text:
# PSAT Math: Rational Expressions and Equations

A rational expression is simply a ratio (or fraction) of polynomials. In other words, it is a fraction with a polynomial as the numerator and another polynomial as the denominator. The rules that govern fractions and polynomials also govern rational expressions, so if you know these well, you’ll be in good shape when you encounter one on Test Day.

There are a few important tidbits to remember about rational expressions; these are summarized here. They are also true for rational equations.

• For an expression to be rational, the numerator and denominator must both be polynomials.
• Like polynomials, rational expressions are also designated certain degrees based on the term with the highest variable exponent sum. For instance, the expression $\frac{1\phantom{\rule{.2em}{0ex}}-\phantom{\rule{.2em}{0ex}}2x}{3{x}^{2}+\phantom{\rule{.2em}{0ex}}3}$ has a first-degree numerator and a second-degree denominator.
• Because rational expressions by definition can have polynomial denominators, they will often be undefined for certain values. For example, the expression $\frac{x\phantom{\rule{.2em}{0ex}}-\phantom{\rule{.2em}{0ex}}4}{x\phantom{\rule{.2em}{0ex}}+\phantom{\rule{.2em}{0ex}}2}$ is defined for all values of x except –2. This is because when x = –2, the denominator of the expression is 0, which would make the expression undefined.
• Factors in a rational expression can be cancelled when simplifying, but under no circumstances can you do the same with individual terms. Consider, for instance, the expression $\frac{{x}^{2}-\phantom{\rule{.2em}{0ex}}x\phantom{\rule{.2em}{0ex}}-\phantom{\rule{.2em}{0ex}}6}{{x}^{2}+\phantom{\rule{.2em}{0ex}}5x\phantom{\rule{.2em}{0ex}}+\phantom{\rule{.2em}{0ex}}6}$. Many students will attempt to cancel the x 2, x, and 6 terms to give$\frac{1\phantom{\rule{.2em}{0ex}}-\phantom{\rule{.2em}{0ex}}1\phantom{\rule{.2em}{0ex}}-\phantom{\rule{.2em}{0ex}}1}{1\phantom{\rule{.2em}{0ex}}+\phantom{\rule{.2em}{0ex}}5\phantom{\rule{.2em}{0ex}}+\phantom{\rule{.2em}{0ex}}1}=\frac{–1}{7}$, which is never correct. Don’t even think about trying this on Test Day.
• Like fractions, rational expressions can be proper or improper. A proper rational expression has a lower-degree numerator than denominator $\left(\text{e}.\text{g}.,\frac{1\phantom{\rule{.2em}{0ex}}-\phantom{\rule{.2em}{0ex}}x}{{x}^{2}+\phantom{\rule{.2em}{0ex}}3}\right)$, and an improper one has a higher-degree numerator than denominator $\left(\text{e}.\text{g}.,\frac{{x}^{2}+\phantom{\rule{.2em}{0ex}}3}{1\phantom{\rule{.2em}{0ex}}-\phantom{\rule{.2em}{0ex}}x}\right)$. The latter can be simplified using polynomial long division.

### PSAT Math Practice: Solving Rational Equations

Rational equations are just like rational expressions except for one difference: They have an equal sign. They follow the same rules as rational expressions. The steps you take to solve the more friendly-looking linear equations apply to rational equations as well.

When solving rational equations, beware of extraneous solutions, solutions derived that don’t satisfy the original equation. This happens when the derived solution causes 0 in the denominator of any of the terms in the equation (because division by 0 is not possible).

Take the equation $\frac{1}{x\phantom{\rule{.2em}{0ex}}+\phantom{\rule{.2em}{0ex}}4}+\frac{1}{x\phantom{\rule{.2em}{0ex}}-\phantom{\rule{.2em}{0ex}}4}=\frac{8}{\left(x\phantom{\rule{.2em}{0ex}}+\phantom{\rule{.2em}{0ex}}4\right)\left(x\phantom{\rule{.2em}{0ex}}-\phantom{\rule{.2em}{0ex}}4\right)}$, for instance.

After multiplying both sides by the common denominator (x + 4)(x − 4), you have (x − 4) + (x + 4) = 8. Solving for x yields 2x = 8 which simplifies to x = 4. However, when 4 is substituted for x, you get 0 in the denominator of both the second and third terms of the equation, so 4 is an extraneous solution. Therefore, this equation is said to have no solution.

### Note

Whenever you encounter an equation with variables in a denominator or under a radical, make sure you check the solutions by plugging the values back into the original equation.

1. Which value of x satisfies the equation $\frac{4}{x}+\frac{2}{x\phantom{\rule{.2em}{0ex}}-\phantom{\rule{.2em}{0ex}}8}=\frac{–2}{{x}^{2}\phantom{\rule{.2em}{0ex}}-\phantom{\rule{.2em}{0ex}}8x}$ ?

A. –1
B. 1
C. 4
D. 5

Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

Strategic Thinking Math Scratchwork

Step 1: Read the question, identifying and organizing important information as you go

You’re asked to determine which value of x satisfies the equation.

Step 2: Choose the best strategy to answer the question

Start by identifying a common denominator. By factoring x out of the denominator of the term on the right side of the equation, you’ll see that the common denominator across all three terms is x(x − 8). Multiply the entire equation by this expression, then distribute properly and solve for x.

${x}^{2}\phantom{\rule{.2em}{0ex}}\text{–}\phantom{\rule{.2em}{0ex}}8x\phantom{\rule{.2em}{0ex}}\to \phantom{\rule{.2em}{0ex}}x\left(x\phantom{\rule{.2em}{0ex}}–\phantom{\rule{.2em}{0ex}}8\right)$

$x\left(x\phantom{\rule{.2em}{0ex}}–\phantom{\rule{.2em}{0ex}}8\right)\left(\frac{4}{x}\text{+}\frac{2}{x\phantom{\rule{.2em}{0ex}}–\phantom{\rule{.2em}{0ex}}8}\text{=}\frac{–2}{x\left(x\phantom{\rule{.2em}{0ex}}–\phantom{\rule{.2em}{0ex}}8\right)}\right)$
$\begin{array}{rcl}4\left(x\phantom{\rule{.2em}{0ex}}–\phantom{\rule{.2em}{0ex}}8\right)\phantom{\rule{.2em}{0ex}}\text{+}\phantom{\rule{.2em}{0ex}}2x& =& –2\\ 4x\phantom{\rule{.2em}{0ex}}–\phantom{\rule{.2em}{0ex}}32\phantom{\rule{.2em}{0ex}}\text{+}\phantom{\rule{.2em}{0ex}}2x& =& –2\\ 6x& =& 30\\ x& =& 5\end{array}$

Step 3: Check that you answered the right question

Because “no solution” is not a choice, you can be confident that 5 is not an extraneous solution. Choice (D) is therefore correct.


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2019
Том 71
№ 11

# Bodenchuk V. V.

Articles: 2
Article (Ukrainian)

### Exact Values of Kolmogorov Widths for the Classes of Analytic Functions. II

Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1011-1018

It is shown that the lower bounds of the Kolmogorov widths $d_{2n}$ in the space $C$ established in the first part of our work for the function classes that can be represented in the form of convolutions of the kernels $${H}_{h,\beta }(t)={\displaystyle \sum_{k=1}^{\infty}\frac{1}{ \cosh kh} \cos \left(kt-\frac{\beta \pi }{2}\right),\kern1em h>0,\kern1em \beta \in \mathbb{R},}$$ with functions $φ ⊥ 1$ from the unit ball in the space $L_{∞}$ coincide (for all $n ≥ nh$) with the best uniform approximations of these classes by trigonometric polynomials whose order does not exceed $n − 1$. As a result, we obtain the exact values of widths for the indicated classes of convolutions. Moreover, for all $n ≥ nh$, we determine the exact values of the Kolmogorov widths $d_{2n-1}$ in the space $L_1$ of classes of the convolutions of functions $φ ⊥ 1$ from the unit ball in the space $L_1$ with the kernel $H_{h,β}$.

Article (Ukrainian)

### Exact Values of Kolmogorov Widths for the Classes of Analytic Functions. I

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 719-738

We prove that the kernels of analytic functions of the form $${H}_{h,\beta }(t)={\displaystyle \sum_{k=1}^{\infty}\frac{1}{ \cosh kh} \cos \left(kt-\frac{\beta \pi }{2}\right),}h>0,\beta \in \mathbb{R},$$ satisfy Kushpel’s condition $C_{y,2n}$ starting from a certain number $n_h$ explicitly expressed via the parameter $h$ of smoothness of the kernel. As a result, for all $n ≥ n_h$ , we establish lower bounds for the Kolmogorov widths $d_{2n}$ in the space $C$ of functional classes that can be represented in the form of convolutions of the kernel $H_{h,β}$ with functions $φ⊥1$ from the unit ball in the space $L_{∞}$.

format: plain


text:
# π Places

π (say ‘pie’) is the lower case Greek letter symbol that stands for a number defined as the ratio of the circumference (C) of a circle divided by its diameter (d).

Circumference means the distance around something, like a circle or a polygon. A synonym would be perimeter.

A diameter is the length of a line that is drawn starting from any point on the circle’s boundary and going through the center point of the circle and then continuing to the opposite boundary point. A radius (r) is the length of a line drawn from the center to any point on the circle’s boundary. It is half the diameter. It’s clearer in the diagram:

As a formula (no numbers, please), we write:

pi = C/d                                                   [1]

If d = 1 inch, then because π has a fixed, constant value, C must be equal to that value of π inches.

A closely related formula involving π involves the Area A, of a Circle and its radius r, (remember? – half of the diameter’s length). The formula is:

pi = A/r^2                                                   [2]

Most descriptions of π state when it was first historically used and trace a series of numerical refinements in its approximated value over the last 4500 years.

A chronology of the computation of π from 2500 BC (calculated as 22/7 ) to the present is shown in the following link: π Chronology

It shows the date, the person, milestone or event and the decimal value of π (or the correct number of decimal digits after 3.) up through October 8, 2014, when π had been computed and verified (anonymously) to 1.33∙10^13 (over 13 Trillion) decimal digits.

## Classifications

It turns out that π is an irrational number. (Johann Heinrich Lambert proved this in 1761.)

This means that we cannot discover a fraction that provides its value exactly. Writing its decimal digits never ends. So over the years, mathematicians have labored to compute π accurately to more and more decimal places using “good” fractions and infinite series where only the first “few” terms are evaluated and summed.

It is also true the π is a transcendental number. A transcendental number is an irrational number that is not an algebraic number. An algebraic number is one that satisfies a polynomial equation with a finite number of terms and with rational coefficients. So, for example, sqrt(2) is an irrational (non-repeating decimal number) and an algebraic number, since it is a solution to the algebraic equation: x^2 = 2

For non-algebraic equations, we have two equations from trigonometry, which use pi/4 radians rather than 45 degrees:

tangent(pi/4) = 1                                                  [3]

and particularly its counterpart:

arctangent(1) = pi/4                                                [4]

is used as a starting point for determining π more precisely.

## Rational (Fractional) Approximations

Methods of approximation started with Archimedes, who, around 250 BC, applied the method of exhaustion with inscribed and circumscribed regular polygons to enclose the lower and upper bounds of area and circumference of the circle. Shown below are inscribed and circumscribed pentagons, hexagons and octagons (n = 5, 6, 8). Starting with a hexagon, he doubled the sides for inner and outer n-gons until n = 96, finding the inner and outer perimeters at each value of n.

Archimedes found bounds for π as:

3 + 10/71 < pi < 3 + 10/70      or      223/71 < pi < 22/7                   [5]

This gives the value of π to 2 decimal places, as 3.14.

