text:
# Unbreakable space smaller than ~

Using the cleveref package for reference ranges, I'd like to change the standard \crefrangeconjunction from _to~ to _-_, where the underscores represent spaces. Issuing

\newcommand{\crefrangeconjunction}{~-~}

I find the spaces are much too large. Defining instead

\newcommand{\crefrangeconjunction}{\,-\,}

is more compact and looks much better in my eyes but has the disadvantage of breaking at line ends.

Is there an unbreakable space smaller than ~?

• Related/duplicate: Thin unbreakable space. – <NAME> May 31 '17 at 12:25
• btw: most typographical guidelines would use n-dashes, not hyphens, for this purpose and would not have spaces (let alone full spaces like ~) around the dash. – Florian May 31 '17 at 12:33
• Related/duplicate: How to typeset a “small” non-breaking space – clemens May 31 '17 at 12:41
• @Florian I'm aware of that. In fact, the cleveref documentation itself says "it is perfectly reasonable to define it [\crefrangeconjunction] to be an endash “--”". Call me crazy, but I really can't stand the look of it, though, so I'll stick with a hyphen for now. :) – Casimir May 31 '17 at 13:00

It's - that's causing the break, not \,.

\documentclass{article}

\newcommand{\unbreakablethinspace}{\leavevmode\,}

\begin{document}

\parbox{0pt}{
a\unbreakablethinspace-\unbreakablethinspace b

a\unbreakablethinspace\mbox{-}\unbreakablethinspace b
}

\end{document}

Hiding - in a box disallows the possibility of breaking at it.

I defined \unbreakablethinspace because \, might give surprises if used at the beginning of a paragraph. You can simply use \, if you're sure it will always go after a word.


text:
Enroll in one of our FREE online STEM summer camps. Space is limited so join now!View Summer Courses

Finding an Indefinite Integral In Exercises 15- 3…

01:19
University of Houston

Need more help? Fill out this quick form to get professional live tutoring.

Get live tutoring
Problem 23

Finding an Indefinite Integral In Exercises 15- 36 , find the indefinite integral and check the result by differentiation.
$\int \frac{1}{x^{5}} d x$

$$\frac{-1}{4 x^{4}}+C$$

Discussion

You must be signed in to discuss.

Video Transcript

very funny. We're taking off the indefinite integral one over x. Yes. Then an accident angle, Alan. So we're gonna add one which becomes X to the negative force power. And they're super bowl of negative. Four is 14 So negative four times 14 wouldn't give us one. The corporation. Thanks. And then we have to add in our seats because any constant would have dropped away. If I want to write this without the negative and fun in negative one more X to the poor, let's see would be impulse trimming way of writing answer.

format: plain


text:
# Article

Full entry | PDF   (0.3 MB)
Keywords:
nonnegative matrices; M-matrices; determinants
Summary:
Suppose that $A$ is an $n\times n$ nonnegative matrix whose eigenvalues are $\lambda = \rho (A), \lambda _2,\ldots , \lambda _n$. Fiedler and others have shown that $\det (\lambda I - A) \le \lambda ^n - \rho ^n$, for all $\lambda > \rho$, with equality for any such $\lambda$ if and only if $A$ is the simple cycle matrix. Let $a_i$ be the signed sum of the determinants of the principal submatrices of $A$ of order $i\times i$, $i = 1,\ldots ,n - 1$. We use similar techniques to Fiedler to show that Fiedler’s inequality can be strengthened to: $\det (\lambda I - A) + \sum _{i = 1}^{n - 1} \rho ^{n - 2i}|a_i|(\lambda - \rho )^i \le \lambda ^n -\rho ^n$, for all $\lambda \ge \rho$. We use this inequality to derive the inequality that: $\prod _{2}^{n}(\rho - \lambda _i) \le \rho ^{n - 2}\sum _{i = 2}^{n}(\rho - \lambda _i)$. In the spirit of a celebrated conjecture due to Boyle-Handelman, this inequality inspires us to conjecture the following inequality on the nonzero eigenvalues of $A$: If $\lambda _1 = \rho (A),\lambda _2,\ldots , \lambda _k$ are (all) the nonzero eigenvalues of $A$, then $\prod _{2}^{k}(\rho - \lambda _i) \le \rho ^{k-2}\sum _{i = 2}^{k}(\rho -\lambda )$. We prove this conjecture for the case when the spectrum of $A$ is real.
References:
[1] S. Ambikkumar and S. <NAME>: Some remarks on a conjecture of Boyle and Handelman. Lin. Alg. Appl. 264 (1997), 63–99. DOI 10.1016/S0024-3795(96)00402-8 | MR 1465857
[2] J. Ashley.: On the Perron-Frobenius eigenvector for nonnegative integral matrices whose largest eigenvalue is integral. Lin. Alg. Appl. 94 (1987), 103–108. DOI 10.1016/0024-3795(87)90081-4 | MR 0902070 | Zbl 0622.15012
[3] <NAME> and R. <NAME>: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia, 1994. MR 1298430
[4] M. Boyle and D. Handelman: The spectra of nonnegative matrices via symbolic dynamics. Annals of Math. 133 (1991), 249–316. DOI 10.2307/2944339 | MR 1097240
[5] M. Fiedler: Untitled private communication. 1982.
[6] J. Keilson and G. Styan: Markov chains and M-matrices: Inequalities and equalities. J.  Math. Anal. Appl. 41 (1973), 439–459. DOI 10.1016/0022-247X(73)90219-9 | MR 0314873
[7] I. Koltracht, M. Neumann and D. Xiao: On a question of Boyle and Handelman concerning eigenvalues of nonnegative matrices. Lin. Multilin. Alg. 36 (1993), 125–140. DOI 10.1080/03081089308818282 | MR 1308915

Partner of


text:
Incrementally adding fastai goodness to your Ignite training

We're going to use the MNIST training code from Ignite's examples directory (as at August 2020), converted to a module.

from migrating_ignite import *

from fastai.vision.all import *

To use it in fastai, we first pull the DataLoaders from the module into a DataLoaders object:

data = DataLoaders(*get_data_loaders(64, 128)).cuda()

We can now create a Learner and fit:

opt_func = partial(SGD, momentum=0.5)
learn = Learner(data, Net(), loss_func=nn.NLLLoss(), opt_func=opt_func, metrics=accuracy)
learn.fit_one_cycle(1, 0.01)

epoch train_loss valid_loss accuracy time
0 0.999266 0.597913 0.856200 00:22

As you can see, migrating from Ignite allowed us to replace 52 lines of code (in run()) with just 3 lines, and doesn't require you to change any of your existing data pipelines, optimizers, loss functions, models, etc. Once you've made this change, you can then benefit from fastai's rich set of callbacks, transforms, visualizations, and so forth.

Note that fastai is very different from Ignite, in that it is much more than just a training loop (although we're only using the training loop in this example) - it is a complete framework including GPU-accelerated transformations, end-to-end inference, integrated applications for vision, text, tabular, and collaborative filtering, and so forth. You can use any part of the framework on its own, or combine them together, as described in the fastai paper.


text:
# How do you find an equation of a line containing the point (3, 2), and perpendicular to the line y - 2 = (2/3)x?

May 21, 2016

$y = - \frac{3}{2} x + \frac{13}{2}$

#### Explanation:

Given -

$y - 2 = \frac{2}{3} x$

Rewrite it -

$y = \frac{2}{3} x + 2$

Its Slope ${m}_{1} = \frac{2}{3}$

Two lines are perpendicular if -

${m}_{1} \times {m}_{2} = - 1$

Find the slope of the perpendicular line

$\frac{2}{3} \times {m}_{2} = - 1$
${m}_{2} = - 1 \times \frac{3}{2} = - \frac{3}{2}$

The perpendicular line is passing through the point $\left(3 , 2\right)$

Its equation is -

$y - {y}_{1} = m 2 \left(x - {x}_{1}\right)$
$y - 2 = - \frac{3}{2} \left(x - 3\right)$
$y = - \frac{3}{2} x + \frac{9}{2} + 2$
$y = - \frac{3}{2} x + \frac{13}{2}$


text:
Free Version
Moderate

# Multiple Math Methods in a Statement

APCSA-VWVKGG

What range of values could num possibly have after this statement is executed?

double num = Math.pow(Math.random()*10,Math.sqrt(Math.pow(2, 0))*2);
System.out.println(num);

A

Any decimal number between 0 (inclusive) and 100 (exclusive): [0, 100).

B

Any decimal number between 0 (inclusive) and 100 (inclusive): [0, 100].

C

Any decimal number between 0 (exclusive) and 100 (exclusive): (0, 100).

D

Any decimal number between 0 (inclusive) and 10 (exclusive): [0, 10).

E

Any decimal number between 0 (inclusive) and 10 (inclusive): [0, 10].


text:
# Teaching Kids Programming – Number of Common Factors (Brute Force Algorithm + Greatest Common Divisor)

Teaching Kids Programming – Number of Common Factors (Brute Force Algorithm + Greatest Common Divisor) | ninjasquad

Teaching Kids Programming: Videos on Data Structures and Algorithms

Given two positive integers a and b, return the number of common factors of a and b.

An integer x is a common factor of a and b if x divides both a and b.

Example 1:
Input: a = 12, b = 6
Output: 4
Explanation: The common factors of 12 and 6 are 1, 2, 3, 6.

