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# meaning of standard error in statistics Cortez, Florida

McHugh. The two concepts would appear to be very similar. Because of random variation in sampling, the proportion or mean calculated using the sample will usually differ from the true proportion or mean in the entire population. In statistics, a sample mean deviates from the actual mean of a population; this deviation is the standard error.

more... However, while the standard deviation provides information on the dispersion of sample values, the standard error provides information on the dispersion of values in the sampling distribution associated with the population A quantitative measure of uncertainty is reported: a margin of error of 2%, or a confidence interval of 18 to 22. Please review our privacy policy.

Analytical evaluation of the clinical chemistry analyzer Olympus AU2700 plus Automatizirani laboratorijski nalazi određivanja brzine glomerularne filtracije: jesu li dobri za zdravlje bolesnika i njihove liječnike? The researchers report that candidate A is expected to receive 52% of the final vote, with a margin of error of 2%. Bence (1995) Analysis of short time series: Correcting for autocorrelation. JSTOR2340569. (Equation 1) ^ James R.

Consider, for example, a researcher studying bedsores in a population of patients who have had open heart surgery that lasted more than 4 hours. The standard error depends on three factors: N: The number of observations in the population. The Standard Error of the estimate is the other standard error statistic most commonly used by researchers. If one survey has a standard error of $10,000 and the other has a standard error of$5,000, then the relative standard errors are 20% and 10% respectively.

They report that, in a sample of 400 patients, the new drug lowers cholesterol by an average of 20 units (mg/dL). When there are fewer samples, or even one, then the standard error, (typically denoted by SE or SEM) can be estimated as the standard deviation of the sample (a set of Trading Center Sampling Error Sampling Standard Deviation Sampling Distribution Non-Sampling Error Representative Sample Sample Heteroskedastic Central Limit Theorem - CLT Next Up Enter Symbol Dictionary: # a b c d e These assumptions may be approximately met when the population from which samples are taken is normally distributed, or when the sample size is sufficiently large to rely on the Central Limit

Assumptions and usage Further information: Confidence interval If its sampling distribution is normally distributed, the sample mean, its standard error, and the quantiles of the normal distribution can be used to Secondly, the standard error of the mean can refer to an estimate of that standard deviation, computed from the sample of data being analyzed at the time. See unbiased estimation of standard deviation for further discussion. GraphPad Statistics Guide The SD and SEM are not the same The SD and SEM are not the same Feedback on: GraphPad Statistics Guide - The SD and SEM are not

For each sample, the mean age of the 16 runners in the sample can be calculated. The resulting interval will provide an estimate of the range of values within which the population mean is likely to fall. Scenario 2. Br J Anaesthesiol 2003;90: 514-6. [PubMed]2.

The standard error falls as the sample size increases, as the extent of chance variation is reduced—this idea underlies the sample size calculation for a controlled trial, for example. Then subtract the result from the sample mean to obtain the lower limit of the interval. This is important because the concept of sampling distributions forms the theoretical foundation for the mathematics that allows researchers to draw inferences about populations from samples. In this scenario, the 2000 voters are a sample from all the actual voters.

ISBN 0-8493-2479-3 p. 626 ^ a b Dietz, David; Barr, Christopher; Çetinkaya-Rundel, Mine (2012), OpenIntro Statistics (Second ed.), openintro.org ^ T.P. We can estimate how much sample means will vary from the standard deviation of this sampling distribution, which we call the standard error (SE) of the estimate of the mean. BMJ 1994;309: 996. [PMC free article] [PubMed]4. Similarly, the sample standard deviation will very rarely be equal to the population standard deviation.

They may be used to calculate confidence intervals. Because the 9,732 runners are the entire population, 33.88 years is the population mean, μ {\displaystyle \mu } , and 9.27 years is the population standard deviation, σ. As an example of the use of the relative standard error, consider two surveys of household income that both result in a sample mean of \$50,000. It will be shown that the standard deviation of all possible sample means of size n=16 is equal to the population standard deviation, σ, divided by the square root of the

It is calculated by squaring the Pearson R. It is an even more valuable statistic than the Pearson because it is a measure of the overlap, or association between the independent and dependent variables. (See Figure 3).     Standard error of the mean This section will focus on the standard error of the mean. An Introduction to Mathematical Statistics and Its Applications. 4th ed.

The sample mean will very rarely be equal to the population mean. The standard deviation of the age for the 16 runners is 10.23, which is somewhat greater than the true population standard deviation σ = 9.27 years. The SD you compute from a sample is the best possible estimate of the SD of the overall population. For a value that is sampled with an unbiased normally distributed error, the above depicts the proportion of samples that would fall between 0, 1, 2, and 3 standard deviations above

In this scenario, the 400 patients are a sample of all patients who may be treated with the drug. Infect Immun 2003;71: 6689-92. [PMC free article] [PubMed]Articles from The BMJ are provided here courtesy of BMJ Group Formats:Article | PubReader | ePub (beta) | PDF (46K) | CitationShare Facebook Twitter These numbers yield a standard error of the mean of 0.08 days (1.43 divided by the square root of 312). Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.

Note: The Student's probability distribution is a good approximation of the Gaussian when the sample size is over 100. Edwards Deming. The standard error of the mean (SEM) (i.e., of using the sample mean as a method of estimating the population mean) is the standard deviation of those sample means over all Standard error is a statistical term that measures the accuracy with which a sample represents a population.

A practical result: Decreasing the uncertainty in a mean value estimate by a factor of two requires acquiring four times as many observations in the sample. It is rare that the true population standard deviation is known. As will be shown, the standard error is the standard deviation of the sampling distribution. Available at: http://damidmlane.com/hyperstat/A103397.html.

This formula may be derived from what we know about the variance of a sum of independent random variables.[5] If X 1 , X 2 , … , X n {\displaystyle The sample mean x ¯ {\displaystyle {\bar {x}}} = 37.25 is greater than the true population mean μ {\displaystyle \mu } = 33.88 years.


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# How to extract data from OpenFOAM log files with regular expressions and PyFoam

When you need to extract specific values from OpenFOAM log files.

### Step-by-step guide

You will need to have PyFoam and gnuplot installed, as well as a file containing the details of the extracted fields.

1. Firstly, there needs to be a file inside the folder of case or in the same folder as the log file. The name of this file needs to be customRegexp in order for the PyFoam command to automatically detect it.

Inside the file you need to specify the values you wish to capture. For example in the file below we want to capture the values that are inside the log files and are in the format of the regular expression provided. Regular expressions are very strict and need to match exactly in order to get the results. All the Regexp-Groups (surrounded by “( )") is used as data for plotting and in this case there is only one. %f% in this case acts as a variable since this is a value that is going to change inside the log file and we need to capture all of these different values.

Name and theTitle are used for naming the output file and the legend of the figure. Finally, titles is also used for the X-Y axis legends of the data lines found during fitting. These names are used in the order in which they are found in the regular expression, but in our case it is just one.

CpuTimePerTimeStep
{
expr "Time step CPU time: (%f%)";
name "CPU_Time_per_time_step";
theTitle "CPU Time per time step";
titles
(
"time_step"
);
type regular;
}

2. Execute the following command to get a plot that updates whenever there are changes. When finished the user has to stop the process (Ctrl + C).

pyFoamPlotWatcher.py --write-files --single-data-files-only --hardcopy log.pimpleFoam

pyFoamPlotWatcher.py is an executable that should be available if PyFoam is installed on the system. It provides a lot of options, for changing the gnuplot terminal, where to find the regular expression file, what to plot and many more. Here we are using three options: the first one (write-files) is used to create files of the parsed data that the user can later access and the second one is used so that there is a single file containing all the different values instead of creating a new file for each new value. The third option will create png (can be modified) files at the end of the process.

For example if we have a log file like this log.pimpleFoam Download and parse it with the command above a folder with the name log.pimpleFoam.analyzed will be created in the same folder and a file named cpu_time_per_time_step will contain all of the values with respect to time. This file will keep updating until the user decides to stop the process. Furthermore, at the end of the process png files are created automatically with the use of the hardcopy option on the command above that look like the figure below.

### Attachments

Here you can find the log.pimpleFoam-file used as an example above:

format: plain


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## Moduli space, heights and isospectral sets of metrics.(English)Zbl 0703.58010

Sémin. Équations Dériv. Partielles 1989-1990, Exp. No. 2, 6 p. (1990).
The author defines a $$C^{\infty}$$-topology on nonisometric classes of metrics and a uniform metric having constant Gauss curvature. It is established that an isospectral set of closed Riemannian 2-manifolds is compact in the $$C^{\infty}$$-topology.
Reviewer: H.D.Pande

### MSC:

 58D15 Manifolds of mappings 58D29 Moduli problems for topological structures 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
Full Text:


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# How to find standard deviation/error of average cumulative values?

