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## 4.4 Statistical Issues When Comparing Two Systems

The previous sections have concentrated on estimating the performance of a system through the execution of a single simulation model. The running of the model requires the specification of the input variables (e.g. mean time between arrivals, service distribution, etc.) and the structure of the model (e.g. FIFO queue, process flow, etc.) The specification of a set of inputs (variables and/or structure) represents a particular system configuration, which is then simulated to estimate performance. To be able to simulate design configurations, you may have to build different models or you may be able to use the same model supplied with different values of the program inputs. In either situation, you now have different design configurations that can be compared. This allows the performance of the system to be estimated under a wide-variety of controlled conditions. It is this ability to easily perform these what-if simulations that make simulation such a useful analysis tool. Figure 4.59 represents the notion of using different inputs to get different outputs.

Naturally, when you have different design configurations, you would like to know which configurations are better than the others. Since the simulations are driven by random variables, the outputs from each configuration (e.g. $$Y^1, Y^2$$) are also random variables. The estimate of the performance of each system must be analyzed using statistical methods to ensure that the differences in performance are not simply due to sampling error. In other words, you want to be confident that one system is statistically better (or worse) than the other system.

The techniques for comparing two systems via simulation are essentially the same as that found in books that cover the statistical analysis of two samples (e.g. ). This section begins with a review of these methods. Assume that samples from two different populations (system configurations) are available:

$X_{11}, X_{12},\ldots, X_{1 n_1} \ \ \text{a sample of size n_1 from system configuration 1}$

$X_{21}, X_{22},\ldots, X_{2 n_2} \ \ \text{a sample of size n_2 from system configuration 2}$

The samples represent a performance measure of the system that will be used in a decision regarding which system configuration is preferred. For example, the performance measure may be the average system throughput per day, and you want to pick the design configuration that has highest throughput.

Assume that each system configuration has an unknown population mean for the performance measure of interest, $$E[X_1] = \theta_1$$ and $$E[X_2] = \theta_2$$. Thus, the problem is to determine, with some statistical confidence, whether $$\theta_1 < \theta_2$$ or alternatively $$\theta_1 > \theta_2$$. Since the system configurations are different, an analysis of the situation of whether $$\theta_1 = \theta_2$$ is of less relevance in this context.

Define $$\theta = \theta_1 - \theta_2$$ as the mean difference in performance between the two systems. Clearly, if you can determine whether $$\theta >$$ 0 or $$\theta <$$ 0 you can determine whether $$\theta_1 < \theta_2$$ or $$\theta_1 > \theta_2$$. Thus, it is sufficient to concentrate on the difference in performance between the two systems.

Given samples from two different populations, there are a number of ways in which the analysis can proceed based on different assumptions concerning the samples. The first common assumption is that the observations within each sample for each configuration form a random sample. That is, the samples represent independent and identically distributed random variables. Within the context of simulation, this can be easily achieved for a given system configuration by performing replications. For example, this means that $$X_{11}, X_{12},\ldots, X_{1 n_1}$$ are the observations from $$n_1$$ replications of the first system configuration. A second common assumption is that both populations are normally distributed or that the central limit theorem can be used so that sample averages are at least approximately normal.

To proceed with further analysis, assumptions concerning the population variances must be made. Many statistics textbooks present results for the case of the population variance being known. In general, this is not the case within simulation contexts, so the assumption here will be that the variances associated with the two populations are unknown. Textbooks also present cases where it is assumed that the population variances are equal. Rather than making that assumption it is better to test a hypothesis regarding equality of population variances.

The last assumption concerns whether or not the two samples can be considered independent of each other. This last assumption is very important within the context of simulation. Unless you take specific actions to ensure that the samples will be independent, they will, in fact, be dependent because of how simulations use (re-use) the same random number streams. The possible dependence between the two samples is not necessarily a bad thing. In fact, under certain circumstance it can be a good thing.

The following sections first presents the methods for analyzing the case of unknown variance with independent samples. Then, we focus on the case of dependence between the samples. Finally, how to use to do the work of the analysis will be illustrated.

### 4.4.1 Analyzing Two Independent Samples

Although the variances are unknown, the unknown variances are either equal or not equal. In the situation where the variances are equal, the observations can be pooled when developing an estimate for the variance. In fact, rather than just assuming equal or not equal variances, you can (and should) use an F-test to test for the equality of variance. The F-test can be found in most elementary probability and statistics books (see ).

The decision regarding whether $$\theta_1 < \theta_2$$ can be addressed by forming confidence intervals on $$\theta = \theta_1 - \theta_2$$. Let $$\bar{X}_1$$, $$\bar{X}_2$$, $$S_1^2$$, and $$S_2^2$$ be the sample averages and sample variances based on the two samples (k = 1,2):

$\bar{X}_k = \dfrac{1}{n_k} \sum_{j=1}^{n_k} X_{kj}$

$S_k^2 = \dfrac{1}{n_k - 1} \sum_{j=1}^{n_k} (X_{kj} - \bar{X}_k)^2$

An estimate of $$\theta = \theta_1 - \theta_2$$ is desired. This can be achieved by estimating the difference with $$\hat{D} = \bar{X}_1 - \bar{X}_2$$. To form confidence intervals on $$\hat{D} = \bar{X}_1 - \bar{X}_2$$ an estimator for the variance of $$\hat{D} = \bar{X}_1 - \bar{X}_2$$ is required. Because the samples are independent, the computation of the variance of the difference is:

$Var(\hat{D}) = Var(\bar{X}_1 - \bar{X}_2) = \dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}$

where $$\sigma_1^2$$ and $$\sigma_2^2$$ are the unknown population variances. Under the assumption of equal variance, $$\sigma_1^2 = \sigma_2^2 =\sigma^2$$, this can be written as:

$Var(\hat{D}) = Var(\bar{X}_1 - \bar{X}_2) = \dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2} = \sigma^2 (\dfrac{1}{n_1} + \dfrac{1}{n_2})$

where $$\sigma^2$$ is the common unknown variance. A pooled estimator of $$\sigma^2$$ can be defined as:

$S_p^2 = \dfrac{(n_1 - 1)S_1^2 + (n_2 - 1)S_2^2}{n_1 + n_2 - 2}$

Thus, a ($$1 - \alpha$$)% confidence interval on $$\theta = \theta_1 - \theta_2$$ is:

$\hat{D} \pm t_{1-(\alpha/2) , v} s_p \sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$

where $$v = n_1 + n_2 - 2$$. For the case of unequal variances, an approximate ($$1 - \alpha$$)% confidence interval on $$\theta = \theta_1 - \theta_2$$ is given by:

$\hat{D} \pm t_{1-(\alpha/2) , v} \sqrt{S_1^2/n_1 + S_2^2/n_2}$

where $v = \Biggl\lfloor \frac{(S_1^2/n_1 + S_2^2/n_2)^2}{\frac{(S_1^2/n_1)^2}{n_1 +1} + \frac{(S_2^2/n_2)^2}{n_2 + 1}} - 2 \Biggr\rfloor$

Let $$[l, u]$$ be the resulting confidence interval where $$l$$ and $$u$$ represent the lower and upper limits of the interval with by construction $$l < u$$. Thus, if $$u < 0$$, you can conclude with ($$1 - \alpha$$)% confidence that $$\theta = \theta_1 - \theta_2 < 0$$ (i.e. that $$\theta_1 < \theta_2$$). If $$l > 0$$, you can conclude with ($$1 - \alpha$$)% that $$\theta = \theta_1 - \theta_2 > 0$$ (i.e. that $$\theta_1 > \theta_2$$). If $$[l, u]$$ contains 0, then no conclusion can be made at the given sample sizes about which system is better. This does not indicate that the system performance is the same for the two systems. You know that the systems are different. Thus, their performance will be different. This only indicates that you have not taken enough samples to detect the true difference. If sampling is relatively cheap, then you may want to take additional samples in order to discern an ordering between the systems.

Two configurations are under consideration for the design of an airport security checkpoint. A simulation model of each design was made. The replication values of the throughput per minute for the security station for each design are provided in Table 4.2.

Table 4.2: Comparing two designs
Design 1 Design 2
1 10.98 8.93
2 8.87 9.82
3 10.53 9.27
4 9.40 8.50
5 10.10 9.63
6 10.28 9.55
7 8.86 9.30
8 9.00 9.31
9 9.99 9.23
10 9.57 8.88
11 8.05
12 8.74
$$\bar{x}$$ 9.76 9.10
$$s$$ 0.74 0.50
$$n$$ 10 12

Assume that the two simulations were run independently of each other, using different random numbers. Recommend the better design with 95% confidence.

According to the results:

$\hat{D} = \bar{X}_1 - \bar{X}_2 = 9.76 - 9.1 = 0.66$

In addition, we should test if the variances of the samples are equal. This requires an $$F$$ test, with $$H_0: \sigma_{1}^{2} = \sigma_{2}^{2}$$ versus $$H_1: \sigma_{1}^{2} \neq \sigma_{2}^{2}$$. Based on elementary statistics, the test statistic is: $$F_0 = S_{1}^{2}/S_{1}^{2}$$. The rejection criterion is to reject $$H_0$$ if $$F_0 > f_{\alpha/2, n_1-1, n_2 -1}$$ or $$F_0 < f_{1-(\alpha/2), n_1-1, n_2 -1}$$, where $$f_{p, u, v}$$ is the upper percentage point of the $$F$$ distribution. Assuming a 0.01 significance level for the $$F$$ test, we have $$F_0 = (0.74)^{2}/(0.50)^{2} = 2.12$$. Since $$f_{0.005, 9, 11} = 5.54$$ and $$f_{0.995, 9, 11} = 0.168$$, there is not enough evidence to conclude that the variances are different at the 0.01 significance level. The value of $$f_{p, u, v}$$ can be determined in as F.INV.RT(p, u, v) within Excel. Note also that $$f_{1-p, u, v} = 1/f_{p, v, u}$$. In $$R$$, the formula is $$f_{p, u, v} = qf(1-p, u,v)$$, since $$R$$ provides the quantile function, not the upper right tail function.

