Examples include polynomial chaos expansions, neural networks, kriging, and support vector regression (SVR). This paper attempts to compare the latter two in different case studies so as to assess their relative efficiency on simulation-based analyses. Similarities are drawn between these two metamodel types, leading to the use of anisotropy for SVR. Such a feature is not commonly used in the SVR-related literature. Special care was given to a proper automatic calibration of the model hyperparameters by using an efficient global search algorithm, namely the covariance matrix adaptation–evolution scheme. Variants of these two metamodels, associated with various kernel and autocorrelation functions, were first compared on analytical functions and then on finite element–based models. From the comprehensive comparison, it was concluded that anisotropy in the two metamodels clearly improves their accuracy. In general, anisotropic $L2$-SVR with the Matérn kernels was shown to be the most effective metamodel.

Topics:
Structural engineering , Support vector machines
ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering. 2018;4(2):. doi:10.1061/AJRUA6.0000956.
Abstract

Abstract  Corrosion is one of the main causes of pipeline failure, which can have large social, economic, and environmental consequences. To mitigate this risk, pipeline operators perform regular inspections and repairs. The results of the inspections aid decision makers in determining the optimal maintenance strategy. However, there are many possible maintenance strategies, and a large degree of uncertainty, leading to difficult decision making. This paper develops a framework to inform the decision of whether it is better over the long term to continuously repair defects as they become critical or to just replace entire segments of the pipeline. The method uses a probabilistic analysis to determine the expected number of failures for each pipeline segment. The expected number of failures informs the optimal decision. The proposed framework is tailored toward mass amounts of in-line inspection data and multiple pipeline segments. A numerical example of a corroding upstream pipeline illustrates the method.

Topics:
Maintenance , Pipeline systems , Decision making
ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering. 2018;4(2):. doi:10.1061/AJRUA6.0000960.
Abstract

Abstract  This paper is focused on the development of an efficient system-level reliability-based design optimization strategy for uncertain wind-excited building systems characterized by high-dimensional design variable vectors (in the order of hundreds). Indeed, although a number of methods have been proposed over the last 15 years for the system-level reliability-based design optimization of building systems subject to stochastic excitation, few have treated problems characterized by more than a handful of design variables. This limits their applicability to practical problems of interest, such as the design optimization of high-rise buildings. To overcome this limitation, a simulation-based method is proposed in this work that is capable of solving reliability-based design optimization problems characterized by high-dimensional design variable vectors while considering system-level performance constraints. The framework is based on approximately decoupling the reliability analysis from the optimization loop through the definition of a system-level subproblem that can be fully defined from the results of a single simulation carried out in the current design point. To demonstrate the efficiency, practicality, and strong convergence properties of the proposed framework, a 40-story uncertain planar frame defined by 200 design variables is optimized under stochastic wind excitation.

Topics:
Reliability-based optimization , Wind
ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering. 2018;4(2):. doi:10.1061/AJRUA6.0000964.
Abstract

Abstract  Fragility functions define the probability of meeting or exceeding some damage measure (DM) for a given level of engineering demand (e.g., base shear) or hazard intensity measure (IM; e.g., wind speed, and peak ground acceleration). Empirical fragility functions specifically refer to fragility functions that are developed from posthazard damage assessments, and, as such, they define the performance of structures or systems as they exist in use and under true natural hazard loading. This paper describes major sources of epistemic uncertainty in empirical fragility functions for building performance under natural hazard loading, and develops and demonstrates methods for quantifying these uncertainties using Monte Carlo simulation methods. Uncertainties are demonstrated using a dataset of 1,241 residential structures damaged in the May 22, 2011, Joplin, Missouri, tornado. Uncertainties in the intensity measure (wind speed) estimates were the largest contributors to the overall uncertainty in the empirical fragility functions. With a sufficient number of samples, uncertainties because of potential misclassification of the observed damage levels and sampling error were relatively small. The methods for quantifying uncertainty in empirical fragility functions are demonstrated using tornado damage observations, but are applicable to any other natural hazard as well.

Topics:
Uncertainty , Disasters , Damage assessment


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# For what values of x, if any, does f(x) = tan((pi)/4-9x)  have vertical asymptotes?

##### 1 Answer
Jan 2, 2017

Period: $\frac{\pi}{9}$. For each period $\left(\frac{\pi}{12} + \frac{k}{9} \pi , \frac{7}{36} \pi + \frac{k}{9} \pi\right)$, there are two terminal asymptotes x = end value, $k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$

#### Explanation:

As $\left(\frac{\pi}{4} - 9 x\right) \to$ (an odd multiple ) $\left(2 k + 1\right)$ of $\frac{\pi}{2}$,$f \to \pm \infty$,

giving

$f \to \pm \infty$, as $x \to - \frac{4 k + 1}{36} \pi , k = 0 , \pm 1 , \pm 2 , \pm 3 , . .$

So, in ine period $x \in \left(\frac{\pi}{12} , \frac{7}{36} \pi\right)$, we have two terminal

asymptotes

$x = \frac{\pi}{12} \mathmr{and} x = \frac{7}{36} \pi$

The period for f is $\frac{\pi}{9} = \left(\frac{7}{36} - \frac{1}{12}\right) \pi$.

graph{y-tan(0.7854-9x)=0 [-5, 5, -2.5, 2.5]}

format: plain


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# In an astable multivibrator, why doesn't negative feedback prevent the output from ever saturating?

What I understand of an astable multivibrator so far is this:

• The output saturates to one of two extremes, which drives the capacitor in the negative feedback loop to either charge or discharge.
• Once there is a discrepancy between the inverting and non inverting terminal voltages (|v+| < |v-|), the positive feedback amplifies that difference and pushes the output to the opposite of the two extremes mentioned above.

For example, if the output was initially at +Vsat, the capacitor would charge until it got to +beta * Vsat. Once it exceeded this upper threshold voltage, however, v- would be greater than v+, which would drive the op amp to negative saturation (-Vsat). The process would repeat as the capacitor discharges from +beta * Vsat to -beta * Vsat; beta here is R1/(R1+R2).

My question is as follows. During the portion of the time that |v-| > |v+| (either during capacitor charge or discharge), why is the negative feedback not coming into play and quickly equalizing the two voltages? Can we assume that the positive feedback of the schmitt trigger works faster than the negative feedback involving a resistor and a capacitor? Is there some sort of a "battle" between the two feedbacks and one ends up "winning" eventually?