I am a science student, however unfortunately a complete newbie in math and statistics. I am measuring soil respiration for my master thesis project and have a certain amount of carbon respired that I have measured over 30 day period. I plotted the respired carbon as a function of time in days and it looks similar to this:

Each data point is a sum of the measured value at a certain day plus value of the previous measurement at a certain day (time). Now, I know how to calculate and report error bars for a basic data, but these are sums of values (cumulative values), so I am wondering how to estimate standard error or deviation for this type of data. How would I calculate and report error bars for my data in this situation?

P.S. This graph is just an example, not my actual data, but it is essentially the same.

$\text{Var}(X_1+X_2) = \text{Var}(X_1)+\text{Var}(X_2)+2\text{Cov}(X_1,X_2)$ and the $k-$variable case can be obtained recursively from that.
$f_n$ = $F_n$ - $F_{n-1}$
Where $f_n$ denotes the basic data and $F_n$ is the cumulative value


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# The network wave equation  (Page 7/11)

 Page 7 / 11

Many of the concepts from the single-string case carry over to networks.

## Assembling theAndMatrices

We begin by describing the notation of the information represented by our data structures. We denote the $i$ th node to have $xyz$ position vector ${p}_{i}$ . For each node at ${p}_{i}$ , there is an associated set of the indices of connected neighbor nodes ${N}_{i}$ and a set ${C}_{i}$ containing the physical constants ${k}_{ij},{s}_{ij}$ pertaining to the connection between nodes $i$ and $j\in {N}_{i}$ . Since we can divide through by ${\rho }_{i}$ on both sides of the network wave equation, we can assume without loss of generality that the constant ${\rho }_{i}=1$ , and that any data carried by the density ${\rho }_{i}$ is now contained in ${k}_{ij}$ .

Assuming we are given a set of $N$ nodes, along with the $xy$ positions of each node (the $z$ positions are assumed to be 0, such that the web is planar in the $xy$ plane at rest), our first goal is to compute our step sizes ${h}_{ij}$ and orientation vectors ${v}_{ij}$ for connections between two nodes $i$ and $j\in {N}_{i}$ . To account for Dirichlet boundary conditions, we also create an anchored node for each endpoint. In this implementation, if a node has only one neighbor, we assume it is connected to a pinned endpoint whose position is in the opposite direction but the same distance away as the only neighbor (this is required to calculate an inner product). For a node connected to an endpoint, we append the index 0 to ${N}_{i}$ .

Given ${p}_{i}$ , ${N}_{i}$ , ${C}_{i}$ , we proceed as follows

• for each $i$ from $1,2,...,N$
• for all $j\in {N}_{i}$ , calculate ${h}_{ij}={\parallel {p}_{i}-{p}_{j}\parallel }_{2}$
• if the number of elements in any set ${N}_{i}=1$
• create an“endpoint" node at position ${p}_{0}=2{p}_{i}-{p}_{j}$ , with step size ${h}_{i0}={\parallel {p}_{i}-{p}_{0}\parallel }_{2}$
• set ${N}_{i}=\left\{{N}_{i},0\right\}$ to indicate the $i$ th node is connected to an anchor

In practice, we normalize the positions of our nodes such that the web lies within a box of a desired arbitrary size $s$ . We do this by calculating the maximum distance ${d}_{max}$ between the anchored endpoints of a web, then scaling the positions ${p}_{i}$ of every node by the factor $s/{d}_{max}$ . Since the absolute positions of the nodes don't affect our discretization, we don't need to worry about subtracting off the centroid of all the node positions.

With all our variables now in place, we can now proceed to the actual construction of our discretization matrices. This requires knowing $〈{\phi }_{i},,,{\phi }_{j}〉$ , $a\left({\phi }_{i},,,{\phi }_{j}\right)$ , and ${P}_{ij}$ . We deal first with constructing the $M$ matrix, which requires only knowledge of $〈{\phi }_{i},,,{\phi }_{j}〉$ . Just as in the case of finite element on the single string, most of the basis functions ${\phi }_{i}$ and ${\phi }_{j}$ don't share support and their inner products are zero. However, in addition to calculating inner products of regular hat functions, we need to compute the inner products with joint, generalized and nonsymmetric hat functions as well.

Starting with our $M$ matrix, we only need to calculate the inner product $〈{\phi }_{i},,,{\phi }_{j}〉$ for $j=i$ and $j\in {N}_{i}$ . For the diagonal case $j=i$ , we note that our inner product $〈u,,,v〉={\sum }_{i=1}^{L}{\int }_{0}^{{\ell }_{i}}u\left({x}_{i}\right)v\left({x}_{i}\right)\phantom{\rule{0.166667em}{0ex}}d{x}_{i}$ needs only be calculated on the support of ${\phi }_{i}$ . For given $i$ ,

$〈{\phi }_{i},,,{\phi }_{i}〉=\sum _{j\in {N}_{i}}{\int }_{{p}_{i}}^{{p}_{i}+{h}_{j}}{\phi }_{i}^{2}\left(x\right)\phantom{\rule{0.166667em}{0ex}}d{x}_{i}=\sum _{j\in {N}_{i}}\frac{{h}_{ij}}{3}$

The last part is a generalization of our inner product for a uniform grid on a single string. For the off-diagonal case $j\in {N}_{i}$ , $j\ne i$ , the inner product is analogous to the single-string case.

how can chip be made from sand
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no can't
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currently
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nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
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has a lot of application modern world
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ya I also want to know the raman spectra
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please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
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What is STMs full form?
LITNING
scanning tunneling microscope
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text:
# Faraday's and Lenz's law application

1. Jan 31, 2015

### cseil

1. The problem statement, all variables and given/known data
A rectangular coil with a and b sides can rotate around the axis AA' with angular velocity ω. It is in a magnetic field as in figure.

Calculate the flux of B when the coil is orthogonal to the axis AA'. Calculate the $\epsilon_{max}$ and express the position of the coil.

2. Relevant equations
$\epsilon = -\frac{d}{dt} \phi_B$

3. The attempt at a solution

The flux when the coil is orthogonal to the axis AA' is 0.

$\phi_B = NBabcos\theta$

$\theta$ is 90°, so the flux is 0.

Now I calculate the emf.
$\epsilon = -\frac{d}{dt} \phi_B = NBab\omega sin(\omega t)$

The emf is max when $sin(\theta)$ is 1, so when $\theta$ is 90°.

Is it correct my procedure?
If yes, why emf is maximum when flux is 0?

2. Jan 31, 2015

### Staff: Mentor

Since the coil rotates about that axis, it is always orthogonal to it. I assume you mean when the plane of the coil is perpendicular to the surface of the drawing.

Yes.

What matters is not where the emf is maximum but where its rate of change is maximum.

Edit: Oops... I meant flux, not emf, of course.

Last edited: Jan 31, 2015
3. Jan 31, 2015

### cseil

Yes, sorry, I meant that.
Thank you!


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# Is the kernel of any ring homomorphism a subring, according to this definition?

This is an exercise taken verbatim from Birkhoff and MacLane, A Survey of Modern Algebra:

Show that if $\phi: R \rightarrow R'$ is any homomorphism of rings, then the set $K$ of those elements in $R$ which are mapped onto $0 \in R'$ is a subring.

Let me recall from this section of the text the definition of a subring:

... define a subring of a commutative ring $A$ as a subset of $A$ which contains, with any two elements $f$ and $g$, also $f \pm g$ and $fg$, and which also contains the unity of $A$.

It seems to me that the implication in the exercise is false in general. I agree that $a,b \in K$ implies $a \pm b, ab \in K$, but it seems $1 \notin K$ in general. For example, take $\phi$ to be the identity map from $\mathbb{Z}$ to itself, then $K = \{0\}$ and hence $K$ does not contain the unity element. But the definition of subring given here requires that it contain the unity element.

I wonder if I am missing something here, and in case I am correct that the implication doesn't hold, is there some variation of the statement which is true?

• Right. $K$ is an ideal, so $1 \in K \Rightarrow K = R$. Often, a subring is defined to not necessarily include $1$, probably the authors just mixed their definitions up and thought of the one without including the $1$ when posing the exercise. – Daniel Fischer Jul 7 '13 at 1:41
• @DanielFischer: ok, thanks. – AG. Jul 7 '13 at 2:34

## 2 Answers

This is a community wiki answer intended to remove this question from the unanswered queue.

As Daniel Fischer pointed out in the comments, the OP's reasoning is correct, and this was probably just a momentary lapse in consistency with definitions.

For a ring to be a subring it does not need to have the unit element. It only has to have the zero element, closure under addition and multiplication and an additive inverse, meaning a+(some element)=(zero element). If k={0} it has the zero element of the integers, it is closed under Addition and multiplication and 0+(-0)=0.

