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# Kerala Syllabus 7th Standard Maths Solutions Chapter 6 Square and Square Root
## Kerala State Syllabus 7th Standard Maths Solutions Chapter 6 Square and Square Root
### Square and Square Root Text Book Questions and Answers
Triangular numbers Textbook Page No. 80
See the dots arranged in triangles:
How many dots are there in each?
1, 3, 6
How many dots would be there in the next triangle?
Such numbers as 1, 3, 6, 10, … are called triangular numbers.
The first triangular number is 1.
The second is 1 + 2 = 3.
The third is 1 + 2 + 3 = 6.
What is the tenth triangular number?
From the above logic we need to add 10 numbers sum to obtain the 10th triangular number.
To find the 10th triangular number T10 we need to add 1+ 2+3+4+5+6+7+8+9+10 = 55
Therefore, 10th triangular number T10 is 55.
Squares and triangles Textbook Page No. 81
Look at these pictures:
Each square is divided into two triangles.
Let’s translate this into numbers:
4 = 1 + 3
9 = 3 + 6
16 = 6 + 10
Check whether the same pattern continues. What do we see?
All perfect squares after 1 are the sums of two consecutive triangular numbers.
What is the sum of the seventh and eighth triangular numbers?
Yes adding consecutive triangular numbers gives us perfect squares. So, we get the next pattern 9+16 = 25 which in turn is a perfect square.
7th Triangular number can be obtained as T7 = 1+2+3+4+5+6+7 = 28
8th Triangular number can be obtained as T8 = 1+2+3+4+5+6+7+8 = 36
Sum of Seventh and Eighth Triangular Numbers = T7+T8
= 28+36
= 64
Thus, the sum of 7th, 8th Triangular Numbers is 64
Increase and decrease Textbook Page No. 82
Look at this number pattern:
1 = 1
4 = 1 + 2 + 1
9 = 1 + 2 + 3 + 2 + 1
16 = 1 + 2 + 3 + 4 + 3 + 2 + 1
Can you split some more perfect squares like this?
Writing some more perfect squares we have 25 = 1+2+3+4+5+4+3+2+1
36 = 1+2+3+4+5+6+5+4+3+2+1
49 = 1+2+3+4+5+6+7+6+5+4+3+2+1
64 = 1+2+3+4+5+6+7+8+7+6+5+4+3+2+1
Square difference
We have see that
22 = 12 + (1 + 2)
32 = 22 + (2 + 3)
42 = 32 + (3 + 4) and so on.
We can write these in another manner also:
22 – 12 = 1 + 2
32 – 22 = 2 + 3
42 – 32 = 3 + 4
In general, the difference of the squares of two consecutive natural numbers is their sum. Now look at these:
32 – 12 = 9 – 1 = 8
42 – 22 = 16 – 4 = 12
52 – 32 = 25 – 9 = 16
What is the relation between the difference of the squares of alternative natural numbers and their sum?
32 – 12 = (3+1) x 2 = 8
42 – 22 = (4+2) x 2 = 12
52 – 32 = = (5+3) x 2 = 16
The difference between squares of two alternate natural numbers is always even i.e. twice the sum of two numbers that are squared.
Project Textbook Page No. 84
Last digit
Look at the last digit of squares of natural numbers from 1 to 10:
1, 4, 9, 6, 5, 6, 9, 4, 1, 0
Now, look at the last digits of squares of natural numbers from 11 to 20.
Do we have the same pattern?
Let’s look at another thing: Does any perfect square end in 2?
Which are the digits which do not occur at the end of perfect squares?
Is 2637 then a perfect square?
Last digits of squares of natural numbers from 11 to 20 are 1, 4, 9, 6, 5, 6, 9, 4, 1, 0. Yes they have the same pattern
As we observe the digits 2, 3, 7, 8 doesn’t occur at the end of perfect squares.
As we know 2637 ends with digit 7 at the end it is not a perfect square.
To decide that a number is not a perfect square, we need only look at the last digit.
Can we decide that a number is a perfect square from its last digit alone?
Yes, we can decide if a number is perfect square or not by seeing the last digit alone as you can see above.
Rectangle and square
Look at this picture.
Dots in a rectangle.
Can you rearrange the dots to make another rectangle?
Can you rearrange the dots to make a square? Start like this:
How many more are needed to make a square?
How many dots were there in the original rectangle? How many in this square?
What do we see here?
42 = (3 × 5) + 1
Can we do this for all rectangular arrangements?
The numbers here are 3, 4, 5.
So, for this trick to work, what should be the relation between the number of dots in each row and column of the rectangle?
We can write this in numbers as
22 = (1 × 3) + 1
32 = (2 × 4) + 1
42 = (3 × 5) + 1
Try to continue this
52 = (4 x 6) +1
62 = (5 x 7) +1
72 = (6 x 8) +1
82 = (7 x 9) +1
92 = (8 x 10)+1
Square root of a perfect square
784 is a perfect square. What is its square root? 784 is between the perfect squares 400 and 900; and we know that their square roots are 20 and 30. So $$\sqrt{784}$$ is between 20 and 30. Since last digit of 784 is 4, its square root should have 2 or 8 as the last digit. So $$\sqrt{784}$$ is either 22 or 28.
784 is near to 900 than 400. So $$\sqrt{784}$$ must be 28. Now calculate 282 and check.
Given that 1369, 2116, 2209 are perfect squares, find their square roots like this.
Project Textbook Page No. 87
Digit sum
16 is a perfect square and the sum of its digits is 7.
The next perfect square 25 also has digit sum 7.
The digit sum of 36 is 9.
The sum of the digits of the next perfect square 49 is 13. If we add the digits again, the sum is 4. Find the sum of the digit sums (reduced to a single digit number) of perfect squares starting from 1.
Do you see any pattern?
Is 3324 is perfect square?
Given number is 3324
Now Split the number and add each number 3 + 3 + 2 + 4 = 12
As the result is more than 1 number we should add it again 1 + 2 = 3
All possible numbers that are perfect square have roots of either 1, 4, 7, 9
As 3 is not in the list 3324 is not a Perfect Square.
Rows and columns Textbook Page No. 80
Look this picture:
Dots in rows and columns make a rectangle.
How many dots in all?
Did you count the dots one by one?
Can you make other rectangles with 24 dots?
Is any one of these a square?
How many more dots do we need to make a square? Can you remove some dots and make a square? How many?
Can you remove some dots and make a square? How many?
Numbers which can be arranged in squares are called square numbers.
Do you see anything special about of the number of dots making a square?
There are total 4×6=24 dots in all.
No, the dots were counted by multiplying the number of dots in rows and number of dots in column.
None of these is a square.
We need more 12 dots to make a square, 24+12=36 or 62.
Yes, we can remove some dots and make a square.
We need to remove 8 dots to make a square, 24-8=16 or 42.
The number of dots making a square is the sqaure of that number.
Squares
What are the ways in which we can write 36 as the product of two numbers?
2 × 18, 3 × 12, 4 × 9
We can also write
36 = 6 × 6
And we have seen that it can also be shortened as 36 = 62.
36 is 6 multiplied by 6 itself; that is, the second power of 6.
There is another name for this:
36 is the square of 6.
Then what is the square of 5?
What is the square of $$\frac{1}{2}$$?
Square of 5 = 52 = 25
$$\frac{1}{2}$$ × $$\frac{1}{2}$$ = $$\frac{1}{4}$$
Perfect squares
1, 4, 9, 16,… are the squares of the natural numbers. They are called perfect squares.
What is the perfect square after 16?
Why is 20 not a perfect square?
Prime Factorization of 20 can be written as 22 x 51
As 5 is not in pair 20 is not a perfect square.
The next perfect square after 16 is 25.
Let us look at the succession of perfect squares in another way.
To reach 4 from 1, we must add 3.
To reach 9 from 4?
We can state these as
4 – 1 = 3
9 – 4 = 5
16 – 9 = 7
All these differences are odd numbers, right?
So, the difference of two consecutive perfect squares is an odd number.
Let’s write this as,
4=1 + 3
9 = 4 + 5 = 1 + 3 + 5
16 = 9 + 7 = 1 + 3 + 5 + 7
What do we see here?
When we add consecutive odd numbers starting from 1, we get the perfect squares.
This can be seen from these pictures also.
Can you write down the squares of natural numbers upto 20, by adding odd numbers? You can proceed like this
12 = 1
22 = 1 + 3 = 4
32 = 4 + 5 = 9
42 = 9 + 7 = 16
What is the relation between the number of consecutive odd numbers from 1 and their sum?
What is the sum of 30 consecutive odd numbers starting from 1?
Let us assume the arithmetic series 1, 3, 5, 7, 9, 11, 13…..
we know the formula for sum of arithmetic series Sn = n/2[2a+(n-1)d]
Here d = 2
first term a = 1
Substituting the inputs we have Sn= 30/2[2×1+(30-1)2]
= 30/2[2+58]
= 30/2[60]
= 900
Therefore, the sum of 30 consecutive odd numbers starting from 1 is 900.
Tricks with ten Textbook Page No. 82
The square of 10 is 100. What is the square of 100?
In the square of 1000, how many zeros are there after 1?
What about the square of 10000?
What happens to the number of zeros on squaring?
So how do we spot the perfect squares among 10, 100, 1000, 10000 and so on?
Is one lakh a perfect square?
Now find out the squares of 20, 200 and 2000.
Is 400000000 a perfect square?
What if we put in one more zero?
Square of a number containing x zeros will become 2 times number of zeros.
1 lakh is not a perfect square as the number of zeros is not even.
Ten Lakhs is a perfect square as it comes with 6 zeros that are even.
20 = 20 x 20 = 400
200 = 200 x 200 = 40000
2000 = 2000 x 2000 = 4000000
400000000 is a perfect square as the number of zeros 8 is even.
• Find out the squares of these numbers.
• 30
302 = 30 x 30
= 900
The square of 30 is 900.
• 400
4002 = 400 x 400
= 160000
The square of 400 is 160000
• 7000
70002 = 7000 x 7000
= 49000000
The Square of 7000 is 49000000
• 6 × 1025
(6 × 1025)= (6 × 1025) x (6 × 1025)
= 36 x (1025)2
= 36 x 1050
• Find out the perfect squares among these numbers.
• 2500
The given number 2500 can be written as 502
Hence the 2500 is a perfect square number.
• 36000
We cannot write the given number 36000 as square of two equal numbers. Hence it is not a perfect square.
• 1500
We cannot write the given number 1500 as square of two equal numbers. Hence it is not a perfect square.
• 9 × 107
We cannot write the given number 9 × 107 as square of two equal numbers. Hence it is not a perfect square.
• 16 × 1024
The given number 2500 can be written as (4x 1012)2
Hence the given number 16 × 1024 is a perfect square.
Next square
What is the square of 21?
Wait a bit before you start multiplying.
The square of 20 is 400, isn’t it? So to get the square of 21, we need only add an odd number.
Which odd number?
Let’s start from the beginning. We can write
22 = 12 + 3 = 12 + (1 + 2)
32 = 22 + 5 = 22 + (2 + 3)
42 = 32 + 7 = 32 + (3 + 4)
52 = 42 + 9 = 42 + (4 + 5)
and so on. Continuing like this, how do we write 212?
212 = 202 + (20 + 21)
That is,
212 = 400 + 41 = 441
Now we can continue as before with
222 = 441 +43 = 484
and so on.
How do we find out the square of 101?
1002 = 10000
100 + 101 = 201
So, 1012 = 10000 + 201 = 10201
• Find out the squares of these numbers using the above idea.
• 51
Using the above process we write the 512 = 502+(50+51)
= 2500+(50+51)
= 2500+101
= 2601
• 61
It can be written as 612 = 602+(60+61)
= 3600+(60+61)
= 3600+121
= 3721
• 121
The given number is written as 1212 = 1202+(120+121)
= 14400+(120+121)
= 14400+241
= 14641
• 1001
It is written as 10012 = 10002+(1000+1001)
= 10002+(1000+1001)
= 1000000+2001
= 1002001
• Compute the squares of natural numbers from 90 to 100.
902
= 90 x 90
=8100
912
= 91 x 91
= 8281
922
= 92 x 92
= 8464
932
= 93 x 93
= 8649
942
= 94 x 94
= 8836
952
= 95 x 95
= 8025
962
= 96 x96
= 8928
972
= 97 x 97
= 9409
982
= 98 x 98
= 9310
992
= 99 x 99
= 9801
1002
= 100 x 100
= 10,000
Fraction squares Textbook Page No. 83
A fraction multiplied by itself is also a square.
What is the square of $$\frac{3}{4}$$ ?
($$\frac{3}{4}$$)2 = $$\frac{3}{4}$$ × $$\frac{3}{4}$$ = $$\frac{3 \times 3}{4 \times 4}$$ = $$\frac{9}{16}$$
That is,
($$\frac{3}{4}$$)2 = $$\frac{9}{16}$$ = $$\frac{3^{2}}{4^{2}}$$
So to square a fraction, we need only square the numerator and denominator separately.
Now do these problems without pen and paper.
• Find out the squares of these numbers.
• $$\frac{2}{3}$$
$$\frac{2}{3}$$2 = $$\frac{2}{3}$$*$$\frac{2}{3}$$
= $$\frac{2×2}{3×3}$$
= $$\frac{4}{9}$$
• $$\frac{1}{5}$$
$$\frac{1}{5}$$2 = $$\frac{1}{5}$$*$$\frac{1}{5}$$
= $$\frac{1×1}{5×5}$$
= $$\frac{1}{25}$$
• $$\frac{7}{3}$$
$$\frac{7}{3}$$2 = $$\frac{7}{3}$$*$$\frac{7}{3}$$
= $$\frac{7×7}{3×3}$$
= $$\frac{49}{9}$$
• 1$$\frac{1}{2}$$
1$$\frac{1}{2}$$2 = $$\frac{1}{2}$$*$$\frac{1}{2}$$
=1 $$\frac{1×1}{2×2}$$
=1 $$\frac{1}{4}$$
= $$\frac{5}{4}$$
• Which of the fractions below are squares?
• $$\frac{4}{15}$$
Numerator is 4 and Denominator is 15 .
Both are not perfect Square numbers so it is not possible to write the given fraction as squares.
• $$\frac{8}{9}$$
Numerator is 8 and Denominator is 9.
Here 8 is not a perfect Square number and Denominator is a perfect square number. so it is not possible to write the given fraction as squares.
• $$\frac{16}{25}$$
In a given fraction, numerator is 16 and Denominator is 25 which were square numbers of 4 and 5.Hence we can Write as $$\frac{4}{5}$$2
• 2$$\frac{1}{4}$$
When the given 2$$\frac{1}{4}$$ is converted to improper fraction, we get $$\frac{9}{4}$$.
Here the Numerator is 9 and Denominator is 4
We can write the numerator as perfect square i.e. (3)2
Denominator 4 can be written as (2)2
Therefore, 2$$\frac{1}{4}$$ can be written as $$\frac{3}{2}$$2
• 4$$\frac{1}{9}$$
When the given 4$$\frac{1}{9}$$ is converted to improper fraction, we get $$\frac{1}{9}$$.
Here the Numerator is 1 and but the Denominator is 9 as a square of number (3)2
So, we cannot write the given fraction as Squares.
• $$\frac{8}{18}$$
Numerator is 8 and Denominator is 18.
Here numerator and Denominator are not perfect square numbers so there is no chance to write the given fraction as square of numbers.
Decimal squares
What is the square of 0.5?
We know that 52 = 25. How many decimal places would be there in the product 0.5 × 0.5?
Why?
0.5 = $$\frac{5}{10}$$, right?
Can you find out the square of 0.05?
0.052 =0.05×0.05
= 0.0025 Hence four decimal places are there.
0.0025 = $$\frac{25}{1000}$$
You have computed the squares of many natural numbers. Using that table, can you find out the square of 0.15?
• Find out the squares of these numbers.
0.152 = 0.15 x 0.15
= 0.0225
• 1.2
1.22 = 1.2 x 1.2
= 1.44
• 0.12
0.122 = 0.12 x 0.12
= 0.0144
• 0.013
0.0132 = 0.013 x 0.013
= 0.000169
• Which of the following numbers are squares?
• 2.5
We cannot write the number 2.5 as a square.
• 0.25
0.25 = 0.5 x 0.5.
Hence the number 0.25 is written as 0.52
• 0.0016
0.0016 = 0.04 x 0.04.
Hence the number 0.0016 is written as 0.042
• 14.4
We cannot write the number 14.4 as square.
• 1.44
1.44= 1.2×1.2
Hence the number 1.44 is written as 1.22
Square product Textbook Page No. 84
What is 52 × 42?
52 × 42 = 25 × 16 = ……..
There is an easier way:
52 × 42 = 5 × 5 × 4 × 4
= (5 × 4) × (5 × 4)
= 20 × 20
= 400
= (5 x 4) x (5 x 4)
= 20 x 20
= 400
Can you find out the products below like this, without pen and paper?
• 52 × 82
52 × 82 = 5 x 5 x 8 x 8
= (5 x 8) x (5 x 8)
= 40 x 40
= 1600
• 2.52 × 42
2.52 × 42 = 2.5 x 2.5 x 4 x 4
= (2.5 x 4) x (2.5 x 4)
= 10 x 10
= 100
• (1.5)2 × (0.2)2
(1.5)2 × (0.2)2 = 1.5 x 1.5 x 0.2 x 0.2
= (1.5 x 0.2) x (1.5 x 0.2)
= 0.3 x 0.3
= 0.009
What general rule did we use in all these?
The product of the squares of two numbers is equal to the square of their product.
How do we say this in algebra?
x2y2 = (xy)2, for any numbers x, y
let the three numbers be x, y, and z
x2y2z2 = (xyz)2, for any numbers x, y, and z.
Square factors
How do we write 30 as a product of prime numbers?
30 = 2 × 3 × 5
So how do we factorize 900?
900 = 302 = (2 × 3 × 5)2 = 22 × 32 × 52
Similarly, using the facts that 24 = 23 × 3 and 242 = 576, we get
576 = 242 = (23 × 3)2 = (23)2 × 32 = 26 × 32
Can you write each number below and its square as a product of prime powers?
• 35
35 = 5 x 7
Thus 35 can be written as product of prime powers 51 x 71
• 45
45 = 5 x 9
= 5 x 3 x 3
= 51 x 32
Thus, 45 can be written as 51 x 32
• 72
72 = 24 x 3
= 8 x 3 x 3
= 2 x 2 x 2 x 3 x 3
= 23 x 32
Thus, 72 can be written as 23 x 32
• 36
36 = 9 x 4
= 9 x 2 x 2
= 3 x 3 x 2 x 2
= 32 x 22
Thus, 36 can be written as 32 x 22
• 49
49 = 7 x 7
= 72
Thus, 49 can be written as 72
Did you note any peculiarity of the exponents of the factors of the squares?
Reverse computation
We have to draw a square; and its area must be 9 square centimetres.
How do we do it?
The area of a square is the square of the side.
So if the area is to be 9 square centimetres, what should be the side?
To draw a square of area 169 square centimetres, what should be the length of a side?
For that, we must find out which number squared gives 169. Looking up our table of squares, we find 132 = 169. So we must draw a square of side 13 centimetres.
Here, given a number we found out which number it is the square of. This operation is called extracting the square root.
That is, instead of saying the square of 13 is 169, we can say in reverse that the square root of 169 is 13.
Just as we write
132 = 169
as shorthand for statement “the square of 13 is 169”, we write the statement “the square root of 169 is 13” in shorthand form as
$$\sqrt{169}$$ = 13
(the extraction of square root is indicated by the symbol )
Similarly, the fact that the square of 5 is 25 can also be stated, the square root of 25 is 5. In short hand form,
52 = 25
$$\sqrt{25}$$ = 5
In general
For numbers x and y, if x2 = y, then $$\sqrt{y}$$ = x
Now find out the square root of these numbers:
• 100
10 = 100
$$\sqrt{100}$$ = 10
Therefore, square root of 100 is 10
• 256
162 = 256
$$\sqrt{256}$$ = 16
Thus, the square root of 256 is 16.
• $$\frac{1}{4}$$
$$\frac{1}{4}$$ = $$\frac{1}{2}$$2
Thus, the Square root of $$\frac{1}{4}$$ = $$\frac{1}{2}$$
• $$\frac{16}{25}$$
$$\frac{16}{25}$$ = $$\frac{4}{5}$$2
Thus, the square root of $$\frac{16}{25}$$ is $$\frac{4}{5}$$.
• 1.44
1.44 = (1.2)2
Thus, the square root of 1.44 is 1.2
• 0.01
0.01 = (0.1)2
Thus, the square root of 0.01 is 0.1
Square root factors Textbook Page No. 87
How do we find the square root of 1225?
Since a product of squares is the square of the product, we need only write 1225 as a product of squares.
First factorize 1225 into primes:
1225 = 52 × 72
And we can write
52 × 72 = (5 × 7)2 = 352
So, 1225 = 352
From this, we get $$\sqrt{1225}$$ = 35
Let’s take another example. What is the $$\sqrt{3969}$$ ?
As before, we first factorize 3969 into primes.
3969 = 32 × 32 × 72
= (3 × 3 × 7)2
From this, we get $$\sqrt{3969}$$ = 3 × 3 × 7 = 63
Now compute the square roots of these.
• 256
Given number is 256
Firstly factorizing it into primes we have 256 = 2 x 128
= 2 x 2 x 64
= 2 x 2 x 2 x 32
= 2 x 2 x 2 x 2 x 16
= 2 x 2 x 2 x 2 x 2 x 8
= 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2
= (2 x 2) x (2 x 2) x (2 x 2) x (2 x 2)
= 4 x 4 x 4 x 4
= 16 x 16
= (16)2
From this, we get $$\sqrt{256}$$ = 16
• 2025
Given number is 2025
Writing the factorization of 2025 we have
= 3 x 3 x 3 x 3 x 5 x 5
= 9 x 9 x 5 x 5
= (9 x 5)2
= (45)2
Therefore, the square root of $$\sqrt{2025}$$ is 45
• 441
Given number is 441
Writing the factorization of 441 we have
= 3 x 3 x 7 x 7
= 32 x 72
= (3 x 7)2
= (21)2
Therefore, the square root of 441 is 21.
• 921
Writing the factorization of 921 we have 3 x 307
Thus, 921 can’t be written as perfect square and the square root of 921 isn’t a natural number.
• 1089
Given number is 1089
Writing the factorization of it we have 1089 = 3 x 3 x 11 x 11
= 32 x 112
= (3 x 11)2
= (33)2
Therefore, square root of 1089 is 33.
• 15625
Given number is 15625
Writing the factorization of it we have 15625 = 5 x 5 x 5 x 5 x 5 x 5
= 52 x 52 x 52
= (5 x 5 x 5)2
= (125)2
Therefore, the square root of 15625 is 125
• 1936
Given number is 1936
Writing the factorization of 1936 we have 2 x 2 x 2 x 2 x 11 x 11
= (22 x 22 x 112)
= (2 x 2 x 11)2
= 442
Therefore, the square root of 1936 is 44.
• 3025
Given number is 3025
Writing the factorization of 3025 we have 5 x 5 x 11 x 11
= 52 x 112
= (5 x 11)2
= 552
Therefore, the square root of 3025 is 55.
• 12544
Given number is 12544
Writing the factorization of 12544 we have 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 7 x 7
= 28 x 72
= (112)2
Therefore, Square Root of 12544 is 112.
Let’s do it!
Question 1.
The area of a square plot is 1024 square meters. What is the length of its sides?
Given area of Square plot = 1024 Square Meters
Imagine ‘S’ is side of a Square. As we know Area of Square = Side x Side
1024 = Side x Side
1024 = Side2
Side = √1024
Side = 2 5
Hence Length of Square Plot = 32 meters
Question 2.
In a hall, 625 chairs are arranged in rows and columns, with the number of rows equal to the number of columns. The chairs in one row and one column are removed. How many chairs remain?
Total number of chairs = 625
Number of Rows and Columns present are Equal i.e. Rows = Columns , Let it be y.
Rows .Columns=625
y. y =625
y 2= 625
y = √625
y = √(25)2
y = 25
so Number of Rows and Number of Columns =25
When 1 Row and 1 Column is removed as below
Number of Rows =25-1=24
Number of Rows =25-1=24
When the 24 Rows and 24 Columns are arranged = 24 x 24
= 576
Question 3.
The sum of a certain number of consecutive odd numbers, starting with 1, is 5184. How many odd numbers are added?
We know the formula of sum of arithmetic series Sn = n/2[2a+(n-1)d]
The arithmetic series 1, 3, 5, 7, 9, 11, 13 ……
Here the first term a = 1
Number of odd numbers to be added = n
Common difference d = 2
Substituting in the formula above we have
5184 = n/2[2×1+(n-1)2]
5184 = n/2[2+2n-2]
5184 = n/2[2n]
5184 = n2
n = 72
Thus, the number of odd numbers to be added is 72.
Question 4.
The sum of two consecutive natural numbers and the square of the first is 5329. What are the numbers?
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## Elementary Linear Algebra 7th Edition
a basis for the nullspace of $A$ consists of the vector \left[\begin{aligned} 1\\ 2 \end{aligned}\right].
Given the matrix $$A=\left[ \begin{array}{rr}{2} & {-1} \\ {-6} & {3}\end{array} \right]$$ The reduced row echelon form is given by $$\left[ \begin {array}{cc} 1&-\frac{1}{2}\\ 0&0\end {array} \right] .$$ The corresponding system is \begin{aligned} 2x_{1}- x_{2} &=0 \end{aligned}. The solution of the above system is $x_1=t$ and $x_2=2t$. This means that the solution space $Ax = 0$ of consists of all solution vectors of the following form x=\left[\begin{aligned} x_{1}\\ x_{2} \end{aligned}\right]=\left[\begin{aligned} t\\ 2t \end{aligned}\right]=t\left[\begin{aligned} 1\\ 2 \end{aligned}\right]. So, a basis for the nullspace of $A$ consists of the vector \left[\begin{aligned} 1\\ 2 \end{aligned}\right].
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# Integer programming formulation: which algorithms
I have a complex problem that I have simplified coming to a simple integer linear programming formulation.
Given the scalar $K > 0$, the vectors $v(t) \in R^n$ and $b_i \in R^n, \forall i=1,\ldots,K$ are known. In particular $v(t)$ is the measured value (ex: every 10 minutes) while the $b_i$ are fixed (computed ones, previously). Consider that I have: $$K >> n,$$ i.e., I have to identify the presence of $K$ (ex: $K=20$) elements, each one of dimension $n$ (ex: $n=2$).
So, for every fixed $t$, the problem that I have to solve has the following formulation:
\newcommand\norm[1]{\left\lVert#1\right\rVert} \begin{equation*}\begin{aligned} & \underset{x_i}{\text{minimize}} & & \norm{v - \sum_{i=1}^{K} b_i x_i}_{2} \\ & \text{subject to} & & x_i \in \{0,1\},\ \forall i = 1, \ldots, K. \end{aligned} \end{equation*}
Do you know which algorithms solve that formulation? In particular I am looking for an algorithm implemented in Python.
• Comments are not for extended discussion; this conversation has been moved to chat. Feb 10 '17 at 14:50
Contrary to what you wrote in the first sentence of the question, your problem is not an instance of integer linear programming (ILP) and cannot be formulated as an ILP problem.
If you used the $L_1$ norm (instead of the $L_2$ norm), it could be formulated as an ILP problem. You'd introduce additional variables $t_1,\dots,t_n$, add the linear inequalities
$$-t_j \le \left(v - \sum_{i=1}^K b_ix_i \right)_j \le t_j$$
where $(\cdots)_j$ denotes the $j$th coefficient of the vector, and then minimize the objective function $t_1+\dots+t_n$.
With the $L_2$ norm, this is no longer an ILP problem. It is an instance of integer quadratic programming. You could try applying an off-the-shelf solver for mixed-integer quadratic programming (MIQP). However, MIQP is pretty tough in general, so I don't know whether this will be effective.
As another alternative, you could relax your MIQP instance to an instance of quadratic programming or semidefinitive programming, solve for a solution in the reals, and then round each real number to the nearest integer (or use randomized rounding), and hope that the resulting solution is "pretty good". This might be more computationally feasible (as quadratic programming / semidefinitive programming over the reals is easier than MIQP) but there are no guarantees on the quality of the resulting solution; it might be arbitrarily bad.
Your problem seems related to the Closest Vector Problem (CVP) in lattices, which is believed to be hard. Here you have the additional constraint that the coefficients be 0 or 1 (instead of them being arbitrary integers). If $n$ is not too large, you might be able to use existing algorithms, like LLL basis reduction. I don't know whether this will work or not.
You may reparameterize $x_i$ as $x_i = sigmoid(\alpha \cdot z_i), z_i \in \mathbb{R}$, with $\alpha$ being a constant such that $\alpha \gg 1$, and then proceed with your favorite optimization method for finding $z_i$, and from there $x_i$.
This, of course, may present problems, as $x_i$ are no longer bounded in $\{0, 1\}$, but may suffice your use case.
• I had thought to transform the problem in this way, such to have a classical non-linear uncontrained optimization problem, but I still hope that mine is a "known" problem with a known solution. Otherwise (I'll wait until tomorrow i think) I'll mark your answer as the best one. In the meantime: thanks! Feb 7 '17 at 15:17
Another option could be to model your problem as a bayesian linear regression where $x_i$ are random variables that follow a Bernoulli distribution with probability $p_i$, that is, $x_i \sim Bernoulli (p_i)$, and $v \sim \mathcal{N}(x^T b, \sigma^2)$.
The posterior distribution of $p_i$ would enable you to tell the values of $x_i$ that most likely fit with the observed data. These posterior distributions may be inferred via MCMC or variational inference.
The three main frameworks to implement bayesian models in Python are pymc3 (example), edward (example) and pystan (example). The three of them allow straighforward implementation of bayesian linear regression as formulated above.
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# 3.5: Two dimensional systems and their vector fields
Let us take a moment to talk about constant coefficient linear homogeneous systems in the plane. Much intuition can be obtained by studying this simple case. Suppose we have a $$2\, \times \, 2$$ matrix $$P$$ and the system
$\begin{bmatrix} x\\y \end{bmatrix}' = P\begin{bmatrix} x\\y \end{bmatrix}.$
The system is autonomous (compare this section to § 1.6) and so we can draw a vector field (see end of § 3.1). We will be able to visually tell what the vector field looks like and how the solutions behave, once we find the eigenvalues and eigenvectors of the matrix $$P$$. For this section, we assume that $$P$$ has two eigenvalues and two corresponding eigenvectors.
Case 1. Suppose that the eigenvalues of $$P$$ are real and positive. We find two corresponding eigenvectors and plot them in the plane. For example, take the matrix $$\begin{bmatrix} 1&1 \\ 0&2 \end{bmatrix}$$. The eigenvalues are 1 and 2 and corresponding eigenvectors are $$\begin{bmatrix} 1\\0 \end{bmatrix}$$ and $$\begin{bmatrix} 1 \\1 \end{bmatrix}$$. See Figure 3.3.
Now suppose that $$x$$ and $$y$$ are on the line determined by an eigenvector $$\vec{v}$$ for an eigenvalue $$\lambda$$. That is, $$\begin{bmatrix} x \\ y \end{bmatrix} = a \vec{v}$$ for some scalar $$a$$. Then
$\begin{bmatrix} x\\y \end{bmatrix}' = P \begin{bmatrix} x\\y \end{bmatrix} = P(a \vec{v}) = a(P \vec{v}) = a \lambda \vec{v}$
The derivative is a multiple of $$\vec{v}$$ and hence points along the line determined by $$\vec{v}$$. As $$\lambda > 0$$, the derivative points in the direction of $$vec{v}$$ when $$\alpha$$ is positive and in the opposite direction when $$\alpha$$ is negative. Let us draw the lines determined by the eigenvectors, and let us draw arrows on the lines to indicate the directions. See Figure 3.4.
Figure 3.4: Eigenvectors of $$P$$ with directions.
We fill in the rest of the arrows for the vector field and we also draw a few solutions. See Figure 3.5. Notice that the picture looks like a source with arrows coming out from the origin. Hence we call this type of picture a source or sometimes an unstable node.
Case 3. Suppose one eigenvalue is positive and one is negative. For example the matrix $$\begin{bmatrix} 1&1\\0&-2 \end{bmatrix}$$. The eigenvalues are 1 and -2 and corresponding eigenvectors are $$\begin{bmatrix} 1\\0\end{bmatrix}$$and $$\begin{bmatrix}1\\-3\end{bmatrix}$$. We reverse the arrows on one line (corresponding to the negative eigenvalue) and we obtain the picture in Figure 3.7. We call this picture a saddle point.
Figure 3.7: Example saddle vector field with eigenvectors and solutions.
For the next three cases we will assume the eigenvalues are complex. In this case the eigenvectors are also complex and we cannot just plot them in the plane.
Case 4.Suppose the eigenvalues are purely imaginary. That is, suppose the eigenvalues are $$\pm ib$$. For example, let $$P = \begin{bmatrix} 0&1\\-4&0\end{bmatrix}$$. The eigenvalues turn out to be $$\pm 2i$$ and eigenvectors are $$\begin{bmatrix} 1 \\ 2i \end{bmatrix}$$ and $$\begin{bmatrix} 1 \\ -2i \end{bmatrix}$$. Consider the eigenvalue $$2i$$ and its eigenvector $$\begin{bmatrix} 1\\ 2i \end{bmatrix}$$. The real and imaginary parts of $$\vec{v} e^{i 2t}$$ are
$$Re \begin{bmatrix} 1\\2i \end{bmatrix} e^{i2t} = \begin{bmatrix} cos(2t)\\-2sin(2t) \end{bmatrix}$$
$$Im \begin{bmatrix} 1\\2i \end{bmatrix} e^{i2t} = \begin{bmatrix} sin(2t) \\2cos(2t) \end{bmatrix}$$
We can take any linear combination of them to get other solutions, which one we take depends on the initial conditions. Now note that the real part is a parametric equation for an ellipse. Same with the imaginary part and in fact any linear combination of the two. This is what happens in general when the eigenvalues are purely imaginary. So when the eigenvalues are purely imaginary, we get ellipses for the solutions. This type of picture is sometimes called a center. See Figure 3.8.
Case 5. Now suppose the complex eigenvalues have a positive real part. That is, suppose the eigenvalues are $$a \pm ib$$ for some $$a > 0$$. For example, let $$P = \begin{bmatrix} 1&1 \\ -4&1 \end{bmatrix}$$. The eigenvalues turn out to be $$1 \pm 2i$$ and eigenvectors are $$\begin{bmatrix}1\\2i \end{bmatrix}$$ and $$\begin{bmatrix} 1 \\ -2i \end{bmatrix}$$. We take $$1\pm 2i$$ and its eigenvector $$\begin{bmatrix} 1 \\ 2i \end{bmatrix}$$ and find the real and imaginary of $$\vec{v}e^{(1+2i)t}$$ are
$$Re \begin{bmatrix} 1\\2i \end{bmatrix} e^{(1+2i)t} =e^t \begin{bmatrix} cos(2t)\\-2sin(2t) \end{bmatrix}$$
$$Im \begin{bmatrix} 1\\2i \end{bmatrix} e^{(1+2i)t} =e^t \begin{bmatrix} sin(2t) \\2cos(2t) \end{bmatrix}$$
Note the $$e^t$$ in front of the solutions. This means that the solutions grow in magnitude while spinning around the origin. Hence we get a spiral source. See Figure 3.9.
Figure 3.9: Example spiral source vector field.
Case 6. Finally suppose the complex eigenvalues have a negative real part. That is, suppose the eigenvalues are $$-a \pm ib$$ for some $$a > 0$$. For example, let $$P = \begin{bmatrix} -1& -1 \\ 4 & -1\end{bmatrix}$$. The eigenvalues turn out to be $$-1 \pm 2i$$ and eigenvectors are $$\begin{bmatrix} 1\\ -2i \end{bmatrix}$$ and $$\begin{bmatrix} 1\\ 2i \end{bmatrix}$$. We take $$-1-2i$$ and its eigenvector $$\begin{bmatrix} 1\\ 2i \end{bmatrix}$$ and find the real and imaginary of $$\vec{v} e^{(-1-2i)t}$$ are
$$Re \begin{bmatrix} 1\\2i \end{bmatrix} e^{(-1-2i)t} =e^{-t} \begin{bmatrix} cos(2t)\\ 2sin(2t) \end{bmatrix}$$
$$Im \begin{bmatrix} 1\\2i \end{bmatrix} e^{(-1-2i)t} =e^{-t} \begin{bmatrix} -sin(2t) \\2cos(2t) \end{bmatrix}$$
Note the $$e^{-t}$$ in front of the solutions. This means that the solutions shrink in magnitude while spinning around the origin. Hence we get a spiral sink. See Figure 3.10.
We summarize the behavior of linear homogeneous two dimensional systems in Table 3.1.
Table 3.1: Summary of behavior of linear homogeneous two dimensional systems.
Eigenvalues Behavior
real and both positive source / unstable node
real and both negative sink / stable node
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## anonymous one year ago 1. What do you do when you paraphrase something? A. Eliminate unnecessary words from a quote. B. Put an author's words into your own words. C. Cite a quote to make a point clear. D. Use one quote to explain another.
1. horsegirl27
Wrong section. You'll see this is the OpenStudy Feedback section. Go to "choose more subjects" and move this to writing
2. TheSmartOne
$$\bf\huge~~~~\color{#ff0000}{W}\color{#ff2000}{e}\color{#ff4000}{l}\color{#ff5f00}{c}\color{#ff7f00}{o}\color{#ffaa00}{m}\color{#ffd400}{e}~\color{#bfff00}{t}\color{#80ff00}{o}~\color{#00ff00}{O}\color{#00ff40}{p}\color{#00ff80}{e}\color{#00ffbf}{n}\color{#00ffff}{S}\color{#00aaff}{t}\color{#0055ff}{u}\color{#0000ff}{d}\color{#2300ff}{y}\color{#4600ff}{!}\color{#6800ff}{!}\color{#8b00ff}{!}\\\bf ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Made~by~TheSmartOne$$ Hey there!!! Since you are new here, read this legendary tutorial for new OpenStudiers!! http://goo.gl/5pp1u0 @Kayleighwood2 You might not have noticed but you are in the "OpenStudy Feedback" subject. Hover your mouse where it says OpenStudy Feedback and find the correct subject by clicking the 'Find More Subjects' button or simply clicking the correct subject in the list that you have already studied :) Alternately, you can click this subject link to get there faster :) http://www.openstudy.com/study#/groups/English
3. TheSmartOne
$$\color{blue}{\text{Originally Posted by}}$$ @horsegirl27 Wrong section. You'll see this is the OpenStudy Feedback section. Go to "choose more subjects" and move this to writing $$\color{blue}{\text{End of Quote}}$$ and where is this "choose more subjects" located? ;p
4. horsegirl27
hey I'm getting there TSO, give me a sec >.>
5. TheSmartOne
@Elsa213 No direct answers, you should know better. >.> <.<
6. Elsa213
Pffft I gave no direct answers. e.e
7. horsegirl27
=_= guys.
8. Elsa213
Yesh? cx
9. TheSmartOne
10. Elsa213
:O Why are you looking der? e.e
11. horsegirl27
because he's a stalker, duh ;)
12. TheSmartOne
@horsegirl27 still waiting for that explanation though ;p
13. horsegirl27
you already did it for me >.>
14. Elsa213
Tehehehehe
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# Kinematic reconstruction of $Z/H \rightarrow \tau\tau$ decay in proton-proton collisions
Abstract : The knowledge of $\tau$ lepton kinematic and kinematic of the $\tau$ pair in the decay $Z/H \rightarrow \tau\tau$ is essential for various analysis at LHC. However, the reconstruction of the whole kinematic of the $\tau$ decay is a challenging task, since in every $\tau$ decay at least one neutrino is present in the final state which escapes detection. In this paper a kinematic technique (Global Event Fit) to estimate the momentum escaped with neutrinos and hence the full momentum of the $\tau$ lepton pair is described. The algorithm is based on iterative minimization of the likelihood with constrains derived from all available kinematic information on the decay. The method requires the direction of at least one $\tau$ lepton to be well defined and therefore the method can be applied to the decays $Z/H \rightarrow \tau\tau \rightarrow X + a_{1}\nu$ with $a_{1}$ resonance decaying into three charged pions.
Keywords :
Document type :
Preprints, Working Papers, ...
Domain :
https://hal.archives-ouvertes.fr/hal-01806984
Contributor : Inspire Hep <>
Submitted on : Monday, June 4, 2018 - 12:11:10 PM
Last modification on : Wednesday, April 24, 2019 - 5:18:40 AM
### Citation
Vladimir Cherepanov, Alexander Zotz. Kinematic reconstruction of $Z/H \rightarrow \tau\tau$ decay in proton-proton collisions. 2018. ⟨hal-01806984⟩
Record views
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## Electronic fine structure in the nickel carbide superconductor Th$_{2}$NiC$_{2}$ [PDF]
Y. Quan, W. E. Pickett
The recently reported nickel carbide superconductor, body centered tetragonal $I4/mmm$ Th$_2$NiC$_2$ with T$_c$ = 8.5 K increasing to 11.2 K upon alloying Th with Sc, is found to have very fine structure in its electronic spectrum, according to density functional based first principles calculations. The filled Ni 3d band complex is hybridized with C $2p$ and Th character to and through the Fermi level ($E_f$), and a sharply structured density of states arises only when spin-orbit coupling is included, which splits a zone-center degeneracy leaving a very flat band edge lying at the Fermi level. The flat part of the band corresponds to an effective mass $m^*_{z} \rightarrow \infty$ with large and negative $m^*_{x}=m^*_{y}$. Although the region over which the effective mass characterization applies is less than 1% of the zone volume, it yet supplies of the order of half the states at (or just above) the Fermi level. The observed increase of T$_c$ by hole-doping is accounted for if the reference as-synthesized sample is minutely hole-doped, which decreases the Fermi level density of states and will provide some stabilization. In this scenario, electron doping will increase the Fermi level density of states and the superconducting critical temperature. Vibrational properties are presented, and enough coupling to the C-Ni-C stretch mode at 70 meV is obtained to imply that superconductivity is electron-phonon mediated.
View original: http://arxiv.org/abs/1307.0415
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# Motivating Characteristic Classes Using $S^2$
Trying to understand characteristic classes, hoping someone can explain/fit my example below into the wider scheme of things:
Chern's book says
Characteristic classes are the simplest global invariants which measure the deviation of a local product structure from a product structure. They are intimately related to the notion of curvature in differential geometry. In fact, a real characteristic class is a "total curvature," according to a well-defined relationship.
Lets take a small examples - it seems to me that the example of going from
$$S^2 = \{(x,y,z) \in \mathbb{R}^3 | x^2 + y^2 + z^2 = 1 \}$$
to the functions
$$z_{\pm} = \pm \sqrt{1 - x^2 - y^2}$$
can be formulated in terms of fiber bundles, where
$$S^2 = \{(x,y,z) \in \mathbb{R}^3 | x^2 + y^2 + z^2 = 1 \}$$
is the total space $E$,
$$D^1 = \{(x,y) \in \mathbb{R}^2 | x^2 + y^2 \leq 1 \}$$
is the base $B$ of the bundle, the maps
$$z_{\pm} = \pm \sqrt{1 - x^2 - y^2}$$
are sections of the bundle, with fibers of the form $$F_x = \{ ((x,y),z_+(x,y)),((x,y),z_-(x,y))\}$$
(Hope I've set that up about right).
So we have two local product structures $D^1 \times z_+$ and $D^1 \times z_-$ representing the top and bottom of the sphere. My guess is that if we want to measure the deviation of local product structure from global product structure we want to measure how the transition functions between $z_+$ and $z_-$ interact right?
Is all of this about right so far? Can it be cleaned up if so/not?
Bolstering some of this intuition - in the first 3 minutes of this video it seems like they say that characteristic classes arise (in this example) in the form of some function $f(n)$ when you analyze the transition function gluing $z_+$ to $z_-$ on the equator, $$z_- = f(n)z_+$$
I can clarify this with the accompanying notes (notation a bit different):
That's a bit confusing, but trying to use some of it in my example, it seems like they are saying that the $-1$ in
$$z_-(x,y) = -1 z_+(x,y)$$
which can be found as a function of $n$ through something like
$$f(n) = e^{in \pi }$$
(for $n$ odd), in general this function lets us 'characterize' the fibers into 2 'classes', one representing the top of the sphere $S^2$ when $n$ is even giving us $z_+$, another representing the bottom half of the sphere when $n$ is odd giving us $z_-$.
Am I right in saying that this function tells us the bundle splits into two classes so that we've measured the deviation of local product structure from global product structure by noting you need to piece together two classes to form the global object?
How does this baby picture relate to grand statements like
Generally speaking, for a vector bundle on a manifold M, a characteristic class associates a cohomology class of M.
Characteristic classes are constructed as polynomials of the curvature $F = dA + A \wedge A$
In fact, a real characteristic class is a "total curvature," according to a well-defined relationship.
Characteristic classes are subsets of the cohomology classes of the base space and measure the non-triviality or twisting of a bundle. In this sense, they are obstructions which prevent a bundle from being a trivial bundle. Most of the characteristic classes are given by the de Rham cohomology classes.
In other words, if you put vector bundles on the sphere and start constructing things like Chern classes and set up curvatures and stuff, can those things be interpreted using my simple example here?
• Your mappings from the sphere to the disk do not constitute a fibre bundle: Interior points of the disk sit below two points of $S^{2}$, while boundary points sit below only one point. Instead, you want to look at trivial circle bundles over the disk, and ask how the fibres over the boundary in one copy of the disk are glued to fibres over the boundary of the other disk. This amounts to specifying a mapping $S^{1} \to S^{1}$, whose degree is (modulo sign) the first Chern class of the resulting circle bundle over $S^{2}$. (Haven't read your question any further than "Is this right so far?") – Andrew D. Hwang May 18 '16 at 23:05
While $S^2$ is perhaps the most convenient space for thinking about actual topology, some special features of $S^2$ seem to have thrown you for a loop. In particular, you're confusing the so-called clutching construction - which only works over spheres - with a fiber bundle. Andrew Hwang has illustrated why your construction is not a fiber bundle (the fibers at the equator are different), but I'd like to talk about the clutching construction a bit.
Let's remember the most important fact about vector bundles (and more general classes of bundles as well, in fact):
Fact 1: Every vector bundle over a contractible (nice) space is trivial.
You should know how to prove this, and what this result means intuitively. If you don't, please figure it out and then come back. This fact is pretty terrific - it tells us that what really matters in vector bundles are the transition maps!
So let's take some of the simplest examples of non-contractible spaces - the spheres! The spheres are super-convenient because they can be covered by two contractible sets. Write the 2-sphere as $S^2 = N \cup S$, where $N$ and $S$ are the north and south hemispheres. If $E \to S^2$ is a vector bundle (with structure group $G$), then we know $E|_N \to N$ and $E|_S \to S$ are trivial bundles. So all of the "topological information" must be contained in the transition functions $N \cap S \to G$. It's not hard to see that we only care about the space $N \cap S$ and the map $N \cap S \to G$ up to homotopy. But $N \cap S$ is homotopy equivalent to $S^1$ - our favorite topological space of all time!
(A note on structure groups: when I say structure group, you should think "automorphism group of a fiber". So for real vector bundles of rank $k$, this is just $GL_k\mathbb R$, and for complex vector bundles of complex rank $m$, this is just $GL_m\mathbb C$. It turns out that most of the discussion here works for principal $G$-bundles for nice Lie groups $G$.)
All that discussion, once made rigorous, and with a little more work for uniqueness and existence, gives us the following fact. (We assume all vector bundles come with an orientation; otherwise we've got a bit more work to do in the real-vector-bundle case.)
Fact 2, version 1: Vector bundles over $S^2$ are in one-to-one correspondence with homotopy classes of maps $S^1 \to G$.
Hold up - "homotopy classes of maps from $S^1$ into a space sounds familiar: isn't that just the fundamental group? So let's upgrade fact 2 again.
Fact 2, version 2: Vector bundles over $S^2$ are in one-to-one correspondence with elements of $\pi_1(G)$.
But what if we did this construction for $S^n$ instead of $S^2$? (Assume $n \ge 2$.) The same argument would go through, but our equatorial space would be $S^{n-1}$ - so we'd just get higher homotopy groups!
Fact 2, final version: Vector bundles over $S^n$ for $n \ge 2$ are in one-to-one correspondence with elements of $\pi_{n-1}(G)$.
How do we use Fact 2? Well, let's start with complex vector bundles. Any complex vector bundle of rank $k$ has structure group $GL_k\mathbb C$. So we need to find the homotopy groups of this space. It turns out that $U(k)$ is a deformation retract of $GL_k\mathbb C$, and since it's compact it's easier to work with.