Since the variances can be assumed equal, we can use the pooled variance, which is:

\begin{aligned} S_p^2 & = \dfrac{(n_1 - 1)S_1^2 + (n_2 - 1)S_2^2}{n_1 + n_2 - 2}\\ & = \dfrac{(10 - 1)(0.74)^2 + (12 - 1)(0.5)^2}{12 + 10 - 2} \\ & = 0.384\end{aligned}

Thus, a ($$1 -0.05$$)% confidence interval on $$\theta = \theta_1 - \theta_2$$ is:

\begin{aligned} \hat{D} & \pm t_{1-(\alpha/2), v} s_p \sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}} \\ 0.66 & \pm t_{1-0.025 , 20} (\sqrt{0.384}) \sqrt{\dfrac{1}{10} + \dfrac{1}{12}} \\ 0.66 & \pm (2.086)(0.6196)(0.428) \\ 0.66 & \pm 0.553\end{aligned}

where $$v = n_1 + n_2 - 2 = 10 + 12 - 2 = 20$$. Since this results in an interval $$[0.10, 1.21]$$ that does not contain zero, we can conclude that design 1 has the higher throughput with 95% confidence.

The confidence interval can assist in making decisions regarding relative performance of the systems from a statistically significant standpoint. However, if you make a conclusion about the ordering of the system, it still may not be practically significant. That is, the difference in the system performance is statistically significant but the actual difference is of no practical use. For example, suppose you compare two systems in terms of throughput with resulting output $$\bar{X}_1$$ = 5.2 and $$\bar{X}_2$$ = 5.0 with the difference statistically significant. While the difference of 0.2 may be statistically significant, you might not be able to achieve this in the actual system. After all, you are making a decision based on a model of the system not on the real system. If the costs of the two systems are significantly different, you should prefer the cheaper of the two systems since there is no practical difference between the two systems. The fidelity of the difference is dependent on your modeling assumptions. Other modeling assumptions may overshadow such a small difference.

The notion of practical significance is model and performance measure dependent. One way to characterize the notion of practical significance is to conceptualize a zone of performance for which you are indifferent between the two systems. Figure 4.60 illustrates the concept of an indifference zone around the difference between the two systems. If the difference between the two systems falls in this zone, you are indifferent between the two systems (i.e. there is no practical difference).

Using the indifference zone to model the notion of practical significance, if $$u < -\Delta$$, you can conclude confidence that $$\theta_1 < \theta_2$$, and if $$l > \Delta$$, you can conclude with confidence that $$\theta_1 > \theta_2$$. If $$l$$ falls within the indifference zone and $$u$$ does not (or vice versa), then there is not enough evidence to make a confident conclusion. If $$[l,u]$$ is totally contained within the indifference zone, then you can conclude with confidence that there is no practical difference between the two systems.

### 4.4.2 Analyzing Two Dependent Samples

In this situation, continue to assume that the observations within a sample are independent and identically distributed random variables; however, the samples themselves are not independent. That is, assume that the $$(X_{11}, X_{12},$$, $$\ldots, X_{1n_1})$$ and $$(X_{21}, X_{22}, \ldots, X_{2n_2})$$ from the two systems are dependent.

For simplicity, suppose that the difference in the configurations can be implemented using a simple parameter change within the model. For example, the mean processing time is different for the two configurations. First, run the model to produce $$(X_{11}, X_{12}, \ldots, X_{1n_1})$$ for configuration 1. Then, change the parameter and re-executed the model to produce $$(X_{21}, X_{22}, \ldots, X_{2n_2})$$ for configuration 2.

Assuming that you did nothing with respect to the random number streams, the second configuration used the same random numbers that the first configuration used. Thus, the generated responses will be correlated (dependent). In this situation, it is convenient to assume that each system is run for the same number of replications, i.e. $$n_1$$ = $$n_2$$ = n. Since each replication for the two systems uses the same random number streams, the correlation between $$(X_{1,j}, X_{2,j})$$ will not be zero; however, each pair will still be independent across the replications. The basic approach to analyzing this situation is to compute the difference for each pair:

$D_j = X_{1j} - X_{2j} \ \ \text{for} \; j = 1,2,\ldots,n$

The $$(D_1, D_2, \ldots, D_n)$$ will form a random sample, which can be analyzed via traditional methods. Thus, a ($$1 - \alpha$$)% confidence interval on $$\theta = \theta_1 - \theta_2$$ is:

$\bar{D} = \dfrac{1}{n} \sum_{j=1}^n D_j$

$S_D^2 = \dfrac{1}{n-1} \sum_{j=1}^n (D_j - \bar{D})^2$

$\bar{D} \pm t_{1-\alpha/2, n-1} \dfrac{S_D}{\sqrt{n}}$

The interpretation of the resulting confidence interval $$[l, u]$$ is the same as in the independent sample approach. This is the paired-t confidence interval presented in statistics textbooks.

Assume that the two simulations were run dependently using common random numbers. Recommend the better design with 95% confidence.

According to the results:

$\bar{D} = \bar{X}_1 - \bar{X}_2 = 50.88 - 48.66 = 2.22$

Also, we have that $$S_D^2 = (0.55)^2$$. Thus, a ($$1 -0.05$$)% confidence interval on $$\theta = \theta_1 - \theta_2$$ is:

\begin{aligned} \hat{D} & \pm t_{1-(\alpha/2), n-1} \dfrac{S_D}{\sqrt{n}}\\ 2.22 & \pm t_{1-0.025 , 9} \dfrac{0.55}{\sqrt{10}}\\ 2.22 & \pm (2.261)(0.1739)\\ 2.22 & \pm 0.0.393\end{aligned}

Since this results in an interval $$[1.827, 2.613]$$ that does not contain zero, we can conclude that design 1 has the higher cost with 95% confidence.

Of the two approaches (independent versus dependent) samples, the latter is much more prevalent in simulation contexts. The approach is called the method of common random numbers (CRN) and is a natural by product of how most simulation languages handle their assignment of random number streams.

To understand why this method is the preferred method for comparing two systems, you need to understand the method’s affect on the variance of the estimator. In the case of independent samples, the estimator of performance was $$\hat{D} = \bar{X}_1 - \bar{X}_2$$. Since

\begin{aligned} \bar{D} & = \dfrac{1}{n} \sum_{j=1}^n D_j \\ & = \dfrac{1}{n} \sum_{j=1}^n (X_{1j} - X_{2j}) \\ & = \dfrac{1}{n} \sum_{j=1}^n X_{1j} - \dfrac{1}{n} \sum_{j=1}^n X_{2j} \\ & = \bar{X}_1 - \bar{X}_2 \\ & = \hat{D}\end{aligned}

The two estimators are the same, when $$n_1 = n_2 = n$$; however, their variances are not the same. Under the assumption of independence, computing the variance of the estimator yields:

$V_{\text{IND}} = Var(\bar{X}_1 - \bar{X}_2) = \dfrac{\sigma_1^2}{n} + \dfrac{\sigma_2^2}{n}$

Under the assumption that the samples are not independent, the variance of the estimator is:

$V_{\text{CRN}} = Var(\bar{X}_1 - \bar{X}_2) = \dfrac{\sigma_1^2}{n} + \dfrac{\sigma_2^2}{n} - 2\text{cov}(\bar{X}_1, \bar{X}_2)$

If you define $$\rho_{12} = corr(\bar{X}_1, \bar{X}_2)$$, the variance for the common random number situation is:

$V_{\text{CRN}} = V_{\text{IND}} - 2\sigma_1 \sigma_2 \rho_{12}$

Therefore, whenever there is positive correlation $$\rho_{12} > 0$$ within the pairs we have that, $$V_{\text{CRN}} < V_{\text{IND}}$$.

If the variance of the estimator in the case of common random numbers is smaller than the variance of the estimator under independent sampling, then a variance reduction has been achieved. The method of common random numbers is called a variance reduction technique. If the variance reduction results in a confidence interval for $$\theta$$ that is tighter than the independent case, the use of common random numbers should be preferred. The variance reduction needs to be big enough to overcome any loss in the number of degrees of freedom caused by the pairing. When the number of replications is relatively large ($$n > 30$$) this will generally be the case since the student-t value does not vary appreciatively for large degrees of freedom. Notice that the method of common random numbers might backfire and cause a variance increase if there is negative correlation between the pairs. An overview of the conditions under which common random numbers may work is given in (Law 2007).

This notion of pairing the outputs from each replication for the two system configurations makes common sense. When trying to discern a difference, you want the two systems to experience the same randomness so that you can more readily infer that any difference in performance is due to the inherent difference between the systems and not caused by the random numbers.

Use common random numbers when comparing two alternatives to block out the factor of the assigned random numbers and reduce the variance of your estimates for the difference between the system configurations.

In experimental design settings, this is called blocking on a factor. For example, if you wanted to perform and experiment to determine whether a change in a work method was better than the old method, you should use the same worker to execute both methods. If instead, you had different workers execute the methods, you would not be sure if any difference was due to the workers or to the proposed change in the method. In this context, the worker is the factor that should be blocked. In the simulation context, the random numbers are being blocked when using common random numbers.

In the next section, we introduce a modeling situation involving the production of rings. We will then apply the methods of this section to compare two design configurations involving this system.

### References

Law, A. 2007. Simulation Modeling and Analysis. 4th ed. McGraw-Hill.
Montgomery, D. C., and G. C. Runger. 2006. Applied Statistics and Probability for Engineers. 4th ed. John Wiley & Sons.


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## Composite loop connection

engidesk toolbox for the calculation of the stress distribution of a loop connection built from uni-directional rovings

Free

• ### Short description

For almost every load-bearing structure, forces have to be induced or diverted in a way, that the transmission of force is to be configured. The engidesk toolbox at hand allows for the pre-dimensioning of those loop connections in which UD-layers wind around a bolt in form of a parallel loop.