Example 2:
Input: a = 25, b = 30
Output: 2
Explanation: The common factors of 25 and 30 are 1, 5.

Constraints:
1 <= a, b <= 1000

### Number of Common Factors (Bruteforce Algorithm + Greatest Common Divisor)

We can bruteforce the numbers from 2 (inclusive) to the GCD (Greatest Common Divisor) between two numbers.

 1 2 3 4 5 6 7 8 class Solution:     def commonFactors(self, a: int, b: int) -> int:         x = gcd(a, b)         ans = 1         for i in range(2, x + 1):             if x % i == 0:                 ans += 1         return ans
class Solution:
def commonFactors(self, a: int, b: int) -> int:
x = gcd(a, b)
ans = 1
for i in range(2, x + 1):
if x % i == 0:
ans += 1
return ans

And this can be written using one-liner:

 1 return sum(1 for i in range(1, x + 1) if x % i == 0)
return sum(1 for i in range(1, x + 1) if x % i == 0)

The GCD Algorithm (Teaching Kids Programming – Algorithms of Greatest Common Divisor and Least Common Multiples) of two integers:

 1 2 3 4 def gcd(a, b):   while b:     a, b = b, a %b   return a
def gcd(a, b):
while b:
a, b = b, a %b
return a

And it can be done Recursively:

 1 2 3 4 def gcd(a, b):   if b:     return gcd(b, a % b)   return a
def gcd(a, b):
if b:
return gcd(b, a % b)
return a

The time complexity of GCD algorithm is $tex_77988249062c78cf62a5f522b3ef49ef Teaching Kids Programming - Number of Common Factors (Brute Force Algorithm + Greatest Common Divisor) algorithms math python Recursion teaching kids programming youtube video$

GD Star Rating


text:
# Different seasonality patterns

I'm currently trying out several ARIMA models for time series data with seasonality.

The data consists of daily demand for two years. The data has strong weekly seasonality since Fridays and Saturdays have much higher values than remaining days.

The data presents also other types of seasonality since summer and Christmas have always higher values.

So, my data shows two types of seasonality. Is it possible to cope with both patterns in the same model?

$$ARIMA(p,d,q)(p,d,q)_7(p,d,q)_{30}$$

Does the model above makes sense?

• I'm not sure if seasonality of 12 makes sense unless you believe current day can be predicted based on data from 12 days before which is rather unusual. If you want to factor in monthly seasonality, you can simply dummy code the month of the year and pass it on as a regressors. – forecaster Nov 29 '15 at 21:02
• Can be done, in principle, but I do not know of canned software which implements it! You must do your own programming. – kjetil b halvorsen Nov 29 '15 at 21:04
• @forecaster - thanks for noticing. I've should have written 30 days lag and not 12 months lag. I will edit my post. – Eduardo Nov 29 '15 at 21:08
• Ok, 30 days will also be a problem because you have 28 days and 31 days in a month, you could use day of the month and month of the year dummies as a regression variables and capture seasonality. – forecaster Nov 29 '15 at 21:37
• @forecaster, using month of the year may not work since the OP says he only has two years of data. Thus he effectively has only two observations of the seasonal effect for each month. Eduardo, <NAME> has several useful blog posts about modelling seasonality. His blog is here, you may search for "seasonality" and "TBATS". – <NAME> Nov 30 '15 at 19:58


text:
# Number of five-letter words that use letters from the set $\{A, B, C, D, E\}$ and contain exactly three different letters.

I'd like to see if what I did below makes sense.

We are not allowed to have words like $ABCDE$ because the words must contain exactly three different letters, not at least three different letters. So, the possible permutations are $ABCCC, ABBCC, CDEEE, \ldots$

Now it's clear that our problem can be rewritten as

Find the number of five-letter words that use letters from the set $\{X, Y, Z\}$ where $X, Y, Z \in \{A, B, C, D, E\}$ at least once.

So, we have two sets:

Set $1$ (one of the letters repeated three times): $XYZZZ, XXXYZ, YYYXZ$

Set $2$ (two of the three letters repeated twice each): $XYYZZ, XXYZZ, XXYYZ$

Now the number of permutations of $XYZZZ$ is $\binom {5}{1, 1, 3} = \frac {5!}{1! \cdot 1! \cdot 3!} = 20.$ Since there are two more words in the set $1$, the number of all the possible permutations of this set is $3 \cdot \binom {5}{1, 1, 3} = 60.$

The number of permutations of $XYYZZ$ is $\binom {5}{1, 2, 2} = \frac{5!}{1! \cdot 2! \cdot 2!} = 30.$ Since there are two more permutations in the set $2$, the number of all the possible permutations of this set is $3 \cdot \binom {5}{1, 2, 2} = 90.$

Thus, the number of all the words with the given condition is $\binom 52 \left(3\binom {5}{1, 1, 3} + 3\binom {5}{1, 2, 2}\right) = 1500$ where the binomial coefficient is the number of places to put $X, Y, Z$

EDIT:

Here's the model problem:

Find the number of words of a given length from a given set of letters, if each letter must occur at least once in each word.

Solution: $T(m, n) = \sum_{m_i \ge 1} \binom{m}{m_1,\ldots, m_n}$ where $\sum_{i =1}^n m_i = m.$

The problem in my OP has an answer in the back of my book given as $\binom{5}{2}T(5, 3) = 1500.$

• From your examples it seems you want the letter to appear in alphabetic order in your word. Is that a constraint in the question ? Otherwise, of course, you also have to consider words like ZXZYX, which you currently seem to leave aside. – Evargalo May 16 '17 at 15:48
• @Evargalo, no, that's not necessary. – 4chan May 16 '17 at 15:50
• No, your analogy is incorrect. In the problem statement, we'd count ADEEE, well as ABCCC, as distinctly different, but valid for counting. (Note that Given the above examples would, we are using 5 letters, yet in each case, the string meets the original question, but not your alternative approach. – Namaste May 16 '17 at 15:50
• I don't get the downvote. Downvoting an incorrect approach is not a reason for downvoting. Alternatively, note the askers attempt, and effort. We should credit any question that shows effort on the part of the asker, but at the very least, not penalize them. – Namaste May 16 '17 at 15:56
• @N.Shales, yes exactly what I am saying. Thank you! – 4chan May 16 '17 at 17:19

You are getting the right answer, but your explanation of it doesn't sound quite right. I would approach it this way:

First, pick two letters not to use. This can be done in ${5\choose2}$ ways. Next, with the remaining three letters, consider the two cases:

1. Pick one letter to use three times, the other two being used just once. This can be done $3$ ways, and then the letters can be arrange in $5\choose1,1,3$ ways.

2. Pick one letter to use just once, with the other being being used twice each. Again, there are $3$ choices for the singlet, and then $5\choose1,2,2$ ways to arrange the letters.

The total number of words with three different letters is thus

$${5\choose2}\left(3{5\choose1,1,3}+3{5\choose1,2,2}\right)=10(3\cdot20+3\cdot30)=1500$$

• I see where I tripped up. Well, I am still glad I was able to get the main part somewhat correct :) Thank you! – 4chan May 16 '17 at 16:24

A solution by substraction:

• Number of five-letter words with a five-letters alphabet: 5^5

• Number of them with five different letters: 5!

• Number of them with four different letters: $5!*\binom{5}{2}$ (you gotta pick the letter that doesn't appear and the one that appears twice)
• Number of them with two different letters: $2^5*\binom{5}{2}-4*5$ (picking two letters * all the words with a 2-letters alphabet - 5 one-letter words counted 4 times each)
• Number of them with one single letter: 5

Hence the number of five-letter words with exactly three different letters: $5^5-5!-5!*\binom{5}{2}-(2^5*\binom{5}{2}-4*5)-5=1500$

• Good to know because I found out we can also use $n!S(k, n)$ to solve this problem. Thank you. – 4chan May 16 '17 at 16:31


text:
# If C is a fusion category over a field of nonzero characteristic and dim C = 0, is Z(C) ever fusion?

If $C$ is a fusion category and $\dim(C) \neq 0$ (the latter is automatic in characteristic zero, but not in nonzero characteristic), then the Drinfel'd center $Z(C)$ is fusion. More generally, if $C$ is a fusion category and $M$ is a semisimple module category over $C$, then the dual of $C$ over $M$ is fusion if $\dim(C)$ is nonzero. But if $\dim(C)=0$ these results need not hold. For example $\operatorname{Vec}(G)$ in characteristic $p$ when $p | \#G$ acts on $\operatorname{Vec}$ and its dual is $\operatorname{Rep}(G)$ which is not semisimple. Similarly, $Z(\operatorname{Vec}(G))$ is not semisimple when $p | \#G$.

I want to know if $C$ and $Z(C)$ both being semisimple implies that $\dim(C) \neq 0$.

(Feel free to assume everywhere that the base field is algebraically closed if you'd like to.)

