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# Jekyll With Multi-Collection Blog

When we host our blogs on Github, I guess most of us use Jekyll. Before I realize I need a different collection of blogs, I always start with a single collection or category as everybody does. However, I have to face a problem that to make multiple categories of blog into different URLs under the same domain. For example,

http://jilongliao.com/technology

http://jilongliao.com/journey

I searched the web for a little while and figure out this could be easily solved, but also a little challenging when handle some tricky cases with Github policies (we will get there later). Jekyll does a very good job for you to make your blog multi-categories. Simply put a new directory under your Jekyll site root directory would work. Inside that directory it needs to contains an index.html file and _posts folder. The _posts folder is the place you can put all your post belonging to this category in the future. After that, you will have a directory structure like the following.

This would make your site’s URL functioning well, however, it lacks of the ability to filter your blogs. To address this issue, I use the category keyword as the collection separator inside the markdown file. So you might want to add the category information in each of your blog. For instance,

To filter out the different category, you can write whatever ruby code you feel comfortable. I found Adam’s code is very clean and well designed, which I could borrow for this purpose in my site. For convenience purpose, I put the code as bellow. You save this code to your _plugins folder.

Use this Jekyll plugin is very simple, because it makes all the blog a dictionary which indexed by the category. As an example, the code is blow.

After you finish this, you can test it perfectly locally. However when you publish the site to Github, the site will fail to build because the Github does not allow customized plugin in their build system. So we have to hack this a little.

An interesting compromise Github provides is that you can push your _site directly to the master branch where you can play any plugins as you want. So instead of maintain the code in the master branch, I switch it to source branch but use master branch to push _site. This is a little complicated because you need to manipulate the repository between remote and local back-and-forth. So I prepare a bash script that could easily do this job.

so you run this script in bash shell should do the job automatically. Of course, you site will be down for a little while, but it should be back in a few seconds.


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Efficient way of vcf files handling
1
0
Entering edit mode
7.9 years ago
Earendil ▴ 50

Hi all

I have a .tgz file in which there are folders, one for each individual. Each folder contains vcf files, one for each chromosome. It's in this fashion: .tgz has ./individual1/chr1.vcf ./individual1/chr2.vcf etc, ./individual2/chr1.vcf ./individual2/chr2.vcf etc.

I want to create vcf files, one for each chromosome, containing individuals of a population using bcftools merge, so that later I can use the vcf files for an Fst estimation among populations using vcftools.

Having no previous experience for this kind of work, the only thing I can think is to extract the .tgz file, compress each vcf with bgzip and index it with tabix, run bcftools merge for each chromosome and population and finally run the Fst test on vcftools.

But is there a better way to do this? Extracting the .tgz file will chew up a lot of storage space.

vcf bcftools genome vcftools • 2.8k views
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Entering edit mode
7.9 years ago

It's a not-widely-appreciated feature of tar that you can just untar the files you want; so that, for instance, you could do this by chromosome if disk space is the concern:

#!/bin/bash
mkdir -p merged
for chrom in X Y $( seq 1 22 ) do tar xzvf vcfs.tgz individual{1,2,3,4,5}/chr${chrom}.vcf
for individual in individual*
do
cd $individual bgzip chr${chrom}.vcf
tabix -p vcf chr${chrom}.vcf.gz cd .. done bcftools merge -m all --force-samples -O z individual*/chr${chrom}.vcf.gz  > merged/chr${chrom}.vcf.gz rm -f individual*/chr${chrom}.*
done

0
Entering edit mode

I don't see the "tar's under-appreciated feature" part. Really good use of a bash loop and the bash 3.0 {} syntax, but all bash commands can "just work on the files one wants".

1
Entering edit mode

It's been my experience lately that a number of people - including many who are quite proficient in the linux shell environment - don't realize you can just pull out the individual files you want out of a tar file, rather than untarring the whole thing. That may indeed be already familiar to you - I assumed everyone knew that myself until recently.

0
Entering edit mode

Apologies - I see what you're highlighting now. I'd misread OP's question. It is indeed an underused feature, because we rarely have need for that today. In fact, it was only when I had to pick a script from a large project archive on my HPC that I discovered this nifty little feature :-)

0
Entering edit mode

Thank you very much for your answer! I am just a starter on the linux shell environment so I need to explore various ways to do things. I have been experimenting with scripts based on your answer and the storage problem is solved.

To try to optimize it a bit more, would there be a way to make the process faster? With brace and parameter expansion, the tar xzvf command runs anew each time for every different individual/chromosome combination, so it needs to sweep the whole tar file each time. This makes things slow; is there any possible way to avoid it? I could leave bcftools merge aside for the time being if that is a problem.

Thanks again

format: plain


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# Help with Text book answer

1. May 14, 2005

### powp

Hello

I am doing this problem in my text book and I am not sure what is happing in this one step.

$$2y = 360 \sqrt{\pi} - \sqrt{\pi}y$$

Trying to solve for y and this is what they show as the next step

$$(2 + \sqrt{\pi})y = 360 \sqrt{\pi}$$

Where does this $$(2 + \sqrt{\pi})$$ come from?? where did the other y go and the $$-\sqrt{\pi}$$???

2. May 14, 2005

### Nylex

The $$(2 + \sqrt{\pi})y$$ came from adding $$\sqrt{\pi}y$$ to both sides and then factoring out a y from the left hand side. Have you not learnt about collecting terms??

$$2y = 360 \sqrt{\pi} - \sqrt{\pi}y$$

$$2y + \sqrt{\pi}y = 360\sqrt{\pi} + \sqrt{\pi}y - \sqrt{\pi}y$$

$$(2y + \sqrt{\pi})y = 360\sqrt{\pi}$$

Last edited: May 14, 2005
3. May 14, 2005

### Jameson

They added $$\sqrt{\pi}y$$ to both sides.

Now on the left side of the equation we get $$2y + \sqrt{\pi}y$$

and on the right we get $$360\sqrt{\pi}$$

Now on the left side, they factored out the y so you can solve for it.

$$2y + \sqrt{\pi}y = y(2 + \sqrt{\pi})$$

Now putting all of this info together we get

$$(2 + \sqrt{\pi})y = 360 \sqrt{\pi}$$

and $$y = \frac{(360 \sqrt{\pi})}{(2 + \sqrt{\pi})}$$

Jameson

Last edited: May 14, 2005


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Javier Múgica de Rivera javier at digi21.net
Wed Oct 25 18:15:52 CEST 2006

Taco Hoekwater escribió:
>
> Hi Javier,
>
> Let me start by saying this: If your font is set up correctly for your
> language, not all characters really 'have to' be active. Inputenc.sty
> makes them active so that it can support many encodings, but there is
> no real need for that.
>
> If (for instance) é is in slot 233 of your input encoding, and you use
> a font encoding that has it in slot 233 as well, then it can simply be
> a 'letter', and you could happily use it in a control sequence.
Yes, I know that, but it is very likeky, specialy in some non-latin
scripts, that your input encoding does not match the font encoding. You
may argue that you can remap fonts, but be realistic, how many people
actually now how to do that (and besides, it is still much simpler if
you let the inputenc/fontenc packages do the thing).
> In the future, even with a different font encoding, the need for all
> these characters to be active will disappear completely because future
> versions of pdfTeX will have a separation between characters and font
> glyphs, with a tunable remapping stage inbetween.
That is really good news.

Cheers,
Javier A.


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# Thread: Variance calc with weighted coefficients

1. ## Variance calc with weighted coefficients

I want to understand the calculation of the variance of a sum of rvs where each variable is weighted using a constant coefficient. I post the full question and answer here and I highlight the areas that I am struggling to follow.

Question.

$Y_1,... Y_n$ is a sample from a population with mean $\mu$ and variance $\sigma$.

The sample is not random, $Cov(Y_i,Y_j)={\rho}\sigma^2$ for $i\neq{j}$. Let $U=\Sigma_{i=1}^na_iY_i$.

(a) Give the condition on a constants $a_1,...a_n$ for U to be unbiased estimator of $\mu$.
(b) Under this condition, calculate MSE(U).