Let's start with the easiest case: $k = 1$, so line bundles. $U(1)$ is another way to think about our favorite topological space - it's just $S^1$. So line bundles over $S^n$ for $n \ge 3$ are in one-to-one correspondence with elements of $\pi_{n-1}(S^1)$ - which is trivial (lift to the universal cover). So we've proved something nontrivial:
Fact 3: Every complex line bundle over $S^n$ for $n \ge 3$ is trivial.
But the real magic happens at $n = 2$. Here we get a classification by elements of $\pi_1(S^1) = \mathbb Z$. You've constructed some of these bundles by hand; all you need to know is that the Chern class is just the degree (or winding number) of the transition map $S^1 \to S^1$.
As an aside, Fact 2 is the main reason why physicists care about homotopy groups of $SO(k)$, $U(k)$, and $SU(k)$. We can reduce the (noncompact) $GL_k^+\mathbb R$ case of oriented real vector bundles to the (compact) $SO(k)$ calculation; there are several questions on this site that detail that equivalence. In the complex case, we can reduce (noncompact) $GL_k\mathbb C$ to (compact) $U(k)$. Then we can use a little algebraic topology (the long exact sequence in homotopy of a fibration) to reduce one step further to $SU(k)$.
Let's step back and think about the general theory of characteristic classes for a bit. Here's the most general recipe I know:
Fix $G$ a Lie group, and consider $G$-bundles over a space $X$. Construct $BG$, the classifying space of $G$, and compute its cohomology $H^{\bullet}(BG)$. The classifying space has the property that $G$-bundles over $X$ give rise to maps $X \to BG$, and two such maps are homotopic if and only if the $G$-bundles they arise from are isomorphic. Hence given any $G$-bundle $P \to X$ we get a map $X \stackrel{P}{\rightarrow} BG$; if $P$ is the trivial bundle then the induced map $X \stackrel{P}{\rightarrow} BG$ is null-homotopic.
Now take cohomology; this gives us a map $P^{\bullet}: H^{\bullet}(BG) \to H^{\bullet}(X)$. Then characteristic classes of $P$ are just the images in $H^{\bullet}(X)$ of generators of $H^*(BG)$. And if $P$ is trivial then the map is zero on cohomology so all characteristic classes must vanish - this is how characteristic classes provide an obstruction to a bundle being trivial.
For a concrete example, take complex line bundles, so $G = GL_1\mathbb C \simeq U(1)$. It turns out that $BU(1) = \mathbb{CP}^{\infty}$, and $H^{\bullet}(\mathbb{CP}^{\infty}) = \mathbb Z[x]$ as a ring, where $x$ has degree 2. So when we map into $H^{\bullet}(X)$, the image of $x$ is something in $H^2(X)$ - it's just $c_1(P)$, the first Chern class of your bundle!
In addition, all the higher Chern classes vanish in $H^{\bullet}(BU(1))$, just as the axioms would predict. And if you look at the cohomology of $BU(n)$ for larger $n$, you get the first $n$ Chern classes $c_1, \cdots, c_n$ as generators.
For (oriented) real vector bundles the story is similar. At a first pass, it's easier to work over $\mathbb{Z}/2$, and then $H^{\bullet}(BSO(n);\mathbb{Z}/2) = \mathbb{Z}/2[w_2, \cdots, w_n]$. (Here $w_1$ must vanish because we assume our bundles are oriented, hence orientable.) And if we do the hard work of lifting to integer coefficients, we get all those goofy Pontrjagin and Euler classes.
The last story that you're asking about - getting characteristic classes from curvature of connections - is called Chern-Weil theory. I don't know this story well enough to write a good post on it, but there are quite a few references that you can find via your favorite search engine. (There are even some posts on M.SE about it that should give you good starting places.)
• Any favourite texts to learn everything you just mentioned? I would very much like to have this much knowledge! – snulty May 25 '16 at 17:01
• Some combination of Milnor-Stasheff Characteristic Classes, Hatcher's book project Vector Bundles and K-Theory, the nLab, M.SE/MO, and other sources I can't remember. But the first two are really excellent. – Thurmond May 26 '16 at 20:07
• @Thurmond This is probably a stupid question, but, regarding Fact 2, Version 2, which element of $\pi_1(SO(2)) \cong \pi_1(S^1) \cong \mathbb{Z}$ corresponds to the the tangent bundle of $S^2$? (Here I'm using the fact that $G = GL_2(\mathbb{R})$ strong deformation retracts onto $SO(2)$ via the Gram-Schmidt process.) – Jeffrey Rolland Dec 24 '16 at 0:08
• (Reading through $\textit{Vector Bundles and K-Theory}$ by Hatcher, on page 22 it appears to say that 2∈ℤ≅[S1→G] is the characteristic class of $T(S^2)$; please feel free to correct me if this is wrong.) (Additional dumb question: Why does the vector field Hatcher chose on page 22 produce the correct clutching function? Why doesn't a vector field where all vector point parallel, forming parallels of lattitude for integral curves, work to produce the clutching function of the tangent bundle as well, for instance?) – Jeffrey Rolland Dec 24 '16 at 1:01
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Questions & Answers
Question
Answers
# Let $R$ be the relation on $Z$ defined by $R=\left\{ (a,b):a,b\in Z,a-b\text{ is an integer} \right\}$ . Find the domain and range of $R$ .
Answer Verified
Hint: We are given that $R$ is the relation on $Z$ defined by $R=\left\{ (a,b):a,b\in Z,a-b\text{ is an integer} \right\}$. So we know that the difference of integers is always integers. Try it, you will get the answer.
Complete step-by-step answer:
Relations and its types of concepts are some of the important topics of set theory. Sets, relations, and functions all three are interlinked topics. Sets denote the collection of ordered elements whereas relations and functions define the operations performed on sets. A relation is said to be asymmetric if it is both antisymmetric and irreflexive or else it is not.
The relations define the connection between the two given sets. Also, there are types of relations stating the connections between the sets.
Sets and relations are interconnected with each other. The relation defines the relation between two given sets.
If there are two sets available, then to check if there is any connection between the two sets, we use relations. In discrete Maths, an asymmetric relation is just opposite to a symmetric relation. In a set A, if one element less than the other satisfies one relation, then the other element is not less than the first one. Hence, less than (<), greater than (>), and minus (-) are examples of asymmetric. We can also say, the ordered pair of set A satisfies the condition of asymmetric only if the reverse of the ordered pair does not satisfy the condition. This makes it different from symmetric relation, where even if the position of the ordered pair is reversed, the condition is satisfied. A relation is said to be asymmetric if it is both antisymmetric and irreflexive or else it is not.
For example, an empty relation denotes none of the elements in the two sets is the same.
It is given in the question that, $a,b\in Z$.
The relation $R$ on $Z$ is defined by
$R=\left\{ (a,b):a,b\in Z,a-b\text{ is an integer} \right\}$
As we know the difference of integers is always integers.
So, Domain $R$= Z. Also, range R$=$ $Z$.
Note: Read the question carefully. Also, you must know the concept behind the relation. You should be familiar with the formulae. Do not make a silly mistake while simplifying. Solve the problem in a step by step manner.
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## General help
### HTTP 500 error after Backup/Restore
HTTP 500 error after Backup/Restore
Hello again,
It has come to my attention that after running backups and restores of quizzes and courses that the HTTP 500 error message appears while trying to do these actions. I've examined the server and database error logs, turned on debugging, and searched the Moodle support forums yet I'm not a step closer to identifying the cause or this problem or solving this.
This is for running Moodle v.2.5.6 and an upgrade to a present version of Moodle is forthcoming.
Any help is appreciated.
Thank you.
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Re: HTTP 500 error after Backup/Restore
Some more information: I've examined the .mbz files that are used for this attempted backup/restore and found that the .xml files within are non-problematic. So the problem, I would assume, rests more with the Moodle database but I'm still unsure.
Any help is appreciated.
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Re: HTTP 500 error after Backup/Restore
Hello again,
We've recently upgraded to v.3.0.9 but our backup/restore problem has carried over across to this newer version. Furthermore, the HTTP 500 error message is still the only error message even after turning debugging on. I've checked the server logs and found no clues that could aid in resolving this.
Any help is appreciated.
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Re: HTTP 500 error after Backup/Restore
Probably would help to disclose some more info ... php version, mysql version as well a Operating system. Am Linux person myself and use command line a lot, so wonder if you'd like to try this to see if there is something that displays ...
In moodlecode/admin/cli/ there is a php script that backs up individual courses with the defaults as set by Moodle UI in backups. Try runnning it for a front page backup which is course ID 1 and save the backup outside of Moodle file system.
php backup.php --courseid=1 --destination=/root/
Even without debugging turned on one might see something of the routine itself.
If successful, then it suggest configuration of web service might be the issue.
'spirit of sharing', Ken
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Re: HTTP 500 error after Backup/Restore
Hi Ken,
I've ran the backup.php script and it did provide a *.mbz file output that looks appropriate but we've received another HTTP 500 error when trying to restore this.
It's looking more like our web service indeed may be the cause of concern here.
We are using:
- PHP v.5.6.30
- IIS 7.0
- MS SQL Management Studio 2008
How should we proceed from here?
Thanks again.
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Re: HTTP 500 error after Backup/Restore
Not an IIS person myself ... and think you are correct ... 500's are general catch all and there could be all sorts of reasons. ownerships/permissions on moodledata/temp/backup/?
That assumes that the backup.php script did create a valid backup file ... mbz.
One way to see about valid .mbz ... un-archive the backup .mbz into a test directory. One should see a 'moodle_backup.xml' file at the root of the 'test' directory. That file is like the crontoller for restores.
Turn on debugging to developer and try the restore again - might get something ... although 500's will even stop that.
Check IIS server logs ... not only web service but anything you can think of that involves creating files/directories in moodledata/temp/backup/ .... that includes the levels of access to also remove or change ownerships in that directory.
'spirit of sharing', Ken
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Re: HTTP 500 error after Backup/Restore
Hi again,
I've de-archived the .mbz file and it does look like that this backup from the command line php backup.php worked properly. I do see the moodle_backup.xml file and it is populated with data.
I've then checked the folder permissions and IIS does indeed have permissions for all folders in question.
Next, I have the server log files open and I do see that there are some entries within that note HTTP 500 for the 'sc-status' header.
In the moodledata/temp/backup folder there are log files here, all from 8/15 (when we did the command line backup) and these are all 0 KB in filesize with a few folders also listed for the same date/time.
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Re: HTTP 500 error after Backup/Restore
Un-archived backup sounds good.
Those .log files with 0 byte size normal - means a successfull backup. Any .log file with more than 0 bytes, you can open and inspect. What you'll find is references to a 'level' like 100,200,600,700,900 .... think 1000 is success.
If there are directories that have long filenames ... those were being used when building the backup file. They should have been un-linked and then removed upon successful backups. If there are some remaining, that could be the error 500 thing.
It is safe to manually erase those directories/folders. Do so before your next attempt at backups.
The 'sc-status' header is the clue:
Those should provide a number ... which you might find in:
https://support.microsoft.com/en-us/help/943891/the-http-status-code-in-iis-7-0--iis-7-5--and-iis-8-0
Like I said, don't run IIS so .... time for you to dig a little more.
Is your PHP a 64 bit version? Need 64 bit .... even if the backup files don't appear to be large there could be 'large chunks of data' which 32bit PHP couldn't handle.
'spirit of sharing', Ken
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Re: HTTP 500 error after Backup/Restore
Hi again,
In the error logs following the '500' that has been popping is 0, which refers to the sc-substatus column header. This suggests to me that the error would be an HTTP 500.0 error and the link provided states that this is a '500.0 - Module or ISAPI error occurred'.
The directories within the moodledata/temp/backup have been removed. I'll check this folder again after the next backup attempt.
As I recall this is PHP v.5.6.30 64-bit. Furthermore, I've been told that these courses backed up (or attempted to be backed up) are small in file-size.
Thanks again.
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Re: HTTP 500 error after Backup/Restore
Unless a true Windows person pops in here to 'save the day', I could be confusing the issue because, as I've stated before, don' t run IIS. Had your initial posting in this thread stated the issue was on Windows, I probably wouldn't have responded - not having first hand experience with Windows servers and Moodle .... not since NT 4.0.
Please learn to use whatever search tool you like to dig more.
In a quickie search, did hit on tha fact that logs show more than just Module or ISAPI error if one digs into what that means and even suggest testing 'locally' ... ie, from the desktop of the server itself.
Here's one such hit:
https://serverfault.com/questions/381872/why-would-i-get-a-500-internal-server-error-iis7-fastcgimodule-with-php
That may/may not apply to your situation ... more food for thought?
Guess we'll have to wait for a Windows admin to jump in here.
'spirit of sharing', Ken
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Re: HTTP 500 error after Backup/Restore
Hello again,
I've enabled the Handler Mappings in IIS to enable php-cgi and isapi.dll and then further gave Windows permissions to php-cgi.exe as specified in the link but; unfortunately, our results were identical.
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Re: HTTP 500 error after Backup/Restore
Small update here. We're now seeing partial success now as our recent testing shows that one course was able to be backed up but another cannot. The unsuccessful attempt still results in the HTTP 500 error.
I'll post more details as they become available.
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Re: HTTP 500 error after Backup/Restore
One further item to note is that the error log has been giving the sc-win32-status error code of 258. Looking around this code seems to indicate it is for a WAIT_TIMEOUT error along with the HTTP 500.0. Further searches have not indicated how to resolve this, however.
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Re: HTTP 500 error after Backup/Restore
I've looked up for further resolutions to resolving this HTTP 500 and (likely) Wait Timeout error but nothing is effective. I've further adjusted IIS settings such as script time-out to a higher value, and set error messages to display within the browser, and I'm still not closer to a solution.
If there is anyone familiar with this problem and IIS/Windows your help is appreciated.
Thanks.
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Re: HTTP 500 error after Backup/Restore
Another update, still not closer to a solution however.
We've managed to get a proper error message displayed and what appears is:
Module FastCgiModule ExecuteRequestHandler PHP 0x80070102
We've then updated the Activity Timeout and Request Timeout to 300 seconds but the page still gave this HTTP 500 error.
And in the IIS logs this HTTP 500 error is reflected in the error message:
2017-09-06 17:45:20 10.0.10.206 POST /backup/backup.php - 80 - 10.33.10.172 Mozilla/5.0+(Windows+NT+10.0;+Win64;+x64)+AppleWebKit/537.36+(KHTML,+like+Gecko)+Chrome/60.0.3112.113+Safari/537.36 http://mysite.com/backup/backup.php training.mysite.com 500 0 258 303359
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Re: HTTP 500 error after Backup/Restore
Partial success now.
I've updated the FastCGI and applicationHost.config settings and managed to get proper outputs of course backups. These settings were for the Activity Timeout, Idle Timeout, and Request Timeout (I'm still a little unsure on which exactly is the most relevant here) and the Instance Max Requests.
This has enabled me to complete about a half-dozen backups but others would go on for a prolonged period of time and in moodledata\temp\backup I would see the working files/folders in production here but work to a certain point, stall, and then wait out the remaining time limits.
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Re: HTTP 500 error after Backup/Restore
I've also been looking in the DB at the mdl_files table and see that is has about 1/3 or its remaining KB capacity. Should I clear this table or assign more memory space her to get past these stalled backups?
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Re: HTTP 500 error after Backup/Restore
Think I'd Leave the mdl_files table alone. '1/3 remaining KB capacity'?????? ' Pardon my ignorance, but I don't re-call ever having to be concerned about such a thing. Now config of mysql for data size and number of instances, yes but that's the entire DB ... not just certain tables.
Are you absolutely sure that web service user can: create new files, copy files, delete files in moodledata/temp/backup/ as well as copy what could be *large* .mbz files from the build area to destination ... which, if left with defaults, would be moodledata/filedir/
IF backups/restores have been failing might be due to php settings ... time for a script to run, memory a script can consume. Are there any long directory names in moodledata/temp/backup/? Are there any larger than 0 byte .log files in same directory.
'spiirt of sharing', Ken
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Re: HTTP 500 error after Backup/Restore
Hi Ken,
1) In mdl_files I was examining the possibility whether the table had exceeded its memory limits or not and whether that is relevant to what I've been experiencing.
2) Yes, IIS webservice does have full control and permissions to read, write, modify, execute, etc. the moodledata folder, subfolders, and its contents.
3) Earlier today I've managed to backup a large course of roughly 140 MB whereas others occupy low MB or fewer in file size. Even smaller courses would timeout after the set period.
4) The moodledata/temp/backup folder does contain log files with 0 KB in size. The folders are larger than 0 KB and both the files and folders contain the long string of letters and numbers as their names.
5) I've been examining a series of PHP and FastCGI settings that could be relevant here yet successful backups are random.
Thank you once again.
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Re: HTTP 500 error after Backup/Restore
Successful backups/restores leave only a .log file. Anything else and it's failed. If you went into one of those failed course backups one will see .xml file. Looking at those and knowing what a course has in it might help track this issue down.
To avoid confusion, remove anything in /moodledata/temp/backup/ Then attempt to backup again one of the courses that failed. That way you have only one attempt to sort through.
148M mbz isn't really a 'large course' .... have some servers that multiple G's .... like 20+.
What counts is heavy processing .... is there anything common to the courses that fail? Like a bunch of quizzes/test?
Let's remember that while you are backing up courses others are using the site. So it's possible your backup added with the usage by users at the same time could cause server slow down and a backup to 'time out'. Plus there are Moodle task/cron jobs. Depending upon site, task/cron jobs could spike from time to time.
How about web service error logs again ... same errors today 500's? Did you track those down?
BTW, if a web based backup fails, one can do the same course ID via commands line using the backup.php script in moodlecode/admin/cli/
Try that ... should take web service out of the loop. And, the CLI script might show/reveal a little more about errors, too.
'spirit of sharing', Ken
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Re: HTTP 500 error after Backup/Restore
Hi again,
1) For the incomplete/failed backups I've looked into their folders within moodledata/temp/backup and the .xml files do have data populated within.
2) I have removed this folder of any existing folders or .log files and it does not contribute to the success of a backup.
3) I have inquired about any common quizzes or other course element that may be common across all these failed backup attempts. I'm presently awaiting a response from the users that regularly perform the backups.
4) Yes, I do see the HTTP 500s and HTTP 200s (for successes) in the IIS logs.
5) The command line backups do work but for the users that regularly do these backups it is impractical, and they would much rather prefer to handle it via Moodle. I have informed them that this is an option and can be done if needed while work continues to remedy this existing problem.
Thank you once again for your continued assistance.
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Re: HTTP 500 error after Backup/Restore
In response to:
1 - size of the xml files gives you clues as to how heavy proceesing course backup might be ... large quiz.xml file ... lot's of quizzes .... large files.xml ... lots of files.
2 - did say manually removing would fix the problem ... just gives you a target to look at.
5 - indicates that it is truly an IIS issue then.
How long did the command line backup take? That takes php time outs out of the loop.
Do the same course you backed up via the command line via web interface.
IF it fails, inspect the moodledata/backup/temp/longnamedirectory's .log file.
That is text and can open to inspect with notepad. Backups build a plan then execute the plan recording the stage in the .log file. To what stage did the backup get? 100, 200, 300, ?
Have asked this before .... don't re-call if you reponded or not:
what is max time for a php script to run?
what is max memory any script can consume?
Both of those found in Site Admin -> Server -> PHP Info.
You might need to increase those values.
Also, the backup.php script in admin/cli/ has an option to give the path to where the backup is to be eventually copied from moodledata/temp/backup/ When you ran the command line backup on that course did you provide a path? If NOT then the cli script uses defaults as set in the backup of courses via the Moodle Admin UI.
Like I've said before ... I don't run Windows.
Guess there's are no Windows experts in these forums - hmmmmm .... what does that tell ya?
'spirit of sharing', Ken
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Re: HTTP 500 error after Backup/Restore
Hi again,
- The temporary folders do include quizzes and they do take up the bulk of the file size in these temporary folders. To answer an earlier question, these quizzes are not commonly used across several courses and each quiz is unique.
- I've tried backing up using both methods for a course that repeatedly did not backup. The Moodle approach ran the full duration (ten minutes, 600 seconds) whereas the command line approach will state '== Performing backup. . . ==' and then wait indefinitely.
- Examining the server logs, the Moodle approach did run POST /backup/backup.php and did provide two HTTP 200 messages and one HTTP 500 message after timeout.
- When I run the command line script I do specify a destination - D:\Moodle\ but no completed backups are relocated here.
- Opening the .log files in notepad (or any other text editor) comes up blank.
In the phpinfo.php display, the following is listed:
- max_execution_time = 600
- memory_limit=128M
Thank you once again for your continued assistance.
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Re: HTTP 500 error after Backup/Restore
On a failed backup, for whatever reason, there should still be a larger than 0 byte .log file. As the backup process progresses that .log file is written to .... at each stage of backup. Take it you did change the extension from .log to .txt before attempting to open with NotePad.
What are ownerships/permissions on moodledata ... on temp ... on backups directories?
When you run the command line version and give a destination does D:\Moodle\ ownerships/permisisons allow PHP to *copy* the built 'backup-whatever.mbz' file from moodledata/temp/backup/longfilename build directory?
Where the CLI might fail in Linux via command line is when running as apache user and apache user doesn't have access rights to write to destination (that's why I run those as the root user) OR the backup is too large for the copy command. Evidence of that is a .mbz file that remains in the build directory.
Take a screen shot of what the temp build directory looks like for one failed backup and attach in a response.
Really don't think this is a moodle code issue but a IIS config issue from what you've described. Sorry ... 500 errors are hard to track down.
If this issue were rampant with Windows installs running IIS there would be many many postings with 'me too', etc..
'spirit of sharing' .... still looking for some IIS expert to jump in here!
Ken
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Re: HTTP 500 error after Backup/Restore
Hi again,
- I've not changed the files from .log to .txt. Regardless, it still provides a blank if I do change these file extensions.
- moodledata, temp, and backup each have IIS_IUSRS listed with full control of the directories.
- Specifying a destination in the command line approach does place the backup file in the specified location for courses that can be backed up. So I can specify D:\Moodle, or moodledata\temp\backup and it will appear wherever specified.
Would quizzes or some other course content be responsible for this stalling problem? Could there be dependencies on this data which does not enable it to undergo these backups?
Screenshot below:
Thank you again.
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Re: HTTP 500 error after Backup/Restore
Sigh ... again ... 0 byte sized log files indicate a *successful backup*. .log files larger than 0 bytes indicate a failed backup and show the stages of the backup process it was able to complete before failing. Viewing that larger then 0 byte file might give you a clue for what to look for in the course - * see below.
In an eariler posting you said when you ran the command line script and gave a destination the script did not complete ... no .mbz file in the destination directory. Now you say there are .mbz files in desitination?
Too bad Windows can't show realtime processes and activity in directories as they occur ... or fail. :\
Since there is no log file created when the process begins, me thinks that's the point at which if fails ... then times out.
Now to the last question ... yes, there could be a part of the course that's causing the failure, but there should be a contenthash named directory and a .log file begun IF the process can even begin in any attempt.
* IF you suspect it's something in specific courses causes, then the only way I know is to inspect each of the troubled courses and determine what's NOT core. Step through a manual backup where you select the items to be backed up ... excluding the addons ... include only core activities/mods, etc.
IF that completes then what would that show us?
I take it that if you do have addons they are all up to date and current for your version of Moodle.
'spirit of sharing', Ken
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Re: HTTP 500 error after Backup/Restore
To clarify, I've tested the command line approach for courses that have proven to be successfully backed up already and those that presently cannot be backed up, by either method. If the backup performed successfully, the .mbz file would be placed wherever specified. If the backup was not performed successfully there would be no .mbz file placed wherever specified.
I will test another backup attempt and exclude any add-ons or miscellaneous materials and see if this has any difference.
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Re: HTTP 500 error after Backup/Restore
Quick update:
It did work if I uncheck all other backup settings and leave activities and resources. So it looks like the error(s) was within one of these settings.
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Re: HTTP 500 error after Backup/Restore
Another quick update:
If I uncheck the 'Include enrolled users' it will then disable five other fields - Include user role assignment, Include comments, Include badges, Include user completion details, and Include grade history. I've now made successful backups by setting this one field off.
Next, I'll try turning these settings off individually to see where specifically the problem is occurring.
Also, the plugins and add-ons for the site were updated back in August but this problem has existed prior to this.
Thanks.
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Re: HTTP 500 error after Backup/Restore
You are getting closer!!! Progress is good ... to find additional plugins for the site:
Might see if this applies:
Look for any related to quiz - and how many instances of that addon are in the site.
The version you are currently running 2.5.x is soooooo old now the closest thing I have to compare is a 3.0.x
Have re-read your original posting ... going to be upgrading soon ... need to as 2.5.x is very old.
Think I'd map out strategy for upgrading beginning now - right after or even before starting the upgrade 'march'. It will be a 'march' meaning you'll have to upgrade from 2.5.x to at least 2.7.highest and then I'd do some research on what the next version should be.
Especially take stock/notice whatever plugin is causing the issue and check if there is a compatible version for your destination version by searching Moodle plugins site.
Bsst of luck!
'spirit of sharing', Ken
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Re: HTTP 500 error after Backup/Restore
Hi again.
Yes, we are getting closer to resolving this problem.
As mentioned earlier, we've upgraded to Moodle v.3.0.9 and updated the plugins afterwards, but since this problem was preexisting we were unclear whether upgrading Moodle would impact this backup/restore and HTTP 500 problem.
Also, I've tried unchecking the other five backup settings would still result in a backup failure. Only unchecking the 'Include enrolled users' will result in a successful backup now. This leads to my next question of what is included when I select this setting, and how can I distinguish what user/course information goes into a successful or unsuccessful backup.
Thanks again.
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Re: HTTP 500 error after Backup/Restore
Well, guess I missed 'already upgraded to 3.0.9 ...' ... which is also still behind and no longer getting fixes and updates for anything ... but you know that.
I do re-call there being issues with certain role conflcts (superuser/course creator?) on restores, but not on backups - and not error 500's.
Windows systems use 'Administrator' as super user ... there is a config.php line that dictates 'admin' in Moodle being admin in moodle ... supposedly not to confuse WinDoze whose system admin might have changed 'Administrator' to 'admin'.
$CFG->admin = 'admin'; Dunno if that would be a factor or not ... again ... don't do Windoze. Approach to backing up .... check to see who on the system has superuser/admin rights. Write them down. Go into troubled courses. See if those users are also assigned as course creators or Teachers in the course. Remove those users .... leaving only the students. Try a backup. What are you using for authentication? On Windoze would assume LDAP. Are there any users in troubled courses that are authenticating with something other than LDAP? Moodle either does all users or none ... nothing in between - no user select list. IF one can get a .mbz backup and restore fails ... might be due to roll conflict. I've had to un-gizip an old backup, edit users.xml removing the user ID'd as having a conflicting role (Moodle doesn't tell you which user ... just that there is a role conflict), then re-gzipped the backup making sure moodle_backup.xml is at the root of the gzip compressed file. Then, and only then, would the course restore. Still however, that situation didn't throw a 500 error. ????? what happens if you make a no user ... no students no work no assignments no files ... no teacher ... backup of a troubled course? What addons do you have installed? Am nearing ropes end here and some of what I might suggest just might send you down a rabbit hole ... illusive 'wabbit' (Elmer Fudd here). Shame shame on Windows experts in these forums. 'spirit of sharing', Ken 'spirit of sharing', Ken Average of ratings: - Re: HTTP 500 error after Backup/Restore Hello again. Sorry for the delay. I've gotten sidetracked. 1) v.3.0.9 is the present version in use and plans to modernize again will commence in the near future. 2) The$CFG->admin variable has never changed from 'admin' and I don't suspect this would be the cause of the problem as other errors would likely have appeared prior to this backup problem.
3) Yes, we are using LDAP on this site for login authentication. I will investigate whether this has any effect on enrolled users in courses during the backup process.
4) A troubled course will successfully backup if I remove the 'Include enrolled users' setting during the backup initial settings page. All other settings are inconsequential.
Thank you again.
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Re: HTTP 500 error after Backup/Restore
Hello again,
I've gotten side-tracked again but I've been able to give Moodle some more attention today.
Since the 'Include Enrolled Users' checkbox is a factor here for the success or failure of a backup, which tables in particular are being included in this backup? This way I can run some queries to better identify where specifically this user data is faulting the backup process.
I have discussed LDAP as a possible factor in this situation and we're in the middle of devising some action to investigate this further. This could possibly lead to disabling LDAP-based user accounts and see if this has any effect in getting proper course backups.
Thanks.
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Re: HTTP 500 error after Backup/Restore
Will say this ... you are relentless! (compliment) .... not sure that you'll find fault with what the tables for logs contain for users or there would be other issues present.
Having said that, the issue really might be too much data per user might be causing this issue.
Anyhoo,
SHOW COLUMNS FROM mytable FROM mydb;
is a MySQL 5.7 command to show really how a table is structured ... column names/attributes.
So if using MySQL plugin your DB replacing mydb above and take a gander at:
mysql> show tables like '%log%';
which will output more than one table but the two to view are: mdl_log and mdl_logstore_standard_log
You might want to get a heavy user ID so you can query the last table above to see just how many entries there are for just one heavy user.
OR you could add the code suggested to skip course ID 1. ;)
'spirit of sharing', Ken
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Re: HTTP 500 error after Backup/Restore
Hi again,
I've looked at the log and storage tables and they are indeed large but not so large that I would expect them to interfere with the backups here. I've adjusted the php.ini file to provide additional variable handling but this doesn't seem to do the trick. Any other ways we can increase the volume of user data into these backups?
Thanks again.
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Re: HTTP 500 error after Backup/Restore
Hi again,
There are many logs to view here (hundreds of thousands) and I've looked at increasing my PHP variable limit but to no avail.
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Re: HTTP 500 error after Backup/Restore
Been thinking some more on this (dangereous) ... what you might have to do is upgrade the 2.5.x to highest 2.5.x and then to 2.6.highest as that's where the new compression (.gzips rather than zips) for backups were first introduced. Prior to 2.6 and that change/use of new compression many many sites were having problems with backups ... Windows servers primarily due to 32 bit version of PHP but even Linux servers (CentOS as an example) had issues with the tail end of backup process where the code used *copy* rather than *move* of the completed .mbz file to destination *IF* the backup file was 4 Gig or larger (operating system limitation on copy command).
To be on the safe side, clone your site and do the upgrades to 2.6.x highest on the clone. First thing to check, of course, is one of those troubled courses and backups.
May not have been your doing, but a take away here might be don't allow Moodle code to get that far behind in the future. Not saying one has to get the latest/greatest ... it's good to stay *near* the 'leading edge' but not so far that one is on the 'bleeding edge'. It's open source software thus, even if marked/tagged as 'stable', anyone using .0 or latest and greatest of any version of Moodle is really an 'omicron tester' (that's my humble opinion, of course).
2 cent advice ... spirit of sharing', Ken
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Some publishers, particularly the IEEE, require that the columns on the last page of an article are balanced, so it looks pretty. The problem is that the break is usually needed in the middle of the bibliography, for which less layout-control is available. Fortunately, there are some specific specific solutions for various cases, and one which works for most: flushend.
The IEEEtran BibTeX style fortunately provides a specific command to break the references after a given number of entries.
``````\IEEEtraggeratref{XX}
``````
In the lucky cases where the last page contains something more than the bibliography, one can fiddle with the space in which the text is laid out, and shorten it to force a column break.
`\enlargethispage{-Xcm}`
Problems arise when using something different than the IEEEtran BibTeX style (including the BibLaTeX IEEE style), and the bibliography is such that the page that one get a chance to enlarge is not, actually, the last page.
A recent package cater for all needs: flushend. Simply including it in the preamble is sufficient to make LaTeX render the last page with roughly balanced columns, regardless of their contents. Pretty neat!
`\usepackage{flushend}`
Update 2016-05-09: When using the biblatex-ieee package, the last line of the last citation may be flushed too much to the left with `flushend`. Adding the `keeplastbox` option when loading the packages fixes this.
`\usepackage[keeplastbox]{flushend}`
## 2 thoughts on “Balancing the last page in twocolumn LaTeX documents”
1. When I using \usepackage{flushend} in my preamble, the last page just turns to blank. Do you know why?
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# Re: [tlaplus] Re: TLC choose a new unique element everytime
Leslie actually meant to write "with", not "when". The latter is a synonym for "await" and is used for synchronization.
Regards,
Stephan
> On 14 Jul 2017, at 21:46, Shuhao Wu <shu...@xxxxxxxxxxxx> wrote:
>
> Thank you for pointing me to the "when" construct. I'll study it in greater details next.
>
> For the TLC error, the PlusCal algorithm is on my previous email and the model values are:
>
> Records <- [ model values ] {r1, r2, r3}
> N <- 5
> defaultInitValue <- [ model value ]
> NoRecord <- [ model value ] (This is actually in the definition override)
>
> Under the Invariants under the "What to check?" section, I'm checking DataOK.
>
> If there's anything else I can provide, I'll be happy to do so.
>
> Thanks again,
> Shuhao
>
>> On 2017-07-14 03:15 PM, Leslie Lamport wrote:
>> First of all, when reporting a TLC error, please include the relevant model
>> values that produce the error.
>> mention. Some operators in the TLC module, including RandomElement, are
>> not mathematics. TLA+ specs should be mathematics. Therefore,
>> RandomElement should not be used in a TLA+/PlusCal spec. Randomness is
>> relevant for obtaining statistics. You are apparently using RandomElement
>> to introduce nondeterminism. If you look at any examples of PlusCal
>> algorithms, you will see that nondeterminism is expressed with the *when*
>> construct.
>> Leslie
>>> On Friday, July 14, 2017 at 11:06:52 AM UTC-7, Shuhao Wu wrote:
>>> Hello,
>>>
>>> I've been trying to specify a simple problem where one process writes
>>> arbitrary data to a source datastore and a log and a second process
>>> follows and applies that log to a different datastore (similar to MySQL
>>> binlogs). The corresponding Pluscal implementation is shown at the end
>>> of the email. The implementation is supposed to show a violation of the
>>> DataOK invariance as lfread and lfwrite are two steps rather than one
>>> atomic steps.
>>>
>>> The idea is that the DataUpdater process will write an entry into
>>> source. This is done via the source[currentI] :=
>>> RandomElement(PossibleRecords). In TLC, I have to specify
>>> PossibleRecords as a set of finite number of model values. This does not
>>> seem like it's "correct" so to speak, as it is possible for
>>> RandomElement(PossibleRecords) = RandomElement(PossibleRecords) if two
>>> of the same values happened to be picked.
>>>
>>> Is there a better way to specify this for TLC? I feel this may be
>>> impossible as I can't give TLC an infinite set. *In which case: is there
>>> a better way to model this so TLC can work on it?
>>>
>>> Furthermore, it seems like even if I use RandomElement, the operation of
>>> source[currentI] := RandomElement(PossibleRecords) causes TLC to
>>> somehow loses traces. If I run the below algorithm in TLC with
>>> Invariance of DataOK, I will get:
>>>
>>> Invariant DataOK is violated.
>>> Failed to recover the initial state from its fingerprint.
>>> This is probably a TLC bug(1).
>>>
>>> I see that people have reported this in the past
>>> after reading the thread, I'm not quite sure if there is a workaround
>>> for this. Does anyone know?
>>>
>>> Thanks,
>>> Shuhao
>>>
>>> ----------------------------------------------------------------------------
>>>
>>>
>>> EXTENDS TLC, Integers, Sequences
>>>
>>> CONSTANT N, Records
>>>
>>> NoRecord == CHOOSE r : r \notin Records
>>>
>>> PossibleRecords == Records \cup {NoRecord}
>>>
>>> (***************************************************************************
>>>
>>> --algorithm foo {
>>> variables
>>> source = [k \in 1..N |-> NoRecord],
>>> target = [k \in 1..N |-> NoRecord],
>>> log = <<>>
>>> ;
>>>
>>> process (DataUpdater = "DataUpdater")
>>> variable currentI = 1;
>>> {
>>> duloop: while (currentI <= N) {
>>> source[currentI] := RandomElement(PossibleRecords);
>>> log := Append(log, source[currentI]);
>>> currentI := currentI + 1;
>>> }
>>> }
>>>
>>> process (LogFollower = "LogFollower")
>>> variable currentR;
>>> {
>>> lfloop: while (Len(target) < N) {
>>> lfwrite: target[Len(log)] := currentR;
>>> }
>>> }
>>> }
>>> ***************************************************************************)
>>>
>>>
>>> DataOK == (\A self \in ProcSet: pc[self] = "Done") => source = target
>>>
>
> --
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> For more options, visit https://groups.google.com/d/optout.
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# Stieltjes constants
The area of the blue region converges on the Euler–Mascheroni constant, which is the 0th Stieltjes constant.
In mathematics, the Stieltjes constants are the numbers $\gamma_k$ that occur in the Laurent series expansion of the Riemann zeta function:
$\zeta(s)=\frac{1}{s-1}+\sum_{n=0}^\infty \frac{(-1)^n}{n!} \gamma_n \; (s-1)^n.$
The zero'th constant $\gamma_0 = \gamma = 0.577\dots$ is known as the Euler–Mascheroni constant.
## Representations
The Stieltjes constants are given by the limit
$\gamma_n = \lim_{m \rightarrow \infty} {\left\{\sum_{k = 1}^m \frac{\ln^n k}{k} - \frac{\ln^{n+1} \! m}{n+1}\right\}}.$
(In the case n = 0, the first summand requires evaluation of 00, which is taken to be 1.)
Cauchy's differentiation formula leads to the integral representation
$\gamma_n = \frac{(-1)^n n!}{2\pi} \int_0^{2\pi} e^{-nix} \zeta\left(e^{ix}+1\right) dx.$
Various representations in terms of integrals and infinite series are given in works of Jensen, Franel, Hermite, Hardy, Ramanujan, Ainsworth, Howell, Coppo, Connon, Coffey, Choi, Blagouchine and some other authors.[1][2][3][4][5][6] In particular, Jensen-Franel's integral formula, often erroneously attributed to Ainsworth and Howell, states that
$\gamma_n \,=\,\frac{1}{2}\delta_{n,0}+\,\frac{1}{i}\!\int\limits_0^\infty \! \frac{dx}{e^{2\pi x}-1} \left\{ \frac{\ln^n(1-ix)}{1-ix} - \frac{\ln^n(1+ix)}{1+ix} \right\}\,, \qquad\quad n=0, 1, 2,\ldots$
where δn,k is the Kronecker symbol (Kronecker delta).[5][6] Among other formulae, we find
$\gamma_n \,=\,-\frac{\pi}{2(n+1)}\! \int\limits_{-\infty}^{+\infty} \frac{\ln^{n+1}\!\big(\frac{1}{2}\pm ix\big)}{\cosh^2\!\pi x}\, dx \qquad\qquad\qquad\qquad\qquad\qquad n=0, 1, 2,\ldots$
$\begin{array}{l} \displaystyle \gamma_1 =-\left[\gamma -\frac{\ln2}{2}\right]\ln2+\,i\!\int\limits_0^\infty \! \frac{dx}{e^{\pi x}+1} \left\{ \frac{\ln(1-ix)}{1-ix} - \frac{\ln(1+ix)}{1+ix} \right\}\, \\[6mm] \displaystyle \gamma_1 = -\gamma^2 - \int\limits_0^\infty \left[\frac{1}{1-e^{-x}}-\frac{1}{x}\right] e^{-x}\ln x \, dx \end{array}$
see.[1][5][7]
As concerns series representations, a famous series implying an integer part of a logarithm was given by Hardy in 1912[8]
$\gamma_1\,=\, \frac{\ln2}{2}\sum_{k=2}^\infty \frac{(-1)^k}{k} \,\lfloor \log_2{k}\rfloor\cdot \big(2\log_2{k} - \lfloor \log_2{2k}\rfloor\big)$
Israilov[9] gave semi-convergent series in terms of Bernoulli numbers $B_{2k}$
$\gamma_m\,=\,\sum_{k=1}^n \frac{\,\ln^m\! k\,}{k} - \frac{\,\ln^{m+1}\! n\,}{m+1} - \frac{\,\ln^m\! n\,}{2n} - \sum_{k=1}^{N-1} \frac{\,B_{2k}\,}{(2k)!}\left[\frac{\ln^m\! x}{x}\right]^{(2k-1)}_{x=n} + \theta\cdot\frac{\,B_{2N}\,}{(2N)!}\left[\frac{\ln^m\! x}{x}\right]^{(2N-1)}_{x=n} \,,\qquad 0<\theta<1$
Oloa and Tauraso[10] showed that series with harmonic numbers may lead to Stieltjes constants
$\begin{array}{l} \displaystyle \sum_{n=1}^\infty \frac{\,H_n - (\gamma+\ln n)\,}{n} \,=\, \,-\gamma_1 -\frac{1}{2}\gamma^2+\frac{1}{12}\pi^2 \\[6mm] \displaystyle \sum_{n=1}^\infty \frac{\,H^{(2)}_n - (\gamma+\ln n)^2\,}{n} \,=\, \,-\gamma_2 -2\gamma\gamma_1 -\frac{2}{3}\gamma^3+\frac{5}{3}\zeta(3) \end{array}$
Blagouchine[6] obtained slowly-convergent series involving unsigned Stirling numbers of the first kind $\left[{\cdot \atop \cdot}\right]$
$\gamma_m\,=\,\frac{1}{2}\delta_{m,0}+ \frac{\,(-1)^m m!\,}{\pi} \sum_{n=1}^\infty\frac{1}{\,n\cdot n!\,} \sum_{k=0}^{\lfloor\!\frac{1}{2}n\!\rfloor}\frac{\,(-1)^{k}\cdot\left[{2k+2\atop m+1}\right] \cdot\left[{n\atop 2k+1}\right]\,} {\,(2\pi)^{2k+1}\,}\,,\qquad m=0,1,2,...,$
as well as semi-convergent series with rational terms only
$\gamma_m\, =\,\frac{1}{2}\delta_{m,0}+(-1)^{m} m!\cdot\!\sum_{k=1}^{N}\frac{\,\left[{2k\atop m+1}\right]\cdot B_{2k}\,}{(2k)!} \,+\, \theta\cdot\frac{\,(-1)^{m} m!\!\cdot \left[{2N+2\atop m+1}\right]\cdot B_{2N+2}\,}{(2N+2)!}\,,\qquad 0<\theta<1$
where m=0,1,2,... Several other series are given in works of Coffey.[2][3]
## Asymptotic growth
The Stieltjes constants satisfy the bound
$\big|\gamma_n\big|\,\leqslant\, \begin{cases} \displaystyle \frac{2\,(n-1)!}{\pi^n}\,,\qquad & n=1, 3, 5,\ldots \\[3mm] \displaystyle \frac{4\,(n-1)!}{\pi^n}\,,\qquad & n=2, 4, 6,\ldots \end{cases}$
given by Berndt in 1972.[11] Better bounds were obtained by Lavrik, Israilov, Matsuoka, Nan-You, Williams, Knessl, Coffey, Adell, Saad-Eddin, Fekih-Ahmed and Blagouchine (see the list of references given in[6]). One of the best estimations, in terms of elementary functions, belongs to Matsuoka:
$|\gamma_n| < 10^{-4} e^{n \ln \ln n}\,,\qquad n\geqslant5$
As concerns estimations resorting to non-elementary functions, Knessl, Coffey[12] and Fekih-Ahmed[13] obtained quite accurate results. For example, Knessl and Coffey give the following formula that approximates the Stieltjes constants relatively well for large n.[12] If v is the unique solution of
$2 \pi \exp(v \tan v) = n \frac{\cos(v)}{v}$
with $0 < v < \pi/2$, and if $u = v \tan v$, then
$\gamma_n \sim \frac{B}{\sqrt{n}} e^{nA} \cos(an+b)$
where
$A = \frac{1}{2} \ln(u^2+v^2) - \frac{u}{u^2+v^2}$
$B = \frac{2 \sqrt{2\pi} \sqrt{u^2+v^2}}{[(u+1)^2+v^2]^{1/4}}$
$a = \tan^{-1}\left(\frac{v}{u}\right) + \frac{v}{u^2+v^2}$
$b = \tan^{-1}\left(\frac{v}{u}\right) - \frac{1}{2} \left(\frac{v}{u+1}\right).$
Up to n = 100000, the Knessl-Coffey approximation correctly predicts the sign of γn with the single exception of n = 137.[12]
## Numerical values
The first few values are:
n approximate value of γn OEIS 0 +0.5772156649015328606065120900824024310421593359 A001620 1 −0.0728158454836767248605863758749013191377363383 A082633 2 −0.0096903631928723184845303860352125293590658061 A086279 3 +0.0020538344203033458661600465427533842857158044 A086280 4 +0.0023253700654673000574681701775260680009044694 A086281 5 +0.0007933238173010627017533348774444448307315394 A086282 6 −0.0002387693454301996098724218419080042777837151 A183141 7 −0.0005272895670577510460740975054788582819962534 A183167 8 −0.0003521233538030395096020521650012087417291805 A183206 9 −0.0000343947744180880481779146237982273906207895 A184853 10 +0.0002053328149090647946837222892370653029598537 A184854 100 −4.2534015717080269623144385197278358247028931053 × 1017 1000 −1.5709538442047449345494023425120825242380299554 × 10486 10000 −2.2104970567221060862971082857536501900234397174 × 106883 100000 +1.9919273063125410956582272431568589205211659777 × 1083432
For large n, the Stieltjes constants grow rapidly in absolute value, and change signs in a complex pattern.
Further information related to the numerical evaluation of Stieltjes constants may be found in works of Keiper,[14] Kreminski,[15] Plouffe[16] and Johansson.[17] The latter author provided values of the Stieltjes constants up to n = 100000, accurate to over 10000 digits each. The numerical values can be retrieved from the LMFDB [1].
## Generalized Stieltjes constants
### General information
More generally, one can define Stieltjes constants γn(a) that occur in the Laurent series expansion of the Hurwitz zeta function:
$\zeta(s,a)=\frac{1}{s-1}+\sum_{n=0}^\infty \frac{(-1)^n}{n!} \gamma_n(a) \; (s-1)^n.$
Here a is a complex number with Re(a)>0. Since the Hurwitz zeta function is a generalization of the Riemann zeta function, we have γn(1)=γn The zero'th constant is simply the digamma-function γ0(a)=-Ψ(a),[18] while other constants are not known to be reducible to any elementary or classical function of analysis. Nevertheless, there are numeorous representations for them. For example, there exists the following asymptotic representation
$\gamma_n(a) \,=\, \lim_{m\to\infty}\left\{ \sum_{k=0}^m \frac{\ln^n (k+a)}{k+a} - \frac{\ln^{n+1} (m+a)}{n+1} \right\}\,, \qquad\; \begin{array}{l} n=0, 1, 2,\ldots\, \\[1mm] a\neq0, -1, -2, \ldots \end{array}$
due to Berndt and Wilton. The analog of Jensen-Franel's formula for the generalized Stieltjes constant is the Hermite formula[5]
$\gamma_n(a) \,=\,\left[\frac{1}{2a}-\frac{\ln{a}}{n+1} \right]\ln^n\!{a} -i\!\int\limits_0^\infty \! \frac{dx}{e^{2\pi x}-1} \left\{ \frac{\ln^n(a-ix)}{a-ix} - \frac{\ln^n(a+ix)}{a+ix} \right\} \,, \qquad\; \begin{array}{l} n=0, 1, 2,\ldots\, \\[1mm] a\neq0, -1, -2, \ldots \end{array}$
Generalized Stieltjes constants satisfy the following recurrent relationship
$\gamma_n(a+1) \, =\, \gamma_n(a) - \frac{\,\ln^n\! a\,}{a} \,, \qquad\; \begin{array}{l} n=0, 1, 2,\ldots\, \\[1mm] a\neq0, -1, -2, \ldots \end{array}$
as well as the multiplication theorem
$\sum_{l=0}^{n-1} \gamma_p\!\left(\! a+\frac{l}{\,n\,} \right) =\, (-1)^p n \! \left[\frac{\ln n}{\,p+1\,} - \Psi(an) \right]\!\ln^p\! n \,+\, n\sum_{r=0}^{p-1}(-1)^r \binom{p}{r} \gamma_{p-r}(an) \cdot \ln^r\!{n}\,, \qquad\qquad n=2, 3, 4,\ldots$
where $\binom{p}{r}$ denotes the binomial coefficient (see[19] and,[20] pp. 101–102).
### First generalized Stieltjes constant
The first generalized Stieltjes constant has a number of remarkable properties.