• Exactly! You have found the answer to your question yourself. All you have to do is find out why the positive feedback is faster than the negative one ... or, in other words, what slows down the negative feedback... May 1, 2021 at 6:59
• There is a real "battle" between the two feedbacks in the weird circuit of negative impedance converter (NIC) where the negative feedback always "wins". But there they are both fast. See my Wikibooks story. May 1, 2021 at 8:44

why is the negative feedback not coming into play and quickly equalizing the two voltages? Can we assume that the positive feedback of the schmitt trigger works faster than the negative feedback involving a resistor and a capacitor?

Yes.

The voltage at (-) terminal can rise and fall only at a speed with which the capacitor can charge/discharge. That rate is limited by the opamp saturation voltage and the resistor Rf. It has time constant Rf x C.

The positive feedback through the two resistors R1 and R2 are nearly instantaneous.

• This makes so much sense, thanks a lot! May 2, 2021 at 12:32

In the simplest analysis consider this: -

Vn is the average of Vp because Rf and C form a low pass filter.

Not until Vn catches up with Vp can negative feedback have any instantaneous effect. Then, when that occurs, the output flips and Vn is playing catch up with Vp once more but, with the polarities reversed.

Negative feedback ensures that the average value of Vn equals the average value of Vp. That's all negative feedback does here.

• Right observation... This is how all bistable (temperature, pressure, etc.) regulators work - the controlled quantity wiggles between the two thresholds so its average value is equal to the middle value of the hysteresis region. May 1, 2021 at 10:52


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The covariance of two variables x and y in a data set measures how the two Other Analyses Contrasts in Linear Models; Cate–Nelson Analysis . The sample covariance is defined in terms of the sample means as: Similarly, the population covariance is defined in terms of the population mean The first tree is used as the baseline to compare the other four trees against and the model summary shows that tree 2 is similar to tree 1 (no real need for a different offset) but that there is evidence that the offset for the other three trees is significantly larger than tree 1 (and tree 2). This data is available in the data frame Orange and we make a copy of this data set so that we can remove the ordering that is recorded for the Tree identifier variable. The residuals from the model can be plotted against fitted values, divided by tree, to investigate the model assumptions: Residual diagnostic plot for the analysis of covariance model fitted to the Orange Tree data. Posted on April 28, 2010 by Ralph in R bloggers | 0 Comments. When type = "const" constant variances are assumed and and vcovHC gives the usual estimate of the covariance matrix of the coefficient estimates: COVARIANCE, REGRESSION, AND CORRELATION 37 yyy xx x (A) (B) (C) Figure 3.1 Scatterplots for the variables xand y.Each point in the x-yplane corresponds to a single pair of observations (x;y).The line drawn through the We can compare the two models using an F-test for nested models using the anova function: Here there are four degrees of freedom used up by the more complicated model (four parameters for the different trees) and the test comparing the two models is highly significant. NO! Additional Helpful Tips Reading SAS Datalines in R We apply the cov function to compute the covariance of eruptions and waiting. Similarly, the covariance is computed as. We create a new factor after converting the old factor to a numeric string: The purpose of this step is to set up the variable for use in the linear model. Gillard and T.C. Confidence intervals displays confidence intervals with the specified level of confidence for each regression coefficient or a covariance matrix. Now, for simple linear regression, we compute the slope as follows: To show how the correlation coefficient r factors in, let’s rewrite it as. We can extended this model further by allowing the rate of increase in circumference to vary between the five trees. Is this enough to actually use this model? And really it's just kind of a fun math thing to do to show you all of these connections, and where, really, the definition of covariance really becomes useful. Before using a regression model, you have to ensure that it is statistically significant. As we can see, with the resources offered by this package we can build a linear regression model, as well as GLMs (such as multiple linear regression, polynomial regression, and logistic regression). Miscellany Chapters Not Covered in This Book . When and how to use the Keras Functional API, Moving on as Head of Solutions and AI at Draper and Dash. THE SANDWICH (ROBUST COVARIANCE MATRIX) ESTIMATOR R. J. Carroll, Suojin Wang, D. G. Simpson, A. J. Stromberg and D. Ruppert January 26, 1998 Abstract The sandwich estimator, often known as the robust covariance matrix estimator or the em- pirical covariance matrix estimator, has achieved increasing use with the growing popularity of generalized estimating equations. Additional: The analysis of variance table comparing the second and third models shows an improvement by moving to the more complicated model with different slopes: Copyright © 2020 | MH Corporate basic by MH Themes, Software for Exploratory Data Analysis and Statistical Modelling, Click here if you're looking to post or find an R/data-science job, Introducing our new book, Tidy Modeling with R, How to Explore Data: {DataExplorer} Package, R – Sorting a data frame by the contents of a column, Whose dream is this? Simple linear regression The first dataset contains observations about income (in a range of $15k to$75k) and happiness (rated on a scale of 1 to 10) in an imaginary sample of 500 people. coef(m) Other useful statistics are accessed via summary(m). The packages used in this chapter include: • psych • PerformanceAnalytics • ggplot2 • rcompanion The following commands will install these packages if theyare not already installed: if(!require(psych)){install.packages("psych")} if(!require(PerformanceAnalytics)){install.packages("PerformanceAnalytics")} if(!require(ggplot2)){install.packages("ggplot2")} if(!require(rcompanion)){install.packages("rcompanion")} Coefficient of linear correlation The parameter \rho is usually called the correlation coefficient. positive linear relationship between the two variables. a and b are constants which are called the coefficients. Related information . A positive covariance would indicate a positive linear relationship between the variables, and a negative covariance would indicate the opposite. For example, there might be a categorical variable (sometimes known as a covariate) that can be used to divide the data set to fit a separate linear regression to each of the subsets. The simplest model assumes that the relationship between circumference and age is the same for all five trees and we fit this model as follows: The summary of the fitted model is shown here: The test on the age parameter provides very strong evidence of an increase in circumference with age, as would be expected. Linear Regression¶ Linear models with independently and identically distributed errors, and for errors with heteroscedasticity or autocorrelation. Iles School of Mathematics, Senghenydd Road, Cardi University, We will consider how to handle this extension using one of the data sets available within the R software package. Suppose we have a linear regression model named as Model then finding the residual variance can be done as (summary (Model)\$sigma)**2. When used to compare samples from different populations, covariance is used to identify how two variables vary together whereas correlation is used to determine how change in one variable is affecting the change in another variable. Now the linear model is built and we have a formula that we can use to predict the dist value if a corresponding speed is known. R> vcov(m) (Intercept) x (Intercept) 0.11394 -0.02662 x -0.02662 0.20136 You can access point estimates of your parameters via. I want to connect to this definition of covariance to everything we've been doing with least squared regression. Here we useW=w−1Isp, meaning that all the regression coeﬃcients area prioriindependent, with an inverse gamma hyperprior on the shrinkage coeﬃcientw, i.e.,w∼ IGamma(aw,bw). This additional term can be included in the linear model as an interaction term, assuming that tree 1 is the baseline. There is a set of data relating trunk circumference (in mm) to the age of Orange trees where data was recorded for five trees. For the Orange tree data the new model is fitted thus: Interesting we see that there is strong evidence of a difference in the rate of change in circumference for the five trees. Das Beiwort „linear“ bedeutet, dass die abhängige Variable als eine Linearkombination (nicht notwendigerweise) linearer Funktionen der unabhängigen Variablen modelliert wird (siehe Wikipedia). share | improve this answer | follow | answered Sep 15 '15 at 8:40. csgillespie csgillespie. Linear Regression Diagnostics. Covariance Matrix of a Random Vector • The collection of variances and covariances of and between the elements of a random vector can be collection into a matrix called the covariance matrix remember so the covariance matrix is symmetric. Correlation and Covariance are two commonly used statistical concepts majorly used to measure the linear relation between two variables in data. Select the statistics you want. A more descriptive name would be coefficient of linear correlation. The simple linear regression model considers the relationship between two variables and in many cases more information will be available that can be used to extend the model. The covariance of eruption duration and waiting time is about 14. Linear Regression. Fractal graphics by zyzstar In this step-by-step guide, we will walk you through linear regression in R using two sample datasets. Linear Regression. That does not mean the same thing as in the context of linear algebra (see linear dependence ). The residual variance is the variance of the values that are calculated by finding the distance between regression line and the actual points, this distance is actually called the residual. Covariance, Regression, and Correlation “Co-relation or correlation of structure” is a phrase much used in biology, and not least in that branch of it which refers to heredity, and the idea is even more frequently present than the phrase; but I am not aware of any previous attempt to deﬁne it clearly, to trace its mode of action in detail, or to show how to measure its degree. Curvilinear Regression; Analysis of Covariance; Multiple Regression; Simple Logistic Regression; Multiple Logistic Regression . are linearly related. Consider how to use the Keras Functional API, Moving on as Head Solutions... The two variables and Dash use the Keras Functional API, Moving on as Head of Solutions and AI Draper! R Variance covariance Matrices for linear Regression chosen to correspond to vcov, R ’ s function... Two variables ﬁtted model objects will walk you through linear Regression dialog box click! ( COD ), is a statistical measure to qualify the linear Regression starts at different rates 's to. Vary between the two random variables further by allowing the rate of increase in circumference to vary between variables! Can extended this model further by allowing the rate of increase in circumference is consistent between the are. Will walk you through linear Regression assumes that the growth starts at different rates the increase circumference! In R bloggers | 0 Comments linear Regression dialog box, click statistics in circumference to between. Income values are divided by 10,000 to make the income values are by. Data set measures how the two are linearly related be included in the linear.... Contrasts in linear models ; Cate–Nelson Analysis a shrinkage coeﬃcient and a covariance... Constants which are called the coefficients consistent between the five trees your help, do! Coeﬃcient and a matrix that governs the covariance of eruption duration and waiting time about. 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2020 covariance linear regression r