### A Cool Procedure And Shell Script For a better Fraction for π

Using the circumference formula [1], we can create a procedure based on one suggested by Rhett Allain (See:
Best Fraction For π to calculate a fraction that represents π for a given number of digits.

For reference, suppose πref is set to 3.14159265358979, which is to 14 decimal places. We use this 12 step procedure:

1. Set πref = 3.14159265358979
2. Set (1/πref) = 0.31830988618379
3. Set loopcount = 500
4. Set C = 22
5. Set D = 7
6. Set Percentage Error = 0.001
7. Is loopcount = 0 ?
Then output BestC, BestD, BestC/BestD, Error, BestLoop
and stop

8. Set πest = C/D
9. Calculate Absolute Value of E = | (πref – πest) / πref |
via E = | (πref – πest) ∙ (1/πref) |
10. Is E < Error ?
Then set BestC = C, set BestD=D, set Error = E,
and set Bestloop = 500 – loopcount

11. Is πest < πref ?
Then add 1 to C and go to step 12
Otherwise add 1 to D and go to step 12

12. Subtract 1 from loopcount and go back to step 7 and compute new πest

Here is the bash shell script that I wrote to program this algorithm. It takes slightly under 7.5 seconds (what’s your hurry?) to finish 500 iterations (on a 2010 vintage 32-bit MacBook Pro) and 3.55 seconds on a 64-bit GNU/Linux Server.

The script successfully finds the fraction

355/113 = 3.141592920353982                                              [6]

a close approximation to the reference π that is accurate to 6 decimal places. The percentage error using this fraction is 0.0000084913679%

In this script, the Linux utility bc is needed to offer/preserve 15 decimal place precision in the computations it performs. The shell facilities for operating with numbers using anything other than positive integers are minimal.

#! /bin/bash
USAGE="Usage: pifraction.bash"
# Version 1.0 by arkay on 1/17/2015
# Version 1.1 by arkay on 1/18/2015
#   Computed (1/Pi_ref) once and used it to multiply for
#   % difference.

# Set Starting values for variables
Pi_ref="3.14159265358979"
Pi_ref_inv="0.31830988618379"
loopcount=500
C=22
D=7
Error="0.001"
# loop 500 times
until (( loopcount < 1 ))
do
# Calculate Pi.est
Pi_est=$(echo "scale=15;$C/$D" | bc -l) # Calculate the percentage error E E=$(echo "scale=15; ($Pi_ref -$Pi_est)*$Pi_ref_inv" | bc -l) if [ "${E%%[0-9.]*}" == "-" ]  #Apply Absolute Value:
# extract first character of E, either "–" or ""
then
E=$(echo "scale=15; (-1)*$E" | bc -l)
fi
# Compare E with current minimum % Error
T=$( echo "scale=15;$E<$Error" | bc -l ) if [ "$T" -eq "1" ] # bc returns 1 if inequality true
then
BestC=$C; BestD=$D; Error=$E; BestLoop=$(expr 500 - $loopcount) fi # Compare Pi_est with Pi_ref S=$( echo "scale=15; $Pi_est<$Pi_ref" | bc -l )
if [ "$S" -eq "1" ] # bc returns 1 if inequality true then (( C += 1)) else (( D += 1)) fi (( loopcount -= 1 )) done # Produce the results BestCoverD=$(echo "scale=15; $BestC/$BestD" | bc -l)
echo "BestC=$BestC BestD=$BestD BestCoverD=$BestCoverD" echo "Pi_ref=$Pi_ref  Error=$Error BestLoop=$BestLoop"
exit   # Normal stopping point

# End of pifraction.bash

Output results from this shell script are shown as:
BestC=355 BestD=113 BestCoverD=3.141592920353982
Pi_ref=3.14159265358979 Error=.000000084913679 BestLoop=439

In my next post about π, I will write about greater decimal place approximators and Interesting π activities to try.

Oh, one last thing…
This is a delightful Math Trick attributed to Martin Gardner:

Write all 26 letters of the alphabet, but start with the letter J as shown:

JKLMNOPQRSTUVWXYZABCDEFGHI

Then, remove all the letters that have vertical symmetry as shown:

JKL   N   PQRS              Z   BCDEFG

Now, count the letters that remain in each subset: 3 1 4 1 6.

# Factorials For Fun

Start from the beginning. A factorial is a specific multiplication function applied to a positive integer value and all of its positive descendants. True but vague and tough to digest.

A factorial means to multiply a consecutive sequence of descending natural numbers.

Below, n is any positive integer (equal to a natural number), being multiplied:

n! = n·(n–1)·(n–2)·…·3·2·1                                                                                [1]

The symbol for a factorial is the exclamation point: ! . n! is pronounced ‘en’ factorial (math fans) or ‘en’ bang (programmers) or shriek out n (literalists and comedians).

Here are some examples:                      3! = 3·2·1 = 6

10! = 10·9·8·7·6·5·4·3·2·1 = 3628800

{ Sudden question: In 1 second, how much would  9!  be? }

1! = 1

To see the first 100 factorials, please admire the following table by N. J. A. Sloane:   First 100 Factorials

These integer results become huge as n increases. In a future post, I will write about how to determine the number of digits of n! without calculating the factorial and visually counting the digits.

Because mathematicians revere economy in writing, n! can be defined for any positive integer n by two rules:

n! = n·(n–1)!
and                                                                                [2]
1! = 1

meaning, n! is the product of n and the factorial of the next lower value shown as (n – 1).

In the case of the second rule, this only works if we also have

0! = 1                                                                                [3]

Since by rule 1 (and I know this may be a head-scratcher…)

1! = 1·0! = 1                                                                                [4]

To find the value of n! , the first rule is applied repeatedly, as shown:

n! = n·(n–1)! = n·(n–1)·(n-2)! = … =
n
·(n–1)·(n-2)··3·2·1! =                                                                                [5]
n
·(n–1)·(n-2)··3·2·1

At this point, the original definition of n! in [1] has been recreated.

This [2] is called a recursive definition of n! . It has been said anonymously that, “in order to understand recursion, one must first understand recursion”.

In Computer Programming, n! can be determined via an iterative set of instructions, involving looping. Also, n! can be determined via a recursively called function or program, which calls itself repeatedly.

The Rosetta Code.org Wiki for factorial lists approximately 179 different programming languages with both iterative and recursive factorial code segments. It’s quite impressive – See:
Factorial Programs and Functions – Rosetta Code

Within this listing are programs that are part of the Unix/Linux facilities: awk, bash, bc, dc, JavaScript, m4, perl, php, python, R, ruby, and Unix shells (sh, ksh, zsh). These programs do not require a compiler. In future posts, these will be examined more closely.

There’s much more to factorials, which I will defer to further posts .

It has been suspected that mathematicians invented the factorial symbol (!) as a way of making mathematics look more exciting; after all, as the value of n increases, its factorial increases incredibly fast.


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You are reading solutions / Python

# Python close() File – Open and Close Files Properly

For anyone working with Python, interacting with files will be unavoidable. The simplest method for opening and closing files is below.

file = open('example_file.txt')  # open the example file
file.close()                     # close it

One of the most common mistakes people make when using files in Python is not closing them afterward. In this lesson, we'll look at some issues caused by leaving files open and how we can avoid these by using the established best practice.

## Why Close Files?

Not closing a file after an interaction is not only a waste of program resources but could also prevent other file interactions and may lead to inconsistencies when running your scripts on different Python versions. Leaving files open is also widely regarded as poor practice, and it's always best to get into good habits early on.

## Example 1: Not closing correctly

Imagine you've got a file, called file_1.txt, store in the same directory as your Python script. This file contains the text "This is the file 1 text".

You want to read the file into Python, display the text, and append a new line to mark that you have read the text file. This process is shown (incorrectly) below:

file = open('file_1.txt','r')        # file is opened in read (r) mode

file = open('file_1.txt', 'a')       # file is opened in append (a) mode
file.write(' and we have read it.')

file = open('file_1.txt','r')        # file is opened in read (r) mode

file.close()
Out:
This is file 1 text
This is file 1 text and we have read it.

The file was closed correctly, right? Wrong.

In this example, we have only closed the file after the most recent open. For some Python versions, the code used might not record the appended text correctly as the file wasn't closed after completing the operation. To demonstrate this, observe what happens if we try to close each file individually at the end as a solution:

f1 = open('file_1.txt','r')

f2 = open('file_1.txt', 'a')
f2.write(' and we have read it.')

f3 = open('file_1.txt','r')

f1.close()
f2.close()
f3.close()
Out:
This is file 1 text
This is file 1 text

We can see that appending to the file from f2 isn't reflected in f3.

Coding like this could also potentially corrupt a file, preventing it from being read or used again in the future. This example is simple, but if you were writing a script that appended extra information to a long file filled with valuable customer data or data that's taken months to web scrape, file corruption would become a costly problem.

To fix this example, we would need to make sure to close the file after writing before trying to read it again:

f1 = open('file_1.txt','r')
f1.close()

f2 = open('file_1.txt', 'a')
f2.write(' and we have read it.')
f2.close()

f3 = open('file_1.txt','r')
f3.close()
Out:
This is file 1 text
This is file 1 text and we have read it.

An easier alternative to manually handling the .close() for each file is to use a context manager.

## Example 2: Context Managers

To close files property, the most straightforward solution that follows best practices is to use what's called a with statement whenever opening a file. This pattern is also known as using a context manager, a fundanmental concept in Python.

Below is an example of using a with statement to make sure files are closed:

with open('file_1.txt','r') as f:

with open('file_1.txt', 'a') as f:
f.write(' and we have read it.')

with open('file_1.txt','r') as f:
print(f.read())
Out:
This is file 1 text and we have read it.
This is file 1 text and we have read it. and we have read it.

The updated code provides the same functionality with less coding required. Using a with open() statement will automatically close a file once the block has completed.

Not only will using a context manager free you from having to remember to close files manually, but it will also make it much easier for others reading your code to see precisely how the program is using the file.

Using with open() is often the right choice, but there may be situations where your project may call for manually closing files.

### Context manager scoping errors

When using a context manager, there are a few errors that could occur. One example of this is a ValueError: I/O operation on closed file. Consider the code below:

with open('file_2.txt', 'a') as f:
message = 'This is the file 2 text'
f.write(message)
Out:
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-5-2deecae51d65> in <module>
1 with open('file_2.txt', 'w') as f:
2     message = 'This is the file 2 text'
----> 3 f.write(message)

ValueError: I/O operation on closed file.

This error is occurs because we've used a write() file operation outside of with open() scope. Since the context manager automatically closes the file after executing the indented code, Python cannot perform any subsequent file operations. We can easily fix this problem by keeping everything under the with open() scope, like so:

with open('file_2.txt', 'a') as f:
message = 'This is the file 2 text'
f.write(message)

The new script executes successfully, writing the message to the file before closing it.

## Summary

Anyone working with Python will most likely need to work with files extensively. An essential part of file interaction is making sure to close them when you have finished!

Leaving files open can cause many problems, especially in large projects that rely on a wide range of files.

As we've discussed, a simple way to avoid the complications that leaving files open can potentially cause is to open your files using a context manager, which automatically closes files for you. Following this pattern will save you a job while preventing wasted resources and version inconsistencies. Most importantly, using with statements will help make your code look clean and professional.


text:
## anonymous 5 years ago i am doing imaginary numbers i=-1 x(x+4)-=12. the answer is x=-2+ or - 2i sq. root 2

1. anonymous

$x(x+4)-12 the answer is -2 \pm2 i \sqrt{2}$

2. anonymous

how do i get to this answer?


text:
# Synopsis: Mirror, Mirror in Free Space

A Fabry-Pérot cavity exhibits the same behavior when an atom replaces a mirror.