### Details

Release

Last Update

Current version

Price

Licence

Language

01.10.2015

07.03.2016

1.1

Free

MIT-Licence

English, German

### Screenshots

Example Dashboard

• ### Assumptions of the model:

Forces remain constant over the loop's width

There is no friction effect between bolt and loop

No consideration of residual stress caused by thermical and swell effects

The calculation is carried out according to the method of elasto-statics, whereat utilizing a single infinitesimal element, 3 equation systems (balance, equation of elasticity & geometrical relations) can be derived and solved. Calculating the tangential stress works according to the following formula:

$\sigma_{ t }(r)=-p_{ i }*E_{ v }*\left( \frac{ r_{ i }^{ 1+E_{ v } } }{ r_{ i }^{ 2*E_{ v } }-r_{ a }^{ 2*E_{ v } }} *r^{ E_{ v }-1 }*-\frac{ r_{ i }^{ 1-E_{ v } } }{r_{ i }^{ -2*E_{ v } }-r_{ a }^{-2*E_{ v }} }*r^{ -E_{ v }-1 }\right)$

The average tangential stress results from:

$\bar{\sigma}_{ t } =\frac{ \frac{ F }{2 } }{b*\left( r_{ a }-r_{ i } \right) }$

The calculation of the radial stress takes place via the following formula:

$\sigma_{ r }(r)=\frac{ -p_{ i }*r_{ i }^{ 1+E_{ v } } }{ r_{ i }^{ 2*E_{ v } }-r_{ a }^{ 2*E_{ v } }}+\frac{ -p_{ i }*r_{ i }^{ 1-E_{ v } } }{ r_{ i }^{ -2*E_{ v } }-r_{ a }^{ -2*E_{ v } } }*r^{ -E_{ v }-1 }$

In case the loop is supported sideways, a triaxial state of stress prevails. The calculation of the axial stress in longitudinal direction of the sideways supported bolt is carried out according to the following formula:

$\sigma_{ z }(r_{ i })=\left( \nu_{ \perp\perp }*\frac{ \sigma_{ r }(r_{ i }) }{E_{ \perp } } + \nu_{ \perp\parallel }*\frac{ \sigma_{ t }(r_{ i }) }{ E_{ \parallel } }\right)*E_{ \perp }$

Annotation: The experimental proof of prototyping is mandatory, due to the fact that details in the design might heavily influence the load-bearing capacity of the connection. Most importantly, since the structural durability cannot be predicted reliably with the help of FE-methods as well.

• ### Name

engidesk GmbH

Since the company’s formation as RWTH Aachen university spin-off in 2012, engidesk GmbH represents a modern and dynamic software house based in Aachen, Germany. The highly committed team of young professionals around the founders Dr. Lars Lambrecht and Marc Branscheid constantly enhances its uniquely innovative development environment for research and development engineers. The engidesk software enables companies & research facilities to make better use of their technical know-how by harmonising and standardising complex calculations across work groups and entire companies. Unique and easy-to-use interfaces allow for the seamless combination and merger of any calculation know-how to single, directly applicable modules – regardless of the original developing system. Above that, the engidesk market place provides knowledge holders with the possibility to commercialise valuable know-how by putting up problem oriented – all compatible – calculation sequences for sale.

format: plain


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# Singular value decomposition

Formally, the singular value decomposition of an ${\displaystyle m\times n}$ complex matrix M is a factorization of the form ${\displaystyle \mathbf {U\Sigma V^{*}} }$, where U is an ${\displaystyle m\times m}$ complex unitary matrix, ${\displaystyle \mathbf {\Sigma } }$ is an ${\displaystyle m\times n}$ rectangular diagonal matrix with non-negative real numbers on the diagonal, and V is an ${\displaystyle n\times n}$ complex unitary matrix. The diagonal entries ${\displaystyle \sigma _{i}=\Sigma _{ii}}$ of ${\displaystyle \mathbf {\Sigma } }$ are known as the singular values of M. The columns of U and the columns of V are called the left-singular vectors and right-singular vectors of M, respectively.


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Course overview

The notes contain a substantial amount of snippets of code that are fully self-contained within the chapter in which they are included. This allows understanding of how the methods and theory translate neatly to the practice. The software employed in the course is the statistical language R and its most common IDE (Integrated Development Environment) nowadays, RStudio. Prior basic knowledge of both is assumed.1

The Shiny interactive apps on the notes can be downloaded and run locally, which in particular allows inspection of their codes. Check out this GitHub repository for the sources.

We will employ several packages that are not included within R by default. These can be installed as:

# Installation of required packages
packages <- c("ks", "mvtnorm", "nor1mix", "rgl", "misc3d", "viridis",
"manipulate", "geometry", "numDeriv", "OceanView", "ISLR",
"emstreeR", "circular", "maps", "MASS", "microbenchmark",
"np", "locfit", "latex2exp", "dgof", "goftest", "nortest",
"boot", "energy")
install.packages(packages)

The notes make explicit mention of the package to which a function belongs by using the operator ::, unless when the use of the functions of a package is very repetitive and that package is loaded. You can load all the packages by running:

# Load packages
lapply(packages, library, character.only = TRUE)

1. Among others: basic programming in R, ability to work with objects and data structures, ability to produce graphics, knowledge of the main statistical functions, and ability to run scripts in RStudio.↩


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# Is y=x^2 + 3x - 15 a function?

Yes, it is (in the usual sense that $y$ is a function of $x$).
For any value of $x$, the expression gives only one (never two or more) values for $y$.
Here is the graph on $y = {x}^{2} + 3 x - 15$. As you can see, it passes the vertical line test.
(There are many times that 2 values of $x$ give the same value of $y$, but that is ok for a function.)


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### A quantized-diffusion model for rendering translucent materials

##### Content of review 1, reviewed on November 07, 2013

Let $\tau_0:=\frac{1}{s}\tau_1$, then $\tau_i=s^{i-1}\tau_1$ and $v_i=D(\tau_{i+1}+\tau_i)=s^i v_0$ $$\int\limits_{0}^{\infty} G_{3D}(2D\tau,r)e^{-\mu_a\tau}d\tau \approx \sum\limits_{i=0}^{k-1}\int\limits_{\tau_i}^{\tau_{i+1}} G_{3D}(2D\tau,r)e^{-\mu_a\tau}d\tau = \sum\limits_{i=0}^{k-1}(\tau_{i+1}-\tau_i) G_{3D}(D(\tau_{i+1}+\tau_i),r)e^{-\mu_a\frac{\tau_{i+1}+\tau_i}{2}} = \frac{s-1}{s+1}\sum\limits_{i=0}^{k-1}(\tau_{i+1}+\tau_i) G_{3D}(D(\tau_{i+1}+\tau_i),r)e^{-\mu_a\frac{\tau_{i+1}+\tau_i}{2}} = \frac{s-1}{s+1}\sum\limits_{i=0}^{k-1}\frac{s^i v_0}{D} G_{3D}(s^i v_0,r)e^{-\mu_a\frac{s^i v_0}{2D}}$$

Given $s=\frac{1+\sqrt{5}}{2}$, $\frac{s-1}{s+1} \approx 0.236068$ rather than $0.240606$ in the paper. Can someone tell me how to get that magic number? And why $v_0$ is specified in terms of $\mu_a$? Thanks for your kindness.


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# Determine the position function given the velocity function

Show 40 post(s) from this thread on one page
Page 1 of 2 12 Last
• Dec 1st 2012, 10:11 AM
JDS
Determine the position function given the velocity function
The Velocity function is:

v(t) = 3e-t - 2 and the initial position is s(0) = 0.

Here are my calculations:

s(t) = 3t(e-t) - 2t + C

Since s(0) = 0 we have c=0, Thus,

s(t) = 3t(e-t) - 2t + 0

Did I do this one correct?
• Dec 1st 2012, 10:39 AM
rousseau
Re: Determine the position function given the velocity function
There looks to be a mistake involving the integral of e^-t
• Dec 1st 2012, 10:40 AM
JDS
Re: Determine the position function given the velocity function
I thought as much, I was not sure how to go about doing that
• Dec 1st 2012, 10:52 AM
JDS
Re: Determine the position function given the velocity function
so then would this be correct? s(t) = 3t(et) - 2t + 0
• Dec 1st 2012, 11:11 AM
Plato
Re: Determine the position function given the velocity function
Quote:

Originally Posted by JDS
so then would this be correct? s(t) = 3t(et) - 2t + 0

No it should be $s(t)=-3e^{-t}-2t+3.$ WHY?
• Dec 1st 2012, 11:16 AM
rousseau
Re: Determine the position function given the velocity function
Can you do the derivative of $e^{-x}$? if so think backwards and try and figure out what the integral would be.
• Dec 1st 2012, 11:18 AM
JDS
Re: Determine the position function given the velocity function
I do not understand.... in the question it says s(0) = 0 so would c not equal 0? and I also do not understand how you got -3e^-t.....It looks to me like the derivative of that would not be 3e^-t.......perhaps I need more coffee :(
• Dec 1st 2012, 11:21 AM
rousseau
Re: Determine the position function given the velocity function
I suggest going back to basics and just looking up what the derivative is of a function of the type [e^{ax}] and then reasoning backwards to figure out what the integral, knowing that integration is the opposite of differentiation.
• Dec 1st 2012, 11:27 AM
Plato
Re: Determine the position function given the velocity function
Quote:

Originally Posted by JDS
I do not understand.... in the question it says s(0) = 0 so would c not equal 0?

Look at reply #5. What is $s(0)=~?$
• Dec 1st 2012, 11:28 AM
JDS
Re: Determine the position function given the velocity function
thanks, to both of you, and I did just that rousseau, sometimes all this calculus just jiggles around for me lol, but I do see, now, how the derivative of e^-x would in fact equal -e^(-x)..thanks! so that now makes since.....

-3e^-t - 2t + C However I still am unsure why you got a 3 in place of the C(Constant) when the constant was defined as 0 (at least for the starting point)

Sorry if I sound like an idiot, teaching calculus to oneself is quite a task!
• Dec 1st 2012, 11:33 AM
JDS
Re: Determine the position function given the velocity function
correction, I guess I am no longer teaching myself, you are all teaching me, which is greatly appreciated!!!
• Dec 1st 2012, 11:36 AM
JDS
Re: Determine the position function given the velocity function
Quote:

Originally Posted by Plato
Look at reply #5. What is $s(0)=~?$

From what I can tell, s(0) = 0

does it not?

hmmmm....I just had a thought...but I could be wrong..... does it matter what C is as long as it is any constant?
• Dec 1st 2012, 11:49 AM
Plato
Re: Determine the position function given the velocity function
Quote:

Originally Posted by JDS
From what I can tell, s(0) = 0 does it not? I just had a thought...but I could be wrong..... does it matter what C is as long as it is any constant?