• Not if a subring has the unit element it just means it is a ring with identity, but rings don't have to have identity or a multiplicative inverse at that – Helper Oct 7 '15 at 17:13
• The definition in the book used by the OP requires a subring to contain the unity of the ambient ring. – Daniel Fischer Oct 7 '15 at 17:13


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What are “maps” between proper classes?

When defining a functor (between categories), I am usually told that it assigns to each object of the source category an object of the target category. I do not find this very satisfactory since we are dealing with proper classes here. Judging by the definition, it must be possible to have the concept of a "map" between proper classes. I would like to know what exactly that is and how it is defined.

I have attempted to read some books on set theory in search for an answer, but they all treat classes very briefly and never mention the possibility of having anything like a map between two of them. I would be just as happy if you could point me to a book where this is explained.

-
This is becoming a very much non-research question. I think perhaps we can close the question. On the other hands, foundational questions on MO seem to be generally below research level and have mostly educational value, which would be a reason to keep this one. – Andrej Bauer Apr 28 '11 at 8:35
A class is given by a formula (which it defines). If $C,D$ are classes, then a map $C \to D$ is simply a class $f$ which defines a "map" $C \to D$ in the obvious sense: For all $x \in C$, i.e. all elements satisfying the formula defining $C$, there is exactly one $y \in D$, i.e. an element satisfying the formula defining $D$, such that $y = f(x)$, i.e. $(x,y)$ satisfies the formula defining $f$. As an exercise, prove that $V \to V, x \mapsto x + 1 := x \cup \{x\}$ is a map, where $V$ is the universe. – Martin Brandenburg Apr 28 '11 at 8:46
My problem was that I simply did not know you could easily define ordered pairs of classes. – Jesko Hüttenhain Apr 28 '11 at 8:51
-1. I agree with Andrej Bauer. – Sergei Tropanets Apr 30 '11 at 14:16

http://en.wikipedia.org/wiki/Ordered_pair#Morse_definition

Definition:
A relation $R$ is functional if and only if for all ordered pairs $\langle x,z\rangle$ and $\langle y,w\rangle$ in $R$, if $x=y$ then $z=w$.

Definition: If $R$ is a relation, $\operatorname{Range}(R) = \{y : (\exists x)(\langle x,y\rangle \in R)\}$.

Definition: A map is an ordered pair $\langle R,C\rangle$ such that $R$ is a functional relation and $\operatorname{Range}(R) \subseteq C$.

-
I suppose it wouldn't hurt to mention that all of what you've written applies to classes equally well as to sets. – Andrej Bauer Apr 28 '11 at 8:12
I suppose you're right. – Ricky Demer Apr 28 '11 at 8:15
So, I am guessing, this is only possible if I use Morse-Kelley and not with ZFC alone? My problem was that I thought $C\times D$ was not really defined if $C$ and $D$ were proper classes. – Jesko Hüttenhain Apr 28 '11 at 8:18
It's possible with ZFC as well. Of couse $C \times D$ is defined for classes, just use the same definition as for sets. – Andrej Bauer Apr 28 '11 at 8:34
Well, now, that helped clarify a lot. Thank you both a lot, that trivial little miscomprehension caused me years of agony. I would really like to accept both answers, but apparently you can't do that. – Jesko Hüttenhain Apr 28 '11 at 8:37

Your question sounds like your preferred foundation is set theory, so let me speak in terms of set theory. A map $f : A \to B$ between sets is a functional relation, i.e., a subset $f \subseteq A \times B$ satisfying:

1. Totality: $\forall x \in A . \exists y \in B . (x,y) \in f$
2. Single-valuedness: $\forall x \in A . \forall y, z \in B . ((x,y) \in f \land (x,z) \in f \implies y = z)$.

We usually write $f(x) = y$ instead of $(x,y) \in f$.

The same definition applies to classes. A map $F : C \to B$ between classes $C$ and $D$ is a subclass of $C \times D$ which is total and single-valued.

Exercise (allowed since this is not a research question): the domain and codomain of a function $F : C \to D$ cannot be recovered from the functional relation $F$. (If $C$ and $F$ are empty, how do we recover $D$?) Therefore, the object part of a functor must be a triple $(C,D,F)$ rather than just $F$. But how can we form ordered triples of classes?

-
Wait, can't the domain be recovered from the relation by Totality? I mean, if this really works for classes the same way it does for sets. Then, a functor would only have to be a pair $(F,D)$ and I just learned that that's easily possible. – Jesko Hüttenhain Apr 28 '11 at 8:43
Yes, you can record just $D$, but why would you do such an ugly thing? – Andrej Bauer Apr 28 '11 at 8:44
Well then, beauty it is! If you can have ordered pairs of classes, however, it should not be a problem to have any finite ordered tuple of classes, right? (So in particular, triples) – Jesko Hüttenhain Apr 28 '11 at 8:49
Right, but please don't confused the product $C \times D$ of two classes with an ordered pair $(C,D)$ of classes. That's what the exercise was for. – Andrej Bauer Apr 28 '11 at 9:19
I kan't sspell. – Andrej Bauer Apr 28 '11 at 9:20

Instead of MK set theory + Morse ordered pair definition, is ARC set theory (F.A.Muller, "Sets, Classes, and Categories", 2001, Bibliography PDF) with the usual Kuratowski ordered pair an option?

ARC supposedly proves the existence of the $n$-th power-class of the set universe V for any $n \in \mathbb{N}$, and all so-called "good" classes provably exist, good classes being the class of all sets and "the powerclass and the union-class of a good class, and the union-class, the intersectionclass, the complement-class, the pair-class, the ordered pair-class, and the Cartesian product-class of any finite number of members of one good class".

According to the cited paper, ARC is consistent relative to ZFC plus a strongly inaccessible cardinal axiom. I think that this set theory looks quite nice, so I'm wondering why it didn't take off at all. The formal proofs should be in Muller's PhD thesis, which I don't have access to.

-


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# Getting the limit

• October 15th 2010, 07:52 AM
wonderboy1953
Getting the limit
Reviewed my textbook on limits. It went over various types including the difference between limits that go to infinity. A problem occurred to me that the book didn't cover so I'm posting this as a puzzle.

Let's define this function: g(x) = exp(x) - ln(x) and let x approach infinity. What value does g(x) take on when x approaches infinity? (I'm wondering whether using the series approach will solve this problem).
• October 15th 2010, 12:02 PM
$y=e^x,\;\;\;x=log_ey$

Hence $y=log_ex$ is the mirror-image of $y=e^x$ through the line $y=x$

The functions differ by ever-increasing magnitude
• October 15th 2010, 02:50 PM
Unbeatable0
Using a "series approach":

$e^x \ge 1+x+\frac{x^2}{2}$ and $\ln x \le x+1$ (for $x\ge 1$) we get

$e^x-\ln x \ge (1+x+\frac{x^2}{2}) - (x+1) = \frac{x^2}{2}$

from which the limit follows, since $\lim_{x\rightarrow\infty} \frac{x^2}{2} = \infty$

Another approach:

$\lim_{x\rightarrow\infty} e^x-\ln x = \lim_{x\rightarrow\infty} (1-\frac{\ln x}{e^x}) e^x$

and since $\lim_{x\rightarrow\infty} 1-\frac{\ln x}{e^x} = 1$ and

$\lim_{x\rightarrow\infty} e^x = \infty$ we get the desired result.
• October 15th 2010, 10:38 PM
CaptainBlack
Quote:

Originally Posted by wonderboy1953
Reviewed my textbook on limits. It went over various types including the difference between limits that go to infinity. A problem occurred to me that the book didn't cover so I'm posting this as a puzzle.

Let's define this function: g(x) = exp(x) - ln(x) and let x approach infinity. What value does g(x) take on when x approaches infinity? (I'm wondering whether using the series approach will solve this problem).

Since exp(x) grows faster than any power of x, and log(x) grows more slowly than any non-zero power of x their difference grows faster than any power of x.

CB
• October 16th 2010, 08:30 AM
wonderboy1953
Quote:

Originally Posted by CaptainBlack
Since exp(x) grows faster than any power of x, and log(x) grows more slowly than any non-zero power of x their difference grows faster than any power of x.

CB

That's my line of reasoning CB. I was just looking for a more rigorous line of thought on what I suspected.
• October 16th 2010, 11:29 PM
CaptainBlack
Quote:

Originally Posted by wonderboy1953
That's my line of reasoning CB. I was just looking for a more rigorous line of thought on what I suspected.

It is actually rigorous, I was just expressing known big-O results in colloquial English.