Electromagnetic fields change with boundary conditions around them. For example, a cavity made up of two end mirrors alters the field inside it.

In a paper in Physical Review Letters, Gabriel Hétet and colleagues at the University of Innsbruck in Austria report an experiment in which one mirror of a cavity is replaced by a barium ion $30$ cm away, trapped in free space by an applied electric field. The presence of the single mirror alters the field around the atom, which also acts like a mirror and coherently reflects incident light. The experiment demonstrates how the presence of the mirror alters the way the atom couples to the laser light and changes the atomic coupling constant. The group observed that the setup behaves just like a simple cavity with two parallel mirrors, called a Fabry-Pérot cavity.

This part-cavity, part-free-space setup—an advance in the field of cavity quantum electrodynamics—can be used for storage and retrieval of single photons from the atom, necessary for quantum communication protocols. – Sonja Grondalski

### Announcements

More Announcements »

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Researchers have developed a computer algorithm that doesn’t solve problems but instead creates them for the purpose of evaluating quantum computers. Read More »

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By bathing a superconducting qubit in squeezed light, researchers have been able to confirm a decades-old prediction for the resulting phase-dependent spectrum of resonance fluorescence. Read More »

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### Synopsis: Polarons Drive a Magneto-Optical Effect

A surprisingly large magneto-optical response occurs when mobile electrons in a cooled material become trapped by their interaction with the surrounding lattice. Read More »


text:
## A community for students. Sign up today

Here's the question you clicked on:

## anonymous 5 years ago Find the probability that a point X, selected randomly on line LQ, is on the given segment. 1)Line LM 2)line NP 3)line OQ 4)line MQ

• This Question is Closed
1. anonymous

2. anonymous

its upside down sorry about that

3. anonymous

The probability would be the ratio of the segment lengths to the total. Let $S_{AB} = The\ length\ of\ a\ segment\ from\ A\ to\ B$ $P_{AB} = The\ probability\ for\ a\ random\ point\ to\ be\ on\ the\ segment\ AB$ $\rightarrow P_{LM} = S_{LM}/S_{LQ}$ $\rightarrow P_{NP} = S_{NP}/S_{LQ}$ $\rightarrow P_{OQ} = S_{OQ}/S_{LQ}$ $\rightarrow P_{MQ} = S_{MQ}/S_{LQ}$ etc.

4. anonymous

The probability of x being on a line segment is the length of the segment divided by the total length of the line. Which part of this is confusing?

5. anonymous

so the first one would be 2/17

6. anonymous

Nearly. 2 is the length of LM, but the length of LQ is not 17.

7. anonymous

so 2/24

8. anonymous

Yes.

9. anonymous

Np=12/24 OQ= 12/24 MQ= 19/24

10. anonymous

MQ is 22 units in length. Otherwise those numbers are correct.

11. anonymous

okay thank you

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text:
# Crystal Structure of the Interleukin-4/Receptor α Chain Complex Reveals a Mosaic Binding Interface

@article{Hage1999CrystalSO,
title={Crystal Structure of the Interleukin-4/Receptor $\alpha$ Chain Complex Reveals a Mosaic Binding Interface},
author={T. Hage and W. Sebald and P. Reinemer},
journal={Cell},
year={1999},
volume={97},
pages={271-281}
}
• Published 1999
• Biology, Medicine
• Cell
Interleukin-4 (IL-4) is a principal regulatory cytokine during an immune response and a crucial determinant for allergy and asthma. IL-4 binds with high affinity and specificity to the ectodomain of the IL-4 receptor alpha chain (IL4-BP). Subsequently, this intermediate complex recruits the common gamma chain (gamma c), thereby initiating transmembrane signaling. The crystal structure of the intermediate complex between human IL-4 and IL4-BP was determined at 2.3 A resolution. It reveals a… Expand
189 Citations
The Structure of the Interleukin-15α Receptor and Its Implications for Ligand Binding*
• Medicine, Biology
• Journal of Biological Chemistry
• 2006
The model shows that, rather than the familiar hydrophobic forces dominating the interaction interface between cytokines and their cognate receptors, the interaction between the IL-15 andIL-15Rα complex involves a large network of ionic interactions, which is essential for the biological effects of this important cytokine. Expand
Structure of the Quaternary Complex of Interleukin-2 with Its α, ß, and γc Receptors
• Biology
• Science
• 2005
The structure of γc provides a rationale for loss-of-function mutations found in patients with X-linked severe combined immunodeficiency diseases (X-SCID) and provides a framework for other γ c-dependent cytokine-receptor interactions and for the engineering of improved IL-2 therapeutics. Expand
Molecular and Structural Basis of Cytokine Receptor Pleiotropy in the Interleukin-4/13 System
The crystal structures of the complete set of type I and type II type I receptor heterodimer signals with different potencies in response to IL-4 versus IL-13 suggest that the extracellular cytokine-receptor interactions are modulating intracellular membrane-proximal signaling events. Expand
Structural reorganization of the interleukin-7 signaling complex
• C. McElroy, +4 authors S. Walsh
• Chemistry, Medicine
• Proceedings of the National Academy of Sciences
• 2012
An unliganded receptor structure in the common gamma-chain (γc) family of receptors and cytokines is reported and a molecular mechanism can be used to explain recently discovered IL-7– and γc-independent gain-of-function mutations inIL-7Rα from B- and T-cell acute lymphoblastic leukemia patients. Expand
Functional epitope of common γ chain for interleukin-4 binding
• Biology
• 2002
The alanine-scanning mutational analysis results provide a basis for elucidating the molecular recognition mechanism in the IL-4 receptor system and a paradigm for other γc-dependent cytokine receptor systems. Expand
Structure, binding, and antagonists in the IL-4/IL-13 receptor system.
• Biology, Medicine
• Biochimica et biophysica acta
• 2002
Detailed knowledge of the site of interaction of IL-4 andIL-4Ralpha has been gained by structure analysis of the complex of these two proteins and through functional studies employing mutants of IL -4 and its receptor subunits, and the hitherto elusive goal of designing small molecular mimetics may be feasible. Expand
Solution structure of human IL-13 and implication for receptor binding.
• Chemistry, Medicine
• Journal of molecular biology
• 2001
Based on the available structural and mutational data, an IL-13/IL-4Ralpha model and a sequential mechanism for forming the signaling heterodimer is proposed forIL-13. Expand
Structure analysis of the IL-5 ligand-receptor complex reveals a wrench-like architecture for IL-5Rα.
The structure demonstrates that for steric reasons, homodimeric IL-5 can bind only one receptor molecule, even though two equivalent receptor-binding sites exist, and advances in molecular understanding provided by this structure are of greatest value. Expand
Solution structure of interleukin-13 and insights into receptor engagement.
• Biology, Medicine
• Journal of molecular biology
• 2001
The first experimentally determined high-resolution structure of human interleukin-13 (IL-13) is reported, significantly different from an earlier homology model, which could have led to improper estimation of receptor interaction surfaces and design of mutational experiments. Expand
The high-affinity interaction of human IL-4 and the receptor alpha chain is constituted by two independent binding clusters.
• Chemistry, Medicine
• Journal of molecular biology
• 2002
IL4-BP, the extracellular binding domain of the IL-4 receptor alpha chain, contains a high-affinity binding epitope for IL-4 (K(D) 150 pM). Previous results on the crystal structure of theExpand

#### References

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The interleukin‐4 site‐2 epitope determining binding of the common receptor γ chain
• Biology
• 1998
The present results are in accordance with a two-step-dimerisation mechanism forIL-4 receptor activation, where solute IL-4 at physiological concentrations binds first via the high-affinity site 1 to the α chain only, since the affinity of IL- 4 site 2 for γc is too low. Expand
Crystals of a 1:1 complex between human interleukin-4 and the extracellular domain of its receptor alpha chain.
• Chemistry, Medicine
• European journal of biochemistry
• 1998
The crystals are isomorphous to that of the complex with normal IL-4 and therefore can be used to solve the crystallographic phase problem by the method of multiple anomalous diffraction (MAD). Expand
A mixed-charge pair in human interleukin 4 dominates high-affinity interaction with the receptor alpha chain.
• Biology, Medicine
• Proceedings of the National Academy of Sciences of the United States of America
• 1997
It is revealed that the high-affinity binding of IL-4 originates from a continuous patch of a few mostly polar or charged amino acid side chains located on helices A and C, which provides the basis for an understanding of molecular recognition in cytokine receptor complexes and for anIL-4 antagonist design. Expand
Global and Local Determinants for the Kinetics of Interleukin‐4/Interleukin‐4 Receptor α Chain Interaction
• Chemistry
• 1996
An engineered interleukin-4-binding protein (IL4-BP) representing the extracellular domain of the human interleukin-4 (IL-4) receptor α chain was expressed in Sf9 cells. The purified IL4-BP wasExpand
Global and local determinants for the kinetics of interleukin-4/interleukin-4 receptor alpha chain interaction. A biosensor study employing recombinant interleukin-4-binding protein.
• Medicine
• European journal of biochemistry
• 1996
The results indicate that the high-affinity binding of IL-4 to its receptor (Kd approximately 100 pM) is mainly the result of an unusually high association rate. Expand
The interleukin-2 and interleukin-4 receptors studied by molecular modelling.
• Biology, Medicine
• Structure
• 1994
The models offer structural explanations for observed behaviour such as the effects of mutation of the A- and D-helices of the cytokines and may be of use in the identification of residues which may interact in the ligand-receptor interfaces, and which would be worthy of further investigation. Expand
The interleukin-2 receptor gamma chain: its role in the multiple cytokine receptor complexes and T cell development in XSCID.
The molecular identification of the gamma chain brought a grasp of the structures and functions of the cytokine receptor and an in-depth understanding of the cause of human XSCID. Expand
Interleukin-2 receptor gamma chain: a functional component of the interleukin-4 receptor.
The observation thatIL-2R gamma is a functional component of the IL-4 receptor, together with the finding that IL- 2R gamma associates with theIL-7 receptor, begins to elucidate why deficiency of this common gamma chain has a profound effect on lymphoid function and development, as seen in X-linked severe combined immunodeficiency. Expand
Sharing of the interleukin-2 (IL-2) receptor gamma chain between receptors for IL-2 and IL-4.
The results suggest that theIL-2 receptor gamma chain is functionally involved in the IL-4 receptor complex, which is an indispensable subunit for IL-2 binding and intracellular signal transduction. Expand
A model of the complex between interleukin‐4 and its receptors
• Biology, Medicine
• Proteins
• 1995
A three‐dimensional model of interleukin‐4 (IL‐4) bound to one molecule each of the high‐ and low‐affinity receptors (IL‐4R and IL‐2Rγ) was built, using the crystal structure of the complex of humanExpand


text:
My watch list
my.chemeurope.com

# Magnetization

Magnetization is a property of some materials (e.g. magnets) that describes to what extent they are affected by magnetic fields, and also determines the magnetic field that the material itself creates. Magnetization is defined as the amount of magnetic moment per unit volume. The origin of the magnetic moments that create the magnetization can be either microscopic electric currents corresponding to the motion of electrons in atoms, or the spin of the electrons.

In some materials (e.g., ferromagnets) magnetization can exist even without an external magnetic field (spontaneous magnetization). In other types of materials, magnetization is induced only when an external magnetic field is present. Magnetization is not always homogeneous within a body, but rather a function of position.