It matters very much if we are given a initial position to start with.
We were given, $s(0)=$. So the position is 0 if t=0.
• Dec 1st 2012, 12:20 PM
JDS
Re: Determine the position function given the velocity function
okay, so then you are saying that s(3) = 3 = t which makes that the constant?
• Dec 1st 2012, 12:21 PM
JDS
Re: Determine the position function given the velocity function
I understand that s(0) = 0 is the starting point, but how did we arrive at the constant being equal to 3, that is the part Im just not quite getting
Show 40 post(s) from this thread on one page
Page 1 of 2 12 Last


text:
# Most efficient way to replace/ update the values multiple rows and columns of a Dataset

Suppose we wish to replace the cols "b" and "c" of rows 3 to 5 in the following dataset to b and c respectively:

dataset = Dataset[{
<|"a" -> 1, "b" -> "x", "c" -> {1}|>,
<|"a" -> 2, "b" -> "y", "c" -> {2, 3}|>,
<|"a" -> 3, "b" -> "z", "c" -> {3}|>,
<|"a" -> 4, "b" -> "x", "c" -> {4, 5}|>,
<|"a" -> 5, "b" -> "y", "c" -> {5, 6, 7}|>,
<|"a" -> 6, "b" -> "z", "c" -> {}|>}]

1. What is the most efficient way to do it at scale, i.e. assuming the replacements are going to be across multiple/ all rows in large datasets.
2. Is is possible to do it inline, i.e., in the current dataset, without having to recreate a new dataset?
3. It would be preferrable if the updates are computed (i.e., not constants as shown in this toy example)
• It's not really an answer but two things I would mention: 1) I've found Dataset to not be very handy for working with data. I tend to process data as associations and only create a Dataset at the end when I'm going to do queries / analysis. 2) Associations support structural sharing so #2 may not actually be a concern (see youtu.be/0ZnZlAKMZdg?t=843 ) -- obviously I don't know everything about what you're doing but if it's just about memory usage then it should be ok.
– Dan
Dec 10, 2018 at 2:51

You can use ReplacePart directly on the dataset but it seems a tedious/inefficient way of doing this. It is scalable as long as you can use patterns or Alternative to select the positions you want to replace. I expect someone else will be able to provide a more efficient method that works directly on the dataset without first converting to a list of associations.

dataset=ReplacePart[dataset, {{3 | 4 | 5, "b"} -> b, {3 | 4 | 5, "c"} -> c}]

• This does not perform an in-place modification. If you evaluate dataset after the ReplacePart, it is unchanged. Dec 10, 2018 at 1:01
• @RohitNamjoshi good point. I'll leave the answer up even though it does not provide in-place modification Dec 10, 2018 at 5:21

I mentioned it in the comments but basically I like to keep it pretty simple and while I'm not sure if it is the fastest way to do it it is nice and clean and the memory usage is not as bad as it would seem due to Mathematica's good choice of persistent data structures to build Associations on.

In[1]:= MemoryInUse[] / 1000000 // N
Out[1]= 96.5472
In[2]:= data = ResourceData["US State Income"]
In[3]:= MemoryInUse[] / 1000000 // N
Out[3]= 117.604
(* Updating a field *)
In[4]:= data2 = Dataset[<|"State"->#State, "Income"->#Income/2.0|>&/@Normal@data]
In[5]:= MemoryInUse[] / 1000000 // N
Out[5]= 117.885
(* Appending a field *)
In[6]:= data3 = Dataset[<|#, "IncomeSquared"->#Income^2|>&/@Normal@data]
In[7]:= MemoryInUse[] / 1000000 // N
Out[7]= 118.574

Actually, I don't now how to do that with a Dataset but with nested Lists and Association, in-place modification is possible:

assoc = Normal[dataset];
assoc[[3 ;; 5, {"b", "c"}]] = ConstantArray[{b, c}, 3];
assoc // Dataset

So, if you don't need any specific properties of Dataset, I'd suggest to stick with nested Associations.

• Im trying to manipualte a real-world dataset and am exploring using Dataset in MMA for its GroupBy and other convenience features like plotting by columns etc. Dec 9, 2018 at 18:10


text:
# GRE Scores (and GMAT, MCAT, and LSAT scores!)

#### How’d I do? An overly simplified look at GRE scores.

[Note : This module may take a little while to load, so be patient!]

When preparing for the GRE it can be hard to figure out what all the numbers mean. We’re used to seeing scores like 93 (or maybe more like 63!), B–, or 3.2, and, over the course of growing up, we kind of have a feel for what those mean.

But the GRE’s not like that.

In a way, that’s because it’s very hard to figure out what’s a hard question and what’s an easy one. Think about it : the people writing the GRE have spent years understanding the topics that they are writing about, kind of like you’ve spent years spelling. Imagine how difficult it would be to pick a word which is just hard enough to spell that not everyone can spell it, but some people can. Or an easy word : a word that exactly 15% of people can spell. Not too hard? Then think about having to pick a whole ton of words of varying degrees of difficulty so that you can figure out just how good a speller someone is. Now you’ve gotta find a hundred words that exactly 77% of people (say) can spell correctly. See the problem?

Very roughly speaking, the way the GRE makers do this is simply by writing questions (not thinking about their exact difficulty) and putting them on experimental sections of tests that aren’t graded. Then they can look at how many people get those right/wrong, and try to figure out a good collection of questions.

Unfortunately, that’s not all they do. Once they give the tests, they then standardize the grades by looking at how well everyone else taking the test did. Some people who took the SAT several times (which is made similarly) will tell you that one of the times they took the exam was much harder than another (or easier); they may be right. Some GREs are harder than others. To make up for it, the grading system tries to put everyone on one common funky scale, regardless of how hard the test was.

Specifically, they aim to scale the test so that the grades form a specific distribution; that’s why when you see the test, it’s no good seeing the score itself—you have to know what percentile that score corresponds to. The wikipedia article on the subject gives tables for each section: quantitative, verbal, and writing.

It turns out that each of these distributions is more or less unique, but many of them are roughly normal (i.e. they can be approximated by a bell curve). We can reverse engineer the data from Wikipedia to find good approximations to the distribution they use. These are the distributions below. Enjoy!

By setting the Max value to a score (and leaving Min at the bottom), you can compute the percentile of a certain score.

The GMAT is composed of four parts – integrated reasoning, analytical writing, quantitative and verbal. Using the data from mba.com, we again fit distributions to the percentiles. It turns out only the Verbal distribution is very normal; the analytical writing distribution and the quantitive distribution are particularly bad. To account for this, we model the former using a beta distribution, and continue to model the others with mixture normals.

## And the MCAT scores?

The MCAT has three sections – physical sciences, biological sciences, and verbal reasoning – which each range in scores from 1 to 15. Using the 2012 data from the AAMC, the distributions look like…

## The LSAT Scores

Using the LSAT score percentile data from alphascore.com, the distribution looks quite bell shaped.

# Random triangles

At a basic level, a random triangle is simply a triangle whose corners are three random points on a piece of paper.

Mathematically speaking, a few decisions have to be made characterize exactly how the random point selection works. Think of it this way : should every place on the piece of paper be equally likely, or should the middle of the page be more likely to be selected than near the borders?

In this module, we assume that the points are coming from a bivariate normal distribution with unit variances and correlation $\rho$.

## Play with random triangles!

The following module generates bunches of random triangles using the bivariate normal distribution with correlation coefficient $\rho$. The red triangles are obtuse, and the green triangles are acute (the likelihood of seeing a right triangle is 0, so it doesn’t get a color.) You can change $\rho$ with the slider under the module. What happens as $\rho$ approaches -1 or 1?

In Professor Strang’s lecture he discusses what the triangles look like in “triangle space”. The basic idea is that every triangle has three angles which sum to $180^{\circ}$, call them $\alpha$, $\beta$, and $\gamma$. Every triangle is therefore represented by a single point in the “triangle space”. Further, the triangle space itself can be broken into four regions.

In the diagram below, the regions of the triangle are colored according to the kinds of triangles which are “zoned” to those regions : the red regions represent obtuse triangles, and the green region represents acute triangles. Notice that as $\rho$ approaches -1 or 1, all of the triangles get pulled towards the corners. Why?


text:
# SCM Repository

[rmetrics] View of /pkg/randtoolbox/man/runifInterface.Rd
 [rmetrics] / pkg / randtoolbox / man / runifInterface.Rd

# View of /pkg/randtoolbox/man/runifInterface.Rd

Tue Apr 30 12:56:05 2019 UTC (6 weeks, 6 days ago) by dutangc
File size: 6132 byte(s)
remove old URL, update argument checking, disable sscanf for a CRAN release
\name{runifInterface}
\alias{runifInterface}
\alias{set.generator}
\alias{put.description}
\alias{get.description}
\title{Functions for using runif() and rnorm() with randtoolbox generators}
\description{
These functions allow to set some of the random number generators from randtoolbox
package to be used instead of the standard generator in the functions, which use
random numbers, for example \code{runif()}, \code{rnorm()}, \code{sample()}
and \code{set.seed()}.
}
\usage{
set.generator(name=c("WELL", "MersenneTwister", "default"),
parameters=NULL, seed=NULL, \dots, only.dsc=FALSE)

put.description(description)

get.description()
}
\arguments{
\item{name}{A character string for the RNG name.}
\item{parameters}{A numeric or character vector describing a specific RNG from the
family specified by the \code{name}.}
\item{seed}{A number, whose value is an integer between \code{0} and \code{2^32 - 1},
to be used as a seed.}
\item{\dots}{Arguments describing named components of the vector of parameters,
if argument \code{parameters} is missing or \code{NULL}.}
\item{only.dsc}{Logical. If \code{TRUE}, a description of the specified RNG is returned
and the generator is not initialized.}
\item{description}{A list describing a specific RNG as created by
\code{set.generator(,only.dsc=TRUE)} or \code{get.description()}.}
}
\details{
Random number generators provided by R extension packages are set using
\code{RNGkind("user-supplied")}. The package \pkg{randtoolbox} assumes that
this function is not called by the user directly and is called from
the functions \code{set.generator()} and \code{put.description()}.