• Malmsten's identity (reflection formula for the first generalized Stieltjes constants): the reflection formula for the first generalized Stieltjes constant has the following form
$\gamma_1 \biggl(\frac{m}{n}\biggr)- \gamma_1 \biggl(1-\frac{m}{n} \biggr) =2\pi\sum_{l=1}^{n-1} \sin\frac{2\pi m l}{n} \cdot\ln\Gamma \biggl(\frac{l}{n} \biggr) -\pi(\gamma+\ln2\pi n)\cot\frac{m\pi}{n}$
where m and n are positive integers such that m<n. This formula has been long-time attributed to Almkvist and Meurman who derived it in 1990s.[21] However, very recently Blagouchine found that this identity, albeit in a slightly different form, was first obtained by Carl Malmsten in 1846.[5][22]
• Rational arguments theorem: the first generalized Stieltjes constant at rational argument may be evaluated in a quasi closed-form via the following formula
$\begin{array}{ll} \displaystyle \gamma_1 \biggl(\frac{r}{m} \biggr) =& \displaystyle \gamma_1 +\gamma^2 + \gamma\ln2\pi m + \ln2\pi\cdot\ln{m}+\frac{1}{2}\ln^2\!{m} + (\gamma+\ln2\pi m)\cdot\Psi\!\left(\!\frac{r}{m}\!\right) \\[5mm] \displaystyle & \displaystyle\qquad +\pi\sum_{l=1}^{m-1} \sin\frac{2\pi r l}{m} \cdot\ln\Gamma \biggl(\frac{l}{m} \biggr) + \sum_{l=1}^{m-1} \cos\frac{2\pi rl}{m}\cdot\zeta''\!\left(\! 0,\,\frac{l}{m}\!\right) \end{array}\,,\qquad\quad r=1, 2, 3,\ldots, m-1\,.$
due also to Blagouchine in.[5][18] An alternative proof was later proposed by Coffey.[23]
• Finite summations: there are numerous summation formulae for the first generalized Stieltjes constants. For example
$\begin{array}{ll} \displaystyle \sum_{r=0}^{m-1} \gamma_1\!\left(\! a+\frac{r}{\,m\,} \right) =\, m\ln{m}\cdot\Psi(am) - \frac{m}{2}\ln^2\!m + m\gamma_1(am)\,,\qquad a\in\mathbb{C}\\[6mm] \displaystyle \sum_{r=1}^{m-1} \gamma_1\!\left(\!\frac{r}{\,m\,} \right) =\, (m-1)\gamma_1 - m\gamma\ln{m} - \frac{m}{2}\ln^2\!m \\[6mm] \displaystyle \sum_{r=1}^{2m-1} (-1)^r \gamma_1 \biggl(\!\frac{r}{2m} \!\biggr) \,=\, -\gamma_1+m(2\gamma+\ln2+2\ln m)\ln2\\[6mm] \displaystyle \sum_{r=0}^{2m-1} (-1)^r \gamma_1\biggl(\!\frac{2r+1}{4m} \!\biggr) \,=\, m\left\{4\pi\ln\Gamma \biggl(\frac{1}{4} \biggr) - \pi\big(4\ln2+3\ln\pi+\ln m+\gamma \big)\!\right\}\\[6mm] \displaystyle \sum_{r=1}^{m-1} \gamma_1 \biggl(\!\frac{r}{m}\!\biggr) \!\cdot\cos\dfrac{2\pi rk}{m} \,=\, -\gamma_1 + m(\gamma+\ln2\pi m) \ln\!\left(\!2\sin\frac{\,k\pi\,}{m}\!\right) +\frac{m}{2} \left\{\zeta''\!\left(\! 0,\,\frac{k}{m}\!\right)+ \, \zeta''\!\left(\! 0,\,1-\frac{k}{m}\!\right) \! \right\}\,, \qquad k=1,2,\ldots,m-1 \\[6mm] \displaystyle \sum_{r=1}^{m-1} \gamma_1\biggl(\!\frac{r}{m} \!\biggr) \!\cdot\sin\dfrac{2\pi rk}{m} \,=\,\frac{\pi}{2} (\gamma+\ln2\pi m)(2k-m) - \frac{\pi m}{2} \left\{\ln\pi -\ln\sin\frac{k\pi}{m} \right\} + m\pi\ln\Gamma \biggl(\frac{k}{m} \biggr) \,, \qquad k=1,2,\ldots,m-1 \\[6mm] \displaystyle \sum_{r=1}^{m-1} \gamma_1 \biggl(\!\frac{r}{m} \!\biggr)\cdot\cot\frac{\pi r}{m} =\, \displaystyle \frac{\pi }{6} \Big\{\!(1-m)(m-2)\gamma + 2(m^2-1)\ln2\pi - (m^2+2)\ln{m}\Big\} -2\pi\!\sum_{l=1}^{m-1} l\!\cdot\!\ln\Gamma\!\left(\! \frac{l}{m}\!\right) \\[6mm] \displaystyle \sum_{r=1}^{m-1} \frac{r}{m} \cdot\gamma_1 \biggl(\!\frac{r}{m} \!\biggr) =\, \frac{1}{2}\left\{\!(m-1)\gamma_1 - m\gamma\ln{m} - \frac{m}{2}\ln^2\!{m}\! \right\} -\frac{\pi}{2m}(\gamma+\ln2\pi m) \!\sum_{l=1}^{m-1} l\!\cdot\! \cot\frac{\pi l}{m} -\frac{\pi}{2} \!\sum_{l=1}^{m-1} \cot\frac{\pi l}{m} \cdot\ln\Gamma\biggl(\!\frac{l}{m} \!\biggr) \end{array}$
For more details and further summation formulae, see.[5][20]
• Some particular values: some particular values of the first generalized Stieltjes constant at rational arguments may be reduced to the gamma-function, the first Stieltjes constant and elementary functions. For instance,
$\gamma_1\!\left(\!\frac{1}{\,2\,}\!\right) = - 2\gamma\ln2 - \ln^2\!2 + \gamma_1\,= \,-1.353459680\ldots$
At points 1/4, 3/4 and 1/3, values of first generalized Stieltjes constants were independently obtained by Connon[24] and Blagouchine[20]
$\begin{array}{l} \displaystyle \gamma_1\!\left(\!\frac{1}{\,4\,}\!\right) =\, 2\pi\ln\Gamma\!\left(\!\frac{1}{\,4\,} \! \right) - \frac{3\pi}{2}\ln\pi - \frac{7}{2}\ln^2\!2 - (3\gamma+2\pi)\ln2 - \frac{\gamma\pi}{2}+\gamma_1 \,=\,-5.518076350\ldots \\[6mm] \displaystyle \gamma_1\!\left(\!\frac{3}{\,4\,} \! \right) =\, -2\pi\ln\Gamma\!\left(\!\frac{1}{\,4\,}\! \right) + \frac{3\pi}{2}\ln\pi - \frac{7}{2}\ln^2\!2 - (3\gamma-2\pi)\ln2 + \frac{\gamma\pi}{2}+\gamma_1 \,=\,-0.3912989024\ldots \\[6mm] \displaystyle \gamma_1\!\left(\!\frac{1}{\,3\,} \! \right) = \, - \frac{3\gamma}{2}\ln3 - \frac{3}{4}\ln^2\!3 + \frac{\pi}{4\sqrt{3\,}}\left\{\ln3 - 8\ln2\pi -2\gamma +12 \ln\Gamma\!\left(\!\frac{1}{\,3\,} \! \right) \!\right\} + \,\gamma_1 \, = \,-3.259557515\ldots \end{array}$
At points 2/3, 1/6 and 5/6
$\begin{array}{l} \displaystyle \gamma_1\!\left(\!\frac{2}{\,3\,} \! \right) = \, - \frac{3\gamma}{2}\ln3 - \frac{3}{4}\ln^2\!3 - \frac{\pi}{4\sqrt{3\,}}\left\{\ln3 - 8\ln2\pi -2\gamma +12 \ln\Gamma\!\left(\!\frac{1}{\,3\,} \! \right) \!\right\} + \,\gamma_1 \, = \,-0.5989062842\ldots \\[6mm] \displaystyle \gamma_1\!\left(\!\frac{1}{\,6\,} \! \right) = \, - \frac{3\gamma}{2}\ln3 - \frac{3}{4}\ln^2\!3 - \ln^2\!2 - (3\ln3+2\gamma)\ln2 + \frac{3\pi\sqrt{3\,}}{2}\ln\Gamma\!\left(\!\frac{1}{\,6\,}\! \right) \\[5mm] \displaystyle\qquad\qquad\quad - \frac{\pi}{2\sqrt{3\,}}\left\{3\ln3 + 11\ln2 + \frac{15}{2}\ln\pi + 3\gamma \right\}+\, \gamma_1 \, =\,-10.74258252\ldots\\[6mm] \displaystyle \gamma_1\!\left(\!\frac{5}{\,6\,} \! \right) = \, - \frac{3\gamma}{2}\ln3 - \frac{3}{4}\ln^2\!3 - \ln^2\!2 - (3\ln3+2\gamma)\ln2 - \frac{3\pi\sqrt{3\,}}{2}\ln\Gamma\!\left(\!\frac{1}{\,6\,}\! \right) \\[6mm] \displaystyle\qquad\qquad\quad + \frac{\pi}{2\sqrt{3\,}}\left\{3\ln3 + 11\ln2 + \frac{15}{2}\ln\pi + 3\gamma \right\}+\, \gamma_1 \, =\,-0.2461690038\ldots \end{array}$
such values were calculated by Blagouchine.[20] To the latter author are also due
$\begin{array}{ll} \displaystyle \gamma_1\biggl(\!\frac{1}{5} \!\biggr)=& \displaystyle\!\!\! \gamma_1 + \frac{\sqrt{5}}{2}\!\left\{\zeta''\!\left(\! 0,\,\frac{1}{5}\!\right) + \zeta''\!\left(\! 0,\,\frac{4}{5}\!\right)\!\right\} + \frac{\pi\sqrt{10+2\sqrt5}}{2} \ln\Gamma \biggl(\!\frac{1}{5} \!\biggr) \\[5mm] & \displaystyle + \frac{\pi\sqrt{10-2\sqrt5}}{2} \ln\Gamma \biggl(\!\frac{2}{5} \!\biggr) +\left\{\!\frac{\sqrt{5}}{2} \ln{2} -\frac{\sqrt{5}}{2} \ln\!\big(1+\sqrt{5}\big) -\frac{5}{4}\ln5 -\frac{\pi\sqrt{25+10\sqrt5}}{10} \right\}\!\cdot\gamma \\[5mm] & \displaystyle - \frac{\sqrt{5}}{2}\left\{\ln2+\ln5+\ln\pi+\frac{\pi\sqrt{25-10\sqrt5}}{10}\right\}\!\cdot\ln\!\big(1+\sqrt{5}) +\frac{\sqrt{5}}{2}\ln^2\!2 + \frac{\sqrt{5}\big(1-\sqrt{5}\big)}{8}\ln^2\!5 \\[5mm] & \displaystyle +\frac{3\sqrt{5}}{4}\ln2\cdot\ln5 + \frac{\sqrt{5}}{2}\ln2\cdot\ln\pi+\frac{\sqrt{5}}{4}\ln5\cdot\ln\pi - \frac{\pi\big(2\sqrt{25+10\sqrt5}+5\sqrt{25+2\sqrt5} \big)}{20}\ln2\\[5mm] & \displaystyle - \frac{\pi\big(4\sqrt{25+10\sqrt5}-5\sqrt{5+2\sqrt5} \big)}{40}\ln5 - \frac{\pi\big(5\sqrt{5+2\sqrt5}+\sqrt{25+10\sqrt5} \big)}{10}\ln\pi\\[5mm] & \displaystyle = -8.030205511\ldots \\[6mm] \displaystyle \gamma_1\biggl(\!\frac{1}{8} \!\biggr) =& \displaystyle\!\!\!\gamma_1 + \sqrt{2}\left\{\zeta''\!\left(\! 0,\,\frac{1}{8}\!\right) + \zeta''\!\left(\! 0,\,\frac{7}{8}\right)\!\right\} + 2\pi\sqrt{2}\ln\Gamma \biggl(\!\frac{1}{8} \!\biggr) -\pi \sqrt{2}\big(1-\sqrt2\big)\ln\Gamma \biggl(\!\frac{1}{4} \!\biggr) \\[5mm] & \displaystyle -\left\{\!\frac{1+\sqrt2}{2}\pi+4\ln{2} +\sqrt{2}\ln\!\big(1+\sqrt{2}\big) \!\right\}\!\cdot\gamma - \frac{1}{\sqrt{2}}\big(\pi+8\ln2+2\ln\pi\big)\!\cdot\ln\!\big(1+\sqrt{2}) \\[5mm] & \displaystyle - \frac{7\big(4-\sqrt2\big)}{4}\ln^2\!2 + \frac{1}{\sqrt{2}}\ln2\cdot\ln\pi -\frac{\pi\big(10+11\sqrt2\big)}{4}\ln2 -\frac{\pi\big(3+2\sqrt2\big)}{2}\ln\pi\\[5mm] & \displaystyle = -16.64171976\ldots \\[6mm] \displaystyle \gamma_1\biggl(\!\frac{1}{12} \!\biggr) =& \displaystyle\!\!\!\gamma_1 + \sqrt{3}\left\{\zeta''\!\left(\! 0,\,\frac{1}{12}\!\right) + \zeta''\!\left(\! 0,\,\frac{11}{12}\right)\!\right\} + 4\pi\ln\Gamma \biggl(\!\frac{1}{4} \!\biggr) +3\pi \sqrt{3}\ln\Gamma \biggl(\!\frac{1}{3} \!\biggr) \\[5mm] & \displaystyle -\left\{\!\frac{2+\sqrt3}{2}\pi+\frac{3}{2}\ln3 -\sqrt3(1-\sqrt3)\ln{2} +2\sqrt{3}\ln\!\big(1+\sqrt{3}\big) \!\right\}\!\cdot\gamma \\[5mm] & \displaystyle - 2\sqrt3\big(3\ln2+\ln3 +\ln\pi\big)\!\cdot\ln\!\big(1+\sqrt{3}) - \frac{7-6\sqrt3}{2}\ln^2\!2 - \frac{3}{4}\ln^2\!3 \\[5mm] & \displaystyle + \frac{3\sqrt3(1-\sqrt3)}{2}\ln3\cdot\ln2 + \sqrt3\ln2\cdot\ln\pi -\frac{\pi\big(17+8\sqrt3\big)}{2\sqrt3}\ln2 \\[5mm] & \displaystyle +\frac{\pi\big(1-\sqrt3\big)\sqrt3}{4}\ln3 -\pi\sqrt3(2+\sqrt3)\ln\pi = -29.84287823\ldots \end{array}$
as well as some further values.
### Second generalized Stieltjes constant
The second generalized Stieltjes constant is much less studied than the first constant. Blagouchine showed that, similarly to the first generalized Stieltjes constant, the second generalized Stieltjes constant at rational argument may be evaluated via the following formula
$\begin{array}{rl} \displaystyle \gamma_2 \biggl(\frac{r}{m} \biggr) = \, \gamma_2 + \frac{2}{3}\!\sum_{l=1}^{m-1} \cos\frac{2\pi r l}{m} \cdot\zeta'''\!\left(\!0,\,\frac{l}{m}\!\right) - 2 (\gamma+\ln2\pi m)\! \sum_{l=1}^{m-1} \cos\frac{2\pi r l}{m} \cdot\zeta''\!\left(\!0,\,\frac{l}{m}\!\right) \\[6mm] \displaystyle \quad + \pi\!\sum_{l=1}^{m-1} \sin\frac{2\pi r l}{m} \cdot\zeta''\!\left(\!0,\,\frac{l}{m}\!\right) -2\pi(\gamma+\ln2\pi m)\! \sum_{l=1}^{m-1} \sin\frac{2\pi r l}{m} \cdot\ln\Gamma \biggl(\frac{l}{m} \biggr) - 2\gamma_1 \ln{m} \\[6mm] \displaystyle\quad - \gamma^3 -\left[(\gamma+\ln2\pi m)^2-\frac{\pi^2}{12}\right]\!\cdot\! \Psi\!\biggl(\frac{r}{m} \biggr) + \frac{\pi^3}{12}\cot\frac{\pi r}{m} -\gamma^2\ln\big(4\pi^2 m^3\big) +\frac{\pi^2}{12}(\gamma+\ln{m}) \\[6mm] \displaystyle\quad - \gamma\big(\ln^2\!{2\pi} +4\ln{m}\cdot\ln{2\pi}+2\ln^2\!{m}\big) -\left\{\!\ln^2\!{2\pi}+2\ln{2\pi}\cdot\ln{m}+\frac{2}{3}\ln^2\!{m}\!\right\}\!\ln{m} \end{array}\,,\qquad\quad r=1, 2, 3,\ldots, m-1\,.$
A similar result was later obtained by Coffey by another method.[23]
## References
1. ^ a b Marc-Antoine Coppo. Nouvelles expressions des constantes de Stieltjes. Expositiones Mathematicae, vol. 17, pp. 349-358, 1999.
2. ^ a b Mark W. Coffey. Series representations for the Stieltjes constants, arXiv:0905.1111
3. ^ a b Mark W. Coffey. Addison-type series representation for the Stieltjes constants. J. Number Theory, vol. 130, pp. 2049-2064, 2010.
4. ^ Junesang Choi. Certain integral representations of Stieltjes constants, Journal of Inequalities and Applications, 2013:532, pp. 1-10
5. ^ a b c d Iaroslav V. Blagouchine. Expansions of the generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only, arXiv:1501.00740
6. ^ Math StackExchange: A couple of definite integrals related to Stieltjes constants
7. ^ G. H. Hardy. Note on Dr. Vacca's series for γ, Q. J. Pure Appl. Math. 43, pp. 215–216, 2012.
8. ^ M. I. Israilov. On the Laurent decomposition of Riemann's zeta function [in Russian]. Trudy Mat. Inst. Akad. Nauk. SSSR, vol. 158, pp. 98-103, 1981.
9. ^ Math StackExchange: A closed form for the series ...
10. ^ Bruce C. Berndt. On the Hurwitz Zeta-function. Rocky Mountain Journal of Mathematics, vol. 2, no. 1, pp. 151-157, 1972.
11. ^ a b c Charles Knessl and Mark W. Coffey. An effective asymptotic formula for the Stieltjes constants. Math. Comp., vol. 80, no. 273, pp. 379-386, 2011.
12. ^ Lazhar Fekih-Ahmed. A New Effective Asymptotic Formula for the Stieltjes Constants, arXiv:1407.5567
13. ^ J.B. Keiper. Power series expansions of Riemann ζ-function. Math. Comp., vol. 58, no. 198, pp. 765-773, 1992.
14. ^ Rick Kreminski. Newton-Cotes integration for approximating Stieltjes generalized Euler constants. Math. Comp., vol. 72, no. 243, pp. 1379-1397, 2003.
15. ^ Simon Plouffe. Stieltjes Constants, from 0 to 78, 256 digits each
16. ^ Fredrik Johansson. Rigorous high-precision computation of the Hurwitz zeta function and its derivatives, arXiv:1309.2877
17. ^ a b Math StackExchange: Definite integral
18. ^ Donal F. Connon New proofs of the duplication and multiplication formulae for the gamma and the Barnes double gamma functions, arXiv:0903.4539
19. ^ a b c d
20. ^ V. Adamchik. A class of logarithmic integrals. Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, pp. 1-8, 1997.
21. ^ Math StackExchange: evaluation of a particular integral
22. ^ a b Mark W. Coffey Functional equations for the Stieltjes constants, arXiv:1402.3746
23. ^ Donal F. Connon The difference between two Stieltjes constants, arXiv:0906.0277
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# Variance after scaling and summing: One of the most useful facts from statistics
What do $R^2$, laboratory error analysis, ensemble learning, meta-analysis, and financial portfolio risk all have in common? The answer is that they all depend on a fundamental principle of statistics that is not as widely known as it should be. Once this principle is understood, a lot of stuff starts to make more sense.
Here’s a sneak peek at what the principle is.
Don’t worry if the formula doesn’t yet make sense! We’ll work our way up to it slowly, taking pit stops along the way at simpler formulas are that useful on their own. As we work through these principles, we’ll encounter lots of neat applications and explainers.
This post consists of three parts:
• Part 1: Sums of uncorrelated random variables: Applications to social science and laboratory error analysis
• Part 2: Weighted sums of uncorrelated random variables: Applications to machine learning and scientific meta-analysis
• Part 3: Correlated variables and Modern Portfolio Theory
## Part 1: Sums of uncorrelated random variables: Applications to social science and laboratory error analysis
Let’s start with some simplifying conditions and assume that we are dealing with uncorrelated random variables. If you take two of them and add them together, the variance of their sum will equal the sum of their variances. This is amazing!
To demonstrate this, I’ve written some Python code that generates three arrays, each of length 1 million. The first two arrays contain samples from two normal distributions with variances 9 and 16, respectively. The third array is the sum of the first two arrays. As shown in the simulation, its variance is 25, which is equal to the sum of the variances of the first two arrays (9 + 16).
from numpy.random import randn
import numpy as np
n = 1000000
x1 = np.sqrt(9) * randn(n) # 1M samples from normal distribution with variance=9
print(x1.var()) # 9
x2 = np.sqrt(16) * randn(n) # 1M samples from normal distribution with variance=16
print(x2.var()) # 16
xp = x1 + x2
print(xp.var()) # 25
This fact was first discovered in 1853 and is known as Bienaymé’s Formula. While the code example above shows the sum of two random variables, the formula can be extended to multiple random variables as follows:
If $X_p$ is a sum of uncorrelated random variables $X_1 .. X_n$, then the variance of $X_p$ will be $$\sigma_{p}^{2} = \sum{\sigma^2_i}$$ where each $X_i$ has variance $\sigma_i^2$.
What does the $p$ stand for in $X_p$? It stands for portfolio, which is just one of the many applications we’ll see later in this post.
### Why this is useful
Bienaymé’s result is surprising and unintuitive. But since it’s such a simple formula, it is worth committing to memory, especially because it sheds light on so many other principles. Let’s look at two of them.
#### Understanding $R^2$ and “variance explained”
Psychologists often talk about “within-group variance”, “between-group variance”, and “variance explained”. What do these terms mean?
Imagine a hypothetical study that measured the extraversion of 10 boys and 10 girls, where extraversion is measured on a 10-point scale (Figure 1. Orange bars). The boys have a mean extraversion of 4.4 and the girls have a mean extraversion 5.0. In addition, the overall variance of the data is 2.5. We can decompose this variance into two parts:
• Between-group variance: Create a 20-element array where every boy is assigned to the mean boy extraversion of 4.4, and every girl is assigned to the mean girl extraversion of 5.0. The variance of this array is 0.9. (Figure 1. Blue bars).
• Within-group variance: Create a 20-element array of the amount each child’s extraversion deviates from the mean value for their sex. Some of these values will be negative and some will be positive. The variance of this array is 1.6. (Figure 1. Pink bars).
Figure 1: Decomposition of extraversion scores (orange) into between-group variance (blue) and within-group variance (pink).
If you add these arrays together, the resulting array will represent the observed data (Figure 1. Orange bars). The variance of the observed array is 2.5, which is exactly what is predicted by Bienaymé’s Formula. It is the sum of the variances of the two component arrays (0.9 + 1.6). Psychologists might say that sex “explains” 0.9/2.5 = 36% of the extraversion variance. Equivalently, a model of extraversion that uses sex as the only predictor would have an $R^2$ of 0.36.
#### Error propagation in laboratories
If you ever took a physics lab or chemistry lab back in college, you may remember having to perform error analysis, in which you calculated how errors would propagate through one noisy measurement after another.
Physics textbooks often say that standard deviations add in “quadrature”, which just means that if you are trying to estimate some quantity that is the sum of two other measurements, and if each measurement has some error with standard deviation $\sigma_1$ and $\sigma_2$ respectively, the final standard deviation would be $\sigma_{p} = \sqrt{\sigma^2_1 + \sigma^2_2}$. I think it’s probably easier to just use variances, as in the Bienaymé Formula, with $\sigma^2_{p} = \sigma^2_1 + \sigma^2_2$.
For example, imagine you are trying to estimate the height of two boxes stacked on top of each other (Figure 2). One box has a height of 1 meter with variance $\sigma^2_1$ = 0.01, and the other has a height of 2 meters with variance $\sigma^2_2$ = 0.01. Let’s further assume, perhaps optimistically, that these errors are independent. That is, if the measurement of the first box is too high, it’s not any more likely that the measurement of the second box will also be too high. If we can make these assumptions, then the total height of the two boxes will be 3 meters with variance $\sigma^2_p$ = 0.02.
Figure 2: Two boxes stacked on top of each other. The height of each box is measured with some variance (uncertainty). The total height is the the sum of the individual heights, and the total variance (uncertainty) is the sum of the individual variances.
There is a key difference between the extraversion example and the stacked boxes example. In the extraversion example, we added two arrays that each had an observed sample variance. In the stacked boxes example, we added two scalar measurements, where the variance of these measurements refers to our measurement uncertainty. Since both cases have a meaningful concept of ‘variance’, the Bienaymé Formula applies to both.
## Part 2: Weighted sums of uncorrelated random variables: Applications to machine learning and scientific meta-analysis
Let’s now move on to the case of weighted sums of uncorrelated random variables. But before we get there, we first need to understand what happens to variance when a random variable is scaled.
If $X_p$ is defined as $X$ scaled by a factor of $w$, then the variance $X_p$ will be $$\sigma_{p}^{2} = w^2 \sigma^2$$ where $\sigma^2$ is the variance of $X$.
This means that if a random variable is scaled, the scale factor on the variance will change quadratically. Let’s see this in code.
from numpy.random import randn
import numpy as np
n = 1000000
baseline_var = 10
w = 0.7
x1 = np.sqrt(baseline_var) * randn(n) # Array of 1M samples from normal distribution with variance=10
print(x1.var()) # 10
xp = w * x1 # Scale this by w=0.7
print(w**2 * baseline_var) # 4.9 (predicted variance)
print(xp.var()) # 4.9 (empirical variance)
To gain some intuition for this rule, it’s helpful to think about outliers. We know that outliers have a huge effect on variance. That’s because the formula used to compute variance, $\sum{\frac{(x_i - \bar{x})^2}{n-1}}$, squares all the deviations, and so we get really big variances when we square large deviations. With that as background, let’s think about what happens if we scale our data by 2. The outliers will spread out twice as far, which means they will have even more than twice as much impact on the variance. Similarly, if we multiply our data by 0.5, we will squash the most “damaging” part of the outliers, and so we will reduce our variance by more than a factor of two.
While the above principle is pretty simple, things start to get interesting when you combine it with the Bienaymé Formula in Part I:
If $X_p$ is a weighted sum of uncorrelated random variables $X_1 ... X_n$, then the variance of $X_p$ will be $$\sigma_{p}^{2} = \sum{w^2_i \sigma^2_i}$$ where each $w_i$ is a weight on $X_i$, and each $X_i$ has its own variance $\sigma_i^2$.
The above formula shows what happens when you scale and then sum random variables. The final variance is the weighted sum of the original variances, where the weights are squares of the original weights. Let’s see how this can be applied to machine learning.
### An ensemble model with equal weights
Imagine that you have built two separate models to predict car prices. While the models are unbiased, they have variance in their errors. That is, sometimes a model prediction will be too high, and sometimes a model prediction will be too low. Model 1 has a mean squared error (MSE) of \$1,000 and Model 2 has an MSE of \$2,000.
A valuable insight from machine learning is that you can often create a better model by simply averaging the predictions of other models. Let’s demonstrate this with simulations below.
from numpy.random import randn
import numpy as np
n = 1000000
actual = 20000 + 5000 * randn(n)
errors1 = np.sqrt(1000) * randn(n)
print(errors1.var()) # 1000
errors2 = np.sqrt(2000) * randn(n)
print(errors2.var()) # 2000
# Note that this section could be replaced with
# errors_ensemble = 0.5 * errors1 + 0.5 * errors2
preds1 = actual + errors1
preds2 = actual + errors2
preds_ensemble = 0.5 * preds1 + 0.5 * preds2
errors_ensemble = preds_ensemble - actual
print(errors_ensemble.var()) # 750. Lower than variance of component models!
As shown in the code above, even though a good model (Model 1) was averaged with an inferior model (Model 2), the resulting Ensemble model’s MSE of \$750 is better than either of the models individually. The benefits of ensembling follow directly from the weighted sum formula we saw above, $\sigma_{p}^{2} = \sum{w^2_i \sigma^2_i}$. To understand why, it’s helpful to think of models not as generating predictions, but rather as generating errors. Since averaging the predictions of a model corresponds to averaging the errors of the model, we can treat each model’s array of errors as samples of a random variable whose variance can be plugged in to the formula. Assuming the models are unbiased (i.e. the errors average to about zero), the formula tells us the expected MSE of the ensemble predictions. In the example above, the MSE would be which is exactly what we observed in the simulations. (For a totally different intuition of why ensembling works, see this blog post that I co-wrote for my company, Opendoor.) ### An ensemble model with Inverse Variance Weighting In the example above, we obtained good results by using an equally-weighted average of the two models. But can we do better? Yes we can! Since Model 1 was better than Model 2, we should probably put more weight on Model 1. But of course we shouldn’t put all our weight on it, because then we would throw away the demonstrably useful information from Model 2. The optimal weight must be somewhere in between 50% and 100%. An effective way to find the optimal weight is to build another model on top of these models. However, if you can make certain assumptions (unbiased and uncorrelated errors), there’s an even simpler approach that is great for back-of-the envelope calculations and great for understanding the principles behind ensembling. To find the optimal weights (assuming unbiased and uncorrelated errors), we need to minimize the variance of the ensemble errors $\sigma_{p}^{2} = \sum{w^2_i \sigma^2_i}$ with the constraint that $\sum{w_i} = 1$. It turns out that the variance-minimizing weight for a model should be proportional to the inverse of its variance. When we apply this method, we obtain optimal weights of $w_1$ = 0.67 and $w_2$ = 0.33. These weights give us an ensemble error variance of which is significantly better than the$750 variance we were getting with equal weighting.
This method is called Inverse Variance Weighting, and allows you to assign the right amount of weight to each model, depending on its error.
Inverse Variance Weighting is not just useful as a way to understand Machine Learning ensembles. It is also one of the core principles in scientific meta-analysis, which is popular in medicine and the social sciences. When multiple scientific studies attempt to estimate some quantity, and each study has a different sample size (and hence variance of their estimate), a meta-analysis should weight the high sample size studies more. Inverse Variance Weighting is used to determine those weights.
## Part 3: Correlated variables and Modern Portfolio Theory
Let’s imagine we now have three unbiased models with the following MSEs:
• Model 1: MSE = 1000
• Model 2: MSE = 1000
• Model 3: MSE = 2000
By Inverse Variance Weighting, we should assign more weight to the first two models, with $w_1=0.4, w_2=0.4, w_3=0.2$.
But what happens if Model 1 and Model 2 have correlated errors? For example, whenever Model 2’s predictions are too high, Model 3’s predictions tend to also be too high. In that case, maybe we don’t want to give so much weight to Models 1 and 2, since they provide somewhat redundant information. Instead we might want to diversify our ensemble by increasing the weight on Model 3, since it provides new independent information.
To determine how much weight to put on each model, we first need to determine how much total variance there will be if the errors are correlated. To do this, we need to borrow a formula from the financial literature, which extends the formulas we’ve worked with before. This is the formula we’ve been waiting for.
If $X_p$ is a weighted sum of (correlated or uncorrelated) random variables $X_1 ... X_n$, then the variance of $X_p$ will be $$\sigma_{p}^{2} = \sum\limits_{i} \sum\limits_{j} w_i w_j \sigma_i \sigma_j \rho_{ij}$$ where each $w_i$ and $w_j$ are weights assigned to $X_i$ and $X_j$, where each $X_i$ and $X_j$ have standard deviations $\sigma_i$ and $\sigma_j$, and where the correlation between $X_i$ and $X_j$ is $\rho_{ij}$.
There’s a lot to unpack here, so let’s take this step by step.
• $\sigma_i \sigma_j \rho_{ij}$ is a scalar quantity representing the covariance between $X_i$ and $X_j$.
• If none of the variables are correlated with each other, then all the cases where $i \neq j$ will go to zero, and the formula reduces to $\sigma_{p}^{2} = \sum{w^2_i \sigma^2_i}$, which we have seen before.
• The more that two variables $X_i$ and $X_j$ are correlated, the more the total variance $\sigma_{p}^{2}$ increases.
• If two variables $X_i$ and $X_j$ are anti-correlated, then the total variance decreases, since $\sigma_i \sigma_j \rho_{ij}$ is negative.
• This formula can be rewritten in more compact notation as $\sigma_{p}^{2} = \vec{w}^T\Sigma \vec{w}$, where $\vec{w}$ is the weight vector, and $\Sigma$ is the covariance matrix (not a summation sign!)
If you skimmed the bullet points above, go back and re-read them! They are super important.
To find the set of weights that minimize variance in the errors, you must minimize the above formula, with the constraint that $\sum{w_i} = 1$. One way to do this is to use a numerical optimization method. In practice, however, it is more common to just find weights by building another model on top of the base models
Regardless of how the weights are found, it will usually be the case that if Models 1 and 2 are correlated, the optimal weights will reduce redundancy and put lower weight on these models than simple Inverse Variance Weighting would suggest.
### Applications to financial portfolios
The formula above was discovered by economist Harry Markowitz in his Modern Portfolio Theory, which describes how an investor can optimally trade off between expected returns and expected risk, often measured as variance. In particular, the theory shows how to maximize expected return given a fixed variance, or minimize variance given a fixed expected return. We’ll focus on the latter.
Imagine you have three stocks to put in your portfolio. You plan to sell them at time $T$, at which point you expect that Stock 1 will have gone up by 5%, with some uncertainty. You can describe your uncertainty as variance, and in the case of Stock 1, let’s say $\sigma_1^2$ = 1. This stock, as well Stocks 2 and 3, are summarized in the table below:
Stock ID Expected Return Expected Risk ($\sigma^2$)
1 5.0 1.0
2 5.0 1.0
3 5.0 2.0
This financial example should remind you of ensembling in machine learning. In the case of ensembling, we wanted to minimize variance of the weighted sum of error arrays. In the case of financial portfolios, we want to minimize the variance of the weighted sum of scalar financial returns.
As before, if there are no correlations between the expected returns (i.e. if Stock 1 exceeding 5% return does not imply that Stock 2 or Stock 3 will exceed 5% return), then the total variance in the portfolio will be $\sigma_{p}^{2} = \sum{w^2_i \sigma^2_i}$ and we can use Inverse Variance Weighting to obtain weights $w_1=0.4, w_2=0.4, w_3=0.2$.
However, sometimes stocks have correlated expected returns. For example, if two of the stocks are in oil companies, then one stock exceeding 5% implies the other is also likely to exceed 5%. When this happens, the total variance becomes
as we saw before in the ensemble example. Since this includes an additional positive term for $w_1 w_2 \sigma_1 \sigma_2 \rho_{1,2}$, the expected variance is higher than in the uncorrelated case, assuming the correlations are positive. To reduce this variance, we should put less weight on Stocks 1 and 2 than we would otherwise.
While the example above focused on minimizing the variance of a financial portfolio, you might also be interested in having a portfolio with high return. Modern Portfolio Theory describes how a portfolio can reach any abitrary point on the efficient frontier of variance and return, but that’s outside the scope of this blog post. And as you might expect, financial markets can be more complicated than Modern Portfolio Theory suggests, but that’s also outside scope.
## Summary
That was a long post, but I hope that the principles described have been informative. It may be helpful to summarize them in backwards order, starting with the most general principle.
If $X_p$ is a weighted sum of (correlated or uncorrelated) random variables $X_1 ... X_n$, then the variance of $X_p$ will be $$\sigma_{p}^{2} = \sum\limits_{i} \sum\limits_{j} w_i w_j \sigma_i \sigma_j \rho_{ij}$$ where each $w_i$ and $w_j$ are weights assigned to $X_i$ and $X_j$, where each $X_i$ and $X_j$ have standard deviations $\sigma_i$ and $\sigma_j$, and where the correlation between $X_i$ and $X_j$ is $\rho_{ij}$. The term $\sigma_i \sigma_j \rho_{ij}$ is a scalar quantity representing the covariance between $X_i$ and $X_j$.
If none of the variables are correlated, then all the cases where $i \neq j$ go to zero, and the formula reduces to $$\sigma_{p}^{2} = \sum{w^2_i \sigma^2_i}$$ And finally, if we are computing a simple sum of random variables where all the weights are 1, then the formula reduces to $$\sigma_{p}^{2} = \sum{\sigma^2_i}$$
# Using your ears and head to escape the Cone Of Confusion
One of coolest things I ever learned about sensory physiology is how the auditory system is able to locate sounds. To determine whether sound is coming from the right or left, the brain uses inter-ear differences in amplitude and timing. As shown in the figure below, if the sound is louder in the right ear compared to the left ear, it’s probably coming from the right side. The smaller that difference is, the closer the sound is to the midline (i.e the vertical plane going from your front to your back). Similarly, if the sound arrives at your right ear before the left ear, it’s probably coming from the right. The smaller the timing difference, the closer it is to the midline. There’s a fascinating literature on the neural mechanisms behind this.
Inter-ear loudness and timing differences are pretty useful, but unfortunately they still leave a lot of ambiguity. For example, a sound from your front right will have the exact same loudness differences and timing differences as a sound from your back right.
Not only does this system leave ambiguities between front and back, it also leaves ambiguities between top and down. In fact, there is an entire cone of confusion that cannot be disambiguated by this system. Sound from all points along the surface of the cone will have the same inter-ear loudness differences and timing differences.
While this system leaves a cone of confusion, humans are still able to determine the location of sounds from different points on the cone, at least to some extent. How are we able to do this?
Amazingly, we are able to do this because of the shape of our ears and heads. When sound passes through our ears and head, certain frequencies are attenuated more than others. Critically, the attenuation pattern is highly dependent on sound direction.
This location-dependent attenuation pattern is called a Head-related transfer function (HRTF) and in theory this could be used to disambiguate locations along the cone of confusion. An example of someone’s HRTF is shown below, with frequency on the horizontal axis and polar angle on the vertical axis. Hotter colors represent less attenuation (i.e. more power). If your head and ears gave you this HRTF, you might decide a sound is coming from the front if it has more high frequency power than you’d expect.
HRTF image from Simon Carlile's Psychoacoustics chapter in The Sonification Handbook.
This system sounds good in theory, but do we actually use these cues in practice? In 1988, Frederic Wightman and Doris Kistler performed an ingenious set of experiments (1, 2) to show that that people really do use HRTFs to infer location. First, they measured the HRTF of each participant by putting a small microphone in their ears and playing sounds from different locations. Next they created a digital filter for each location and each participant. That is to say, these filters implemented each participant’s HRTF. Finally, they placed headphones on the listeners and played sounds to them, each time passing the sound through one of the digital filters. Amazingly, participants were able to correctly guess the “location” of the sound, depending on which filter was used, even though the sound was coming from headphones. They were also much better at sound localization when using their own HRTF, rather than someone else’s HRTF.
Further evidence for this hypothesis comes from Hofman et al., 1998, who showed that by using putty to reshape people’s ears, they were able to change the HRTFs and thus disrupt sound localization. Interestingly, people were able to quickly relearn how to localize sound with their new HRTFs.
Image from Hofman et al., 1998.
A final fun fact: to improve the sound localization of humanoid robots, researchers in Japan attached artificial ears to the robot heads and implemented some sophisticated algorithms to infer sound location. Here are some pictures of the robots.
Their paper is kind of ridiculous and has some questionable justifications for not just using microphones in multiple locations, but I thought it was fun to see these principles being applied.
# Hyperbolic discounting — The irrational behavior that might be rational after all
When I was in grad school I occasionally overheard people talk about how humans do something called “hyperbolic discounting”. Apparently, hyperbolic discounting was considered irrational under standard economic theory.
I recently decided to learn what hyperbolic discounting was all about, so I set out to write this blog post. I have to admit that hyperbolic discounting has been pretty hard for me to understand, but I think I now finally have a good enough handle on it to write about it. Along the way, I learned something interesting: Hyperbolic discounting might be rational after all.
## Rational and irrational discounting
##### Rationality of hyperbolic discounting
The problem with this story is that it only works if you assume that the interest rate is constant. In the real world, the interest rate fluctuates.
Before taking on the fluctuating interest rate scenario, let’s first take on a different assumption that is still somewhat simplified. Let’s assume that the interest rate is constant but we don’t know what it is, just as we didn’t know what the hazard rate was in the previous interpretation. With this assumption, the justification for hyperbolic discounting becomes similar to the explanation in the blue and pink plots above. When you do a probability-weighted average over these decaying exponential curves, you get a hyperbolic function.
The previous paragraph assumed that the interest rate was constant but unknown. In the real world, the interest rate is known but fluctuates over time. Farmer and Geanakoplos (2009) showed that if you assume that interest rate fluctuations follow a geometric random walk, hyperbolic discounting becomes optimal, at least asymptotically as $\tau \rightarrow \infty$. In the near future, you know the interest rate with reasonable certainty and should therefore discount with an exponential curve. But as you look further into the future, your uncertainty about the interest rate increases and you should therefore discount with a hyperbolic curve.
Is the geometric random walk a process that was cherry picked by the authors to produce this outcome? Not really. Newell and Pizer (2003) studied US bond rates in the 19th and 20th century and found that the geometric random walk provided a better fit than any of the other interest rate models tested.
## Summary
When interpreting discounting as a survival function, a hyperbolic discounting function is rational if you introduce uncertainty into the hazard parameter via an exponential prior (Souza, 2015). When interpreting the discount rate as an interest rate, a hyperbolic discounting function is asymptotically rational if you introduce uncertainty in the interest rate via a geometric random walk (Farmer and Geanakoplos, 2009).
# Religions as firms
I recently came across a magazine that helps pastors manage the financial and operational challenges of church management. The magazine is called Church Executive.
Readers concerned about seasonal effects on tithing can learn how to “sustain generosity” during the weaker summer months. Technology like push notifications and text messages is encouraged as a way to remind people to tithe. There is also some emphasis on messaging, as pastors are told to “make sure your generosity-focused sermons are hitting home with your audience”.
Churches need money to stay active, and it’s natural that pastors would want to maintain a healthy cash flow. But the brazen language of Church Executive reminded me of the language of profit-maximizing firms. This got me thinking: What are the other ways in which religions act like a business?
This post is my attempt to understand religions as if they were businesses. This isn’t a perfect metaphor. Most religious leaders are motivated by genuine beliefs, and few are motivated primarily by profit. But it can still be instructive to view religions through the lens of business and economics, if only as an exercise. After working through this myself, I feel like I have a better understanding of why religions act the way they do.
### Competition
As with any business, one of the most pressing concerns of a religion is competition. According to sociologist Carl Bankston, the set of religions can be described as a marketplace of competing firms that vie for customers. Religious consumers can leave one church to go to another. To hedge their bets on the afterlife, some consumers may even belong to several churches simultaneously, in a strategy that has been described as “portfolio diversification”.
One way that a religion can ward off competitors is to prohibit its members from following them. The Bible is insistent on this point, with 26 separate verses banning idolatry. Other religions have been able to eliminate competition entirely by forming state-sponsored monopolies.
### Pricing
Just like a business, religions need to determine how to price their product. According to economists Laurence Iannaccone and Feler Bose, the optimal pricing strategy for a religion depends on whether it is proselytizing or non-proselytizing.
Non-proselytizing religions like Judaism and Hinduism earn much of their income from membership fees. While exceptions are often made for people who are too poor to pay, and while donations are still accepted, the explicit nature of the membership fees help these religions avoid having too many free riders.
Proselytizing religions like Christianity are different. Because of their strong emphasis on growth, they are willing to forgo explicit membership fees and instead rely more on donations that are up to the member’s discretion. Large donations from wealthy individuals can cross-subsidize the membership of those who make smaller donations. Even free riders who make no donations at all may be worthwhile, since they may attract more members in the future.
### Surge Pricing
Like Uber, some religions raise the price during periods of peak demand. While attendance at Jewish synagogue for a regular Shabbat service is normally free, attendance during one of the High Holidays typically requires a payment for seating, in part to ensure space for everyone.
Surge pricing makes sense for non-proselytizing religions such as Judaism, but it does not make sense for proselytizing religions such as Christianity, which views the higher demand during peak season as an opportunity to convert newcomers and to reactivate lapsed members. Thus, Christian churches tend to expand seating and schedule extra services during Christmas and Easter, rather than charging fees.
### Product Quality
Just as business consumers will pay higher prices for better products, consumers of polytheistic religions will pay higher “prices” for gods with more wide-ranging powers. Even today, some American megachurches have found success with the prosperity gospel, which emphasizes that God can make you wealthy.
Of course, not all religious consumers will prefer the cheap promises of the prosperity gospel. For many religions, product quality is defined primarily by community, a sense of meaning, and in some cases the promise of an afterlife.
A good business should be constantly updating its product to fix bugs and to respond to changes in consumer preference or government regulation. Some religions do the same thing, via the process of continuous revelation from their deity. Perhaps no church exemplifies this better than the Church of Jesus Christ of Latter-day Saints.
For most of the history of the Mormon Church, individuals of African descent were prohibited from serving as priests. By the 1960s, as civil rights protests against the church received media attention, the policy became increasingly untenable. On June 1, 1978, Mormon leaders reported that God had instructed them to update the policy and allow black priests. This event was known as the 1978 Revelation on the Priesthood.
In the late 19th Century, when the Mormon Church was under intense pressure from the US Government regarding polygamy, the Church president claimed to receive a revelation from Jesus Christ asking him to prohibit it. This revelation, known as the 1890 Revelation, overturned the previous 1843 Revelation which allowed polygamy.
While frequent updates usually make sense in business, they don’t always make sense in religion. Most religions have a fairly static doctrine, as the prospect of future updates undermines the authority of current doctrine.
### Growth and marketing
Instead of focusing only on immediate profitability, many businesses invest in user growth. As mentioned earlier, many religions are willing to cross-subsidize participation from new members, especially young members, with older members bearing most of the costs.
Christianity’s concept of a heaven and hell encouraged its members to convert their friends and family. In some ways, this is reminiscent of viral marketing.
### International expansion
Facebook and Netflix both experienced rapid adoption, starting with a U.S. audience. But as U.S. growth began to slow down, both companies needed to look towards international expansion.
A similar thing happened with the Mormon church. By the 20th century, U.S. growth was driven only by increasing family sizes, so the church turned towards international expansion.
The graph below shows similar US and international growth curves for Netflix and the Church of Jesus Christ of Latter-day Saints.[1,2,3,4]
### Branding
Like any company, most religions try to maintain a good brand. But unlike businesses, most religions do not have brand protection, and thus their brands can be co-opted by other religions. Marketing from Mormons and from Jehovah’s Witnesses tends to emphasize the good brand of Jesus Christ, even though most mainstream Christians regard these churches as heretical.
One of the most interesting risks to brands is genericide, in which a popular trademark becomes synonymous with the general class of product, thereby diluting its distinctive meaning. Famous examples of generic trademarks include Kleenex and Band-Aid. Amazingly, genericide can also happen to religious deities. The ancient Near East god El began as a distinct god with followers, but gradually became a generic name for “God” and eventually merged with the Hebrew god Yahweh.
### Mergers and spin-offs
In business, companies can spin off other companies or merge with other companies. But with rare exceptions, religions only seem to have spin-offs. Why do religions hardly ever merge with other religions? My guess is that since there is no protection for religious intellectual property, religions can acquire the intellectual property of another religion without requiring a merger. Religions can simply copy each other’s ideas.
Another reason that religious mergers are rare is that religions are strongly tied to personal identity and tap into tribal thinking. When WhatsApp was acquired, its leadership was happy to adopt new identities as Facebook employees. But it is far less likely that members of, say, the Syriac Catholic Church would ever tolerate merging into the rival Syriac Maronite Church, even if it might provide them with economies of scale and more political power.
On Twitter, I asked why there are so few religious mergers and got lots of interesting responses. People pointed out that reconciliation of doctrine could undermine the authority of the leaders, and that there is little benefit from economies of scale. Others noted that religious mergers aren’t that rare. Hinduism and Judaism may have began as mergers of smaller religions, many Christian traditions involve mergers with religions they replaced, and that even today Hinduism continues to be a merging of various sects.
It’s worth repeating that economic explanations aren’t always great at describing the conscious motivations of religious individuals, who generally have sincere beliefs. Nevertheless, economic reasoning does a decent job of predicting the behavior of systems, and it’s been pretty interesting to learn how religion is no exception.
# Part 2: A bipartisan list of people who argue in good faith
In Part 1, I posted a bipartisan list of people who are bad for America. Those people present news stories that cherry pick the worst actions from the other side so that they can get higher TV ratings and more social media points.
Here in Part 2, I post a list of people who don’t do that, at least for the most part. This isn’t a list of centrists. If anything, it is a more politically diverse list than the list in Part 1. This is a list of people who usually make good-faith attempts to persuade others about their point of view.
• Megan McArdle (Twitter, Bloomberg) – Moderately libertarian ideas presented to a diverse audience
• Noah Smith (Twitter, Bloomberg) – Center-left economics
• Ross Douthat (Twitter, NYT) – Social conservatism presented to a left-of-center audience
• Noam Chomsky (Website)
• Conor Friedersdorf (The Atlantic)
• Ben Sasse — Has the third-most conservative voting record in the Senate but never caricatures the other side and is very concerned about filter bubbles.