-
This is far from an answer, but on page 3 of "From Subfactors to Categories and Topology II," Mueger suggests that "there is little hope of proving semisimplicity" of the center when the dimension is 0. Of course, this does nothing to rule out the possibly that the center might just happen to be semisimple in some anomalous cases. He also points out that in any case the center could never be modular, as modular categories necessarily have nonzero dimension. –  <NAME> Aug 11 '11 at 5:56
That means any counterexample would have to be nonspherical, right? Hence there couldn't be any known counterexamples. (Or maybe I'm missing something about what modularity means in nonzero characteristic.) Probably that can be bumped up to a proof by looking at the sphericalization (at least outside of characteristic 2). –  <NAME> Aug 11 '11 at 6:23
It appears so. In the paper "Categorical centers and Reshetikhin-Turaev invariants," Bruguieres and Virelizier show that the center of a spherical category is (not necessarily semisimple) modular even when the dimension is noninvertible, although it seems that some essential pieces of the proof are relegated to a paper that is still forthcoming. –  <NAME> Aug 11 '11 at 18:53
That said, your desired result is probably less hard than proving that all fusion categories are spherical. Drinfeld, Gelaki, Nikshych, and Ostrik have managed to prove quite a lot about not-necessarily-spherical categories (including that the center of a fusion category is nondegenerate, the nonspherical analogue of modular), albeit only over an algebraically closed field of characteristic zero. One might hope that combining the BV and DGNO viewpoints would give such a result for nonspherical categories with noninvertible dimensions, but I don't want to speculate too much about it. –  <NAME> Aug 11 '11 at 19:01
I think we've worked out that Z(C) fusion implies $\mathrm{dim} C \neq 0$. I will post an answer soon when the proof is posted. –  <NAME> Jun 8 '13 at 4:23


text:
# Integer valued polynomials and fast equdistribution in number fields

• <NAME>
• Tuesday 05 February 2019, 14:30-15:30
• MR13.

In his early work on integer valued polynomials Bhargava introduced the notion of a p-ordering. Let A be Dedekind domain with the fraction field k and let $\frac p$ be a prime ideal. Roughly speaking, a $\frac p$-ordering in A is a sequence that equidistributes modulo powers of $\frac p$ as fast as possible. Using $\frac p$-orderings Bhargava defined an analogue of the factorial function and constructed generating sets of the modules of degree n integer valued polynomials in $k[X]$. Of particular importance are the sequences which are $\frac p$-orderings for all primes $\frac p$ at the same time. We call them simultaneous p-orderings. Bhargava asked which Dedekind rings admit such sequences. For a long time the answer was not even known in the particular case of rings of integers of global fields. In a recent joint work with <NAME> we prove that the only number field k whose ring of integers O_k admits a simultaneous p-ordering is Q. The result follows from a stronger statement that puts an obstacle on the simultaneous equidistribution of finite subsets of O_k modulo all primes at the same time. Our proof relies on effective bounds on the number of solutions of certain degree 2 norm inequalities that we establish using Baker’s bounds theorem on linear forms in logarithms.

This talk is part of the Number Theory Seminar series.


text:
# Is there a linear relationship between x and y coordinates

### Year 8 Plotting linear relationships

The position of any point in the plane can be represented by an ordered pair of numbers (x, y). These ordered pairs are called the coordinates of the point. To find the y-intercept, we substitute 0 for x in the equation, because we know that every point on the y-axis has an x-coordinate of 0. Once we do that, we can. single solution of a linear equation in two variables can in the Euclidean plane with a Cartesian coordinate system.

And it also goes through the origin. And it makes sense that it goes through an origin. Because in a proportional relationship, actually when you look over here, zero over zero, that's indeterminate form, and then that gets a little bit strange, but when you look at this right over here, well if X is zero and you multiply it by some constant, Y is going to need to be zero as well.

So for any proportional relationship, if you're including when X equals zero, then Y would need to be equal to zero as well. And so if you were to plot its graph, it would be a line that goes through the origin. And so this is a proportional relationship and its graph is represented by a line that goes through the origin. Now let's look at this one over here, this one in blue. So let's think about whether it is proportional.

And we could do the same test, by calculating the ratio between Y and X. So it's going to be, let's see, for this first one it's going to be three over one, which is just three.

Then it's gonna be five over two. Five over two, well five over two is not the same thing as three. So already we know that this is not proportional. We don't even have to look at this third point right over here, where if we took the ratio between Y and X, it's negative one over negative one, which would just be one. Let's see, let's graph this just for fun, to see what it looks like. When X is one, Y is three. When X is two, Y is five. X is two, Y is five. And when X is negative one, Y is negative one.

• Proportional relationships: graphs

When X is negative one, Y is negative one. And I forgot to put the hash mark right there, it was right around there. And so if we said, okay, let's just give the benefit of the doubt that maybe these are three points from a line, because it looks like I can actually connect them with a line.

Then the line would look something like this. The line would look something like this.

## Linear equations in the coordinate plane

So notice, this is linear. This is a line right over here. But it does not go through the origin. So if you're just looking at a relationship visually, linear is good, but it needs to go through the origin as well for it to be proportional relationship. And you see that right here. This is a linear relationship, or at least these three pairs could be sampled from a linear relationship, but the graph does not go through the origin. And we see here, when we look at the ratio, that it was indeed not proportional.

So this is not proportional. Now let's look at this one over here. Let's look at what we have here. So I'll look at the ratios.

## Linear equation

So for this first pair, one over one, then we have four over two, well we immediately see that we are not proportional. And then nine over three, it would be three. So clearly this is not a constant number here. We don't always have the same value here, and so this is also not proportional. But let's graph it just for fun.

### Linear equation - Wikipedia

When X is one, Y is one. When X is two, Y is four. This actually looks like the graph of Y is equal to X squared. When you are dealing with data points plotted on a coordinate plane, a negative slope indicates a negative correlation and the steeper the slope, the stronger the negative correlation.

Consider working in your vegetable garden. If you have a flat of 18 pepper plants and you can plant 1 pepper plant per minute, the rate at which the flat empties out is fairly high, so the absolute value of m is a greater number and the line is steeper.

If you can only plant 1 pepper plant every 2 minutes, you still empty out the flat, but the rate at which you do so is lower, the absolute value of m is low, and the line is not as steep. Zero Slope When there is no change in y as x changes, the graph of the line is horizontal.

A horizontal line has a slope of zero. Undefined Slope When there is no change in x as y changes, the graph of the line is vertical. You could not compute the slope of this line, because you would need to divide by 0. These lines have undefined slope. Lines with the Same Slope Lines with the same slope are either the same line, or parallel lines. In all three of these lines, every 1-unit change in y is associated with a 1-unit change in x.

All three have a slope of 1. Solving Two-Step Linear Equations with Rational Numbers When a linear equation has two variables, as it usually does, it has an infinite number of solutions. Each solution is a pair of numbers x,y that make the equation true. Solving a linear equation usually means finding the value of y for a given value of x. To find ordered pairs of solutions for such an equation, choose a value for x, and compute to find the corresponding value for y.

Students may be asked to make tables of values for linear equations. These are simply T-tables with lists of values for x with the corresponding computed values for y. Two-step equations involve finding values for expressions that have more than one term. The terms in an expression are separated by addition or subtraction symbols.

To find a value for y given a value for x, substitute the value for x into the expression and compute. First, find the value of the term that contains x, then find the value of the entire expression.

This requires mirroring operations balancing on each side of the equation until y is by itself on the one side of the equation, set equal to an expression involving x. You can manipulate the equation in this way because of the equality properties: This equation is not in slope-intercept form. There are two ways to put it in slope-intercept form.


text:
Programming with Python¶

Exercises: Strings and generators¶

Problem 1. Write a function copy_rev(s) that takes a string s and returns a new string that consists of an original string s followed by its reverse copy.

For example, copy_rev("aBcd") should return "aBcddcBa".

Problem 2. Write functions

• copy_to_len_whole(s, l) that returns the maximum length string made of copies of s (only the whole s, no slicing) with the length at most l,
• copy_to_len_sliced(s, l) that returns the maximum length string made of copies of s with the length exactly l; if the length of s doesn't match, it ends with the beginning slice of s.

For example,

• copy_to_len_whole("12345", 17) should return "123451234512345" (length 15);
• copy_to_len_sliced("12345", 17) should return "12345123451234512" (length 17).

You may assume that s is not an empty string and that l >= 0.

Problem 3. Write functions

• fade_copies_left(s) that returns the string made by concatenating s with its left slices of decreasing sizes, i.e., with the s without the last character, then with s without the last two characters, etc;
• fade_copies_right(s) that returns the string made by concatenating s with its right slices of decreasing sizes, i.e., with the s without the first character, then with s without the first two characters, etc.

For example,

• fade_copies_left("abcde") should return "abcdeabcdabcaba";
• fade_copies_right("abcde") should return "abcdebcdecdedee".

Problem 4. We define words as sequences of non-space characters separated by a single space.

Write a function last_word(s) that returns the last word in s, i.e., the substring beginning after the last occurence of the space character " " in s all the way to the end. If the string s ends with a space, the function returns an empty string "". If the string contains no spaces, the whole string is its last word.

Further, write a main() function that loads an integer n and then n strings which it prints sorted by the last word.

Problem 5. We define words as sequences of non-space characters separated by a sequence of consecutive spaces (one or more of them).

Write a function second_word(s) that returns the second word in s. Words are mutually separated by one or more spaces. You may assume that s will not start with a space.

For example, in the string "This is a sentence." the second word is "is" even though there are 5 spaces before it.


text:
How should the diameter of the wire used be chosen so to minimize the heat dissipated in windings ? Check the details in question.

The turns of a solenoid is designed to provide a given magnetic flux density along its axis are wound to fill the space between 2 concentric cylinder of fixed radii.How should the diameter of the wire used be chosen so to minimize the heat dissipated in windings ?