(a) is not a problem, just need to calculate a if $E(U)=\mu$ which gives

$E(U)=\Sigma_{i=1}^na_iE(Y_i)=\mu\Sigma_{i=1}^na_i$ and $\Sigma_{i=1}^na_i=1$ is the condition.

(b) given (a) is met, then MSE(U)=Var(U):

$Var(U)=Var(\Sigma_{i=1}^na_iY_i)=E[\Sigma_{i=1}^na_i(Y_i-\mu)\Sigma_{j=1}^na_j(Y_j-\mu)]$ (1)

and starting from (2) I find it tricky to follow the solution:

$=\Sigma_{i=1}^n\Sigma_{j=1}^na_ia_jCov(Y_i, Y_j)$ (2)

$=\Sigma_{i=1}^na_i^2Var(Y_i)+\Sigma_{j=1}^n\Sigma_ {i\neq{j}}Cov(Y_i, Y_j)$ (3)

It looks to me like a sum of all entries of the variance matrix of an nx1 vector... but I cannot yet move beyond that.

Then it gets OK again, a straightforward substitution

$=\sigma^2\Sigma_{i=1}^na_i^2+\rho\sigma^2\Sigma_{j =1}^n\Sigma_{i\neq{j}a_ia_j}$ (4).

I'd appreciate a bit of clarification on manipulations in (2) and (3)...

2. I'm confused as to the covariance of Y's being $\sigma^2$
that means the correlation is 1.
The variance is just $\sigma^2\sum_{i=1}^n\sum_{j=1}^n a_ia_j$

3. Originally Posted by matheagle
I'm confused as to the covariance of Y's being $\sigma^2$
that means the correlation is 1.
I misspelt rho as ro! Latext didn't recognise it. Sorry! I edited. It's $\rho\sigma^2$

4. I've just realised I messed up the final answer too, it should read:

$\sigma^2\Sigma_{i=1}^na_i^2+\rho\sigma^2\Sigma_{j\ neq{i}}a_ia_j$

Anyways, I had a bright idea to look up "variance of a sum of correlated variables" and I've figure it out

$Var(\Sigma_{i=1}^na_iY_i)=\Sigma_{i=1}^n\Sigma_{j= 1}^nCov(a_iY_i,a_jY_j)$ (per Wiki),

which also includes $Var(a_iY_i)=Cov(a_iY_i,a_iY_i)$ when i=j:

$=\Sigma_{i=1}^nVar(a_iY_i)+\Sigma_{i=1}^n\Sigma_{j \neq{i}}Cov(a_iY_i,a_iY_i)$

$=\Sigma_{i=1}^na_i^2Var(Y_i)+\Sigma_{i=1}^n\Sigma_ {j\neq{i}}a_ia_jCov(Y_i,Y_i)$ (we can take constants outside)

$=\Sigma_{i=1}^na_i^2\sigma^2+\Sigma_{i=1}^n\Sigma_ {j\neq{i}}a_ia_j\rho\sigma^2$ (substibute info given in the question)

Finally, taking common factors outside the summation sign

$=\sigma^2\Sigma_{i=1}^na_i^2+\rho\sigma^2\Sigma_{i =1}^n\Sigma_{j\neq{i}}a_ia_j$ as per the final book answer.

Strange though that I didn't use the fact that from (a) that $\Sigma_{i=1}^na_i=1$

5. Originally Posted by Volga
Strange though that I didn't use the fact that from (a) that $\Sigma_{i=1}^na_i=1$
If $\sum a_i \ne 1$ then you would have to tack on the square of the bias, so you are using it implicitly when you jump straight into calculating the variance.


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Is there a motivic Cauchy integral formula?

Let $R$ be a complete dvr with fraction field $K$ and residue field $k$, and let $X, Y$ be two smooth projective $R$-schemes with isomorphic generic fibers.

Is it true that $[X_k]=[Y_k]$ in $K_0(\text{Var}_k)$?

Recall that $K_0(\text{Var}_k)$ is the so-called Grothendieck ring of varieties, namely, the free Abelian group on $k$-varieties, modulo the relation $[X]=[Y]+[X\setminus Y]$ for $Y\subset X$ closed.

That's the short version; let me say why I would call this a Cauchy integral formula and say what I know.

First of all, why is this a Cauchy integral formula? In this analogy, I'm thinking of a morphism $X\to Y$ as a function $Y(T)\to K_0(\text{Var}_T)$, sending a point $y$ to $X_y$. You're supposed to think of $[X_K]$ as a function $\text{Spec}(K)$ (a puntured disk), and the question asks if we can recover the value at the central point $\text{Spec}(k)$ from this data. The complex-analytic analogue is exactly the Cauchy integral formula.

Let me give a slightly non-trival example. A trivial family of Hirzebruch surfaces $X_n$ may degenerate to either $X_n$ or $X_{n+2}$. But of course $[X_n]=[X_{n+2}]=(\mathbb{L}+1)^2$.

Some other examples: if $X_K, Y_K$ are isomorphic as polarized varieties and one of them is not ruled, all is good by Matsusaka-Mumford. In particular, if $X, Y$ are canonically polarized, the answer to my question is affirmative.

Likewise suppose $X_K, Y_K$ have trivial canonical bundle. Then $X_k, Y_k$ have trivial canonical bundle as well (at least if $K$ has characteristic zero) and are birationally equivalent, by spreading out the isomorphism $X_K\simeq Y_K$. But birational Calabi-Yau's have the same class in (some mild localization of) $K_0(\text{Var}_k)$, by Kontsevich's motivic integration results.

A slightly better version of the question is:

Let $X, Y$ be smooth projective $R$-schemes with $[X_K]=[Y_K]$. Does this imply $[X_k]=[Y_k]$?

This would give some kind of specialization map $K_0(\text{Var}_K)\to K_0(\text{Var}_k)$, at least if $\text{char}(K)=0$.

ADDED: The result is also true for curves and surfaces. For curves this is easy; for surfaces, observe that $X_k, Y_k$ are birationally equivalent. Thus there is a surface $Z$ obtained by blowing up $X_k, Y_k$ at some number of points, so $$[X_k]+n\mathbb{L}=[Z]=[Y_k]+m\mathbb{L}.$$ But $n=m$ because $X_k, Y_k$ have equal Euler characteristic, so $[X_k]=[Y_k]$.

• I suspect this is not true. Begin with $F$ a finite field, $n$ and $d$ positive integers, and $X_F$, $Y_F$ two degree $d$, smooth hypersurfaces in $\mathbb{P}^n_F$ such that $X_F$ has an $F$-point, yet $Y_F$ has no $F$-point. Let $[X_F]$ and $[Y_F]$ be the corresponding $F$-points in the projective space $\mathbb{P}^M_F$ parameterizing degree $d$ hypersurfaces in $\mathbb{P}^n$. Let $\nu_X:P_X\to \mathbb{P}^M_F$, resp. $\nu_Y:P_Y\to \mathbb{P}^M_F$ be the blowing up at $[X_F]$, resp. at $[Y_F]$. (continued) – Jason Starr Jun 4 '15 at 0:42
• Let $A_X$, resp. $A_Y$ be the local ring of $P_X$, resp. $P_Y$, at the generic point of the exceptional divisor. Of course $A_X$ is isomorphic to $A_Y$. Moreover, the fraction fields of $A_X$ and $A_Y$ are both $F(\mathbb{P}^M_F)$. The pullbacks over $\text{Spec}(A_X)$, resp. $\text{Spec}(A_Y)$ of the universal family $Z$ in $\mathbb{P}^M_F\times_F \mathbb{P}^n_F$ have isomorphic generic fibers. Thus, let $R$ be the completion of these two DVRs, and let $X_R$ and $Y_R$ be the pullback of the universal family $Z$ in $\mathbb{P}^n_R$. (continued) – Jason Starr Jun 4 '15 at 0:45
• The generic fibers of $X_R$ and $Y_R$ are both the base change of the generic fiber of $Z$. Yet the closed fiber $X_k$ has a $k$-point, whereas the closed fiber $Y_k$ has no $k$-point. If I correctly recall the Lang-Nishimura Theorem, I believe existence of rational points should distinguish elements in the Grothendieck ring of varieties. – Jason Starr Jun 4 '15 at 0:49
• CORRECTION. I realize now that the generic fibers $X_K$ and $Y_K$ in my example are not $K$-isomorphic. Rather, there is a an automorphism $\sigma$ of $K$, and $X_K$ is $K$-isomorphic to $\sigma^*Y_K$. If $d>n+1$, then degree $d$ hypersurfaces in $\mathbb{P}^n_R$ have ample canonical bundle, so if the generic fibers are $K$-isomorphic, then also the special fibers are $K$-isomorphic. – Jason Starr Jun 4 '15 at 3:41
• You might find this paper helpful:arxiv.org/pdf/math/0107134.pdf – Wille Liou Jun 4 '15 at 23:51

I believe the answer is yes in the residue characteristic zero case. I will show it for a mild localisation of the Grothendieck ring (inverting $[L]$ and $[L]-1$). This may also follow from work of Denef and Loeser on the motivic nearby finer.