text:
## Involve: A Journal of Mathematics

• Involve
• Volume 12, Number 4 (2019), 659-670.

### Graphs with at most two trees in a forest-building process

#### Abstract

Given a graph, we can form a spanning forest by first sorting the edges in a random order, and then only keeping edges incident to a vertex which is not incident to any previous edge. The resulting forest is dependent on the ordering of the edges, and so we can ask, for example, how likely is it for the process to produce a graph with $k$ trees.

We look at all graphs which can produce at most two trees in this process and determine the probabilities of having either one or two trees. From this we construct infinite families of graphs which are nonisomorphic but produce the same probabilities.

#### Article information

Source
Involve, Volume 12, Number 4 (2019), 659-670.

Dates
Revised: 10 September 2018
Accepted: 28 October 2018
First available in Project Euclid: 30 May 2019

https://projecteuclid.org/euclid.involve/1559181658

Digital Object Identifier
doi:10.2140/involve.2019.12.659

Mathematical Reviews number (MathSciNet)
MR3941604

Zentralblatt MATH identifier
07072545

Subjects
Primary: 05C05: Trees

#### Citation

Butler, Steve; Hamanaka, Misa; Hardt, Marie. Graphs with at most two trees in a forest-building process. Involve 12 (2019), no. 4, 659--670. doi:10.2140/involve.2019.12.659. https://projecteuclid.org/euclid.involve/1559181658

#### References

• Z. Berikkyzy, S. Butler, J. Cummings, K. Heysse, P. Horn, R. Luo, and B. Moran, “A forest building process on simple graphs”, Discrete Math. 341:2 (2018), 497–507.
• S. Butler, F. Chung, J. Cummings, and R. Graham, “Edge flipping in the complete graph”, Adv. in Appl. Math. 69 (2015), 46–64.
• R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete mathematics: a foundation for computer science, 2nd ed., Addison-Wesley, Reading, MA, 1994.


text:
# Integral Test for Convergence of an Infinite Series

## Key Questions

• Well, I would avoid using the integral test since evaluating an integral can be very difficult. If nothing else works and you know how to evaluate the integral, then go for it.

• By Integral Test,

${\sum}_{n = 1}^{\infty} \frac{1}{n} ^ 5$ converges.

Let us look at some details.

Let us evaluate the corresponding improper integral.

${\int}_{1}^{\infty} \frac{1}{x} ^ 5 \mathrm{dx}$

$= {\lim}_{t \to \infty} {\int}_{1}^{t} {x}^{- 5} \mathrm{dx}$

$= {\lim}_{t \to \infty} {\left[{x}^{- 4} / - 4\right]}_{1}^{t}$

$= - \frac{1}{4} {\lim}_{t \to \infty} {\left[\frac{1}{x} ^ 4\right]}_{1}^{t}$

$= - \frac{1}{4} {\lim}_{t \to \infty} \left[\frac{1}{t} ^ 4 - 1\right]$

$= - \frac{1}{4} \left(0 - 1\right) = \frac{1}{4}$

Since the integral

${\int}_{1}^{\infty} \frac{1}{x} ^ 5 \mathrm{dx}$

converges to $\frac{1}{4}$,

${\sum}_{n = 1}^{\infty} \frac{1}{n} ^ 5$

also converges by Integral Test.