CB
• October 17th 2010, 03:56 AM
$e^x>2x^2lnx$
$\displaystyle\lim_{x \to \infty}\left(e^x-lnx\right)\;>\;\lim_{x \to \infty}\left(2x^2-1\right)lnx$
$lnx>1$ for $x>e$


text:
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Page owner: Isaac Farley   |   Last updated 2020-October-06


text:
## The Honorable Sylvester Turner

State representative and lawyer Sylvester Turner was born on September 27, 1954 in Acres Homes, Texas. His mother was a maid at the Rice Hotel and his father, a commercial painter. Turner was raised with eight brothers and sisters. In 1973, he graduated as the valedictorian of Klein High School. Four years later, Turner received his B.A. degree in political science from the University of Houston, after which he attended Harvard Law School, where he received his J.D. degree in 1980.

Turner was hired at the Houston-based law firm Fulbright & Jaworski. After three years, Turner left and formed his own law firm with partner Barry M. Barnes. Barnes & Turner specialize in corporate and commercial law. In 1984, Turner ran for a Harris County Commissioner seat, but he lost to El Franco Lee. In 1988, he won the seat in the Texas House of Representatives for District 139, a mostly minority district. Turner also taught at the Thurgood Marshall School of Law at Texas Southern University, the South Texas College of Law, and at the University of Houston Law School’s continuing legal education program. He also ran for the mayor of Houston twice, once in 1991 where he lost in a hotly contested race, and again in 2003, where he lost to Bill White. In 2003, Turner became the Speaker Pro Tempore in the Texas House of Representatives, a post he held until 2009. His major legislative accomplishment, HB 109, expanded access to the children’s health insurance program and was passed in 2007.

Turner sits on the State Affairs committee and is the Vice Chair of the Appropriations Committee. He is also on the Subcommittee on the Current Fiscal Condition. He is a member of Brookhollow Baptist Church and has one daughter, Ashley Paige Turner.

Sylvester Turner was interviewed by The HistoryMakers<\em> on August 15, 2012.

Accession Number

A2012.156

Sex

Male

Interview Date

8/15/2012

Last Name

Turner

Maker Category
Marital Status

Divorced

Occupation
Organizations
Schools

Harvard Law School

University of Houston

Klein Forest High School

Garden City Elementary and Junior High School

Klein Intermediate School

First Name

Sylvester

Birth City, State, Country

Houston

HM ID

TUR07

Favorite Season

Spring, Summer

State

Texas

Favorite Vacation Destination

South Africa

Favorite Quote

I Can Do All Things Through Christ That Strengthens Me.

Bio Photo
Speakers Bureau Region State

Texas

Birth Date

9/27/1954

Birth Place Term
Speakers Bureau Region City

Houston

Country

United States

Favorite Food

Spaghetti, Ox Tails

Short Description

Mayor, state representative, and lawyer The Honorable Sylvester Turner (1954 - ) represented district 139 in the Texas House of Representatives from 1988 to 2016, when he became the mayor of Houston, Texas. He also founded the law firm of Barnes and Turner LLP.

Employment

Texas House of Representatives

Barnes & Turner

University of Houston

South Texas College of Law

Texas Southern University

Fulbright & Jaworski

City of Houston, Texas

Favorite Color

Blue

Timing Pairs
0,0:380,8:1604,36:7724,163:8156,170:10316,223:18750,252:22422,313:26358,341:27285,351:50377,577:57466,610:77669,792:78024,798:78734,811:79799,830:80083,835:92624,969:94016,988:106892,1136:140006,1522:141014,1533:144934,1619:167844,1904:173690,1994:180690,2087:181880,2120:191646,2255:208620,2463:208990,2469:213334,2656:219318,2808:259770,3011:266990,3079:281530,3243:285824,3263:286164,3269:292840,3443:299198,3492:302405,3528:304170,3536:304548,3549:304764,3554:305142,3563:308916,3627:313108,3672:313888,3683:317554,3751:325020,3807:325460,3812:366638,4401:367070,4416:370785,4464:371050,4470:371262,4475:372240,4499$0,0:27348,354:28086,361:28988,373:33062,403:35174,434:39650,465:53194,619:54118,632:54454,637:72446,774:91845,962:94811,990:97624,1076:123530,1376:123850,1381:129690,1507:141350,1624:142946,1649:165488,1935:186730,2255:204944,2400:212130,2475:212410,2481:212802,2490:233200,2640 DAStories Tape: 1 Story: 1 - Slating of The Honorable Sylvester Turner's interview Tape: 1 Story: 2 - The Honorable Sylvester Turner lists his favorites Tape: 1 Story: 3 - The Honorable Sylvester Turner describes his mother's family background Tape: 1 Story: 4 - The Honorable Sylvester Turner describes his father's family background Tape: 1 Story: 5 - The Honorable Sylvester Turner remembers working with his dad Tape: 1 Story: 6 - The Honorable Sylvester Turner describes his parents' personalities and who he takes after Tape: 1 Story: 7 - The Honorable Sylvester Turner talks about his roots in Chappell Hill, Texas Tape: 1 Story: 8 - The Honorable Sylvester Turner lists his siblings Tape: 1 Story: 9 - The Honorable Sylvester Turner remembers the Acres Homes community in Houston, Texas Tape: 2 Story: 1 - The Honorable Sylvester Turner talks about his experiences of school integration in Houston, Texas, pt. 1 Tape: 2 Story: 2 - The Honorable Sylvester Turner talks about his experiences of school integration in Houston, Texas, pt. 2 Tape: 2 Story: 3 - The Honorable Sylvester Turner talks about African American political representation in Houston, Texas Tape: 2 Story: 4 - The Honorable Sylvester Turner describes the African American community in Houston, Texas Tape: 2 Story: 5 - The Honorable Sylvester Turner describes the sights, sounds and smells of his childhood Tape: 2 Story: 6 - The Honorable Sylvester Turner remembers the Bethel Baptist Church in Houston, Texas Tape: 2 Story: 7 - The Honorable Sylvester Turner recalls his influential teachers Tape: 2 Story: 8 - The Honorable Sylvester Turner remembers being bused to an all-white school Tape: 3 Story: 1 - The Honorable Sylvester Turner describes his experiences of school integration Tape: 3 Story: 2 - The Honorable Sylvester Turner talks about his early ambitions Tape: 3 Story: 3 - The Honorable Sylvester Turner describes the demographics of Klein High School in Houston, Texas Tape: 3 Story: 4 - The Honorable Sylvester Turner recalls his influences at the Bethel Baptist Church in Houston, Texas Tape: 3 Story: 5 - The Honorable Sylvester Turner talks about his community in Houston, Texas Tape: 3 Story: 6 - The Honorable Sylvester Turner remembers his valedictorian speech Tape: 3 Story: 7 - The Honorable Sylvester Turner remembers his father's death Tape: 3 Story: 8 - The Honorable Sylvester Turner recalls his decision to attend the University of Houston in Houston, Texas Tape: 4 Story: 1 - The Honorable Sylvester Turner recalls his early aspiration to become a lawyer Tape: 4 Story: 2 - The Honorable Sylvester Turner talks about his decision to attend the Harvard Law School Tape: 4 Story: 3 - The Honorable Sylvester Turner remembers his friendships with Leroy Hassell Tape: 4 Story: 4 - The Honorable Sylvester Turner talks about his social life at Harvard Law School Tape: 4 Story: 5 - The Honorable Sylvester Turner remembers Derrick A. Bell, Jr. Tape: 4 Story: 6 - The Honorable Sylvester Turner talks about the faculty of the Harvard Law School Tape: 4 Story: 7 - The Honorable Sylvester Turner remembers his club football team at Harvard University Tape: 4 Story: 8 - The Honorable Sylvester Turner recalls hearing a female preacher for the first time Tape: 5 Story: 1 - The Honorable Sylvester Turner recalls his internship at Fulbright and Jaworski LLP in Houston, Texas Tape: 5 Story: 2 - The Honorable Sylvester Turner describes a memorable legal case Tape: 5 Story: 3 - The Honorable Sylvester Turner remembers founding Barnes and Turner LLP Tape: 5 Story: 4 - The Honorable Sylvester Turner remembers losing his first political campaign Tape: 5 Story: 5 - The Honorable Sylvester Turner remembers his campaign for the Texas House of Representatives Tape: 5 Story: 6 - The Honorable Sylvester Turner recalls his election to the Texas House of Representatives Tape: 5 Story: 7 - The Honorable Sylvester Turner talks about his interest in healthcare reform Tape: 5 Story: 8 - The Honorable Sylvester Turner recalls arguing a civil suit against the Phillips Petroleum Company Tape: 6 Story: 1 - The Honorable Sylvester Turner remembers his first campaign for the mayoralty of Houston, Texas Tape: 6 Story: 2 - The Honorable Sylvester Turner talks about the aftermath of the 1991 mayoral election in Houston, Texas Tape: 6 Story: 3 - The Honorable Sylvester Turner remembers Lee P. Brown Tape: 6 Story: 4 - The Honorable Sylvester Turner talks about politics in Texas Tape: 6 Story: 5 - The Honorable Sylvester Turner talks about political redistricting in Texas Tape: 6 Story: 6 - The Honorable Sylvester Turner describes his legislative achievements Tape: 6 Story: 7 - The Honorable Sylvester Turner remembers his second campaign for the mayoralty of Houston, Texas Tape: 6 Story: 8 - The Honorable Sylvester Turner remembers the passage of Texas House Bill 109 Tape: 7 Story: 1 - The Honorable Sylvester Turner remembers meeting President Barack Obama Tape: 7 Story: 2 - The Honorable Sylvester Turner describes his hopes and concerns for the African American community Tape: 7 Story: 3 - The Honorable Sylvester Turner talks about his work in the Acres Homes section of Houston, Texas Tape: 7 Story: 4 - The Honorable Sylvester Turner reflects upon his legacy Tape: 7 Story: 5 - The Honorable Sylvester Turner talks about his family Tape: 7 Story: 6 - The Honorable Sylvester Turner reflects upon his life Tape: 7 Story: 7 - The Honorable Sylvester Turner describes how he would like to be remembered DASession 1$1