• Julia Galef (Twitter) – Has some great advice for understanding the other side
• Nicky Case (Twitter)
• Fareed Zakaria (Washington Post) – Center-left foreign policy
• Eli Lake (Twitter, Bloomberg) – Hawkish foreign policy
• Kevin Drum (Mother Jones) – Center-left blogger who writes in good faith
• John Carl Baker (Twitter) – One of the few modern socialists I have found who avoids in-group snark.
• Michael Dougherty (Twitter, The Week)
• Reihan Salam (Twitter, NRO)
• Avik Roy (Twitter, NRO) – Conservative health care
• Ezra Klein (Vox, early days at the American Prospect) – While at the American Prospect, Ezra did an amazing job trying to persuade people about the benefits of Obamacare. Vox, the explainer site that he started, sometimes slips into red meat clickbait. But to its credit, Vox has managed to reach a wide audience with mostly explainer content.
Reading the people on this list with an open mind will broaden your worldview.
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Are Tate twists of t-positive motives positive with respect to the Voevodsky's homotopy t-structure?
Let $X$ be a Voevodsky's motif (over a perfect field) that belongs to the positive part of the homotopy $t$-structure (i.e. its cohomology as an object of $D^-(ShSmCor)$ is zero in negative degrees). Is the same true for $X(1)$?
This seems to be a difficult question, and I will try to express my ideas about it. The Beilinson-Soule conjecture predicts that the answer is positive for $X=\mathbb{Z}(j)$, $j\ge 0$. Moreover, for any geometric motif $X$ this conjecture together with Poincare duality yields that there exists a constant $c_X$ such that $X(i)[c_X]$ is $t$-positive for any $i\ge 0$.
Besides, one can certainly consider motives either with $\mathbb{Z}/l\mathbb{Z}$-coefficients or with $\mathbb{Q}$-ones in this question. For the first of this possibilities, one can replace the Beilinson-Soule conjecture with the Bloch-Kato one (so the statements above become unconditional). Yet the Bloch-Kato conjecture does not seem to answer my question.
I would be deeply grateful for any ideas or (counter)examples for my question.
More generaly, it would be interesting to understand how much negative cohomology can the tensor product of two $t$-positive motives have.
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tk_choose.files
0th
Percentile
Choose a List of Files Interactively
Use a Tk file dialog to choose a list of zero or more files interactively.
Keywords
file
Usage
tk_choose.files(default = "", caption = "Select files",
multi = TRUE, filters = NULL, index = 1)
Arguments
default
which filename to show initially.
caption
the caption on the file selection dialog.
multi
whether to allow multiple files to be selected.
filters
two-column character matrix of filename filters.
index
unused.
Details
Unlike file.choose, tk_choose.files will always attempt to return a character vector giving a list of files. If the user cancels the dialog, then zero files are returned, whereas file.choose would signal an error.
The format of filters can be seen from the example. File patterns are specified via extensions, with "*" meaning any file, and "" any file without an extension (a filename not containing a period). (Other forms may work on specific platforms.) Note that the way to have multiple extensions for one file type is to have multiple rows with the same name in the first column, and that whether the extensions are named in file chooser widget is platform-specific. The format may change before release.
Value
A character vector giving zero or more file paths.
Note
A bug in Tk 8.5.0--8.5.4 prevented multiple selections being used.
file.choose, tk_choose.dir
library(tcltk) # NOT RUN { Filters <- matrix(c("R code", ".R", "R code", ".s", "Text", ".txt", "All files", "*"), 4, 2, byrow = TRUE) Filters if(interactive()) tk_choose.files(filter = Filters) # }
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# Number of integer solutions of $a^2+b^2=10c^2$
Find the number of integer solutions of the equation $$a^2+b^2=10c^2$$.
I can only get by inspection that $$a=3m, b=m,c=m$$ satisfies for any $$m \in Z$$.
Is there a formal logic to find all possible solutions? Any hint?
Also i tried taking $$a=p^2-q^2$$, $$b=2pq$$ and $$10c^2=(p^2+q^2)^2$$
which gives $$\frac{p^2+q^2}{c}=\sqrt{10}$$ which is invalid, since a rational can never be an irrational.
• The Titel of your post is inaproppiate. The number o Solution is infinite as you already mention in your Post. You Wanze to find all solutions Jun 23, 2021 at 14:27
Let us consider the circle $$C:x^2+y^2=10$$. The question you asked is equivalent to finding all the rational points on this circle. Clearly, $$(1,3)$$ is one such point.
We will project from the point $$(1,3)$$ to the $$Y$$-axis. Let $$(0,t)$$ be the point of intersection of the line $$L$$ through $$(1,3)$$ and a point $$(x,y)$$ on the circle. Since $$L$$ passes through $$(0,t)$$ and $$(1,3)$$, its equation is \begin{align*} &x=\frac{y-t}{3-t}\\ \end{align*}
Now, the point $$(x,y)$$ is on the line $$L$$ as well as the circle $$C$$. So, $$\begin{equation*} \left (\frac{y-t}{3-t} \right )^2+y^2=10 \end{equation*}$$ Solving this for $$y$$ and using $$x=\frac{y-t}{3-t}$$, we can get expressions for $$x$$ and $$y$$ in terms of $$t$$ only.
Now, since the $$Y$$-axis is a rational line, the rational points on circle must be mapped to rational points on the line. Also, from the geometry of this approach, it is clear that this gives us all the rational points (which means we get all the solutions of the equation you asked for).
Actually, for any given conic, if we know at least one rational point, we can get all others by this projection method. So, this is a very general approach. As you can see, in this case, we will get a very ugly expression for $$t$$- maybe a better choice of the initial rational point could have given better results. For a more detailed study, refer to chapter 1 section 1 of Rational Points on Elliptic Curves. They deal with the same problem, only with $$x^2+y^2=1$$ which gives a simpler expression of $$t$$.
We first transform the given equation $$a^2+b^2=10c^2$$ as follows:
Consider $$x=\frac{a}{c}, y=\frac{b}{c}$$ (assuming we are interested in $$c \neq 0$$ case). Then we have to find rational points on the circle $$x^2+y^2=10.$$ Now $$(x,y)=(3,1)$$ is an obvious solution. To find other rational solutions, consider the line that passes through $$(3,1)$$ and has slope $$m$$. It is given by $$y=mx+(1-3m).$$ Consider the intersection of this line with the circle $$x^2+y^2=10$$. We can find the intersection from $$x^2+(mx+(1-3m))^2=10 \implies (1+m^2)x^2+2m(1-3m)x+(1-3m)^2-10=0.$$ But instead of solving the quadratic, we can argue that if two roots are $$x_1$$ and $$x_2$$, then we have $$x_1=3$$, so by means of Viete formula etc. we have $$x_1+x_2=-\frac{2m(1-3m)}{1+m^2} \implies x_2=\frac{3m^2-2m-3}{1+m^2}.$$ Now if $$m$$ is rational then $$x_2$$ is also rational and so will $$y_2$$ be, because $$y_2=\frac{1-m^2-6m}{1+m^2}.$$
Now for $$m=\frac{p}{q}$$, where $$p,q \in \Bbb{Z}$$ and $$q \neq 0$$, we can have $$\color{blue}{a=3p^2-2pq-3q^2, \quad b=q^2-p^2-6pq, \quad c=p^2+q^2}$$ Now the main question is: this combined with your answer (which can be obtained if we allow $$q=0$$) is that the totality of all solutions?
• If there would be another solution (a,b) of x^2+y^2=10, then we would to consider the line through by (a,b) and (3,1). In this case there would be a slope m (that is computed imposing b=ma +(1-m) ) such that (a,b) is the intersection of the line with slope m passing in the point (3,1) with the circle. Thus your construction identifies all the solutions, right? Jun 22, 2021 at 20:52
• Could you explain why did you choose a line through $(3,1)$ only? Jun 22, 2021 at 20:57
• The reason you choose a line through a known point is that it restricts you down to one parameter (the slope) and let’s you use Vieta formula to get a rational solution without radicals. In general, conics aren’t guaranteed to go through any rational points but you can use one rational solution to find others.
– Eric
Jun 22, 2021 at 21:19
• @Umeshshankar In more complicated situations we may need more than one parametrization to get all primitive integral solutions of an indefinite ternary. In this case, just one suffices, proof added. Jun 23, 2021 at 3:00
• @FedericoFallucca Yes that's the idea to ensure that all rational solutions are accounted for. Jun 23, 2021 at 7:48
$$\color{magenta}{a=3p^2-2pq-3q^2, \quad b=-p^2-6pq+q^2, \quad c=p^2+q^2}$$
or, same thing
$$\color{green}{a=3x^2-2xy-3y^2, \quad b=-x^2-6xy+y^2, \quad c=x^2+y^2}$$
When $$p,q$$ are coprime, the primes that can still divide $$\gcd(a,b,c)$$ are $$2$$ and $$5.$$ The proof of all (primitive, integral) solutions is just showig that these don't matter.
When $$x,y$$ are both odd, all three of $$a,b,c$$ are divisible by $$2,$$ and we need to worry about whether half the triple is represented by the given parametrization. Well taking $$p = \frac{x-y}{2} \; , \; \; q = \frac{x+y}{2} \; , \; \;$$
$$3 p^2 - 2pq -3q^2 = \frac{1}{2} \left( -x^2 - 6 xy + y^2 \right)$$ $$- p^2 - 6pq +q^2 = \frac{-1}{2} \left( 3x^2 - 2 xy -3 y^2 \right)$$ $$p^2 +q^2 = \frac{1}{2} \left( x^2 + y^2 \right)$$
$$5$$ is the other possibility . This happens when $$2x+y \equiv x - 2y \equiv 0 \pmod 5.$$
Taking $$p = \frac{2x+y}{5} \; , \; \; q = \frac{x-2y}{5} \; , \; \;$$
$$3 p^2 - 2pq -3q^2 = \frac{1}{5} \left( 3x^2 - 2 xy -3 y^2 \right)$$ $$- p^2 - 6pq +q^2 = \frac{1}{5} \left( -x^2 - 6 xy + y^2 \right)$$ $$p^2 +q^2 = \frac{1}{5} \left( x^2 + y^2 \right)$$
$$\bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc$$
The example I like to show is solving $$2(x^2 + y^2 + z^2) - 113(yz + zx + xy)=0,$$ four "recipes," $$\left( \begin{array}{r} x \\ y \\ z \end{array} \right) = \left( \begin{array}{r} 37 u^2 + 51 uv + 8 v^2 \\ 8 u^2 -35 uv -6 v^2 \\ -6 u^2 + 23 uv + 37 v^2 \end{array} \right)$$
$$\left( \begin{array}{r} x \\ y \\ z \end{array} \right) = \left( \begin{array}{r} 32 u^2 + 61 uv + 18 v^2 \\ 18 u^2 -25 uv -11 v^2 \\ -11 u^2 + 3 uv + 32 v^2 \end{array} \right)$$
$$\left( \begin{array}{r} x \\ y \\ z \end{array} \right) = \left( \begin{array}{r} 38 u^2 + 45 uv + 4 v^2 \\ 4 u^2 -37 uv -3 v^2 \\ -3 u^2 + 31 uv + 38 v^2 \end{array} \right)$$
$$\left( \begin{array}{r} x \\ y \\ z \end{array} \right) = \left( \begin{array}{r} 29 u^2 + 63 uv + 22 v^2 \\ 22 u^2 -19 uv -12 v^2 \\ -12 u^2 -5 uv + 29 v^2 \end{array} \right)$$
For all four recipes, $$x^2 + y^2 + z^2 = 1469 \left( u^2 + uv + v^2 \right)^2$$
$$\bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc$$
$$x^2+y^2=nz^2\tag{1}$$ In general, if equation $$(1)$$ has one integer solution $$(x,y,z)=(x_0,y_0,z_0)$$ then there exists an infinitely many integer solutions.
Substitute $$x=t+x_0, y=t+y_0, z=at+z_0$$ to equation $$(1)$$, then we get
$$t = \frac{2(x_0+y_0-nz_0a)}{-2+na^2}$$
Let $$a=\frac{p}{q}$$ with p,q are integers, hence we get a parametric solution $$(x,y,z)=(x_0np^2-2qnz_0p+2q^2y_0, y_0np^2-2qnz_0p+2q^2x_0, nz_0p^2-2qx_0p-2qy_0p+2q^2z_0).$$
Example of $$x^2+y^2=10z^2.$$
Let $$n=10, (x_0,y_0,z_0)=(3,1,1).$$
$$(x,y,z)=(15p^2-10qp+q^2, 5p^2-10qp+3q^2, 5p^2-4qp+q^2).$$
Solving the $$C$$-function of Euclid's formula $$\quad A=m^2-k^2,\quad B=2mk,\quad C=m^2+k^2\quad$$ for $$(k), \space$$ we can find Pythagorean triples for any given $$C$$-values, if they exist, that are primitive, doubles, or square multiples of primitives. This will not find, for example $$(9,12,15)\space$$ or $$(15,20,25),$$ but it will find $$(3,4,5),\space (6,8,10),\space (12,16,20),\space (27,36,45), \space$$ etc. We begin with the following formula. Any $$m$$-value that yields an integer $$k$$-value indicates a valid $$(m,k)$$ pair for generating a Pythagorean triple.
$$$$C=m^2+k^2\implies k=\sqrt{C-m^2}\\ \text{for}\qquad \bigg\lfloor\frac{ 1+\sqrt{2C-1}}{2}\bigg\rfloor \le m \le \lfloor\sqrt{C-1}\rfloor$$$$ The lower limit ensures $$m>k$$ and the upper limit ensures $$k\in\mathbb{N}.$$
Here is an example for $$C=40\implies 10c=4$$ where $$c$$ is the one shown in the OP equation.
$$C=40\implies \bigg\lfloor\frac{ 1+\sqrt{80-1}}{2}\bigg\rfloor=4 \le m \le \lfloor\sqrt{40-1}\rfloor=6\\ \land \quad m\in\{6\}\Rightarrow k\in\{2\}\\$$ $$F(6,2)=(32,24,40)\implies (32,24,10\times 4)$$
This method will not find all Pythagorean triples that match the criteria but it will find an infinite number of triples that do such as:
$$c=1\longrightarrow (8,6,10\times 1)\\ c=2\longrightarrow (12,16,10\times 2)\\ c=4\longrightarrow (32,24,10\times 4)\\ c=9\longrightarrow (72,54,10\times 9)\\$$
Note that any multiple of a triple found also yields a valid triple so $$3\times (8,6,10)\longrightarrow (24,18,10\times3)$$ and provides the "missing" $$c=3$$ triple in the list above. The combination of the two will find all Pythagorean triples where the $$c$$ in $$10c$$ is an integer except for the most unusual case like $$(3,1,10\times 1)$$ mentioned in another post.
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The objective function gives the quantity that is to be maximized (or minimized), and the constraints determine the set of feasible solutions. Quadratic programming is a subfield of nonlinear optimization which deals with quadratic optimization problems subject to optional boundary and/or general linear equality/inequality constraints: Quadratic programming problems can be solved as general constrained nonlinear optimization problems. A typical example would be taking the limitations of materials and labor, and then determining the "best" production levels for maximal profits under those conditions. Linear programming is the method of considering different inequalities relevant to a situation and calculating the best value that is required to be obtained in those conditions. Core Imports CenterSpace. Simplex algorithm is based in an operation called pivots the matrix what it is precisely this iteration between the set of extreme points. py or add a shebang to the top of your script file #!/usr/bin/env python. Francisco Alvarez shows us an example of linear programming in Python: The first two constraints, x1 ≥ 0 and x2 ≥ 0 are called nonnegativity constraints. because it has certain limitations and these are following:. Welcome to PyMathProg¶. In this tutorial, you. Constraints differ from the common primitives of other programming languages in that they do not specify a step or sequence of steps to execute but rather the properties of a solution to be found. These are problems in which you have a quantity, depending linearly on several variables, that you want to maximize or minimize subject to several constraints that are expressed as linear inequalities in the same variables. LINEAR PROGRAMMING PROBLEM (LPP) TOPIC: COST MINIMIZATION 2. To obtain the solution to this Linear Program, we again write a short program in Python to call PuLP's modelling functions, which will then call a solver. Machine Learning for Healthcare Using Python, TensorFlow, and R. solver = pywraplp. I understand the hand-wave that makes dictionary building linear (though I have a hard time with even that). Algorithms and Data Structures Fall 2007 Robert Sedgewick and Kevin Wayne Department of Computer Science Princeton University Princeton, NJ 08544. about standard form? The main reason that we care about standard form is that this form is the starting point for the simplex method, which is the primary method for solving linear programs. The AI Programming with Python Nanodegree program is comprised of content and curriculum to support two (2) projects. Well j is the square root of -1 and as python supports complex numbers and we learn to solve quadratics with complex roots a linear equation solver ought to handle complex coefficents. You may want to predict continous values. It is used by the pure mathematician and by the mathematically trained scien-tists of all. There are many modules for Machine Learning in Python, but scikit-learn is a popular one. This exercise was done using Numpy library functions. Determinant of a square matrix. Why linear programming is a very important topic? Alot of problemscan be formulated as linear programmes, and There existefficient methodsto solve them or at least givegood approximations. ij dollars. We want to give a short example of how to solve a linear programming problem with Python. Welcome to IBM® Decision Optimization CPLEX® Modeling for Python. This book assumes you know a little bit about Python or programming in general. In this context, the function is called cost function, or objective function, or energy. A procedure called the simplex method may be used to find the optimal solution to multivariable problems. We create two arrays: X (size) and Y (price). Students will learn about the simplex algorithm very soon. Why is Python slow I Interpreted, not compiled. Analysis Namespace CenterSpace. For this example, we will be using the pandas and sci-kit learn libraries in Python in order to both calculate and visualize the linear regression in Python. The constraints you have are a linear combination of the decision variables. The GNU Linear Programming Kit (glpk) is a very versatile Mixed Integer Linear Programming solver that is especially well suited for teaching and research purposes. There exist several ILP solvers, free or commercial, that offer a java interface. The cost of producing each unit of X is:. Linear: while loop and everything stops running -program runs forever -program gives an answer but different than expected Types Python Programs 1. The following are links to scientific software libraries that have been recommended by Python users. • Binding a variable in Python means setting a name to hold a reference to some object. For non-unit-demand bidders, performs linear programming to minimize estimated v_ij for each item j given Av = b, where each A(r, :) indicates items in bundle and b(r) indicates corresponding bid. Solve linear least-squares problem. Details of model can be found in: Wilson JM. Recommended Python Training – DataCamp. There is a wide variety of free and commercial libraries for linear programming. They have 600 notebooks, 500 folders and 400 pens in stock, and they plan on packing it in two different forms. describe the characteristics of an LP in terms of the objective, decision variables and constraints, formulate a simple LP model on paper,. However, I found this Python library called pulp that provides a nice interface to glpk and other libraries. Alternative formulations of a flow-shop scheduling problem. In which we show how to use linear programming to approximate the vertex cover problem. However, he has only $1200 to spend and each acre of wheat costs$200 to plant and each acre of rye costs $100 to plant. Details and examples for functions, symbols, and workflows. Unfortunately, in Python there is no single official package that supports this solution. It then took around 100 ms to solve problems of moderate size. Regression is a statistical way to establish a relationship between a dependent variable and a set of independent variable(s). linprog¶ scipy. More advanced optimization tools don’t work off of spreadsheets, but instead require you to model your problem in a the form of a series of linear formulas. pyplot module in use. Moreover, we will understand the meaning of Linear Regression and Chi-Square in Python. As an example, we suppose that we have a set of affine functions $$f_i({\bf x}) = a_i + {\bf b}_i^\top {\bf x}$$, and we want to make all of them as small as possible, that is to say, to minimize their maximum. Solution Display Some browsers (including some versions of Internet Explorer) use a proportional width font (like Geneva or Times) in text boxes. Please write a program to print some Python built-in functions documents, such as abs(), int(), raw_input(). Luckily, we can use one of the many packages designed for precisely this purpose, such as pulp, PyGLPK, or PyMathProg. How to specify an IF-THEN constraint with an Integer Linear Programming (ILP) solver How to specify an IF-THEN constraint with an Integer Linear Programming (ILP) solver How to specify an IF-THEN constraint with an Integer Linear Programming (ILP) solver. A model in which the objective cell and all of the constraints (other than integer constraints) are linear functions of the decision variables is called a linear programming (LP) problem. This book will teach you how to make graphical computer games in the Python programming language using the Pygame library. Demand for employees with AI skills is skyrocketing and Python is one of the most widely used languages in Artificial Intelligence. It is a set of routines written in ANSI C and organized in the form of a callable library. Quadratic programming is a subfield of nonlinear optimization which deals with quadratic optimization problems subject to optional boundary and/or general linear equality/inequality constraints: Quadratic programming problems can be solved as general constrained nonlinear optimization problems. If there are points. The AI Programming with Python Nanodegree program is comprised of content and curriculum to support two (2) projects. Update: a much better solution is to use CVXOPT. Gurobi is the most powerful mathematical optimization solver out there. Constraint programming is a programming paradigm where relations between variables can be stated in the form of constraints. Actually, linear programming can be done graphically only in two or three variables, linear programming in more than three variables requires the use of special algorithms, one of which is the simplex algorithm, which can be found in any text on linear programming. You might be familiar with algebraic modeling languages such as AMPL, AIMMS, and GAMS. Mathematical optimization deals with the problem of finding numerically minimums (or maximums or zeros) of a function. However, many relationships in data do not follow a straight line, so statisticians use nonlinear regression instead. They are provided to bring the reader up to speed in the part of Python we use in the book. IMSL Numerical Libraries - linear, quadratic, nonlinear, and sparse QP and LP optimization algorithms implemented in standard programming languages C, Java, C#. Welcome to IBM® Decision Optimization CPLEX® Modeling for Python. One of the critical steps in solving a linear program, or working with systems of inequalities in any context, is to graph them and find the feasible region. Using drop-in interfaces, you can replace CPU-only libraries such as MKL, IPP and FFTW with GPU-accelerated versions with almost no code changes. Linear programming is not a programming language like C++, Java, or Visual Basic. Problem Statement. Graphing Linear Inequalities with Python Here is a practical example of the matplotlib. A Linear Equation is an equation for a line. It is a set of routines written in ANSI C and organized in the form of a callable library. Linear regression is a prediction method that is more than 200 years old. A software engineer puts the mathematical and scientific power of the Python programming language on display by using Python code to solve some tricky math. Why is Python slow I Interpreted, not compiled. APM Python - APM Python is free optimization software through a web service. Python result_status = solver. The library provides a wide range of mathematical routines such as random number generators, special functions and least-squares fitting. REGRESSION Linear Regression Datasets. You’ve been learning about data science and want to get rocking immediately on solving some problems. The crux of the matter is the linear program. Punctually, I'm trying to understand how you'll code something like the. The code of the article can be found here. NTRODUCTIONI British Standard Glossary of terms (3811:1993) defined maintenance as the combination of all technical and administrative actions, including supervision actions, intended to retain an item in, or restore it to, a state in which it can perform a required function. I am going to use a Python library called Scikit Learn to execute Linear Regression. Linear programming is the method of considering different inequalities relevant to a situation and calculating the best value that is required to be obtained in those conditions. Overview This tutorial uses PyCharm as the IDE. Python For Data Science Cheat Sheet Pandas Basics Learn Python for Data Science Interactively at www. SageMath is listed as a Python environment, because technically it is one. He has to plant at least 7 acres. Python | Linear Programming in Pulp Linear Programming (LP) , also known as linear optimization is a mathematical programming technique to obtain the best result or outcome, like maximum profit or least cost, in a mathematical model whose requirements are represented by linear relationships. The Python Optimization Modeling Objects (Pyomo) package described in this paper represents a fourth strategy, where a high level programming language is used to formulate a problem that can be solved by optimizers written in low-level lan-guages. Linear Search. There are many modules for Machine Learning in Python, but scikit-learn is a popular one. In simpler terms, we try to optimize (to maximize or minimize) a function denoted in linear terms and bounded by linear constraints. The up-to-date code, along some documentation, can be found here. 1 Linear Programming Relaxations An integer linear program (abbreviated ILP) is a linear program (abbreviated LP) with the additional constraints that the variables must take integer values. The time (in minutes) to process one unit of each product on each machine is shown below:. Reading CSV Files with Pandas. Digital Transformation Technical Leaders Program. The other constraints are then called the main constraints. Before we continue to focus topic i. It is widely used in mathematics, and to a lesser extent in business, economics, and for some engineering problems. (Integer) Linear Programming in Python. IMSL Numerical Libraries - linear, quadratic, nonlinear, and sparse QP and LP optimization algorithms implemented in standard programming languages C, Java, C#. Linear Programming A method used to find optimal solutions such as maximum or minimum profits Steps: 1. Constrained quadratic programming. Let's write those up now: import pandas as pd import numpy as np import matplotlib. With this library, you can quickly and easily add the power of optimization to your application. Also known as half search method, logarithmic chop, or binary chop. If you don’t know how to program, you can learn by downloading the. Number Crunching and Related Tools. Python runs on Windows, Linux/Unix, Mac OS X. MIDACO is suitable for problems with up to several hundreds to some thousands of optimization variables and features parallelization in Matlab, Python, R, C/C++ and Fortran. Linear Programming in Python with CVXOPT In a previous post , I compared the performances of two Linear Programming (LP) solvers, COIN and GLPK, called by a Python library named PuLP. Visual Basic code F# code IronPython code Back to QuickStart Samples. Comprehensive documentation for Mathematica and the Wolfram Language. PYTHON is a general-purpose interpreted, interactive, object- oriented, and high level programming language. network warrior network guide to networks rar. Also, we will look at Python Linear Regression Example and Chi-square example. Journal of the Operational Research Society (1989) 40:395-399. Linear programming was revolutionized when CPLEX software was created over 20 years ago: it was the first commercial linear optimizer on the market written in the C language, and it gave operations researchers unprecedented flexibility, reliability and performance to create novel optimization algorithms, models, and applications. Functional Programming. This page attempts to collect information and links pertaining to the field of Operations Research, which includes problems in Linear Programming, Integer Programming, Stochastic Programming, and other Optimization methods in python. Linear programming can be applied to various fields of study. IMSL Numerical Libraries - linear, quadratic, nonlinear, and sparse QP and LP optimization algorithms implemented in standard programming languages C, Java, C#. mating the running time of programs by allowing us to avoid dealing with constants that are almost impossible to determine, such as the number of machine instructions that will be generated by a typical C compiler for a given source program. June 4th, 2017. A binary tree is a tree data structure in which each node has at most two children. An example of model equation that is linear in parameters Y = a + (β1*X1) + (β2*X22) Though, the X2 is raised to power 2, the equation is still linear in beta parameters. If it is found then we print the location at which it occurs, otherwise the list doesn't contain the element we are searching. Solver('simple_lp_program', pywraplp. The book is accompanied by about fifty programs written in Python and Perl that generate concrete Integer Linear Programming formulations for many of the biological problems in the book. Formalizing The Graphical Method17 4. To understand this example, you should have the knowledge of following Python programming topics:. There are seven steps. Optimization with PuLP¶. The model, which is of the simplex type, is restricted by systematic development plans, production and stockpiling abilities, and available resource. , are to be optimized. Greetings, Earthling! Welcome to The Hitchhiker’s Guide to Python. ) directories. The goal and constraints require linear relationships to have the math work in your favor. Python has a nice package named PuLP which can be used to solve optimization problems using Linear programming. You can begin learning Python and using PuLP by looking at the content below. – Python’s syntax is very clean and naturally adaptable to expressing mathematical programming models. Now that you know what Linear and Binary Search methodologies are, let us look at how these searches would work on a list of numbers. Here you will get program for linear search in python. lib: data for: a set of test problems in MPS format. The LP technique will determine optimum values for the process design variables, so as to achieve minimum cost. This article shows two ways to solve linear programming problems in SAS: You can use the OPTMODEL procedure in SAS/OR software or. They provide help with statistics on the topics such as SPSS, STATA, Linear programming, Normal distribution, Data Analysis, Data Research & Data Mining etc. “Linear algebra is at the heart of how the car learns to drive itself,” says Jamthe. It also publishes articles that give significant applications of matrix theory or linear algebra to other. To obtain the solution to this Linear Program, we again write a short program in Python to call PuLP's modelling functions, which will then call a solver. Screenshots from my Jupyter notebook are shown below: Step 1 - Import relevant packages. Since it's introduction in release R2014a, we've had several blog posts now showing some applications of intlinprog, the mixed-integer linear programming (MILP) solver found in the Optimization Toolbox. Solve a linear system of equations. FORMULATING LINEAR PROGRAMMING PROBLEMS One of the most common linear programming applications is the product-mix problem. This imports numpy, which is a linear algebra library. The main features of LiPS are: LiPS is based on the efficient implementation of the modified simplex method that solves large scale problems. There are many libraries in the Python ecosystem for this kind of optimization problems. Linear programming is a technique to solve optimization problems whose constraints and outcome are represented by linear relationships. So the assumption is satisfied in this case. … Continue reading A Simple Interior Point Linear Programming Solver in Python. Let’s see what this means. 4 A Linear Programming Problem with no solution. All the variables are non-negative Each constraint can be written so the expression involving the variables is less than or equal to a non-negative constant. Another way to use a linear program to solve an optimization problem is to transform a new problem into a problem for which we already have a linear program solution—this is a reduction. This is the origin and the two non-basic variables are x 1 and x 2. Number Crunching and Related Tools. Each project will be reviewed by the Udacity reviewer network. Moreover, we will understand the meaning of Linear Regression and Chi-Square in Python. Set up the initial tableau. Linear Programming Basics. Linear Programming Suppose you are given: I A matrix A with m rows and n columns. It is a special case of mathematical programming. Linear Programming Problems Steve Wilson. This two-language approach leverages the flexibility of the high-level lan-. Find a length-n vector ~x such that A~x ~b and so that ~c ~x := Xn j=1 c jx j is as large as possible. [SciPy-User] Linear Programming via Simplex Algorithm. There are seven steps. ˜2 10 PDF from the pdf() function in the scipy. As a differential and algebraic modeling language, it facilitates the use of advanced modeling and solvers. Set up the initial tableau. To each linear program there is associated another linear program called its \dual". Calculates, or predicts, a future value by using existing values. CVXOPT is a Python library for convex optimization. It relies on the technique of traversing a list from start to end by exploring properties of all the elements that are found on the way. Write the initial tableau of Simplex method. NTRODUCTIONI British Standard Glossary of terms (3811:1993) defined maintenance as the combination of all technical and administrative actions, including supervision actions, intended to retain an item in, or restore it to, a state in which it can perform a required function. Here, the objective function is x1 + x2. Linear Programming (LP) Linear programming, simply put, is the most widely used mathematical programming technique. Rutgers University CS111 Programming exams with solutions. Linear programming (LP), involves minimizing or maximizing a linear objective function subject to bounds, linear equality, and inequality constraints. Linear programming was revolutionized when CPLEX software was created over 20 years ago: it was the first commercial linear optimizer on the market written in the C language, and it gave operations researchers unprecedented flexibility, reliability and performance to create novel optimization algorithms, models, and applications. Punctually, I'm trying to understand how you'll code something like the. When the preprocessing finishes, the iterative part of the algorithm begins until the stopping criteria are met. It builds on and extends many of the optimization methods of scipy. Constraints in linear programming problems are seldom all of the “less-than-or-equal-to” (≤) vari- ety seen in the examples thus far. Linear Programming and CPLEX Ting-Yuan Wang Advisor: Charlie C. Linear programming. In this post I intend to explain what a Linear Program (LP) is, and how to solve an LP problem using Karmarkar's Algorithm implemented in Python. PuLP: Algebraic Modeling in Python PuLP is a modeling language in COIN-OR that provides data types for Python that support algebraic modeling. Gaussian Elimination and Linear Programming). If you don’t know how to program, you can learn by downloading the. PuLP only supports development of linear models. In particular, these are some of the core packages:. Then modify the example or enter your own linear programming problem in the space below using the same format as the example, and press "Solve. Nonlinear Programming problem are. Linear Programming And Network Flows Solution Manual Download. Deep Learning Book Series 2 4 Linear Dependence And Span. # Create an optimizer with the desired parameters. This chapter discusses simple linear regression analysis while a subsequent chapter focuses on multiple linear regression analysis. Contribute to coin-or/pulp development by creating an account on GitHub. Linear Search, Binary Search and other Searching Techniques By Prelude Searching for data is one of the fundamental fields of computing. Discover the best Linear Programming in Best Sellers. MAXIMIZATION PROBLEMS. Graphing Linear Inequalities with Python Here is a practical example of the matplotlib. Deriving the dual from the primal is a purely mechanical procedure. PyMathProg is an easy and flexible mathematical programming environment for Python. Dynamic programming is an optimization approach that transforms a complex problem into a sequence of simpler problems; its essential characteristic is the multistage nature of the optimization procedure. A farmer is going to plant apples and bananas this year. Solving this problem is called linear programming or linear optimization. This is always a highlight of the teaching period as I get to see the awesome things my students have come up with. We estimate that students can complete the program in three (3) months working 10 hours per week. The AI Programming with Python Nanodegree program is comprised of content and curriculum to support two (2) projects. Matlab is not free, but, while you are a student at OSU, you have access to Matlab through the College of Engineering. Figure 1: Schematic of an Oil Refinery. Their examples are crystal clear and. Coefficients of the linear objective function to be minimized. Tag: Linear Programming (4) Linear Programming and Discrete Optimization with Python using PuLP - May 8, 2019. Deriving the dual from the primal is a purely mechanical procedure. Note: The rows of A represent deterministic strategies for rowboy, while columns of A represent deterministic strategies for colgirl. How to specify an IF-THEN constraint with an Integer Linear Programming (ILP) solver How to specify an IF-THEN constraint with an Integer Linear Programming (ILP) solver How to specify an IF-THEN constraint with an Integer Linear Programming (ILP) solver. Develop the technical leaders of tomorrow by growing scientists and engineers into digital scientists and engineers in a program that combines training, apprenticeship, and solving their business problems. linear programming decoder to output vectors that do not correspond to codewords; such vectors are known as pseudocodewords. Due to the widespread use of Linear. NET Matrix Library for VB. Linear Programming With Python - DZone. Journal of the Operational Research Society (1989) 40:395-399. Linear regression example with Python code and scikit-learn Now we are going to write our simple Python program that will represent a linear regression and predict a result for one or multiple data. gz, 129K) Sparse Linear Programming in Fortran77 (by Jacek Gondzio). PySP: Modeling and Solving Stochastic Programs in Python Jean-Paul Watson · David L. Download Python Linear Programming Modeler for free. Solve a linear system of equations. The constraints you have are a linear combination of the decision variables. At that time I never heard of Data Science. The following code produces valid solutions, but when your vector$b\$ changes you have to. Python runs on Windows, Linux/Unix, Mac OS X. “Linear algebra is at the heart of how the car learns to drive itself,” says Jamthe. VisualBasic ' A. A simple example of two-stage recourse is the following:. The linear program we start with is typically called the \primal". LINEAR PROGRAMMING PROBLEM (LPP) TOPIC: COST MINIMIZATION 2. This section provides you a brief description about Linear Queue in Data Structure Tutorial with Algorithms, Syntaxes, Examples, and solved programs, Aptitude Solutions and Interview Questions and Answers. We want to give a short example of how to solve a linear programming problem with Python. assert result_status == pywraplp. As implied by "linear", the objective function for such a problem is a linear combination of the decision variables. using the module gurobipy. A mechanics company can produce 2…. , are to be optimized. Linear programming can be applied to various fields of study. The plan of the paper is as follows. Each project will be reviewed by the Udacity reviewer network. For example, ERP5 uses linear programming to determine resource capacities. If the slack is zero, then the corresponding constraint is active. Hi all, I'm new to this group so I don't know if this question has been posted before, but does anyone knows about linear/integer programming routines in Python that are available on the web, more specifically of. With the start of school approaching, a store is planning on having a sale on school materials. A binary search can be more efficient than a linear search. After completing this unit, you should be able to. Linear programming is one of the most common optimization techniques. The rst two steps put. Thanks to Discretelizard for pointing this out to me. Linear programming is a simple technique where we depict complex relationships through linear functions and then find the optimum points. The important word in previous sentence is depict. word, txt, pdf, ppt, kindle, zip, and rar. What Is An Efficient Algorithm To Solve A Large 10 6 Linear. Learn matrix inversion, solving systems of linear equations, and elementary linear algebra using NumPy and SciPy in this video tutorial by Charles Kelly. Overview This is a tutorial about some interesting math and geometry connected with. A survey of linear programming tools was conducted to identify potential open-source solvers. Note that this is the most crucial step as all the subsequent steps depend on our analysis here. The constraints you have are a linear combination of the decision variables. Linear programming solves problems of the. The plan of the paper is as follows. The Simplex algorithm is a popular method for numerical solution of the linear programming problem. First Linear Regression Example in Python We believe it is high time that we actually got down to it and wrote some code! So, let's get our hands dirty with our first linear regression example in Python. mathematical programs. Predicting Housing Prices with Linear Regression using Python, pandas, and statsmodels In this post, we'll walk through building linear regression models to predict housing prices resulting from economic activity. Linear Programming. through PYTHON. !Magic algorithmic box. Linear programming is a simple technique where we depict complex relationships through linear functions and then find the optimum points. http://wiki. If our set of linear equations has constraints that are deterministic, we can represent the problem as matrices and apply matrix algebra. Python is ideally suited to handle linear programming problems. – Python’s syntax is very clean and naturally adaptable to expressing mathematical programming models. The following are links to scientific software libraries that have been recommended by Python users. Linear programming (LP), involves minimizing or maximizing a linear objective function subject to bounds, linear equality, and inequality constraints. It is widely used in business and economics. We have seen that we are at the intersection of the lines x 1 = 0 and x 2 = 0. solver = pywraplp. In all other cases, linear programming problems are solved through matrix linear algebra. The real relationships might be much more complex – but we can simplify them to linear relationships. Chen Department of Electrical and Computer Engineering University of Wisconsin-Madison. Write the initial tableau of Simplex method. Linear Programming, also sometimes called linear optimisation, involves maximising or minimising a linear objective function, subject to a set of linear inequality or equality constraints. Python was created out of the slime and mud left after the great flood. ARTIFICIAL AND SURPLUS VARIABLES. Learn at your own pace from top companies and universities, apply your new skills to hands-on projects that showcase your expertise to potential employers, and earn a career credential to kickstart your new career. Linear programming is a simple technique where we depict complex relationships through linear functions and then find the optimum points. With this library, you can quickly and easily add the power of optimization to your application. In the term linear programming, programming refers to mathematical pro-gramming. Regression analysis is almost always performed by a computer program, as the equations are extremely time-consuming to perform by hand. Francisco Alvarez shows us an example of linear programming in Python: The first two constraints, x1 ≥ 0 and x2 ≥ 0 are called nonnegativity constraints. Rowboy pays colgirl a. Comprehensive documentation for Mathematica and the Wolfram Language. If you're new to Octave, I'd recommend getting started by going through the linear algebra tutorial first. Linear programming is a beautiful area of mathematics with a lot of elegance that makes use of linear algebra without anyone ever needing to know about it. SageMath is listed as a Python environment, because technically it is one. Programming Exercise 1: Linear Regression. This equivalence allows us to solve a Sudoku puzzle using any of the many freely available ILP solvers; an implementation of a solver (in Python 3) which follows the formulation described in this post can be found found here. Pyomo is less terse than GLPK MathProg or AMPL as it must be parsed as Python. Prosciutto cotto, Emmental. Several years of exams with solutions. We begin by reducing the input linear program to a spe-. The initial tableau of Simplex method consists of all the coefficients of the decision variables of the original problem and the slack, surplus and artificial variables added in second step (in columns, with P 0 as the constant term and P i as the coefficients of the rest of X i variables), and constraints (in rows). If that means using an external solver that comes as a stand-alone application, don’t avoid it just because you are lazy to learn how to do it. Solve a linear system of equations. Since it's introduction in release R2014a, we've had several blog posts now showing some applications of intlinprog, the mixed-integer linear programming (MILP) solver found in the Optimization Toolbox.
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# Logging into your new instance “in the cloud” (Mac version)¶
OK, so you’ve created a running computer. How do you get to it?
The main thing you’ll need is the network name of your new computer. To retrieve this, go to the instance view and click on the instance, and find the “Public DNS”. This is the public name of your computer on the Internet.
Copy this name, and connect to that computer with ssh under the username ‘ubuntu’, as follows.
Next, start Terminal (in Applications... Utilities...) and type:
chmod og-rwx ~/Downloads/amazon.pem
to set the permissions on the private key file to “closed to all evildoers”.
Then type:
ssh -i ~/Downloads/amazon.pem ubuntu@ec2-???-???-???-???.compute-1.amazonaws.com
Here, you’re logging in as user ‘ubuntu’ to the machine ‘ec2-174-129-122-189.compute-1.amazonaws.com’ using the authentication key located in ‘amazon.pem’ on your Desktop.
Note, you have to replace the stuff after the ‘@’ sign with the name of the host; see the red circle in:
At the end you should see text and a prompt that look like this:
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# How to bring beauty to this pgfplot? [closed]
We are making a plot as follows:
\documentclass{standalone}
\usepackage{pgfplots}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
\def\xmin{0}
\def\xmax{10}
\def\ymin{0}
\def\ymax{7}
\draw[style=help lines, ystep=1, xstep=1] ;
\draw (-.25,-.25) node[auto] {0};
\draw[->] (\xmin,\ymin) -- (\xmax,\ymin) node[right] {$x$};
\draw[->] (\xmin,\ymin) -- (\xmin,\ymax) node[above] {$f(x)$};
\def\intersectX1{2}
\def\intersectY1{4/5+3}
\def\intersectX2{7}
\def\intersectY1{9/5+3}
\def\intersectX{4.76}
\def\intersectY{4.26}
\def\QPX{4}
\def\QPY{5}
\draw[color=red,smooth] plot [domain=0:8] (\x,{((\x-4)^2)/5+3)});
\draw[dashed] (2,0) node[below] {$x_1$} -- (2,3.8) node[up,left] {$f(x_1)$};
\draw[dashed] (7,0) node[below] {$x_1$} -- (7,4.8) node[up,right] {$f(x_2)$};
\draw[dashed,orange] (4,0) node[below] {$\alpha\,x_1+(1-\alpha)x_2$} -- (4,3) node[below] {$f(\alpha\,x_1+(1-\alpha)x_2)$};
\draw[color=black,dashedd] (4,3) -- (4,4.2) node[up] {$\alpha f(x_1) + (1-\alpha)f(x_2)$};
\draw[color=blue] (2,3.8) -- (7,4.8);
\end{tikzpicture}
\end{document}
We want to achieve:
1. The labels $f(x_1)$, ... needs to be smaller but more clear -- by bringing more color in them.
2. Display that : $\alpha f(x_1) + (1-\alpha)f(x_2) \ge f (\alpha x_1 + (1-\alpha) x_2)$. Just the picture need to tell the story.
3. Any idea to beautify this plot is very welcome.
-
## closed as too localized by Jake, Claudio Fiandrino, Marco Daniel, diabonas, zerothFeb 17 '13 at 16:01
This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question.
Why do you load pgfplots and don't use it? pgfplots provides a enhanced axis environment and addplot command for advanced plotting. See manual – bloodworks Feb 16 '13 at 13:29
Try to avoid text over graphics? The orange color might be too light and another color might better accompany the other colors? – N.N. Feb 16 '13 at 13:31
Number 3 is too subjective for SE, please read the FAQ on don't-ask: "avoid asking subjective questions" – Camil Staps Feb 16 '13 at 13:32
Maybe you could use pgfplots legends to put move some formulas further away from the plots? – N.N. Feb 16 '13 at 14:03
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# 笔记--c++临时对象与const
c++中规定临时对象只能赋给const的引用
const A& a = A() //合法
A& a = A() //非法
It is likely to lead to unexpected behaviour. Imagine something first like this:
Code:
std::string & make_upper( std::string & s ){ std::transform( s.begin(), s.end(), s.begin(), ::toupper ); return s;}
Reasonable enough implementation. Now first suppose you tried to pass in char * instead.
Code:
char text[] = "Hello world";make_upper( char_text );
std::string has an implicit constructor from const char * (and thus char *) so it might create a temporary which can be used for this function, but if that were allowed, the std::string makes a copy so you would only be modifying the copy and the changes would be lost.
In case you think that this should apply only to implicit constructors and not temporaries returned from functions, there are a lot of cases whereby objects are constructed through a helper function because these functions can resolve the template parameters (needed for the class) so you don't need to supply them to the function. (bind1st and make_pair are examples of such functions).
Note that if the class is a wrapper for a non-const reference or pointer, you can work around the situation by using a const reference to the buffer. For example in class to input to a vector.
Code:
template< T >class vec_wrapper_t{ std::vector< T > & m_v;public: vec_wrapper_t( std::vector< T > & v ) : m_v ( v ) { } void input( std::istream & is ) const { // implement input to the vector }};template < typename T >vec_wrapper_t< T > vec_wrapper( std::vector< T > & v ){ return vec_wrapper_t< T >( v );}template < typename T >std::istream & operator>>( std::istream & is, const vec_wrapper_t & vw ){ vw.input( is ); return is;}
Above illustrated
1. That you pass in vec_wrapper_t as const reference (or value) even to the overload of operator >> from istream
2. vec_wrapper is a typical function that resolves the template parameter for you and returns a temporary.
3. with the code above, if v is a vector and is is an input stream you can write is >> vec_wrapper( v ) in your code because of the fact that it's a const reference.
4. In reality vec_wrapper_t is going to have other features, (in my case it has the delimiters used for inputting). You can't get operator>> to write these because you vec_wrapper_t is temporary, i.e. anything you wrote to it would be lost and for the next vector you tried to write to, it would not be there.
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# Retrieving the non-relativistic formulas for electric and magnetic fields from relativistic formulas
I was checking the formulas for the electric and magnetic fields components E and B given in this link from Wikipedia: https://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity#Transformation_of_the_fields_between_inertial_frames
By the end of the section, they convert these formulas from the relativistic to the non-relativistic aspect by just approximating that the Lorentz factor becomes very small, since $$v<. I was thinking if there is any way to derive these formulas by applying a Taylor expansion on the gamma-factor. Is this possible, and if yes, could someone write me how any of these vectors would transform to non-relativistic approach? I already know how to do the Taylor expansion of the gamma factor, but afterward, I cannot figure out the calculations. Any idea or help is very appreciated.
• Non-relativistic doesn’t mean $v/c$ is absent. The gamma factor gives $(v/c)^2$ terms, which you can ignore. – G. Smith Jan 14 at 18:39
• You’re being too literal. There is a $\mathbf{v}$ in each equation, but no $v^2$. Sometimes there is no $c$, and sometimes there is $c^2$; this is just because of the use of non-Gaussian units. In Gaussian units, every $v$ would be divided by $c$. – G. Smith Jan 14 at 19:12
• There is no point in using a Taylor series on gamma if you want equations accurate only to first order in $v$. That is why they approximated gamma as 1. – G. Smith Jan 14 at 19:17
• A nonrelativistic transformation usually keeps first order in $v$, but not higher, because this is similar to the Galilean transformation $x’=x-vt$. – G. Smith Jan 14 at 19:19
• In fact, taking the Galilean limit of electromagnetism is far more subtle. There are actually two different self-consistent ways to do it. To learn more just search it up on this site, it’s been discussed before. – knzhou Jan 14 at 22:19
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# B Inertia (and, to some extent, circular motion again)
1. Jan 2, 2017
### jds10011
I often hear inertia used as an explanation in areas where it seems to make intuitive sense, but appears to me to be inconsistent with the definition of inertia as just depending on an object's mass. I offer three examples (they're very similar):
Example 1: An elevator
Suppose an elevator begins at rest and then accelerates upward. The rider has naturally brought a bathroom scale to stand on in the car. Of course, the scale indicates that the normal force on the rider has increased from just the magnitude of the rider's weight. Often this is explained by saying: "The rider was at rest, and therefore due to inertia had a tendency to remain at rest. By attempting to remain at rest, the rider exerted a greater-than-usual force on the floor of the car (scale). You can see this also by the fact that the rider's knees buckled slightly as the elevator started to move. In fact, we can see that it is a result of the person's inertia by substituting a more massive person -- for the same acceleration of the car, the more massive person pushes down harder on the scale, since they have more inertia." The issue that I have with this explanation is that if the elevator were now given a greater acceleration, the person would exert greater force on the floor/scale. However, their inertia has not changed, so it seems like a poor explanation for the phenomena.