Random number generators in \pkg{randtoolbox} are represented at the R level by a list
containing mandatory components \code{name, parameters, state} and possibly an
optional component \code{authors}. The function \code{set.generator()} internally
creates this list from the user supplied information and then runs \code{put.description()}
on this list, which initializes the generator. If the generator is initialized, then the
function \code{get.description()} may be used to get the actual state of the generator,
which may be stored in an R object and used later in \code{put.description()} to continue
the sequence of the random numbers from the point, where \code{get.description()}
was called. This may be used to generate several independent streams of random numbers
generated by different generators.

The component \code{parameters} is a character or a numeric vector, whose structure
is different for different types of the generators. This vector may be passed
to \code{set.generator()}, if it is prepared before its call, however, it is
also possible to pass its named components via the \code{...} parametr of
\code{set.generator()} and the vector \code{parameters} is created internally.
If the vector \code{parameters} is not supplied and the arguments in \code{...}
are not sufficient to create it, an error message is produced.

\subsection{Linear congruential generators}{
Parameters for the linear congruential generators (\code{name="congruRand"})
are integers represented either as a character or a numeric vector. The
components are
\describe{
\item{mod}{The modulus.}
\item{mult}{The multiplier.}
\item{incr}{The increment.}
}
}

\subsection{WELL generators}{
Parameters for the WELL generators is a character vector with components
\describe{
\item{order}{The number of bits in the internal state.
Possible values are 512, 521, 607, 800, 1024, 19937, 21701, 23209, 44497.}
\item{version}{The version letter "a", "b", or "c" to be appended to the order.}
}
The concatenation of \code{order} and \code{version} should belong to
\code{"512a", "521a", "521b", "607a", "607b", "800a", "800b", "1024a", "1024b",
"19937a", "19937b", "19937c", "21701a", "23209a", "23209b", "44497a", "44497b"}.
When order and version are specified in \code{...} parametr of \code{set.generator()},
then the parameter \code{order} is optional and if missing, it is assumed that the
parameter \code{version} contains also the number of bits in the internal state
and the combination belongs to the list above.
}

\subsection{Mersenne Twister 2002 generator}{
Parameters for the Mersenne Twister 2002 is a character vector with components
\describe{
\item{initialization}{Either "init2002" or "array2002". The initialization to be used.}
\item{resolution}{Either 53 or 32. The number of random bits in each number.}
}
}

}
\value{
\code{set.generator()} with the parameter \code{only.dsc=TRUE} and
\code{get.description()} return the list describing a generator.
\code{put.description()} with the parameter \code{only.dsc=TRUE} (the default)
and \code{put.description()} return \code{NULL}.
}
\seealso{
}
\author{
Petr Savicky and Christophe Dutang
}
\examples{
#set WELL19937a
set.generator("WELL", version="19937a", seed=12345)
runif(5)

#Store the current state  and generate 10 random numbers
storedState <- get.description()
x <- runif(10)

\dontrun{
#Park Miller congruential generator
set.generator(name="congruRand", mod=2^31-1, mult=16807, incr=0, seed=12345)
runif(5)
setSeed(12345)
congruRand(5, dim=1, mod=2^31-1, mult=16807, incr=0)

# the Knuth Lewis RNG
set.generator(name="congruRand", mod="4294967296", mult="1664525", incr="1013904223", seed=1)
runif(5)
setSeed(1)
congruRand(5, dim=1, mod=4294967296, mult=1664525, incr=1013904223)
}

#Restore the generator from storedState and regenerate the same numbers
put.description(storedState)
x == runif(10)

# generate the same random numbers as in Matlab
set.generator("MersenneTwister", initialization="init2002", resolution=53, seed=12345)
runif(5)
# [1] 0.9296161 0.3163756 0.1839188 0.2045603 0.5677250
# Matlab commands rand('twister', 12345); rand(1, 5) generate the same numbers,
# which in short format are   0.9296    0.3164    0.1839    0.2046    0.5677

#Restore the original R setting
set.generator("default")
RNGkind()

}
\keyword{distribution}


text:
Multiplying and dividing by 10, 100, 1000 | Maths Made Easy

# Multiplying and dividing by 10, 100, 1000

Level 1-3

## Multiplying and dividing by 10, 100, 1000

When we multiple by 10, 100 or 1000 we move the decimal place to the right. When we divide by 10, 100 or 1000 we move the decimal place to the left.

Make sure you are happy with the following topics before continuing.

## Skill 1: Multiplying by 10, 100, 1000

When multiplying by $10, 100, 1000$ move the decimal place to the right, the same number of places as zeros in the number.

Multiplying by $10$ – decimal place $1$ space to the right

Multiplying by $100$ – decimal place $2$ space to the right

Multiplying by $1000$ – decimal place $3$ space to the right

Example: Calculate $327 \times 100$

Here we need to more the decimal place 2 spaces to the right, writing zeros in any new places created.

After moving the decimal place 2 spaces to the right and add the zeros in the spaces created, this gives the answers $32700$

$327 \times 100 = 32700$

KS3 Level 1-3

## Skill 2: Dividing by 10, 100, 1000

When Dividing by $10, 100, 1000$ move the decimal place to the left, the same number of places as zeros in the number.

Multiplying by $10$ – decimal place $1$ space to the left

Multiplying by $100$ – decimal place $2$ space to the left

Multiplying by $1000$ – decimal place $3$ space to the left

Example: Calculate $1234 \div 1000$

Here we need to more the decimal place 3 spaces to the left.

The decimal place is moved three places to the left.

This gives the answer to be $1.234$

$1234 \div 1000 = 1.234$

KS3 Level 1-3

## GCSE Maths Revision Cards

(242 Reviews) £8.99

### Example Questions

Here we are multiplying by $100$ this will mean we move the decimal place $2$ places to the right.

$17\times100 = 1700$

Here we are multiplying by $1000$ so we must move the decimal place 3 places to the right.

$219.65\times1000= 219650$

As we are dividing by $100$ we must move the decimal place $2$ places to the left.

$3101 \div 100 = 31.01$

As we are dividing by $1000$ we must move the decimal place 3 places to the left.

$12\div1000= 0.012$

### Learning resources you may be interested in

We have a range of learning resources to compliment our website content perfectly. Check them out below.


text:
# Underactuated Mechanical Systems - CiteSeerX

Underactuated Mechanical Systems - CiteSeerX

2. Consider Euler's equations. ˙. J1 +( 1.

Let’s analyse by adding sin(x) to cos(x), which I’ve highlighted in magenta: What you may have noticed is how sin(x) + cos(x) is similar to the expansion for e^x. ∫ cos = cos sin 2 2 Without Euler's identity, this integration requires the use of integration by parts twice, followed by algebric manipulation. 2008-06-10 · Proof of Euler's identity depends on taylor series for e^x, sin x, cos x. Taylor series for sin x depends on the fact that d (sin x)/dx = cos x, d(cos x)/dx = -sin x and so on. Proof of the fact that d(sin x)/dx = cos x, depends on the relation. sin (A+B) = sin A cos B + cos A sin B. so you see, euler's theorem involves this result to be Se hela listan på science4all.org In short, $$e^{ix} = \cos(ix) + i\sin(ix)$$!

2021-01-08 · Verify Euler’s identity for cos θ using Euler’s formula. Buy Find launch.

### Abraham Kuyper - Wikidocumentaries

Respondents to a Physics World poll called the identity "the most profound mathematical statement ever written", "uncanny and sublime", "filled with cosmic beauty" and "mind-blowing". Công thức Euler là một công thức toán học trong ngành giải tích phức, được xây dựng bởi nhà toán học người Thụy Sĩ Leonhard Euler.Công thức chỉ ra mối liên hệ giữa hàm số lượng giác và hàm số mũ phức. Understanding cos(x) + i * sin(x). The equals sign is overloaded. Sometimes we mean "set  As a consequence of Euler's formula, the sine and cosine functions can be represented as.

Euler's formula is eⁱˣ=cos(x)+i⋅sin(x), and Euler's Identity is e^(iπ)+1=0. See how these are obtained from the Maclaurin series of cos(x), sin(x), and eˣ. This is one of the most amazing things in all of mathematics! We can use Euler’s theorem to express sine and cosine in terms of the complex exponential function as s i n c o s 𝜃 = 1 2 𝑖 𝑒 − 𝑒 , 𝜃 = 1 2 𝑒 + 𝑒 . Using these formulas, we can derive further trigonometric identities, such as the sum to product formulas and formulas for expressing powers of sine and cosine and products Proofs Euler's formula using the MacLaurin series for sine and cosine. Introduces Euler's identify and Cartesian and Polar coordinates. The same result can be obtained by using Euler's identity to expand into and negating the imaginary part to obtain , where we used also the fact that cosine is an even function () while sine is odd ().
Survey report template word

· Let us start with Euler's Identity: · eix = cos x + i sin x (1) · Replace  Feb 24, 2006 eix = cos x + i sin x. QED Corollary: De Moivre's Formula (cos x + isin x)n = cos(nx )  Euler's formula states that for any real number x : where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted cis x (" c osine plus i s ine").

double-angle formula sub.
Turkisk ek


text:
Extending Vector Fields.

1. Nov 20, 2007

WWGD

Hi: I am going over Lee's Riemm. mflds, and there is an exercise that asks:

Let M<M' (< is subset) be an embedded submanifold.

Show that any vector field X on M can be extended to a vector field on M'.

Now, I don't know if he means that X can be extended to the _whole_ of

M', because otherwise, there is a counterexample:

dt/t on (0,1) as a subset of IR cannot be extended to the whole of IR.

Anyone know?.

What Would Gauss Do?

2. Nov 20, 2007

Hurkyl

Staff Emeritus
Are you sure M is not supposed to be a closed submanifold?

3. Nov 21, 2007

WWGD

Hi. I did not see this stated, and I cannot see if it is being assumed somehow.

Only conditions I saw where that M<M' , and M embedded submanifold of M',

and a V.Field is defined in M.

Re the second issue, of extensions, I guess these extensions are local, tho

not necessarily global, right?. The answer would seem to be yes pretty

clearly, but my intuition has failed me before.

4. Nov 21, 2007

Chris Hillman

Consider the function with constant value one on (0,1). How many ways can you extend it to R?

5. Nov 23, 2007

WWGD

Thanks, Chris, I am not sure I get the hint; there are uncountably many ways,

for a continuous extension, most obvious extension being f==1 in IR (attach

a line segment of slope m, m in (-oo,oo) , maybe

smaller cardinality for smooth extensions . For smoothness I would imagine

some combination of e^-x 's attached to both ends, or maybe some other

bump functions ( Lee does not specify if X is C^1 , or C^k, C^oo).