The real meat of the argument will be the weak factorisation theorem.

First, by standard algebraization arguments + removal of singularities, we can reduce to the following problem:

Let $X$ and $Y$ be smooth varieties both mapping to a curve $C$, with the fibrations isomorphic away from a point $x$ in $C$, and both smooth over a neighbourhood of $x$. Show that $[X_x]=[Y_x]$.

To solve this, consider the involution of $K^0(Var)[L^{-1}]$ that sends $[X]$ to $[X] /[L]^{d}$ for $[X]$ a smooth projective variety of dimension $d$. We can check that this is well-defined using the blow-up relations of Franziska Bittner/Heinloth: For $Y \to X$ a blow-up with centre $Z$ of codimension $c$ and exceptional divisor $Z$, $$[X] -[L]^c [Z]= [Y]- [L][E]$$ follows from $$[X]-[Z]=[Y]-[E]$$ and $$([L]^c-1)[Z]) = ([L]-1)[Y],$$ both obvious.

Now applying the involution to the known relation $$[X]-[X_x]=[Y]-[Y_x],$$ we obtain $$[X]-[L][X_x]=[Y]-[L][Y_x].$$ Subtracting, we obtain $$\left([L]-1\right) X_x = \left([L]-1\right) [Y_x]$$ and we win by dividing by $[L]-1$.

I claim the division by $[L]$ and $[L]-1$ in this argument may be removed if desired, to obtain an identity in the unlocalised Grothendieck ring. When working with varieties of dimension $\leq d$, rather than dividing $[X]$ by $[L]^{\dim X}$ we may multiply by $[L]^{d-\dim X}$ to achieve the same effect. This removes the need to adjoin $[L]^{-1}$. Furthermore, rather than working with the involution directly, observe that it only changes a class by a multiple of $[L]-1$ and work with this change divide by $[L]-1$, which is well-defined because we can remove an extra factor of $[L]-1$ from our argument for the relation. This removes the need to adjoin $([L]-1)^{-1}$.

One cool thing about this argument is it tells you something also in the case where $X_x$ is singular - something like "the integral over $X_x$ of the motivic nearby fiber of Denef and Loeser depends only on the $X_\eta$".

• Great! I think this works in the equicharacteristic, algebraically closed residue field case. (Thinking through the "standard algebraization arguments" you allude to, I think you need a field extension.) This is indeed very nice. It would be good to work through exactly how the application of Bittner uses weak factorization etc. here, i.e. what is this argument saying about equidecomposability of the two special fibers? – Daniel Litt Oct 15 '16 at 15:53
• @DanielLitt I had an earlier, more complicated version of the argument that made these issues more clear. Basically you can get from $X$ to $Y$ by a series of blow-ups and blow-downs during which $X$ and $Y$ remain smooth and $X_x$ and $Y_x$ are simple normal crossings divisors (this is one version of weak factorization). You can check that the invariant of a normal crossing divisor which is the sum of the Grothendieck classes of the strata, with the codimension $c$ strata weighted by $(1-[L])^c$, is preserved by blowup, so it is the same for $X_x$ and $Y_x$. – Will Sawin Oct 15 '16 at 19:54
• Great! This gives a new proof of Kontsevich's theorem that birat'l CYs have the same class in the Grothendieck ring, in the hyperkahler case, I think. – Daniel Litt Oct 16 '16 at 17:25
• It might be worth writing out the details of the specialization map you describe and seeing if one can do anything else with it... – Daniel Litt Oct 16 '16 at 17:27


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# Chi-Square StatisticHow to Calculate It /.

The Chi Square Test Excel Function will calculate the distribution of Chi square in Excel. The distribution is commonly used for studying the variation in percentage across samples. In financial analysis, the function can be useful in finding out the variations in an individual’s assumptions. The chi-square distribution also called the chi-squared distribution is a special case of the gamma distribution; A chi square distribution with n degrees of freedom is equal to a gamma distribution with a = n / 2 and b = 0.5 or β = 2. Let’s say you have a random sample taken from a normal distribution. With the pasted range still selected, use one of the following procedures, as appropriate for the version of Excel that you are running: In Microsoft Office Excel 2007, click the Home tab, click Format in the Cells group, and then click AutoFit Column Widths. In Excel 2003, point to Column on the Format menu, and then click AutoFit Selection.

The chi-square test gives an indication of whether the value of the chi-square distribution, for independent sets of data, is likely to have occurred by chance alone. Function Description The Excel CHISQ.TEST function performs the chi-square test on two supplied data sets of observed and expected frequencies, and returns the probability that the differences between the sets are simply due to. Chi-square Distribution Table d.f.995.99.975.95.9.1.05.025.01 1 0.00 0.00 0.00 0.00 0.02 2.71 3.84 5.02 6.63 2 0.01 0.02 0.05 0.10 0.21 4.61 5.99 7.38 9.21.

In probability theory and statistics, the chi distribution is a continuous probability distribution. It is the distribution of the positive square root of the sum of squares of a set of independent random variables each following a standard normal distribution, or equivalently, the distribution of the Euclidean distance of the random variables. Statistical tables: values of the Chi-squared distribution. Normal distributions are bell shaped that's why they are sometimes called bell curves, and have a symmetrical distribution with one single peak. Computing normal distribution curves is a time consuming process by hand. However, with Excel 2007, you can make an Excel chart of the normal distribution in minutes.

## Chi-square Distribution Table - University of Queensland.

24/02/2019 · Did you know that you can use pivot tables to easily create a frequency distribution in Excel? You can also use the Analysis Toolpak to create a histogram. Remember, our data set consists of 213 records and 6 fields. Order ID, Product, Category, Amount, Date and Country. First, insert a. Breve rassegna delle principali alcune funzioni definite in EXCEL e OpenOffice-Calc per l'analisi statistica dei dati. Il nome delle funzioni cambia a seconda della lingua con cui si è istallato Office OpenOffice. L'elenco dei nomi delle funzioni con la traduzione in Italiano e Inglese si trova qui. Density, distribution function, quantile function and random generation for the chi-squared \\chi^2\ distribution with df degrees of freedom and optional non-centrality parameter ncp. 11/06/2019 · The chi-square distribution allows you to test hypotheses about a population variance. In this video, learn how to use Excel to carry out this test.is now LinkedIn Learning! To accesscourses again, please join LinkedIn Learning. Advanced Excel Formulas You Must Know Advanced Excel Formulas Must Know These advanced Excel formulas are critical to know and will take your financial analysis skills to the next level. Advanced Excel functions you must know. Learn the top 10 Excel formulas every world-class financial analyst uses on a regular basis.

Interactive Graph of the Poisson Distribution in Excel 2010 and Excel 2013 Overview of the Chi- Square Distribution. The Chi-Square distribution is a family of continuous probability distributions. This is evidenced by the smooth shape of the above graph of a Chi-Square distribution’s PDF Probability Density Function curve. How to test for goodness of fit using Excel chi-square. These tests can also be used to check whether observed data fit a certain distribution.

illustrate the use of Excel’s Data Table function and standard formulas to perform the Chi-square test for independence. The Chi-square test is an integral part in introductory statistics. Simulation using Excel is used to generate many random samples and calculate the p-value of the test. The empirical distribution of the statistic is also. You can see a comparison of our results with those given by Excel. Other statistical calculators. See our Binomial distribution calculator which calculates a table of the binomial distribution for given parameters and displays graphs of the distribution function, fx, and cumulative distribution.