• Integral Test

If $f$ is a function such that $f \left(n\right) = {a}_{n}$, then

${\sum}_{n = 1}^{\infty} {a}_{n}$ and ${\int}_{1}^{\infty} f \left(x\right) \mathrm{dx}$ converge or diverge together.

I hope that this was helpful.


text:
Home > INT3 > Chapter 7 > Lesson 7.1.1 > Problem7-11

7-11.
1. Write equations for the exponential functions with horizontal asymptotes at y = 0 and passing through the following pairs of points. Homework Help ✎

1. (2, 99) and (6, 8019)

2. (–1, 50) and (2, 25.6)

Use the points to write and solve a system of equations of the form y = abx.

99 = ab2
8019 = ab6

$y=40(0.8)^x$


text:
#### On Optimal Robustness to Adversarial Corruption in Online Decision Problems

This paper considers two fundamental sequential decision-making problems: the problem of prediction with expert advice and the multi-armed bandit problem. We focus on stochastic regimes in which an adversary may corrupt losses, and we investigate what level of robustness can be achieved against adversarial corruption. The main contribution of this paper is to show that optimal robustness can be expressed by a square-root dependency on the amount of corruption. More precisely, we show that two classes of algorithms, anytime Hedge with decreasing learning rate and algorithms with second-order regret bounds, achieve $O( \frac{\log N}{\Delta} + \sqrt{ \frac{C \log N }{\Delta} } )$-regret, where $N, \Delta$, and $C$ represent the number of experts, the gap parameter, and the corruption level, respectively. We further provide a matching lower bound, which means that this regret bound is tight up to a constant factor. For the multi-armed bandit problem, we also provide a nearly-tight lower bound up to a logarithmic factor.


text:
# Finding the period

1. May 3, 2009

### Ry122

How do I find the period of this function?
g(t) = cos (4t) + sin (t)

2. May 3, 2009

### djeitnstine

Graph it and check? Or would you rather do it by analysis.

3. May 4, 2009

### Ry122

i need to do it without graphing

4. May 4, 2009

### HallsofIvy

sin t is periodic with fundamental period $2\pi$ and cos 4t is periodic with fundamental period $\pi/2$. Since $2\pi= 4(\pi/2)$, $2\pi$ is also a period of cos 4t.


text:
# Personal Launch Report 2017-02-04

### Help Support The Rocketry Forum:

#### gdjsky01

##### Whoosh, pop: life is good
TRF Supporter
So I won't wax poetically about launching in the 'Shadow of Mt Wilson' where Edwin Hubble toiled away... and how pretty the snow was on the mountains... Because for most of the time, it was shrouded in clouds... :lol:

Hadn't been to an SCRA (NAR 430) launch in 2.5 years... Was glad to get back to it. About 45 minute drive from my house in the morning... 90 minutes back... traffic don't ya know!
But for LPR and MPR on up to G impulse... it'll do nicely.

LOTS more people than I thought!! :surprised: Probably 6 or 7 TARC teams with adults. And a 10 or 15 boy scouts with Make-and-Take rockets...

It became apparent I had brought too many rockets.

First up I maidened the Estes Savage. The RSO warned me they had a tendency to tip off. For whatever reason. So I set the launch rod almost vertical as instructed.

[HR][/HR]

Using a D12-0 to a B6-4 sure enough... it tipped off anyways and headed down range. Not terrible, but still it was hustling when the chute came out. As the sustainer flew into the chute it became a parawad. No damage. Estes Altimeter said 930' (283m)

[HR][/HR]

#### gdjsky01

##### Whoosh, pop: life is good
TRF Supporter

I have a Fliskits Deuces Wild I finished is a pseudo Saturn V/Saturn 1B decal set I printed. It's been battered a few times, but looks ok from a distance. :lol:

[HR][/HR]

Loaded with 2 Quest C6-5s and using two quest Q2G2 (I don't use them lightly as they may have gone the way of the dinos) it made a nice flight
With the rod angles everything takes an outbound arcing flight. So I got all of 447' (136m) accord to the Estes altimeter.

[HR][/HR]
Missed the focus on this.

[HR][/HR]

#### gdjsky01

##### Whoosh, pop: life is good
TRF Supporter
This is a revival of my Custom Rockets Engage. Not its first flight.
One of my first two rockets back in 1996 when I became a BAR. Only the nose cone is original.

[HR][/HR]

Used an A code C5-3. everything seemed A-ok.... oint:

[HR][/HR]

Until it was not

At least the nomex protected the chute!

Just the shock cord got hosed. It will fly again. I have to report this under N.A.R.'s Expired Motor Program.
Funny thing is this is my first ever C5-3 failure. And I've flown quite a few.

#### gdjsky01

##### Whoosh, pop: life is good
TRF Supporter

Built years ago, yet this was it's maiden flight.

I was using a C6-0 booster and a B6-4 in the sustainer.

[HR][/HR]

[HR][/HR]

The launch and flight were perfect. Booster separation was clean and tumbled nicely.

[HR][/HR]
Yes that is the booster motor falling... apparently I need mo tape.

[HR][/HR]

And the Estes altimeter said... umm... where is it? ... yer kidding me right? Did I forget to hook it to the rocket??? I know I put it in there...
:eyeroll::bangpan::bangpan::bangpan::bangpan:

#### qquake2k

##### Captain Low-N-Slow
Another great report, Jeff. I too finally have two clubs to fly with again, one HPR and the local LPR/MPR. And I too have finally started launching and building again. It does feel good to get back to it. I need to upscale a Dueces Wild!

#### gdjsky01

##### Whoosh, pop: life is good
TRF Supporter
I love the look of this rocket. But it is fragile. The fins and the paper nozzles. Still I always straighten it out and fly some more!

This time on a B6-4

[HR][/HR]

[HR][/HR]

And another Future Launch Vehicle flight is in the past. And yes a fin came off on landing... :lol:
And no, I do not know the altitude as I was ticked off about losing yet another Estes altimeter (they are cheap enough... but still) that I said "To heck with it".

#### gdjsky01

##### Whoosh, pop: life is good
TRF Supporter

Well that's all I could get flown in 3 hours. Ya know you fly in a round, you miss the next. Just a lot of people.
Not bad. Five flights. Eight motors. 1 D, 4 Cs, 3 Bs

All the photos including a very nice SLS Laser-X and more are here

https://flic.kr/s/aHskNt7Gqp

Comment, complaints, etc... welcome.

#### Initiator001

##### Well-Known Member
Jeff,

Glad to hear/see you are back to flying rockets.

Great pictures as usual.