## Magnetization in Maxwell's equations

The behavior of magnetic fields ($\mathbf{B}$, $\mathbf{H}$), electric fields ($\mathbf{E}$, $\mathbf{D}$), charge density ($\rho\,$), and current density ($\mathbf{J}$) is described by Maxwell's equations. The role of the magnetization is described below.

#### Relations between B, H and M

The magnetization defines the auxiliary magnetic field $\mathbf{H}$ as

$\mathbf{B} = \mu_0\mathbf{(H + M)}$

which is convenient for various calculations.

A relation between $\mathbf{M}$ and $\mathbf{H}$ exists in many materials. In diamagnets and paramagnets, the relation is usually linear:

$\mathbf{M} = \chi_m\mathbf{H}$

where $\chi_m\,$ is called the volume magnetic susceptibility.

In ferromagnets there is no one-to-one correspondence between $\mathbf{M}$ and $\mathbf{H}$ because of hysteresis.

#### Magnetization current

The magnetization $\mathbf{M}$ makes a contribution to the current density $\mathbf{J}$, known as the magnetization current or bound current:

$\mathbf{J_m} = \nabla\times\mathbf{M}$

so that the total current density that enters Maxwell's equations is given by

$\mathbf{J} = \mathbf{J_f} + \nabla\times\mathbf{M} + \frac{\partial\mathbf{P}}{\partial t}$

where $\mathbf{J_f}$ is the electric current density of free charges, the second term is the contribution from the magnetization, and the last term is related to the electric polarization $\mathbf{P}$.

#### Magnetostatics

In the absence of free electric currents and time-dependent effects, Maxwell's equations describing the magnetic quantities reduce to

$\mathbf{\nabla\cdot H} = - \nabla\cdot\mathbf{M} \qquad \mathbf{\nabla\times H} = 0$

These equations can be easily solved in analogy with electrostatic problems where

$\mathbf{\nabla\cdot E} = \frac{\rho}\epsilon_0 \qquad \qquad \qquad \mathbf{\nabla\times E} = 0$

In this sense $-\epsilon_0\nabla\cdot\mathbf{M}$ plays the role of a "magnetic charge density" analogous to the electric charge density $\rho\,$.

Magnetization is volume density of magnetic moment. That is: if a certain volume has magnetization $\mathbf{M}$ then volume element dV has magnetic moment of $d \mathbf{m} = \mathbf{M} dV$.

## Types of magnetism

### Diamagnetism

This is the most common magnetic behavior. The diamagnetic magnetization is proportional and opposing to the applied magnetic field. All materials present a diamagnetic response, although it may be shadowed by stronger magnetic behaviors. Diamagnetism can be explained by the normal response of the orbiting electrons considering the Lenz's law. This is a weak form of magnetism that is non permanent and persists only while external field is applied. The magnitude of induced magnetic moment is very small and in a direction opposite to that of applied field. Therefore, relative permeability is less than 1 and magnetic susceptibility is negative. When placed between the poles of a strong electromagnet, diamagnetic materials are pushed out towards the region where the field is weaker.

### Paramagnetism

Paramagnetic materials present a magnetization that is proportional to the applied field and reinforces it. This arises from the existence of magnetic dipoles in the material. Paramagnetism varies inversely with temperature and is characterized by the material's saturation magnetization.

### Superparamagnetism

Superparamagnetic materials are paramagnetic materials whose magnetization saturates at very large fields. They are obtained using magnetic nanoparticle aggregates with large net magnetic moments. Each particle is a single magnetic domain. Consequently, the alignment of spins under applied field is no longer impeded by domain walls. Above the blocking temperature, thermal vibrations randomly fluctuate the net spins, cancelling one another and the net moment of the collective particles is zero at zero field (no coercive field). If a magnetic field is applied, the particles will align producing a net moment. This behavior is characteristic of paramagnetic materials, but the difference is that each nanoparticle has a large net moment, so the saturation of magnetization occurs at very large fields of several teslas.

### Ferromagnetism

Ferromagnetic materials present a magnetization much larger than other materials. Ferromagnetism arises from the strong coupling between the neighboring magnetic dipoles in the material. Ferromagnetic materials can present spontaneous magnetization, and this gives rise to the hysteresis loops. Ferromagnetic materials can be characterized by their permeability, Curie temperature (temperature of the phase change to paramagnetic behavior), coercive field (field strength needed to demagnetize the material) and remnant magnetization (magnetization at zero external field).


text:
Welcome to symthic forums! We would love if you'd register!
You don't have to be expert in bit baking, everyone is more than welcome to join our community.

You are not logged in.

Hey! If this is your first visit on symthic.com, also check out our weapon damage charts.
Currently we have charts for Battlefield 3, Call of Duty: Black Ops 2, Medal of Honor: Warfighter and Call of Duty: Modern Warfare 3

He's watching...

Posts: 1,791

Date of registration
: May 5th 2012

Platform: PC

Location: Tucson

Battlelog:

Reputation modifier: 11

Monday, February 2nd 2015, 6:55am

### Quoted from "iota-09"

so, today i SHOULD have gone playing, but at the very last second the game was disbanded,and i couldn't take photos or videos of the field, so i'll have to make a comparison with an image i found:

http://i59.tinypic.com/2je03m.jpg

imagine if there were no logs or pieces of wood at the side of the way, think of the way itself being large only 1/5 of what is depicted if not less(also with some broken logs that block it in some places, and you have to vault or crouch to continue walking), and imagine if there was half the trees and a lot more mud, that's basically what most of the field looks like, and also why i need better gloves and boots.

Looks like the woods in the midwestern United States. Go with a leather palmed motorcycle glove and it'll take much more abuse than a Mechanix style glove without the price tag of Kevlar/Nomex "tactical" gloves. Trust me on that.

Something like these:

Icon Compound Mesh Short Gloves - RevZilla

Goaskin leather palm with reinforced areas, mesh texture back to keep your hands cool, and it's armored to protect you from shots to the hand without being bulky. Right around $50. Simple, cheap, and effective. If I didn't like the gloves I already have for the motorcycle I'd probably get those myself. All bike, all the time! ### Spoiler Pinkie Posts: 7,780 Date of registration : Feb 25th 2012 Platform: PC Location: italy Battlelog: Reputation modifier: 19 Monday, February 2nd 2015, 11:42am ### Quoted from "Watcher-45" ### Quoted from "iota-09" so, today i SHOULD have gone playing, but at the very last second the game was disbanded,and i couldn't take photos or videos of the field, so i'll have to make a comparison with an image i found: http://i59.tinypic.com/2je03m.jpg imagine if there were no logs or pieces of wood at the side of the way, think of the way itself being large only 1/5 of what is depicted if not less(also with some broken logs that block it in some places, and you have to vault or crouch to continue walking), and imagine if there was half the trees and a lot more mud, that's basically what most of the field looks like, and also why i need better gloves and boots. Looks like the woods in the midwestern United States. Go with a leather palmed motorcycle glove and it'll take much more abuse than a Mechanix style glove without the price tag of Kevlar/Nomex "tactical" gloves. Trust me on that. Something like these: Icon Compound Mesh Short Gloves - RevZilla Goaskin leather palm with reinforced areas, mesh texture back to keep your hands cool, and it's armored to protect you from shots to the hand without being bulky. Right around$50. Simple, cheap, and effective. If I didn't like the gloves I already have for the motorcycle I'd probably get those myself.

good i'll see if i can find them in italy.
"I'm just a loot whore."

### stuff mostly unrelated to BF4 that interests nobody

bf4
on 13/05/2016
23rd M320FB user on pc(13/05/16)
rush mode score RANK:2794 TOP:2% OUT OF:215398
obliteration mode scoreRANK:994 TOP:1% OUT OF:159466
handgun medals RANK:2236 TOP:2% OUT OF:143874
longest headshot RANK:9512 TOP:4% OUT OF:257589
recon score RANK:10871 TOP:4% OUT OF:274899
general score per minute RANK:10016 TOP:4% OUT OF:294774

bf3
31/3/2012 4:58:

Revives per assault minute RANK: 6019 TOP: 3%
Headshots / kill percentage RANK:25947 TOP:13%
MVP ribbons RANK:18824 TOP:11%

*= 6 if we not count the EOD BOT headshots

### Quoted from "CobaltRose"

@kataklism

ARGUMENT DESTROYED 100

ENEMY KILLED [REASON] JSLICE20 100

WRITING SPREE STOPPED 500

http://i.imgur.com/4X0321O.gif

Posts: 848

Date of registration
: Dec 8th 2013

Platform: PC

Battlelog:

Reputation modifier: 12

Wednesday, February 4th 2015, 7:43am

So... since my paintball marker basically takes airsoft accessories (it has a front Delta Ring for screw-on hand guards and standard buttstock tube), I will ask here.

I am moving my helical magazine from its current "underslung" location (a la PP-19 Bizon) to a direct feed on the side of my marker. That means now I have room on the hand guard front bottom rail for a foregrip. I never ran a foregrip before, so you guys might know better. Same as so many BF4 attachment questions:

or

As much as I harp on using Suppressor + Ergo for my BF4 DMRs, I have recently switched over to running AFG + HB / AFG + Suppressor and my accuracy went up by ~10%. I ended up being much more able to shoot through suppression (success rate went from what felt like 10% to now perhaps 50%), comfortable firing rate while engaging ranged targets went up (33% reduction in V-Recoil per shot is nothing to sneeze at), ADS - Moving for DMRs with mid-range optics and no Ergo turn out to be not as bad as previously thought, and DMR hipfiring is so crappy that even Ergo doesn't help enough.

Still, that's BF4. What about real life?

PS rail space is no issue as I have a 9in barrel shroud.

Salt Miner

Posts: 3,389

Date of registration
: Mar 19th 2014

Platform: Xbox One

Battlelog:

Reputation modifier: 15

Wednesday, February 4th 2015, 9:21am

Some highlights from the 250 or so pictures my friend took of our game on Saturday! First pic is me.

Who has fun, wins.

I'll be there... around every corner... in every empty room...

Posts: 3,446

Date of registration
: Nov 8th 2013

Platform: PC

Location: Boston's Trashcan, USA

Battlelog:

Reputation modifier: 14

Wednesday, February 4th 2015, 7:00pm

@BleedingUranium
No face mask? That's playing a bit dangerous <.<

Anyways: getting a rubber trench knife/tanto. OFC now the issue is finding a holster big enough to fit it because it has a massive 7" blade on it...
Who knows what evil lurks in the hearts of men?

### Quoted from "Spectacle"

I once tried to burn a bridge, but due to the half-assed levolution implementation it kept standing up just fine.

### Quoted from "DeathOfTheDodo"

Compared to the MTAR, the PDR has the TTK of your average pack of cigarettes.

The World Champion, on the Tactical Light:

### Quoted from "The World Champion"

"Oh don't mind me, I'm just out of ammo and I'll just POCKET SUN!!!!"

### Quoted from "Sheepnub"

What can't be changed can be banned.

### Quoted from "Gummybear"

I'm completely serious. Well, seriously insane actually

### Quoted from "xESxMaver1ck911"

If only more people left the wheel chair, adjustable hospital bed, and crutches behind and played HC. (aka Regen, Minimap, Killcam)

### Quoted from "hunturk"

Natalia Poklonskaya:
Putin the cute in prosecute since 2014.

### Quoted from "DeathOfTheDodo"

I can't look at my own avatar without having to pee. GG Me. GG.

### Quoted from "DeathOfTheDodo"

I swear to god if you reply with a picture of the AEK I'll mail you a buttplug shaped like one.

### Quoted from "Nick 30075"

The one time when "it's only three inches" is a good thing.