DATape

5$5 DAStory 2$8

DATitle
The Honorable Sylvester Turner describes a memorable legal case
The Honorable Sylvester Turner recalls arguing a civil suit against the Phillips Petroleum Company
Transcript
Now, is there a memorable case from that period of time that you can tell us about?$$I guess it's, it's one in particular. The plaintiff was a guy by the name of Willie Harris [ph.]. I guess it is memorable since I still remember it, and that's been years ago. But, anyway, Willie was an entrepreneur, African American, and he was in his company's truck, and he was coming over the Ship, Ship Channel Bridge [Sam Houston Tollway Ship Channel Bridge, Houston, Texas]. And this 18-wheeler hit him, and he was seriously injured, and he sued the 18-wheeler. I represented the company. And, and I made him an offer through his attorney, and he did not, he did not accept the offer to settle. It end up--it went to trial. I made him another offer, and his attorney did not accept it. And, quite frankly, you know, I had, I had much more to give, okay. And, but, and so, we went to trial. During the trial, I made him another offer, attorney didn't accept it. And then, the attorney came to me during the trial, and asked me, was the offer still on the table? And I said, "Well, if you accept it now." Now, mind you, I had a lot more to give. And in many ways, I said to myself, the attorney is crazy as hell (unclear) to be accepting--I mean, I represent my client, so if, you know, and so, I say, "Yeah, if you, if you accept it now, it's on the table." This is during the course of trial. And he went over and talked to Willie, and I could, and I could kind of hear and see the exchange, where Willie was not liking the offer. And his attorney kept talking to him, kept talking to him, kept talking. And Willie finally relented and said, "Okay." And the attorney came to me and said, "We'll accept." And in my mind, I was saying, "You're crazy as hell but, okay, no problem." So, when he stood before the judge to announce that the case had settled, and the judge said, "All parties in agreement?" I said, you know, "It's the best terms for the defendant, judge, yes, I'm in agreement." Asked the other attorney, the attorney said, "Yes." And the judge asked Willie Harris. "Mr. Harris, are you in agreement with the settlement?" And he kind of said, "Oh, well," and said, "You should, well, you don't have to--are you in agreement with it?" And attorney, his attorney looked at him, and he finally said, "Yeah, yeah." And she said, "Okay, all parties in agreement. This case is dismissed." It's over. So, I was packing up, and Willie comes over to me. And he said, "Mr.," he said, "Mr. Turner [HistoryMaker Sylvester Turner], you know, I'm hurt, you know, I'm hurt, and this does not cover me for my injuries," and stuff like that. And I said, I said, "Mr. Harris, I'm not your attorney. I represent, I represent my client, and I did my job." And he said, "But, brother, you know, I'm--," he said, "Brother, you know, I'm hurt." I said, "Mr. Harris, I'm not your attorney. I represent my client. I did my job." And, and my client and I got up, and we walked out. That one, that one stands out because it's one of those deals that, yeah, you know, he had a poor lawyer. Had a poor lawyer, but it's not a case where I can be the lawyer for my client, and be the lawyer for his client as well. Okay. Now, subsequently, a few years later, I'm no longer at Fulbright [Fulbright and Jaworski LLP], and now I'm in my own shop [Barnes and Turner LLP; Barry Barnes and Associates PLLC, Houston, Texas]. Willie comes to me, and became my client, you know, but that one stands out. And, and, and because it's nothing like having a good lawyer. It's nothing like having somebody that's going to advocate for you, and fight for you, and get everything that's on the--that's potentially is on the table for you. Nothing like having a good lawyer. And in his case, his lawyer fell short, and he paid the price.$$I heard such cases before when cold--cold aspects of law sometimes, you know, the people don't know. They--$$You know--$$--don't give, give a thing (unclear).$$Right, but you can't be, you know, the way the system is designed, you know, I can't be the lawyer for my client, and be the lawyer for you at the same time. And my job is, as a lawyer is to represent my client, and represent my client zealously, and do the best I can, so but it points out the importance of having quality representation, and not only quality representation, you've got to have people who are willing to advocate for you.$$Okay.$$And if you don't have that, you'll fall short.Now, in 1989, you sued Phillips Petroleum [Phillips Petroleum Company; Phillips 66]. That's when the big Phillips plant explosion--$$Um-hm.$$--okay, that's the Phillips plant explosion case. Tell us about that.$$(Cough) I represented Janet Little. She was an employee at Phillips Petroleum. Interesting story on how we met--I was speaking at a, at a church association banquet in--I want to say, in Sealy, Texas. And she and her parents were in the audience. Later, goes the Phillips Petroleum explos- explosion. And her mom calls me here at the firm and says--she introduced herself, Ms. Foy, and she says, "My daughter has been seriously burnt. And there are a lot of lawyers that are around here at (unclear). But she asked me to call you because she wanted, she wanted the lawyer that spoke at the banquet, and that was you." And then, we--I met with them and signed on, and represented her, and I had a very favorable outcome. She's been a client with this firm ever since. From the proceeds, her father [Charles H. Foy] was a pastor in Dickinson, Texas. And from part of the proceeds, she, she built, she constructed a new church in Dickinson [Mount Carmel Missionary Baptist Church] and paid for it herself, which is one of the, one of the largest churches now in Dickinson, Texas. You know, it was, it was, it just started the ball, the ball just--things just started changing in the, in the life of the firm.$$So, the plant was caused by some negligence of Phillips?$$Yeah, they were, they were negligent and then caused the explosion. And I represent Anna Brooks [ph.] and her, and a couple of other people. Ironically, the people that were defending, the lawyers that were defending Phillips came from Fulbright and Jaworski [Fulbright and Jaworski LLP]. And one of, and one of my mentors, Blake Tartt, was the lead attorney.$$That's, that's interesting.$$Yeah. And we were in, we were in a conference room which it was a settlement meeting. And we were talking and, you know, and Blake says, "Sylvester [HistoryMaker Sylvester Turner], are we going to get this case settled?" And I said, "I hope so, Mr. Tartt." He would call me Sylvester and I called him Mr. Tartt 'cause I'd looked up to him. And then, he asked me, how much was I asking for. And I, I wrote him a note on a sheet, on a sheet of paper, and I forwarded it to him. And he crossed it out, and sent a note back and, and I told him, I said, "If I accepted this, you would, you would lose all respect for me, and I would not be the, the student that you had taught well." So, I crossed it out, and sent him another note. And he said, "Done."


text:
I'm trying to print an architectural model of a house with 2 floors and a room on the roof. It was designed in SketchUp by a friend. I removed inner walls and floors to print it upside down to get just the outer look of that building. All is good in SketchUp and in Meshlab (converted .skp to .3ds) and even after exporting it to .stl format. But when I slice it and select preview, it is adding a floor on its own. Please take a look at the attached screenshots to get a better idea of what I am referring to. Any idea what's causing this and how it can be solved?

(Using Ultimaker Cura 4.1.0 for slicing at 0.2 mm layer height)

• Welcome to 3dPrinting.SE! Sep 14 '19 at 12:45
• I don't think your STL file is okay, it appears that you removed the floors, but there might be some problems with the normals of the faces. Maybe you can fix the file through NetFabb.
– 0scar
Sep 14 '19 at 12:59
• Check the setting in the slicer for adding supports. Your overhand will be difficult to print. Sep 14 '19 at 13:36
• you can, in the preview, disable viewing the normal material and just display supports. Could you share a picture of that, so we get a better clue what type of print cura generates for you? Sep 15 '19 at 9:24
• When I printed a similar house model designed in SketchUp, the generated .stl was of horrible quality. Windows 10's build in "3D Builder" has a very powerful .stl repair tool built in, which got rid of the defects.
– towe
Sep 16 '19 at 8:02

SketchUp has a problem with generating the normals of an STL in the right way. As you see in the slicer, the model claims, that the "tops" are overhangs, which means that the whole model is "inside out" with the normals.