Example 2: A ball on a string
Suppose a person whirls a ball on a string around in a circle at a constant speed. There is tension in the string, so clearly the person is pulling inward on the string, and the ball is pulling outward on the string (yes, even though there is no outward force on the ball, there is on the string). Many are surprised that the ball has reason to pull outward on the string, or that the person must pull inward. Often this is explained by saying: "The ball has inertia -- the tendency to continue in straight-line motion. In order to make it go around in a circle, rather than continue in a straight line, the person must use the string to change the direction of the ball's motion, which the ball resists as it tries to go in a straight line. Again, we can see this is the result of the ball's inertia by substituting a ball of greater mass -- for the same speed of revolution, the person and the ball must pull harder on the string." As before, the issue that I have with this explanation is that if the ball were now given a greater speed, the person and the ball would both pull harder on the string. However, the ball's inertia has not changed, so it seems like a poor explanation for the phenomena.
Example 3: A person in a gravitron ride (or a towel in the clothes dryer):
Suppose the ride travels at a constant speed. A rider (inside) is against the outer wall of the drum. The person is surprised that they seem to be pushing against the wall (in fact, they can stand horizontally on the wall if the ride goes fast enough). Often this is explained exactly as with the ball on the string -- "The person's inertia means they want to continue in a straight line, so they keep running into the wall, which then responds to this contact force by exerting its third law pair, the normal force, on the person. Again, we can see this is the result of the rider's inertia by substituting a rider of greater mass -- for the same speed of revolution, the new rider pushes harder on the wall, and the wall responds with a greater normal force." As before, the issue that I have with this explanation is that if the ride were now given a greater speed, the person would exert greater force on the wall (and the wall would exert greater force on the rider). However, their inertia has not changed, so it seems like a poor explanation for the phenomena.
How would you revise these explanations? Or is there a different issue here? Thanks!
2. Jan 3, 2017
### Orodruin
Staff Emeritus
Inertia is the resistance to acceleration. The force is proportional to the inertia only if you keep the same kinematic setup. This is well understood. If you change acceleration the force will also increase in proportion to the inertia.
3. Jan 3, 2017
### vela
Staff Emeritus
As Newton told us, $F = ma$. The force doesn't depend on the mass alone.
4. Jan 3, 2017
### jds10011
I agree. Hence, the reason that I don't see the concept of inertia as fully explaining these scenarios. The same mass offers greater "resistance" to greater accelerations, but this isn't called inertia.
5. Jan 3, 2017
### Orodruin
Staff Emeritus
This statement makes no sense. The ratio between the force and the acceleration is the inertia in all of the cases you mentioned.
6. Jan 3, 2017
### jds10011
So, for example, the ball on the string. If I try to whirl it around faster, the ball pulls harder on the string, i.e. it resists the faster circular motion more than it did the slower one. Yet, its inertia is unchanged.
7. Jan 3, 2017
### Drakkith
Staff Emeritus
Get away from using "normal" language and try to look at it from a standpoint of the equations. The ball isn't resisting the change in its direction of motion any more or less than at any other time. In all cases its mass determines how quickly it is acceletated under a force, or how much force must be applied for any given acceleration. There is little ambiguity there.
8. Jan 3, 2017
### PeroK
Circular motion at constant speed, unlike linear motion at constant speed, requires constant acceleration (towards the centre of the motion). So, the "faster" the circular motion the greater the force that is required. Unlike linear motion where once you have reached a certain speed it takes essentially no force to maintain that speed.
By the way, "inertia" is not a term I've ever used. The term "mass" does the job.
9. Jan 3, 2017
### Orodruin
Staff Emeritus
Yes, because by increasing the speed you have increased the acceleration according to $a = v^2/r$ which holds for any circular motion at constant speed. Therefore you need more force - all in accordance with $F = ma$.
10. Jan 3, 2017
### jds10011
I agree with most of this. And yes, I am specifically trying to get at an issue of language here. It seems like substituting the more massive ball is analogous to whirling the less massive ball faster -- in both cases, as I hold the end of the string, I can feel an increased tension, which I would attribute to the ball's resistance to changes in its motion (inertia). Yet, substituting the more massive ball is clearly changing the inertia (I don't think this is disputed), whereas increasing the rotation speed isn't (I don't think this is disputed). However, would you not say that in both cases the ball's resistance to changes in its motion has increased? If I want to break the string, let's say, I can accomplish this either by using such a massive ball that the string is incapable of sufficiently changing its motion even at a low speed, or I can do this by whirling the smaller mass so fast that the string is again incapable of sufficiently changing its motion. It still seems like the former is clearly a result of inertia, and the latter is clearly not.
11. Jan 3, 2017
### Orodruin
Staff Emeritus
No. We are saying that in the case when you spin it faster the ball's change in motion has increased. If you can change the motion more by applying the same force, clearly you have less inertia. Again, inertia tells you how much force you need for a given rate of change in the motion.
12. Jan 3, 2017
### jds10011
So, the fact that the ball applies more force to the string when it is spun faster is not an indication that it is providing more resistance to its speed changing? In other words, suppose I am holding the string blindfolded when the ball is set in motion by a friend. I will feel some tension force. Now the setup is changed by either giving the ball a larger speed or by changing the mass of the ball and keeping the same speed. I will now feel a larger tension force. As I am blindfolded, I don't know which has occurred. Are you saying I can't tell based on the increased tension force that the ball's resistance to changes in its motion has changed?
Or, in the gravitron (drum) ride, if a rider asks why they are slamming into the outer wall harder the faster the ride goes, the answer is not related to the fact that they are more resistant to changes in their speed at higher speeds? (And yes, I know they "shouldn't" be, but bear with me, and I appreciate the help.)
I guess what I keep going back to is the old trick question about a stick being swung to hit a small 100 gram block on a frictionless horizontal surface. The question asks "If I apply a 1000N horizontal contact force to the block, by N3 does the block respond to me with 1000N?" The answer is "No, because you'll never be able to apply a 1000N force in such a scenario. The tiny mass doesn't resist the change in motion sufficiently for you to be able to remain in contact with it beyond a certain applied force -- far less than 1000N. Whatever force you do succeed in applying will be applied back to you by the block in accordance with N3."
It seems that the ball on the string is responding to the string pulling it inward with an equally large outward force on the string in accordance with N3, and if I increase the speed by increasing the inward pull, it responds by increasing its outward pull on the string. This is what I am interpreting as an increase in resistance, and I think where I am tripping myself up. I think you are saying that its resistance to changes in its motion is solely based on its mass (inertia), and therefore to accomplish a larger change in motion means a larger force (N2). I understand this from the perspective of the person applying the force. I think the issue is just that when I think of the ball pulling outward on the string, I am at a loss to explain why it is doing that harder at higher speeds from its perspective.
I think I'm getting close, and I appreciate your help and patience.
13. Jan 3, 2017
### PeroK
Your confusion, in general, is due to using an inexact concept of inertia as "resistance to motion" and attributing a characteristic of the motion to inertia of the object (a property of the object itself). The two examples here are
1) rate of acceleration is a characteristic of the motion (not of the object): increase the rate of acceleration and you increase the required force (but you do not change the inertia of the object).
2) Speed of circular motion is a characteristic of the motion: increase the circular speed and you increase the required centripetal force (but you do not change the inertia of the object).
You could add a third, which is an object moving on a rough surface. It's harder to increase the speed on the rough surface, so you could attribute this to increased inertia, But, it's easier to reduce the speed of the object, so you could attribute this to decreased inertia. In this way, you would have a variable inertia that depends on whether you are trying to increase or decrease the speed of the object.
This would also pass your blindfold test. Blindfolded you might interpret the friction as an increase or decrease in inertia of the object.
In an extreme case, someone might glue the object to the surface and you'd attribute this to the ball having gained inertia. This example highlights the problem of using an airy-fairy notion of inertia instead of mass. The mass of the object is the same, it's just glued to a surface. But, in a woolly way, by being glued down its resistance to motion has increased.
14. Jan 3, 2017
### jds10011
This is a really helpful set of explanations. Much appreciated. I certainly can see that a block on a surface with friction would resist changes in motion based on the amount of friction AND its mass. I guess the question then becomes that I can easily see the friction force as causing this, and would look to a free-body diagram to explain it. With the ball on the string, though, I don't have any other forces (we can assume the experiment is done in a vacuum, if needed) on the ball other than its weight and the tension on the string. Thus, I don't really see why a faster speed causes the ball to pull outward on the string harder than it would for a slower speed. This is where I was saying that the concept of inertia seems to be misapplied to explain this.
15. Jan 3, 2017
### PeroK
You're the one misapplying inertia by insisting it's a woolly concept separate from mass. If it's not mass, what is it? It can't be mass x acceleration, as that's force. The beauty of Newton's 2nd law is that all motion boils down to $F = ma$, a simple relationship involving three fundamental quantities. There is no room in this equation for "inertia", "resistance", "recalcitrance" or "temporary lassitude". Those are things exhibited by students, not moving objects!
16. Jan 3, 2017
### vela
Staff Emeritus
Why are you consistently ignoring the acceleration of the ball? When the ball moves faster around the circle, its velocity changes at a greater rate, which requires a bigger force to cause it.
17. Jan 3, 2017
### Drakkith
Staff Emeritus
How so? As you increase the ball's speed, the force required to keep it moving in a circle increases as well. So the resistance to motion hasn't changed, but the rate of change of its motion has. The same "resistance", but a greater rate of change, and hence a larger force.
18. Jan 3, 2017
### jds10011
100% in agreement. I began this discussion by saying I have often been told these concepts are explained by inertia, and yet, it doesn't seem to be consistent with "just mass" to me.
When I whirl the ball on the string faster, the ball pulls outward on the string harder. Given that it is not inertia that causes this additional "resistance", what is it? And yes, with the blindfold test I do believe we've established that I feel the same "resistance" in both the case of the added mass OR the case of the increased velocity.
19. Jan 4, 2017
### Drakkith
Staff Emeritus
There is no increased resistance in your example, there is just a greater force since the "rate of change" of the motion is greater. Increased force is not the same thing as increased inertia.
20. Jan 4, 2017
### jds10011
I agree 100% that inertia has not increased. Yet, the tension increases, just as it would if the mass (inertia) were increased rather than the speed. It appears that in the case of the increased mass, we would explain the tension increase by saying that the object has more of a tendency to continue in a straight line, and is therefore "straining at the leash" more than before. However, when we increase the speed instead, we again see the tension increase ("straining at the leash"), but we now seem to fall back on just saying, well, F=ma, so since a increased, F increased (and, yes, this is not disputed). In other words, we seem not to have a property of the object to ascribe its tendency to "strain at the leash" more in this situation. Are we saying that the mass SOLELY pulls outward on the string as an N3 reaction to being pulled by the string? Why is it ok to ascribe its outward pull on the string to a property of the object (inertia) ONLY if we aren't discussing why the outward pull increases when the rotation speed is increased?
This is what is bugging me -- let's go back to the gravitron (drum) ride. If I'm in the ride, smashing against the wall, and you ask me why I keep hitting the wall, it's OK to say "Well, I am in motion, so inertia means I have a tendency to keep moving in a straight line. The wall keeps getting in my way." Now, if you give me a heavy backpack to wear, and I'm hitting the wall harder, it's OK to say "See, now my inertia has increased due to the added mass. Now I have even more tendency to go in a straight line, so I smash into the wall harder." Yet, if instead of the backpack, the ride's rotation speed is increased, and again I start hitting the wall harder, it would clearly be incorrect to ascribe this to my (unchanged) inertia, despite the fact that the effect on me is the same -- I smash into the wall harder. If you now ask me why I'm smashing into the wall harder, must I say "actually, now it is the wall smashing me harder in an attempt to get me to revolve faster"? Can I not explain my action by any means other than as an N3 reaction?
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# 7.3.3 Processing steps
The results of all-sky classification were obtained through the following steps.
1. 1.
Crossmatch of Gaia with literature to identify objects of known classes (Section 7.3.3).
2. 2.
Selection of catalogues to crossmatch and their prioritisation (in case of conflictual information on the same objects).
3. 3.
Filtering of sources not satisfying simple statistics (such as colour, magnitude, literature period, amplitude, skewness, and Abbe value computed on magnitudes sorted in time as well as in phase) that are typical of class ownership, while allowing for a large range of possible distance, extinction, and reddening.
4. 4.
Resampling of sources for a more representative distribution in the sky, in the number of FoV transits, and in magnitude.
5. 5.
Pipeline run of the Statistics module on time series pre-processed as described in Section 7.2.3.
6. 6.
Generation and selection of classification attributes (Section 7.3.3).
7. 7.
Training of a multi-stage classifier with optimized parameters.
8. 8.
Application of the multi-stage classifier to the Gaia data.
9. 9.
Improvement of the training set (sources and attributes) including high-confidence classifications and iterating steps 3–6 (Section 7.3.3).
10. 10.
Training of the improved multi-stage classifier with optimized parameters (Section 7.3.3).
11. 11.
Pipeline run of the Statistics and the Classification modules on time series pre-processed as described in Section 7.2.3.
12. 12.
Training of contamination-cleaning classifiers and their application to the results of the previous step, for RR Lyrae stars, Cepheids, and SX Phoenicis/$\delta$ Scuti stars (Section 7.3.3).
13. 13.
Definition of classification scores of the published results (Section 7.3.3).
14. 14.
Assessment of completeness and contamination of the published results (Section 7.3.4).
## Classes
The training set included objects of the classes targeted for publication in Gaia DR2 (listed in bold) as well as other types to reduce the contamination of the published classification results. The full list of object classes, with labels (used in the rest of this section) and corresponding descriptions, follows below.
1. 1.
ACEP: Anomalous Cepheids.
2. 2.
ACV: $\alpha^{2}$ Canum Venaticorum-type stars.
3. 3.
ACYG: $\alpha$ Cygni-type stars.
4. 4.
ARRD: Anomalous double-mode RR Lyrae stars.
5. 5.
BCEP: $\beta$ Cephei-type stars.
6. 6.
BLAP: Blue large amplitude pulsators.
7. 7.
CEP: Classical ($\delta$) Cepheids.
8. 8.
CONSTANT: Objects whose variations (or absence thereof) are consistent with those of constant sources (Section 7.2.3).
9. 9.
CV: Cataclysmic variables of unspecified type.
10. 10.
DSCT: $\delta$ Scuti-type stars.
11. 11.
ECL: Eclipsing binary stars.
12. 12.
ELL: Rotating ellipsoidal variable stars (in close binary systems).
13. 13.
FLARES: Magnetically active stars displaying flares.
14. 14.
GCAS: $\gamma$ Cassiopeiae-type stars.
15. 15.
GDOR: $\gamma$ Doradus-type stars.
16. 16.
MIRA: Long period variable stars of the $o$ (omicron) Ceti type (Mira).
17. 17.
OSARG: OGLE small amplitude red giant variable stars.
18. 18.
QSO: Optically variable quasi-stellar extragalactic sources.
19. 19.
ROT: Rotation modulation in solar-like stars due to magnetic activity (spots).
20. 20.
RRAB: Fundamental-mode RR Lyrae stars.
21. 21.
RRC: First-overtone RR Lyrae stars.
22. 22.
RRD: Double-mode RR Lyrae stars.
23. 23.
RS: RS Canum Venaticorum-type stars.
24. 24.
SOLARLIKE: Stars with solar-like variability induced by magnetic activity (flares, spots, and rotational modulation).
25. 25.
SPB: Slowly pulsating B-type stars.
26. 26.
SXARI: SX Arietis-type stars.
27. 27.
SXPHE: SX Phoenicis-type stars.
28. 28.
SR: Long period variable stars of the semiregular type.
29. 29.
T2CEP: Type-II Cepheids.
## Crossmatch with literature
Training-set objects are selected from Gaia sources crossmatched with objects associated with known classes in the literature. In order to increase the reliability of crossmatch results, a set of metrics was used in the comparison of Gaia and literature sources, always including the angular separation, and whenever possible also the time-series median magnitude in the $G$ band, the $G_{\mathrm{BP}}-G_{\mathrm{RP}}$ colour, as well as time series quantities characterising the amplitude of variations in the $G$ band such as the range or standard deviation. Such metrics were combined in a multi-dimensional distance which was minimised in an iterative process in order to allow for the tuning of empirical relations between the Gaia and literature photometric quantities (affected in particular by the different bandwidth coverage and sensitivity). The best matches were projected onto planes for all combinations of crossmatch metrics to inspect the corresponding distributions and reduce the chance of mis-matches by applying thresholds to exclude dubious outliers and excessive tails of the distributions. Although this approach sacrificed completeness in some cases, it was considered appropriate for training purposes, given the large number of sources available.
In order to sample as many regions of the sky as possible, cover most of the range of Gaia magnitudes, and include a large number of variability types, a multitude and variety of catalogues were selected from a larger set, following general reliability considerations, and prioritised in case of conflicting classifications for the same sources. The full list of catalogues employed in the training sets are presented in Table 7.1, including references and crossmatch metrics. Among the over seven hundred fifty thousand crossmatched objects available for training, only a small sample (of about 33 thousand sources) was vetted to train classifiers (Section 7.3.3), leaving many reliable crossmatches for the validation of results (Section 7.3.4).
## Classification attributes
About one hundred fifty attributes were computed to characterise sources with photometric (and some astrometric) time series features. Each classifier (described in Section 7.3.3) was tested with a varying number of attributes (e.g., Guyon and Elisseeff 2003) and a subset of 40 attributes represented the union of attributes used by all classifiers. The employed classification attributes are defined below, with units quoted in brackets after the attribute name (unless the attribute is dimensionless).
1. 1.
ABBE: The Abbe value (von Neumann 1941, 1942) computed from the magnitudes of FoV transits in the $G$ band.
2. 2.
BP_MINUS_RP_COLOUR (mag): The possibly reddened colour index from the median magnitudes in the $G_{\rm BP}$ and $G_{\rm RP}$ bands.
3. 3.
BP_MINUS_G_COLOUR (mag): The possibly reddened colour index from the median magnitudes in the $G_{\rm BP}$ and $G$ bands.
4. 4.
DENOISED_UNBIASED_UNWEIGHTED_KURTOSIS_MOMENT (mag${}^{4}$): The sample-size unbiased and unweighted kurtosis central moment of FoV transit magnitudes in the $G$ band, denoised assuming Gaussian uncertainties (Rimoldini 2014).
5. 5.
DENOISED_UNBIASED_UNWEIGHTED_VARIANCE (mag${}^{2}$): The sample-size unbiased and unweighted variance of FoV transit magnitudes in the $G$ band, denoised assuming Gaussian uncertainties (Rimoldini 2014).
6. 6.
DURATION (d): The duration of the time series from the first to the last FoV transit observation in the $G$ band.
7. 7.
G_MINUS_RP_COLOUR (mag): The possibly reddened colour index from the median magnitudes in the $G$ and $G_{\rm RP}$ bands.
8. 8.
G_VS_TIME_IQR_ABS_SLOPE (mag d${}^{-1}$): The unweighted interquartile range of the absolute values of magnitude changes per unit time between successive FoV transits in the $G$ band.
9. 9.
G_VS_TIME_MAX_SLOPE (mag d${}^{-1}$): The unweighted 95th percentile of magnitude changes per unit time between successive FoV transits in the $G$ band.
10. 10.
G_VS_TIME_MEDIAN_ABS_SLOPE (mag d${}^{-1}$): The unweighted median of the absolute values of magnitude changes per unit time between successive FoV transits in the $G$ band.
11. 11.
IQR_BP (mag): The unweighted interquartile magnitude range of FoV transits in the $G_{\rm BP}$ band.
12. 12.
IQR_RP (mag): The unweighted interquartile magnitude range of FoV transits in the $G_{\rm RP}$ band.
13. 13.
LOG_QSO_VAR: The decadic logarithm of the reduced chi-square of FoV transit magnitudes in the $G$ band with respect to a parameterised quasar variance model, represented by $\log_{10}(\chi^{2}_{\mathrm{QSO}}/\nu)$ in Butler and Bloom (2011); see Rimoldini et al. (in preparation) for details on the parameter values for the Gaia data.
14. 14.
LOG_NONQSO_VAR: The decadic logarithm of the reduced chi-square of FoV transit magnitudes in the $G$ band not to follow a parameterised quasar variance model, represented by $\log_{10}(\chi^{2}_{\mathrm{False}}/\nu)$ in Butler and Bloom (2011); see Rimoldini et al. (in preparation) for details on the parameter values for the Gaia data.
15. 15.
MAD_G (mag): The unweighted median absolute deviation from the median magnitude of FoV transits in the $G$ band.
16. 16.
MAX_ABS_SLOPE_HALFDAY (mag d${}^{-1}$): The maximum value of the magnitude ranges of FoV transits in the $G$ band within sliding windows of half a day, divided by the time span of the $G$-band observations within such sliding windows.
17. 17.
MEAN_G (mag): The unweighted arithmetic mean magnitude of FoV transits in the $G$ band.
18. 18.
MEAN_BP (mag): The unweighted arithmetic mean magnitude of FoV transits in the $G_{\rm BP}$ band.
19. 19.
MEAN_RP (mag): The unweighted arithmetic mean magnitude of FoV transits in the $G_{\rm RP}$ band.
20. 20.
MEDIAN_ABS_SLOPE_HALFDAY (mag d${}^{-1}$): The unweighted median of the magnitude ranges of FoV transits in the $G$ band within sliding windows of half a day, divided by the time span of the $G$-band observations within such sliding windows.
21. 21.
MEDIAN_ABS_SLOPE_ONEDAY (mag d${}^{-1}$): The unweighted median of the magnitude ranges of FoV transits in the $G$ band within sliding windows of one day, divided by the time span of the $G$-band observations within such sliding windows.
22. 22.
MEDIAN_G (mag): The unweighted median magnitude of FoV transits in the $G$ band.
23. 23.
MEDIAN_BP (mag): The unweighted median magnitude of FoV transits in the $G_{\rm BP}$ band.
24. 24.
MEDIAN_RANGE_HALFDAY_TO_ALL: The unweighted median of the magnitude ranges of FoV transits in the $G$ band within sliding windows of half a day, divided by the $G$-band magnitude range of the full time series.
25. 25.
MEDIAN_RP (mag): The unweighted median magnitude of FoV transits in the $G_{\rm RP}$ band.
26. 26.
NONQSO_PROB: A quantity distributed according to the null-hypothesis distribution of $\chi^{2}_{\mathrm{QSO}}$, given the data, for non-quasar objects, computed from a parameterised quasar variance model with magnitudes of FoV transits in the $G$ band, related to $P(\chi^{2}_{\mathrm{QSO}}|x,\mbox{not quasar})$ in Butler and Bloom (2011); see Rimoldini et al. (in preparation) for details on the parameter values for the Gaia data.
27. 27.
NORMALISED_CHI_SQUARE_EXCESS: The difference between the chi-square of FoV transit magnitudes in the $G$ band and the mean of the chi-square distribution expected for constant objects (i.e., the number of degrees of freedom), normalised by the standard deviation of the chi-square distribution of constant objects (i.e., the square root of twice the number of degrees of freedom).
28. 28.
OUTLIER_MEDIAN_G: The absolute difference between the most outlying FoV transit magnitude with respect to the median magnitude in the $G$ band, normalised by the uncertainty of the most outlying measurement.
29. 29.
PARALLAX (mas): The parallax value of the source derived from a preliminary astrometric solution (Section 7.2.2).
30. 30.
PROPER_MOTION (mas yr${}^{-1}$): The proper motion of the source projected in the sky derived from a preliminary astrometric solution (Section 7.2.2).
31. 31.
PROPER_MOTION_ERROR_TO_VALUE_RATIO: The ratio between the estimated projected proper motion uncertainty and the projected proper motion value of the source, derived from a preliminary astrometric solution (Section 7.2.2).
32. 32.
RANGE_G (mag): The magnitude range of FoV transits in the $G$ band.
33. 33.
REDUCED_CHI2_G: The reduced chi-square of FoV transit magnitudes in the $G$ band.
34. 34.
SIGNAL_TO_NOISE_STDEV_OVER_RMSERR_G: The ratio between the sample-size biased unweighted standard deviation of FoV transit magnitudes in the $G$ band and the root-mean-square of their uncertainties.
35. 35.
SKEWNESS_G: The sample-size unbiased and unweighted skewness central moment of FoV transit magnitudes in the $G$ band, normalised by the third power of the unbiased unweighted standard deviation of the same time-series measurements.
36. 36.
SKEWNESS_PERCENTILE_5: A robust measure of the skewness of the magnitude distribution of FoV transits in the $G$ band, computed as $(P_{95}+P_{5}-2\,P_{50})/(P_{95}-P_{5})$ where $P_{n}$ is the $n$th unweighted percentile.
37. 37.
STETSON_G: The single-band Stetson variability index (Stetson 1996) computed from the magnitudes of FoV transits in the $G$ band, pairing observations within 0.1 days.
38. 38.
STETSON_G_BP: The double-band Stetson variability index (Stetson 1996) computed from the magnitudes of FoV transits in the $G$ and $G_{\rm BP}$ bands, pairing observations in different bands within 0.001 days.
39. 39.
TRIMMED_RANGE_G (mag): The magnitude range between the 5th and 95th unweighted percentiles of FoV transits in the $G$ band.
40. 40.
TRIMMED_RANGE_RP (mag): The magnitude range between the 5th and 95th unweighted percentiles of FoV transits in the $G_{\rm RP}$ band.
## Classification models
A hierarchical structure of Random Forest (Breiman 2001) classifiers identified objects in progressively more detailed (groups of) classes. For Gaia DR2, we focused on high-amplitude variable stars, so objects with negligible or low amplitude variations were first separated from the high amplitude ones, which were then split into the types and subtypes of interest by subsequent classifiers.
Every Random Forest classifier was configured with unlimited depths and with a minimum number of instances per class at the leafs set to one. Other configuration parameters (number of trees nTree and number of tested attributes mTry to best split the data at a given node of a tree), the training-set classes to identify (specified by the labels defined in Section 7.3.3), and the selected attributes (described in Section 7.3.3) are listed below for each classifier. Aggregations of types are denoted by connecting single type labels with an underscore (unless indicated otherwise in brackets).
1. 1.
Random Forest classifier configured with nTree=400 and mTry=10.
1. (a)
Training set:
1. i.
14 684 CONSTANT;
2. ii.
3885 LOW_AMPLITUDE_VARIABLE (ACV, ACYG, BCEP, low-amplitude DSCT_GDOR, ELL, FLARES, GCAS, GDOR, OSARG, ROT, SOLAR_LIKE, SPB, SXARI);
3. iii.
14 999 OTHER_VARIABLE (ACEP, ARRD, BLAP, CEP, CV, DSCT, ECL, MIRA, QSO, RRAB, RRC, RRD, RS, SR, SXPHE, T2CEP).
2. (b)
Attributes: BP_MINUS_G_COLOUR, BP_MINUS_RP_COLOUR,
DENOISED_UNBIASED_UNWEIGHTED_VARIANCE, DURATION, G_MINUS_RP_COLOUR, G_VS_TIME_MEDIAN_ABS_SLOPE, IQR_BP, IQR_RP, LOG_NONQSO_VAR, LOG_QSO_VAR, MAD_G, MEDIAN_ABS_SLOPE_ONEDAY, MEDIAN_BP, MEDIAN_G, MEDIAN_RP,
NONQSO_PROB, NORMALISED_CHI_SQUARE_EXCESS, OUTLIER_MEDIAN_G,
RANGE_G, REDUCED_CHI2_G, SIGNAL_TO_NOISE_STDEV_OVER_RMSERR_G,
SKEWNESS_PERCENTILE_5, STETSON_G, STETSON_G_BP, and TRIMMED_RANGE_RP.
2. 2.
Random Forest classifier configured with nTree=321 and mTry=4 (not relevant to the classification results published in Gaia DR2, but still described for details on the objects of low-amplitude types employed).
1. (a)
Training set:
1. i.
363 ACV_ACYG_BCEP_GCAS_SPB_SXARI (combination of poorly represented low-amplitude objects characterized by multiperiodic, pulsating, rotating, or irregular light variations);
2. ii.
866 DSCT_GDOR_LOW_AMPLITUDE (DSCT, GDOR, and DSCT-GDOR hybrids with low amplitude variations);
3. iii.
397 ELL;
4. iv.
996 OSARG;
5. v.
1247 SOLARLIKE_FLARES_ROT.
2. (b)
Attributes: BP_MINUS_RP_COLOUR, DURATION, G_MINUS_RP_COLOUR, IQR_RP,
LOG_QSO_VAR, MEAN_BP, MEAN_G, PARALLAX, PROPER_MOTION.
3. 3.
Random Forest classifier configured with nTree=336 and mTry=3.
1. (a)
Training set: 10 BLAP, 711 CEP_ACEP_T2CEP, 518 CV, 1326 DSCT_SXPHE, 3861 ECL, 1945 MIRA_SR, 1996 QSO, 4108 RRAB_RRC_RRD_ARRD, and 500 RS.
2. (b)
Attributes: ABBE, BP_MINUS_RP_COLOUR,
DENOISED_UNBIASED_UNWEIGHTED_VARIANCE, G_MINUS_RP_COLOUR,
G_VS_TIME_MAX_SLOPE, MEAN_G, MEAN_RP, MEDIAN_ABS_SLOPE_ONEDAY,
MEDIAN_RANGE_HALFDAY_TO_ALL, NORMALISED_CHI_SQUARE_EXCESS,
PARALLAX, PROPER_MOTION, PROPER_MOTION_ERROR_TO_VALUE_RATIO,
RANGE_G, and SKEWNESS_G.
4. 4.
Random Forest classifier configured with nTree=202 and mTry=3.
1. (a)
Training set: 2922 RRAB, 969 RRC, 197 RRD, and 20 ARRD.
2. (b)
Attributes: BP_MINUS_RP_COLOUR,
DENOISED_UNBIASED_UNWEIGHTED_KURTOSIS_MOMENT,
G_VS_TIME_IQR_ABS_SLOPE, G_VS_TIME_MAX_SLOPE,
NORMALISED_CHI_SQUARE_EXCESS, STETSON_G, and TRIMMED_RANGE_G.
5. 5.
Random Forest classifier configured with nTree=135 and mTry=3.
1. (a)
Training set: 99 ACEP, 455 CEP, and 157 T2CEP.
2. (b)
Attributes: BP_MINUS_RP_COLOUR, DURATION, LOG_NONQSO_VAR,
LOG_QSO_VAR, MAX_ABS_SLOPE_HALFDAY, MEAN_G,
MEDIAN_ABS_SLOPE_HALFDAY, and MEDIAN_RP.
## Semi-supervised classification
Semi-supervised classification was applied to constant objects, RR Lyrae stars, and long period variables, in order to improve their representation in the training set as follows.
1. 1.
High-confidence classifications of such classes were selected as candidate training sources.
2. 2.
Candidate training objects were filtered by the statistics mentioned in item 3 of Section 7.3.3, except for the literature period and the Abbe value computed on phase-sorted magnitudes (not available for results classified without period computation).
3. 3.
Filtered candidate training objects were selected to cover regions in the sky and/or magnitude intervals that lacked proper representation in the training set.
## Contamination cleaning
The contamination of preliminary classification results was reduced with the help of dedicated classifiers applied to RR Lyrae stars, Cepheids, and SX Phoenicis/$\delta$ Scuti stars, separately for each type, as follows.
1. 1.
Samples of true positives and false positives (according to crossmatched objects) were selected from the candidates of the previous classification stage.
2. 2.
Classification attributes were generated and selected.
3. 3.
A binary classifier of true positives versus false positives (in similar amounts) was trained and optimized.
4. 4.
The preliminary classification candidates (above some minimal level of classification probability depending on the type) were processed by the binary classifier (item 3) and objects classified as true positives with a minimum probability of 50 per cent were retained.
## Classification score
The results of the contamination-cleaning classifiers are associated with classification scores which express the confidence of the classifier given the training set, thus such scores should not be interpreted as true probabilities. The scores of Gaia DR2 classification results are obtained by linearly mapping the internal classifier probabilities to values within a range from zero to one (from the weakest to the strongest candidate), for each variability type.
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# Size reconstructibility of graphs
Groenland, C
Guggiari, H
Scott, A
5 August 2020
## Journal:
Journal of Graph Theory
## Last Updated:
2021-07-26T10:21:55+01:00
2
96
## DOI:
10.1002/jgt.22616
326-337
## abstract:
The deck of a graph $G$ is given by the multiset of (unlabelled) subgraphs
$\{G-v:v\in V(G)\}$. The subgraphs $G-v$ are referred to as the cards of $G$.
Brown and Fenner recently showed that, for $n\geq29$, the number of edges of a
graph $G$ can be computed from any deck missing 2 cards. We show that, for
sufficiently large $n$, the number of edges can be computed from any deck
missing at most $\frac1{20}\sqrt{n}$ cards.
905470
Submitted
Journal Article
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# Codecademy Curriculum Markdown Style Guide
## Description/Purpose
Codecademy's Markdown rendering supports Github Flavored Markdown. (Here's a quick reference you can use.)
## Code Blocks
Code blocks support syntax highlighting for the following languages:
• C++: cpp
• C#: cs
• CSS: css
• HTML: html
• Java: java
• JavaScript: js
• PHP: php
• Python: py
• R: r
• Ruby: rb
• Sass: scss
• Shell/Program Output/Unhighlighted:
• SQL: sql
Note that code blocks must contain syntactically-valid code snippets for the syntax highlighting to be consistent.
Will render correctly:
function doSomething() {
// statements go here
}
Will NOT render correctly:
function doSomething() {
...
}
## Tables
Markdown tables used to be created using the narrative-table-container wrapper. Now it can be created without. For example:
| A | B | Ouput |
| --- | --- | --- |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
will result in:
A B Ouput
0 0 0
0 1 1
1 0 1
1 1 1
Note that the number of columns with dashes, under the table headers, should be the same as the number of columns.
## LaTeX
tex allows TeX directives to be written into instructional narrative.
x^2 + y^2 = 1
will result in:
y = \sqrt{1 - x^2}
will result in:
\sum\limits_{n=1}^{\infty} 2^{-n} = 1
will result in:
## Keyboard Input
Keyboard inputs can be enclosed by the <kbd> tag:
<kbd>control</kbd> + <kbd>c</kbd>
will result in:
control + c.
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×
# List of Wiki Articles I've Contributed to
Here's a list of wiki articles I've contributed to on Brilliant.
Now I didn't write all these articles all by myself. Many members of the community also contributed to them and made them as they are today. So if you see an article you can help improve, by all means do that!
I hope you'll enjoy reading these articles and learn something from them. I hope these will inspire you to contribute the wiki yourself.
Cheers!
Mursalin
Note by Mursalin Habib
2 years, 1 month ago
Sort by:
Wow Mursalin! Your contributions are $$\color{Red}{\textbf{AWESOME!}}$$ · 2 years, 1 month ago
@megh choksi I agree that the comment made was extremely discouraging, and was out of line with the Brilliant community. I have since deleted it. I am been in communication with Mursalin, and am impressed with the quality of his contributions. We have featured several of his wikis, like the Induction and Strong Induction Wikis. I believe that the community looks forward to reading more of his articles.
Just in case, I would like to point out that even though the comment was from a "Calvin Lin", that is not my account. You can recognize staff accounts by the yellow STAFF tag that appears next to the name. Staff · 2 years ago
Sir I know you can't demotivate anyone · 2 years ago
Ah, I wanted to contribute to the FMID section of General Diophantine equations but instead ended up contributing in the wrong section. I contributed to the one in the quadratic diophantine equation section and am now realizing that I should have contributed to the one in the general diophantine equations section. Perhaps I will add more problems in the context of quadratic diophantine equations. · 2 years, 1 month ago
We will eventually combine these 2 wikis. They make sense to be presented on one page, as they essentially cover the same ideas, even though the types of equations are slightly different.
The quadratic diophantine equations page looks great, thanks! Staff · 2 years ago
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Annual percentage change in incidence
Bimi
New Member
Hello everyone
Could you please give an idea to calculate annual percentage change in incidence in stata??
Bimi
New Member
Thankyou for reply but I was seeking help to calculate annual percentage change in incidence for time series analysis
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# Question 2 If the random variable X has the following cumulative distribution function, find the cumulative...
###### Question:
Question 2 If the random variable X has the following cumulative distribution function, find the cumulative distribution function for Z vX. x < -1, x< 0, Fx(x) 1/3, 1,
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# usual tensor product of chain complexes of sheaves and flatness
Let $Sh(\mathcal{O}$ be the category of sheaves of ${\mathcal{O}}_X$-modules over a scheme $X$. Also let $Ch(R)$ be the category of chain complexes of (left) $R$-modules. We know that in both of $Ch(R)$ and $Ch(O)$(the category of chain complexes of sheaves of $\mathcal{O}_X$-modules), one can consider the usual tensor product of chain complexes and ${\rm Hom}$ functor. But it is known that the usual tensor product dose not characterize flatness as it dose in the category of modules. In paricula we have a complex$X$ with $X\bigotimes -$ exact and yet $X$ is not flat. Actually if we know only that $X^n$ is flat then $X\bigotimes -$ exact.
This motivated Enochs and Rozas to define a new tensor product of chain complexes which characterize flatness in $Ch(R)$ (Enochs and Rozas, Tensor Product of Chain Complexes, Math Jor of Okayam Univ, 1997).
Question: Can the usual tensor product of chain complexes of sheves characterize flatness in $Ch(\mathcal{O})$? If the answer is No, how can we define a new tensor product to deduce that $F$ is flat if and only if $F\bigotimes -$ is exact?
-
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# Darboux Theorem
Prove that the set $T = \{(x,y) \in I \times I : x is a connected subset of $\Re^2$ with the standard topology.
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# Tag Info
15
This is a temperamental difference more than a physical one, but I feel like this question deserves an answer with a lot less formalism than what Urs is using. The physical point that you should never lose sight of is that gauge symmetries are not symmetries at all: they don't map one state to another one, but instead identify a priori different states as ...
12
The mystery here should disappear once one realizes that the BRST complex -- being a dg-algebra -- is the formal dual to a space , namely to the "homotopically reduced" phase space. For ordinary algebras this is more familiar: the algebra of functions $\mathcal{O}(X)$ on some space $X$ is the "formal dual" to $X$, in that maps $f : X \to Y$ correspond to ...
11
Dear Student, as Moshe says, the reason why the Faddeev Popov ghosts "decouple" is that they're designed to decouple. More precisely, they - and all the formulae that depend on them - are designed so that the excitations of these ghosts, as well as unphysical excitations of more ordinary physical fields - such as the time-like and longitudinal components ...
11
Ghostly Lie algebra cohomology Let $\mathfrak{g}$ be our Lie algebra and $V_\rho$ a representation space with representation map $\rho : \mathfrak{g} \to \mathrm{End}(V_\rho)$. $V_\rho$ is, by the action through the representation, naturally a $\mathfrak{g}$-module (people missing the ring structure in $\mathfrak{g}$ - just embed it into the universal ...
8
You seem to be talking about the "old covariant quantization" in which $L_n$ for positive $n$ and $(L_0-a)$ annihilate physical ket states $|\psi\rangle$, right? It's analogous to the Gupta-Bleuler quantization http://en.wikipedia.org/wiki/Gupta-Bleuler_quantization which was a standard procedure used already in electromagnetism. The idea is that the ...
7
The main answer to the question is that the full generator $$L_n = L^{(\mathrm{m})}_n+L^{(\mathrm{g})}_n$$ in Bosonic string theory is a sum of a matter part with normal ordering constant $a=1$, $$L^{(\mathrm{m})}_n=\frac{1}{2} \eta_{\mu\nu}\sum_m:\alpha_{n-m}^{\mu}\alpha_m^{\nu}:-\hbar a\delta_n^0,$$ and a ghost part $$L^{(\mathrm{g})}_m=\sum_n(m-n):... 7 J.W. van Holten's "Aspects of BRST Quantization" arXiv:hep-th/0201124 might be what you're looking for... 6 I) First of all, note that although gauge theory and BRST formulation originally only referred to Yang-Mills theory (and hence QED), they nowadays apply to general theories with so-called local gauge symmetry, cf. e.g. this Phys.SE post. The Lagrangian and Hamiltonian BRST formalism are known as Batalin-Vilkovisky (BV) formalism and Batalin-Fradkin-... 5 As you already wrote, the (3/2)\partial^2 c term is needed for the current to be a one-form i.e. (1,0) tensor field; see also page 131 of Polchinski's String Theory, volume 1. This means that if you compute the OPE$$ T(z) j^{BRST}(0)\sim \dots, $$you want to get$$\dots \sim \frac{1}{z^2} j^{BRST}(0)+\frac{1}{z}\partial j^{BRST}(0), $$see e.g. ... 5 A modern treatment of this subject can be found in Segreev's book on the Kahler geometry of loop spaces also available online. This line of research started with the seminal work of Bowick and Rajeev: The holomorphic geometry of the closed bosonic string theory and Diff S^1/S^1 (Spires) (and independently Kirillov and Yuriev (Please see the reference in ... 5 This is an interesting question. And I know that Lubos has already written a nice and complete answer but I think I can add some of my own words here. I'll concentrate here on the non-abelian YM case, but the results here are quite general. The idea is that the BRST quantum action can be written as: S(qu) = S(cl) + \int\; d^4x\; s\Psi \end{... 5 I) Let us first clarify the left and right derivatives. Left derivatives is explained between eq. (15.8.9) and (15.8.10) in Ref. 1. A left derivative means a derivative that acts from the left. E.g. if F= \chi G, where G does not depend on \chi, then \frac{\delta_LF}{\delta\chi}=G. Similarly, a right derivative acts from the right. E.g. if F= G\chi... 5 Your mechanism of "one ghost falling in" is a different way of talking about the "evaporation of FP ghosts by a black hole". So do black holes evaporate ghosts? This is not a well-defined question because FP ghosts are unphysical, too. They're just a mathematical method to deal with gauge symmetries, in this case the diffeomorphism symmetry. In this sense, ... 4 I) The gauge-fixed pure Maxwell action is$$\tag{1} S[A,c,\bar{c}]~=~\int \! d^4x~ {\cal L} $$with Lagrangian density^1$$\tag{2} {\cal L}~=~{\cal L}_0 -\frac{\chi^2}{2\xi}-d_{\mu}\bar{c}~d^{\mu}c, \qquad {\cal L}_0~:=~-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}, \qquad \chi~:=~d_{\mu} A^{\mu}, \qquad \xi~>~0,$$consisting of (i) the Maxwell term, (ii) ... 4 The LSZ formula definitely applies to theories with massless gauge bosons, like QED and QCD. The S-matrix is given by the LSZ formula, which relates the former to correlation functions, which are in turn given by a path integral. The LSZ formula assumes asymptotic in- and out-states for particles of interest. In the path integral formalism, gauges can be ... 3 On the torus there are two real moduli \tau_1 and \tau_2, and two conformal Killing vectors corresponding to translations. This means that you need two insertions of b and two insertions of c in order to saturate the zero mode path integral. 3 The reason for the discrepancy is that the BRST operator is not self adjoint \Omega^{\dagger} \ne \Omega, This can easily be seen from its action on the ghosts:$$\delta c^{a} = \epsilon \{\Omega, c^a\} = i \epsilon f^{a}_{bc} c^b c^c\delta \bar{c}^{a} = \epsilon \{\Omega, \bar{c}^a\} = - i \epsilon b^a\delta b^{a} = \epsilon \{\Omega, b^a\} = ...
3
answer for questions $1$,$2$, and $3a$ 1) Looking at $2.7.22$ to $2.7.24$ (and also $2.7.18abc$), one define the ghost number $N^g = \frac{-1}{2\pi i} \int_0^{2 \pi}dz :b(z)c(z):$ , and that all the operators $c_n$ increase the ghost number by one. $[N^g,c_m] = c_m$, So the field $c(z)$, made of operators $c_m$, increase the ghost number by one unit.So $c^\... 3 "Something is conserved for an action" simply means that the action carries a zero overall value of "something" (for an additive quantity). In this case, the action has$N_{gh}=0$. It follows that the equations of motion derived from the action imply$dN_{gh}/dt=0$. Most typically, they imply$\partial_\mu J^\mu_{gh}=0$i.e. the local continuity law for a ... 3 I did not furnish all the details because it would be too long, but I give some hints at the end of the answer. I have used the formulae$:T^g: ~= ~:2(\partial c) b + c(\partial b):$and$:\frac{1}{2}cT^g: ~= ~:bc \partial c:$, when there is an ambiguity in the calculus. We begin by : $$j_B = cT^m+:\frac{1}{2}:cT^g:+\frac{3}{2}\partial^2c=cT^m+:bc\partial ... 3 The central charge counts the number of degrees of freedom only for matter fields living on a flat manifold (or supermanifold in the case of superstrings). An example where this counting argument fails for matter fields is the case of strings moving on a group manifold G whose central charge is given by the Gepner-Witten formula: c = \frac{k\mathrm{dim}(... 3 On one hand, by including the Lautrup-Nakanishi field B^a, we have an off-shell BRST formulation, i.e. we can prove the nilpotency of the BRST transformation without using the (Euler-Lagrange) equations of motion. On the other hand, for some applications, a simpler on-shell BRST formulation (where the Lautrup-Nakanishi field B^a has been integrated out/... 3 I) Since total divergence terms do not contribute to Euler-Lagrange (EL) equations, cf. e.g. this Phys.SE post, one could just integrate the Faddeev-Popov \bar{c}c term by part so that there are no more than first derivatives present and the standard form of the EL equations applies. II) Alternatively, in the presence of higher derivatives, the EL ... 3 Since R[A] is gauge invariant, the variation of R[A] is zero when A^a_{\mu} undergoes the infinitesimal gauge transformation A^a_{\mu}\rightarrow A^a_{\mu} + \epsilon (D_{\mu}\alpha)^a where \alpha^a is any Lie algebra valued field and \epsilon is an infinitesimal parameter. The variation of R[A] under this gauge transformation is$$0=\delta R ... 3 Caveat: The first part of this answer takes a very technical stance on the BRST procedure, and additionally works with a finite-dimensional phase space for convenience. It could appear quite far from the understanding of ghosts in the average application of BRST transformations or ghosts as a tool. The general conception of ghosts There are many ... 2 I thought at first this question was more general involving the nature of the Virasoro algebra. As a result the first two paragraphs are boiler plate discussions on that. The actual question is addressed in the third paragraph. The vector fields$T^m~=~z^{m+1}\partial z$satisfy the Witt algebra or Virasoro algebra without central extension for each index ... 2 Kugo and Ojima's work was one of the major breakthroughs in understanding the role of BRST in the quantization of gauge theories. Historically BRST was discovered in the path integral formalism. The understanding of this theory as a cohomology theory started from the Kugo and Ojima's work. Now, the action is BRST invariant with and without the Gaussian ... 2 I got it.$\epsilon$anticommutes with$b_A$. 2 In your case, I think that$\eta$is playing the role of$\epsilon$in David Bar Moshe 's answer, that is :$\eta\$ is a real Lorentz-scalar Grassmann variable. So, you will have : $$\delta_\Omega \bar\psi= \overline{\delta_\Omega\psi} = \overline {i\eta\psi} = -i \eta \bar\psi = i \bar\psi \eta$$
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Bosonic path integrals : $$Z = \int D\phi ~e^{-i \large \int ~ dx [\frac{1}{2}\phi (\square+m^2)\phi]}$$ or Femionic path integrals (like Fadeev-Popov ghosts) : $$Z = \int D\eta D \tilde \eta ~e^{-i \large \int ~ dx [\tilde \eta^a \square \eta^a]}$$ are not mathematically well-defined, because of the presence of the imaginary unit in the exponential. ...
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# Inducing homomorphisms on localizations of rings/modules
I'm trying to work out Exercise 2.6 in Commutative Algebra by Eisenbud, which asks to prove the Chinese Remainder Theorem for commutative rings.
Exercise: Let $R$ be a commutative ring, and let $I_1,\ldots,I_d$ be pairwise comaximal ideals. Prove that $R/\left(\bigcap_{k=1}^d I_k\right) \simeq \prod_{k=1}^d R/I_k$, via the map $\varphi: R\rightarrow \prod R/I_k$ given by $r \mapsto (r+I_1,\ldots,r+I_d)$.
The kernel of $\varphi$ is precisely $\bigcap_{k=1}^d I_k$, so the goal of the argument is to show that $\varphi$ is surjective. Eisenbud hints to use his Corollary 2.9, which states the following.
Lemma: Let $\varphi:M\rightarrow N$ be an $R$-module homomorphism. Then $\varphi$ is surjective if and only if the induced map $\varphi_{\mathfrak{m}} : M_{\mathfrak{m}} \rightarrow N_\mathfrak{m}$ is surjective for every maximal ideal $\mathfrak{m}$ of $R$.
(Specifically, the map $\varphi_\mathfrak{m}$ is given by $m/u \mapsto \varphi(m)/u$.)
I have a few problems with this.
1. The lemma is a statement about module homomorphisms, but the $\varphi$ appearing in the Chinese Remainder Theorem is not a module homomorphism (it's a ring homomorphism). I tried to change the lemma to work for ring homomorphisms, but ran into the following problem.