All I can think of when I think of immersed submanifolds is slice coordinates, tho

this does not seem to make sense for 1-manifolds like (0,1) in IR.

Am I on the right track?.

6. Nov 23, 2007

Chris Hillman

Mrmph... never mind the hint, I was looking at the wrong textbook in Lee's excellent trilogy, my mistake! Unfortunately you are using the one I don't have, but can you state the what exercise you are attempting? I might be able to obtain a copy next week. I expect we will be able to figure it out!

Last edited: Nov 23, 2007
7. Nov 23, 2007

WWGD

Yes, this is Lee's Riemannian mflds, problem 2.3, part b, p.15 in my edition.

Thanks.


text:
# 7. The reaction of acetic acid with ammonia is very rapid (it is an acid-reaction, after...

###### Question:

7. The reaction of acetic acid with ammonia is very rapid (it is an acid-reaction, after all) and follows the simple rate law shown in the following equation. Purpose a mechanism that is consistent with this rate law. Show curly arrows for both forward and reverse reaction. rate = k/H,C-C-0-H) NHS

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##### Polnt) Find y as function of t If100y'" 100y' + 6ly = = 0,y(0) = 4 y (0) = 9. y(t) e^((1/214)(4cos(((sqrt39)/10Jt)+(70/sqrt(39) Jsin(((sqrt39)/10)4))
polnt) Find y as function of t If 100y'" 100y' + 6ly = = 0, y(0) = 4 y (0) = 9. y(t) e^((1/214)(4cos(((sqrt39)/10Jt)+(70/sqrt(39) Jsin(((sqrt39)/10)4))...
##### The diagram below shows . voltaic cell with Fe in 1 M Fe2- (aq) and Pb in 1 MPb2- (aq) The standard reduction potentials are the following:Fez- 2e 3Eo = -0.45VPb2+ + 2e-PbEo = -0.13VNaNOPbFe2+Pb2+
The diagram below shows . voltaic cell with Fe in 1 M Fe2- (aq) and Pb in 1 MPb2- (aq) The standard reduction potentials are the following: Fez- 2e 3 Eo = -0.45V Pb2+ + 2e- Pb Eo = -0.13V NaNO Pb Fe2+ Pb2+...
##### The assooulion of four polypeptides (2 L chains and 2 H chains) formn an antibody Is an example ofstructure2" struclure tertiary structure qualumnary slruciuru
The assooulion of four polypeptides (2 L chains and 2 H chains) formn an antibody Is an example of structure 2" struclure tertiary structure qualumnary slruciuru...
##### Required: a. Firm A has a margin of 8%, sales of $630,000, and ROl of 16.8%.... Required: a. Firm A has a margin of 8%, sales of$630,000, and ROl of 16.8%. Calculate the firm's average total assets. b. Firm B has net income of $246,400, turnover of 1.1, and average total assets of$1,600,000. Calculate the firm's sales, margin, and ROI. c. Firm C has net income of $43,... 2 answers ##### Consider the differential equation (1 + x^2 )(5 âˆ’ x) ^3 y'' +tan(x)y' âˆ’ 1 /(xâˆ’8)y = 0.Please don't attempt to solvethis differential equation.(a) This D. E. has a series solution centered at x = 0 What isthe open interval of convergence? Give a reason for youranswer.(b) Is this equation guaranteed to have a series or a Frobeniusseries solution centered at x = 8? Why or why not?(c) Is this equation guaranteed to have a series or a Forbeniusseries solution centered at x Consider the differential equation (1 + x^2 )(5 âˆ’ x) ^3 y'' + tan(x)y' âˆ’ 1 /(xâˆ’8)y = 0.Please don't attempt to solve this differential equation. (a) This D. E. has a series solution centered at x = 0 What is the open interval of convergence? Give a reason for y... 5 answers ##### An economist obtains random sample of 800 urban households and finds tnat 338 0f them have Internet Jccess He obtains random sample 750 rural households and finos that 292 of them have Internet access Find 9971 confidence interval for the difference between [he proportion of urbun households that have Internet access and the proportion of rural households that have Internet accessThe following data represent the cost of a one night stay in Hampton Inn Hotels and La Quinta Inn Hotels for a random An economist obtains random sample of 800 urban households and finds tnat 338 0f them have Internet Jccess He obtains random sample 750 rural households and finos that 292 of them have Internet access Find 9971 confidence interval for the difference between [he proportion of urbun households that ha... 1 answer ##### 4. A firm has Debt-Equity ratio of 1.2 and Total Assets of$2 million. What must...
4. A firm has Debt-Equity ratio of 1.2 and Total Assets of $2 million. What must be total debt? 5. A firm has sales of$355,000, net income of $28,000, and dividends of$12,500. Total debt is $73,000 and Total equity is$95,000. If the firm grows at the SGR, issues no new equity, and maintains its D...
##### 3 Define biholomorphism conformal mapping bijective holomorphic function) H 3 D. (Note that for questions like this just giving the definition of the map is not enough; you also have to justify Your answer_ Remark: This example is done in Stein-Shakarchi , Chapter 8, Section 1.1.)
3 Define biholomorphism conformal mapping bijective holomorphic function) H 3 D. (Note that for questions like this just giving the definition of the map is not enough; you also have to justify Your answer_ Remark: This example is done in Stein-Shakarchi , Chapter 8, Section 1.1.)...
##### A. Determine the points where the curve 7x+2y2 _y = has a vertical tangent line_Does the curve have any horizontal tangent lines?The curve has a vertical tangent line at the point(s) (Type an ordered pair: Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression:)
a. Determine the points where the curve 7x+2y2 _y = has a vertical tangent line_ Does the curve have any horizontal tangent lines? The curve has a vertical tangent line at the point(s) (Type an ordered pair: Use a comma to separate answers as needed. Use integers or fractions for any numbers in the ...
##### Repeat Example $5.1,$ this time using a horizontal/vertical coordinate system.
Repeat Example $5.1,$ this time using a horizontal/vertical coordinate system....
##### Compute the standard deviation of the five monthly returns for PG&E: 1.25 percent, −1.50 percent, 4.25...
Compute the standard deviation of the five monthly returns for PG&E: 1.25 percent, −1.50 percent, 4.25 percent, 3.75 percent, and 1.98 percent. Multiple Choice 1.876 percent 1.946 percent 2.046 percent 2.287 percent...
##### Also the given answers might not be correct so don't mind them — HOV NAI 2K...
also the given answers might not be correct so don't mind them — HOV NAI 2K 385k Ž 4K rok to Vout W & 20k lou OX oßiß2 = 100, VA = 00 a) Find the overall Voltage gaina out = 3909 W b) find the current gain Contain, Rin and Rout νς € Repeat exercise a, if...
##### Identify the points where the curve has a vertical tangent line. Select all that apply:2_ 1;) 26Sx31 2
Identify the points where the curve has a vertical tangent line. Select all that apply: 2_ 1;) 2 6 Sx 31 2...
##### 18. You start at point A You walk 3 miles at 25 degrees: How far east are VOU of point A?How far north are yoU of point A?19. You leave your house and travel for minutes at an average velocity of 5 m/s. How far have you traveled?20. You leave your house, accelerating 10 m/s over 20 seconds_ How far have you traveled?
18. You start at point A You walk 3 miles at 25 degrees: How far east are VOU of point A? How far north are yoU of point A? 19. You leave your house and travel for minutes at an average velocity of 5 m/s. How far have you traveled? 20. You leave your house, accelerating 10 m/s over 20 seconds_ How f...
##### Cumtipiele)14.6.43Find the Iinearization L(x,Y,z) of the function f(xy,z) = JR+y+2 at the points (5,0,0), (7,4,0), and (1,4,8)
cumtipiele) 14.6.43 Find the Iinearization L(x,Y,z) of the function f(xy,z) = JR+y+2 at the points (5,0,0), (7,4,0), and (1,4,8)...
Question 21 (2 points) Use the Graph below to answer the following question: $100 Sus P (thousands) SW$40 510 530 Dus D 510 0 100 190 250 . 335 Q Q thousands World Semiconductor Market U.S. Domestie Semiconductor Market [2 points) 21. Consumer surplus after the implementation of the $10 tariff is: ... 5 answers ##### Determine the following probabilities a: fnz6 N-10, and A =,7 find P(X = 5) b. If n=5,N=8 and A = 4 find C. P(X = 2). If n=5,N=11 and A = 4, find P(X = 0). d. If nz4, N=ll, and A=4, find P(X= 4) Determine the following probabilities a: fnz6 N-10, and A =,7 find P(X = 5) b. If n=5,N=8 and A = 4 find C. P(X = 2). If n=5,N=11 and A = 4, find P(X = 0). d. If nz4, N=ll, and A=4, find P(X= 4)... 2 answers ##### Physics - Unit V Challenge Problem No matter how bad your day is going, here's proof that things could be A LOT worse Physics - Unit V Challenge Problem No matter how bad your day is going, here's proof that things could be A LOT worse. This is a bricklayer's accident report that was printed in the newsletter of the English equivalent of the Workers' Compensation Board. So here, thanks to John Sedgwick,... 4 answers ##### SECTION A ContinuedLet G = (y? + 1) + yz/ + (x+2)k and let be the curve (5) parameterised by which Y = and between the points (0,-2,2) and (7,71,5) . Evaluate the Iine integral: ( G SECTION A Continued Let G = (y? + 1) + yz/ + (x+2)k and let be the curve (5) parameterised by which Y = and between the points (0,-2,2) and (7,71,5) . Evaluate the Iine integral: ( G... 1 answer ##### Problem 16.89 . Part A lf link CD has an angular velocity ofwCD-7 rad/s , detemine... Problem 16.89 . Part A lf link CD has an angular velocity ofwCD-7 rad/s , detemine the velocity of point E on link BC at the instant θ = 42".(Figure 1) Enter the r and y components of the velocity separated by a comma. vec mis Submit Request Answer Figure 1 af 1 ▼ Part B Determine th... 1 answer ##### 20 On January 2, 2015, a company purchased a delivery truck (Truck 1) for$45,000 cash....
20 On January 2, 2015, a company purchased a delivery truck (Truck 1) for $45,000 cash. The truck had an estimated useful life of seven years and an estimated salvage value of$3,000. The straight-line method of depreciation has been used. Prepare the journal entries to record depreciation expense a...
##### MAA MAThEwaTicaL AssocIatioX OF Americawebwork 2020_fall_Jones_mth23 6.1_introduction_to_random_variables_and_probability_distributions6.1 Introduction to Random Variables and Probabil Problem 3Provlous ProblemProblem LlstNext Problempolnt)9 /10[1712 [Pk)01020.40105 Given Ihe dlscrete probabllity distribuilon above, determine the following:(a) P(? > I Or ? < 9) =P(x < 12) =P(t =8) =Note: You can earn partial credit on thts problem.
MAA MAThEwaTicaL AssocIatioX OF America webwork 2020_fall_Jones_mth23 6.1_introduction_to_random_variables_and_probability_distributions 6.1 Introduction to Random Variables and Probabil Problem 3 Provlous Problem Problem Llst Next Problem polnt) 9 /10[1712 [Pk)01020.40105 Given Ihe dlscrete probabl...
##### Convert the following points from rectangular form to polar form Please use an and radians for the angle Drav A picture_Itdoes helpll (0,-5) <(1-)Find the slope of the tangent line to the graph ofcos30 when 0 = 23/3.Find the area of the region inside the cunve You may want to sketch the regionsZcosu but outside55 33 "I77197
Convert the following points from rectangular form to polar form Please use an and radians for the angle Drav A picture_Itdoes helpll (0,-5) <(1-) Find the slope of the tangent line to the graph of cos30 when 0 = 23/3. Find the area of the region inside the cunve You may want to sketch the region...
##### 10.A segment of one strand from a DNA molecule has the sequence 5'TCCATGAGTTGA-3': What is the sequence of nucleotides in the opposite _ or complementary, DNA chain?
10.A segment of one strand from a DNA molecule has the sequence 5'TCCATGAGTTGA-3': What is the sequence of nucleotides in the opposite _ or complementary, DNA chain?...
##### Horace has been mowing his neighbors' yards for his work to charity: He several years He donates all the mioney the nevjhbors give hm ! wants t0 test the hypotheses Ho 25 versus: 25 , where Ihe Irue mcan amount 0l money donated per job Assume neighbors donate independently ol each Olher arkl (hal (he Amoutis MI = Wox futlly distributed with an unknown u. A 95% confidence Interval lor (ourci t0 be 28 _ 4.1. Which ot Ihe tollowrg} conclusions can be drawn on the basis of (his conlidence inter
Horace has been mowing his neighbors' yards for his work to charity: He several years He donates all the mioney the nevjhbors give hm ! wants t0 test the hypotheses Ho 25 versus: 25 , where Ihe Irue mcan amount 0l money donated per job Assume neighbors donate independently ol each Olher arkl (h...