To fix that, you'd need to open the file in a program that can fix that by recalculating the normals. I suggest blender:

• Open Blender
• Delete the cube by clicking on it and pressing the delete button
• Import the STL via Drag & Drop or the menu
• Change to edit mode in the top left corner
• A to select everything
• Ctrl + N to recalculate normals
• N then under Mesh Display > Normals hook Faces

Now, your model should look like a hedgehog, each face having a blue line poking out.

It also should show if there are any faces remaining, which can be selected and deleted manually by clicking it, so that only it is highlighted, and then pressing delete


text:
[This article was first published on Posts on R Lover ! a programmer, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

Well, it’s election day here in America and the last thing in the world I want
to talk about is politics. Hope you found a way to vote. Meanwhile, while we
wait let me (hopefully) amuse and perhaps educate you with some information
about another topic that we all care deeply about. A vaccine for COVID-19.

The other day I was in my car listening to the radio when I heard Olivier Knox
interviewing Dr. Moncef Slaoui
about progress towards developing a vaccine for COVID-19. Since I was driving it
was difficult to pay attention to exact details but I remember being very
surprised at some numbers I thought I heard quoted. Had to do with how few cases
they were going to use for making their first decision. I’d heard all about the large number of participants in the trials…

Moderna, the biotechnology firm partnering with the National Institutes of Health to develop a coronavirus vaccine, announced Thursday that it has fully enrolled its trial, with 30,000 participants — more than a third of whom are minorities. Washington Post

And I had been sort of paying attention to the methodology knowing that
participants were injected and then simply went about living their normal
lives…

Much of the debate over the timeline for a vaccine has been fueled by the unpredictability of the clock in each vaccine trial, which ticks forward according to the accumulation of covid-19 cases among study participants… Moderna is likely to have 53 covid-19 cases among participants by November — enough for a first look at the data — with sufficient safety data available just before Thanksgiving… Before they authorize a vaccine, regulators will require that the vaccine be at least 50 percent effective, that there be at least five severe cases of covid-19 among people who receive a placebo and that there be at least two months of follow-up on half the study participants. Washington Post

53 out of 30,000 just seems like an awfully small number, so it started me
wondering about confidence and credibility in a statistical sense and was the
germ seed for this post. I’m no epidemiologist or bio-statistician but I like to
think I have a decent grasp on the basics so off we go, with no claim that these
are the exact methods they’ll follow but at least an informed look at the
basics.

### The Very Basics

If you have some background in statistics the next section may be a little
boring for you but bear with me I promise an opportunity to delve more deeply
later and revisit more complex issues like the the differences between
frequentist and bayesian methods later.
I’ve explored this topic before
but there’s plenty of new stuff below. Let’s just get the basics under our belts.

If you’ve ever taken an introductory course in probability and statistics you’ve
likely been exposed to the statistician’s love of coin flips. This is another
instance where at it’s simplest comparing our vaccine trials to coin flips can
be instructive. Although overall the study is a
randomized controlled trial
(actually several of them) on a very large scale in many ways our understanding
of these first 53 cases can be modeled as a set of coin flips. For each of the
53 people who are confirmed to have contracted covid-19 they either got the
vaccine or a placebo (heads or tails).

R let’s us quickly get a sense of what the probability is that of the 53 people
in the trial who contracted covid-19 what if only the minimum of 5 were given
the actual vaccine and not the placebo? For those of you who don’t like
scientific notation that’s thirty-one billion, eight hundred fifty-nine million,
nine hundred thousand probability against you doing that randomly flipping coins.

got_covid <- 53
vaccinated_got_covid <- 5
placebo_got_covid <- 53 - vaccinated_got_covid
dbinom(vaccinated_got_covid, got_covid, .5)
## [1] 3.18599e-10
vaccinated_got_covid <- 19
placebo_got_covid <- 53 - vaccinated_got_covid
dbinom(vaccinated_got_covid, got_covid, .5)
## [1] 0.01321525

Even if 19 people who received the true vaccine got covid the probability is
still less than 2% that the vaccine versus placebo doesn’t matter that it’s a
50/50 chance.

Let’s talk about effectiveness. We want our vaccine to be at least 50%
effective. We can operationalize that most simply by
$\frac{placebo~cases - vaccinated~cases}{placebo~cases}$
So for our current example where 19 of the 53
$\frac{placebo~cases - vaccinated~cases}{placebo~cases} = (34-19)/34 = 0.4411765$
which is very close to what we need. Let’s display it graphically for the range
between 5 and 26 vaccinated cases and dispense with one other pesky issue. So
far we have been ignoring the fact that there aren’t 53 people involved in the
trial, there are 30,000. We don’t really want to lose sight of this even though
it makes very little difference to our math in most cases. Let’s calculate
effectiveness both with and without 30,000 in the denominator and even track
rounding error in our little tibble.

library(dplyr)
library(ggplot2)
library(kableExtra)
theme_set(theme_bw())

effectiveness <-
tibble::tibble(vaccinated = 5:26) %>%
mutate(placebo = 53 - vaccinated,
effectiveness = (placebo - vaccinated) / placebo * 100,
placeborate = placebo / (15000 - placebo),
vaccinatedrate = vaccinated / (15000 - vaccinated),
pctratedifference = (placeborate - vaccinatedrate) / placeborate * 100,
rounding = effectiveness - pctratedifference)

effectiveness %>%
kbl(digits = c(0,0,2,4,4,2,2),
caption = "Effectiveness as a function of positive COVID cases") %>%
kable_minimal(full_width = FALSE,
position = "left") %>%
add_header_above(c("Cases (53)" = 2, " " = 1, "Infection Rate" = 2, "% difference in rate" = 1, " " = 1))
Table 1: Effectiveness as a function of positive COVID cases
Cases (53)
Infection Rate
% difference in rate
vaccinated placebo effectiveness placeborate vaccinatedrate pctratedifference rounding
5 48 89.58 0.0032 0.0003 89.61 -0.03
6 47 87.23 0.0031 0.0004 87.27 -0.03
7 46 84.78 0.0031 0.0005 84.82 -0.04
8 45 82.22 0.0030 0.0005 82.27 -0.04
9 44 79.55 0.0029 0.0006 79.59 -0.05
10 43 76.74 0.0029 0.0007 76.80 -0.05
11 42 73.81 0.0028 0.0007 73.86 -0.05
12 41 70.73 0.0027 0.0008 70.79 -0.06
13 40 67.50 0.0027 0.0009 67.56 -0.06
14 39 64.10 0.0026 0.0009 64.16 -0.06
15 38 60.53 0.0025 0.0010 60.59 -0.06
16 37 56.76 0.0025 0.0011 56.82 -0.06
17 36 52.78 0.0024 0.0011 52.84 -0.06
18 35 48.57 0.0023 0.0012 48.63 -0.06
19 34 44.12 0.0023 0.0013 44.17 -0.06
20 33 39.39 0.0022 0.0013 39.45 -0.05
21 32 34.38 0.0021 0.0014 34.42 -0.05
22 31 29.03 0.0021 0.0015 29.07 -0.04
23 30 23.33 0.0020 0.0015 23.37 -0.04
24 29 17.24 0.0019 0.0016 17.27 -0.03
25 28 10.71 0.0019 0.0017 10.73 -0.02
26 27 3.70 0.0018 0.0017 3.71 -0.01
effectiveness %>%
ggplot(aes(x = vaccinated)) +
geom_line(aes(y = effectiveness)) +
geom_line(aes(y = pctratedifference), color = "green") +
geom_hline(aes(yintercept = 50)) +
scale_y_continuous(labels = scales::label_percent(scale = 1),
breaks = seq.int(0, 100, 10)) +
scale_x_continuous(breaks = seq.int(2, 26, 2)) +
xlab("Number of positive COVID-19 outcomes among those who received the vaccine") +
ylab("Effectiveness as a percentage") +
ggtitle("Vaccine effectiveness as a function of outcomes (53 total cases)")