12/04/2019 · Normal distribution graph in excel is a continuous probability function. It is a common method to find the distribution of data. A formula has been found in excel to find a normal distribution which is categorized under statistical functions. This is completely depending on the mean and standard. A central chi-squared distribution with n degrees of freedom is the same as a Gamma distribution with shape a = n/2 and scale s = 2. Hence, see dgamma for the Gamma distribution. The central chi-squared distribution with 2 d.f. is identical to the exponential distribution with rate 1/2: χ^2_2 = Exp1/2, see dexp.

Nella teoria delle probabilità la distribuzione chi quadrato o chi-quadro è la distribuzione di probabilità della somma dei quadrati di variabili aleatorie normali indipendenti. In statistica viene particolarmente utilizzata per l'omonimo test di verifica d'ipotesi test χ 2. The chi-square distribution is used in many cases for the critical regions for hypothesis tests and in determining confidence intervals. Two common examples are the chi-square test for independence in an RxC contingency table and the chi-square test to determine if the standard deviation of a population is equal to a pre-specified value. 09/12/2019 · When you’re working with small samples in Excel — less than 30 or 40 items — you can use what’s called a student t-value to calculate probabilities rather than the usual z-value, which is what you work with in the case of normal distributions. Excel provides six t-distribution functions. T.DIST: Left-tail Student t. In probability theory and statistics, the noncentral chi-square distribution or noncentral chi-squared distribution, noncentral distribution is a generalization of the chi-square distribution. It often arises in the power analysis of statistical tests in which the null distribution is perhaps asymptotically a chi-square distribution; important examples of such tests are the likelihood.

1 Il termine “chi-quadro” si usa per indicare contemporaneamente la distribuzione di probabilita, una parti-colare tecnica di inferenza statistica e un indice statistico. Alcuni autori usano il simbolo χ2 chi greca minuscola per indicare la distribuzione di probabilita e X2 chi greca maiuscola. distribution: The degrees of freedom of a t-distribution are the number of squares of independent normal random variables that go into making up the chi-squared distribution occurring under the radical in the denominator of the t random variable Z U k. To see what a t-distribution looks like, we can use the four standard normal.

Basic directions on how to use Google Spreadsheet and Excel to calculate and display Chi-Square results for bivariate proportional data. \n The Chi-Square Distribution using. Below is a graph of the Chi distribution for the above example. \n. Note that our value of 12.99 gives us the probability value that we calculated using the. How To Fit Distributions in Excel. Excel has become the de facto standard application for data analysis and presentation across a variety of industries, so if you deal with random data of any kind, chances are your data is stored in Excel workbooks.


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# Tag Info

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Call the number of days until expiration the "freshness" of the cookie. A cookie with freshness $1$ is here (and can be eaten) today, but will be gone tomorrow; a cookie with freshness $n+1$ will have freshness $n$ tomorrow. If all the cookies' freshnesses have the same parity, then this will persist for the entire game: one player (Eve) will only see ...

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Methods Starting from the base url http://www.cross-tables.com/annotated.php?a=1 I used a combination of Python's urllib, multiprocessing and BeautifulSoup to extract the first 10000 games. The games were parsed and turned into numpy 15x15 Boolean matrices. The matrices were then turned into graphs making an edge if two adjacent cells in the matrix were ...

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In 3-dimensional chess, it is possible to force checkmate starting with a finite number of rooks. As this fact still appears to be open, I'll post a method of forcing checkmate with 96 rooks, even though it should be clear that this is not optimal. You can remove some of the rooks from the method I'll give below, but I am aiming for a simple explanation of ...

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An optimum board just before the last number is played on 4x4: $$\begin{array} {|c|c|c|c|} \hline & 256 & 512 & 65536 \\ \hline 4 & 128 & 1024 & 32768 \\ \hline 8 & 64 & 2048 & 16384 \\ \hline 16 & 32 & 4096 & 8192 \\ \hline \end{array}$$ ... and then a 4 luckily falls into the empty square giving the ...

15

Update: The code below has a small, but significant oversight. I was unaware that a stalemate would not be counted the same way as a board with insufficient pieces to play and this changes the answer. @Winther has fixed the bug and reran the simulations. That said, there is still value to the code being posted so I'll leave it up for anyone else to repeat ...

13

EDIT: I found a bug in the code given in Hooked's answer (which means that my original reanalysis was also flawed): one also have to check for insufficient material when assessing a draw, i.e. int(board.is_stalemate()) should be replaced with int(board.is_insufficient_material() or board.is_stalemate()) This changes things quite a bit. The probabillity ...

13

The point is that in the game Hex, it never hurts to have an extra piece on the board. So, suppose there is a strategy for the second player, but you are stuck with being the first player. What should you do? Well, you can place a stone on the board, and then pretend in your own mind that it isn't there! In other words, you are imagining that the other ...

10

There are several standard texts. You may want to start with a short essay, "The History of Combinatorial Game Theory", by Richard J. Nowakowski. The paper lists the main texts you may want to consult. For a more exhaustive list of references, see "Combinatorial games: Selected bibliography with a succinct GourmetIntroduction", by Aviezri Fraenkel. Thomas ...

9

$X$ takes $4$ $Y$ picks some amount: $c$ $X$ then takes $6-c$ $Y$ has no option but to take the last one.

8

This is known as Wythoff's game. The losing positions are $(\lfloor n \varphi\rfloor, \lfloor n \varphi^2\rfloor)$ where $\varphi$ is the golden ratio. An interesting way of presenting this game is to have the players move a Queen on a chessboard, which you can find at cut-the-knot here.

8

It turns out that the maximum possible diameter uses the full hundred tiles! My friend Carl illustrated this in 2008 in a game he made up, here: http://www.cross-tables.com/annotated.php?u=2493#0# . Thanks for the question!

8

There is some discussion of this topic at the very end of the second edition (2000) On Numbers and Games by Conway. He describes work by himself, Simon Norton, and Martin Kruskal to define integration. According to the description, it looked good for a while, producing workable logarithm function in terms of the integral of $x^{-1}$, but then got stuck, and ...

8

This is a most intriguing problem and took many, many, many thinking hours to solve, I even wrote a script to better understand it. It is possible for the sinners (players) to win every time. The number of players playing is irrelevant and the number of numbers they choose per turn is irrelevant as long as the players get to choose more numbers than the ...

7

In short, because — as emphasized by the last phrase of your bolded passage — having extra pieces on the board in Tic-Tac-Toe is never bad. It's not just 'having more squares' vs. 'having fewer squares'; it's that if position A is exactly position B with an extra X on it, then position A is always at least as good for player X as position B is. ...