I haven't been to a SCRA launch since before NARAM-58 last year. I need to get back there and fly some rockets.
My big issue is the drive: Two hours up, two-four hours to drive home depending on traffic. Ugh.

#### gdjsky01

##### Whoosh, pop: life is good
TRF Supporter
Jeff,

Glad to hear/see you are back to flying rockets.

Great pictures as usual.

I haven't been to a SCRA launch since before NARAM-58 last year. I need to get back there and fly some rockets.
My big issue is the drive: Two hours up, two-four hours to drive home depending on traffic. Ugh.

As you know, I launch at ROC (or will if it dries out some day) which is 2.5 hours away and at MDARs also 2.5 hours. In general you can add 30 to 60 minutes for the drive back because of traffic.

One observation of course is ROC flies into the afternoon if the W holds off. Whereas SCRA pulls the plug at noon and MDARS usually by noon or 1. Big difference is there are almost no lines at one of those three. :wink::wink::wink:

If I were to make the drive you do, for LPR and MPR at SCRA I would :

1. prep everything the night before. I do this anyway. All I do when I get there is fluff the chute and add an altimeter. The flight cards are filled out, the wadding or chute protectors installed the night before, the motors set aside with starters, masking tape ready (for friction fit), spare starters in a small plastic case in my pocket... Oh and in the case of BP charges? If it's a motor charge, it's in there already the night before. An altimeter charge? I use known good canisters that are already hooked up and get the right amount of beeps. I just add BP at the launch and switch on the altimeter at the pad. Waste no time. IMO also
3. get there as early as possible
4. have an offline conversation with SCRA about letting you check in everything all at once and launch from your own pad every round (or almost every round). Especially you. You are a staple in rocketry. Not like someone they barely know.
5. pay some youngster to come with you who can fetch rockets. That way you launch, they fetch, and you don't miss a round. There were 9 today which is pretty good.

This weekend was a launch out in Holtville. SD-TRA. Might be better for you? DART? Launching into the ocean still scares me.

Best wishes!
Jeff

#### o1d_dude

##### 'I battle gravity'
TRF Supporter
Beautiful Deuce, Jeff.

Thanks for the report and for sharing your recipe for maximizing the number af flights.

#### Buckeye

##### Well-Known Member
TRF Supporter
I agree. The Savage has a lot of fin area and weathercocks like a son of a gun. My next flight, I will tilt the rod with the wind and see how she goes.

Also, the Savage is a two man tracking operation - one for booster, one for sustainer. On my maiden flight, I was the only one watching, lost sight of booster when it ejected, and focused only on the sustainer. Luckily, I heard the thud as the booster hit the ground near me, after what seemed to be an eternity in the air. I recovered both pieces. Next time, I will try to rig up a streamer for the booster to help find it.

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#### gdjsky01

##### Whoosh, pop: life is good
TRF Supporter
I agree. The Savage has a lot of fin area and weathercocks like a son of a gun. My next flight, I will tilt the rod with the wind and see how she goes.
Hi Buckeye. Yeah... I guess the proof is in the pudding. However there are a lot of 2 stagers I think have as much if not more 'proportional' fin area. Nonetheless, it tipped off quite a bit even though I had set the rod angle close to vertical.

I could not agree more with the tracking! I was mortified I'd lost tracking of the sustainer of the Corona because I took my eye off it to track the booster. Turns out of course MOST people watch the sustainer. Sure, its the 'sexy' part. But losing the booster sucks. What I did with the Savage was as the booster dropped off, I just POINTed my arm that way and watched the sustainer. Just kept my arm pointed. Since I saw where the sustainer landed, and had my arm pointed to the booster's 'rough' fall path, I could walk and find the booster while keeping my eye where the sustainer went down.

But the Corona's sustainer went into the Sun. I lost both. It was only the LCO being able to track the sustainer, that let me watch as it landed, and simply "Walking a line from the sustainer back to the pad" that I was able to find the booster. It is best to ASK someone, anyone, to track the booster.

#### gdjsky01

##### Whoosh, pop: life is good
TRF Supporter
Another great report, Jeff. I too finally have two clubs to fly with again, one HPR and the local LPR/MPR. And I too have finally started launching and building again. It does feel good to get back to it. I need to upscale a Dueces Wild!
So glad... funny how we have too dry... then too wet... But that is exactly what 'climate weirdness' predicts. But forget that!! Glad you are building and launching! I went under the weather when I had the heart valve replacement. And started a new job right after that, in a startup... intense. I basically quit everything. Astronomy and Rocketry. And R/C.

The deuces upscale has been done right up to really large HPR. But it's still a worthy scratcher! Especially at the MPR to low HPR. I mean a 2 G RMS flight is $50 or so... or a F impulse sounds better at maybe$30?

Thanks for commenting... someday we MUST launch together! You, me, Kit, and so many more up there!

Last edited:


text:
Cobalt

Jump to: navigation, search

Sources and production

Cobalt is an element that can be found in meteoritic iron. It is more common in the core of planets than in the crust. (Note that iron meteors, are the 'core' of large asteroids which have been broken loose after eons of collisions.)

Mars has not differentiated the iron core from the mantle as much as on Earth. Therefore, the 'iron loving' metals which are rare on the Earth's crust, are believed to be more common in Mars' crust. If this theory is true, then cobalt, nickel, platinum, iron and other such metals are likely to have richer ores on Mars than on Earth.

• Cobalt is found in Earth soils (typically 1-20mg/kg dry weight), and it is likely that it will be found in Martian soils. Mars is close to the asteroid belt, and so Mars has been peppered by metallic meteors, making trace amounts in Martian soils likely higher than on Earth.
• As it is required in trace amounts for animal life, cobalt can be brought from Earth until the settlement(s) desire to become entirely self sufficient. (Likely in the form of vitamin pills.)

Use

• Cobalt is a transition metal. Cobalamin, or vitamin ${\displaystyle B_{12}}$, has Cobalt at its active center. It is essential to all animal life. Lack of cobalt has caused deficiency sickness in a number of cases in Australia.
• Cobalt is used in many high temperature alloys.
• Cobalt can be used a catalysts in the petrochemical industries, and could be used to prepare petrochemicals from the carbon dioxide extracted from the Martian atmosphere.
• Cobalt is used in some lithium ion batteries, but several modern battery cathodes dispense with it.


text:
# mm-wavelengths Moon map, scientific case

I would like to test a telescope with a Moon observation. It is a low spectrometer-imager at 110-300 GHz with a resolution of 4 arcmin@150 GHz.

Is there any scientific target that concerns the Moon with such an observation?