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# complex number example

## 19 Jan complex number example

Complex Numbers in Polar Form. Imaginary Numbers when squared give a negative result. Python converts the real numbers x and y into complex using the function complex(x,y). Output: Square root of -4 is (0,2) Square root of (-4,-0), the other side of the cut, is (0,-2) Next article: Complex numbers in C++ | Set 2 This article is contributed by Shambhavi Singh.If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. In spite of this it turns out to be very useful to assume that there is a number ifor which one has (1) i2 = −1. Just for fun, let's use the method to calculate i2, We can write i with a real and imaginary part as 0 + i, And that agrees nicely with the definition that i2 = −1. And here is the center of the previous one zoomed in even further: when we square a negative number we also get a positive result (because. r is the absolute value of the complex number, or the distance between the origin point (0,0) and (a,b) point. A complex number is a number of the form a + bi, where a and b are real numbers, and i is an indeterminate satisfying i = −1. That is, 2 roots will be 180° apart. are actually many real life applications of these "imaginary" numbers including The initial point is $3-4i$. The Complex class has a constructor with initializes the value of real and imag. Complex numbers are algebraic expressions which have real and imaginary parts. Add Like Terms (and notice how on the bottom 20i − 20i cancels out! How to Add Complex numbers. ): Lastly we should put the answer back into a + bi form: Yes, there is a bit of calculation to do. Overview: This article covers the definition of A complex number, then, is made of a real number and some multiple of i. If a solution is not possible explain why. This complex number is in the fourth quadrant. In the previous example, what happened on the bottom was interesting: The middle terms (20i − 20i) cancel out! \blue{12} - \red{\sqrt{-25}} & \red{\sqrt{-25}} \text{ is the } \blue{imaginary} \text{ part} Complex Numbers - Basic Operations. In the following example, division by Zero produces a complex number whose real and imaginary parts are bot… It means the two types of numbers, real and imaginary, together form a complex, just like a building complex (buildings joined together). Also i2 = −1 so we end up with this: Which is really quite a simple result. \blue 3 + \red 5 i & Examples and questions with detailed solutions. Consider again the complex number a + bi. Therefore, all real numbers are also complex numbers. In general, if we are looking for the n-th roots of an equation involving complex numbers, the roots will be 360^"o"/n apart. The natural question at this point is probably just why do we care about this? The "unit" imaginary number (like 1 for Real Numbers) is i, which is the square root of −1, And we keep that little "i" there to remind us we need to multiply by √−1. This will make it easy for us to determine the quadrants where angles lie and get a rough idea of the size of each angle. Subtracts another complex number. = 3 + 1 + (2 + 7)i 5. \\\hline = 3 + 4 + (5 − 3)i Complex mul(n) Multiplies the number with another complex number. 8 (Complex Number) Complex Numbers • A complex number is a number that can b express in the form of "a+b". \\\hline Learn more at Complex Number Multiplication. $$. The coeﬃcient determinant is 1+i 2−i 7 8−2i = (1+i)(8−2i)−7(2−i) = (8−2i)+i(8−2i)−14+7i = −4+13i 6= 0 . Complex numbers have their uses in many applications related to mathematics and python provides useful tools to handle and manipulate them. In addition to ranging from Double.MinValue to Double.MaxValue, the real or imaginary part of a complex number can have a value of Double.PositiveInfinity, Double.NegativeInfinity, or Double.NaN. We do it with fractions all the time. each part of the second complex number. Here, the imaginary part is the multiple of i. For instance, an electric circuit which is defined by voltage(V) and current(C) are used in geometry, scientific calculations and calculus. We also created a new static function add() that takes two complex numbers as parameters and returns the result as a complex number. A complex number like 7+5i is formed up of two parts, a real part 7, and an imaginary part 5. Operations on Complex Numbers, Some Examples. Identify the coordinates of all complex numbers represented in the graph on the right. (including 0) and i is an imaginary number. This complex number is in the 2nd quadrant. = + ∈ℂ, for some , ∈ℝ In this example, z = 2 + 3i. COMPLEX NUMBER Consider the number given as P =A + −B2 If we use the j operator this becomes P =A+ −1 x B Putting j = √-1we get P = A + jB and this is the form of a complex number. Therefore a complex number contains two 'parts': note: Even though complex have an imaginary part, there In what quadrant, is the complex number$$ 2- i $$? It is a plot of what happens when we take the simple equation z2+c (both complex numbers) and feed the result back into z time and time again. (which looks very similar to a Cartesian plane). \blue 9 - \red i & This rule is certainly faster, but if you forget it, just remember the FOIL method. But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers. On this plane, the imaginary part of the complex number is measured on the 'y-axis', the vertical axis; the real part of the complex number goes on the 'x-axis', the horizontal axis; When we add complex numbers, we can visualize the addition as a shift, or translation, of a point in the complex plane. Just use "FOIL", which stands for "Firsts, Outers, Inners, Lasts" (see Binomial Multiplication for more details): Example: (3 + 2i)(1 + 7i) = (3×1 − 2×7) + (3×7 + 2×1)i = −11 + 23i. Well, a Complex Number is just two numbers added together (a Real and an Imaginary Number). For example, solve the system (1+i)z +(2−i)w = 2+7i 7z +(8−2i)w = 4−9i. When we combine a Real Number and an Imaginary Number we get a Complex Number: Can we make up a number from two other numbers? Calcule le module d'un nombre complexe. We will here explain how to create a construction that will autmatically create the image on a circle through an owner defined complex transformation. 1. . electronics. In most cases, this angle (θ) is used as a phase difference. Converting real numbers to complex number. The general rule is: We can use that to save us time when do division, like this: 2 + 3i4 − 5i×4 + 5i4 + 5i = 8 + 10i + 12i + 15i216 + 25. \begin{array}{c|c} are examples of complex numbers. But it can be done. To display complete numbers, use the − public struct Complex. • In this expression, a is the real part and b is the imaginary part of complex number. But they work pretty much the same way in other fields that use them, like Physics and other branches of engineering. WORKED EXAMPLE No.1 Find the solution of P =4+ −9 and express the answer as a complex number. complex numbers – ﬁnd the reduced row–echelon form of an matrix whose el-ements are complex numbers, solve systems of linear equations, ﬁnd inverses and calculate determinants. Examples and questions with detailed solutions on using De Moivre's theorem to find powers and roots of complex numbers. If a 5 = 7 + 5j, then we expect 5 complex roots for a. Spacing of n-th roots. Sure we can! A complex number can be written in the form a + bi Where. You know how the number line goes left-right? Nombres, curiosités, théorie et usages: nombres complexes conjugués, introduction, propriétés, usage Let 2=−බ ∴=√−බ Just like how ℝ denotes the real number system, (the set of all real numbers) we use ℂ to denote the set of complex numbers. But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers. De Moivre's Theorem Power and Root. Real Number and an Imaginary Number. The trick is to multiply both top and bottom by the conjugate of the bottom. Multiply top and bottom by the conjugate of 4 − 5i : 2 + 3i4 − 5i×4 + 5i4 + 5i = 8 + 10i + 12i + 15i216 + 20i − 20i − 25i2. Given a ... has conjugate complex roots. Complex numbers which are mostly used where we are using two real numbers. To extract this information from the complex number. Nearly any number you can think of is a Real Number! If the real part of a complex number is 0, then it is called “purely imaginary number”. Complex div(n) Divides the number by another complex number. Extrait de l'examen d'entrée à l'Institut indien de technologie. Create a new figure with icon and ask for an orthonormal frame. If a is not equal to 0 and b = 0, the complex number a + 0i = a and a is a real number. So, a Complex Number has a real part and an imaginary part. oscillating springs and Creation of a construction : Example 2 with complex numbers publication dimanche 13 février 2011. A Complex Number is a combination of a Solution 1) We would first want to find the two complex numbers in the complex plane. 57 Chapter 3 Complex Numbers Activity 2 The need for complex numbers Solve if possible, the following quadratic equations by factorising or by using the quadratic formula. If a n = x + yj then we expect n complex roots for a. This article gives insight into complex numbers definition and complex numbers solved examples for aspirants so that they can start with their preparation. Real World Math Horror Stories from Real encounters. Complex Numbers (Simple Definition, How to Multiply, Examples) \\\hline It is just the "FOIL" method after a little work: And there we have the (ac − bd) + (ad + bc)i pattern. complex numbers. Well let's have the imaginary numbers go up-down: A complex number can now be shown as a point: To add two complex numbers we add each part separately: (3 + 2i) + (1 + 7i) In what quadrant, is the complex number$$ -i - 1 $$? \blue{12} + \red{\sqrt{-3}} & \red{\sqrt{-3}} \text{ is the } \blue{imaginary} \text{ part} We know it means "3 of 8 equal parts". We will need to know about conjugates in a minute! = 4 + 9i, (3 + 5i) + (4 − 3i) Complex numbers The equation x2 + 1 = 0 has no solutions, because for any real number xthe square x 2is nonnegative, and so x + 1 can never be less than 1. , fonctions functions. This complex number is in the 3rd quadrant. Addition and subtraction of complex numbers: Let (a + bi) and (c + di) be two complex numbers, then: (a + bi) + (c + di) = (a + c) + (b + d)i (a + bi) -(c + di) = (a -c) + (b -d)i Reals are added with reals and imaginary with imaginary. The color shows how fast z2+c grows, and black means it stays within a certain range. • Where a and b are real number and is an imaginary. In the following video, we present more worked examples of arithmetic with complex numbers. If b is not equal to zero and a is any real number, the complex number a + bi is called imaginary number. = 7 + 2i, Each part of the first complex number gets multiplied by So, to deal with them we will need to discuss complex numbers. Ensemble des nombres complexes Théorème et Définition On admet qu'il existe un ensemble de nombres (appelés nombres complexes), noté tel que: contient est muni d'une addition et d'une multiplication qui suivent des règles de calcul analogues à celles de contient un nombre noté tel que Chaque élément de s'écrit de manière unique sous la […] Here is an image made by zooming into the Mandelbrot set, a negative times a negative gives a positive. The fraction 3/8 is a number made up of a 3 and an 8. complex numbers of the form$$ a+ bi $$and how to graph by using these relations. 6. where a and b are real numbers So, a Complex Number has a real part and an imaginary part. But just imagine such numbers exist, because we want them. The real and imaginary parts of a complex number are represented by Double values. Complex Numbers (NOTES) 1. Argument of Complex Number Examples. Instead of polynomials with like terms, we have the real part and the imaginary part of a complex number. The answer is that, as we will see in the next chapter, sometimes we will run across the square roots of negative numbers and we’re going to need a way to deal with them. For, z= --+i We … And Re() for the real part and Im() for the imaginary part, like this: Which looks like this on the complex plane: The beautiful Mandelbrot Set (pictured here) is based on Complex Numbers. Step by step tutorial with examples, several practice problems plus a worksheet with an answer key Complex numbers are built on the concept of being able to define the square root of negative one. Example 2 . Example 1) Find the argument of -1+i and 4-6i. Complex numbers, as any other numbers, can be added, subtracted, multiplied or divided, and then those expressions can be simplified. Visualize the addition $3-4i$ and $-1+5i$. Complex numbers multiplication: Complex numbers division: \frac{a + bi}{c + di}=\frac{(ac + bd)+(bc - ad)i}{c^2+d^2} Problems with Solutions. An complex number is represented by “ x + yi “. Some sample complex numbers are 3+2i, 4-i, or 18+5i. Complex Numbers and the Complex Exponential 1. \\\hline Complex numbers are often represented on a complex number plane (which looks very similar to a Cartesian plane). 3 roots will be 120° apart.$$ For the most part, we will use things like the FOIL method to multiply complex numbers. \end{array} These are all examples of complex numbers. 2. Double.PositiveInfinity, Double.NegativeInfinity, and Double.NaNall propagate in any arithmetic or trigonometric operation. Example: z2 + 4 z + 13 = 0 has conjugate complex roots i.e ( - 2 + 3 i ) and ( - 2 – 3 i ) 6. For example, 2 + 3i is a complex number. April 9, 2020 April 6, 2020; by James Lowman; Operations on complex numbers are very similar to operations on binomials. I'm an Electrical Engineering (EE) student, so that's why my answer is more EE oriented. Python complex number can be created either using direct assignment statement or by using complex function. You need to apply special rules to simplify these expressions with complex numbers. With this method you will now know how to find out argument of a complex number. 11/04/2016; 21 minutes de lecture; Dans cet article Abs abs. A conjugate is where we change the sign in the middle like this: A conjugate is often written with a bar over it: The conjugate is used to help complex division. Complex numbers are often represented on a complex number plane Table des matières. Example. Complex numbers are often denoted by z. In what quadrant, is the complex number $$2i - 1$$? Interactive simulation the most controversial math riddle ever! 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Next: Phase Analysis of a Up: Basic Properties of a Previous: Orthogonality, Reality, and Uniqueness   Contents   Index