7

Here is an example of a game where the second player (B) has a winning strategy. It is more comfortable to describe it using multisets, this should lead to no confusion. Our base set has $3\times a_i$ and $6\times b_i$ for $i=0,1,2$, thus a total of $27$ elements. The winning sets are (with indices mod $3$) the following. ($3\times a_i);\ (3\times a_i,\ ... 7 Fun question. I don't have enough karma to add this as a comment, so I'll offer it up as an answer, although it is (currently) an incomplete one. Some comments The reference you link to for the$N=3$case does not say that there are$255,168$valid 3-by-3 tic-tac-toe boards, but that there are$255,168$distinct 3-by-3 tic-tac-toe games. That is, if ... 7 Here's one option of a type you didn't suggest. Take any symmetric game. Modify the game as follows: After the first player's first move, give the second player the option to continue the game as usual, or to switch places, taking the first player's position and giving the (former) first player the next move as second player. Assuming that ties are ... 6 The Sprague-Grundy theory should kill this problem pretty dead; it solves any game of a large class that includes this game. Without getting into the theoretical details of the theory: You can tabulate the "value" of each position, where the "value" is a non-negative integer A position with no legal moves has value 0. A position with legal moves to one or ... 6 This PDF might be a useful starting point. It also mentions a book Fair Game: How to Play Impartial Combinatorial Games, by Richard K. Guy. I’ve not seen it, but it was published by COMAP; the combination of author and publisher suggests that it might be well worth looking into. (There is of course the massive Winning Ways, by Berlekamp, Conway, & Guy, ... 6 You can do it generically. Treat the case of the multiplier separately. So there's 11C5 (eleven choose 5) ways with no multiplier. You want the average. It's just 5 times the average of those eleven numbers. Then there's 11C4 ways with the multiplier. Let the multiplier be k. The average for the other 4 will be 4 times the average of the eleven. So, if A's ... 6 For a$6 \times 3$board, it's impossible. Consider the board below, and the squares, marked with an$\times$. $$\begin{matrix} \cdot & \cdot & * & \cdot & \cdot & \cdot \\ \times & \cdot & \cdot & \cdot & \times & \cdot \\ \cdot & \cdot & * & \cdot & \cdot & \cdot \\ \end{matrix}$$ The only ... 6 The minimum number of switches required is$100$. This puzzle can be turned into a linear algebra problem over${\Bbb Z}/2{\Bbb Z}$by letting the vector$v\in({\Bbb Z}/2{\Bbb Z})^{100}$record which switches are pressed ($1$if pressed,$0$if not), the vector$w\in({\Bbb Z}/2{\Bbb Z})^{100}$record which lights are initially on ($1$if on,$0$if off), ... 6 Astoundingly enough, this has already been studied. And I'm almost embarrassed to say that I'm familiar with the result. I used to freecell a lot. And FYI, 11982 is the impossible Frecell game. But I recommend entering in games -1, -2, -3, etc too. So here are some stats from some studies of freecell. Firstly, the depth of the aces, i.e. how many cards ... 6 Each of the 16 pawns can move at most 6 times and there are 30 captures possible. Therefore$(16\cdot6+30)\cdot 50=6300$is a rough upper bound (for example, not all pawns can make it to the opposite line without sometimes capturing - which would mean that sometimes pawn move and capture occur together). 6 You seem to be inferring some unintended patterns in the truncated list as it was given. The LHS is simply supposed to be a sample of dyadic fractions which fall short of pi, chosen so that the supremum of the sample is precisely pi, and similarly the RHS a sample of dyadic fractions strictly greater than pi chosen so that the infimum is pi. 6 So, to interpret your question, you have two different procedures for filling out a table of numbers which has a definite top and left edge, but continues indefinitely down and to the right. First is Don't fill in any cell before all the positions directly above and directly to the left of it are filled. Write in each cell the smallest nonnegative ... 5 In nim (in fact, in any finite combinatorial game), the following facts are true: (1) From a losing position, all moves lead to a winning position. (2) From a winning position, there is at least one move to a losing position. (The terms "winning position" and "losing position" are to be interpreted from the point of view of the person whose turn it is to ... 5 Decision version of Minesweeper is NP-complete, because it is possible to encode logic gates on the minefield (reference). So one should not expect any nice criteria that determine solvability of a given field. 5 Every ordinal number$\alpha$has a canonical copy as a surreal number$\hat\alpha$, whose left set has as members$\hat\beta$for every$\beta\lt\alpha$, and whose right set is empty. Succinctly, $$\hat\alpha=\{\ \{\hat\beta\mathrel{:} \beta\lt\alpha\ \}\mathrel{|} \}.$$ One can prove by induction that$\alpha\mapsto\hat\alpha\$ is isomorphism of the ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
### Prompt Cards

These two group activities use mathematical reasoning - one is numerical, one geometric.

### Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

### Exploring Wild & Wonderful Number Patterns

EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.

# Two Primes Make One Square

## Two Primes Make One Square

Try with the squares of the numbers between $4$ and $20$.

Did you find any square numbers which cannot be made by adding two prime numbers together?

### Why do this problem?

This problem brings together two important classes of numbers: primes and squares. By working with them in an investigative way, learners will become more familiar with their properties and also discover some interesting number facts. This investigation will encourage children to be persistent as they work at mathematics and it gives you an opportunity to highlight different methods of approach.

### Possible approach

As a starter, ask the class to find all the different $3$-digit numbers they can make from the digits $1$, $3$ and $5$, then to talk in pairs about which are prime. This should bring up some divisibility rules and remind pupils of the properties of prime numbers, so that in fact they will conclude that none of them are prime. You could also use the interactive teaching programme number grid (developed as part of the National Numeracy Strategy in England) to discuss prime numbers further by, for example, shading all the numbers that are multiples of $2$, $3$, $5$ etc.

Pose the question and if necessary discuss square numbers. You may like to ask pupils to try a few on mini-whiteboards as a start to check that they understand the task, and then leave them to explore the challenge. It would be worth saying that you are not expecting learners to find solutions straight away - they may need to persevere!

A mini-plenary after $10$ minutes or so could focus on different ways of approaching this task. Some children might make lists of the square numbers first and work that way, others might start with pairs of prime numbers. It would be worth sharing these different strategies and drawing attention to any useful systems that the pupils have developed, for example starting with the smallest square number and working upwards.

As they work, you could invite children to write up their sums on the board or on a wall for everyone to see. This would give a focus for a plenary as you can talk about the square numbers that haven't been made. Are they impossible to make or is it just that no-one has found a way yet? Why?

### Key questions

What are the squares of the numbers from $4$ to $20$?
What happens when you add two prime numbers together?
What happens if you add two different prime numbers together?
Why can't you make that square number?

### Possible extension

Some children might like to explore all the different ways that each square number can be made. How do they know they have them all?

### Possible support

Many pupils will find that calculators, number squares and multiplication squares will be useful tools. You could encourage children to make a list of the first six square numbers and the primes up to $36$.


text:
The 34th International Symposium on Lattice Field Theory (Lattice 2016)

Jul 24 – 30, 2016
Highfield Campus, University of Southampton
Europe/London timezone

Isovector Axial Charge with Current Improvement

Jul 26, 2016, 8:00 PM
1h
Highfield Campus, University of Southampton

Highfield Campus, University of Southampton

Highfield Campus, Southampton SO17 1BJ, UK

Speaker

Dr YIBO Yang (university of Kentucky)

Description

We employ dimension 4 operators to improve the local vector and axial vector currents, as the leading order approximation of the lattice conserved current, and then calculate the nucleon iso-vector axial coupling $g^3_A$ using overlap valence on Domain Wall Fermion sea. Using the equality of $g^3_A$ from $A_i$ and $A_4$ components of the axial-vector current as a normalization condition in addition to axial Ward identity, we find two to three percent increase of $g^3_A$ towards the experimental value. The excited state contamination has been taken into account with three time separations between the source and sink. The improved axial charges $g_A^{IM}(24I)=1.188(7)$, $g_A^{IM}(32I)=1.177(9)$ are obtained on $24^3\times 64$ and $32^3\times 64$ lattices at the unitary point where the pion masses are $330$ MeV and $300$ MeV respectively.

Primary author

Dr Jian Liang (University of Kentucky)

Co-authors

Prof. Andrei Alexandru (The George Washington University) Prof. Keh-Fei Liu (University of Kentucky) Raza Sufian (University of Kentucky, United States) Prof. Terrence Draper (University of Kentucky) Dr YIBO Yang (university of Kentucky)

Presentation materials

There are no materials yet.


text:
## Syllabus for IIIT Hyderabad  DDEE  is given below.

1. General Aptitude : Logical reasoning, graphic intelligence, language analysis

2. Mathematics syllabus

Unit I. Relations and functions

1. Relations and Functions:
Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, inverse of a function. Binary operations.

Unit-II: Algebra

1. Matrices:
Concept, notation, order, equality, types of matrices, zero matrix,transpose of a matrix, symmetric and skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simpleproperties of addition, multiplication and scalar multiplication. Non-commutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).

2. Determinants:
Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, cofactors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.

Unit-III: Calculus

1. Continuity and Differentiability:
Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit function.Concept of exponential and logarithmic functions and their derivative. Logarithmic differentiation. Derivative of functions expressed in parametric forms. Second order derivatives. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretations.

2. Applications of Derivatives:
Applications of derivatives: rate of change, increasing/decreasing functions, tangents & normals, approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).