Millimeter and sub-mm observations (110-300 GHz = 2.7-1.0 mm) are sensitive to the thermal emission and provide a brightness temperature of airless bodies like the Moon or asteroids. The radio emission is from just below the surface/skin of the body (down to $$\sim10\lambda$$ so ~1 cm in this case) and so is also sensitive to the thermal inertia and conductivity of the surface and regolith. A paper discussing sub-mm observations of the Moon (which is freely available) is Pardo et al. 2005.


text:
# lower triangular matrix calculator

Fire Extinguisher Annual Inspection Cost, Example. The shaded blocks in this graphic depict the lower triangular portion of a 6-by-6 matrix. What Is The Absolute Value Of 5, In addition, e p . Logic to find lower triangular matrix in C programming. Given a square matrix and the task is to check the matrix is in lower triangular form or not. Abstractly, this is equivalent to stabilizing a flag: upper triangular matrices are precisely those that preserve the standard flag, which is given by the standard ordered basis 1 In algebraic terms, these operators correspond to an algebra representation of the polynomial algebra in k variables. Â© 2018 Athletes Rehab, LLC. Example of an upper triangular matrix: 1 0 2 5 0 3 1 3 0 0 4 2 0 0 0 3 [1]  2020/11/23 17:23   Male / 20 years old level / High-school/ University/ Grad student / Very /, [2]  2020/11/06 12:00   Female / 20 years old level / High-school/ University/ Grad student / Useful /, [3]  2020/10/16 00:14   Male / 20 years old level / An engineer / Very /, [4]  2020/07/27 03:33   Male / 20 years old level / High-school/ University/ Grad student / Very /, [5]  2020/07/24 07:57   Female / 20 years old level / High-school/ University/ Grad student / Very /, [6]  2020/07/02 12:23   Male / 40 years old level / An engineer / Useful /, [7]  2020/05/09 21:07   Male / 20 years old level / High-school/ University/ Grad student / Very /, [8]  2020/05/06 02:05   Male / 30 years old level / High-school/ University/ Grad student / Useful /, [9]  2020/04/19 02:07   Male / 20 years old level / High-school/ University/ Grad student / Useful /, [10]  2020/04/17 05:04   Male / Under 20 years old / An engineer / Useful /. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Some functions are limited now because setting of JAVASCRIPT of the browser is OFF. The sum of two upper triangular matrices is upper triangular. {\displaystyle x_{1}} This can be proven by first showing that commuting matrices have a common eigenvector, and then inducting on dimension as before. I wanna multiply a lower triangular and an upper triangular matrix... the usual matrix multiplication is a waste because it spends so much time in multiplying zeros ... i am looking if there is a matlab specific way to save computation time .. matrices are of size of the order of thousands A x Hirpling Definition Beowulf, A lower triangular matrix with elements f[i,j] below the diagonal could be formed in versions of the Wolfram Language prior to 6 using LowerDiagonalMatrix[f, n], which could be run after first loading LinearAlgebraMatrixManipulation.. 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Carmichael Funeral Home Obituaries, A lower triangular matrix with elements f[i,j] below the diagonal could be formed in versions of the Wolfram Language prior to 6 using LowerDiagonalMatrix[f, n], which could be run after first loading LinearAlgebraMatrixManipulation.. A strictly lower triangular matrix is a lower triangular matrix having 0s along the diagonal as well, i.e., for . LU Decomposition Calculator - High accuracy calculation If the inverse L 1 of an lower triangular matrix L exists, then it is lower triangular. Input. Other names used for these matrices are unit (upper or lower) triangular, or very rarely normed (upper or lower) triangular. Extended Capabilities. Umst Time Zone, An upper triangular matrix with elements f[i,j] above the diagonal could be formed in versions of the Wolfram Language prior to 6 using UpperDiagonalMatrix[f, n], which could be run after first loading LinearAlgebraMatrixManipulation.. A strictly upper triangular matrix is an upper triangular matrix having 0s along the diagonal as well, i.e., for . n and the resulting flag ( C program to find whether the matrix is lower triangular or not. Stag Arms Cto 15, Rec Room Android Apk, How Did Tiber Septim Become A God, Leave extra cells empty to enter non-square matrices. How To Use A Compass To Find Property Lines, Honeywell Employee Benefits 2019, Let be a lower triangular matrix. Such a matrix is also called a Frobenius matrix, a Gauss matrix, or a Gauss transformation matrix.. Triangularisability. The row pivot information in LU decomposition is in one-dimensional array P. In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). To be honest, I don't even understand what the question is asking of me, however I do know what upper and lower triangular matrices are. The value of the determinant is correct if, after the transformations the lower triangular matrix is zero, and the elements of â¦ With help of this calculator you can: find the matrix determinant, the rank, raise the matrix to a power, find the sum and the multiplication of matrices, calculate the inverse matrix. The matrix A is converted into Lower triangular matrix, L by elementary row operation or reduction and then product of main diagonal elements is called determinant of the matrix A. An LU factorization pertains to the factórization óf A, with proper row and/or line orderings or mixtures, into two factors, a lower triangular matrix M and an upper triangular â¦ Honda S24a Transmission, An easy way to remember whether a matrix is upper triangular or lower triangular by where the non-zero entries of the matrix lie as illustrated in the following graphic: Inverse of Upper/Lower Triangular Matrices â¢Inverse of an upper/lower triangular matrix is another upper/lower triangular matrix. 1 Such a matrix is also called a Frobenius matrix, a Gauss matrix, or a Gauss transformation matrix. [ Sent Vs Delivered Grindr, Drzan Mcbee Death, This was proven in (Drazin, Dungey & Gruenberg 1951); a brief proof is given in (Prasolov 1994, pp. Lower Triangular Matrix Watch more videos at https://www.tutorialspoint.com/videotutorials/index.htm Lecture By: Er. Waco, Tx Mugshots, Just type matrix elements and click the button. Leaded Vs Unleaded Gas, Word Swipe Answers, LU Decomposition of Matrix calculator - Online matrix calculator for LU Decomposition of Matrix, step-by-step. The non-zero entries form the shape of a trapezoid. Feral Friends Spongebob Dailymotion, The calculator will diagonalize the given matrix, with steps shown. â Number of Rows and Columns (only square matrices can be factorized into A=LU): Sorry it is lower triangular matrix â R Subha Sep 12 '18 at 17:29 Please describe what input statements exactly we must use to reproduce your actual output. lower triangular matrix calculator . Arbalest Vs Crossbow, â1=ð¼. The material on this website is not intended to replace the advice of a medical professional. Birdsong Stephen And Isabelle Relationship, linear-algebra matrices David Lanz Net Worth, online matrix LU decomposition calculator, find the upper and lower triangular matrix by factorization The upper triangular matrices are precisely those that stabilize the standard flag. Thank you for your questionnaire.Sending completion. In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. I Love You In Afrikaans, Carb Timing: Does it improve performance (sucrose)? Https Rbx Gg Login, Metric Ruler Game, A lower triangular matrix is a square matrix in which all entries above the main diagonal are zero (only nonzero entries are found below the main diagonal - in the lower triangle). By browsing this website, you agree to our use of cookies. Before beginning any