Mar 7, 2018

This is what I get.

Explanation:

We are required to fill the cross sectional area between between two concentric cylinders of fixed radii with wire of diameter $d$.

$\implies$ The cross sectional area to be filled is fixed.
Cross section of wire $= \frac{\pi {d}^{2}}{4}$
$\implies$ Cross section of wire $\propto {d}^{2}$
$\therefore$Number of turns per unit length $n \propto {d}^{-} 2$ ........(1)

Resistance of one turn of wire $\propto {\text{Area of cross section of wire}}^{-} 1$
$\implies$ Resistance of per unit length of wire $R \propto n {d}^{-} 2$

$R \propto {d}^{-} 4$ .......(2)

We know that magnetic flux density produced $B = \mu I n$
$\implies$ For the fixed magnetic field $I \propto {n}^{-} 1$
From (1) above becomes

$I \propto {d}^{2}$ .....(3)

Heat dissipated in a resistance $= {I}^{2} R$
Using (2) and (3)
Heat dissipated in a resistance $\propto {\left({d}^{2}\right)}^{2} \left({d}^{-} 4\right)$

$\implies$Heat dissipated in a multilayered solenoid, as described above, is independent of diameter of wire.


text:
# 17 Distributions and models

So far, you have learnt to ask a RQ, identify different ways of obtaining data, design the study, collect the data describe the data, summarise data graphically and numerically, and understand the decision-making process.

In this chapter, you will learn about distributions and models to describe the distribution of populations and samples. You will learn to:

• describe distributions.
• describe populations using normal distributions.
• use $$z$$-scores to compute probabilities related to normal distributions.
• use $$z$$-scores to ‘work backwards’ from probabilities for normal distributions.


text:
Reference documentation for deal.II version Git 2276b94619 2020-02-28 10:11:55 +0100
SolutionTransfer< dim, VectorType, DoFHandlerType >::Pointerstruct Struct Reference

Detailed Description

template<int dim, typename VectorType = Vector<double>, typename DoFHandlerType = DoFHandler<dim>> struct SolutionTransfer< dim, VectorType, DoFHandlerType >::Pointerstruct

All cell data (the dof indices and the dof values) should be accessible from each cell. As each cell has got only one user_pointer, multiple pointers to the data need to be packetized in a structure. Note that in our case on each cell either the vector<unsigned int> indices (if the cell will be refined) or the vector<double> dof_values (if the children of this cell will be deleted) is needed, hence one user_pointer should be sufficient, but to allow some error checks and to preserve the user from making user errors the user_pointer will be 'multiplied' by this structure.

Definition at line 535 of file solution_transfer.h.

The documentation for this struct was generated from the following files:


text:
Objective of the Course:

Copyright law has evolved as a response to the change in the new technology and hence it is complex in character.


text:
# Token List¶

## 1inch (s1INCH)¶

Decimals: 18

Tracks the price of 1inch (1INCH) through price feeds supplied by an oracle.

## Aave (sAAVE)¶

Decimals: 18

Tracks the price of Aave (AAVE) through price feeds supplied by an oracle.

## Alphabet (sGOOG)¶

Decimals: 18

Tracks the price of Alphabet (GOOG) through price feeds supplied by an oracle.

## Amazon (sAMZN)¶

Decimals: 18

Tracks the price of Amazon (AMZN) through price feeds supplied by an oracle.

## Apple (sAAPL)¶

Decimals: 18

Tracks the price of Apple (AAPL) through price feeds supplied by an oracle.

## Australian Dollars (sAUD)¶

Decimals: 18

Tracks the price of Australian Dollars (AUD) through price feeds supplied by an oracle.

## Binance Coin (sBNB)¶

Decimals: 18

Tracks the price of Binance Coin (BNB) through price feeds supplied by an oracle.

## Bitcoin (sBTC)¶

Decimals: 18

Tracks the price of Bitcoin (BTC) through price feeds supplied by an oracle.

Decimals: 18

Tracks the price of Cardano (ADA) through price feeds supplied by an oracle.

## Centralised Exchange Index (sCEX)¶

Decimals: 18

Index of: sCEX

Token Symbol Units Initial Weight
Binance Coin BNB 12.88000 29.15%
Crypto com CRO 1,303.31000 24.3%
OKB OKB 25.72000 16.97%
UNUS SED LEO LEO 94.54000 13.42%
Huobi Token HT 23.06000 11.73%
FTX FTT 9.36000 3.56%
KuCoin Shares KCS 9.18000 0.87%

Tracks the price of the index: Centralised Exchange Index (CEX) through price feeds supplied by an oracle. This index is made up of the following assets and weights: 12.88 of undefined (Binance Coin), 1303.31 of undefined (Crypto com), 94.54 of undefined (UNUS SED LEO), 23.06 of undefined (Huobi Token), 9.36 of undefined (FTX), 25.72 of undefined (OKB), 9.18 of undefined (KuCoin Shares).

Decimals: 18

Tracks the price of Chainlink (LINK) through price feeds supplied by an oracle.

## Coinbase (sCOIN)¶

Decimals: 18

Tracks the price of Coinbase (COIN) through price feeds supplied by an oracle.

## Compound (sCOMP)¶

Decimals: 18

Tracks the price of Compound (COMP) through price feeds supplied by an oracle.

## Curve DAO Token (sCRV)¶

Decimals: 18

Tracks the price of Curve DAO Token (CRV) through price feeds supplied by an oracle.

## Dash (sDASH)¶

Decimals: 18

Tracks the price of Dash (DASH) through price feeds supplied by an oracle.

## DeFi Index (sDEFI)¶

Decimals: 18

Index of: sDEFI

Token Symbol Units Initial Weight
Aave AAVE 4.72000 15%
Synthetix Network Token SNX 87.20000 15%
Uniswap UNI 100.71000 15%
Maker MKR 0.54000 10%
Balancer BAL 14.84000 5%
Compound COMP 1.55000 5%
Curve DAO Token CRV 254.94000 5%
Kyber Network KNC 334.06000 5%
Ren REN 680.59000 5%
SushiSwap SUSHI 43.11100 5%
UMA UMA 26.06000 5%
yearn.finance YFI 0.01604 5%
1inch 1INCH 71.83700 2.5%
Bancor BNT 90.66100 2.5%

Tracks the price of the index: DeFi Index (DEFI) through price feeds supplied by an oracle. This index is made up of the following assets and weights: 1.55 of undefined (Compound), 0.54 of undefined (Maker), 4.72 of undefined (Aave), 26.06 of undefined (UMA), 87.2 of undefined (Synthetix Network Token), 680.59 of undefined (Ren), 100.71 of undefined (Uniswap), 334.06 of undefined (Kyber Network), 254.94 of undefined (Curve DAO Token), 43.111 of undefined (SushiSwap), 0.01604 of undefined (yearn.finance), 14.84 of undefined (Balancer), 90.661 of undefined (Bancor), 71.837 of undefined (1inch).

## EOS (sEOS)¶

Decimals: 18

Tracks the price of EOS (EOS) through price feeds supplied by an oracle.

## Ether (sETH)¶

Decimals: 18

Tracks the price of Ether (ETH) through price feeds supplied by an oracle.

## Ethereum Classic (sETC)¶

Decimals: 18

Tracks the price of Ethereum Classic (ETC) through price feeds supplied by an oracle.

## Euros (sEUR)¶

Decimals: 18

Tracks the price of Euros (EUR) through price feeds supplied by an oracle.

## FTSE 100 Index (sFTSE)¶

Decimals: 18

Tracks the price of FTSE 100 Index (FTSE100) through price feeds supplied by an oracle.

Decimals: 18

Tracks the price of Facebook (FB) through price feeds supplied by an oracle.

## Gold Ounce (sXAU)¶

Decimals: 18

Tracks the price of Gold Ounce (XAU) through price feeds supplied by an oracle.

## Inverse Aave (iAAVE)¶

Decimals: 18

Inverse of: sAAVE

Entry Point Upper Limit Lower Limit
$406$609 $203 Inversely tracks the price of Inverse Aave (AAVE) through price feeds supplied by an oracle. The entry point is$406 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $609 (i.e. when Inverse Aave's value reaches$203) or its lower limit of $203 (i.e. when Inverse Aave’s value reaches$609). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iAAVE with different limits.

## Inverse Binance Coin (iBNB)¶

Decimals: 18

Inverse of: sBNB

Entry Point Upper Limit Lower Limit
$253$379.50000 $126.50000 Inversely tracks the price of Inverse Binance Coin (BNB) through price feeds supplied by an oracle. The entry point is$253 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $379.5 (i.e. when Inverse Binance Coin's value reaches$126.5) or its lower limit of $126.5 (i.e. when Inverse Binance Coin’s value reaches$379.5). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iBNB with different limits.

## Inverse Bitcoin (iBTC)¶

Decimals: 18

Inverse of: sBTC

Entry Point Upper Limit Lower Limit
$58,785$88,177.50000 $29,392.50000 Inversely tracks the price of Inverse Bitcoin (BTC) through price feeds supplied by an oracle. The entry point is$58785 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $88177.5 (i.e. when Inverse Bitcoin's value reaches$29392.5) or its lower limit of $29392.5 (i.e. when Inverse Bitcoin’s value reaches$88177.5). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iBTC with different limits.