### Quoted from "RRR3186"

How do you guys control your FAMAS burst length since it fires its mag in just 1,5 seconds))
I consider each magazine a burst. :/

### Quoted from "ViperFTW"

This is Symthic, we don't do "feels" around here

Cannot into Rankine

Posts: 1,234

Date of registration
: Dec 19th 2013

Platform: PC

Location: 45.4000° N, 75.6667° W

Battlelog:

Reputation modifier: 8

Wednesday, February 4th 2015, 7:56pm

here you got

### REP

Apparently we do have a facebook... You should go ahead and like it

If only people would talk back it wouldn't be so lonely.[In B4 I get overwhelmed]

"Talk sense to a fool and he calls you foolish." - Euripides, Bacchae

"You are not entitled to your opinion. You are entitled to your informed opinion. No one is entitled to be ignorant."- Harlan Ellison

### Quoted from "LeGarcon"

Hardline is a fun and sometimes silly Cops and Robbers sorta thing and I think that's great. Or it would be if it didn't suck.

### Quoted from "Rezal"

intel best ethernet controllers

ammo regen pls

### Quoted from "tehmoriz"

TL;DR: all vehicles in BF4 are screwed up in some way or another.

### Quoted from "Aenonar"

I know I shouldn't be surprised that something like that even exists but....

### Quoted from "NoctyrneSAGA"

Salt Miner

Posts: 3,389

Date of registration
: Mar 19th 2014

Platform: Xbox One

Battlelog:

Reputation modifier: 15

Thursday, February 5th 2015, 12:23am

### Quoted from "CobaltRose"

@BleedingUranium
No face mask? That's playing a bit dangerous <.<

I'm not the only one, and I've never seen it be a problem. I've been hit in the face a few times, nothing special really; it doesn't hit any harder than any other bare skin, like arms or wherever (I took off the tunic later and had my N7 t-shirt under the vest).
Who has fun, wins.

Pinkie

Posts: 7,780

Date of registration
: Feb 25th 2012

Platform: PC

Location: italy

Battlelog:

Reputation modifier: 19

Thursday, February 5th 2015, 2:09am

### Quoted from "CobaltRose"

@BleedingUranium
No face mask? That's playing a bit dangerous <.<

I'm not the only one, and I've never seen it be a problem. I've been hit in the face a few times, nothing special really; it doesn't hit any harder than any other bare skin, like arms or wherever (I took off the tunic later and had my N7 t-shirt under the vest).

the problem isn't if they hit you on the face, it's if they hit you on the teeth... i run with a neck-heater, most times i forget to bring it up, but it's stille better than nothing, i don't want to lose my teeth for as bad as they are.
"I'm just a loot whore."

### stuff mostly unrelated to BF4 that interests nobody

bf4
on 13/05/2016
23rd M320FB user on pc(13/05/16)
rush mode score RANK:2794 TOP:2% OUT OF:215398
obliteration mode scoreRANK:994 TOP:1% OUT OF:159466
handgun medals RANK:2236 TOP:2% OUT OF:143874
longest headshot RANK:9512 TOP:4% OUT OF:257589
recon score RANK:10871 TOP:4% OUT OF:274899
general score per minute RANK:10016 TOP:4% OUT OF:294774

bf3
31/3/2012 4:58:

Revives per assault minute RANK: 6019 TOP: 3%
Headshots / kill percentage RANK:25947 TOP:13%
MVP ribbons RANK:18824 TOP:11%

*= 6 if we not count the EOD BOT headshots

### Quoted from "CobaltRose"

@kataklism

ARGUMENT DESTROYED 100

ENEMY KILLED [REASON] JSLICE20 100

WRITING SPREE STOPPED 500

http://i.imgur.com/4X0321O.gif

I'll be there... around every corner... in every empty room...

Posts: 3,446

Date of registration
: Nov 8th 2013

Platform: PC

Location: Boston's Trashcan, USA

Battlelog:

Reputation modifier: 14

Thursday, February 5th 2015, 2:59am

### Quoted from "CobaltRose"

@BleedingUranium
No face mask? That's playing a bit dangerous <.<
I'm not the only one, and I've never seen it be a problem. I've been hit in the face a few times, nothing special really; it doesn't hit any harder than any other bare skin, like arms or wherever (I took off the tunic later and had my N7 t-shirt under the vest).
wasn't thinking about that: I was thinking more about the fact that you could get a tooth shot out. Then again, I had braces so I kinda don't wanna fuck up the pearly whites while they're all in line

@Krolman
Who knows what evil lurks in the hearts of men?

### Quoted from "Spectacle"

I once tried to burn a bridge, but due to the half-assed levolution implementation it kept standing up just fine.

### Quoted from "DeathOfTheDodo"

Compared to the MTAR, the PDR has the TTK of your average pack of cigarettes.

The World Champion, on the Tactical Light:

### Quoted from "The World Champion"

"Oh don't mind me, I'm just out of ammo and I'll just POCKET SUN!!!!"

### Quoted from "Sheepnub"

What can't be changed can be banned.

### Quoted from "Gummybear"

I'm completely serious. Well, seriously insane actually

### Quoted from "xESxMaver1ck911"

If only more people left the wheel chair, adjustable hospital bed, and crutches behind and played HC. (aka Regen, Minimap, Killcam)

### Quoted from "hunturk"

Natalia Poklonskaya:
Putin the cute in prosecute since 2014.

### Quoted from "DeathOfTheDodo"

I can't look at my own avatar without having to pee. GG Me. GG.

### Quoted from "DeathOfTheDodo"

I swear to god if you reply with a picture of the AEK I'll mail you a buttplug shaped like one.

### Quoted from "Nick 30075"

The one time when "it's only three inches" is a good thing.

### Quoted from "RRR3186"

How do you guys control your FAMAS burst length since it fires its mag in just 1,5 seconds))
I consider each magazine a burst. :/

### Quoted from "ViperFTW"

This is Symthic, we don't do "feels" around here

Cannot into Rankine

Posts: 1,234

Date of registration
: Dec 19th 2013

Platform: PC

Location: 45.4000° N, 75.6667° W

Battlelog:

Reputation modifier: 8

Thursday, February 5th 2015, 3:46am

### Quoted from "CobaltRose"

@BleedingUranium
No face mask? That's playing a bit dangerous <.<
I'm not the only one, and I've never seen it be a problem. I've been hit in the face a few times, nothing special really; it doesn't hit any harder than any other bare skin, like arms or wherever (I took off the tunic later and had my N7 t-shirt under the vest).
wasn't thinking about that: I was thinking more about the fact that you could get a tooth shot out. Then again, I had braces so I kinda don't wanna fuck up the pearly whites while they're all in line

@Krolman

I only needed to wear braces for 9 months , and if I remember someday I will show my sniper.
Apparently we do have a facebook... You should go ahead and like it

If only people would talk back it wouldn't be so lonely.[In B4 I get overwhelmed]

"Talk sense to a fool and he calls you foolish." - Euripides, Bacchae

"You are not entitled to your opinion. You are entitled to your informed opinion. No one is entitled to be ignorant."- Harlan Ellison

### Quoted from "LeGarcon"

Hardline is a fun and sometimes silly Cops and Robbers sorta thing and I think that's great. Or it would be if it didn't suck.

### Quoted from "Rezal"

intel best ethernet controllers

ammo regen pls

### Quoted from "tehmoriz"

TL;DR: all vehicles in BF4 are screwed up in some way or another.

### Quoted from "Aenonar"

I know I shouldn't be surprised that something like that even exists but....


text:
# Lesson 3Building the Perfect SquareDevelop Understanding

## Jump Start

Graph each function.

## Learning Focus

Find the square of a binomial expression.

Recognize a perfect square trinomial.

Create perfect squares from partial areas.

Find relationships between terms in a perfect square trinomial.

How can we use models to find equivalent expressions for perfect squares?

## Open Up the Math: Launch, Explore, Discuss

Optima has a quilt shop where she sells many colorful quilt blocks for people who want to make their own quilts. She has quilt designs that are made so that they can be sized to fit any bed. She bases her designs on quilt squares that can vary in size, so she calls the length of the side for the basic square , and the area of the basic square is the function . In this way, she can customize the designs by making bigger squares or smaller squares.

### 1.

If Optima adds to the side of the square, what is the area of the square?

When Optima draws a pattern for the square in problem 1, it looks like this:

### 2.

Use both the diagram and the equation to explain why the area of the quilt block square, is also equal to .

The customer service representatives at Optima’s shop work with customers and write up the orders based on the area of the fabric needed. As you can see from problem 2, there are two ways that customers can call in and describe the area of the quilt block. One way describes the length of the sides of the block, and the other way describes the areas of each of the sections of the block.

For each of the following quilt blocks, draw the diagram of the block and write two equivalent equations for the area of the block.

### 3.

Block with side length .

### 4.

Block with side length .

### 5.

What patterns do you notice when you relate the diagrams to the two expressions for the area?

### 6.

Optima likes to have her little dog, Clementine, around the shop. One day, Clementine got a little hungry, and started to chew up the orders. When Optima found the orders, one of them was so chewed up that there were only partial expressions for the area remaining. Help Optima by completing each of the following expressions for the area so that they describe a perfect square. Then, write the two equivalent equations for the area of the square.

### 7.

#### a.

If is a perfect square, what is the relationship between and ?

#### b.

What process can be used to find and complete the square, given ?

#### c.

Will this strategy work if is negative? Why or why not?

#### d.

Will the strategy work if is an odd number? What happens to if is odd?

### 8.

One of the new customer service representatives thinks she doesn’t need to draw diagrams anymore because she found a great shortcut. She writes . Examine her thinking. Is she right, or does she need to revise her strategy? What suggestions would you make, and how would they help?

Prove that , either algebraically or using a diagram.

## Takeaways

If is a perfect square,

The square of a binomial:

An example is:

## Adding Notation, Vocabulary, and Conventions

Binomial:

Trinomial:

Squaring a binomial:

## Lesson Summary

In this lesson, we connected area models for multiplication to show how to multiply binomials to get a perfect square trinomial. We learned to recognize a perfect square trinomial by looking for a relationship between the second and third terms. We also worked to create a perfect square when given the first two terms of a trinomial.

## Retrieval

### 1.

Find the -intercept and the -intercept, and graph the equation.

-intercept:

-intercept:

Each of the following equations has just one intercept; find it and state whether it is an -intercept or - intercept.


text:
## Taiwanese Journal of Mathematics

### Proof of a Conjecture of Farkas and Kra

Nian Hong Zhou

#### Abstract

In this paper we prove a conjecture of Farkas and Kra, which is a modular equation involving a half sum of certain modular form of weight $1$ for congruence subgroup $\Gamma_1(k)$ with any prime $k$. We prove that their conjecture holds for all odd integers $k \geq 3$. A new modular equation of Farkas and Kra type is also established.

#### Article information

Source
Taiwanese J. Math., Volume 23, Number 6 (2019), 1317-1326.