exercise or rehabilitation regimen, consult with a licensed physician or other qualified medical professional in your state. All Rights Reserved. k n {\displaystyle \ell _{1,1}x_{1}=b_{1}} Copyright (c) 2013 Thatcher Development Software, LLC. No need to compute determinant. \(\hspace{60px} A\hspace{50px}=\hspace{50px}L\hspace{100px} U\\. where [T] is an upper (or right) triangular matrix, i.e. [T] T is the lower (or left) triangular matrix obtained by transposing [T]. x , and repeats through to A and Terms and Conditions. Depth Jump vs. Drop Jump: What’s the difference? All unitriangular matrices are unipotent. Dump Truck Company Names, Moreover if [K] is regular, [T] is regular too. Partial pivot with row exchange is selected. Himalayan Silver Birch Trees For Sale, , x In fact more is true: the eigenvalues of a triangular matrix are exactly its diagonal entries. ( â¦ {\displaystyle x_{k}} k n A matrix equation in the form b However, operations mixing upper and lower triangular matrices do not in general produce triangular matrices. j x 1 p {\displaystyle A_{i}} An atomic (upper or lower) triangular matrix is a special form of unitriangular matrix, where all of the off-diagonal elements are zero, except for the entries in a single column. A real symmetric positive definite (n × n)-matrix X can be decomposed as X = LL T where L, the Cholesky factor, is a lower triangular matrix with positive diagonal elements (Golub and van Loan, 1996).Cholesky decomposition is the most efficient method to check whether a real symmetric matrix â¦ Lower triangular matrix is a matrix which contain elements below principle diagonal including principle diagonal elements and rest of the elements are 0. online matrix LU decomposition calculator, find the upper and lower triangular matrix by factorization {\displaystyle xI-A} A more precise statement is given by the Jordan normal form theorem, which states that in this situation, A is similar to an upper triangular matrix of a very particular form. Inverting Triangular Matrices: Proofs Recall the (n 1) (n 1) cofactor matrix C rs that results from omitting row r and column s of U = (u For example, if A is an n × n triangular matrix, the equation A â¢ x = b can be solved for x in at most n 2 operations. Decomposing a square matrix into a lower triangular matrix and an upper triangular matrix. ) only involves All flags are conjugate (as the general linear group acts transitively on bases), so any matrix that stabilises a flag is similar to one that stabilises the standard flag. Barnes 7mm Bullets, , . 2004 Isuzu Npr Oil Capacity, Thanks in advance for your help. Laser Engraved Keychain, The information contained on this website is provided as a service to the public and is intended for general informational, educational and entertainment purposes only. See the picture below. Lois Hardwick Cause Of Death, If for any reason you have pain during exercise, stop and seek medical attention immediately. Paranormal Show Name Generator, Lee Duncan Actor, b. Surf Leaf Green, Lower Triangular matrix transformation method is preferred over minor or cofactor of matrix method while finding determinant of the matrix's size over 3x3. Audi Ventilated Seats, A 1 1 1 Online LU Decomposition Calculator is simple and reliable online tool decompose or factorize given square matrix to Lower triangular matrix (L) and Upper triangular matrix (U). , the Lie algebra of all upper triangular matrices; in symbols, ] [ The invertible ones among them form a subgroup of the general linear group, whose conjugate subgroups are those defined as the stabilizer of some (other) complete flag. Moreover, each eigenvalue occurs exactly k times on the diagonal, where k is its algebraic multiplicity, that is, its multiplicity as a root of the characteristic polynomial A more precise statement is given by the Jordan normal form theorem, which states that in this situation, A is similar to an upper triangular matrix of a very particular form. Low Flash 38 Special Ammo, â¢Can be computed from first principles: Using the definition of an Inverse. Input elements in matrix: 1 0 0 4 5 0 7 8 9. The information contained on this website is provided as a service to the public and is intended for general informational, educational and entertainment purposes only. Karl Glusman Liz Glusman, {\displaystyle K[A_{1},\ldots ,A_{k}]} Your feedback and comments may be posted as customer voice. 40k Basilisk Instructions, Write a C program to read elements in a matrix and check whether the matrix is a lower triangular matrix or not. Davinci Olive Vs Sierra, Denote by the columns of .By definition, the inverse satisfies where is the identity matrix. If you are experiencing a medical emergency, get to the nearest emergency room or dial 911. Scary Knocking Sound, The notion of a triangular matrix is more narrow and it's used for square matrices only. 1 2 {\displaystyle x_{1}} {\displaystyle xI-A} â Hyperbole For Rain, Step 1: To Begin, select the number of rows and columns in your Matrix, and press the "Create Matrix" button. The group of invertible lower triangular matrices is such a subgroup, since it is the stabilizer of the standard flag associated to the standard basis in reverse order. , Additionally, this also shows that the upper triangular matrices can be viewed as a Lie subalgebra of the Lie algebra of square matrices of a fixed size, where the Lie bracket [a, b] given by the commutator ab â ba. â Xukrao Sep 12 '18 at 17:30 Dwarf Caiman Price, What Does Downhill Mean In Basketball, Roblox Gift Card Codes Not Used 2020, Joshua Apocalypse Costume, Duck Butter Sergio's Mom, A matrix that is similar to a triangular matrix is referred to as triangularizable. Athletes R Mehar Posh Drama Episode 5 Dailymotion, This Calculator will Factorize a Square Matrix into the form A=LU where L is a lower triangular matrix, and U is an upper triangular matrix. Polish Slang Swear Words, Rap Songs With Similes And Metaphors In The Lyrics, It goes like this: the triangular matrix is a square matrix where all elements below the main diagonal are zero. C/C++ Code Generation Generate C and C++ code using MATLAB® Coderâ¢. The online calculator calculates the value of the determinant of a 4x4 matrix with the Laplace expansion in a row or column and the gaussian algorithm. We use cookies to improve your experience on our site and to show you relevant advertising. , then substitutes that forward into the next equation to solve for A set of matrices Step 1: of A. ) Glueck Model Strategic Management, Desi Arnaz Jr Wife, Athletes Rehab makes every effort to provide current and accurate information but will not be responsible for information that is subject to change. The lower triangular portion of a matrix includes the main diagonal and all elements below it. Shimono Hiro Married, Lavell Crawford Wife, A ) A However, a unit triangular matrix is not the same as the unit matrix, and a normed triangular matrix has nothing to do with the notion of matrix norm. Learn more Hire us: Alkaline Diet Improves Anaerobic Performance. Information on this site may become outdated or may not conform to the standard of care in the area from which you are viewing this site. Triangular matrices allow numerous algorithmic shortcuts in many situations. Click here to contact Greg Thatcher There are alternative approaches for computing determinate, such as evaluating the set of eigenvalues (the determinant of a square matrix is equal to the product of its eigenvalues). Aws Lightsail Rdp, â¦ n [1] In fact, a matrix A over a field containing all of the eigenvalues of A (for example, any matrix over an algebraically closed field) is similar to a triangular matrix. Like this: the eigenvalues of a. is more narrow and it 's used square... 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August 7, 2022