mean(effectiveness$rounding) # that's .04% not 4.0% ## [1] -0.04559179 So in the best possible scenario (remember we require that at least 5 people who got the vaccine contract COVID before we can run the numbers) our effectiveness is ~90%. We need the number of people who received the vaccine and still contracted COVID to be 17 or less. ### Testing our confidence as a frequentist Now we know what we have to do to convince ourselves about effectiveness let’s address what we can do to be confident that our results are not a fluke. We know that low probability events occur. I don’t want to repeat myself so if you want a little background on frequentist methodology please see this earlier post. If you want one of many cautionary tails about the limits of NHST then please see this post. Having said all that, frequentist methods are certainly the most prevalent and likely to be applied to the warpspeed data so let’s see what we come up with. Let’s start with the simplest possible test we could use. Let’s build a simple matrix (table) of our results and call it dat. We’ll assume the 30,000 participants are equally divided between receiving the vaccine and the placebo. We’ll pretend 19 folks who got the real vaccine later developed COVID and 34 did not. We’ll grab just the first column dat[,1] and put that in an object called covid_gof. Our hypothesis is that the vaccine matters that it helps prevent being infected. In NHST terms our null hypothesis is that there is no difference in the number of people who will get infected. In essence it might as well be a coin toss, 50/50, equal numbers of people will get infected whether they received the vaccine or the placebo. We can use the $$\chi^2$$ goodness of fit test. For clarity we’ll remove the continuity correction and overtly specify 50/50 odds. Since the default chisq.test test results are very terse we’ll use lsr::goodnessOfFitTest on the same data (expressed as vector that is a factor) to make it clear what we’re doing. dat <- matrix(c(34, 15000 - 34, 19, 15000 - 19), nrow = 2, byrow = TRUE) rownames(dat) <- c("Placebo", "Vaccine") colnames(dat) <- c("COVID", "No COVID") dat ## COVID No COVID ## Placebo 34 14966 ## Vaccine 19 14981 covid_gof <- dat[,1] chisq.test(covid_gof, correct = FALSE, p = c(.5, .5)) ## ## Chi-squared test for given probabilities ## ## data: covid_gof ## X-squared = 4.2453, df = 1, p-value = 0.03936 outcomes <- factor(c(rep("Vaccine", 19), rep("Placebo", 34))) lsr::goodnessOfFitTest(outcomes) ## ## Chi-square test against specified probabilities ## ## Data variable: outcomes ## ## Hypotheses: ## null: true probabilities are as specified ## alternative: true probabilities differ from those specified ## ## Descriptives: ## observed freq. expected freq. specified prob. ## Placebo 34 26.5 0.5 ## Vaccine 19 26.5 0.5 ## ## Test results: ## X-squared statistic: 4.245 ## degrees of freedom: 1 ## p-value: 0.039 Either way $$\chi^2$$ = 4.245283 and p = 0.0393595. Therefore we reject the null and can have some “confidence” that our results aren’t simply a fluke. That seems a little too simple and seems to ignore the fact that we actually have data for not 53 people but 30,000 people. So let’s make things more complex. There are actually many variants of a $$\chi^2$$ test. Instead of goodness of fit let’s use $$\chi^2$$ test of independence or association using all four cells of our little matrix. Here our null hypothesis is that our variables are independent of one another, that whether you get COVID has no association to whether you received the vaccine or the placebo. A subtle distinction perhaps but worth it if for no other reason than to exploit the additional data. To make it even more fun there are numerous functions in r to run the test. From base r, where there are at least two variants chisq.test and prop.test to specialized packages like epiR that focus on epidemiology. Notice that they all express the same basic notions $$\chi^2$$ = 4.2527963 and p = 0.0391858, even though they present an array of additional information. chisq.test(dat, correct = FALSE) ## ## Pearson's Chi-squared test ## ## data: dat ## X-squared = 4.2528, df = 1, p-value = 0.03919 prop.test(dat, correct = FALSE) ## ## 2-sample test for equality of proportions without continuity ## correction ## ## data: dat ## X-squared = 4.2528, df = 1, p-value = 0.03919 ## alternative hypothesis: two.sided ## 95 percent confidence interval: ## 0.0000496578 0.0019503422 ## sample estimates: ## prop 1 prop 2 ## 0.002266667 0.001266667 epiR::epi.2by2(dat = as.table(dat), method = "cross.sectional", conf.level = 0.95, units = 1, outcome = "as.columns") ## Outcome + Outcome - Total Prevalence * Odds ## Exposed + 34 14966 15000 0.00227 0.00227 ## Exposed - 19 14981 15000 0.00127 0.00127 ## Total 53 29947 30000 0.00177 0.00177 ## ## Point estimates and 95% CIs: ## ------------------------------------------------------------------- ## Prevalence ratio 1.79 (1.02, 3.14) ## Odds ratio 1.79 (1.02, 3.14) ## Attrib prevalence * 0.00 (0.00, 0.00) ## Attrib prevalence in population * 0.00 (-0.00, 0.00) ## Attrib fraction in exposed (%) 44.12 (2.08, 68.11) ## Attrib fraction in population (%) 28.30 (-2.75, 49.97) ## ------------------------------------------------------------------- ## Test that OR = 1: chi2(1) = 4.253 Pr>chi2 = 0.04 ## Wald confidence limits ## CI: confidence interval ## * Outcomes per population unit Notice that epiR::epi.2by2 provides confirmation of several important pieces of data we generated earlier in our very first table results. If you consult the row for 19 & 34, prevalence matches 0.00227 & 0.00127 the columns labeled “infection rate” and “Attrib fraction in exposed (%) = 44.12” matches our “effectiveness” column. ### Credible? Incredible? A bayesian approach to how confident we are The “problem” with a frequentist’s approach is that the “testing” framework is rather contorted and you really can’t make the statements that you want to make. We want to use the language of probability theory to say things like “there is an 90% chance that the vaccine is effective, which of course means there’s a 10% chance that it doesn’t”. As I have written before and many others have written elsewhere a “p value” and a “confidence interval” don’t give you that capability. Don’t get me wrong, the approach can be useful, is still the most frequent and dominant approach, but I don’t find it the best approach. So let’s approach our warpspeed data using several different tools and allow ourselves the joy (okay I know that sounds geeky) to make probabilistic statements about what the data tell us. The rest of this post will be all about using bayesian tools. Since I’m always a fan of making use of existing packages and code that do what I want them to do I did some exploration looking to see what was available. One of the first things I stumbled upon was a very nice function bayes.prop.test contained in a package called bayesian_first_aid, sadly but not surprisingly there was no CRAN version and development appears to have stopped circa 2015. Not surprising because there has been a lot more code published for Bayesian methods since then and I know how much work it takes to keep up a package. There are others I’ll demonstrate a little later but the appeal of this function is it’s simplicity. It’s a great place to start the code was easy to follow so I just grabbed the framework and updated it for my own needs. The Github code is here. It’s fiendishly easy to set-up and run. We’ll go back to our original dat object and divide it up by column. Column 1 those who got COVID by condition “Placebo” and “Vaccine” and column 2 those who didn’t. To keep this from being tl;dr I’m not going to go into too much detail on jags and runjags. If you need a good tutorial consult a reference like DBDA. I’ll only highlight the key points. For purposes of this post I’m going to use a flat, uninformed prior in all cases (this time the specific line is theta[i] ~ dbeta(1, 1)). One of the nice aspects of bayesian inference is you can express your prior thinking/knowledge mathematically, then let the data inform your thinking. Given the vaccine has already been involved in significant phase I and II trials it wouldn’t be unusual to have a prior that expressed at least a little confidence that it had some effect. But we’ll pretend we know nothing and that any outcome is equally likely. We’re back to flipping coins again. We’re going to explore the probabilities that the percentage of positive cases among those who received the vaccine is the same as the percentage of positive cases among those who received the vaccine. The model is constructed so that we could conceivably have more than two possibilities (e.g. easy to extend to a case where we had two different vaccines + a placebo or two different doses of the same vaccine plus placebo). That is very likely to be modeled later or when more data are in. We’ll “monitor” results for the infection rate placebo and vaccine (theta[1] and theta[2] respectively). As well as the raw predictions of how many people (x_pred[1] and x_pred[2]), not strictly necessary but fun to watch. This is a simple model with just four data elements we need to enter but we’ll still run it in 4 chains spread across 4 cores. If you’re following along I’m suppressing some messages about not choosing different random seeds per chain (run_jags will pick some), and that we used the same initial value for all chains (also not a worry). You should also run plot(my_results) to check the diagnostics for convergence and auto-correlation. I did, they’re fine but to save screen real estate I won’t include them. # Setting up the data dat ## COVID No COVID ## Placebo 34 14966 ## Vaccine 19 14981 got_covid <- dat[,1] not_covid <- dat[,2] got_covid ## Placebo Vaccine ## 34 19 not_covid ## Placebo Vaccine ## 14966 14981 # The model string written in the JAGS language model_string <- "model { for(i in 1:length(got_covid)) { got_covid[i] ~ dbinom(theta[i], not_covid[i]) theta[i] ~ dbeta(1, 1) x_pred[i] ~ dbinom(theta[i], not_covid[i]) } }" my_results <- runjags::run.jags(model_string, sample = 10000, n.chains=4, method="parallel", monitor = c("theta", "x_pred"), data = list(got_covid = got_covid, not_covid = not_covid)) ## Calling 4 simulations using the parallel method... ## Following the progress of chain 1 (the program will wait for all chains ## to finish before continuing): ## Welcome to JAGS 4.3.0 on Wed Nov 4 15:34:07 2020 ## JAGS is free software and comes with ABSOLUTELY NO WARRANTY ## Loading module: basemod: ok ## Loading module: bugs: ok ## . . Reading data file data.txt ## . Compiling model graph ## Resolving undeclared variables ## Allocating nodes ## Graph information: ## Observed stochastic nodes: 2 ## Unobserved stochastic nodes: 4 ## Total graph size: 9 ## . Reading parameter file inits1.txt ## . Initializing model ## . Adapting 1000 ## -------------------------------------------------| 1000 ## ++++++++++++++++++++++++++++++++++++++++++++++++++ 100% ## Adaptation successful ## . Updating 4000 ## -------------------------------------------------| 4000 ## ************************************************** 100% ## . . . Updating 10000 ## -------------------------------------------------| 10000 ## ************************************************** 100% ## . . . . Updating 0 ## . Deleting model ## . ## All chains have finished ## Simulation complete. Reading coda files... ## Coda files loaded successfully ## Calculating summary statistics... ## Calculating the Gelman-Rubin statistic for 4 variables.... ## Finished running the simulation my_results ## ## JAGS model summary statistics from 40000 samples (chains = 4; adapt+burnin = 5000): ## ## Lower95 Median Upper95 Mean SD Mode MCerr ## theta[1] 0.0015847 0.0023125 0.0031218 0.0023357 0.00039503 -- 2.5904e-06 ## theta[2] 0.00077049 0.0013122 0.001919 0.0013345 0.00029786 -- 1.9645e-06 ## x_pred[1] 18 34 50 34.949 8.3756 33 0.048986 ## x_pred[2] 8 20 32 20.036 6.3042 19 0.037185 ## ## MC%ofSD SSeff AC.10 psrf ## theta[1] 0.7 23255 -0.0023991 1.0001 ## theta[2] 0.7 22988 0.0022903 1.0001 ## x_pred[1] 0.6 29234 -0.0046137 1.0001 ## x_pred[2] 0.6 28743 0.0047744 1.0001 ## ## Total time taken: 1.3 seconds # plot(my_results) ## diagnostics were checked. Okay that all looks quite complex, what do we do now? Let’s first investigate the things we care most about. Remember theta[1] & theta[2] represent our infection rates theta[1] = 0.0023357 is our rate among those who got the placebo and theta[2] = 0.0013345 is our rate among those who got the vaccine. Those are the mean values and the medians are similar. The x_pred values are estimates of the case counts. We can use tidybayes::tidy_draws to extract the results of our 40,000 chains and pipe it through select to get the columns we want with the names we’d like. As much as I like Greek $$\theta$$ gets old after awhile. At the same time we can compute via a mutate statement what we really want to know which is the % difference in infection rates which we’ll put in a column called diff_rate. Now when we pass this cleaned up data to bayestestR::describe_posterior(results1) we get back a table that is a little easier to read. Focus on just the line for diff_rate. results1 <- tidybayes::tidy_draws(my_results) %>% select(placebo_rate = theta[1], vaccine_rate = theta[2], placebo_cases = x_pred[1], vaccine_cases = x_pred[2]) %>% mutate(diff_rate = (placebo_rate - vaccine_rate) / placebo_rate * 100) glimpse(results1) ## Rows: 40,000 ## Columns: 5 ##$ placebo_rate   0.00319585, 0.00300531, 0.00293815, 0.00228591, 0.00238…
## $vaccine_rate 0.00108651, 0.00131015, 0.00134464, 0.00137949, 0.00112… ##$ placebo_cases  52, 47, 41, 36, 28, 21, 37, 34, 31, 34, 48, 37, 40, 33,…
## $vaccine_cases 9, 20, 23, 22, 11, 28, 21, 13, 20, 22, 27, 24, 15, 30, … ##$ diff_rate      66.002472, 56.405496, 54.235148, 39.652480, 52.911123, …
bayestestR::describe_posterior(results1, ci = 0.95)
## # Description of Posterior Distributions
##
## Parameter     | Median |           95% CI |      pd |        89% ROPE | % in ROPE
## ---------------------------------------------------------------------------------
## placebo_rate  |  0.002 | [ 0.002,  0.003] | 100.00% | [-0.100, 0.100] |       100
## vaccine_rate  |  0.001 | [ 0.001,  0.002] | 100.00% | [-0.100, 0.100] |       100
## placebo_cases | 34.000 | [20.000, 52.000] | 100.00% | [-0.100, 0.100] |         0
## vaccine_cases | 20.000 | [10.000, 34.000] | 100.00% | [-0.100, 0.100] |         0
## diff_rate     | 43.316 | [ 7.715, 71.077] |  97.97% | [-0.100, 0.100] |         0