Dates
Revised: 23 January 2019
Accepted: 10 March 2019
First available in Project Euclid: 13 March 2019

https://projecteuclid.org/euclid.twjm/1552442420

Digital Object Identifier
doi:10.11650/tjm/190301

Mathematical Reviews number (MathSciNet)
MR4033547

Zentralblatt MATH identifier
07142975

#### Citation

Zhou, Nian Hong. Proof of a Conjecture of Farkas and Kra. Taiwanese J. Math. 23 (2019), no. 6, 1317--1326. doi:10.11650/tjm/190301. https://projecteuclid.org/euclid.twjm/1552442420

#### References

• G. E. Andrews, A simple proof of Jacobi's triple product identity, Proc. Amer. Math. Soc. 16 (1965), 333–334.
• H. M. Farkas and I. Kra, Theta Constants, Riemann Surfaces and the Modular Group: An introduction with applications to uniformization theorems, partition identities and combinatorial number theory, Graduate Studies in Mathematics 37, American Mathematical Society, Providence, RI, 2001.
• ––––, On theta constant identities and the evaluation of trigonometric sums, in: Complex Manifolds and Hyperbolic Geometry (Guanajuato, 2001), 115–131, Contemp. Math. 311, Amer. Math. Soc., Providence, RI, 2002.
• E. T. Whittaker and G. N. Watson, A Course of Modern Analysis: An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions, Fourth edition, Cambridge University Press, New York, 1962.


text:
# Simulating or generating datasets for testing modelling methodologies

I'm interested to know what a good practice is when generating/simulating data for testing or comparing modelling methods. I'm focusing on linear models and measuring prediction accuracy and measuring parameter estimate accuracy.

I'm looking at prediction accuracy (PSE) in terms of cross validated squared prediction error and parameter estimate accuracy in terms of $\|\beta-\hat{\beta}\|$ (MSE).

I have attempted a bunch of simulations using what I believe to be a 'sound' method, but I am not happy with the results. I'm guessing I'm overlooking something. The issue is that adjacent parameter estimates are highly correlated with each other and I cannot figure out why. Another issue is that while I can generate datasets that show a large difference in parameter estimate accuracy, the prediction performance will be very similar. I am not sure why this happens, I would expect the methods with improved MSE to have much improved PSE over the other methods. Here's a typical example: http://ohyur.com/random/typical.png

I will outline what I have attempted.

• Set the true parameter estimates, $\beta$
• Set correlation structure for the predictor variables
• Generate the predictors with specified correlation structure, $X$. I've tried using mvrnorm() in R to generate N(0,1) columns and mvrunif() function to generate U(-1,1) columns. Both with the specified correlation structure.
• Generate the response using $Y=\mu + X\beta+\epsilon$, where $\epsilon$ is $N(0,\sigma$).

To me, that sounds like a reasonably method of generating data for a linear model.

Here is an example of what the generated data might look like: http://ohyur.com/random/data.png ($\mu=3,$ $\beta=(3,2,-3,5,2,-3,5,5)$, $\sigma=15$, $\text{cor}(x_i,x_j)=0.9^{|i-j|}$

Now in my simulations, in each replication I'm generating data according to the above steps. Then fitting some models, estimating the CV PSE and calculating the MSE and storing the results. Here is a boxplot of the results for 3 different modelling methods over 100 replications. http://ohyur.com/random/psemse.png

Now, most interestingly to me, Here is a pairwise plot of the estimated parameters for a full OLS model. Why are the adjacent predictors correlated? http://ohyur.com/random/estimates.png

One thing I'm coverned about is that in the real world, some predictors are more correlated with the response than others, and not just due to effect size. This isn't really accounted for in my setup. But I'm really just after the simplest possible 'sound' method.

Any insight/comments or recommendations on how to properly generate data for testing liner models would be greatly appreciated.

Not an answer, but more of an extended comment:

Regarding your last figure: your predictors are random and correlated, so why wouldn't you expect the corresponding estimators to be correlated?

A possible flaw with your simulation approach is the fact that your predictors are random. In ordinary linear models (the setting that you seem to be interested in), they are assumed to be fixed. When you evalute your methods for random predictors, you are actually evaluating them in an errors-in-variables setting, where the OLS estimator is neither unbiased nor consistent! Thus the results may be misleading... Not sure if this explains the MSE-PSE-inconsistency though.

• I will try checking out the hat matrix now. I tried an example with fairly uncorrelated predictors ($\rho_{ij}=0.25^{|i-j|}$) and got the following: The data: ohyur.com/random/lowdata.png The estimates: ohyur.com/random/lowestimates.png . So even while the pairwise correlations in the other data are much stronger than these, it is still only the adjacent predictors that are correlated.


text:
ISSN 1000-1239 CN 11-1777/TP

### An Efficient Rainbow Table Structure and Faster Cryptanalytic Time-Memory Trad-off Attack

Gu Chunxiang, Zhu Yuefei, Zheng Yonghui, and Li Zheng

1. (The PLA Information Engineering University, Zhengzhou 450002)
• Online:2014-05-15

Abstract: The time-memory trade-off algorithm is a method for quickly inverting a one-way function using pre-computed tables. Since its first introduction by Hellman, many variants and their analysis results have appeared. Oechslin suggested the so-called rainbow tables which made the method efficient and effective for cryptanalytic attacks. The Checkpoints method is a technique that reduces the number of table lookups performed by the algorithm. In this paper, we propose a new method using a novel design of pre-computed table structure. Compared with the rainbow method, the new design achieves distinct reduction in term of the storage requirement. We apply the method on the example of 8-character possible password composed from a character set of 95 different typeable characters. The number of records stored in the storage can increase with about 70% in our example compared with the rainbow method, which also leads to the increase of 7.8% to 15.6% (for different chain lengths) in terms of the success rate of attack if the storage requirement and the computational complexity remain the same. What’s more, our method can further use Checkpoints technique to reduce the complexity of on line computation with 15% to 20% by setting a checkpoint in the middle position of the chain.


text:
Can we make it equal to x?

$$\sqrt{6 +\sqrt{6 +\sqrt{6 + \ldots}}}$$. This is the famous question. I have to calculate it's value. I found somewhere to the solution to be putting this number equal to a variable $$x$$. That is, $$\sqrt{6 +\sqrt{6 +\sqrt{6 + \ldots}}} = x$$.

Then we square both the sides. $$6 +{\sqrt{6 +\sqrt{6 + \ldots}}} = x^2$$.

Then we replace the square root thing with $$x$$.

$$6 + x = x^2$$ and solve the equation to get the answer as 3.

But I have a doubt that what type of number is this? Is it a real number or not? And if it isn't, how can we perform mathematical operations on it?

• It's the limit of a sequence of real numbers, so, if it exists, it's real. – Gerry Myerson Nov 14 '18 at 11:49
• $$3=\sqrt{6+3}=\sqrt{6+\sqrt{6+3}}=\cdots$$ – lab bhattacharjee Nov 14 '18 at 11:55
• Such expressions are called Nested Radicals – Yadati Kiran Nov 14 '18 at 11:56
• You have proven that if the sequence converges, then it converges to 3. The next step would be to prove that it actually does converge. – user3482749 Nov 14 '18 at 11:56
• @GerryMyerson True. However, it's worth noting that one can easily make a mistake from similar (but incorrect) reasoning. For example, a limit of a sequence of rational numbers is not necessarily rational (but it is real!). – Eff Nov 14 '18 at 11:56

Why care must be taken

Equating any such infinite expression to a real number must be done with a hint of caution, because the expression need not have a real value. For example, setting $$x = 1-1+1-1+...$$ is not correct, since one sees that $$1-x = 1+(-1+1-1+1) = x$$, so $$x = \frac 12$$ which is absurd from an addition point of view : whenever you truncate the series, it always has value either $$1$$ or $$0$$, so where does $$\frac 12$$ comes from? With this logic, it is safe to say $$1-1+1-...$$ does not evaluate to any finite real number.

However, once we can confirm that the result of such an expression is real and well defined, then we can play with them as we do with real numbers.

To confirm that $$\sqrt{6+\sqrt{6+\sqrt{6+...}}}$$ is a finite real number, we need the language of sequences.

I won't go very far in, but essentially, if we define a sequence of reals by $$x_1 = \sqrt 6$$ and $$x_{n+1} = \sqrt{6+x_n}$$, then $$x_ 2 = \sqrt{6+\sqrt 6}$$, $$x_3 = \sqrt{6+\sqrt{6+\sqrt 6}}$$, and eventually, $$x_n$$ resembles more and more the expression that we are given to evaluate.

EDITED : I have modified the steps required for showing that $$x_n$$ is a convergent sequence, to the real number $$3$$.

• It is easy to see that $$x_n$$ is bounded. It is clearly positive for all $$n$$, and can be shown to be bounded by $$3$$ above by induction.

• $$a_n$$ is an increasing sequence can also be shown easily. Any bounded increasing sequence is convergent.

• Once convergence is shown, we can then assume that $$\lim x_n = L$$ exists, and then use continuity to take limits on $$x_{n+1} = \sqrt{6+x_n}$$ to see that $$L = \sqrt{6+L}$$. But $$L \geq 0$$ must happen. Thus, $$L=3$$ is the limit, and hence the value of the expression.

Versatility of sequences

To add to this, sequences also offer versatility. A similar question may be asked : what is: $$6+\frac{6}{6+\frac{6}{6+\frac{6}{6+...}}}$$

What we do here is the same : use the language of sequences, by defining $$x_1 = 6$$ and $$x_{n+1} = 6 + \frac{6}{x_n}$$. Once again, we can check convergence i.e. that this quantity is a finite real number(But on this occasion, the sequence rather oscillates around the point of convergence before converging). Next, we can use limits to deduce that if $$L$$ is the value then it satisfies $$L = 6+ \frac 6L$$, which gives one reasonable candidate, $$3+\sqrt{15}$$. So, this expression is actually equal to $$3+\sqrt{15}$$.

It's not easy all the time!

However, the approach using sequences doesn't always give immediate rewards. For example, you could ask for the following : $$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+...}}}}$$

which also looks like a nested radical.Can we find a sequence which, for large $$n$$, looks like this expression? Try to write one down, which you can work with.

Anyway, the answer to the above expression is $$3$$! To see this, we need to use "reverse nesting" : $$3 = \sqrt{9} = \sqrt{1+2\cdot 4} = \sqrt{1+2\sqrt{16}} = \sqrt{1+2\sqrt{1+15}} \\ = \sqrt{1+2\sqrt{1+3\sqrt{25}}} = \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{36}}}} \\= \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{49}}}}} =...$$

And just breathe in, breathe out. Ramanujan, class ten I believe.

EDIT

The nested radical method is wrong, from a rigorous point of view, for the reason pointed out in the comments. However, there is a rigorous proof here. The proof by Yiorgos Smyrlis is brilliant.

Note that the "nested radical" method can be used for the earlier problem as well, by $$3 = \sqrt{6+3} = \sqrt{6+\sqrt{6+3}} = ...$$, but this is unrigorous, only providing intuition. You can try to see if something intuitive can be derived for the continued fraction.

• "Anyway, the answer to the above expression is 3!" Thought you meant 3 factorial at first :) – Aaron Stevens Nov 14 '18 at 20:12
• @AaronStevens $3! \neq 3$! – Todd Sewell Nov 14 '18 at 20:27
• @ToddSewell Yes I know this. – Aaron Stevens Nov 14 '18 at 20:29
• The "reverse nesting" approach seems suspicious. Any number $x$ has a finite expansion $x = \sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + 4\sqrt{1 + 5\sqrt{C}}}}}$ for some constant $C$, which we can keep extending arbitrarily as long as we don't care what happens to $C$. In this way you can justify that the infinite radical is equal to anything. – Misha Lavrov Nov 14 '18 at 23:59
• @MishaLavrov The "nested radical" is wrong, for the reasons you have mentioned. I should add that in the answer above. Indeed, there is a way of working with the above quantity with the help of sequences. I shall add a link to the answer above. – астон вілла олоф мэллбэрг Nov 15 '18 at 6:02

Let $$a_1= \sqrt{6}$$ and $$a_{n+1}=\sqrt{6+a_n}$$ for $$n \ge 1$$.