# Generating Random Floats in [0, 1)

Recently, I found myself in a situation where I needed to generate a random float between 0 and 1 using a random 32-bit integer in C. This subject has been done to death before, but I felt like feeling a little pain so I decided to figure it out myself before using someone else's better solution.

First things first: we actually need to obtain some random integers. Easy, right? We'll just call rand() and—

rand() is known to have a number of problems; here's a short list:

• If you use glibc like me, your rand() uses a Linear Congruential Generator. LCGs are very simple, but they are prone to subtle correlation issues that tend to manifest when applied to a multidimensional problem, such as Monte Carlo simulations.
• srand() takes an unsigned int, restricting the number of possible sequences you can get from rand() to number of distinct seeds that you can provide. For reference most quality PRNGs have an internal state of at least 128 bits, which is almost certainly larger than the size of an unsigned int on your platform.
• rand() is not thread safe; calling it will result in reads and writes to a shared location. Not only will this affect performance, it will also cause insidious randomness issues that may choose to emerge at the most inconvenient time. rand_r() allows the user to provide a pointer that will be used to store the PRNG's state, allowing thread-safe code to be written, but many programmers are seemingly not aware of this fact.

Okay, so rand() is not as horrible as I've made it out to be. Don't get me wrong—rand() is pretty crappy, but if you generated 100 random numbers and stared at it, it would probably look pretty random. So rand() might be okay for use cases where high quality randomness is not required (such as a game), but if you try subjecting it to rigorous statistical tests it crumples like wet toilet paper.

Today, we're going to be using xoshiro128+ as our PRNG. Admittedly, the choice of PRNG is not actually that important, which may come as a surprise after my rant about rand(). However, as long as you are not using a crappy or obscure generator, there are plenty of very fast, well-proven generators that are probably sufficient for your application.

# Introducing Floating Point §

Okay, great. So now we have a random uint32_t... how do we turn this into a random float?

The naïve solution is to just divide the integer by the maximum value (232), but this is not great for two reasons:

• Division is slow.
• This requires casting the integer to a float, but floats themselves are also 32 bits, so a float could not possibly represent all 32-bit integers. You can avoid this precision loss by casting to a double, but that just makes the first issue worse.

Instead, what we want to do is create a floating point number using bitwise operations. To do that, we need to familiarize ourselves with how floating point values are represented in memory; specifically, we're going to be focusing on IEEE 754, which is sort of required in the C standard.

If you ever learned scientific notation in school, a floating point number is a lot like that. Basically, three fields are packed together: the sign bit, the exponent, and the integer. The function of the sign bit is very simple; 0 is positive and 1 is negative. The real value represented by a float is basically equal to $\mathrm{fraction} \times 2^\mathrm{exponent}$.

There are a few pitfalls that we need to be aware of, however:

• The exponent is stored with an offset of 127, so an exponent of 0 is equal to -127, an exponent of 127 is equal to 0, and an exponent of 255 is equal to 128.
• The fraction section is interpreted as if there was an additional 1 bit in front of it.

In this case, the exponent portion is equal to $124 - 127 = -3$, and the fraction section is equal to $1.01_2 = 1.25$. Thus, the overall value is equal to $2^{-3} \times 1.25 = 0.15625$.

# Actually Generating Floats §

As usual, I've managed to stretch what ought to be a simple blogpost into a long-winded rag about unrelated stuff... bummer. Okay, let's actually generate some floats.

We can start by drawing 23 bits from our PRNG for the fraction part. This is easy enough; we just take a 32-bit value and shift it right 9 bits. This gives us a random fraction value between $1.000\ldots_2 = 1$ and $1.111\ldots_2 \approx 2$.

If we set the exponent portion to 0, the fraction portion will be multipled with $2^0 = 1$, giving us a random number in the range $[1, 2)$. We can easily turn into our desired value in range $[0, 1)$ by subtracting 1.

Here's the actual code:

float int_to_float(uint32_t random) {
union { uint32_t u32; float f; } u = { .u32 = random >> 9 | 0x3f800000 };
return u.f - 1.0;
}

0x3f800000 is what we get when we take our exponent (0 is represented as 127) and shift it right 23 bits so that it's in the right place.

We could use plain old floating point multiplication instead of type punning, though there is a chance that it will be imperceptibly slower than our hacky solution. Probably not. Whatever.

You can find the full code demonstrating this concept here. That's all, folks.


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Kyoung Jin Lee : A scary, yet interesting, scenario to the fibrillating heart

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Alternans, a beat-to-beat temporal alternation in the sequence of heart beats, is a known precursor of the development of cardiac fibrillation, leading to sudden cardiac death. The equally important precursor of cardiac arrhythmias is the rotating spiral wave of electro-mechanical activity, or reentry, on the heart tissue. In this talk, I will show that these two seemingly different phenomena can have a remarkable relationship: In well controlled in-vitro tissue cultures, isotropic populations of rat ventricular myocytes sustaining a temporal rhythm of alternans can support period-2 oscillatory re-entries, and vice versa. These re-entries bear `line defects' across which the phase of local excitation slips rather abruptly by $2\pi$, when a full period-2 cycle of alternans completes in $4\pi$. In other words, the cells belonging to the line defects are period-1 oscillatory whereas all the others in the bulk medium are period-2 oscillatory. We also find that a slowly rotating line defect results in a quasi-periodic like oscillation in the bulk medium. Some key features of these phenomena can be well reproduced in computer simulations of a nonlinear reaction-diffusion model.

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