Decimals: 18

Entry Point Upper Limit Lower Limit
$0.97700$1.46600 $0.48900 Inversely tracks the price of Inverse Cardano (ADA) through price feeds supplied by an oracle. The entry point is$0.977 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $1.466 (i.e. when Inverse Cardano's value reaches$0.488) or its lower limit of $0.489 (i.e. when Inverse Cardano’s value reaches$1.465). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iADA with different limits.

## Inverse Centralised Exchange Index (iCEX)¶

Decimals: 18

Inverse of: sCEX

Entry Point Upper Limit Lower Limit
$4,946$7,419 $2,437 Index of: sCEX Inversely tracks the price of Inverse Centralised Exchange Index (CEX) through price feeds supplied by an oracle. The entry point is$4946 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $7419 (i.e. when Inverse Centralised Exchange Index's value reaches$2473) or its lower limit of $2437 (i.e. when Inverse Centralised Exchange Index’s value reaches$7455). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iCEX with different limits.

Decimals: 18

Entry Point Upper Limit Lower Limit
$28.15000$42.22500 $14.07500 Inversely tracks the price of Inverse Chainlink (LINK) through price feeds supplied by an oracle. The entry point is$28.15 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $42.225 (i.e. when Inverse Chainlink's value reaches$14.075) or its lower limit of $14.075 (i.e. when Inverse Chainlink’s value reaches$42.225). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iLINK with different limits.

## Inverse Compound (iCOMP)¶

Decimals: 18

Inverse of: sCOMP

Entry Point Upper Limit Lower Limit
$393$589.50000 $196.50000 Inversely tracks the price of Inverse Compound (COMP) through price feeds supplied by an oracle. The entry point is$393 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $589.5 (i.e. when Inverse Compound's value reaches$196.5) or its lower limit of $196.5 (i.e. when Inverse Compound’s value reaches$589.5). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iCOMP with different limits.

## Inverse Dash (iDASH)¶

Decimals: 18

Inverse of: sDASH

Entry Point Upper Limit Lower Limit
$241.50000$362.25000 $120.75000 Inversely tracks the price of Inverse Dash (DASH) through price feeds supplied by an oracle. The entry point is$241.5 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $362.25 (i.e. when Inverse Dash's value reaches$120.75) or its lower limit of $120.75 (i.e. when Inverse Dash’s value reaches$362.25). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iDASH with different limits.

## Inverse DeFi Index (iDEFI)¶

Decimals: 18

Inverse of: sDEFI

Entry Point Upper Limit Lower Limit
$13,382$20,073 $6,691 Index of: sDEFI Inversely tracks the price of Inverse DeFi Index (DEFI) through price feeds supplied by an oracle. The entry point is$13382 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $20073 (i.e. when Inverse DeFi Index's value reaches$6691) or its lower limit of $6691 (i.e. when Inverse DeFi Index’s value reaches$20073). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iDEFI with different limits.

## Inverse EOS (iEOS)¶

Decimals: 18

Inverse of: sEOS

Entry Point Upper Limit Lower Limit
$4.06000$6.09000 $2.03000 Inversely tracks the price of Inverse EOS (EOS) through price feeds supplied by an oracle. The entry point is$4.06 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $6.09 (i.e. when Inverse EOS's value reaches$2.03) or its lower limit of $2.03 (i.e. when Inverse EOS’s value reaches$6.09). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iEOS with different limits.

## Inverse Ether (iETH)¶

Decimals: 18

Inverse of: sETH

Entry Point Upper Limit Lower Limit
$1,548$2,322 $774 Inversely tracks the price of Inverse Ether (ETH) through price feeds supplied by an oracle. The entry point is$1548 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $2322 (i.e. when Inverse Ether's value reaches$774) or its lower limit of $774 (i.e. when Inverse Ether’s value reaches$2322). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iETH with different limits.

## Inverse Ethereum Classic (iETC)¶

Decimals: 18

Inverse of: sETC

Entry Point Upper Limit Lower Limit
$11.46000$17.19000 $5.73000 Inversely tracks the price of Inverse Ethereum Classic (ETC) through price feeds supplied by an oracle. The entry point is$11.46 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $17.19 (i.e. when Inverse Ethereum Classic's value reaches$5.73) or its lower limit of $5.73 (i.e. when Inverse Ethereum Classic’s value reaches$17.19). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iETC with different limits.

## Inverse Litecoin (iLTC)¶

Decimals: 18

Inverse of: sLTC

Entry Point Upper Limit Lower Limit
$179.50000$269.25000 $89.75000 Inversely tracks the price of Inverse Litecoin (LTC) through price feeds supplied by an oracle. The entry point is$179.5 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $269.25 (i.e. when Inverse Litecoin's value reaches$89.75) or its lower limit of $89.75 (i.e. when Inverse Litecoin’s value reaches$269.25). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iLTC with different limits.

## Inverse Monero (iXMR)¶

Decimals: 18

Inverse of: sXMR

Entry Point Upper Limit Lower Limit
$222.90600$334.36000 $111.45300 Inversely tracks the price of Inverse Monero (XMR) through price feeds supplied by an oracle. The entry point is$222.906 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $334.36 (i.e. when Inverse Monero's value reaches$111.45) or its lower limit of $111.453 (i.e. when Inverse Monero’s value reaches$334.359). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iXMR with different limits.

## Inverse Perpetual Oil Futures (iOIL)¶

Decimals: 18

Inverse of: sOIL

Entry Point Upper Limit Lower Limit
$45.08700$67.63000 $22.54300 Inversely tracks the price of Inverse Perpetual Oil Futures (OIL) through price feeds supplied by an oracle. The entry point is$45.087 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $67.63 (i.e. when Inverse Perpetual Oil Futures's value reaches$22.54) or its lower limit of $22.543 (i.e. when Inverse Perpetual Oil Futures’s value reaches$67.631). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iOIL with different limits.

Decimals: 18

Inverse of: sDOT

Entry Point Upper Limit Lower Limit
$34.61800$51.92800 $17.30900 Inversely tracks the price of Inverse Polkadot (DOT) through price feeds supplied by an oracle. The entry point is$34.618 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $51.928 (i.e. when Inverse Polkadot's value reaches$17.308) or its lower limit of $17.309 (i.e. when Inverse Polkadot’s value reaches$51.927). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iDOT with different limits.

## Inverse Ren (iREN)¶

Decimals: 18

Inverse of: sREN

Entry Point Upper Limit Lower Limit
$1.26000$1.89000 $0.63000 Inversely tracks the price of Inverse Ren (REN) through price feeds supplied by an oracle. The entry point is$1.26 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $1.89 (i.e. when Inverse Ren's value reaches$0.63) or its lower limit of $0.63 (i.e. when Inverse Ren’s value reaches$1.89). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iREN with different limits.

## Inverse Ripple (iXRP)¶

Decimals: 18

Inverse of: sXRP

Entry Point Upper Limit Lower Limit
$0.48240$0.72360 $0.24120 Inversely tracks the price of Inverse Ripple (XRP) through price feeds supplied by an oracle. The entry point is$0.4824 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $0.7236 (i.e. when Inverse Ripple's value reaches$0.2412) or its lower limit of $0.2412 (i.e. when Inverse Ripple’s value reaches$0.7236). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iXRP with different limits.

## Inverse TRON (iTRX)¶

Decimals: 18

Inverse of: sTRX

Entry Point Upper Limit Lower Limit
$0.04540$0.06810 $0.02270 Inversely tracks the price of Inverse TRON (TRX) through price feeds supplied by an oracle. The entry point is$0.0454 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $0.0681 (i.e. when Inverse TRON's value reaches$0.0227) or its lower limit of $0.0227 (i.e. when Inverse TRON’s value reaches$0.0681). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iTRX with different limits.

## Inverse Tezos (iXTZ)¶

Decimals: 18

Inverse of: sXTZ

Entry Point Upper Limit Lower Limit
$3.73000$5.59500 $1.86500 Inversely tracks the price of Inverse Tezos (XTZ) through price feeds supplied by an oracle. The entry point is$3.73 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $5.595 (i.e. when Inverse Tezos's value reaches$1.865) or its lower limit of $1.865 (i.e. when Inverse Tezos’s value reaches$5.595). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iXTZ with different limits.

## Inverse Uniswap (iUNI)¶

Decimals: 18

Inverse of: sUNI

Entry Point Upper Limit Lower Limit
$25.38200$38.07300 $12.69100 Inversely tracks the price of Inverse Uniswap (UNI) through price feeds supplied by an oracle. The entry point is$25.382 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $38.073 (i.e. when Inverse Uniswap's value reaches$12.691) or its lower limit of $12.691 (i.e. when Inverse Uniswap’s value reaches$38.073). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iUNI with different limits.

## Inverse yearn.finance (iYFI)¶

Decimals: 18

Inverse of: sYFI

Entry Point Upper Limit Lower Limit
$35,368$53,052 $17,684 Inversely tracks the price of Inverse yearn.finance (YFI) through price feeds supplied by an oracle. The entry point is$35368 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $53052 (i.e. when Inverse yearn.finance's value reaches$17684) or its lower limit of $17684 (i.e. when Inverse yearn.finance’s value reaches$53052). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Synthetix.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iYFI with different limits.

## Japanese Yen (sJPY)¶

Decimals: 18

Tracks the price of Japanese Yen (JPY) through price feeds supplied by an oracle.

## Litecoin (sLTC)¶

Decimals: 18

Tracks the price of Litecoin (LTC) through price feeds supplied by an oracle.