It is easy to see, by induction, that $$(a_n)$$ is increasing and that $$0 \le a_n \le 3$$ for all $$n$$. Hence $$(a_n)$$ is convergent. If $$x$$ is the limit of $$(a_n)$$, then we have $$x^2-x-6=0$$, thus $$x=3$$ or $$x=-2$$. Since all $$a_n \ge 0$$, we have $$x=3.$$

• Thanks for making sure this series is convergent. – Aayush Aggarwal Nov 14 '18 at 12:45


text:
## Abstract

Control calculation ping-pong is the process of iteratively improving a statistical analysis by comparing results from two independent analysis approaches until agreement. We use the daff package to simplify the comparison of the two results and illustrate its use by a case study with two statisticians ping-ponging an analysis using dplyr and SQL, respectively.

Clip art is based on work by Gan Khoon Lay available under a CC BY 3.0 US license.

## Introduction

If you are a statistician working in climate science, data driven journalism, official statistics, public health, economics or any related field working with real data, chances are that you have to perform analyses, where you know there is zero tolerance for errors. The easiest way to ensure the correctness of such an analysis is to check your results over and over again (the iterated 2-eye principle). A better approach is to pair-program the analysis by either having a colleague read through your code (the 4-eye principle) or have a skilled colleague completely redo your analysis from scratch using her favorite toolchain (the 2-mindset principle). Structured software development in the form of, e.g. version control and unit tests, provides valuable inspiration on how to ensure the quality of your code. However, when it comes to pair-programming analyses, surprisingly many steps remain manual. The daff package provides the equivalent of a diff statement on data frames and we shall illustrate its use by automatizing the comparison step of the control calculation ping-pong process.

## The Case Story

Ada and Bob have to calculate their country’s quadrennial official statistics on the total income and number of employed people in fitness centers. A sample of fitness centers is asked to fill out a questionnaire containing their yearly sales volume, staff costs and number of employees. The present exposition will for the sake of convenience ignore the complex survey part of the problem and just pretend that the sample corresponds to the population (complete inventory count).

### The Data

After the questionnaire phase, the following data are available to Ada and Bob.

Id Version Region Sales Volume Staff Costs People
01 1 A 23000 10003 5
02 1 B 12200 7200 1
03 1 NA 19500 7609 2
04 1 A 17500 13000 3
05 1 B 119500 90000 NA
05 2 B 119500 95691 19
06 1 B NA 19990 6
07 1 A 19123 20100 8
08 1 D 25000 100 NA
09 1 D 45860 32555 9
10 1 E 33020 25010 7

Here Id denotes the unique identifier of the sampled fitness center, Version indicates the version of a center’s questionnaire and Region denotes the region in which the center is located. In case a center’s questionnaire lacks information or has inconsistent information, the protocol is to get back to the center and have it send a revised questionnaire. All Ada and Bob now need to do is aggregate the data per region in order to obtain region stratified estimates of:

• the overall number of fitness centres
• total sales volume
• total staff cost
• total number of people employed in fitness centres

However, the analysis protocol instructs that only fitness centers with an annual sales volume larger than or equal to €17,500 are to be included in the analysis. Furthermore, if missing values occur, they are to be ignored in the summations. Since a lot of muscle will be angered in case of errors, Ada and Bob agree on following the 2-mindset procedure and meet after an hour to discuss their results. Here is what each of them came up with.

Ada loves the tidyverse and in particular dplyr. This is her solution:

ada <- fitness %>% na.omit() %>% group_by(Region,Id) %>% top_n(1,Version) %>%
group_by(Region) %>%
filter(Sales Volume >= 17500) %>%
summarise(NoOfUnits=n(),
Sales Volume=sum(Sales Volume),
Staff Costs=sum(Staff Costs),
People=sum(People))
ada
## # A tibble: 4 x 5
##   Region NoOfUnits Sales Volume Staff Costs People
##   <chr>      <int>          <int>         <int>  <int>
## 1 A              3          59623         43103     16
## 2 B              1         119500         95691     19
## 3 D              1          45860         32555      9
## 4 E              1          33020         25010      7

### Bob

Bob has a dark past as a relational database management system (RDBMS) developer and, hence, only recently experienced the joys of R. He therefore chooses a no-SQL-server-necessary SQLite within R approach. The hope is that in big data situation this might be a little more speedy than base R:

library(RSQLite)
db <- dbConnect(SQLite(), dbname = file.path(filePath,"Test.sqlite"))
##Move fitness data into the ad-hoc DB
dbWriteTable(conn = db, name = "fitness", fitness, overwrite=TRUE, row.names=FALSE)
##Query using SQL
bob <- dbGetQuery(db, "
SELECT Region,
COUNT(*) As NoOfUnits,
SUM([Sales Volume]) As [Sales Volume],
SUM([Staff Costs]) AS [Staff Costs],
SUM(People) AS People
FROM fitness
WHERE [Sales Volume] > 17500 GROUP BY Region
")
bob
##   Region NoOfUnits Sales Volume Staff Costs People
## 1   <NA>         1        19500        7609      2
## 2      A         2        42123       30103     13
## 3      B         2       239000      185691     19
## 4      D         2        70860       32655      9
## 5      E         1        33020       25010      7

Update: An alternative approach with less syntactic overhead would have been the sqldf package, which has a standard SQLite backend and automagically handles the import of the data.frame into the DB using the RSQLite pkg.

##Load package
suppressPackageStartupMessages(library(sqldf))
##Ensure SQLite backend.
options(sqldf.driver = "SQLite")
##Same query as before
bob <- sqldf("
SELECT Region,
COUNT(*) As NoOfUnits,
SUM([Sales Volume]) As [Sales Volume],
SUM([Staff Costs]) AS [Staff Costs],
SUM(People) AS People
FROM fitness
WHERE [Sales Volume] > 17500 GROUP BY Region
")

Even shorter is the direct use of SQL chunks in knitr using the variable db as connection and using the chunk argument output.var=bob:

SELECT Region,
COUNT(*) As NoOfUnits,
SUM([Sales Volume]) As [Sales Volume],
SUM([Staff Costs]) AS [Staff Costs],
SUM(People) AS People
FROM fitness
WHERE [Sales Volume] > 17500 GROUP BY Region

### The Ping-Pong Phase

After Ada and Bob each have a result, they compare their resulting data.frame using the daff package, which was recently presented by @edwindjonge at the useR! 2017 conference.

library(daff)
diff$get_data() ## @@ Region NoOfUnits Sales Volume Staff Costs People ## 1 +++ <NA> 1 19500 7609 2 ## 2 -> A 3->2 59623->42123 43103->30103 16->13 ## 3 -> B 1->2 119500->239000 95691->185691 19 ## 4 -> D 1->2 45860->70860 32555->32655 9 ## 5 E 1 33020 25010 7 After Ada’s and Bob’s serve, the two realize that their results just agree for the region E. Note: Currently, daff has the semi-annoying feature of not being able to show all the diffs when printing, but just n lines of the head and tail. As a consequence, for the purpose of this post, we overwrite the printing function such that it always shows all rows with differences. ##Small helper function for better printing print.data_diff <- function(x) x$get_data() %>% filter(@@ != "")

diff %>% print()
##    @@ Region NoOfUnits   Sales Volume   Staff Costs People
## 1 +++   <NA>         1          19500          7609      2
## 2  ->      A      3->2   59623->42123  43103->30103 16->13
## 3  ->      B      1->2 119500->239000 95691->185691     19
## 4  ->      D      1->2   45860->70860  32555->32655      9

The two decide to first agree on the number of units per region.

diff$get_data() %>% filter(@@ != "") %>% select(@@, Region, NoOfUnits) ## @@ Region NoOfUnits ## 1 +++ <NA> 1 ## 2 -> A 3->2 ## 3 -> B 1->2 ## 4 -> D 1->2 One obvious reason for the discrepancies appears to be that Bob has results for an extra <NA> region. Therefore, Bob takes another look at his management of missing values and decides to improve his code by: #### Pong Bob SELECT Region, COUNT(*) As NoOfUnits, SUM([Sales Volume]) As [Sales Volume], SUM([Staff Costs]) AS [Staff Costs], SUM(People) AS People FROM fitness WHERE ([Sales Volume] > 17500 AND REGION IS NOT NULL) GROUP BY Region ## @@ Region NoOfUnits Sales Volume Staff Costs People ## 1 -> A 3->2 59623->42123 43103->30103 16->13 ## 2 -> B 1->2 119500->239000 95691->185691 19 ## 3 -> D 1->2 45860->70860 32555->32655 9 #### Ping Bob Better. Now the NA region is gone, but still quite some differences remain. Note: You may at this point want to stop reading and try yourself to fix the analysis - the data and code are available from the github repository. #### Pong Bob Now Bob notices that he forgot to handle the duplicate records and apparently misunderstood the exact definition of the €17,500 exclusion limit. His massaged SQL query looks as follows: SELECT Region, COUNT(*) As NoOfUnits, SUM([Sales Volume]) As [Sales Volume], SUM([Staff Costs]) AS [Staff Costs], SUM(People) AS People FROM (SELECT Id, MAX(Version), Region, [Sales Volume], [Staff Costs], People FROM fitness GROUP BY Id) WHERE ([Sales Volume] >= 17500 AND REGION IS NOT NULL) GROUP BY Region ## @@ Region NoOfUnits Sales Volume Staff Costs People ## 1 ... ... ... ... ... ... ## 2 -> D 1->2 45860->70860 32555->32655 9 #### Ping Ada Comparing with Ada, Bob is sort of envious that she was able to just use dplyr’s group_by and top_n functions. However, daff shows that there still is one difference left. By looking more carefully at Ada’s code it becomes clear that she accidentally leaves out one unit in region D. The reason is the too liberate use of na.omit, which also removes the one entry with an NA in one of the not so important columns. However, they discuss the issue, if one really wants to include partial records or not, because summation in the different columns then is over a different number of units. After consulting with the standard operation procedure (SOP) for these kind of surveys they decide to include the observation where possible. Here is Ada’s modified code: ada2 <- fitness %>% filter(!is.na(Region)) %>% group_by(Region,Id) %>% top_n(1,Version) %>% group_by(Region) %>% filter(Sales Volume >= 17500) %>% summarise(NoOfUnits=n(), Sales Volume=sum(Sales Volume), Staff Costs=sum(Staff Costs), People=sum(People)) (diff_final <- diff_data(ada2,bob3)) %>% print() ## @@ Region NoOfUnits ... Staff Costs People ## 1 ... ... ... ... ... ... ## 2 -> D 2 ... 32655 NA->9 #### Pong Ada Oops, Ada forgot to take care of the NA in the summation: ada3 <- fitness %>% filter(!is.na(Region)) %>% group_by(Region,Id) %>% top_n(1,Version) %>% group_by(Region) %>% filter(Sales Volume >= 17500) %>% summarise(NoOfUnits=n(), Sales Volume=sum(Sales Volume), Staff Costs=sum(Staff Costs), People=sum(People,na.rm=TRUE)) diff_final <- diff_data(ada3,bob3) ##Check if the results really are the same length(diff_final$get_data()) == 0
## [1] TRUE

Finally, their results agree and they move on to production and their results are published in a nice report.

## Conclusion

As shown, the ping-pong game is quite manual and particularly annoying, if at some point someone steps into the office with a statement like Btw, I found some extra questionnaires, which need to be added to the analysis asap. However, the two now aligned analysis scripts and the corresponding daff-overlay could be put into a script, which is triggered every time the data change. In case new discrepancies emerge as length(diff\$get_data()) > 0, the two could then be automatically informed.