3. Integrals:
Integration as inverse process of differentiation. Integration of a variaty of functions by substitution, by partial fractions and by parts, only simple integrals of the type to be evaluated. Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

4. Applications of the Integrals:
Applications in finding the area under simple curves, especially lines, areas of circles/ parabolas/ellipses (in standard form only), area between the two above said curves (the region should be clearly identifiable).

5. Differential Equations:
Definition, order and degree, general and particular solutions of a differential equation. Formation of differential equation whose general solution is given. Solution of differential equations by method of separation of variables, homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type: dy/dx+ py = q, where p and q are functions of x.

Unit-IV: Vectors and  Three -Dimensional Geomatry

1. Vectors:
Vectors and scalars, magnitude and direction of a vector. Direction cosines/ratios of vectors. Types of vectors (equal, unit, zero, arallel and collinear vectors), position vector of a point, negativeof a vector, components of a vector, addition of vectors, ultiplication of a vector by a scalar, position vector of a point ividing a line segment in a given ratio. Scalar (dot) product of vectors, projection of a vector on a line. Vector (cross) product of
vectors.

2. Three – Dimensional Geometry:
Direction cosines/ratios of a line joining two points. Cartesian and vector equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane. Angle between (i) two lines, (ii) two planes. (iii) a line and a plane. Distance of a point from a plane.

Unit-V: Linear Programming

1. Linear Programming:
Introduction, definition of related terminology such as constraints, objective function, optimization, different types of linear programming (L.P.) problems, mathematical formulation of L.P. problems, graphical method of solution for problems in two variables, feasible and infeasible regions, feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

Unit-VI: Probability

1. Probability:
Multiplication theorem on probability. Conditional probability, independent events, total probability, Baye’s theorem, Random variable and its probability distribution, mean and variance of random variable. Repeated independent (Bernoulli)trials and Binomial distribution.


text:
URI Online Judge | 2771

# Average

By Yuri Cardoso Santamarina, UFU Brazil

Timelimit: 1

Joãozinho is finishing his first semester in college and in one of the subjects he will decide his grade, based on the notes of the N tests he did during the semester. The final grade of a student is based on the arithmetic average of 3 different notes. But Joãozinho is not worried about what the highest final grade is, but what K-th highest final grade can be obtained. It is guaranteed that there is a K-th note and remember, it is possible to have more than one final note with the same value and they must be considered different.

## Input

The input contains no more than 10 test cases and ends with EOF.
The first line of a test case contains two integers N and K ($$3 \leq N \leq 100$$, $$1 \leq K \leq (N * (N - 1) * (N - 2))/6$$).
The second line contains N integers Ai ($$1 \leq Ai \leq 10^5$$) representing the grade obtained in the i-th test.

## Output

The output must contain a floating-point number formatted to one decimal place.

 Input Sample Output Sample 4 2 1 2 3 4 2.7


text:
Last updated: 25/05/2017

### 2016

• E. Kuhl, “Introduction to Neuromechanics.” 2016.
[Bibtex]
@Conference{workshop-kuhl2016a,
author = {Kuhl, E.},
title = {Introduction to Neuromechanics},
year = {2016},
eventdate = {2016-08-28},
venue = {Lehrstuhl f\"{u}r Technische Mechanik, Friedrich-Alexander Universit\"{a}t Erlangen-N\"{u}rnberg},
url = {http://biomechanics.stanford.edu/Intro_to_neuromechanics_16},
abstract = {Our brain is not only our softest, but also our least well- understood organ. Floating in the cerebrospinal fluid, embedded in the skull, it is almost perfectly isolated from its mechanical environment. Not surprisingly, most brain research focuses on the electrical rather than the mechanical characteristics of brain tissue. Recent studies suggest though, that the mechanical environment plays an important role in modulating brain function. Neuromechanics has traditionally focused on the extremely fast time scales associated with dynamic phenomena on the order of milliseconds. The prototype example is traumatic brain injury where extreme loading rates cause intracranial damage associated with a temporary or permanent loss of function. Neurodevelopment, on the contrary, falls into the slow time scales associated with quasi-static phenomena on the order of months. A typical example is cortical folding, where compressive forces between gray and white matter induce surface buckling. To understand the role of mechanics in neuroanatomy and neuromorphology, we begin this course by discussing the brains's anatomy and correlate our observations to neurophysiology. We discuss morphological abnormalities including lissencephaly and polymicrogyria and illustrate their morphological similarities with neurological disorders including schizophrenia and autism. Then, we address the role of mechanics during brachycephaly, plagiocephaly, tumor growth, and hydrocephalus. Last, we explore the mechanics of traumatic brain injury with special applications to shaken baby syndrome},
owner = {Jean-Paul Pelteret},
timestamp = {2017.05.15},
}
• P. Steinmann, D. Lupascu, and F. Endres, “2$^{\text{nd}}$ Seminar on Ferroic Functional Materials & 12$^{\text{th}}$ International Workshop on Direct and Inverse Problems in Piezoelectricity.” 2016.
[Bibtex]
@Conference{workshop-steinmann2016a,
author = {Steinmann, P. and Lupascu, D. and Endres, F.},
title = {2$^{\text{nd}}$ Seminar on Ferroic Functional Materials \& 12$^{\text{th}}$ International Workshop on Direct and Inverse Problems in Piezoelectricity},
year = {2016},
eventdate = {2016-10-04},
venue = {Lehrstuhl f\"{u}r Technische Mechanik, Friedrich-Alexander Universit\"{a}t Erlangen-N\"{u}rnberg},
url = {http://www.ltm.uni-erlangen.de/veranstaltungen/FERROICS2016/ferroics.shtml},
abstract = {The joint seminar is dedicated to the diverse fields of ferroic materials including ferroelectricity, ferromagnetism and combined multiferroic materials. It covers mathematical modeling, experimental methods and industrial applications of ferroic materials and devices.
Aim of the combined seminar is to contribute to open problems in:
- numerical simulation (direct and inverse analysis)
- discussion of fatigue, damage, cracks and reliability
- manufacturing processes and
- performance of piezoelectric devices
The overall intention of the workshop is to bring together experts from different fields of research such as engineering, applied mathematics, and materials science, as well as from the industrial sector.},
owner = {Jean-Paul Pelteret},
timestamp = {2017.05.15},
}

### 2015

• W. Bangerth and B. Turcksin, “Fifth deal.II Users and Developers Workshop.” 2015.
[Bibtex]
@Conference{workshop-bangerth2015a,
author = {Bangerth, W. and Turcksin, B.},
title = {Fifth deal.II Users and Developers Workshop},
year = {2015},
eventdate = {2015-08-03},
url = {http://dealii.org/workshop-2015/},
owner = {Jean-Paul Pelteret},
timestamp = {2017.05.15},
}
• D. Schubert and J. Kaschta, “TA – Anwendertreffen Rheologie.” 2015.
[Bibtex]
@Conference{workshop-schubert2015a,
author = {Schubert, D. and Kaschta, J.},
title = {TA - Anwendertreffen Rheologie},
year = {2015},
eventdate = {2015-10-01},
venue = {Lehrstuhl f\"{u}r Technische Mechanik, Friedrich-Alexander Universit\"{a}t Erlangen-N\"{u}rnberg},
url = {http://www.tainstruments.com/pdf/Erlangen_Programm2015.pdf},
owner = {Jean-Paul Pelteret},
timestamp = {2017.05.15},
}