## Microsoft (sMSFT)¶

Decimals: 18

Tracks the price of Microsoft (MSFT) through price feeds supplied by an oracle.

## Monero (sXMR)¶

Decimals: 18

Tracks the price of Monero (XMR) through price feeds supplied by an oracle.

## Netflix (sNFLX)¶

Decimals: 18

Tracks the price of Netflix (NFLX) through price feeds supplied by an oracle.

## Nikkei 225 Index (sNIKKEI)¶

Decimals: 18

Tracks the price of Nikkei 225 Index (NIKKEI225) through price feeds supplied by an oracle.

## Perpetual Oil Futures (sOIL)¶

Decimals: 18

Tracks the price of Perpetual Oil Futures (OIL) through price feeds supplied by an oracle.

Decimals: 18

Tracks the price of Polkadot (DOT) through price feeds supplied by an oracle.

## Pound Sterling (sGBP)¶

Decimals: 18

Tracks the price of Pound Sterling (GBP) through price feeds supplied by an oracle.

## Ren (sREN)¶

Decimals: 18

Tracks the price of Ren (REN) through price feeds supplied by an oracle.

## Ripple (sXRP)¶

Decimals: 18

Tracks the price of Ripple (XRP) through price feeds supplied by an oracle.

## Silver Ounce (sXAG)¶

Decimals: 18

Tracks the price of Silver Ounce (XAG) through price feeds supplied by an oracle.

## South Korean Won (sKRW)¶

Decimals: 18

Tracks the price of South Korean Won (KRW) through price feeds supplied by an oracle.

## Swiss Franc (sCHF)¶

Decimals: 18

Tracks the price of Swiss Franc (CHF) through price feeds supplied by an oracle.

## Synthetix (SNX)¶

Decimals: 18

The Synthetix Network Token (SNX) gets staked as collateral to back Synths and entitles stakers to receive fees generated by Synth trades on Synthetix.Exchange.

## THORChain (sRUNE)¶

Decimals: 18

Tracks the price of THORChain (RUNE) through price feeds supplied by an oracle.

## TRON (sTRX)¶

Decimals: 18

Tracks the price of TRON (TRX) through price feeds supplied by an oracle.

## Tesla (sTSLA)¶

Decimals: 18

Tracks the price of Tesla (TSLA) through price feeds supplied by an oracle.

## Tezos (sXTZ)¶

Decimals: 18

Tracks the price of Tezos (XTZ) through price feeds supplied by an oracle.

## US Dollars (sUSD)¶

Decimals: 18

Tracks the price of a single US Dollar (USD). This Synth always remains constant at 1.

Decimals: 18


text:
### Pizza Portions

My friends and I love pizza. Can you help us share these pizzas equally?

### Pies

Grandma found her pie balanced on the scale with two weights and a quarter of a pie. So how heavy was each pie?

### Red Balloons, Blue Balloons

Katie and Will have some balloons. Will's balloon burst at exactly the same size as Katie's at the beginning of a puff. How many puffs had Will done before his balloon burst?

# Balance of Halves

### Why do this problem?

Whether this problem is done with a balance or with pencil and paper, there is more to it than just doing addition. Although addition and subtraction are involved, you are likely to find the children using all sorts of language associated with these two operations and one of the main points here is to establish a balance or equivalence as well as "equals". Some pupils are not so readily used to coming across situations like $3 + 5 = 6 + 2$, for example. You could use this activity to introduce some algebraic ideas, by encouraging the class to articulate what they are trying to find in a general sense. For example "$7$ balances two other numbers", or "$7$ = 'something' add 'something'"or "$7$ = ? $+$ ?". This will help them to get the idea that they are finding different numbers which fit this criterion - not just one answer.

### Key questions

How might you start?
How might you record your thinking?
How do you know you've got all the possibilities?

### Possible extension

As the question suggests, some learners might like to look at a balance involving other fractions.

### Possible support

Children might like to try Getting the Balance instead.


text:
GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 13 Oct 2019, 20:24

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# The Erythrovirus412, which affects farmed Atlantic salmon

Author Message
Manager
Joined: 04 Oct 2018
Posts: 159
Location: Viet Nam
The Erythrovirus412, which affects farmed Atlantic salmon  [#permalink]

### Show Tags

28 Jan 2019, 00:57
2
00:00

Difficulty:

75% (hard)

Question Stats:

48% (01:58) correct 52% (02:09) wrong based on 132 sessions

### HideShow timer Statistics

The Erythrovirus412, which affects farmed Atlantic salmon, infects human beings by adhering to red blood cells. The virus elevates the body’s white blood cell count, indicating that the body is fighting an infection. Surprisingly, when a group of recently-infected people were tested, the proportion of white blood cells compared to red blood cells in their bodies was similar to that of an uninfected person.

Which of the following, if true, most helps to explain the surprising finding?

A - An infected body can sometimes attempt to stimulate its immune system by producing an unusually high number of white blood cells.

B - Ingestion of infected salmon by people whose immune system is already weak can destroy a significant proportion of red blood cells in the body.

C - The Nucleic Acid Test (NAT) used to detect the virus has revealed that the virus embeds itself in the plasma constituting the lining of a cell in addition to circulating through blood cells.

D - The virus multiplies by increasing the number of its host cells.

E - Most people infected by the virus develop fever two weeks after being infected, but show no other symptoms of the disease.

_________________
"It Always Seems Impossible Until It Is Done"
Math Expert
Joined: 02 Aug 2009
Posts: 7945
Re: The Erythrovirus412, which affects farmed Atlantic salmon  [#permalink]

### Show Tags

29 Jan 2019, 07:49
1
Skyline393 wrote:
The Erythrovirus412, which affects farmed Atlantic salmon, infects human beings by adhering to red blood cells. The virus elevates the body’s white blood cell count, indicating that the body is fighting an infection. Surprisingly, when a group of recently-infected people were tested, the proportion of white blood cells compared to red blood cells in their bodies was similar to that of an uninfected person.

Which of the following, if true, most helps to explain the surprising finding?

A - An infected body can sometimes attempt to stimulate its immune system by producing an unusually high number of white blood cells.

B - Ingestion of infected salmon by people whose immune system is already weak can destroy a significant proportion of red blood cells in the body.

C - The Nucleic Acid Test (NAT) used to detect the virus has revealed that the virus embeds itself in the plasma constituting the lining of a cell in addition to circulating through blood cells.

D - The virus multiplies by increasing the number of its host cells.

E - Most people infected by the virus develop fever two weeks after being infected, but show no other symptoms of the disease.

Let us rework on the para first..

E-virus attaches to RBC and results in decrease of WBC of the infected person.
But the ratio of RBC:WBC in infected person is found to be SAME as that in uninfected person.

Logically the $$\frac{RBC}{WBC}$$ should decrease in infected person as the denominator, that is WBC, is increasing.
So, what is happening here??
--- The choice should tell us that.
But, logically $$\frac{RBC}{WBC}$$ will remain same even with increase of WBC when
WBC decreases due to some other reason
RBC too increases in the infected person

Let us check the choices for above two reasons or anything else that solves the paradox.

A - An infected body can sometimes attempt to stimulate its immune system by producing an unusually high number of white blood cells.
A further increase in WBC,.. complicates the issue further... exactly opposite of what we are looking for

B - Ingestion of infected salmon by people whose immune system is already weak can destroy a significant proportion of red blood cells in the body.
A decrease in RBC in few people,..... firstly deals with a small set of people and secondly even if we accept it, it is exactly opposite of what we are looking for

C - The Nucleic Acid Test (NAT) used to detect the virus has revealed that the virus embeds itself in the plasma constituting the lining of a cell in addition to circulating through blood cells.
We are not interested in the way the virus behaves.... Out of scope

D - The virus multiplies by increasing the number of its host cells.
YES, something which talks of an increase of host cells, and what are the host cells here---- RBC. That means RBC increase and this is what we are looking for. A simultaneous increase in RBC keeps the proportion of RBC and WBC same

E - Most people infected by the virus develop fever two weeks after being infected, but show no other symptoms of the disease.
The point is of count of RBC and WBC and not the symptoms

_________________
Re: The Erythrovirus412, which affects farmed Atlantic salmon   [#permalink] 29 Jan 2019, 07:49
Display posts from previous: Sort by


text:
### Refine search

Refine with date range

• Search for dark matter in events with missing transverse momentum and a higgs boson decaying to two photons in pp collisions at 's POT. 1/2'=8  TeV with the ATLAS Detector (2015)

Acesso online ao documento

or search this record in
• Measurement of the correlation between flow harmonics of different order in lead-lead collisions at 's IND. NN' POT. 1/2'=2.76 TeV with the ATLAS detector (2015)

Acesso online ao documento

or search this record in
• Search for new light gauge bosons in Higgs boson decays to four-lepton final states in pp collisions at 'S POT. 1/2'=8  TeV with the ATLAS detector at the LHC (2015)

Acesso online ao documento

or search this record in
• Modelling Z'SETA''tau''tau' processes in ATLAS with 'tau'-embedded Z'SETA''mü''mü' data (2015)

Acesso online ao documento

or search this record in
• Study of (W/Z)'H' production and Higgs boson couplings using 'H''SETA' W'W POT. *' decays with the ATLAS detector (2015)