2. Even if $\varphi:R\rightarrow S$ is a ring homomorphism, how does $\varphi_\mathfrak{m} : R_\mathfrak{m} \rightarrow S_\mathfrak{m}$ make sense? I get that you can localize $R$ at a maximal ideals, but what does "$S_\mathfrak{m}$" mean when $\mathfrak{m}$ is a maximal ideal of $R$? (Note this post, which actually gives the failure of the lemma for ring homomorphisms -- except in that context we localize $S$ at a prime ideal of $S$, and then localize $R$ at the preimage of that prime ideal.)
Regarding the second remark above, I was thinking that one could view $S$ as an $R$-module by $rs:=\varphi(r)s$, and then consider $S_\mathfrak{m}$ as a localization of an $R$-module by a maximal ideal. But then in order for $\varphi_\mathfrak{m}:R_\mathfrak{m}\rightarrow S_\mathfrak{m}$ to be a ring homomorphism, one has to decide how $S_\mathfrak{m}$ is a ring and then define how to induce a ring homomorphism on localization. This seems like a lot for Eisenbud to sweep under the rug when he says to use the Lemma to solve the exercise. So as far as I understand, the above lemma isn't even relevant to the exercise.
Am I missing something here? Is there a clever way to indirectly apply the lemma?
• $\varphi$ is a homomorphism of $R$-modules. – Zev Chonoles Feb 20 '14 at 5:09
• Oh ... right. Thanks. – Ehsaan Feb 20 '14 at 5:18
Note that the ring $R$, and each of the rings $R/I_k$, is also an $R$-module. It is easy to check that the map $\varphi$ is an $R$-module homomorphism.
By the definition of "comaximal", any given maximal ideal $M\subset R$ will contain at most one $I_k$.
For any given maximal ideal $M\subset R$ and any $I_k$, we have $(R/I_k)_M\cong R_M/(I_k)_M$ because localization commutes with taking quotients. If $M$ contains $I_k$, then $(I_k)_M\neq R_M$, whereas if $M$ does not contain $I_k$, we have $(I_k)_M=R_M$ because $I_k\cap (R\setminus M)\neq\varnothing$.
In either case, we see that the localized map $$\small\varphi_M:R_M\to \left(\prod R/I_k\right)_M\cong \prod R_M/(I_k)_M\cong \begin{cases} R_M/(I_k)_M &\text{if I_k is contained in M,}\\ 0 & \text{if none of the I_k's are contained in M} \end{cases}$$ is just a quotient map from the ring $R_M$, hence surjective.
(I leave it to you to check that the composition of $\varphi_M$ with all of these isomorphisms actually does result in the quotient map $R_M\to R_M/(I_k)_M$.)
P.S. This is actually the first time I've seen this approach to proving the map in the Chinese remainder theorem is surjective. I like it a lot, though there are proofs which don't need any localization. Consider the approach taken in Atiyah-Macdonald:
• Thanks! Atiyah-McDonald's proof has the advantage of working for noncommutative rings too, right? I've seen that one, but Eisenbud's exercise looked pretty slick. – Ehsaan Feb 20 '14 at 18:08
• Could you explain why any given maximal ideal M⊂R will contain at most one Ik.? – Bob Feb 12 '17 at 0:20
• Bob: Since the $I_k$ are comaximal, we have $I_k+I_j=R$ for $k \neq j$. If $I_k, I_j \subset M$ for $i \neq j$, then $R \subset M$, which contradicts the definition of maximal ideal. – Massimo Feb 12 '17 at 17:58
• If $M$ does not contain $I_k$, we have $(I_k)_M=R_M$ because $I_k∩(R∖M)\neq∅$. Why does this reason imply $(I_k)_M=R_M$ – Bob Feb 13 '17 at 14:03
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# Week 12 - Random Algorithms
So normally when we talk about algorithms, we can think about them as a deterministic black box: we put input of some kind into the box, maybe a graph, or a list of integers, or the coordinates of points on a plane, and we get back an answer based on that input: a traversal tree of the graph, or the sorted order of the integers, or the convex hull of the points. For algorithms like this, with a given input we're guaranteed that 1) the output of the algorithm will always be the same (and correct, if the algorithm is correct) and 2) the time taken to run the algorithm, the number of computational steps, will always be the same. Of course, the output and run time will change with different inputs --- ask an algorithm to sort a list of 10 elements and it'll run faster than when asked to sort a list of 10,000,000 --- but if given the same input, we expect the same answer and the same runtime.
What we're going to talk about today are a separate class of algorithms, called randomized algorithms, that take, in addition to the input data, non-deterministic, random numbers, and in which one or both of those guarantees are lost. So if we lose guaranteed output (and sometimes even guaranteed correctness!) or guaranteed run time, and we need to supply random numbers as well, why would we want to use randomized algorithms at all?
Well, there are two upsides: 1) Often randomized algorithms are conceptually simpler, and thus simpler to implement, than deterministic algorithms 2) In exchange for losing guarantees about worst-case performance, well-designed randomized algorithms can give much improved expected performance.
#### Generating Random Numbers
Before I go into more detail about algorithms, let's talk a bit about the extra input into randomized algorithms --- random numbers. When I talk about random numbers today, I mean uniform deviates - numbers that lie within a specified range, with any one number in the range just as likely as any other and there are no correlations between successive random numbers. Where can we get random numbers from?
The first option is to gather random numbers from inherently unpredictable physical sources. Some examples of this strategy include coin flipping, dice rolling, radioactive decay, microwave background radiation, etc. This provides (if the physical source is well chosen) truly random numbers, but has the downside that generation of random numbers is relatively costly and slow --- you can't provoke radioactive decay on demand, you just have to wait, and rolling dice takes time and effort as well.
Because of this, nearly always when we use random numbers in computer algorithms, what we're really using are pseudo-random numbers --- generated not by a physical process, but rather by a deterministic "random number generator". I won't go into detail about these other than to give a couple of warnings:
1. You should never, literally never try to design your own random number generator or modify another implementation. E.g., don't try to do clever things to make a generator "more random" like swap bits around, or combine multiple calls, etc.
2. You should think long and hard --- very hard --- before trying to implement your own random number generator from a known (and tested!) design.
3. You should almost always use the built-in random generator provided by your language, and when you do, every time you need a new random number, you need to make a new call to the generator. One number, one call.
The reason for these warnings is that random number generators are very, very easy to get very, very wrong. Errors in your random number generator can lead to serious problems in your algorithms! A couple of potential pitfalls: random number generators have periods --- they repeat after some number of calls, and poor implementation can lead to a short period. Gathering 10,000,000 "random" numbers from a generator with a period of 32,000 is obviously a problem. Poor choice of seeds --- the initial start to a random number generator --- can lead to non-random behavior. A common, but particularly bad, option is to seed with some property of the computer. Usually these seeds are not nearly diverse enough, leading to the same problem.
#### Case Study in Bad RNG
Back in the late 90s, when online poker started getting popular, there was a particular poker site that was trying to build its userbase. To convince users that its shuffles were fair, it publicly published its shuffling algorithm:
• start with an ordered deck of cards
• initialize a RNG with a seed taken from the system clock (a 32-bit integer)
• for each position in the deck, swap the card in that position with a random card from the deck
There are a couple of problems with this: first, the algorithm doesn't produce all shuffled decks with equal probability. The correct algorithm for that is to swap each card randomly with a card at a position greater than or equal to its original position, rather than with any card in the deck. (Take some time and prove that this is the case!) More troubling, though, was the random number generator. The period on the RNG was on the order of $2^{32}$, or a billion. But the number of total shuffled decks is much larger, $52!$, meaning that many decks will simply never occur as no seed will result in that particular shuffle. This isn't great, but a billion possible shuffled decks is still more than we probably need. But the implementation used seeded the RNG with a system call --- the number of milliseconds since midnight on the server clock. There are only a limited number of milliseconds in a day, so the total number of seeds was reduced to only 86 million. But wait --- this is a server clock. If we know where the server is physically located, then we can make a pretty good guess about what the server clock time is, and so we can reduce the number of seeds down to a few hundred thousand (if we can get the server clock time to within a few minutes). This is small enough to do a complete search. So once we're dealt our hand, we can brute force all possible RNG seeds, pick those that would give us the hand that we see, and then we know the complete shuffle of the deck, including all of our opponents' hands.
The lesson is: think very carefully about the limitations of the random number generator you're using.
#### Las Vegas vs. Monte Carlo
Randomized algorithms are usually classified based on which of those guarantees they choose to give up. Monte Carlo algorithms give up guaranteed correctness: they have a fixed running time, but might get the wrong answer. They usually work by randomly generating a fixed number of possible answers and then report the best one. Las Vegas algorithms give up guaranteed running time: they randomly generate possible answers until they get the correct answer. Note that if you have a way to test for an answer's correctness, converting a Monte Carlo algorithm to a Las Vegas one is trivial --- just keep generating answers until one tests as correct.
Let's talk about some times you might think about using randomness and go over some example algorithms.
#### Lots of Good-Enough Answers
A very common situation is a problem where we're asked not for the best answer, but rather just for an answer that satisfies the constraints. Take, for example, a problem where you're given a list of 1,000,000 positive integers less than 1,000 and asked to find a pair of identical integers. There are $10^{12}$ possible pairs of integers, so we can't possibly test them all. But how common are pairs of identical integers? Obviously it will depend on what the numbers in the list actually are. We might have a list full of identical numbers, say, where they're all 10. In that case, every possible pair will be an identical pair. On the other hand, we might have a list with every number equally represented, where there are 1,000 copies of each number. In that case, each random pair will have a 1 in 1000 chance of being identical, which is the worst case.
#### Avoiding Bad Input
Another common use for randomization is in avoiding worst-case input. Let's look at quicksort. This sort works by choosing a pivot from the array and adjusting the array so that all elements less than the pivot are at lower indices than the pivot, and all those greater than the pivot are at higher indices. We then recursively sort on the two separate upper and lower sections. On average, this runs in $O(n lg n)$ time, but in the worst case, it can run in the much slower $O(n^2)$ time. If, for example, we choose the pivot as the first element in the array, then when given an already-sorted array to sort, each pivot won't actually divide the problem into two smaller pieces, giving us very poor performance. For many problems, worst-case inputs are not particularly likely. For programming challenges, though, you're almost guaranteed to see a worst-case input. So how can we avoid this problem? Well, rather than simply choosing the pivot as the first element, randomly pick the pivot from all the elements in the array. It's still possible to get a poor performance, but there's no specific input that could be provided to guarantee bad performance. Any time you have an algorithm that performs well in general but poorly on specific kinds of input, think about how randomization might help you avoid the problem.
#### Simplifying an Implementation
Let's look at the minimum-cut problem in a graph. Now, we didn't manage to get to this problem in the graph theory lecture, so let me describe the problem. Let's take some graph $G$ with $E$ edges and $V$ vertices. A cut of the graph is a division of the vertices of the graph into two subsets, and the size of the cut is the number of edges between the distinct subsets. So when we ask for a minimum cut, we want to find a division of the graph so that there's a minimum number of edges between the two divisions. There exist deterministic algorithms for this problem, but they're complicated enough that I won't cover them today. Instead let's think about randomized algorithms.
One way we might think of is to generate random subsets of vertices and measure the size of the cut. Suppose that there's exactly one minimum cut of size $k$. There are $2^{V-1} - 1$ total possible cuts, and only one of them is correct, so we would have to generate a very large number of random subsets in order to get a good chance of finding the minimum --- large enough that it would probably be faster to just loop through all the subsets (or, of course, use one of the faster deterministic algorithms). But there's another way we can generate cuts using a method called graph contraction.
In this method, rather than generating an initial random cut, we repeatedly choose random edges and contract them, replacing the nodes connected by that edge with a supernode, so that all edges to the two nodes go instead to the supernode, removing any self-loops, but keeping multiedges. We repeat this procedure until we have only two supernodes remaining. Each of these represents a subset of the vertices, and the number of edges between them is the size of the cut they represent. How likely is it that we'll be able to find the minimum cut?
Well, we have $k$ edges of the minimum cut. In order to construct this cut using this contraction algorithm, none of the edges chosen to contract must be one of the $k$ edges present in the minimum cut. The first edge chosen to contract will avoid the minimum cut with probability $1 - k / E$. The minimum degree of the graph must be at least $k$, so there must be at least $Vk / 2$ edges, giving the probability of avoiding the minimum cut on the first contraction as at least $1 - 2 / V$. After contraction, the same solution recurs on a graph with one fewer vertex, so the probability of avoiding the minimum cut after fully contracting the graph is:
$p \geq \prod_{i=0}^{V-3} (1 - \frac{2}{V - i}$,
which simplifies down to $V \choose 2$. So with a polynomial, rather than an exponential, number of random choices, we can get a very high probability of the correct answer!
#### Lab Section
This week's homework problem is Google Code Jam Round 1B 2012 - Equal Sums.
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Wavelet Series Representation and Geometric Properties of Harmonizable Fractional Stable Sheets
11 Mar 2019 · Ayache Antoine, Shieh Narn-Rueih, Xiao Yimin ·
Let $Z^H= \{Z^H(t), t \in \R^N\}$ be a real-valued $N$-parameter harmonizable fractional stable sheet with index $H = (H_1, \ldots, H_N) \in (0, 1)^N$. We establish a random wavelet series expansion for $Z^H$ which is almost surely convergent in all the H\"older spaces $C^\gamma ([-M,M]^N)$, where $M>0$ and $\gamma\in (0, \min\{H_1,\ldots, H_N\})$ are arbitrary... One of the main ingredients for proving the latter result is the LePage representation for a rotationally invariant stable random measure. Also, let $X=\{X(t), t \in \R^N\}$ be an $\R^d$-valued harmonizable fractional stable sheet whose components are independent copies of $Z^H$. By making essential use of the regularity of its local times, we prove that, on an event of positive probability, the formula for the Hausdorff dimension of the inverse image $X^{-1}(F)$ holds for all Borel sets $F \subseteq \R^d$. This is referred to as a uniform Hausdorff dimension result for the inverse images. read more
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Can a single card have multiple activations on a stack? The anion, cation, or radical is stabilized by declocaliztion. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Alkenes, having the weakest pi bonds, are easiest to reduce. q In contrast, the conjugate base of an alcohol, an alkoxide anion, is not resonance stabilized at all, i.e., the negative charge is fully localized upon the oxygen atom. So borane preferentially forms the complex with the more basic carbonyl oxygen of the carboxylic acid. 2. (We previously encountered this same idea when considering the relative acidity and basicity of phenols and aromatic amines in section 7.4). ), Virtual Textbook of Organic Chemistry, Prof. Steven Farmer (Sonoma State University). The answer lies in $$\Delta G$$; whatever the electrostatic effects are doing to balance between $$\Delta H$$ and $$\Delta S$$, it is $$\Delta G$$ that determines the equilibrium constant and $$\Delta G$$ quite consistently follows the predictions of simple electrostatic considerations. How can I better handle 'bad-news' talks about people I don't care about? For example, the solvolysis of the 1-methylheptyl sulfonate, $$5$$, in dilute water solution proceeds 70 times slower when sufficient sodium dodecanyl sulfate $$\left( \overset{\oplus}{\ce{Na}} \overset{\ominus}{\ce{O}} \ce{SO_3C_{12}H_{25}} \right)$$ is added to provide about twice as many dodecanyl sulfate ions in the micelle state as there are molecules of $$5$$ present: This slowing of the solvolysis reaction by the alkyl sulfate requires that $$5$$ be almost completely imprisoned by the micelles, because that part of $$5$$ free in water would hydrolyze rapidly. 2) Conjugated double bonds. q Rather, borane is electrophilic, since the boron has a vacant 2p AO.
This reaction converts the neutral, electrophilic borane to a negatively charged hydride reagent, in which the hydrogens are now quite nucleophilic! 3. The acidity of the carboxyl group arises, at least in part, from the polar nature of the carbonyl group, the polarity of which can be ascribed to contributions of the structure. The electrophilic carbonyl carbon of an aldehyde or ketone makes it reactive toward metal hydrides, which contain nucleophilic hydrogen. q Higher molecular weight carboxylic acids, tend to have less water solubility because of the dominance of the non-polar part of the molecule, which is “hydrophobic”. They react with alkyl halides to form ester.
Carboxylic acids easily dissociate into a carboxylate anion and a positively charged hydrogen ion (proton), much more readily than alcohols do (into an alkoxide ion and a proton), because the carboxylate ion is stabilized by resonance. How can a chess game with clock take 5 hours? Why and how carboxylate ion has greater stability than carboxylic acid?
Is it safe to have two separate circuits hooked to the same outlet? Carboxylic acids have a carbonyl pi bond and, further, they have additional resonance stabilization. The result is strong resonance stabilization of the carboxyl group via the interaction between these two groups. First of all, it would have been clearer if it's a misprint or not if you had attached a screenshot of the wording in your textbook (you still can). q If the alcohol is not such that a large excess can be used, the equilibrium can be driven to the right by the removal of water (the LeChatelier principle) via distillation or the use of a dehydrating agent.
When two or more structures that differ only in the positions of valence electrons can be drawn for a molecule or ion, it means that its valence electrons are delocalized, or spread over more than two atoms. Addition/elimination is a typical route for substitution processes in carboxylic acids, esters, amides, and other analgous acyl compounds in which carbon has the +3 oxidation state. All rights reserved. These are the carbonyl oxygen, the carbonyl carbon, and the alcohol type oxygen. And this is really important for ions because in this case, I can share out that negative charge over two different oxygen atoms.
Since we have two carboxylate sites in the oxalate anion, there are four total possible resonance structures as shown below. Solvent polarity is often measured by the ability to dissolve salts or to provide solvation stabilization for ions. Only one structure can be drawn for an alkoxide ion, but two structures can be drawn for a carboxylate ion.
Unlike the reduction of ester, the reduction of carboxylate is different, due to the lack of the leaving group and the relatively electron-rich carbon atom (due to the negative charge on the oxygen atoms). © copyright 2003-2020 Study.com. You can't really compare ions to molecules in terms of "stability". The sequence of ease of reducibility is alkenes > aldehydes, ketones>carboxylic acids, aromatics.
q In the next chapter we will see that the reverse reaction, the conversion of an ester to a carboxylic acid can be carried out in either acidic or basic solution. To learn more, see our tips on writing great answers. c. Decarboxylation - these are reactions in which the $$\ce{R-C}$$ bond is broken in such a way that $$\ce{CO_2}$$ is lost and $$\ce{R-H}$$ is formed. q The simplest carboxylic acid is methanoic acid, the next ethanoic acid, etc. The activated hydride is then able to react very efficiently at this activated electrophilic center. Both of these have already been discussed rather extensively. The carboxyl carbon is automatically numbered as the. In the carboxylate anion the two contributing structures have equal weight in the hybrid, and the C–O bonds are of equal length (between a double and a single bond).
Please remember,however, that, unlike aldehydes, ketones are not easy to oxidize to carboxylic acids. This is because the experimental result is that the acidity of the carboxylic acid is actually increased. Methanoic acid and almost all the substituted ethanoic acids are stronger than ethanoic acid. When carboxylate salts are put into nonpolar solvents, reversed micelles often are formed, where the polar parts of the molecules are on the inside and the nonpolar parts are on the outside.
The inductive effect of the substituent makes the acid stronger or weaker (relative to the unsubstituted acid), depending on whether the substituent is electron-attracting or electron-donating relative to hydrogen. Watch the recordings here on Youtube!
1. Simultaneously, it converts the carboxylic acid to its (Lewis) conjugate acid (a postively charged, oxonium ion), which (like the corresponding Bronsted conjugate acid) has much more carbocation character at the carbonyl carbon. The stabilization is substantial and carboxylic acids are more stable than would be expected, from summing up their bond energies, by fully $$18 \: \text{kcal mol}^{-1}$$. Alkenes, having the weakest pi bonds, are easiest to reduce. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Essentially an acid-catalyzed addition of an alcohol to the carbonyl group of the carboxylic acid. Best approach to safely bump up version of classes. What does the December 8th deadline mean for the certification of the results of the Electoral College? How can I seal a gap between floor joist boxes and foundation?
The conversion of a carboxylic acid to an ester therefore does not involve either oxidation or reduction. Primarily the resonance stabilization of the conjugate base of a carboxylic acid, i.e., the carboxylate anion. basic sites. In particular, the other structures have charge separation, which is an energy-increasing factor. 4.
, i.e., the negative charge is fully localized upon the oxygen atom. In this molecule there is no carboxylate anion, no dioxallyl system and no $\ce{\Psi_2}$ with a node at $\ce{C_2}$. The substitution reaction does not, however, occur via an S, to the carbonyl group in a manner analogous to the addition of an alcohol to an aldehyde or ketone carbonyl group (hemiacetal formation), followed by an. When one uses hydroxide anion as a base catalyst (or whatever other basic catalyst is used), the preferred reaction is not addition to the carbonyl group, but deprotonation of the carboxyl group to give the carboxylate conjugate base, which is highly unreactive because of both its resonance stabilization and its negative charge. Essentially, the conjugate acid of a ketone has partial positive charge on just two atoms, the carbonyl oxygen and the carbonyl carbon, while the conjugate acid of the carboxylic acid has the postive charge delocalized on three atoms. they are weak acids.
The stem consists of the name of the alkane containing the same number of carbon atoms, except that the terminal –e of the alkane is dropped (e.g., methane becomes methan-). If conditions are controlled carefully, the aldehyde can be isolated in good yield. In the case of the carboxylic acid, the resonance structures are non-equivalent. (see below), while the alcohol oxygen-protonated conjugate acid is not.
The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot.
Can a single card have multiple activations on a stack? The anion, cation, or radical is stabilized by declocaliztion. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Alkenes, having the weakest pi bonds, are easiest to reduce. q In contrast, the conjugate base of an alcohol, an alkoxide anion, is not resonance stabilized at all, i.e., the negative charge is fully localized upon the oxygen atom. So borane preferentially forms the complex with the more basic carbonyl oxygen of the carboxylic acid. 2. (We previously encountered this same idea when considering the relative acidity and basicity of phenols and aromatic amines in section 7.4). ), Virtual Textbook of Organic Chemistry, Prof. Steven Farmer (Sonoma State University). The answer lies in $$\Delta G$$; whatever the electrostatic effects are doing to balance between $$\Delta H$$ and $$\Delta S$$, it is $$\Delta G$$ that determines the equilibrium constant and $$\Delta G$$ quite consistently follows the predictions of simple electrostatic considerations. How can I better handle 'bad-news' talks about people I don't care about? For example, the solvolysis of the 1-methylheptyl sulfonate, $$5$$, in dilute water solution proceeds 70 times slower when sufficient sodium dodecanyl sulfate $$\left( \overset{\oplus}{\ce{Na}} \overset{\ominus}{\ce{O}} \ce{SO_3C_{12}H_{25}} \right)$$ is added to provide about twice as many dodecanyl sulfate ions in the micelle state as there are molecules of $$5$$ present: This slowing of the solvolysis reaction by the alkyl sulfate requires that $$5$$ be almost completely imprisoned by the micelles, because that part of $$5$$ free in water would hydrolyze rapidly. 2) Conjugated double bonds. q Rather, borane is electrophilic, since the boron has a vacant 2p AO.
This reaction converts the neutral, electrophilic borane to a negatively charged hydride reagent, in which the hydrogens are now quite nucleophilic! 3. The acidity of the carboxyl group arises, at least in part, from the polar nature of the carbonyl group, the polarity of which can be ascribed to contributions of the structure. The electrophilic carbonyl carbon of an aldehyde or ketone makes it reactive toward metal hydrides, which contain nucleophilic hydrogen. q Higher molecular weight carboxylic acids, tend to have less water solubility because of the dominance of the non-polar part of the molecule, which is “hydrophobic”. They react with alkyl halides to form ester.
Carboxylic acids easily dissociate into a carboxylate anion and a positively charged hydrogen ion (proton), much more readily than alcohols do (into an alkoxide ion and a proton), because the carboxylate ion is stabilized by resonance. How can a chess game with clock take 5 hours? Why and how carboxylate ion has greater stability than carboxylic acid?
Is it safe to have two separate circuits hooked to the same outlet? Carboxylic acids have a carbonyl pi bond and, further, they have additional resonance stabilization. The result is strong resonance stabilization of the carboxyl group via the interaction between these two groups. First of all, it would have been clearer if it's a misprint or not if you had attached a screenshot of the wording in your textbook (you still can). q If the alcohol is not such that a large excess can be used, the equilibrium can be driven to the right by the removal of water (the LeChatelier principle) via distillation or the use of a dehydrating agent.
When two or more structures that differ only in the positions of valence electrons can be drawn for a molecule or ion, it means that its valence electrons are delocalized, or spread over more than two atoms. Addition/elimination is a typical route for substitution processes in carboxylic acids, esters, amides, and other analgous acyl compounds in which carbon has the +3 oxidation state. All rights reserved. These are the carbonyl oxygen, the carbonyl carbon, and the alcohol type oxygen. And this is really important for ions because in this case, I can share out that negative charge over two different oxygen atoms.
Since we have two carboxylate sites in the oxalate anion, there are four total possible resonance structures as shown below. Solvent polarity is often measured by the ability to dissolve salts or to provide solvation stabilization for ions. Only one structure can be drawn for an alkoxide ion, but two structures can be drawn for a carboxylate ion.
Unlike the reduction of ester, the reduction of carboxylate is different, due to the lack of the leaving group and the relatively electron-rich carbon atom (due to the negative charge on the oxygen atoms). © copyright 2003-2020 Study.com. You can't really compare ions to molecules in terms of "stability". The sequence of ease of reducibility is alkenes > aldehydes, ketones>carboxylic acids, aromatics.
q In the next chapter we will see that the reverse reaction, the conversion of an ester to a carboxylic acid can be carried out in either acidic or basic solution. To learn more, see our tips on writing great answers. c. Decarboxylation - these are reactions in which the $$\ce{R-C}$$ bond is broken in such a way that $$\ce{CO_2}$$ is lost and $$\ce{R-H}$$ is formed. q The simplest carboxylic acid is methanoic acid, the next ethanoic acid, etc. The activated hydride is then able to react very efficiently at this activated electrophilic center. Both of these have already been discussed rather extensively. The carboxyl carbon is automatically numbered as the. In the carboxylate anion the two contributing structures have equal weight in the hybrid, and the C–O bonds are of equal length (between a double and a single bond).
Please remember,however, that, unlike aldehydes, ketones are not easy to oxidize to carboxylic acids. This is because the experimental result is that the acidity of the carboxylic acid is actually increased. Methanoic acid and almost all the substituted ethanoic acids are stronger than ethanoic acid. When carboxylate salts are put into nonpolar solvents, reversed micelles often are formed, where the polar parts of the molecules are on the inside and the nonpolar parts are on the outside.
The inductive effect of the substituent makes the acid stronger or weaker (relative to the unsubstituted acid), depending on whether the substituent is electron-attracting or electron-donating relative to hydrogen. Watch the recordings here on Youtube!
1. Simultaneously, it converts the carboxylic acid to its (Lewis) conjugate acid (a postively charged, oxonium ion), which (like the corresponding Bronsted conjugate acid) has much more carbocation character at the carbonyl carbon. The stabilization is substantial and carboxylic acids are more stable than would be expected, from summing up their bond energies, by fully $$18 \: \text{kcal mol}^{-1}$$. Alkenes, having the weakest pi bonds, are easiest to reduce. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Essentially an acid-catalyzed addition of an alcohol to the carbonyl group of the carboxylic acid. Best approach to safely bump up version of classes. What does the December 8th deadline mean for the certification of the results of the Electoral College? How can I seal a gap between floor joist boxes and foundation?
The conversion of a carboxylic acid to an ester therefore does not involve either oxidation or reduction. Primarily the resonance stabilization of the conjugate base of a carboxylic acid, i.e., the carboxylate anion. basic sites. In particular, the other structures have charge separation, which is an energy-increasing factor. 4.
, i.e., the negative charge is fully localized upon the oxygen atom. In this molecule there is no carboxylate anion, no dioxallyl system and no $\ce{\Psi_2}$ with a node at $\ce{C_2}$. The substitution reaction does not, however, occur via an S, to the carbonyl group in a manner analogous to the addition of an alcohol to an aldehyde or ketone carbonyl group (hemiacetal formation), followed by an. When one uses hydroxide anion as a base catalyst (or whatever other basic catalyst is used), the preferred reaction is not addition to the carbonyl group, but deprotonation of the carboxyl group to give the carboxylate conjugate base, which is highly unreactive because of both its resonance stabilization and its negative charge. Essentially, the conjugate acid of a ketone has partial positive charge on just two atoms, the carbonyl oxygen and the carbonyl carbon, while the conjugate acid of the carboxylic acid has the postive charge delocalized on three atoms. they are weak acids.
The stem consists of the name of the alkane containing the same number of carbon atoms, except that the terminal –e of the alkane is dropped (e.g., methane becomes methan-). If conditions are controlled carefully, the aldehyde can be isolated in good yield. In the case of the carboxylic acid, the resonance structures are non-equivalent. (see below), while the alcohol oxygen-protonated conjugate acid is not.
The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot.
Can a single card have multiple activations on a stack? The anion, cation, or radical is stabilized by declocaliztion. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Alkenes, having the weakest pi bonds, are easiest to reduce. q In contrast, the conjugate base of an alcohol, an alkoxide anion, is not resonance stabilized at all, i.e., the negative charge is fully localized upon the oxygen atom. So borane preferentially forms the complex with the more basic carbonyl oxygen of the carboxylic acid. 2. (We previously encountered this same idea when considering the relative acidity and basicity of phenols and aromatic amines in section 7.4). ), Virtual Textbook of Organic Chemistry, Prof. Steven Farmer (Sonoma State University). The answer lies in $$\Delta G$$; whatever the electrostatic effects are doing to balance between $$\Delta H$$ and $$\Delta S$$, it is $$\Delta G$$ that determines the equilibrium constant and $$\Delta G$$ quite consistently follows the predictions of simple electrostatic considerations. How can I better handle 'bad-news' talks about people I don't care about? For example, the solvolysis of the 1-methylheptyl sulfonate, $$5$$, in dilute water solution proceeds 70 times slower when sufficient sodium dodecanyl sulfate $$\left( \overset{\oplus}{\ce{Na}} \overset{\ominus}{\ce{O}} \ce{SO_3C_{12}H_{25}} \right)$$ is added to provide about twice as many dodecanyl sulfate ions in the micelle state as there are molecules of $$5$$ present: This slowing of the solvolysis reaction by the alkyl sulfate requires that $$5$$ be almost completely imprisoned by the micelles, because that part of $$5$$ free in water would hydrolyze rapidly. 2) Conjugated double bonds. q Rather, borane is electrophilic, since the boron has a vacant 2p AO.
This reaction converts the neutral, electrophilic borane to a negatively charged hydride reagent, in which the hydrogens are now quite nucleophilic! 3. The acidity of the carboxyl group arises, at least in part, from the polar nature of the carbonyl group, the polarity of which can be ascribed to contributions of the structure. The electrophilic carbonyl carbon of an aldehyde or ketone makes it reactive toward metal hydrides, which contain nucleophilic hydrogen. q Higher molecular weight carboxylic acids, tend to have less water solubility because of the dominance of the non-polar part of the molecule, which is “hydrophobic”. They react with alkyl halides to form ester.
Carboxylic acids easily dissociate into a carboxylate anion and a positively charged hydrogen ion (proton), much more readily than alcohols do (into an alkoxide ion and a proton), because the carboxylate ion is stabilized by resonance. How can a chess game with clock take 5 hours? Why and how carboxylate ion has greater stability than carboxylic acid?
Is it safe to have two separate circuits hooked to the same outlet? Carboxylic acids have a carbonyl pi bond and, further, they have additional resonance stabilization. The result is strong resonance stabilization of the carboxyl group via the interaction between these two groups. First of all, it would have been clearer if it's a misprint or not if you had attached a screenshot of the wording in your textbook (you still can). q If the alcohol is not such that a large excess can be used, the equilibrium can be driven to the right by the removal of water (the LeChatelier principle) via distillation or the use of a dehydrating agent.
When two or more structures that differ only in the positions of valence electrons can be drawn for a molecule or ion, it means that its valence electrons are delocalized, or spread over more than two atoms. Addition/elimination is a typical route for substitution processes in carboxylic acids, esters, amides, and other analgous acyl compounds in which carbon has the +3 oxidation state. All rights reserved. These are the carbonyl oxygen, the carbonyl carbon, and the alcohol type oxygen. And this is really important for ions because in this case, I can share out that negative charge over two different oxygen atoms.
Since we have two carboxylate sites in the oxalate anion, there are four total possible resonance structures as shown below. Solvent polarity is often measured by the ability to dissolve salts or to provide solvation stabilization for ions. Only one structure can be drawn for an alkoxide ion, but two structures can be drawn for a carboxylate ion.
Unlike the reduction of ester, the reduction of carboxylate is different, due to the lack of the leaving group and the relatively electron-rich carbon atom (due to the negative charge on the oxygen atoms). © copyright 2003-2020 Study.com. You can't really compare ions to molecules in terms of "stability". The sequence of ease of reducibility is alkenes > aldehydes, ketones>carboxylic acids, aromatics.
q In the next chapter we will see that the reverse reaction, the conversion of an ester to a carboxylic acid can be carried out in either acidic or basic solution. To learn more, see our tips on writing great answers. c. Decarboxylation - these are reactions in which the $$\ce{R-C}$$ bond is broken in such a way that $$\ce{CO_2}$$ is lost and $$\ce{R-H}$$ is formed. q The simplest carboxylic acid is methanoic acid, the next ethanoic acid, etc. The activated hydride is then able to react very efficiently at this activated electrophilic center. Both of these have already been discussed rather extensively. The carboxyl carbon is automatically numbered as the. In the carboxylate anion the two contributing structures have equal weight in the hybrid, and the C–O bonds are of equal length (between a double and a single bond).
Please remember,however, that, unlike aldehydes, ketones are not easy to oxidize to carboxylic acids. This is because the experimental result is that the acidity of the carboxylic acid is actually increased. Methanoic acid and almost all the substituted ethanoic acids are stronger than ethanoic acid. When carboxylate salts are put into nonpolar solvents, reversed micelles often are formed, where the polar parts of the molecules are on the inside and the nonpolar parts are on the outside.
The inductive effect of the substituent makes the acid stronger or weaker (relative to the unsubstituted acid), depending on whether the substituent is electron-attracting or electron-donating relative to hydrogen. Watch the recordings here on Youtube!
1. Simultaneously, it converts the carboxylic acid to its (Lewis) conjugate acid (a postively charged, oxonium ion), which (like the corresponding Bronsted conjugate acid) has much more carbocation character at the carbonyl carbon. The stabilization is substantial and carboxylic acids are more stable than would be expected, from summing up their bond energies, by fully $$18 \: \text{kcal mol}^{-1}$$. Alkenes, having the weakest pi bonds, are easiest to reduce. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Essentially an acid-catalyzed addition of an alcohol to the carbonyl group of the carboxylic acid. Best approach to safely bump up version of classes. What does the December 8th deadline mean for the certification of the results of the Electoral College? How can I seal a gap between floor joist boxes and foundation?
The conversion of a carboxylic acid to an ester therefore does not involve either oxidation or reduction. Primarily the resonance stabilization of the conjugate base of a carboxylic acid, i.e., the carboxylate anion. basic sites. In particular, the other structures have charge separation, which is an energy-increasing factor. 4.
, i.e., the negative charge is fully localized upon the oxygen atom. In this molecule there is no carboxylate anion, no dioxallyl system and no $\ce{\Psi_2}$ with a node at $\ce{C_2}$. The substitution reaction does not, however, occur via an S, to the carbonyl group in a manner analogous to the addition of an alcohol to an aldehyde or ketone carbonyl group (hemiacetal formation), followed by an. When one uses hydroxide anion as a base catalyst (or whatever other basic catalyst is used), the preferred reaction is not addition to the carbonyl group, but deprotonation of the carboxyl group to give the carboxylate conjugate base, which is highly unreactive because of both its resonance stabilization and its negative charge. Essentially, the conjugate acid of a ketone has partial positive charge on just two atoms, the carbonyl oxygen and the carbonyl carbon, while the conjugate acid of the carboxylic acid has the postive charge delocalized on three atoms. they are weak acids.
The stem consists of the name of the alkane containing the same number of carbon atoms, except that the terminal –e of the alkane is dropped (e.g., methane becomes methan-). If conditions are controlled carefully, the aldehyde can be isolated in good yield. In the case of the carboxylic acid, the resonance structures are non-equivalent. (see below), while the alcohol oxygen-protonated conjugate acid is not.
The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot.
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خانه / دستهبندی نشده / carboxylate anion resonance structures
carboxylate anion resonance structures
The activated hydride is then able to react, In the product, which is analogous to the hydrate of a carbonyl compound, the former carboxyl carbon has been reduced to the +2 oxidation state, which is the same as that of a carbonyl compound. Simply recall that the two best resonance structures of the carboxylate anion are equivalent, and therefore provide a maximum resonance stabilization. As such. Most of them has a pK a of approximately 5, which means that they can be deprotonated by many bases, such as sodium hydroxide or sodium bicarbonate. It is also possible to draw a resonance structure where electrons are removed from the ring and both oxygens carry a negative charge. Boron is not a metal, and the hydrogens of borane are not especially hydridic in character.
These are the carbonyl oxygen and the alcohol type oxygen. , containing both a carbonyl functionality and a hydroxyl functionality.
q Further oxidation of a carboxylic acid (which requires very strenuous thermal oxidation conditions) can only yield carbon dioxide. because of the carbocation character of the carbonyl carbon atom) are directly connected. On the other hand, the conversion of a carboxylic acid to an aldehyde or an alcohol is considered to be a reduction. Why do most carboxylic acids have high pKa (~5) in spite of having a conjugate base ion that is stabilized by resonance? The fact that alcohols are far weaker acids than carboxylic acids may be attributed to the lack of stabilization of alkoxide ions compared to that of carboxylate anions. If $$e_1$$ and $$e_2$$ have the same sign, the energy is positive, and with opposite signs the energy is negative. The inductive effect is different from resonance effects discussed in Section 18-2A in that it is associated with substitution on the saturated carbon atoms of the chain. q Simultaneously, it converts the carboxylic acid to its (Lewis) conjugate acid (a postively charged, oxonium ion), which (like the corresponding Bronsted conjugate acid) has much more carbocation character at the carbonyl carbon. Note, first, that borane is not a metal hydride. This produces a succession of electron shifts along the chain, which, for an electron-attracting substituent, increases the acid strength by making it more energetically feasible for the $$\ce{-OH}$$ hydrogen of the carboxyl group to leave as a proton: Many other groups besides halogens exhibit electron-withdrawing acid-enhancing inductive effects. . The other possible mode of transmission of the polar effect of a substituent group is a purely electrostatic one, sometimes called the "field effect", in which the dipole of the substituent produces an electrostatic field at the carboxyl proton, which helps or hinders ionization depending on the way in which the dipole is oriented with respect to the carboxyl group.
Can a single card have multiple activations on a stack? The anion, cation, or radical is stabilized by declocaliztion. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Alkenes, having the weakest pi bonds, are easiest to reduce. q In contrast, the conjugate base of an alcohol, an alkoxide anion, is not resonance stabilized at all, i.e., the negative charge is fully localized upon the oxygen atom. So borane preferentially forms the complex with the more basic carbonyl oxygen of the carboxylic acid. 2. (We previously encountered this same idea when considering the relative acidity and basicity of phenols and aromatic amines in section 7.4). ), Virtual Textbook of Organic Chemistry, Prof. Steven Farmer (Sonoma State University). The answer lies in $$\Delta G$$; whatever the electrostatic effects are doing to balance between $$\Delta H$$ and $$\Delta S$$, it is $$\Delta G$$ that determines the equilibrium constant and $$\Delta G$$ quite consistently follows the predictions of simple electrostatic considerations. How can I better handle 'bad-news' talks about people I don't care about? For example, the solvolysis of the 1-methylheptyl sulfonate, $$5$$, in dilute water solution proceeds 70 times slower when sufficient sodium dodecanyl sulfate $$\left( \overset{\oplus}{\ce{Na}} \overset{\ominus}{\ce{O}} \ce{SO_3C_{12}H_{25}} \right)$$ is added to provide about twice as many dodecanyl sulfate ions in the micelle state as there are molecules of $$5$$ present: This slowing of the solvolysis reaction by the alkyl sulfate requires that $$5$$ be almost completely imprisoned by the micelles, because that part of $$5$$ free in water would hydrolyze rapidly. 2) Conjugated double bonds. q Rather, borane is electrophilic, since the boron has a vacant 2p AO.
This reaction converts the neutral, electrophilic borane to a negatively charged hydride reagent, in which the hydrogens are now quite nucleophilic! 3. The acidity of the carboxyl group arises, at least in part, from the polar nature of the carbonyl group, the polarity of which can be ascribed to contributions of the structure. The electrophilic carbonyl carbon of an aldehyde or ketone makes it reactive toward metal hydrides, which contain nucleophilic hydrogen. q Higher molecular weight carboxylic acids, tend to have less water solubility because of the dominance of the non-polar part of the molecule, which is “hydrophobic”. They react with alkyl halides to form ester.
Carboxylic acids easily dissociate into a carboxylate anion and a positively charged hydrogen ion (proton), much more readily than alcohols do (into an alkoxide ion and a proton), because the carboxylate ion is stabilized by resonance. How can a chess game with clock take 5 hours? Why and how carboxylate ion has greater stability than carboxylic acid?
Is it safe to have two separate circuits hooked to the same outlet? Carboxylic acids have a carbonyl pi bond and, further, they have additional resonance stabilization. The result is strong resonance stabilization of the carboxyl group via the interaction between these two groups. First of all, it would have been clearer if it's a misprint or not if you had attached a screenshot of the wording in your textbook (you still can). q If the alcohol is not such that a large excess can be used, the equilibrium can be driven to the right by the removal of water (the LeChatelier principle) via distillation or the use of a dehydrating agent.
When two or more structures that differ only in the positions of valence electrons can be drawn for a molecule or ion, it means that its valence electrons are delocalized, or spread over more than two atoms. Addition/elimination is a typical route for substitution processes in carboxylic acids, esters, amides, and other analgous acyl compounds in which carbon has the +3 oxidation state. All rights reserved. These are the carbonyl oxygen, the carbonyl carbon, and the alcohol type oxygen. And this is really important for ions because in this case, I can share out that negative charge over two different oxygen atoms.
Since we have two carboxylate sites in the oxalate anion, there are four total possible resonance structures as shown below. Solvent polarity is often measured by the ability to dissolve salts or to provide solvation stabilization for ions. Only one structure can be drawn for an alkoxide ion, but two structures can be drawn for a carboxylate ion.
Unlike the reduction of ester, the reduction of carboxylate is different, due to the lack of the leaving group and the relatively electron-rich carbon atom (due to the negative charge on the oxygen atoms). © copyright 2003-2020 Study.com. You can't really compare ions to molecules in terms of "stability". The sequence of ease of reducibility is alkenes > aldehydes, ketones>carboxylic acids, aromatics.
q In the next chapter we will see that the reverse reaction, the conversion of an ester to a carboxylic acid can be carried out in either acidic or basic solution. To learn more, see our tips on writing great answers. c. Decarboxylation - these are reactions in which the $$\ce{R-C}$$ bond is broken in such a way that $$\ce{CO_2}$$ is lost and $$\ce{R-H}$$ is formed. q The simplest carboxylic acid is methanoic acid, the next ethanoic acid, etc. The activated hydride is then able to react very efficiently at this activated electrophilic center. Both of these have already been discussed rather extensively. The carboxyl carbon is automatically numbered as the. In the carboxylate anion the two contributing structures have equal weight in the hybrid, and the C–O bonds are of equal length (between a double and a single bond).
Please remember,however, that, unlike aldehydes, ketones are not easy to oxidize to carboxylic acids. This is because the experimental result is that the acidity of the carboxylic acid is actually increased. Methanoic acid and almost all the substituted ethanoic acids are stronger than ethanoic acid. When carboxylate salts are put into nonpolar solvents, reversed micelles often are formed, where the polar parts of the molecules are on the inside and the nonpolar parts are on the outside.
The inductive effect of the substituent makes the acid stronger or weaker (relative to the unsubstituted acid), depending on whether the substituent is electron-attracting or electron-donating relative to hydrogen. Watch the recordings here on Youtube!
1. Simultaneously, it converts the carboxylic acid to its (Lewis) conjugate acid (a postively charged, oxonium ion), which (like the corresponding Bronsted conjugate acid) has much more carbocation character at the carbonyl carbon. The stabilization is substantial and carboxylic acids are more stable than would be expected, from summing up their bond energies, by fully $$18 \: \text{kcal mol}^{-1}$$. Alkenes, having the weakest pi bonds, are easiest to reduce. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Essentially an acid-catalyzed addition of an alcohol to the carbonyl group of the carboxylic acid. Best approach to safely bump up version of classes. What does the December 8th deadline mean for the certification of the results of the Electoral College? How can I seal a gap between floor joist boxes and foundation?
The conversion of a carboxylic acid to an ester therefore does not involve either oxidation or reduction. Primarily the resonance stabilization of the conjugate base of a carboxylic acid, i.e., the carboxylate anion. basic sites. In particular, the other structures have charge separation, which is an energy-increasing factor. 4.
, i.e., the negative charge is fully localized upon the oxygen atom. In this molecule there is no carboxylate anion, no dioxallyl system and no $\ce{\Psi_2}$ with a node at $\ce{C_2}$. The substitution reaction does not, however, occur via an S, to the carbonyl group in a manner analogous to the addition of an alcohol to an aldehyde or ketone carbonyl group (hemiacetal formation), followed by an. When one uses hydroxide anion as a base catalyst (or whatever other basic catalyst is used), the preferred reaction is not addition to the carbonyl group, but deprotonation of the carboxyl group to give the carboxylate conjugate base, which is highly unreactive because of both its resonance stabilization and its negative charge. Essentially, the conjugate acid of a ketone has partial positive charge on just two atoms, the carbonyl oxygen and the carbonyl carbon, while the conjugate acid of the carboxylic acid has the postive charge delocalized on three atoms. they are weak acids.
The stem consists of the name of the alkane containing the same number of carbon atoms, except that the terminal –e of the alkane is dropped (e.g., methane becomes methan-). If conditions are controlled carefully, the aldehyde can be isolated in good yield. In the case of the carboxylic acid, the resonance structures are non-equivalent. (see below), while the alcohol oxygen-protonated conjugate acid is not.
The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot.
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1. ## meanless question
0 / 8 = 0
but how come 8 / 0 = error or E.......???
2. Originally Posted by sonymd23
0 / 8 = 0
but how come 8 / 0 = error or E.......???
Let's do an example...
$\displaystyle \frac{16}{8}=2\text{ because }8+8=8\times2=16$
now try zero...
$\displaystyle \frac{8}{0}=\infty\text{ because }0+0+0+0....=0\times\infty\neq8$
so zero goes into eight infinite times, which we call undefined.
~ $\displaystyle Q\!u\!i\!c\!k$
3. Originally Posted by sonymd23
0 / 8 = 0
but how come 8 / 0 = error or E.......???
Because division by zero, is not allowed in math.
Why?
-----
1)The embarassing physics answer because when one number is divided into another number that says how many times a number goes into that number. For example, 4/2 = 2 because 2 goes in 2 times.
However, when you have 8/0 note that no matter how many times you add 0 you get thus thus you never reach 8 thus there is no such number.
2)The improved applied mathematicians answer because when we write 10/2=5 we mean that 2 times 5 is 10, by definition of what division is (opposite of multiplication). Now when you have 0/8 the answer is 0 because 0 times 8 is 0, good. But what about 8/0? We are trying to find a number such when mutiplied by 0 results in 8, but any number multiplied by 0 is 0 how can we ever get 8!!! Futhermore, then what is the answer to 0/0? Do we say it is any number cuz any number times zero is zero, according to this definition.
3)The elegant and perfect pure mathematicians answer I am not going to state it (I have before on this forum) because you would not understand so there would be no purpose. All, I can say when we limit divison for non-zero number we preserve a very important property called divisors of zero. Thus, to preserve this property we limit ourselves.
4. Actually, to be honest, the Physics answer is that is can't be done on a calculator.
-Dan
5. Originally Posted by sonymd23
0 / 8 = 0
but how come 8 / 0 = error or E.......???
Hello, sonymd23,
your question isn't meaningless at all - and very difficult to answer.
Let us assume that the division by zero is allowed. Then you can say that
$\displaystyle \frac{4}{4}=\frac{8}{8}=\frac{-1}{-1}=\frac{0}{0}=1$
$\displaystyle 5\ \cdot \ 0=0$. Divide both sides of the equation by zero and you'll get:
$\displaystyle \frac{5\ \cdot \ 0}{0}=\frac{0}{0}$. According to our assumption the result is:
5 * 1 = 1
So you see that numbers are ambiguous, if you allow the division by zero.