Given our data and 40,000 samples and assuming we had zero prior knowledge or
estimation of effectiveness, then we have a median estimate of the % difference
in infection rates = 43.316. That’s about what
we would expect given our earlier investigation. The columns we really want
to use are the “95% CI” and “pd” columns for diff_rate. CI in a bayesian framework
is credible interval not confidence interval. Since 95% of our chains wind up in
that interval we can say given our data there’s a 95% probability that diff_rate
lies in its range. No, they are not the same thing as a confidence interval.
The Probability of Direction (pd) is an index of effect existence, ranging from
50% to 100%, representing the certainty with which an effect goes in a
particular direction (i.e., is positive or negative). We can be very confident
that the vaccine does have a positive effect.

Plotting the distribution or range of diff_rate may also help the reader
“see” the results. With the data we have (and remember I have been using
a 19/34 split) you can see that while there’s a chance that the vaccine has
no effect the evidence (the data) supports the notion that it does.

plot(bayestestR::hdi(results1\$diff_rate,
ci = c(.89, .95, .99)
)
)

#### Done

I’ve decided to make this a two parter. Please hang in there more bayesian
“magic” follows soon. Probably tomorrow.

me know if you actually use the tools and find them useful.

Extra credit for me for not expressing a political view at any point. Let the data
speak.

Chuck


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# X-Ray Reflectivity Study of the Surface of Liquid Gallium

 Title: X-Ray Reflectivity Study of the Surface of Liquid Gallium Author: Kawamoto, E. H.; Lee, S.; Pershan, Peter S.; Deutsch, M.; Maskil, N.; Ocko, B. M. Note: Order does not necessarily reflect citation order of authors. Citation: Kawamoto, E. H., S. Lee, Peter S. Pershan, M. Deutsch, N. Maskil, and B. M. Ocko. 1993. X-ray reflectivity study of the surface of liquid gallium. Physical Review B 47(11): 6847-6850. Full Text & Related Files: Kawamoto_X-ray.pdf (775.3Kb; PDF) Abstract: X-ray reflectivity from the surface of liquid gallium was measured under ultrahigh vacuum conditions using a novel technique for curved surfaces. The small deviations between the measured and theoretical Fresnel reflectivity for an ideally sharp flat interface for wave-vector transfer $$\lesssim$$0.5 Å$$^{-1}$$ imply an interfacial width for the electron density profile of $$\lesssim$$1.3$$\pm$$0.2 Å. This is consistent with a model of atomic close packing which lacks structure along the surface normal at length scales >10 Å. Published Version: doi:10.1103/PhysRevB.47.6847 Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:10357542 Downloads of this work:


text:
# recursively enumerable of Godel numbering

There are statements for natural number x, like followings

m: "x is even natural number"

n: "x+1 is odd number and x>1"

l: "x is positive integer multiple of two"

m, n, l has same boolean value for x. If m(x) is true, n(x), l(x) is also true. m(x) is false, n(x), l(x) is also false.

Let's define this statements are "equivalent for x." Hence n, l is equivalent for x with m.

Another example

q: "x is natural number and 3.5 < x < 10"

p: "x is integer and 3 < x < 19/2"

We can say "q is equivalent for x with p"

Now, let set M is "all Gödel numbers of the statements that are equivalent for x with statement m"

Set M is infinite set of natural numbers that is dependent for m.

My question: For given m (a statement of x), set M is recursively enumerable?

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@HoCheolSHIN: If your $m$ is the Gödel number of a formula $\mu(x)$, and $\psi$ is any formula with no free variables, then the Gödel number of $\phi(x)\equiv (\mu(x)\leftrightarrow \psi)$ is equivalent to $\mu(x)$ (and should therefore be emitted by your enumeration of $M$) exactly when $\psi$ is a true statement. Therefore, if you had an enumeration of $M$, then you could convert it into an enumeration of all true statements simply filtering for outputs of the form $\mu(x)\leftrightarrow \psi$. – Henning Makholm Mar 8 '13 at 22:22