### 2014

• J. Schröder and P. Wriggers, “CISM course: Advanced Finite Element Technologies.” 2014.
[Bibtex]
@Conference{workshop-schroeder2014a,
author = {Schr\"{o}der, J. and Wriggers, P.},
title = {CISM course: Advanced Finite Element Technologies},
year = {2014},
eventdate = {2014-10-06},
venue = {Udine, Italy},
url = {http://www.cism.it/courses/C1414/},
abstract = {Advanced nite element technolo- gies are essential for the solution of almost all problems in computa- tional mechanics. Many engineering and mathematical approaches of novel nite element methods were developed in recent years in order to improve the capabilities and the reliability of numerical simulations. One of the great attractions of the nite element method is its enor- mous range of applicability. Beside the classical application areas like mechanical, aerospace or civil engi- neering, also elds like e.g. physics, biology and medicine use the mani- fold possibilities of this method. New materials, technologies or produc- tion methods and the optimization of products and components make great demands on the quality and performance of numerical meth- ods for the solution of the arising, mainly nonlinear, problems. These demands led to many interesting developments in a wide eld of nite element methods, e.g. (adaptive) mixed and enhanced nite element methods as well as stabilized or dis- continuous Galerkin schemes. Their application often aims at improved or robust discretizations that are superior to standard nite element methods, which often do not pro- vide reliable solutions for nonlinear problems, and, sometimes, collapse completely.
The objective of the course is to present an overview of the state of research of advanced nite element technologies. The course will cover some of the most important applica- tion areas in modern element tech- nologies ranging from element de- velopment in engineering to math- ematical analysis. The audience is introduced into new methods and technologies under consideration of ef ciency, robustness and perfor- mance. Therefore, the challenging discretization methods will be devel- oped and fundamental mathemati- cal analysis for reliable simulations will be discussed. The special top- ics of the course are mathematical foundations for variational formula- tions, a deep mathematical under- standing of the analytical require- ments of modern FEM approaches for nite deformations and related adaptive strategies, incompress- ible, isotropic or anisotropic mate- rial behavior, the mathematical and numerical treatment of the well- known locking phenomenon and domains with oscillating coef cients. Furthermore, new results within the nite element development and discretization schemes for the ad- equate approximation of all process variables in nonlinear engineering applications under consideration of extreme cases will be discussed.
The course is intended for doctoral and postdoctoral researchers in civil and mechanical engineering, ap- plied mathematics and physics as well as industrial researchers, which are interested in the topic. After the course participants will have a ba- sic knowledge of advanced mixed Galerkin FEM, least-squares FEM, discontinuous Galerkin methods, mathematical analysis (e.g. con- vergence, ellipticity and coercivity, algebraic interpretation and modern adaptive strategies) as well as for- mulations and applications of these methods to nite (in-)elastic prob- lems. The course is a must for all that are interested and/or involved in advanced nite element tech- nologies.},
owner = {Jean-Paul Pelteret},
timestamp = {2017.05.15},
}

### 2013

• L. Dorfmann and R. Ogden, “CISM course: Nonlinear Mechanics of Soft Fibrous Materials.” 2013.
[Bibtex]
@Conference{workshop-dorfmann2013a,
author = {Dorfmann, L. and Ogden, R.},
title = {CISM course: Nonlinear Mechanics of Soft Fibrous Materials},
year = {2013},
eventdate = {2013-07-01},
venue = {Udine, Italy},
url = {http://www.cism.it/courses/C1307/},
abstract = {Soft brous materials are composites with highly exible laments embedded in a soft base matrix. They are found
in many biological tissues and high performance elastomeric polymers. The bers attribute the material with preferred directions thereby optimizing a set of mechanical properties. There are many applications where natural or synthetic bers are used to modify the mechanical response of the base material. Most broadly, biological materials are reinforced with some kind of proteins, such as collagen, silk, bronectin, titin, and many others. Synthetic bers are most often long chemically cross-linked elastomeric laments, which in the unstretched state occupy a random coiled con guration. When stretched, the filaments extend and orient along the loading directions resulting
in a decrease of entropy. Depending on processing, the macromolecular orientation
can be altered resulting in high stiffness and high strength bers. These are used as reinforcements for tires, ropes, cables, ballistic protections used by law enforcement agencies. Biological and engineered reinforced materials are therefore characterized by one or two families of bers creating preferred directions
in the mechanical response. Mathematical models and numerical simulations are a prerequisite in the development, testing and evaluation of the mechanical properties and
play a fundamental role in the analysis of problems that arise in such materials.
The purpose of the course is
to present a state-of-the-art overview of the fundamental theories, established models and ongoing research related to the modeling of these materials. Two approaches
are conventionally used to develop constitutive relations for highly deformable brous materials. According to the phenomenological approach, a strain energy density function can be de ned in terms of strain invariants. The other approach is based on kinetic theories, which treats a fibrous material as a randomly oriented inter-tangled network of long molecular chains bridged by permanent and temporary junctions. At the micro-level, these are associated with chemical crosslinks and active entanglements, respectively. The presentations will include carefully crafted overviews of the fundamental formulation
of the three-dimensional theory from several points of view, and will address their equivalences and differences. We will also include solutions to boundary-value problems which are amenable to experimental veri cation. A further aspect of the course will be the elasticity of laments, stability of equilibrium and thermodynamics of the molecular network theory.
The course is addressed to doctoral students and postdoctoral researchers in mechanical, civil and electrical engineering, materials science, applied physics and applied mathematics, academic and industrial researchers and practicing engineers.},
owner = {Jean-Paul Pelteret},
timestamp = {2017.05.15},
}
• J. Mergheim, “Workshop in Bavarian Academy of Sciences and Humanities: Modelling of self-healing materials.” 2013.
[Bibtex]
@Conference{workshop-mergheim2013a,
author = {Mergheim, J.},
title = {Workshop in Bavarian Academy of Sciences and Humanities: Modelling of self-healing materials},
year = {2013},
eventdate = {2013-09-20},
venue = {Munich, Germany},
url = {http://www.ltm.uni-erlangen.de/forschung/publikationen/jahresberichte/2013.pdf},
owner = {Jean-Paul Pelteret},
timestamp = {2017.05.16},
}

### 2012

• M. Kuttel, J. Gain, C. Laidler, P. Marais, and B. Merry, “Desktop high performance computing.” 2012.
[Bibtex]
@Conference{course-kuttel2012a,
author = {Kuttel, M. and Gain, J. and Laidler, C. and Marais, P. and Merry, B.},
title = {Desktop high performance computing},
year = {2012},
eventdate = {2012-04-23},
venue = {University of Cape Town, South Africa},
owner = {Jean-Paul Pelteret},
timestamp = {2017.05.15},
}


text:
# Error: package architecture (arm) does not match system (armhf)

I installed a beta version of a Debian package on my Raspberry Pi 3 and received the following error:

dpkg: error processing archive remote-syslog2_0.18_arm.deb (--install):
package architecture (arm) does not match system (armhf)
Errors were encountered while processing:
remote-syslog2_0.18_arm.deb

The suggested solution was to add the arm architecture as follows:

dpkg --add-architecture arm

and then re-run the install command:

dpkg -i remote-syslog2_0.18_arm.deb

I did so and the package installed and worked as expected. However, this made me wonder:

• What did the dpkg --add-architecture arm command do?
• What were the consequences of running that command?

After several hours of reading, experimenting and writing this blog post, I think I know. I’ll start by breaking the command down.

Note: The above solution was a work around while the package was in development. The production version is compiled for the armhf architecture.

## dpkg

dpkg is a command line tool for managing Debian packages. To install a package, the dpkg tool needs to recognize that package’s architecture. Supported architectures are in the /var/lib/dpkg/arch file. The following commands display information about the architecture(s) dpkg supports:

• dpkg --print-architecture will display the current machine architecture. There will always be one current machine architecture. It cannot be removed.
• dpkg --print-foreign-architectures will list any architectures that have been added to dpkg. Foreign architectures can be added and removed.
• dpkg -I package_name.deb will display information about a .deb archive. The architecture of a package will be listed in the ‘Architecture’ field.

In the above screenshot the dpkg --print-foreign-architectures command returns nothing because a foreign architecture has not yet been added.

Architecture in this scenario refers to an Application Binary Interface. There are two ways to install a package that does not match the current architecture:

• dpkg --add-architecture arch_name - adds the given architecture to the list of architectures dpkg will process. After running this command do an apt-get update to update the available package lists. If you get errors see the errors section below. Note: running the add-architecture command does not alter the computer’s architecture. It only tells the dpkg tool which architectures it will process.

• dpkg --force architecture -i package_name.deb - installs a Debian package with no architecture or an architecture that does not match the machine’s architecture. The foreign architecture is not added to the list of architectures that dpkg maintains.