(One effect of this result is: You can buy the most expensive car for 1 $, because 245,456.99$ or 1 $, it's the same) Greetings EB 6. ## PH's second point Originally Posted by sonymd23 0 / 8 = 0 but how come 8 / 0 = error or E.......??? Try writing it out in an equation, such as...$\displaystyle \frac{5}{0}=x$multiply both sides by 0$\displaystyle 5=x\times0$solve:$\displaystyle 5=0$as you can see this is incorrect. In fact the method to do this problem is incorrect because$\displaystyle \frac{0}{0}=\infty$let's try it now, with this new information:$\displaystyle \frac{5}{0}=x$multiply both sides by 0$\displaystyle \frac{5\times 0}{0}=x\times0$extend:$\displaystyle 5\frac{0}{0}=0$solve:$\displaystyle 5\infty=0$this way doesn't work either! because 5 times infinity equals infinity, not zero. So as you can see, there is no mathematical way to express$\displaystyle \frac{x}{0}$7. Originally Posted by Quick Try writing it out in an equation, such as...$\displaystyle \frac{5}{0}=x$multiply both sides by 0$\displaystyle 5=x\times0$solve:$\displaystyle 5=0$as you can see this is incorrect. In fact the method to do this problem is incorrect because$\displaystyle \frac{0}{0}=\infty$let's try it now, with this new information:$\displaystyle \frac{5}{0}=x$multiply both sides by 0$\displaystyle \frac{5\times 0}{0}=x\times0$extend:$\displaystyle 5\frac{0}{0}=0$solve:$\displaystyle 5\infty=0$this way doesn't work either! because 5 times infinity equals infinity, not zero. So as you can see, there is no mathematical way to express$\displaystyle \frac{x}{0}$If anyone did such a calculation in front of me.... let me stop there. Never, have I seen such an informal argument. 8. Originally Posted by ThePerfectHacker If anyone did such a calculation in front of me.... let me stop there. Never, have I seen such an informal argument. here's an even more informal argument: Originally Posted by Attempting to Annoy Hacker you can't divide by zero because someone out there says so. 9. Originally Posted by High School Teacher you can't divide by zero because someone out there says so. When I was younger that bothered me too. Because, I never knew the real reason why. But when I started doing formal math and seeing how the numbers and everything was constructed I finally understood and these things never bothered me ever since. For now put faith into mathematicians and try not to find what 0/0 is, because you will not. Same thing as saying, do not try to find$\displaystyle a/b=\sqrt{2}$because you will not be able to. You might not know the reason but you place faith into the mathematicians and never try. 10. Originally Posted by ThePerfectHacker Same thing as saying, do not try to find$\displaystyle a/b=\sqrt{2}$because you will not be able to. You might not know the reason but you place faith into the mathematicians and never try. It's not that I don't have faith in mathematicians, it's that the easiest way for me to remember things is to know why they are. also, I assume you mean that a and b have to be rational numbers. 11. Originally Posted by Quick Originally Posted by ThePerfectHacker Same thing as saying, do not try to find$\displaystyle a/b=\sqrt{2}\$ because you will not be able to. You might not know the reason but you place faith into the mathematicians and never try.
It's not that I don't have faith in mathematicians, it's that the easiest way for me to remember things is to know why they are.
also, I assume you mean that a and b have to be rational numbers.
There are no such integers is sufficient, that there are no such rationals
is implied by there being no such integers.
RonL
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Krishna
0
Step 1: Recall the formulas of volume of cylinder and cone
Volume of cylinder = \pi (r_1)^2 h_1
Volume of cone = \frac{1}{3} \pi (r_2)^2 h_2
h - height
Step 2: Make a note of the given information
Given that,
Height of cone = Height of cylinder
We have show that, volume of cylinder : volume of cone = 3 : 1
Step 3: Showing that volumes of cylinder and cone are in the ratio of 3:1
= \frac{volume\ of\ cylinder}{volume\ of\ cone}
= \frac{\pi r_1^2 h_1}{\frac{1}{3} \pi r_2^2 h_2}
= \frac{\pi h_1}{\frac{1}{3} \pi h_2} \because r_1 = r_2
= \frac{\pi }{\frac{1}{3} \pi} \because h_1 = h_2
= \frac{3}{1}
Hence, their volumes are in the ratio 3 : 1
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# banach fixed point theorem
Let $T:X \to X$ be a map on a complete non-empty metric space. Assume that for all $x$ and $y$ in $X$, $\sum_n d(T^n(x),T^n(y))<\infty$. Then $T$ has a unique fixed point.
guess: I assume that the existence and unicity of a fixed point can be shown directly with the standard Banach fixed point theorem, by a suitable choice of the metric that makes the map $T$ a contraction.
-
@kahen: The fact that $T$ has a fixed point doesn't depend on the metric... – Najib Idrissi Feb 8 '12 at 9:05
Why isn't the following a counterexample?
Let $X=\{0\}\cup\{1/2^n:n\in\mathbb{N}\}$ and endow $X$ with the absolute-value-metric. Then $X$ is complete. Define $T:X\to X$ by $T(0)=1/2$ and $T(1/2^n)=1/2^{n+1}$. Then $T$ has no fixed point. Now $|T^n(1/2^l)-T^n(1/2^m)|\leq1/2^n$ and every point has this form after at least one application of $T$. So the summability condition should hold too.
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Your counterexample shows that a continuity condition is needed in the original problem. If $T$ is not continuous (as in your example) we may have $T^n(x)\to\bar x$ but $T(T^n(x))$ does not converge to $T(\bar x)$. – Julián Aguirre Feb 8 '12 at 11:53
I am assuming map means continuous function. Otherwise Michael Greinecker's answer gives a counterexample.
Existence:
Take any point $x_0 \in X$ and define inductively $x_{n+1} := T(x_n)$. Given $\epsilon>0$ and assuming $m>n>N$, we have $$d(x_n,x_m) \leq \sum_{k=0}^{m-n-1} d(x_{n+k}, x_{n+k+1}) \leq \sum_{k=0}^\infty d(x_{N+k}, x_{N+k+1}) = \sum_{k=N}^\infty d(T^k(x_0),T^k(x_1)) \leq \epsilon$$ for sufficiently large $N$. By completeness there exists $\bar{x}\in X$ with $\lim_{n\to\infty}x_n=\bar{x}$ and by continuity of $T$ we get $$T(\bar{x}) = T(\lim_{n\to \infty} x_n) = \lim_{n\to \infty}T(x_n) = \lim_{n\to \infty} x_{n+1} = \bar{x}.$$
Uniqueness:
Let $\bar{x}$ and $\tilde{x}$ be fixed points of $T$, then $\sum_{k=0}^\infty d(T^k(\bar{x}),T^k(\tilde{x})) = \sum_{k=0}^\infty d(\bar{x},\tilde{x}) < \infty$, so $d(\bar{x},\tilde{x})=0$, hence $\bar{x} = \tilde{x}$.
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Research Article| Volume 9, P304-309, January 2021
# Estimation of missing values in aggregate level spatial data
Open AccessPublished:October 12, 2020
## Abstract
### Background
Data can be missing when a survey fails to collect information from certain regions due to feasibility issues, which can impose problems while performing spatial analysis.
### Objective
The present study aims to estimate missing aggregate level public health spatial data by utilizing the information from neighbouring regions and accounting for spatial autocorrelation inherently present in the data.
### Methodology
Data was simulated for fixed values of various parameters in spatial regression models under low and high autocorrelation scenarios in dependent and independent variables. In dependent variable, 5%–25% of values were assumed to be missing. Stochastic regression imputation using spatial regression models namely spatial lag model, spatial error model, spatial Durbin model, spatial Durbin error model and spatial lag of X model was performed. The performance of these models were also compared using data from Annual Health Survey 2012-13.
### Results
The simulation analysis revealed that for any amount of missing values in the data, irrespective of whether the other variables in the regression model are spatially autocorrelated or not, if autocorrelation in the variable with missing values is high, stochastic regression imputation performed using spatial lag model, spatial Durbin model and spatial Durbin error model gives accurate estimates of missing values. If the autocorrelation is low, in addition to these three models, spatial lag X model was also found to be effective in estimating the missing values.
### Conclusion
The proposed mechanism results in optimal imputation of missing values in spatial data, which can yield complete data useful for public health professionals for effective interventions.
## 1. Introduction
Missing data is a common problem in many public health surveys. The missing data problem occurs when the essential data is accessible for the use of performing analysis, however some part of it is missing from the possession of the analyst.
• Bennett R.J.
• Haining R.P.
• Griffith D.A.
The problem of missing data on spatial surfaces.
Sometimes the data can be missing in the secondary data source when a survey fails to collect information from certain regions due to feasibility or logistic issues. This creates missing at random mechanism.
• Allison P.D.
Missing Data.
Missing values in the data limit the accuracy of the models in prediction of occurrence of any event like disease, death, disability etc., which in turn hinders the success of public health surveillance to target effective interventions.
• Rubin D.B.
Multiple Imputation for Nonresponse in Surveys.
In many situations a public health researcher may be interested to know various health indicators like infant mortality rate, prevalence of anaemia, child marriage rate etc. in a particular population using data from available sources. However, for few populations these estimates may not be available, as surveys are not conducted in these areas due to logistic reasons. Hence it is important to estimate the values of these indices for the missing populations based on available information from nearby populations as these measures are spatially correlated for nearby places. There are many methods to handle missing data and the commonly used are mean imputation, hot-deck imputation, multiple imputation and expectation-maximization imputation.
• Little R.J.
• Rubin D.B.
Statistical Analysis with Missing Data.
However, estimation of missing values in spatial data must be dealt differently due to the interdependence (autocorrelation) among the spatial data points. Several approaches have been developed in this direction,
• Munoz B.
• Lesser V.M.
• Smith R.A.
Applying multiple imputation with geostatistical models to account for item nonresponse in environmental data.
• Li L.
• Li Y.
• Li Z.
Efficient missing data imputing for traffic flow by considering temporal and spatial dependence.
• Griffith D.A.
• Liau Y.T.
Imputed spatial data: cautions arising from response and covariate imputation measurement error.
nonetheless, most of these techniques focus on obtaining best estimate of the parameters of interest rather than estimating missing values in the data.
The present study aims to estimate missing values in spatial data at aggregate level by utilizing the information from neighbouring regions and accounting for spatial autocorrelation inherently present in the data. To account for the expected spatial association in the data, we intended to propose a method that accounts for the spatial structure of the data. The study also focuses on finding the accuracy in estimating missing values at aggregate level using a stochastic imputation technique using various spatial regression models under varied scenarios such as different levels of autocorrelation as well as various proportions of missing values in the data. After an accurate imputation, the complete data thus obtained can be further used for identifying the predictors for the event under study as well as to design appropriate intervention programs.
## 2. Methods
### 2.1 Spatial regression models
Spatial regression models
• LeSage J.P.
• Pace R.K.
Introduction to Spatial Econometrics.
,
• Golgher A.B.
• Voss P.R.
How to interpret the coefficients of spatial models: spillovers, direct and indirect effects.
namely spatial lag model (SLM), spatial error model (SEM), spatial Durbin model (SDM), spatial Durbin error model (SDEM) and spatial lag of X model (SLX) were fitted in order to impute the missing values using the stochastic regression imputation method. The description of spatial regression models under comparison is given below. Let us consider the general nesting spatial model,
• Elhorst J.P.
Spatial Econometrics: From Cross-Sectional Data to Spatial Panels.
$Equation 1.$
(1)
$Equation 2.$
(2)
where $y$ is an $N×1$ vector of values of the dependent variable, W is an $N×N$ dimensional neighbors weights matrix, where all elements $wij>0$ for all neighbouring units $i$ and $j$ $(i≠j)$, and zero otherwise. $X$ is an $N×K$ matrix of $K$ independent variables, $β$ is a $K×1$ vector of regression coefficients, $ρ$ represents the autoregressive scalar parameter in the dependent variable, $θ$ is $K×1$ vector of spatial spill-over parameters, $u$ represents an $N×1$ vector of spatially autocorrelated disturbances, $λ$ represents the autoregressive scalar parameter in the disturbances. The parameter vector $γ$ specifies the correlation between $X$ and the disturbance term vector $u$, $δk$ represents the autocorrelation in the $kth$ independent variable. $vk$ and $ε$ are independent and randomly distributed disturbances following and , and $xk$ is the $k$th column-vector of $X$ for $k=1,…,K$ independent variables.
The reduced form of this model can be written as,
• Rüttenauer T.
Spatial regression models: a systematic comparison of different model specifications using Monte Carlo experiments.
$Equation 3.$
(3)
When $θ=0$ and $λ=0$ in (1), the model takes the form of spatial lag model (SLM). The SLM assumes that the dependent variable in one unit is influenced by the spatially weighted dependent variable of neighbouring units. Spatial error model (SEM) is a another form of spatial model which is formed when $ρ=0$ and $θ=0$. In this specification it is assumed that the spatial association among the units is produced due to the unobserved features, which are either random or consist of a spatial pattern, and are independent of the explanatory variables included in the model. Spatial Durbin model (SDM) specification is formed when the autocorrelation is contributed by both dependent variable and independent variables, i.e. when $λ=0$. A specification which directly models the spatial spill-over effects by including the spatially lagged independent variables into the regression equation is known as the spatial lag of X (SLX) model, formed when $ρ=0$ and $λ=0$. The model which combines the specifications of SEM and SLX is known as the spatial Durbin error model (SDEM) which is formed when $ρ=0$.
### 2.2 Stochastic regression imputation
One of the most simple and commonly used imputation technique in the non-spatial context is stochastic regression imputation. This method uses the ordinary least squares (OLS) regression to predict missing values and adds a normally distributed residual term to each predicted value. It restores loss in variability and bias associated with regression imputation (Little and Rubin 2019). The stochastic regression imputation is based on complete cases and is given as,
$yˆik=∑j=1k−1βˆjxij+zi$
where $zi$ is a random draw from $N(0,σ2)$ and $σ2$ is assume to be constant for given value of $xj$. The variable with missing values $Yk$ is the variable of interest. This variable is considered as the dependent variable of the stochastic regression imputation model and the auxiliary variables $(x1,…,xk−1)$ used to estimate the missing values are regarded as the independent variables.
In the present study, each missing value in dependent variable was estimated by stochastic regression imputation based on above mentioned five different spatial models. The performance of these five spatial models in accurately estimating the missing values were compared using root mean square error. For comparison purpose the stochastic regression based on OLS was also performed for estimating the missing values. Both simulated data and a real data were used to assess the performance of these six models (SLM, SEM, SDM, SDEM, SLX and OLS) in accurately estimating the missing values in aggregate level spatial data. Simulation was done to study the performance of these modes at varying percentage of missing values as well as for varied values of autocorrelation of dependent and independent variables in the model. Finally, the performance of the models were assessed using a real data from Annual Health Survey 2012–13 published by the Ministry of Home Affairs, Government of India.
• General R.
Annual Health Survey-2012–13.
### 2.3 Data simulation
Data simulation was performed using spdep package in R software to generate variables and impute missing data at aggregate level using stochastic regression imputation based on ordinary least squares (OLS), SLM, SEM, SDM, SDEM and SLX. The geographical space chosen was the map of India consisting of nine states namely Bihar, Chhattisgarh, Jharkhand, Madhya Pradesh, Odisha, Rajasthan, Uttarakhand, Uttar Pradesh and Assam aggregated at the district level. From the shape file of India with 284 districts, neighbourhood matrix W was constructed based on queen contiguity weights matrix, which defines the neighbors as those areas that share boundary points as well as some portion of their boundary. Simulated data of size 284 districts included three independent variables for which the regression coefficient vector was fixed arbitrarily at $β=(−33.5−3)T$. The disturbance parameters were generated as an independent and randomly distributed normal variable with mean zero and fixed at $σv2,σε2=1$. The spatial spill-over parameter vector was fixed at $θ=(−11−1)T$. The autocorrelation in disturbance vector $u$ and the omitted variable bias were fixed at $λ={0.05}$ and $γ=(000)T$ respectively. To generate dependent variable with low and high autocorrelation, the scalar parameter $ρ$ was fixed at two values namely $ρ=0.2$ and $ρ=0.8$ respectively. Similarly, the autocorrelation in independent variables was assumed to be $δ=0.1$ and $δ=0.6$ to generate independent variables with low and high autocorrelation respectively. The variable of interest (dependent variable) was finally simulated following the general nesting spatial model given in equation (3). The simulation resulted in four datasets each of size 284 representing four different situations as given below.
• a)
High autocorrelation in both dependent and independent variables
• b)
Low autocorrelation in both dependent and independent variables
• c)
High autocorrelation in dependent variable and low autocorrelation in independent variables
• d)
Low autocorrelation in dependent variable and high autocorrelation in independent variables
The autocorrelation and Pearson's correlation coefficient values of the simulated dependent and independent variables are given in Table 1.
Table 1Spatial autocorrelation and Pearson's correlation coefficient values of the simulated variables.
YX1X2X3
High-HighCorrelation with Y−0.500.63−0.75
Autocorrelation0.890.480.580.65
Low-LowCorrelation with Y−0.470.62−0.58
Autocorrelation0.250.100.110.01
High-LowCorrelation with Y−0.440.49−0.42
Autocorrelation0.770.070.110.05
Low-HighCorrelation with Y−0.560.65−0.56
Autocorrelation0.280.600.530.61
The data was simulated 1000 times for each of the above four scenarios of levels of autocorrelation among the variables. In each simulated dataset a fixed proportion of values in dependent variable was assumed to be missing according to missing at random (MAR) mechanism. In MAR mechanism, probability of missing values in the dependent variable (Y) was based on the values of other variable(s).
• Little R.J.
• Rubin D.B.
Statistical Analysis with Missing Data.
To generate missing as per MAR mechanism in dependent variable, a fixed proportion of locations were randomly selected among those locations where the values of the independent variable $X1$ were very low (less than the first quartile value of$X1$). In these selected locations the values of dependent variable were assumed to be missing. This was repeated for various proportions of missing, 5%, 10%, 15%, 20% and 25%in the dependent variable in each simulated dataset. For each proportion of missing and for each level of autocorrelation in variables, the missing values were estimated using stochastic regression imputation with ordinary least squares method (OLS), SLM, SEM, SDM, SDEM and SLX.
The performance of the above models in accurately estimating the missing values under each scenario were compared based on the computed root mean squared error (RMSE) values. For $i=1tow$ where$i$ are the regions consisting of missing values and $w$ are the total number of regions with values to be imputed, the RMSE for the variable Y is given as,
$RMSEj[Yˆ]=∑i=1wj(Yˆij−Yij)2wj$
where, $Yˆij$ is the imputed value for the variable Y in region $i$, for a given simulated dataset (j = 1, 2, …,1000), for any given model and any fixed proportion of missing in the dataset. $Yij$is the observed value of the variable Y in region $i$ and jth simulated dataset.
The mean of the RMSE (mRMSE) and standard deviation of RMSE (sdRMSE) for a given model and fixed proportion of missing were calculated as,
$mRMSE[Yˆ]=∑j=11000RMSEj[Yˆ]1000$
$sdRMSE[Yˆ]=∑j=11000(RMSEj[Yˆ]−mRMSE[Yˆ])21000−1$
The mRMSE was computed for the imputations performed using each of the five spatial regression models and OLS model, on the data with 5%–25% proportions of missing values in the dependent variable. The mRMSE values were compared between the models and the model with lowest mRMSE in each scenario was considered the best in estimating the missing values in the dependent variable.
### 2.4 Application on data from Annual Health Survey (AHS) 2012-13
In reality, missing values can occur for the regions where data on certain events/diseases cannot be collected due to constraints such as safety, logistics etc. To reflect such situations, the proposed imputation techniques were applied on data obtained from AHS for 284 districts of India. The variables have been chosen in a manner to reflect the relationships in two scenarios, (i) high autocorrelation in dependent and independent variables (high-high) and (ii) low and high autocorrelation in dependent and independent variables (low-high) respectively. For the first scenario, the dependent variable considered was total fertility rate of a district (TFR) and the independent variables considered were percentage of mothers who received any antenatal check-up and unmet need for spacing in a district. In the case of second scenario namely low-low autocorrelation, the dependent variable considered was percentage of women aged 15–19 years who were already mothers/pregnant at the time of survey while independent variables total fertility rate and percentage of female literacy rate in a district. In real data analysis, overall rates of missingness of 5%, 10%, 15%, 20% and 25% of observations were imposed on the dependent variable according to MAR mechanism. Missing values were imposed on the dependent variable using the same procedure as it was done on the simulated data. The variables considered for the analysis and the values of their spatial autocorrelation and Pearson's correlation coefficient are listed in Table 2.
Table 2Spatial autocorrelation and Pearson's correlation coefficient values of the variables retrieved from Annual Health Survey (AHS 2012-13) data.
Correlation with YAutocorrelation
High-High
Y - Total fertility rate0.63
X1 - Mothers who received any antenatal check-up (%)−0.600.58
X2 - Unmet need for spacing (%)0.570.58
Low-High
Y - Women aged 15–19 years already mothers/pregnant at the time of survey (%)0.44
X1 - Total fertility rate−0.130.63
X2 - Female literacy rate0.290.62
## 3. Results
### 3.1 Comparison of various spatial regression models using simulated data
The stochastic regression imputation with all the six models were applied on the simulated data under two scenarios (i) for four types of levels of autocorrelation among the dependent and independent variables as mentioned in Table 1, and (ii) for various proportion of missing values in the dependent variable. Fig. 1 gives the comparison of mean RMSE values for the six models for the above mentioned two scenarios. When the autocorrelation in both dependent and independent variables were high (Fig. 1(a)), it can be noted that the imputation performed using SEM had higher mRMSE compared to that obtained using the remaining five models. The mRMSE value using SEM was even higher for the imputation performed using OLS model, which model does not account for autocorrelation. The least and similar mRMSE in the imputation were produced by SLM, SDM and SDEM. When the autocorrelation in both dependent and independent variables were low (Fig. 1(b)), the performance of imputation using all the models was more or less similar in terms of mRMSE. However, the imputations performed using SDM, SDEM and SLX found better with least mRMSE. In case of high autocorrelation in dependent variable and low in independent variables (Fig. 1(c)), the pattern of mRMSE is similar to that of situation (a), but it can be noted that the mRMSE of imputation using SEM has reduced. Imputation performed using SLM, SDM and SDEM had least mRMSE when the autocorrelation in dependent variable was low and high in independent variables (Fig. 1(d)). In this case, OLS and SEM performed in a similar manner. Overall, the simulation analysis revealed that when autocorrelation is high in the variable with missing values, stochastic regression imputation performed using spatial lag model, spatial Durbin model and spatial Durbin error model gives accurate and almost similar estimates of missing values irrespective of whether the independent variables in the regression model were spatially autocorrelated or not, and this result was consistent for the varied amount of missing values. When the autocorrelation is low in the variable with missing values, in addition to the above three models, the stochastic regression imputation using spatial lag X model was also found to be equally accurate in estimating the missing values. It can also be noted that stochastic regression imputation performed using OLS model and SEM had the highest mRMSE in all the scenarios considered in the simulation study.
The mean and standard deviations of RMSE values obtained for 1000 simulated datasets and for real data analysis in each scenario is given in supplementary material (Table 3 and Table 4).
### 3.2 Comparison of various spatial regression models using real data from AHS 2012-13
Fig. 2 presents a line graph depicting the mRMSE obtained from imputations performed by stochastic regression using different models. Similar to the results obtained from the simulated data, imputation performed using SLM, SDM and SDEM were found to have least RMSE when the autocorrelation was high in both dependent and independent variables. This pattern remained consistent for varying proportions of missing values. The same three models performed well when the data possessed low autocorrelation in dependent variable and high autocorrelation in independent variables. It can also be noted that stochastic regression imputation performed using OLS model had highest mRMSE in all the scenarios considered.
## 4. Discussion and conclusions
Missing data can be a problem especially when the researcher relies on the secondary data for the purpose of analysis. In many cases the data is incomplete, due to various reasons such as feasibility, safety etc. Various missing data imputation techniques are available but they are inefficient if there is interdependence in the data values. Such data requires to be handled differently while performing imputation and several methods have been developed and applied to deal with missing values. Baker et al.
• Baker J.
• White N.
• Mengersen K.
Missing in space: an evaluation of imputation methods for missing data in spatial analysis of risk factors for type II diabetes.
compared mean imputation, imputation using a multivariate normal prior distribution and using a conditional autoregressive prior distribution to impute missing values in incomplete survey data by accounting for spatial autocorrelation. A Bayesian hierarchical modelling framework was developed by Song et al.
• Song C.
• Yang X.
• Shi X.
• Bo Y.
• Wang J.
Estimating missing values in China's official socioeconomic statistics using progressive spatiotemporal Bayesian hierarchical modeling.
to impute missing values in the official socioeconomic statistics dataset of China by considering spatial autocorrelations and temporal trends into account. Likewise, the present study focuses on such scenarios and intends to propose a simple technique of stochastic regression imputation using the spatial regression models rather than the classical ordinary least squares regression. This technique will help to accurately estimate the missing values at aggregate level public health spatial data which can increase the power and precision of the estimates in further analysis.
Though the present study is first of its kind that attempted to estimate the missing values at aggregate level spatial data, the study has few limitations. The performance of the models were evaluated for up to 25% missing proportion only. The mechanism of missing not at random is not considered since the classical stochastic regression imputation technique has been proven to be efficient only under situations when missing is at random. Another aspect which has not been given consideration in the present study is the missing values in categorical variables, which has a scope for future research.
The study shows that stochastic regression imputation performed using spatial lag model, spatial Durbin model, and spatial Durbin error model performed consistently better for varied levels of missing proportions and also for various combinations of autocorrelation in dependent and independent variables. We recommend using stochastic regression imputation based on one of the above three models in order to estimate missing data, especially when the data has inherent spatial autocorrelation. The complete data obtained after using the proposed method can further be used by the public health professionals for effective interventions.
## Funding
The authors acknowledge the funding support for this study by Department of Science and Technology, Government of India (Sanction order No. NRDMS/01/122/015).
## Availability of data and material
The datasets used for data analysis are obtainable from the corresponding author on request.
## Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
## Appendix A. Supplementary data
• Multimedia component 1
• Multimedia component 2
## References
• Bennett R.J.
• Haining R.P.
• Griffith D.A.
The problem of missing data on spatial surfaces.
Ann Assoc Am Geogr. 1984 Mar; 74: 138-156
• Allison P.D.
Missing Data.
Sage Publications, 2001 Aug 13
• Rubin D.B.
Multiple Imputation for Nonresponse in Surveys.
John Wiley & Sons, 2004 Jun 9
• Little R.J.
• Rubin D.B.
Statistical Analysis with Missing Data.
John Wiley & Sons, 2019 Apr 23
• Munoz B.
• Lesser V.M.
• Smith R.A.
Applying multiple imputation with geostatistical models to account for item nonresponse in environmental data.
J Mod Appl Stat Methods. 2010; 9: 27
• Li L.
• Li Y.
• Li Z.
Efficient missing data imputing for traffic flow by considering temporal and spatial dependence.
Transport Res C Emerg Technol. 2013 Sep 1; 34: 108-120
• Griffith D.A.
• Liau Y.T.
Imputed spatial data: cautions arising from response and covariate imputation measurement error.
Spatial Statist. 2020 Feb 3; : 100419
• LeSage J.P.
• Pace R.K.
Introduction to Spatial Econometrics.
Chapman & Hall/CRC, 2009
• Golgher A.B.
• Voss P.R.
How to interpret the coefficients of spatial models: spillovers, direct and indirect effects.
Spatial Demogr. 2016 Oct 1; 4: 175-205
• Elhorst J.P.
Spatial Econometrics: From Cross-Sectional Data to Spatial Panels.
Springer, Heidelberg2014
• Rüttenauer T.
Spatial regression models: a systematic comparison of different model specifications using Monte Carlo experiments.
Socio Methods Res. 2019 Jun 9; 0049124119882467
• General R.
Annual Health Survey-2012–13.
Ministry of Home Affairs, Government of India, New Delhi2012
• Baker J.
• White N.
• Mengersen K.
Missing in space: an evaluation of imputation methods for missing data in spatial analysis of risk factors for type II diabetes.
Int J Health Geogr. 2014 Dec 1; 13: 47
• Song C.
• Yang X.
• Shi X.
• Bo Y.
• Wang J.
Estimating missing values in China's official socioeconomic statistics using progressive spatiotemporal Bayesian hierarchical modeling.
Sci Rep. 2018 Jul 3; 8: 1-3
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## Intermediate Algebra (12th Edition)
$9t^2-\dfrac{25}{16}$
Using $(a+b)(a-b)=a^2-b^2$ or the product of the sum and difference of like terms, the given expression, $\left(3t-\dfrac{5}{4}\right)\left(3t+\dfrac{5}{4}\right) ,$ is equivalent to \begin{array}{l}\require{cancel} (3t)^2-\left(\dfrac{5}{4}\right)^2 \\\\= 9t^2-\dfrac{25}{16} .\end{array}
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# Cubing unities?
Geometry Level 3
${ S }_{ n }=\sum _{ i=1 }^{ n }{ \cot ^{ -1 }{ \left( { i }^{ 2 }+i+1 \right) } }$
Conside the summation above, $$S_n$$. If the value of $${ S }_{ 100 }$$ can be written as $$\cot ^{ -1 }{ \left( \frac { a }{ b } \right) }$$ for coprime positive integers $$a$$ and $$b$$ with $$a>b$$, find the value of $$2(a+b)$$.
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[Solved][Win32] Creating 2 windows and locking them to each other
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Well what I am doing is creating a main window for my map editor, and making another window that I want to lock to the outside edge of my main window... basically if I move my main window my secondary window moves in tandem with it... I have tried just about everything I can think of and have found out that maybe its something simple, some simple little command that I dont know! im using VS express 2008. here is how I create my 2 windows
int WINAPI WinMain(HINSTANCE hInstance,HINSTANCE hPrevInstance,LPSTR lpCmdLine,int nShowCmd)
{
WNDCLASSEX wcx;//create window class
wcx.cbSize=sizeof(WNDCLASSEX);//set the size of the structure
wcx.style=CS_OWNDC | CS_HREDRAW | CS_VREDRAW | CS_DBLCLKS;//class style
wcx.lpfnWndProc=TheWindowProc;//window procedure
wcx.cbClsExtra=0;//class extra
wcx.cbWndExtra=0;//window extra
wcx.hInstance=hInstMain;//application handle
wcx.hbrBackground=(HBRUSH)GetStockObject(BLACK_BRUSH);//background color
wcx.lpszClassName="WINXCLASS";//class name
wcx.hIconSm=NULL;//small icon
//register the window class, return 0 if not successful
if(!RegisterClassEx(&wcx)) return(0);
//create main window
//error check returns true if our main window failed to create
if(!hWndMain) return(0);
//Create our buttons window
WNDCLASSEX wcx2;//create window class
wcx2.cbSize=sizeof(WNDCLASSEX);//set the size of the structure
wcx2.style=CS_DBLCLKS | CS_NOCLOSE;//class style
wcx2.lpfnWndProc=CommandWindowProc;//window procedure
wcx2.cbClsExtra=0;//class extra
wcx2.cbWndExtra=0;//window extra
wcx2.hInstance=hInstMain;//application handle
wcx2.hbrBackground=(HBRUSH)GetStockObject(BLACK_BRUSH);//background color
wcx2.lpszClassName="WinXButtons";//class name
wcx2.hIconSm=NULL;//small icon
//register the window class, return 0 if not successful
if(!RegisterClassEx(&wcx2)) return(0);
//Create our buttons
hWndCommand=CreateWindowEx(NULL,"WinXButtons","Commands Window", WS_VISIBLE| WS_CAPTION | WS_EX_TOOLWINDOW,MainWindowX + 10,0,300,MainWindowY,hWndCommand,hmenu2,hInstMain,NULL);
now this works great both windows function how I want them to, now I tried putting some code in my windows procs for the two windows as such: anytime the main window is moved the position is stored and the secondary window is moved to where it should be but it just ends up disappearing most of the time... I use and take the .left and .top parameters and add my main windows length to .left (its 800)
GetWindowRect(hWndMain, &rect);// get the Main windows rect
SetWindowPos(hWndCommand, HWND_TOP, rect.left+800,rect.top,NULL,NULL,SWP_NOSIZE)
ugh I just wish there was some way I could tell windows to lock this secondary window to my main one... Also another problem, my second window pops up on my task bar this is undesirable but I can live with that, anyone know a solution to either of these problems, or maybe a link to something I can look at. Thanks in advance! [Edited by - yewbie on January 9, 2009 9:35:41 PM]
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Nm I solved this problem, it was updating my variable every move with addition instead of replacing the variable with the new one FTL!
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Rd Unnctn Specification of part produced at Dave'$Metal Fabricators: Day Measurement] 60 66 75 89 92 70 65 52 25 69 72 01-Nov-14 02-Nov-14 03-Nov-14 04-Nov-14 05-Nov-14 06 Nov-14 07-Nov-14 08-Nov-14 09-Nov-14 10-Nov-14 11-Nov-14 12-Nov-14 Detils A ## Answers #### Similar Solved Questions 5 answers ##### 10. Circle the product you would expect to result from the following reaction (4 points)?OHCOzHCOzHCOzHnot pictured 10. Circle the product you would expect to result from the following reaction (4 points)? OH COzH COzH COzH not pictured... 5 answers ##### A newspaper article reported that people spend mean of 6.5 hours per day watching TV with standard deviation of 1.7 hours psychologist would Ilke to conduct interviews with the 20% of the population who spend the most time watching TV. She assumes that the daily time people spend watching TV is normally distributed _ At least how many hours of daily TV watching are necessary for person to be eligible for the interview? Carry your intermediate computations to at least four decimal places_ Round v A newspaper article reported that people spend mean of 6.5 hours per day watching TV with standard deviation of 1.7 hours psychologist would Ilke to conduct interviews with the 20% of the population who spend the most time watching TV. She assumes that the daily time people spend watching TV is norm... 2 answers ##### 5. (Logistic Regression)Consider the logistic regression model of data points (Ti,yi) . Here, the outputs, Y, are binary outcomes. Recall the logistic regression model assigns a probability between 0 and 1 for an input €Bo + B1zp(c) 1 + efo+Biz and this input is then considered as the weight parameter of bernoulli random variable, Y, which can take on values of 0 or 1_ P(Y) = p(x)Y (1 ~P(z)) (~Y) a) Write out the expression for the log-likelihood of the above model for the observed data points 5. (Logistic Regression) Consider the logistic regression model of data points (Ti,yi) . Here, the outputs, Y, are binary outcomes. Recall the logistic regression model assigns a probability between 0 and 1 for an input € Bo + B1z p(c) 1 + efo+Biz and this input is then considered as the weigh... 5 answers ##### QUESTION 5Use synthetic division to find f(k).k=2 +i; f(x) =x3 10 QUESTION 5 Use synthetic division to find f(k). k=2 +i; f(x) =x3 10... 4 answers ##### Let Write {Give the jil of fo5 ) (f 1 iterval ) notation Let Write {Give the jil of fo5 ) (f 1 iterval ) notation... 5 answers ##### Consider the reaction$2 mathrm{SO}_{2}(g)+mathrm{O}_{2}(g) ightarrow 2 mathrm{SO}_{3}(g)$(a) What volume in liters of oxygen gas at STP is required to produce 2 moles of sulfur trioxide gas?(b) What volume in liters of oxygen gas at$25.0^{circ} mathrm{C}$and 1 atm pressure is required to produce 2 moles of sulfur trioxide gas? Consider the reaction$2 mathrm{SO}_{2}(g)+mathrm{O}_{2}(g) ightarrow 2 mathrm{SO}_{3}(g)$(a) What volume in liters of oxygen gas at STP is required to produce 2 moles of sulfur trioxide gas? (b) What volume in liters of oxygen gas at$25.0^{circ} mathrm{C}$and 1 atm pressure is required to produ... 5 answers ##### 1 220 I2 1 ul 8 3 1 H 2 L 1 I 33 IVY 1 1 j 1 I 6 1 L ! 1 1 1 220 I2 1 ul 8 3 1 H 2 L 1 I 33 IVY 1 1 j 1 I 6 1 L ! 1 1... 5 answers ##### Findtor Ihu function y (sin(x))'SolutionNcar Enter dyldt toror Ior yWs have(sin(r)}t(take logarithms ol both sides iha equaikn and u50 the laws ot Iogarlthms ia sin lity)(lind tho derivetiva wlth respect to ~)(iso ate(gve tha (inal Hemg ot #) Find tor Ihu function y (sin(x))' Solution Ncar Enter dyldt tor or Ior y Ws have (sin(r)}t (take logarithms ol both sides iha equaikn and u50 the laws ot Iogarlthms ia sin lity) (lind tho derivetiva wlth respect to ~) (iso ate (gve tha (inal Hemg ot #)... 5 answers ##### Find the kernel of the linear transformation.$$T: R^{2} ightarrow R^{2}, T(x, y)=(x-y, y-x)$$ Find the kernel of the linear transformation. $$T: R^{2} \rightarrow R^{2}, T(x, y)=(x-y, y-x)$$... 5 answers ##### 0 E 8 8 14/9 1 1 aj?1 0 E 8 8 14/9 1 1 aj? 1... 1 answers ##### Consider the following aqueous solutions:(i)$0.20m\mathrm{HOCH}_{2} \mathrm{CH}_{2} \mathrm{OH}$(nonvolatile, nonelectrolyte); (ii)$0.10m\mathrm{CaCl}_{2}$; (iii)$0.12m\mathrm{KBr}$; and (iv)$0.12m\mathrm{Na}_{2} \mathrm{SO}_{4}$. (a) Which solution has the highest boiling point? (b) Which solution has the lowest freezing point? (c) Which solution has the highest water vapor pressure? Consider the following aqueous solutions:(i)$0.20m\mathrm{HOCH}_{2} \mathrm{CH}_{2} \mathrm{OH}$(nonvolatile, nonelectrolyte); (ii)$0.10m\mathrm{CaCl}_{2}$; (iii)$0.12m\mathrm{KBr}$; and (iv)$0.12m\mathrm{Na}_{2} \mathrm{SO}_{4}$. (a) Which solution has the highest boil... 5 answers ##### Tork and Voikina have twro children. One aboynamed Torky and the other is qid rated Vorki. Many years later. Tcrky mcets and manies girl named Morkalina vcho short What are the possibilitics for thc hoight ol thcir otfspring? Hint: Lookat 54 tor Intormation on Torky:Vorki the dalghler ieols zork named Spork; wlro heterozygous Ior LallHow many wIll be tall? How many wlll be shon? How many Will be TT? How many will be Tt? How many will be tt?Totry nas greenhar and Molkalna nas yelon halr . They Tork and Voikina have twro children. One aboynamed Torky and the other is qid rated Vorki. Many years later. Tcrky mcets and manies girl named Morkalina vcho short What are the possibilitics for thc hoight ol thcir otfspring? Hint: Lookat 54 tor Intormation on Torky: Vorki the dalghler ieols zork n... 5 answers ##### The function$f$is one-to-one. (a) Find its inverse function$f^{-1}$and check your answer. (b) Find the domain and the range of f and$f^{-1}$$f(x)= rac{2 x-3}{x+4}$$
the function $f$ is one-to-one. (a) Find its inverse function $f^{-1}$ and check your answer. (b) Find the domain and the range of f and $f^{-1}$ $$f(x)=\frac{2 x-3}{x+4}$$...
##### Review ConstantsA cellist tunes the C-string of her instrument to fundamental frequency of 65.4 Hz. The vibrating porlion of the string is 0.610 mn long and has mass of 14.1 gPart AWith what tension must she stretch it?Express your answer In newtons:AEdSubmitRequest AnswerPart BWhat percent increase in tension is needed t0 increase the frequency from 65.4 Hz to 73.4 Hz, corresponding to rise in pitch from C to D? Express your answer in percents_AEdSubmitRequest Answer
Review Constants A cellist tunes the C-string of her instrument to fundamental frequency of 65.4 Hz. The vibrating porlion of the string is 0.610 mn long and has mass of 14.1 g Part A With what tension must she stretch it? Express your answer In newtons: AEd Submit Request Answer Part B What percent...
##### (1 point) The Normal distribution with mean / = 6.8 and standard deviation 0 = 6 is good description of the Iowa Test vocabulary scores of seventh grade students in Gary' Indiana. This continuous probability model for the score of randomly chosen student. Figure 3. (p 68) pictures the density curve. Call the score of a randomly chosen student X for short:Note: If necessary, userepresentand >= t0 representWrite the event "the student chosen has score 0f 10 or higher' in terms Of
(1 point) The Normal distribution with mean / = 6.8 and standard deviation 0 = 6 is good description of the Iowa Test vocabulary scores of seventh grade students in Gary' Indiana. This continuous probability model for the score of randomly chosen student. Figure 3. (p 68) pictures the density c...
##### Two beakers arc placed in a mall closed container at 25 "C One caning 209 mL of a 0.274 M aqueous solution of CuSO4; the second contains 446 mL ofa 0.133 M aqucous solution of CuSO4: Small amounts of water evaporate from both solutions As timc passcs, the volum of solution the second beaker gradually and that in the first gradually If we wait long enough; what will the fina] volumes and concentrations be?First beakerSecond beakerInitial209 mL0.274 M446 mL0.IJ} MFinalsubmlt Answerquastlon at
Two beakers arc placed in a mall closed container at 25 "C One caning 209 mL of a 0.274 M aqueous solution of CuSO4; the second contains 446 mL ofa 0.133 M aqucous solution of CuSO4: Small amounts of water evaporate from both solutions As timc passcs, the volum of solution the second beaker gra...
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# How can I apply Bayesian Statistics when the number of data that I have is 1?
I want to show why Maximum Likelihood Estimation is not the right choice when the number of data that I have only is 1. For example, if I have an observed data which is heads from a coin toss. I only toss the coin once and apply Maximum Likelihood Estimation to get the probability of heads. If I apply Maximum Likelihood Estimation to my observed data, I will get 1 for the probability of heads up, tossing the coin only once.
How can I apply my prior experience, the probability of heads up is 0.5, to the situation and calculate the Bayesian way of gettting Maximum Likelihood Estimation which is Maximum a Posterior? I am not sure if I should get the mean of the posterior or the MAP.
• A Bayesian approach returns a whole distribution, not an estimator. – Xi'an Jan 17 at 18:06
• @whuber re: rejected edit. Line 7, there is a clear grammatical error ("Likelhood"). As you have rejected this correction, could you please either make a correction or propose that the OP makes it. I believe it was an erroneous rejection. But if not, this may somewhat discourage editing – PsychometStats Jan 18 at 16:05
TL;DR you can, but the result would strongly depend on your choice of prior.
With maximum likelihood, you would be maximizing the likelihood, that in this case is defined in terms of probability mass function $$f$$ of Bernoulli distribution, i.e. binomial distribution with number of trials $$n=1$$, parametrized by probability of success $$\theta$$
$$\hat\theta = \underset{\theta}{\operatorname{arg\,max}} \; f(X|\theta)$$
In Bayesian setting what changes is that instead of looking for point estimate for $$\theta$$, we learn posterior distribution $$\pi(\theta|X)$$, and we start with a prior distribution $$\pi(\theta)$$ for $$\theta$$
$$\pi(\theta|X) \propto f(X|\theta)\,\pi(\theta)$$
when calculating maximum a posteriori point estimate, you would be maximizing the posterior probability
$$\hat\theta = \underset{\theta}{\operatorname{arg\,max}} \; f(X|\theta) \,\pi(\theta)$$
As you can see, what changes, is that we multiply likelihood by prior. In case of binomial distribution, if we choose beta distribution as a prior, then there exists nice, closed-form solution. If as a prior we choose
$$\theta \sim \mathsf{Beta}(\alpha, \beta)$$
then the posterior distribution is
$$\theta|X \sim \mathsf{Beta}(\alpha + x, \beta+ n-x)$$
where $$n=1$$ is the number of trials, and $$x=1$$ is the number of successes. So in your case, the mean of the distribution is
$$E[\theta|X] = \frac{\alpha + 1}{\alpha+1+\beta}$$
For details, you can check the great What is the intuition behind beta distribution? thread. As you can see, choosing different prior parameters $$\alpha$$, $$\beta$$, would lead to different results and would have significant impact on the final estimate. If you want to assume a priori the probability to be something close to $$0.5$$, you need to set $$\alpha$$, $$\beta$$ to same values. For example, if you set $$\alpha=\beta=1$$, you would estimate $$E[\theta|X]$$ to be $$0.66$$, while $$\alpha=\beta=0.5$$ would lead you to estimating it as $$0.75$$. This impact would diminish with growing sample size, but with single sample it would be quite profound. So using Bayesian approach would enable you to estimate something more reasonable then $$\hat\theta=\tfrac{1}{1}=1$$, but how reasonable the estimate would be, would depend on how reasonable your prior was.
As a sidenote, your example is not that uncommon. In fact, it is often the case to use Bayesian estimators for calculating probabilities when we expect to see zero counts. It is commonly used for working with textual data, where we deal with counts of words. Obviously, some words occur very frequent, e.g. "and", "the", while other are pretty rare, e.g. "aardvark". Estimating probabilities for the common words is straightforward, but for rare words we would end up $$\tfrac{0}{n}$$ as estimated probabilities. When using algorithms like Naive Bayes, where we multiply the probabilities by each other, this would lead to zeroing-out everything after plugging-in single zero to the formula, that is why we use Laplace smoothing.
• Thank you for the answer. I just wonder what I should do with the value that I have found using Maximum a Posterior. Just pretending I wonder the probability of heads up with 1 coin tossing experiment using either Bernoulli or Binomial Mass Function, Can I apply the value that I have found using Maximum a Posterior just like I have found the value from Maximum Likelihood Estimation to either Bernoulli or Binomial Mass Function to find the probability of 1 heads up? I just want to make sure if I am thing correctly. – Changhee Kang Jan 22 at 18:13
• @ChangheeKang MAP is a point estimate, like any other point estimate, e.g. MLE. – Tim Jan 22 at 19:46
When the number of data values is less than the number of parameters in a statistical model the statistical model does not usually provide anything useful. You have one datum and one parameter (the probability of heads) and the datum does not convey very much information because it is a dichotomous outcome rather than continuous, so don't expect too much.
The calculation that you need is not based on the maximal likelihood estimate and the prior maximum because, as X'ian points out in his comment, the relevant Bayesian calculation uses the whole range of parameter values (i.e. all probabilities of heads from zero to one).
You have to provide a complete prior probability function for heads. What you say means that its peak is at 0.5, but you have to decide on how wide the function is, and its shape. (The function has Pr(heads) on the x-axis and probability density on the y-axis).
Give the observation of a single heads, the likelihood function is a right triangle with its maximum at Pr(heads)=1 and minimum of 0 at Pr(heads)=0. Multiply that function by your prior probability density function and re-scale the result to have unit integral, and you have your posterior.
Your posterior probability function for Pr(heads) will not differ much from your prior if your prior is a narrow peak, but will differ more substantially if your prior includes a lot of weight at the Pr(heads)=0 end.
Interpretation of the results is straightforward: a. On the basis of the observation alone, the most likely probability of heads is 1, and Pr(heads)=0 can be ruled out. The likelihood of Pr(heads)=0.5 is half as high as the likelihood of Pr(heads)=1, but a different of two-fold is quite trivial in most cases and the likelihood function is very flat and so it is not very discriminating. b. Your prior expectation is that the most probable value for Pr(heads)=0.5. c. The posterior function shows how probable you should think each value of Pr(heads) is. Do not focus only on the maximum, but see what ranges of Pr(heads) are reasonably plausible (for some version of "reasonably" and "plausible"). A credible interval might be useful.
Note that more data can be added by simply multiplying that posterior by a new likelihood function.
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# Question for finding Probability for sum of two Poisson Distribution
A survey of downtown Chicago found that the typical city block had an average of five rats per block. This was true irrespective of whether or not there was construction occurring on that block. If this average is applicable across all city blocks, what is the probability that, in a sample of two city blocks taken together, there will be a total of five or fewer rats for those two blocks?
In my view, it is a Poisson distribution. Here the average is given, i.e., $$\lambda=5$$. (I have looked at the table even.) But I'm not sure how to solve for two city blocks when taken together.
• Iconoclast, welcome to Cross Validated. Your problem seems like a homework problem, so you should tag it as "self-study" and then explain as much of your solution as you can. You are on the right track. You need to find distribution of the sum of two Poisson random variables. Anything you tell us will help us answer, since the answer could involve moment generating functions, or a double sum, depending on what you've been taught so far. To get you started, if you expect 5 rats on one block, how many would you expect on two, given counts per block are independent and identically distributed? – Peter Leopold Sep 15 at 5:31
• @Iconoclast Per your title -- you're not adding the distributions, you're adding the random variables. When they're independent, the operation applied to the pmfs is convolution, rather than addition. – Glen_b -Reinstate Monica Sep 15 at 8:13
Let's suppose your model is correct and in each city block, if you take a survey by un unspecified method, your probability of finding $$n$$ rats is $$P(X=n)$$ where $$X \sim Poisson(\lambda)$$ and $$\lambda = 5$$.