Acesso online ao documento

or search this record in
• Determination of the top-quark pole mass using 'tau''tau BARRA' + 1-jet events collected with the ATLAS experiment in 7 TeV pp collisions (2015)

Acesso online ao documento

or search this record in
• Summary of the searches for squarks and gluinos using 's POT. 1/2'=8 TeV pp collisions with the ATLAS experiment at the LHC (2015)

Acesso online ao documento

or search this record in
• Search for a heavy neutral particle decaying to e'mü', e'tau', or 'mü''tau' in pp collisions at 's POT. 1/2'=8  TeV with the ATLAS detector (2015)

Acesso online ao documento

or search this record in
• Search for supersymmetry in events containing a same-flavour opposite-sign dilepton pair, jets, and large missing transverse momentum in 's POT. 1/2'=8 TeV pp collisions with the ATLAS detector (2015)

Acesso online ao documento

or search this record in
• Search for massive supersymmetric particles decaying to many jets using the ATLAS detector in pp collisions at 's POT. 1/2'=8  TeV (2015)

Acesso online ao documento

or search this record in
• Differential top-antitop cross-section measurements as a function of observables constructed from final-state particles using pp collisions at 's POT. 1/2'=7 TeV in the ATLAS detector (2015)

Acesso online ao documento

or search this record in
• Search for production of WW/WZ resonances decaying to a lepton, neutrino and jets in pp collisions at 's POT. 1/2'=8 TeV with the ATLAS detector (2015)

Acesso online ao documento

or search this record in
• Determination of spin and parity of the Higgs boson in the W'W POT. 1/2''SETA' eν'mü'ν decay channel with the ATLAS detector (2015)

Acesso online ao documento

or search this record in
• Search for W' 'SETA' tb 'SETA' qqbb decays in pp collisions at 's POT. 1/2' = 8 TeV with the ATLAS detector (2015)

Acesso online ao documento

or search this record in
• Search for new phenomena in events with three or more charged leptons in pp collisions at 's POT.1/2'=8 TeV with the ATLAS detector (2015)

Acesso online ao documento

or search this record in
• Measurement of spin correlation in top-antitop quark events and search for top squark pair production in pp collisions at 's POT. 1/2'=8  TeV using the ATLAS detector (2015)

Acesso online ao documento

or search this record in
• Observation of top-quark pair production in association with a photon and measurement of the 'tau''tau' BARRA''gama' production cross section in pp collisions at 's POT. 1/2'=7 TeV using the ATLAS detector (2015)

Acesso online ao documento

or search this record in
• Search for W' 'SETA' tb'BARRA' in the lepton plus jets final state in proton–proton collisions at a centre-of-mass energy of 's POT. 1/2'= 8 TeV with the ATLAS detector (2015)

Acesso online ao documento

or search this record in
• Measurement of the W W + W Z cross section and limits on anomalous triple gauge couplings using final states with one lepton, missing transverse momentum, and two jets with the ATLAS detector at 's POT. 1/2'=7 TeV (2015)

Acesso online ao documento

or search this record in
• Search for the b'b BARRA' decay of the standard model higgs boson in associated (W/Z)H production with the ATLAS detector (2015)

Acesso online ao documento

or search this record in


text:
Browse Dept. of Statistics by Title

• (1989)
This work deals with a decision-theoretic evaluation of p-value rules. A test statistic is judged on the behavior of its p-value with the loss function being an increasing function G of the p-value.

application/pdf

PDF (1MB)
• (1959)

application/pdf

PDF (1MB)
• (1996)
The identifiability and estimability of the parameters for the Unified Cognitive/IRT Model are studies. A calibration procedure for the Unified Model is then proposed. This procedure uses the marginal maximum likelihood ...

application/pdf

PDF (4MB)
• (2010-08-20)
The statistical inference based on the ordinary least squares regression is sub-optimal when the distributions are skewed or when the quantity of interest is the upper or lower tail of the distributions. For example, the ...

application/pdf

PDF (1MB)
• (2000)
Using results from He & Shao (2000), a proof of the consistency and asymptotic normality of item parameter estimates obtained from the Marginal Maximum Likelihood Estimation (Bock & Lieberman, 1970) procedure as both the ...

application/pdf

PDF (5MB)
• (1989)
In many areas of application of statistics one has a relevent parametric family of densities and wishes to estimate the density from a random sample. In such cases one can use the family to generate an estimator. We fix a ...

application/pdf

PDF (4MB)
• (1967)

application/pdf

PDF (1MB)
• (1989)
Many authors, for example, Fisher (1950), Pearson (1938), Birnbaum (1954), Good (1955), <NAME> Folks (1971, 1973), <NAME> (1979), and Koziol, Perlman, and Rasmussen (1988), have studied the problem of combining ...

application/pdf

PDF (7MB)
• (2012-02-01)
Bayesian inference provides a flexible way of combiningg data with prior information. However, quantile regression is not equipped with a parametric likelihood, and therefore, Bayesian inference for quantile regression ...

application/pdf

PDF (446kB)
• (2002)
This thesis presents a progression from theory development to real-data application. Chapter 1 gives a literature review of other psychometric models for formative assessment, or cognitive diagnosis models, as an introduction ...

application/pdf

PDF (9MB)
• (1993)
We consider the problem of regressing a dichotomous response variable on a predictor variable. Our interest is in modelling the probability of occurrence of the response as a function of the predictor variable, and in ...

application/pdf

PDF (6MB)
• (2011-05-25)
The latent class model (LCM) is a statistical method that introduces a set of latent categorical variables. The main advantage of LCM is that conditional on latent variables, the manifest variables are mutually independent ...

application/pdf

PDF (5MB)
• (2011-05-25)
Quantile regression, as a supplement to the mean regression, is often used when a comprehensive relationship between the response variable and the explanatory variables is desired. The traditional frequentists’ approach ...

application/pdf

PDF (374kB)
• (2007)
Clustering and classification have been important tools to address a broad range of problems in fields such as image analysis, genomics, and many other areas. Basically, these clustering problems can be simplified as two ...

application/pdf

PDF (2MB)
• (2000)
To effectively build a regression model with a large number of covariates is no easy task. We consider using dimension reduction before building a parametric or spline model. The dimension reduction procedure is based on ...

application/pdf

PDF (4MB)
• (2000)
Motivated by consulting in infrastructure studies, we consider the estimation and inference for regression models where the response variable is bounded or censored. In these conditions, least squares methods are not ...

application/pdf

PDF (3MB)
• (2006)
The classical approaches to clustering are hierarchical and k-means. They are popular in practice. However, they can not address the issue of determining the number of clusters within the data. In this dissertation, we ...

application/pdf

PDF (2MB)
• (1991)
Consider the model $y\sb{lj} = \mu\sb{l}(t\sb{j})$ + $\varepsilon\sb{lj}$, $l = 1,..,m$ and $j = 1,..,n,$ where $\varepsilon\sb{lj}$ are independent mean zero finite variance random variables. Under the above setting we ...

application/pdf

PDF (5MB)
• (1990)
Two-stage Bayes procedures, also known as Bayes double sample procedures, for estimating the mean of exponential family distributions are given by Cohen and Sackrowitz (1984). In their study, they develop double sample ...

application/pdf

PDF (2MB)
• (2004)
The flexible forms of nonparametric IRT models make test equating more challenging. Though linear equating under parametric IRT models is obvious and appropriate, it might not be appropriate for nonparametric models. Two ...

application/pdf

PDF (3MB)


text:
 <NAME>'s Website - Yet another perspective on things...
60 guests
Rough Hits : 3688998

### More from me...

how did u find my site?

onomastics (seen in code)

"Consistency is the last refuge of the unimaginative." -- <NAME>
\begin{bmatrix} 1 & 0 & \ldots & 0 \\ 0 & 1 & 0 & \vdots \\ \vdots & 0 & \ddots & 0\\ 0 & \ldots & 0 & 1_{n} \end{bmatrix}

# Bars, Places, and things in Fort Wayne, IN

Sunday, 16 July 2006 00:29
Fort Wayne is a nice little city. It actually is the second largest city in Indiana. You can find some other information about the city @ http://www.cityoffortwayne.org/

Places
CinemaCenter
Baan Thai Home (Baan Thai carryout menu) - great food.

I know, its boston, and doesn't belong here.
Salsa Dancing and Lessons in Boston with Havana Club

< Prev  Next >
Last Updated on Saturday, 02 April 2016 23:21


text:
## Functional Programming

### D more popular than any functional language

The Endeavor - Tue, 07/29/2014 - 13:17

All measures of programming language popularity are indirect and controversial. There’s no consensus on the best way to measure popularity, even if we could gather any data we want. Still, measures of popularity based on data, especially combining several kinds of data, are more useful than personal impressions.

According to the normalized comparison chart on langpop.com, at the time of writing this post, D is easily more popular than all functional programming languages combined. Here’s a portion of the chart zooming in on D and functional languages.

This hardly seems possible. As I stated up front, all measures of programming language popularity are crude proxies for what we’d really like to know. But still, this says that by at least one measure, D is more popular than any functional language.

The langpop.com site lets you twiddle the weights assigned to different components of the comparison. And for a wide range of weight choices, D comes out ahead of any functional language.

My preference would be to set the Google Files component should be zero since it’s only looking at web page languages. (The page won’t let you set a component to exactly zero, but you can make it tiny, say 0.001.) Github stats give you a good view into the open source world, but it would be an example of availability bias to assume the whole world of software development looks like Github. Craigslist might be a good window into commercial software development.