## Application Binary Interface

arm is the name of the Debian ABI for the ARM architecture. ABI, or Application Binary Interface, is a low level interface. It defines how software interacts with each other on a given architecture. From Linux System Programming:

… an ABI defines the binary interface between two or more pieces of software on a particular architecture. It defines how an application interacts with itself, how an application interacts with the kernel, and how an application interacts with libraries. Whereas an API ensures source compatibility, an ABI ensures binary compatibility, guaranteeing that a piece of object code will function on any system with the same ABI, without requiring recompilation.

The ABI is intimately tied to the architecture; the vast majority of an ABI speaks of machine-specific concepts, such as particular registers or assembly instructions. Thus, each machine architecture has its own ABI on Linux. In fact, we tend to call a particular ABI by its machine name, such as Alpha, or x86-64. Thus, the ABI is a function of both the operating system (say, Linux) and the architecture (say, x86-64).

arm is the original Debian port of the arm-linux ABI. It is now considered obsolete and newer ports such as armel and armhf have replaced it. armel is the default Debain port for the ARM architecture. armhf is a port that uses the “hard” Floating Point Unit. This means a hardware unit, as opposed to software, performs floating point calculations.

## Remove architecture

First remove any packages that use the architecture. There are two options to remove the packages:

• apt-get remove ".*:arm" - This will remove all arm packages but will leave configuration files.
• apt-get purge ".*:arm" - This will remove all arm  packages plus their configuration files.

Next, use the dpkg --remove-architecture arch_name command to remove the architecture.

Note: the following error will occur if any packages of the given architecture exist:

dpkg: error: cannot remove architecture 'arm' currently in use by the database

The screenshot below shows the errors and results from removing the arm architecture:

After choosing ‘Y’ at the prompt, the package is removed. Now thearm architecture can be removed successfully:

## Errors from adding the new architecture

One of the questions I had was if there were any consequences from adding the arm architecture. The answer is ‘yes’. I received the following error when I attempted to update my Raspberry Pi using the sudo apt-get update command:

Error:
Hit http://archive.raspberrypi.org jessie InRelease
Hit http://mirrordirector.raspbian.org jessie InRelease
W: Failed to fetch http://mirrordirector.raspbian.org/raspbian/dists/jessie/InRelease  Unable to find expected entry 'main/binary-arm/Packages' in Release file (Wrong sources.list entry or malformed file)

W: Failed to fetch http://archive.raspberrypi.org/debian/dists/jessie/InRelease  Unable to find expected entry 'main/binary-arm/Packages' in Release file (Wrong sources.list entry or malformed file)

This error occurred because apt-get tried to get arm packages that do not exist. To resolve this I specified it to only get armhf packages. I did this by modifying the following two files:

• /etc/apt/sources.list.d/raspi.list
• /etc/apt/sources.list

In each file I added [arch=armhf] to the listing.

/etc/apt/sources.list.d/raspi.list

deb [ arch=armhf ] http://archive.raspberrypi.org/debian/ jessie main ui

/etc/apt/sources.list

deb [arch=armhf] http://mirrordirector.raspbian.org/raspbian/ jessie main contrib non-free rpi

In the following screenshot the first file in the diff command contains the modification I made and the second file is the original file.

I discovered one side effect from adding a new architecture, but are there any more? One possibility is the package may not work if it depends on specific architecture attributes. In my case the package worked, but others may not.

## Conclusion

When faced with installing a package that uses a different architecture:

1. First ask if the package can be compiled for the computer’s architecture. This is the preferred solution.
2. If compiling for your architecture is not possible then first try force architecture.
3. If that does not work then try add architecture.

Wow! I learned a lot! It is amazing what lies beneath one command.


text:
# Heat transfer between two surfaces

Suppose I have surface A in contact with surface B, if I apply Fourier's law of heat transfer, which $K$ should I use, $K_a$ or $K_b$?

Essentially asking whether the same block of material heats faster in 300 degree water or 300 degree air or the same.

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I'm going to add some mathematical detail to what akhmeteli has said.

Let's restrict the discussion to one dimension with coordinate $x$, then Fourier's law in differential form says $$q(x) = -k(x) T'(x)$$ where $q(x)$ is the local heat flux, $k(x)$ is the conductivity, and $T(x)$ is the temperature gradient. Notice that Fourier's law show that that a given point, the derivative of the temperature is important, but derivatives of a function depend on the value of that function in a neighborhood of that point, not just the value of the function at that particular point. Therefore, the (not entirely explicit) answer to your question is that you need both $k$'s at a point where two materials with different $k$ are in contact. Now let's see the math.

If you are looking at a point $x_0$ at which two materials with different conductivities are joined, (say $k_a$ corresponds to $x<x_0$ and $k_b$ corresponds to $x>x_0$, then $k(x)$ has a jump discontinuity that can be written with the use of the Heaviside step function $\theta(x)$; $$k(x) = (k_b-k_a)\theta(x-x_0) + k_a$$ Which results in the following differential equation: $$q(x) = -[(k_b-k_a)\theta(x-x_0) + k_a] T'(x)$$ Which you can attempt to solve in a given case. For example, let's consider a steady-state system in which $q(x) = q_0$ is a constant and for which we want to determine the temperature gradient. Let's suppose that this system consists of metal bars joined at the point $x_0$ and whose endpoints are located at $x_0-L$ and $x_0+L$ respectively. Additionally, we assume that these other two endpoints are kept at a temperature $T_0$ In this case, the differential equation we would want to solve for $T(x)$ is $$q_0 = -[(k_b-k_a)\theta(x-x_0) + k_a] T'(x)$$ with the boundary data $$T(x_0-L) = T_0,\qquad T(x_0+L) = T_0$$ The differential equation we want to solve can be rewritten as a set of two equations, one for $x<x_0$ and another for $x>x_0$; $$q_0 = -k_a T_a'(x), \qquad q_0 = -k_b T_b'(x)$$ The general solutions are $$T_a(x) = T_0-\frac{q_0}{k_a} [x-(x_0-L)], \qquad T_b(x) = T_0-\frac{q_0}{k_b} [x-(x_0+L)]$$ and the temperature everywhere except at $x=x_0$ can be written as $$T(x) = (T_b(x) - T_a(x))\theta(x-x_0) + T_a(x)$$ In particular, notice that there is a jump discontinuity in the temperature at $x=x_0$; $$T_b(x_0) -T_a(x_0) = q_0 L\left(\frac{1}{k_b}-\frac{1}{k_a}\right)$$ and this discontinuity depends on both $k$ values, not just the value on a particular side. Note further that if $k_a = k_b$, the the discontinuity disappears as you might intuitively expect!

Hope that helps! Let me know of any typos.

Cheers!

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Thank you a lot for the effort as well as quality. – user20518 Feb 4 '13 at 1:21
No problem @mezhang. I think this is a great question that should get more attention to be honest. It was also educational for me! – joshphysics Feb 4 '13 at 2:06
This answer seems wrong. First of, even if kb==ba your example results in a discontinuity, as the last equation has a sign error in it. More importantly, you're effectively solving both sides as independent problems, not guaranteeing that the law holds at x0. That is how you can solve the example problem in the frist place, even though it should be impossible: Both ends have the same temperature, and yet you assume that heat is transferred. This would require thermal super conductivity. – Stefan Jul 30 '15 at 22:32
@Stefan I agree. There must be an error. I haven't looked at this in a long time, so I'll need to re-read this answer when I have a bit of time to see what I did wrong. – joshphysics Jul 30 '15 at 22:54
No hurry, it took me 2+ years to point it out to you :) I'm pretty sure in an ideal setting T is continuous, but in real live the contact of two bodies is not perfect. There will be a poorly conductive third material (air, glue,...) partically inbetween, causing a steep temperature change microscopically, resulting in a macroscopic discontinuity. I think this is called "Thermal contact conductance". Wikipedia has a nice article on it. – Stefan Jul 31 '15 at 10:55

You should use the Fourier law in its differential form and use the actual temperature in each point. In your case, it is not important that temperature, strictly speaking, is not unambiguously defined at the contact surface (initially). You can still solve the equation of heat transfer (using distributions).

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