### Sturm's Comparison Theorem

When confronted with the regular boundary value problem

 0 (318)

 0 (319) 0

1. How does the oscillatory nature of , a solution to Eq. 3.18, depend on ?
2. Why do the values of permitted by (3.18) and (3.19) form a discrete and semi-infinite sequence

with a smallest eigenvalue and with as ?

The oscillatory nature'' of a solution is expressed qualitatively by the location of the zeroes of its graph. One could also inquire about its behaviour between successive zeroes. However, we shall see that such an inquiry always leads to the same answer: Provided is positive between a pair of successive zeroes, the graph of has only a single maximum (or minimum). This means that can not oscillate between two of its successive zeroes.

Thus the most important issue is the existence and location of the zeroes, which are controlled entirely by the phase of a given solution . This phase is a scalar function from which one directly constructs the solution. It is preferrable to discuss the behavior of the solution in terms of its phase because the key qualitative properties of the latter are very easy to come by. As we shall see, one only needs to solve a first order differential equation, not the second order S-L equation.

However, before establishing and solving this differential equation, let us use the second order S-L differential equation directly to determine how the zeroes of are affected if the parameter is changed. We express this behaviour in terms of the

Sturm Comparison Theorem:

Whenever , then between two zeroes of the nontrivial solution , there lies a zero of .

This theorem demands that one compare the two different solutions

of Eq. 3.18 corresponding two different constants and . The conclusion is obtained in three steps:

Step 1: Multiply these two equations respectively by and , and then form their difference. The result, after cancelling out the term, is

which is the familiar Lagrange identity, Eq.(3.15), in disguise. Upon integration one obtains

If both and satisfy the mixed Dirichlet-Neumann boundary conditions at , then

If is a singular point of the differential Eq.(3.18), is zero. Thus, if and are finite at , then the left hand side vanishes again at the lower limit . Thus both for a regular and for this singular Sturm-Liouville problem we have

Step 2: Now assume that, for some range of values , each of the corresponding solutions satisfying the boundary condition at , oscillates. In other words, as increases beyond , reaches a maximum, then decreases, passes through zero, reaches a minimum, increases and so on. That such a range of -values exists, we shall see later.

Let lie in this range, and let be the first zero of as in Figure 3.4. Consequently,

Step 3. One must now conclude that if , then the zeroes of are more closely spaced than those of . Why must this statement, the Sturm Comparison Theorem, be true?

a) Assume the contrary, i.e., assume that has no zero in . See Figure 3.4. In that case for all . This implies

But , and , so that . This is a contradiction. Hence our assumption was wrong; the function does have a zero in .

b) Now consider the circumstance where has two successive zeroes at and : . In that case one obtains

If does not change sign in , as in Figure 3.6, then we again have a contradiction because if , then , . In other words, the picture in Figure 3.6 is impossible. We conclude, therefore, that must have a zero inside the open interval .

Next: Phase Analysis of a Up: Basic Properties of a Previous: Orthogonality, Reality, and Uniqueness   Contents   Index
Ulrich Gerlach 2010-12-09


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# Math Insight

### Applet: Function iteration with mapping

The effect of repeatedly applying the function $f(x)$ to the starting value $x_0$ is shown both by a graph of the entire trajectory of the values $x_n$ on the left and a graph of the mapping of the function $f$ on the right. As you click the “step” button, the two perspectives illustrate the process of iterating $f$ to obtain subsequent values of $x_n$ using the recursion $x_n = f(x_{n-1})$. The graph at the right illustrates how $f$ produces new values by mapping $x_{n-1}$ into $x_n$. This graph plots the value of $x_{n}$ versus the previous value $x_{n-1}$, along with the function $f(x)$ in blue.

When you first click “step,” green arrows show how to graphically compute $x_{n}$ from $x_{n-1}$ using the graph of $f$. When you click “step” again, a cyan curve shows how to take the new $x_{n}$ from the $y$-axis back to the same value on the $x$-axis, where it will become the old value $x_{n-1}$ for the next iteration. Clicking “step” additional times continues the process to obtain additional values. The graph at the left summarizes the results obtained from right graph, as it displays the trajectory containing all the generated $x_n$, showing the whole sequence by plotting the $x_n$ versus their index $n$. Note that the vertical axis of the left graph and both axes of the right graph plot the space of $x$. The horizontal axis of the left graph plots values of the index $n$. In the left graph, the values of each $x_{n}$ are also marked with horizontal lines and the last value $x_{n}$ is labeled.

You can change the function $f(x)$ by typing a new function in the box. You can change the initial value $x_0$ by typing a new value in the box or dragging the blue point in the left graph. You can zoom the vertical axis of the left graph with the + and - buttons and pan up and down with the buttons labeled by arrows. The right graph simultaneously changes so that both axes include at least the same range of $x$ as the left graph.

Applet file: function_iteration_mapping.ggb