### Are you serious? "What degree?"

Planet Haskell - Tue, 07/29/2014 - 06:00

### A few new papers

Planet Haskell - Tue, 07/29/2014 - 06:00
I’ve just updated my web page with links to some new papers that are now available: “Homotopical Patch Theory” by <NAME>, <NAME>, <NAME>, and <NAME>. To appear, ICFP, Gothenburg, October 2014. We’re also preparing an expanded version with a new appendix containing material that didn’t make the cut for ICFP. (Why do […]

### EclipseFP 2.6.1 released!

Planet Haskell - Tue, 07/29/2014 - 06:00
I've just released EclipseFP 2.6.1. EclipseFP is a set of Eclipse plugins for Haskell development. This is a bug fixing release, mainly for GHC 7.8 support.Release notes can be found here.As usual, download from http://eclipsefp.sf.net/updates.Happy Haskell Hacking!

### CTO / Tech Co-Founder at Capital Match (Full-time)

Planet Haskell - Tue, 07/29/2014 - 06:00

### Parallelism and Concurrency, Revisited

Planet Haskell - Tue, 07/29/2014 - 06:00
To my delight, I still get compliments on and criticisms of my post from three years ago (can it possibly be that long?) on parallelism and concurrency.  In that post I offered a “top down” argument to the effect that these are different abstractions with different goals: parallelism is about exploiting computational resources to maximize efficiency, concurrency is […]

Planet Haskell - Tue, 07/29/2014 - 06:00
Bitemyapp - Meditations on Using Haskell explains why and how those in the trenches use Haskell, by quoting from conversations on an IRC channel.ESo when i found haskell i slingshotted off through dependent and substructural types. Assuming that if a little was good a lot was better. Made it half way through TaPL and found pure type systems, coq, etc.I think the power to weight ratio isn’t there. I find that Haskell gives amazingly expressive types that have amazingpower for the amount of code you tie up in them and that are very resistant to refactoring.If i write agda and refactor I scrap and rewrite everything. If i write haskell, and get my tricky logic bits right?I can refactor it, split things up into classes, play all the squishy software engineering games to get a nice API I want. And in the end if it still compiles I can trust I didn’t screw up the refactoring with a very high degree of assurance.CAdmittedly I’m not playing at the level E is, but this was my experience. I can make sweeping changes to my API, get all the bugs caught by the type system, and still have minimal code impact.BThat is what I was getting at with the tweet about not using dynamically typed langs because I need to be able to prototype quickly and get rapid feedback.I think a lot of my friends thought i was just being trollish. Even just being able to see what would have to change if you changed your design slightly and being able to back it out quickly…

### That's totes my Bag!

Planet Haskell - Tue, 07/29/2014 - 06:00

### Similarity analysis of quilt blocks

Planet Haskell - Tue, 07/29/2014 - 06:00


text:
Helium is pumped into a spherical balloon at a rate of 2 cubic feet per second. How fast is the radius increasing after 2 minutes?
Note: The volume of a sphere is given by V=(4/3)?r3.

Rate of change of radius (in feet per second) =


text:
# Correlation length in d>1 Ising model, at zero temperature

I am studying the renormalization group approach to the Ising model using as a reference Cardy's book "Scaling and renormalization in statistical mechanics". I cannot understand what happens in the zero temperature case (and possibly for $T < T_c$) to the correlation length $\xi$. Here's my point:

Since the zero temperature is a fixed point, it should be either $\xi =0$ or $\infty$ in fact $\xi(\{K'\}) = 1/2 \ \xi(\{K\})$, but at a fixed point $\{K'\} = \{K\}$ (I am using Cardy's notation where $\{K\}$ denotes the set of coupling for the theory).

Now if $\xi$ is finite below the critical temperature (as it is stated in some books), say at $T_0 < T_c$ or equivalently $K_0 > K^*$, it must be zero at zero temperature. This can be deduced in a similar fashion as it is done at the critical temperature (pag. 38). In short, if $n(K)$ is defined as the number of times you have to apply the renormalization group to get from $K_0$ to $K$, then $n(K)$ diverges as $K \to \infty$ (same to say $T \to 0$). Therefore $\xi(K)$ tends to zero (it is halved $n(K)$ times starting from $\xi(K_0)$ ).

On the other hand, it seems to me that it is possible to evaluate exactly the two-point spins correlation functions in the zero temperature limit as follows: $\langle\sigma(0) \sigma(r) \rangle = \sum_{\{\sigma\}} \sigma(0) \sigma(r) \exp(-\beta H[\sigma]) = 2$. In the last passage I have used that at zero temperature only the two spins configurations with lowest energy contribute to the sum, but those are with all spins up or all spins down and $\langle\sigma(0) \sigma(r) \rangle = 1$. Therefore the correlation length is infinite. (and for the argument above it should be infinite for every $T<T_c$).

So where is the error?

• I forgot about that thank you. But nevertheless, <\sigma(r)> = 0 for every r, since again at zero temperature only the above two configurations contribute and so on...Hence <sigma sigma> - <sigma><sigma> = <sigma sigma> = 2. (I had not the time to read your reference yet but I intend to do) – <NAME> Jan 15 '15 at 8:30
• Thank you very much, even tough I do not fully understand why one should say that the state must be pure (and therefore a boundary condition such as the one you said must be imposed), and not a mixture of the two pure, this must be the solution. I'd gladly upvote your answer if you wrote one. – <NAME> Jan 15 '15 at 13:59

The correlation length below the critical temperature can be defined using the rate of exponential decay of the truncated 2-point function, evaluated in a pure state (I'll choose the one induced by the $+$ boundary condition), namely $$\xi_\beta(\vec n) = \lim_{k\to\infty} -\frac1k \log \langle \sigma_0 ; \sigma_{[k\vec n]}\rangle^+_\beta,$$ where $\vec n$ is a unit vector in $\mathbb{R}^d$ and $[k\vec n]$ is the point of $\mathbb{Z}^d$ closest to the point $k\vec n$ in $\mathbb{R}^d$. I used the standard notation $$\langle \sigma_i ; \sigma_j\rangle^+_\beta = \langle \sigma_i \sigma_j\rangle^+_\beta - \langle \sigma_i \rangle^+_\beta\langle \sigma_j\rangle^+_\beta$$ for the truncated 2-point function (the covariance between the spins). Here, $\langle\cdot\rangle^+_\beta$ denotes expectation w.r.t. the (infinite-volume) Gibbs state obtained by taking the thermodynamic limit with $+$ boundary condition and inverse temperature $\beta$.
There are various ways of showing that the correlation length $\xi_\beta$ indeed goes to $0$ as $\beta\uparrow\infty$. Possibly one of the most straightforward (though a bit tedious) is through cluster expansion techniques (see, e.g., section 5.7.4 of this book).
The main point is that the average of the spins become nonzero below the critical temperature, but the fluctuations around this average value become completely uncorrelated in the limit. Morally, in order to correlate the fluctuations of two spins at $i$ and $j$, you must have a Peierls contour surrounding both $i$ and $j$, and the probability of this goes to $0$ exponentially fast in $\beta\|j−i\|$ (when $\beta>\beta_{\rm c}$).
There are several reasons why you must work with a pure state (or, more precisely, an extremal Gibbs measure) above. One is that these are the states corresponding to thermodynamic equilibrium: the only ones for which all macroscopic observables take deterministic values, for example. Note that if you were to consider the state obtained with free (or periodic) boundary condition, you would indeed see that the truncated 2-point function reduces to the usual 2-point function, $$\langle \sigma_i; \sigma_j \rangle^{\rm free}_\beta = \langle \sigma_i \sigma_j \rangle^{\rm free}_\beta$$ and this quantities does not vanish as $\|j-i\|\to\infty$ when $\beta>\beta_{\rm c}$ (instead, it converges to the square of the spontaneous magnetization $m^*(\beta)$). Note also that, even though $\langle \sigma_i\rangle^{\rm free}_\beta = 0$, there is spontaneous magnetization in typical configurations: the expectation is zero only because this spontaneous magnetization is $m^*(\beta)$ or $-m^*(\beta)$ with probability $1/2$. Actually, a typical configuration under this measure will be typical of either the $+$ state or the $-$ state with probability $1/2$, so any natural way you decide to measure the correlation length from configurations under these pure states, should give you the same answer under the free state. The reason you cannot do it through the truncated 2-point function as above is that the expectation starts to mix up the contributions from the two possible macroscopic behaviors described by the pure states, but this does not correspond to anything physically relevant.


text:
Article Contents
Article Contents

# Asymptotic analysis of a simple model of fluid-structure interaction

• This paper is devoted to the asymptotic analysis of simple models of fluid-structure interaction, namely a system between the heat and wave equations coupled via some transmission conditions at the interface. The heat part induces the dissipation of the full system. Here we are interested in the behavior of the model when the thickness of the heat part and/or the heat diffusion coefficient go to zero or to infinity. The limit problem is a wave equation with a boundary condition at the interface, this boundary condition being different according to the limit of the above mentioned parameters. It turns out that some limit problems are dissipative but some of them are non dissipative or their behavior is unknown.
Mathematics Subject Classification: 35B30, 35M10, 35B40, 93D20.

 Citation:

## Other Articles By Authors

• on this site
• on Google Scholar

/