Question 1: Now the two get the same results, do you agree with them?

## # A tibble: 4 x 5
##   Region NoOfUnits Sales Volume Staff Costs People
##   <chr>      <int>          <int>         <int>  <int>
## 1 A              3          59623         43103     16
## 2 B              1         119500         95691     19
## 3 D              2          70860         32655      9
## 4 E              1          33020         25010      7
Question 2: Are you aware of any other good ways and tools to structure and automatize such a process? If so, please share your experiences as a Disqus comment below. Control calculations appear quite common, but little structured code support appears to be available for such processes.

Photo is copyright Johnathan J. Stegeman under the GNU Free Documentation License, version 1.2.


text:
# Math Help - RE: Is this a proof of Pythagorean Theorem?

1. ## RE: Is this a proof of Pythagorean Theorem?

Hi,

Is this a legitimate proof of the Pythagorean theorem?

We know that Pythagorean triples a, b, c, such that a^2 +b^2 = c^2.

We know that that a = 2n+1, b= 2n(n+1), and c = 2n(n+1) + 1 for finding Pythagorean triples.

If you substitute these into the formula a^2 + b^2 = c^2, expand and collect like terms to show they are equal, would this be considered a proof?

Thanks!

2. ## Re: Is this a proof of Pythagorean Theorem?

no, The Pythagorean thrm states the connection between the side lengths of right triangles and the algebraic equation $a^2 + b^2 = c^2$, namely that every right triangle has side lengths satisfying the equation. What you are doing is merely proving that a = 2n+1, b= 2n(n+1), and c = 2n(n+1) + 1 satisfies the algebraic equation $a^2+b^2=c^2$. In order to prove the Pythagorean thrm for positive integer triples satisfying the algebraic equation $a^2 + b^2 = c^2$, show me that you can use the positive numbers in the tuple (a,b,c) to construct a right triangle OR show that every right triangle has side lengths satisfying $a^2 + b^2 = c^2$

3. ## Re: Is this a proof of Pythagorean Theorem?

Yes, but every side length of a triangle can be constructed using these equations, correct? They would, therefore, be "sides" in general.

4. ## Re: Is this a proof of Pythagorean Theorem?

No. Your 'proof' doesn't even mention right angles.

5. ## Re: Is this a proof of Pythagorean Theorem?

Originally Posted by SC313
Yes, but every side length of a triangle can be constructed using these equations, correct? They would, therefore, be "sides" in general.
You need to prove that every side length of a right triangle can be constructed using the equations. If you can do that then your proof would have shown the Pythagorean theorem for Pythagorean triples only. . Don't not be confused by the term "Pythagorean triples", they are only positive integer solutions to the equation $a^2+b^2=c^2$. People use the "Pythagorean" prefix to "triples" to imply that these are using these triples in the context of representing the sides of a right triangle. If these a,b,c don't represent any object, then they are just triples, numbers, that satisfy $a^2+b^2=c^2$ nothing more.


text:
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Question

IF ( G, * ) is an abelian group then discuss correctness of each of the options ...

a)  a = a-1 for all a $\epsilon$ G

b) a2 = a for all a $\epsilon$ G

c) (a*b)2 = a2 * b2 for all a,b $\epsilon$ G

d) G is finite order

My Approach

Abelian Group -- > Commutative Group

A)

Let  (a*b)-1= b-1 * a-1

givent that a-1 = a

Therefore , (a*b)-1 = b * a

Therefore , G is a abelian group   Given statement is correct

B)

a2 = a
IF a*a = a THEN a is identity element (e)

C)

( a*b)= a2 * b2

(a*b)*(a*b) = (a*a) * (b*b)

a* (b*a)*b = a*(a*b)*b

b*a = a*b   // cancelling a using left cancellation law and b using right cancellation law

Therefore G is abelian group Given statement is correct

D)

what are you proving here?
@arjun sir , chechking the correctness of statements given
But you are asked to prove statement 1. Now, you assumed statement 1 true for A and took another assumption and proved G is abelian. It should be done like this: G is abelian, use properties of abelian group and show that statement A is true. Or just give a counter example showing statement A is false.
Ok. sir

D. We have to prove that every |G| is finite for abelian groups.

Since all abelian groups are cyclic i.e a^n = a; for some n.

now consider n is infinite, which implies this is not a cyclic group, hence not an abelian. Therefore n is finite.

Thus |G| = n which is finite.
–1 vote
Youb r right abolutely,then why r u asking ?


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 ORIGINAL ARTICLE Commun Sci Disord. 2008;13(1): 63-85.
 Writing Characteristics of School-Aged Children with Specific Language Impairment Yu Gyung , and Jeong Eunhee Copyright ©2008 The Korean Academy of Speech-Language Pathology and Audiology 유경(Yu Gyung)| 정은희(Jeong Eunhee) Share : ABSTRACT children. The objective of this study was to investigate the characteristics of writing in the school-aged children with specific language impairment (SLI). Methods Seventeen children with SLI and 24 normal children (4th-6th elementary school students) were evaluated with K-WISC Ⅲ and the Test of Problem Solving. We also evaluated their writing knowledge (i.e., propositional knowledge, strategic knowledge, tacit knowledge) from the text which were collected from their writing task with the picture “Farm”. SPSS 13.0 was used to analyze the children’s test results. The two groups were compared in total scores and subtests scores using the Mann-Whitney U test. Results The results were as follows: (1) the total score of propositional knowledge in the SLI children was significantly lower than that in normal children. In the analysis of propositional knowledge, the scores of the subtests, declarative knowledge and text knowledge in the SLI children were significantly lower than those in normal children, but the subtests score of language knowledge did not differ between the two groups. Furthermore, significantly lower scores were observed in the SLI group in theme correlations in the subsets of declarative knowledge, vocabulary diversity in the subsets of text knowledge, and tense in the subsets of language knowledge; (2) in the analysis of the total score of strategic knowledge, there was no significant difference between the two groups, but in the analysis of subset score, the SLI group showed lower scores in the proper length assessment; and (3) in the analysis of the tacit knowledge, there was no significant difference in the total scores and subset scores between the two groups. Discussion & Conclusion The results of this study suggest that there are some differences between writing of normal children and that of children with SLI and thus, there should be a specialized program in treating children with SLI. Further studies regarding the writing knowledge of SLI need to be implemented and specialized intervention programs which are more appropriate to the characteristics of SLI are recommended. Keywords: 단순언어장애아동 | 쓰기지식 | 명제적 지식 | 절차적 지식 | 암묵적 지식 SLI | writing knowledge | propositional knowledge
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# AmplitudeEstimationResult¶

class AmplitudeEstimationResult[Quellcode]

The AmplitudeEstimation result object.

Methods

 combine Any property from the argument that exists in the receiver is updated.

Attributes

circuit_results

Return the circuit results. Can be a statevector or counts dictionary.

Rückgabetyp

Union[ndarray, Dict[str, int], None]

confidence_interval

Return the confidence interval for the amplitude (95% interval by default).

Rückgabetyp

Tuple[float, float]

confidence_interval_processed

Return the post-processed confidence interval (95% interval by default).

Rückgabetyp

Tuple[float, float]

estimation

Return the estimation for the amplitude in $$[0, 1]$$.

Rückgabetyp

float

estimation_processed

Return the estimation for the amplitude after the post-processing has been applied.

Rückgabetyp

float

max_probability

Return the maximum sampling probability.

Rückgabetyp

float

measurements

Return the measurements as integers with their measurement probability.

Rückgabetyp

Dict[int, float]

mle

Return the MLE for the amplitude, in $[0, 1]$.

Rückgabetyp

float

mle_processed

Return the post-processed MLE for the amplitude.

Rückgabetyp

float

num_evaluation_qubits

Returns the number of evaluation qubits.

Rückgabetyp

int

num_oracle_queries

Return the number of Grover oracle queries.

Rückgabetyp

int

post_processing

Return a handle to the post processing function.

Rückgabetyp

Callable[[float], float]

samples

Return the measurement samples with their measurement probability.

Rückgabetyp

Dict[float, float]

samples_processed

Return the post-processed measurement samples with their measurement probability.

Rückgabetyp

Dict[float, float]

shots

Return the number of shots used. Is 1 for statevector-based simulations.

Rückgabetyp

int


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## Overview

The purpose of this lab is to give students more practice with arrays and loops in Java by implementing a "filtering" method, in which parts of an audio clip are removed based on different criteria.

## Code

The code for this assignment can be found in the homework 5 repository. You will be editing the file AudioFiltering.java

## Overview

As we have seen, digital audio can be represented as an array of doubles between -1 and 1, which we refer to as samples. In this assignment, you will fill in a method that takes in a sequence of audio samples x and creates a new array of audio samples y which is a subset of x. There are three options, and you must implement one. Each option consists of the following steps:

1. Count the number of samples in x that are above or below a certain threshold ("energy" for loudness filtering or "zero crossings" for vowel/consonant filtering). This will be the size of the array y
2. Create an array y of the appropriate size.
3. Copy over the values from x to y that exceed a threshold.

## Option 1: Loudness Filtering

Your first option is to fill in the function

public static double[] filterLoudness(double[] x, int win, double cutoff)

This method should return an array y where all of the quiet regions have been removed. The first line of this function has been filled in for you:

 double[] energy = ArrayUtils.getLocalEnergy(x, win);

This line creates an array energy which is parallel to x. In particular, energy[i] holds a number between 0 and 1 which is the mean of the square of the audio samples in x between indices i-win and i+win (Click here to see the implementation details if you're curious, but you can treat that as a black box for the purposes of this assignment). energy[i] is higher if the audio samples in x around index i are louder, and lower otherwise. Let's say, for example, that we have the following audio:

Below is a plot that shows the original samples of x (top plot), followed by the energy array (middle plot). The regions in which the person is speaking have higher energy, and the regions between speech has lower energy. If we choose a cutoff of 0.005 in energy (shown as the dotted line), and we only fill in samples in y that are above this cutoff, then we get the following audio (which is also shown as the bottom plot)

## Option 2: Consonant Filtering

Your second option is to fill in the function

public static double[] filterZCS(double[] x, int win, int maxzcs)

This method should return an array y where all of the vowels have been removed from the audio. The first line of this function has been filled in for you:

 int[] allzcs = ArrayUtils.getZeroCrossings(x, win);

This line creates an array allzcs which is parallel to x. In particular, allzcs[i] holds the number of zero crossings in x between indices i-win and i+win (Click here to see the implementation details if you're curious, but you can treat that as a black box for the purposes of this assignment). Generally, allzcs[i] is higher if x[i] is in the middle of a consonant, and it is lower otherwise. This is because consonants are higher frequency than vowels, which means they have a smaller period (i.e. they go through more cycles in the same amount of time, which means they cross the x axis more). Let's say, for example, that we have the same audio clip:

If we create a new array y in which the zero crossings in a window of size 4001 around x must be at least 150, we get the following audio

Below is a plot that shows how this happens. As before, the original audio is up top, and the filtered audio is at the bottom. The middle plot shows the zero crossings counted in windows of size 4001 around each sample, and a threshold of 150 is drawn as a dotted line. As an example, consider the circled region around the letter "six." The zero crossings peak around s and x, but they dip around i.

## Option 3: Vowel Filtering

You can also choose to create a method which is almost identical to the consonant filtering method, but where you only keep samples if the zero crossings are below a certain amount. Instead of keeping the consonants, this will keep the vowels. For example, if we create a new array y in which the zero crossings in a window of size 4001 around x must be at most 150, we get the following audio

## What To Submit

You must submit the file Lab5.java to Canvas. Please indicate as a comment which of the three options you chose to implement.