Then, according to the hypotheses in you question the probability of observing rats in each block is i.i.d. (independent and identically distributed); for each block we have $$X_i \sim Poisson(5)$$.
If we visit two blocks, the number of rats we are going to see is the random variable $$Y \sim X_1 + X_2$$.
For a property of the Poisson distribution, check your book, $$Y$$ will be again a Poisson distribution $$Y \sim Poisson(\lambda_1 + \lambda_2)$$ which in our case is $$Y \sim Poisson(10)$$.
Finally, you want to find the probability of observing 5 or fewer rats, that is $$P(Y \le 5)$$. You can compute that in the way you prefer. Using R it is
> ppois(5, 10)
[1] 0.06708596
• Nicely explained (+1). But be mindful that it's not always helpful over the long run to give complete and explicit answers to hwk problems, especially when OP has not shown any engagement toward an answer. (Also, I'd say not always unhelpful either.) – BruceET Sep 15 at 6:23
• Hi, thank you, it is the first time I answer in this group, I don't know your habits. I understand your concern, it is true. There is the possiblity the student is asking just to evade thinking. Let's hope this student will check the formula on the book and who know, maybe he will try to do the proof;) – Nicola Mingotti Sep 15 at 7:17
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Three cationic polyelectrolytes with different charge densities were used to condition activated sludge samples from a municipal plant during a period when the sludge characteristics changed strongly. Initially the sludge had high residual turbidity after sedimentation, high filtration resistance, low floc strength and high amounts of extractable extracellular polymers (ECP). These problems could be related to the snow melting that was going on. When that ceased the sludge recovered as shown by decreased residual turbidity, filtration resistance and amount of extractable ECP and increased floc strength.
The polyelectrolytes with 40 and 100 % charge density gave good and similar filterabilities during the whole investigation period. With the 10 % charged polymer the filtration resistance was generally higher and especially so during the snow melting period. The reason is that low charge density polymers flocculate with a bridging mechanism. This gives a flexible floc structure which allows the formation of dense filter cakes and also to some extent preserves the original structure. High charge density polymers give closer contacts between the particles and thus a more inflexible and open structure which is favourable in filtration.
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# What is context free grammer of $L = \{w: n_c(w) \ne n_a(w) + n_b(w)\}$
I can't find out how to find a context free grammar for bellow language, is there any specific way to solve that?
$$L = \{w: n_c(w) \ne n_a(w) + n_b(w)\}$$
$$\Sigma = \{a,b,c\}$$. Consider the two cases:
1. $$n_c(w) > n_a(w) + n_b(w)$$: The idea is to create one more $$c$$ for each $$a$$ or $$b$$ produce in the word: \begin{align} S_c \rightarrow aS_cS_c | bS_cS_c | cS_c | c \end{align}
2. $$n_c(w) < n_a(w) + n_b(w)$$: In this case, we can produce as many $$c$$'s as $$a$$ and $$b$$ combined + 1. \begin{align} S_{ab} \rightarrow aS_{ab} | bS_{ab} | cS_{ab}S_{ab}|a|b \end{align}
The grammar for the required language would be : $$S \rightarrow S_c | S_{ab}$$.
• How produce $cca$? I think you should write \begin{align} S_c \rightarrow aS_cS_c | bS_cS_c |cS_c| c \end{align}
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A journal of IEEE and CAA , publishes high-quality papers in English on original theoretical/experimental research and development in all areas of automation
Volume 7 Issue 1
Jan. 2020
IEEE/CAA Journal of Automatica Sinica
• JCR Impact Factor: 6.171, Top 11% (SCI Q1)
CiteScore: 11.2, Top 5% (Q1)
Google Scholar h5-index: 51， TOP 8
Turn off MathJax
Article Contents
Nezar M. Alyazidi and Magdi S. Mahmoud, "Distributed \begin{document}$H_{2}/H_\infty$\end{document} Filter Design for Discrete-Time Switched Systems," IEEE/CAA J. Autom. Sinica, vol. 7, no. 1, pp. 158-168, Jan. 2020. doi: 10.1109/JAS.2019.1911630
Citation: Nezar M. Alyazidi and Magdi S. Mahmoud, "Distributed \begin{document}$H_{2}/H_\infty$\end{document} Filter Design for Discrete-Time Switched Systems," IEEE/CAA J. Autom. Sinica, vol. 7, no. 1, pp. 158-168, Jan. 2020.
# Distributed $H_{2}/H_\infty$ Filter Design for Discrete-Time Switched Systems
##### doi: 10.1109/JAS.2019.1911630
Funds:
the Deanship of Scientific Research (DSR) at KFUPM through distinguished professorship project 161065
• This paper addresses an infinite horizon distributed $\pmb{H}_{\bf 2}/\pmb{H}_{\bf\infty}$ filtering for discrete-time systems under conditions of bounded power and white stochastic signals. The filter algorithm is designed by computing a pair of gains namely the estimator and the coupling. Herein, we implement a filter to estimate unknown parameters such that the closed-loop multi-sensor accomplishes the desired performances of the proposed $\pmb H_{\bf 2}$ and $\pmb{H_{\bf\infty}}$ schemes over a finite horizon. A switched strategy is implemented to switch between the states once the operation conditions have changed due to disturbances. It is shown that the stability of the overall filtering-error system with $\pmb{H}_{\bf 2}/\pmb{H}_{\bf\infty}$ performance can be established if a piecewise-quadratic Lyapunov function is properly constructed. A simulation example is given to show the effectiveness of the proposed approach.
• Recommended by Associate Editor Mohammed Chadli.
• [1] M. S. Mahmoud, "Distributed estimation based on information-based covariance intersection algorithms", Int. J. Adapt. Control Signal Process, vol. 30, pp.750-778, 2015. https://www.researchgate.net/publication/283958509_Distributed_estimation_based_on_information-based_covariance_intersection_algorithms [2] R. S. Olfati, "Kalman-consensus filter: optimality, stability, and performance" in Proc. 28th Chinese Decision and Control Conf., pp. 7036-7042, 2009. https://www.researchgate.net/publication/224108550_Kalman-Consensus_Filter_Optimality_Stability_and_Performance [3] F. S. Cattivelli and A.H. Sayed, "Diffusion strategies for distributed Kalman filtering and smoothing, " IEEE Trans. Automatic Control, vol. 55, no. 9, pp. 2069-2084, 2010. [4] M. S. Mahmoud, L. H. Xie, and C. Y. Soh, "Robust Kalman filtering for discrete state-delay systems", in Proc. IEEE Control Theory and Applications, vol. 147, no. 6, pp. 613-618, 2000. https://www.researchgate.net/publication/3352360_Robust_Kalman_filtering_for_discrete_state-delay_systems [5] M. S. Mahmou and H. M. Khalid, "Distributed Kalman filtering: a bibliographic review, " IET Control Theory and Applications, vol. 7, no. 4, pp.483-501, 2013. [6] Y. Zhu, Z. You, J. Zhao, K. Zhang, and R. X. Li, "The optimality for the distributed Kalman filtering fusion", Automatica, vol. 37, no. 9, pp. 1489-1493, 2001. [7] I. Petersen and A. Savkin, "Robust Kalman filtering for signals and systems with large uncertainties, " Control Engineering, 2013. [8] J. Hu, L. Xie, and C. Zhang, "Diffusion Kalman filtering based on covariance intersection, " IEEE Trans. Signal Processing, vol. 60, no. 2, pp. 891-902, 2012. [9] B. Shen, Z. Wang, and Y. Hung, "Distributed-consensus filtering in sensor networks with multiple missing measurements: the finite-horizon case, " Automatica, vol. 46, no. 10, pp. 1682-1688, 2010. [10] V. Ugrinovskii, "Distributed robust filtering with consensus of estimates, " Automatica, vol. 47, no. 1, pp. 1-13, 2011. [11] V. Ugrinovskii, "Distributed robust estimation over randomly switching networks using consensus, " Automatica, vol. 49, no. 1, pp. 160-168, 2013. [12] T. Jiang, I. Matei, and J. S. Baras, "A trust-based distributed Kalman filtering approach for mode estimation in power systems", in Proc. 1st Workshop Secure Control Systems (SCS), pp. 1-6, April 12, 2010. [13] S. R. Olfati, "Distributed Kalman filtering for sensor networks." in Proc. IEEE 46th Conf. Decision and Control, pp. 5492-5498, 2007. https://www.researchgate.net/publication/224303218_Distributed_Kalman_filtering_for_sensor_networks [14] C. Jiang, Y. Chen, and K. R. Liu, "Distributed adaptive networks: a graphical evolutionary game-theoretic view, " IEEE Trans. Signal Processing, vol. 61, no. 22, pp.5675-5688, 2013. [15] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control, New Jersey: Prentice hall, vol. 40, 1996. http://d.old.wanfangdata.com.cn/Periodical/zdhxb200301010 [16] B. D. Anderson and J. B. Moore, "Optimal control: linear quadratic methods, " Courier Corporation, Feb. 27, 2007. https://www.researchgate.net/publication/260228778_Optimal_Control_Linear_Quadratic_Method [17] C. Olalla, R. Leyva, A. El Aroudi, and I. Queinnec, "Robust LQR control for PWM converters: an LMI approach, " IEEE Trans. Industrial Electronics, vol. 56, no. 7, pp. 2548-2558, 2009. [18] B. S. Chen and W. Zhang, "Stochastic $H_{2}/H_\infty$ control with state-dependent noise, " IEEE Trans. Automatic Control, vol. 49, no. 1, pp. 45-57, 2004. [19] W. Zhang, Y. Huang, and H. Zhang, "Stochastic $H_{2}/H_\infty$ control for discrete-time systems with state and disturbance dependent noise, " Automatica, vol.43, no. 3, pp. 513-521, 2007. [20] D. Bernstein and M. Haddad, "LQG control with an $H_\infty$ performance bound: a Riccati equation approach, " IEEE Trans. Automatic Control, vol. 34, no. 3, pp. 293-305, 1989. doi: 10.1109/9.16419 [21] C. S. Tseng and B. S. Chen, "A mixed $H_{2}/H_\infty$ adaptive tracking control for constrained non-holonomic systems, " Automatica, vol. 39, no. 6, pp. 1011-1018, 2003. [22] C. Chen, W. Dong and V. Djapic, " Distributed $H_{2}/H_\infty$ filtering over infinite horizon, " Int. J. Adaptive Control and Signal Processing, vol. 32, no. 2, pp. 330-343, 2018. doi: 10.1002/acs.2847 [23] M. S. Mahmoud, Switched Time-Delay Systems: Stability and Control, Boston: Springer, pp. 109-130, 2010. http://d.old.wanfangdata.com.cn/Periodical/xtgcydzjs201001033 [24] F. L. Lewis and V. L. Syrmos, Optimal Control, John Wiley & Sons, 1995. http://d.old.wanfangdata.com.cn/Periodical/lxxb201606015 [25] N. M. Alyazidi, M. S. Mahmoud, and M. I. Abouheaf, "Adaptive critics based cooperative control scheme for islanded microgrids, " Neurocomputing, vol. 272, pp.532-541, 2018.
### Catalog
###### 通讯作者: 陈斌, bchen63@163.com
• 1.
沈阳化工大学材料科学与工程学院 沈阳 110142
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6.8: Baire Categories. More on Linear Maps
We pause to outline the theory of so-called sets of Category I or Category II, as introduced by Baire. It is one of the most powerful tools in higher analysis. Below, $$(S, \rho)$$ is a metric space.
Definition 1
A set $$A \subseteq(S, \rho)$$ is said to be nowhere dense (in $$S)$$ iff its closure $$\overline{A}$$ has no interior points (i.e., contains no globes): $$(\overline{A})^{0}=\emptyset$$.
Equivalently, the set $$A$$ is nowhere dense iff every open set $$G^{*} \neq \emptyset$$ in $$S$$ contains a globe $$\overline{G}$$ disjoint from $$A.$$ (Why?)
Definition 2
A set $$A \subseteq(S, \rho)$$ is meagre, or of Category I (in $$S),$$ iff
$A=\bigcup_{n=1}^{\infty} A_{n},$
for some sequence of nowhere dense sets $$A_{n}$$.
Otherwise, $$A$$ is said to be nonmeagre or of Category II.
$$A$$ is residual iff $$-A$$ is meagre, but $$A$$ is not.
Examples
(a) $$\emptyset$$ is nowhere dense.
(b) Any finite set in a normed space $$E$$ is nowhere dense.
(c) The set $$N$$ of all naturals in $$E^{1}$$ is nowhere dense.
(d) So also is Cantor's set $$P$$ (Problem 17 in Chapter 3, §14); indeed, $$P$$ is closed $$(P=\overline{P})$$ and has no interior points (verify!), so $$(\overline{P})^{0}=P^{0}=\emptyset$$.
(e) The set $$R$$ of all rationals in $$E^{1}$$ is meagre; for it is countable (see Chapter 1, §9), hence a countable union of nowhere dense singletons {$$r_{n}$$}, $$r_{n} \in R.$$ But $$R$$ is not nowhere dense; it is even dense in $$E^{1},$$ since $$\overline{R}=E^{1}$$ (see Definition 2, in Chapter 3, §14). Thus a meagre set need not be nowhere dense. (But all nowhere dense sets are meagre why?)
Examples (c) and (d) show that a nowhere dense set may be infinite (even uncountable). Yet, sometimes nowhere dense sets are treated as "small" or "negligible," in comparison with other sets. Most important is the following theorem.
Theorem $$\PageIndex{1}$$ (Baire)
In a complete metric space $$(S, \rho),$$ every open set $$G^{*} \neq \emptyset$$ is nonmeagre. Hence the entire space $$S$$ is residual.
Proof
Seeking a contradiction, suppose $$G^{*}$$ is meagre, i.e.,
$G^{*}=\bigcup_{n=1}^{\infty} A_{n}$
for some nowhere dense sets $$A_{n}.$$ Now, as $$A_{1}$$ is nowhere dense, $$G^{*}$$ contains a closed globe
$\overline{G}_{1}=\overline{G_{x_{1}}\left(\delta_{1}\right)} \subseteq-A_{1}.$
Again, as $$A_{2}$$ is nowhere dense, $$G_{1}$$ contains a globe
$\overline{G}_{2}=\overline{G_{x_{2}}\left(\delta_{2}\right)} \subseteq-A_{2}, \quad \text { with } 0<\delta_{2} \leq \frac{1}{2} \delta_{1}.$
By induction, we obtain a contracting sequence of closed globes
$\overline{G}_{n}=\overline{G_{x_{n}}\left(\delta_{n}\right)}, \quad \text { with } 0<\delta_{n} \leq \frac{1}{2^{n}} \delta_{1} \rightarrow 0.$
As $$S$$ is complete, so are the $$\overline{G}_{n}$$ (Theorem 5 in Chapter 3, §17). Thus, by Cantor's theorem (Theorem 5 of Chapter 4, §6), there is
$p \in \bigcap_{n=1}^{\infty} \overline{G}_{n}.$
As $$G^{*} \supseteq \overline{G}_{n},$$ we have $$p \in G^{*}.$$ But, as $$\overline{G}_{n} \subseteq-A_{n},$$ we also have $$(\forall n) p \notin A_{n}$$ ; hence
$p \notin \bigcup_{n=1}^{\infty} A_{n}=G^{*}$
(the desired contradiction!).$$\quad \square$$
We shall need a lemma based on Problems 15 and 19 in §7. (Review them!)
lemma
Let $$f \in L\left(E^{\prime}, E\right), E^{\prime}$$ complete. Let $$G=G_{0}(1)$$ be the unit globe in $$E^{\prime}.$$ If $$\overline{f[G]}$$ (closure of $$f[G]$$ in $$E$$ ) contains a globe $$G_{0}=G_{0}(r) \subset E,$$ then $$G_{0} \subseteq f[G].$$
Note. Recall that we "arrow" only vectors from $$E^{\prime}$$ (e.g., $$\overrightarrow{0}),$$ but not those from $$E$$ (e.g., 0).
Proof
Let $$A=f[G] \cap G_{0} \subseteq G_{0}.$$ We claim that $$A$$ is dense in $$G_{0};$$ i.e., $$G_{0} \subseteq \overline{A}.$$ Indeed, by assumption, any $$q \in G_{0}$$ is in $$f[G].$$ Thus by Theorem 3 in Chapter 3, §16, any $$G_{q}$$ meets $$f[G] \cap G_{0}=A$$ if $$q \in G_{0}.$$ Hence
$\left(\forall q \in G_{0}\right) \quad q \in \overline{A},$
i.e., $$G_{0} \subseteq \overline{A},$$ as claimed.
Now fix any $$q_{0} \in G_{0}=G_{0}(r)$$ and a real $$c(0<c<1).$$ As $$A$$ is dense in $$G_{0}$$,
$A \cap G_{q_{0}}(c r) \neq \emptyset;$
so let $$q_{1} \in A \cap G_{q_{0}}(c r) \subseteq f[G].$$ Then
$\left|q_{1}-q_{0}\right|<c r, \quad q_{0} \in G_{q_{1}}(c r).$
As $$q_{1} \in f[G],$$ we can fix some $$\vec{p}_{1} \in G=G_{0}(1),$$ with $$f\left(\vec{p}_{1}\right)=q_{1}.$$ Also, by Problems 19(iv) and 15(iii) in §7, $$c A+q_{1}$$ is dense in $$c G_{0}+q_{1}=G_{q_{1}}(c r)$$. But $$q_{0} \in G_{q_{1}}(cr)$$. Thus
$G_{q_{0}}\left(c^{2} r\right) \cap\left(c A+q_{1}\right) \neq \emptyset;$
so let $$q_{2} \in G_{q_{0}}\left(c^{2} r\right) \cap\left(c A+q_{1}\right),$$ so $$q_{0} \in G_{q_{2}}\left(c^{2} r\right),$$ etc.
Inductively, we fix for each $$n>1$$ some $$q_{n} \in G_{q_{0}}\left(c^{n} r\right),$$ with
$q_{n} \in c^{n-1} A+q_{n-1},$
i.e.,
$q_{n}-q_{n-1} \in c^{n-1} A.$
As $$A \subseteq f\left[G_{0}(1)\right],$$ linearity yields
$q_{n}-q_{n-1} \in f\left[c^{n-1} G_{0}(1)\right]=f\left[G_{0}\left(c^{n-1}\right)\right], \quad n>1.$
Thus for each $$n>1,$$ there is $$\vec{p}_{n} \in G_{0}(c^{n-1})$$, (i.e., $$|\vec{p}_{n}|<c^{n-1})$$ such that $$f(\vec{p}_{n})=q_{n}-q_{n-1}.$$ Now, as $$|\vec{p}_{n}|<c^{n-1}$$ and $$0<c<1$$,
$\sum_{1}^{\infty}\left|\vec{p}_{n}\right|<+\infty;$
so by the completeness of $$E^{\prime}, \sum \vec{p}_{n}$$ converges in $$E^{\prime}$$ (Theorem 1 in Chapter 4, §13). Let $$\vec{p}=\sum_{k=1}^{\infty} \vec{p}_{k};$$ then
\begin{aligned} f(\vec{p}) &=f\left(\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \vec{p}_{k}\right)=\lim _{n \rightarrow \infty} f\left(\sum_{k=1}^{n} \vec{p}_{k}\right) \\ &=\lim _{n \rightarrow \infty} \sum_{k=1}^{n} f\left(\vec{p}_{k}\right) \text { for } f \in L\left(E^{\prime}, E\right). \end{aligned}
But $$f(\vec{p}_{k})=q_{k}-q_{k-1}(k>1),$$ and $$f(\vec{p}_{1})=q_{1};$$ so
$\sum_{k=1}^{n} f\left(\vec{p}_{k}\right)=q_{1}+\sum_{k=2}^{n}\left(q_{k}-q_{k-1}\right)=q_{n}.$
Thus
$f(\vec{p})=\lim _{n \rightarrow \infty} \sum_{k=1}^{n} f\left(\vec{p}_{k}\right)=\lim _{n \rightarrow \infty} q_{n}=q_{0}.$
Moreover, $$\left|\vec{p}_{k}\right|<c^{k-1}(k \geq 1).$$ Thus
$|\vec{p}| \leq \sum_{k=1}^{\infty}\left|\vec{p}_{k}\right|<\sum_{k=1}^{\infty} c^{k-1}=\frac{1}{1-c};$
i.e.,
$\vec{p} \in G_{\overrightarrow{0}}\left(\frac{1}{1-c}\right).$
But $$q_{0}=f(\vec{p});$$ so
$q_{0} \in f\left[G_{\overrightarrow{0}}\left(\frac{1}{1-c}\right)\right].$
As $$q_{0} \in G_{0}(r)$$ was arbitrary, we have
$G_{0}(r) \subseteq f\left[G_{0}\left(\frac{1}{1-c}\right)\right],$
or by linearity,
$G_{0}(r(1-c)) \subseteq f\left[G_{0}(1)\right]=f[G].$
This holds for any $$c \in(0,1).$$ Hence
$f[G] \supseteq \bigcup_{0<c<1} G_{0}(r(1-c))=G_{0}(r). \quad \text {(Verify!)}$
Thus all is proved.$$\quad \square$$
We can now establish an important result due to S. Banach.
Theorem $$\PageIndex{2}$$ (Banach)
Let $$f \in L\left(E^{\prime}, E\right),$$ with $$E^{\prime}$$ complete. Then $$f\left[E^{\prime}\right]$$ is meagre in $$E$$ or $$f\left[E^{\prime}\right]=E,$$ according to whether $$f\left[G_{0}(1)\right]$$ is or is not nowhere dense.
Proof
If $$f\left[G_{0}(1)\right]$$ is nowhere dense in $$E,$$ so also is $$f\left[G_{0}(n)\right], n>0.$$ (Verify by Problems 15 and 19 in §7.) But then
$f\left[E^{\prime}\right]=f\left[\bigcup_{n=1}^{\infty} G_{0}(n)\right]=\bigcup_{n=1}^{\infty} f\left[G_{\overrightarrow{0}}(n)\right]$
is a countable union of nowhere dense sets, hence meagre, by definition.
Now suppose $$f\left[G_{0}(1)\right]$$ is not nowhere dense; so $$\overline{f\left[G_{0}(1)\right]}$$ contains some $$G_{q}(r) \subseteq E.$$ We may assume $$q \in f\left[G_{\overrightarrow{0}}(1)\right]$$ (if not, replace $$q$$ by a close point from $$f\left[G_{0}(1)\right]).$$ Then $$q=f(\vec{p})$$ for some $$\vec{p} \in G_{0}(1).$$ The latter implies
$|-\vec{p}|=|\vec{p}|=\rho(\vec{p}, \overrightarrow{0})<1;$
so
$G_{-\vec{p}}(1) \subseteq G_{\overrightarrow{0}}(2).$
Also, as $$\overline{f\left[G_{\overline{0}}(1)\right]} \supseteq G_{q}(r),$$ translation by $$-q=f(-\vec{p})$$ yields
$\overline{f\left[G_{\overline{0}}(1)\right]}+f(-\vec{p}) \supseteq G_{q}(r)-q=G_{0}(r),$
i.e.,
$G_{0}(r) \subseteq \overline{f\left[G_{-\vec{p}}(1)\right]} \subseteq \overline{f\left[G_{\overrightarrow{0}}(2)\right]}.$
Hence $$\overline{f\left[G_{\overrightarrow{0}}(1)\right]} \supseteq G_{0}\left(\frac{1}{2} r\right)$$ (why?); so, by the Lemma
$f\left[G_{\overrightarrow{0}}(1)\right] \supseteq G_{0}\left(\frac{1}{2} r\right) \text { in } E.$
This implies $$f\left[G_{\overrightarrow{0}}(2 n)\right] \supseteq G_{0}(n r),$$ and so
$f\left[E^{\prime}\right] \supseteq \bigcup_{n=1}^{\infty} G_{0}(n r)=E,$
i.e., $$f\left[E^{\prime}\right]=E,$$ as required. Thus the theorem is proved.$$\quad \square$$
Theorem $$\PageIndex{3}$$ (Open map principle)
Let $$f \in L\left(E^{\prime}, E\right),$$ with $$E^{\prime}$$ and $$E$$ complete. Then the map $$f$$ is open on $$E^{\prime}$$ iff $$f\left[E^{\prime}\right]=E,$$ i.e., iff $$f$$ is onto $$E$$.
Proof
If $$f\left[E^{\prime}\right]=E,$$ then by Theorem 1, $$f\left[E^{\prime}\right]$$ is nonmeagre in $$E,$$ as is $$E$$ itself. Thus by Theorem 2, $$f\left[G_{\overrightarrow{0}}(1)\right]$$ is not nowhere dense, and (1) follows as before. Hence by Problems 15(iii) and 19 in §7, $$f\left[G_{\vec{p}}\right] \supseteq$$ some $$G_{q}$$ whenever $$q=f(\vec{p})$$. (Why?) Therefore, $$G_{\vec{p}} \subseteq A \subseteq E^{\prime}$$ implies
$G_{f(\vec{p})} \subseteq f\left[G_{\vec{p}}\right] \subseteq f[A];$
i.e., $$f$$ maps any interior point $$\vec{p} \in A$$ into such a point of $$f[A].$$ By Problem 8 in §7, $$f$$ is open on $$E^{\prime}.$$
Conversely, if so, then $$f\left[E^{\prime}\right]$$ is an open set $$\neq \emptyset$$ in $$E,$$ a complete space; so by Theorems 1 and 2, $$f\left[E^{\prime}\right]$$ is nonmeagre and equals $$E.$$ (See also Problem 16(ii) in §7.)$$\quad \square$$
Note 1. Theorem 3 holds even if $$f$$ is not one-to-one.
Note 2. If in Theorem 3, however, $$f$$ is bijective, it is open on $$E^{\prime},$$ and so $$f^{-1} \in L\left(E, E^{\prime}\right)$$ by Note 1 in §7. (This is the promised general proof of Corollary 2 in §6.)
Theorem $$\PageIndex{4}$$ (Banach-Steinhaus uniform boundedness principle)
Let $$E^{\prime} b e$$ complete. Let $$\mathcal{N}$$ be a family of maps $$f \in L\left(E^{\prime}, E\right)$$ such that
$\left(\forall x \in E^{\prime}\right)\left(\exists k \in E^{1}\right)(\forall f \in \mathcal{N}) \quad|f(\vec{x})|<k.$
("$$\mathcal{N}$$ is bounded at each $$\vec{x}$$.")
Then $$\mathcal{N}$$ is "norm-bounded," i.e.,
$\left(\exists K \in E^{1}\right)(\forall f \in \mathcal{N}) \quad\|f\|<K,$
with $$\|\quad\|$$ as in §2.
Proof
It suffices to show that $$\mathcal{N}$$ is "uniformly" bounded on some globe,
$\left(\exists c \in E^{1}\right)\left(\exists G=G_{\vec{p}}(r)\right)(\forall f \in \mathcal{N})(\forall \vec{x} \in G) \quad|f(\vec{x})| \leq c.$
For then $$|\vec{x}-\vec{p}| \leq r$$ implies
$2 c>|f(\vec{x})-f(\vec{p})|=|f(\vec{x}-\vec{p})|,$
or (setting $$\vec{x}-\vec{p}=r \vec{y} )|\vec{y}|<1$$ implies
$(\forall f \in \mathcal{N}) \quad|f(\vec{y})|<\frac{2 c}{r} \quad\text {(why?);}$
so
$(\forall f \in \mathcal{N}) \quad\|f\|=\sup _{|\vec{y}| \leq 1}|f(\vec{y})|<\frac{2 c}{r}.$
Thus, seeking a contradiction, suppose (3) fails and assume its negation:
$\left(\forall c \in E^{1}\right)\left(\forall G=G_{\vec{p}}(r)\right)(\exists f \in \mathcal{N})\left(\exists \vec{x} \in G=G_{\vec{p}}(r)\right) \quad|f(\vec{x})|>c.$
Then for $$c=1,$$ we can fix some $$f_{1} \in \mathcal{N}$$ and $$G_{\vec{x}_{1}}\left(r_{1}\right)$$ such that $$0<r_{1}<1$$ and
$\left|f_{1}\left(\vec{x}_{1}\right)\right|>1.$
By the continuity of the norm $$| |$$ , we can choose $$r_{1}$$ so small that
$\left(\forall \vec{x} \in \overline{G_{\vec{x}_{1}}\left(r_{1}\right)}\right) \quad|f(\vec{x})|>1.$
Again by (4), we fix $$f_{2} \in \mathcal{N}$$ and $$\vec{x}_{2} \in G_{\vec{x}_{1}}\left(r_{1}\right)$$ such that $$\left|f_{2}\right|>2$$ on some globe
$\overline{G_{\vec{x}_{2}}\left(r_{2}\right)} \subseteq G_{\vec{x}_{1}}\left(r_{1}\right),$
with $$0<r_{2}<1 / 2.$$ Inductively, we thus form a contracting sequence of closed globes
$\overline{G_{\vec{x}_{n}}\left(r_{n}\right)}, \quad 0<r_{n}<\frac{1}{n},$
and a sequence $$\left\{f_{n}\right\} \subseteq \mathcal{N},$$ such that
$(\forall n) \quad\left|f_{n}\right|>n \text { on } \overline{G_{\vec{x}_{n}}\left(r_{n}\right)} \subseteq E^{\prime}.$
As $$E^{\prime}$$ is complete, so are the closed globes $$\overline{G_{\vec{x}_{n}}\left(r_{n}\right)} \subseteq E^{\prime}.$$ Also, $$0<r_{n}<$$ 1$$/ n \rightarrow 0.$$ Thus by Cantor's theorem (Theorem 5 of Chapter 4, §6), there is
$\vec{x}_{0} \in \bigcap_{n=1}^{\infty} \overline{G_{\vec{x}_{n}}\left(r_{n}\right)}.$
As $$\vec{x}_{0}$$ is in each $$\overline{G_{\vec{x}_{n}}\left(r_{n}\right)},$$ we have
$(\forall n) \quad\left|f_{n}\left(\vec{x}_{0}\right)\right|>n;$
so $$\mathcal{N}$$ is not bounded at $$\vec{x}_{0},$$ contrary to (2). This contradiction completes the proof.$$\quad \square$$
Note 3. Complete normed spaces are also called Banach spaces.
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# Band insulator to Mott insulator transition in 1T-TaS2
## Abstract
1T-TaS2 undergoes successive phase transitions upon cooling and eventually enters an insulating state of mysterious origin. Some consider this state to be a band insulator with interlayer stacking order, yet others attribute it to Mott physics that support a quantum spin liquid state. Here, we determine the electronic and structural properties of 1T-TaS2 using angle-resolved photoemission spectroscopy and X-Ray diffraction. At low temperatures, the 2π/2c-periodic band dispersion, along with half-integer-indexed diffraction peaks along the c axis, unambiguously indicates that the ground state of 1T-TaS2 is a band insulator with interlayer dimerization. Upon heating, however, the system undergoes a transition into a Mott insulating state, which only exists in a narrow temperature window. Our results refute the idea of searching for quantum magnetism in 1T-TaS2 only at low temperatures, and highlight the competition between on-site Coulomb repulsion and interlayer hopping as a crucial aspect for understanding the material’s electronic properties.
## Introduction
Transition-metal di-chalcogenides are layered quasi-two-dimensional materials that not only show prominent potentials for making ultra-thin and flexible devices, but also exhibit rich electronic phases with unique properties1,2,3. 1T-TaS2 is one prominent example. The crystal structure of 1T-TaS2 consists of S–Ta–S sandwiches, which in turn stack through van der Waals interactions. It is structurally undistorted and electronically metallic at high temperatures. With decreasing temperature, it undergoes successive first-order transitions, resulting in the formation of various charge density waves (CDWs) with increasing degree of commensurability4,5,6,7. As shown in Fig. 1a, with cooling, 1T-TaS2 sequentially enters an incommensurate CDW (I-CCDW) phase below 550 K, a nearly commensurate CDW (NC-CDW) phase below 350 K, and finally a commensurate CDW (C-CDW) phase below 180 K. Prominent hysteretic behavior can be observed when comparing the cooling and heating resistivities data. Upon heating, 1T-TaS2 enters triclinic CDW (T-CDW) phase at 220 K, and then the NC-CDW phase at 280 K. The space modulations of different CDW phases are illustrated in the inset of Fig. 1a. Every adjacent 13 Ta-atoms accumulate together, which is called star-of-David (SD) cluster4,5,6,7. Within one hexagonal SD, 12 Ta-atoms pair and form 6 occupied bands, leaving the center Ta atom localized and unpaired alone. As a result, the insulating ground state of 1T-TaS2 has been proposed to be a Mott insulator8,9,10,11,12,13. In the C-CDW phase, the SD clusters cover entire lattice forming commensurate p$$\left( {\sqrt {13} \times \sqrt {13} } \right)$$R13.9° phase. The localized electrons with S = 1/2 spin arrange on a triangular lattice, making this system a promising candidate material for realizing quantum spin liquid14,15,16,17.
Distinct from the possible Mott physics that occurs within one TaS2 layer, interlayer coupling has recently been emphasized as a crucial aspect to understand the insulating property of 1T-TaS218,19,20. Through an interlayer Peierls CDW transition, interlayer staking order forms in the C-CDW phase. First, every two adjacent TaS2 layers accumulate, forming dimerized bilayer structure. The dimerized TaS2 bilayers then stack onto each other forming different types of staking order with different in-plane sliding configurations19,20. Recent scanning tunneling microscopy and transport measurements show possible existence of the stacking order in 1T-TaS221,22. Band calculations show that without considering the strong electronic correlation the interlayer staking order itself can open a band gap at the Fermi energy (EF), yielding an insulating property of 1T-TaS2. Under this scenario, the ground state of 1T-TaS2 is a band insulator rather than a Mott insulator.
The low-energy electronic structure of 1T-TaS2 consists of one half-filled nonbonding Ta 5d band. In the IC-CDW phase, angle-resolved photoemission spectroscopy (ARPES) data shows that one Ta 5d band disperses crossing EF, forming six oval-like circles around the Brillouin zone boundaries (M) (Fig. 1b). This is well consistent with the first-principle band calculations11. When entering the C-CDW phase, the CDW potential folds the Brillouin zone. The Fermi surface reconstructs into multiple disconnected spots, following the p$$\left( {\sqrt {13} \times \sqrt {13} } \right)$$R13.9° periodicity. Band folding and band gap opening split the Ta 5d band into a manifold of narrow subbands. One flat band emerges at ~200 meV below EF (Fig. 1c). Under the Mott scenario, this flat band was attributed to the lower Hubbard band8,9,10,11,12,13. Previous time-resolved ARPES and inverse-photoemission spectroscopy also reported the observation of Mott gap between the flat band and upper Hubbard band23,24,25, supporting the existence of Mott insulating ground state. However, recent ARPES studies show that the flat band is energy dispersive along the out-of-plane momentum (kz) direction19,26,27, which favors the existence of interlayer stacking order. It is then crucial to make a consensus on the detailed temperature evolution of electronic structure in 1T-TaS2. The results would help to understand how the on-site Coulomb repulsion and interlayer hopping play roles in 1T-TaS2.
In this work, we characterize the temperature dependence of electronic and lattice structures of 1T-TaS2 using ARPES and X-ray diffraction (XRD). We find that, at low temperatures, the band dispersion and diffraction peaks show 2π/2c and 2c periodicity, which indicates that the ground state of 1T-TaS2 is a band insulator with interlayer dimerization. More intriguingly, at high temperatures, the system undergoes an insulator-to-insulator transition and enters an intermediate Mott insulating state, which only exists in a narrow temperature region. Our results show that the energy scales of in-plane hopping, on-site Coulomb repulsion, and interlayer hopping are all comparable in 1T-TaS2. The competition between these interactions are responsible for the complex electronic properties of 1T-TaS2.
## Results
### Temperature dependence of electronic structure
To characterize how the flat band forms in the C-CDW phase, we show the detailed temperature dependence of the energy distribution curves (EDCs) taken at the Brillouin zone center (Γ) (Fig. 2). Upon cooling, the spectral weight of the flat band shifts from EF to −200 meV at 193 K, indicating an energy gap opening at EF. In contrast to the cooling process, the temperature evolution of EDCs during the heating process clearly identifies two-phase transitions (Fig. 2d–f). The first phase transition occurs at 217 K, where the peak width narrows and the peak position shifts from −210 to −170 meV. The second phase transition follows at 233 K, where the spectral weight of the flat band shifts to EF.
In comparison with the resistivity data, the gap opening transition at 193 K can be attributed to the NC-CDW to C-CDW transition upon cooling, while the transition at 217 K coincides with the C-CDW to T-CDW transition upon heating. There is a small temperature deviation between the transition temperatures determined by different techniques. This can be explained by the standard deviation of transition temperatures among different samples due to a small sample inhomogeneity (Supplementary Figs. 13). However, the transition at 233 K does not show up in the transport measurements. It also cannot be explained by a sample inhomogeneity, because the temperature-dependent data are well reproducible in many different samples (Supplementary Fig. 1). Our results thereby identify an undiscovered intermediate (I) phase, which exists in a narrow temperature region (217–233 K) at the C-CDW to T-CDW phase transition. For the transition at 217 K, the spectra show insulating behavior on both sides of the transition (Fig. 2d), which is therefore an insulator-to-insulator transition. In the following, we present experimental evidence for distinct motives behind the two insulating states, which suggests that the I phase is, in fact, a Mott insulator.
The temperature evolution of flat band is shown in Fig. 3. The CDW band gap opens and the flat band forms at EF in the NC-CDW and T-CDW phases. Its bandwidth is narrow due to the large scale of the hexagonal SD structure. The flat band opens an energy gap and shifts from EF to −170 meV at 225 K in the I phase. In the C-CDW phase, the flat band shifts to around −200 meV and becomes more dispersive. Meanwhile, there is an energy spread of the flat band towards EF, which is consistent with the peak broadening observed in Fig. 2d, e. Due to the surface sensitivity of ARPES technique, kz is not conserved during the photoemission process28. The ARPES spectra is thus a superposition of the bulk bands at different kz’s. If the band has moderate band dispersion along kz direction, its ARPES spectra is broadened. Such kz broadening effect explains the energy spread of the flat band, which also indicates that the kz band dispersion reconstructs significantly in the C-CDW phase.
To determine the band dispersion of the flat band along the kz direction, we measured the photon energy-dependent data in the I and C-CDW phases. The results are compared in Fig. 4. The flat band is gapped at 225 K. The weak photon energy dependence indicates the two dimensionality of the flat band in the I phase. However, when entering the C-CDW phase, the data become strongly photon energy dependent. The band positions shift from −230 to −90 meV periodically. This shows that the bandwidth along the kz direction is around 140 meV in the C-CDW phase, which is well consistent with the kz broadening effect observed in Fig. 3. The photon energy-dependent data confirm that the kz band dispersion reconstructs strongly at the I to C-CDW phase transition. More intriguingly, the periodicity of the kz dispersion is around 2π/2c in the C-CDW phase, which indicates the existence of an interlayer dimerization.
### Temperature dependence of lattice structure
We then performed the single-crystal XRD measurement along the (0, 0, L) direction to confirm the existence of interlayer stacking order. Indeed, in addition to the regular integer-indexed Bragg peaks (Fig. 5a), a set of half-integer-indexed reflections are observed at 120 K. The fact that the (0, 0, 7/2) reflection is about twice more intense than the (0, 0, 5/2) reflection, i.e., with the intensity roughly proportional to |Q|2, suggests that these additional reflections are caused by c-axis displacive structural modulations that double the unit cell along c (Fig. 5c). The appearance of half-integer reflections in the C-CDW phase is a direct evidence for interlayer dimerization, which has to be caused by, and provide feedback to, interlayer coupling in the electronic structure. It is therefore no surprise that these half-integer reflections occur at 212 K which coincides with the enhancement of kz band dispersion observed by ARPES (Figs. 4 and 5b).
## Discussion
For the C-CDW phase, its intra-layer CDW structure consists of SD clusters with the p$$\left( {\sqrt {13} \times \sqrt {13} } \right)$$R13.9° periodicity. This has been well studied by previous diffraction experiments and also confirmed by the ARPES mapping in Fig. 1b. Here, our XRD and photon energy-dependent ARPES data show that, in addition to the intra-layer CDW, the system forms an interlayer stacking order along the c direction. While the 2c periodicity could be attributed to an inter-layer dimerization, the stacking configuration between each bilayers remains unknown. Theoretical calculations show that the energy differences between different configurations are small19,20. Therefore, different stacking configurations could coexist, and the layers could stack randomly along the c direction. This is consistent with our XRD data. The weak intensity of the half-integer diffraction peaks and the broadening of the CDW diffraction peak (Supplementary Fig. 5) suggest that the interlayer stacking shows certain randomness in the C-CDW phase. Therefore, to determine the stacking configuration experimentally, layer-resolved XRD studies are needed. Nevertheless, indirect evidences could be found by comparing the ARPES data with theoretical band calculations. Our result is more consistent with an alternating stacking configuration, where the predicted band structure is a small-gap band insulator19,20.
Both ARPES and XRD data show the existence of interlayer stacking order in the C-CDW phase. This makes the low temperature ground state of 1T-TaS2 a band insulator with two electrons fully occupied in the flat band near EF. The next question is then the origin of the I phase where the stacking order vanishes. Apparently, the I phase to C-CDW phase transition is an insulator-to insulator transition. Therefore, it cannot be explained by a simple interlayer Peierls CDW transition that is driven by the electronic instability at the Fermi surface. Electronic correlation need be considered. In a bilayer Hubbard band model, considering both the on-site Coulomb repulsion and interlayer hopping, theories show that the electronic system can transit from a Mott insulator to a band insulator when the interlayer hopping increase above a critical value29,30. The electronic states redistribute without closing the energy gap, resulting in an insulator-to-insulator transition. This naturally explains the C-CDW phase to I phase transition observed here (Fig. 5d, e). The I phase is a two-dimensional Mott insulator originated from the on-site Coulomb repulsion (U). When entering the C-CDW phase, the interlayer hopping (t) as characterized by the kz bandwidth is strongly enhanced. As a result, the system transits from a Mott insulator into a band insulator. Consistently, the energy scale of the in-plane hopping (t//) is around 100 meV as determined by the in-plane bandwidth of the flat band (Fig. 3). The Mott gap between the lower and upper Hubbard band is around 400 meV13,18,25, representing the energy scale of U. The energy scale of t is around 140 meV, which is the kz bandwidth of the flat band (Fig. 4). All these experimentally determined parameters are consistent with the theoretical parameters where the band-insulator to Mott-insulator transition could realize29,30.
It is then intriguing to understand the enhancement of t cross the I to C-CDW phase transition. In fact, the reconstruction of electronic structure along c direction is so strong that upon entering the I phase from the C-CDW phase, the distance between adjacent TaS2 layers undergoes an appreciable sudden decrease from 5.928 to 5.902 Å as seen from the (0, 0, 4) reflection (Fig. 5b). At first glance, the larger interlayer distance in the C-CDW phase compared to the I phase is at variance with the larger t (hence stronger kz dispersion) in the C-CDW phase. This unexpected correlation between t and interlayer distance requires further theoretical understanding. One possible explanation is that the complete in-plane commensurability of CDW in the C-CDW phase turns on additional hopping channels between nearby TaS2 layers, which may further affect the average interlayer distance.
The phase transition at 233 K and the I phase have not been observed in both the resistivity and XRD data. Considering that APRES is more sensitive to the electronic state near the sample surface, we could explain this controversy using two different scenarios. On one hand, the I phase is a surface state that does not exist in the bulk layers. This resembles the Mott insulating phase that was observed at the surface of 1T-TaSe231,32. However, this scenario contradicts with some of our observations. For example, the penetration depth of ARPES is normally around or over two layers. If the surface and bulk electronic states are different, we should observe two different sets of bands representing the surface and bulk, respectively. However, our data only show one set of bands, which suggests that the I phase extends along c direction for at least two TaS2 layers. Furthermore, the phase transition at 217 K observed by ARPES is well consistent with the C-CDW transition temperature determined by transport and XRD measurements. The photon energy-dependent data also clearly show the kz dispersion of bands, whose periodicity is well consistent with the bulk lattice parameter. All these results suggest that the ARPES data reflect the bulk electronic properties of 1T-TaS2. On the other hand, the I phase exists in bulk layers, but only emerges in certain layers which are phase separated from the other layers. Therefore, the I phase cannot be picked up by the resistivity and XRD measurements. Under this scenario, the observation of I phase by ARPES should be highly dependent on the cleaved surface. However, this is not the case. The high reproducibility of our ARPES data seems to suggest that the I phase may have a high probability of occurrence near the sample surface.
To examine whether the I phase exists in the bulk or not, layer-resolved transport and XRD measurements are required. The resistivity measurements on thin 1T-TaS2 films show that the transition becomes broad with the decreasing of sample thickness33,34,35, which may suggest the existence of a phase separation. Moreover, the system remains insulating in an ultrathin film, while the C-CDW phase is fully suppressed33,34,35. This is consistent with our results that the interlayer hopping is important for the C-CDW phase. In an ultrathin film where the on-site Coulomb repulsion plays a more dominating role than the interlayer hopping, the insulating property may originate from the intermediate Mott insulating phase observed here.
In summary, we believe that the competition between U (within SD) and t (between neighboring TaS2 layers) is responsible for the complex insulating phases in 1T-TaS2. While the C-CDW phase is a band insulator featuring strong interlayer hoping and dimerization, the I phase is likely a two-dimensional Mott insulator. Although we show that the low-temperature C-CDW phase is not a Mott insulator, the S = 1/2 degrees of freedom could still be realized in the I phase. Further experimental studies are required to characterize possible magnetism in the I phase and verify its occurrence in bulk layers. Meanwhile, if the stacking order in the C-CDW phase can be suppressed, such as by chemical doping or reducing dimensionality, quantum magnetism may still be realized down to the lowest temperature in 1T-TaS2.
## Methods
### Sample growth
High-quality single crystals of 1T-TaS2 were synthesized using chemical vapor transport (CVT) method. By mixing the appropriate ratio of Ta powder and S pieces (2% excess) well, the compound was sealed in a quartz tube with ICl3 as transport agent. The quartz tube was put in the two-zone furnace with thermal gradient between 750 and 850 °C for 120 h, and then quenched in water. The refinement of power XRD data confirms the phase purity of our sample and the trigonal crystal structure of 1T-TaS2 with the P-3m1 space group with the lattice parameters a = b = 3.366 Å, c = 5.898 Å (Supplementary Fig. 6).
### Resistivity measurements
Resistivity data were measured in physical property measurement system (PPMS, Quantum Design, Inc.) utilizing standard four-probe method. The heating and cooling rates are 3 K min−1.
### ARPES measurements
ARPES measurements were performed at Peking University using a DA30L analyzer and a helium discharging lamp. The photon energy of helium lamp is 21.2 eV. Photon energy-dependent measurement was performed at the BL13U beamline in National Synchrotron Radiation Laboratory (NSRL). The overall energy resolution was ~12 meV and the angular resolution was ~0.3°. The crystals were cleaved in situ and measured in vacuum with a base pressure better than 6 × 10−11 mbar. The EF for the samples were referenced to that of a gold crystal attached onto the sample holder by Ag epoxy. For the measurements in a heating process, the samples were first cooled down from 300 to 80 K with a rapid fall of temperature (about 20 K min−1). After adequate cooling for about 10 min at 80 K, the samples were cleaved and heated up to 190 K with a rate about 1.5 K min−1; The samples were then heated up to 300 K slowly with a rate about 0.23 K min−1. For the measurement in a cooling process, The sample was cleaved at 300 K directly and cooled down to 210 K with a rate about 2 K min−1; The sample was then cooled down to 160 K slowly with a rate about 0.5 K min−1. ARPES data were collected during the cooling and heating processes. More experimental details could be found in Supplementary Table 1.
### XRD measurements
XRD data were recorded on a Bruker D8 diffractometer using Cu Kα radiation (λ = 1.5418 Å). We mounted a high-quality single crystal of 1T-TaS2 in a liquid nitrogen cryostat sitting in an Euler cradle to measure the c axis at different temperatures.
The sample was cooled down until 93 K with a rapid fall of temperature (about 10 K min−1) from room temperature naturally. After adequate cooling for about 1 h at 120 K, we heated the sample from 120 to 170 K at 3 K min−1. We then heated the sample to 370 K with a rate about 0.16 K min−1. XRD data were collected during the heating process.
### Experimental reproducibility
The ARPES and resistivity measurements were repeated on different samples. The observations are reproducible in all measured samples with a small variable of transition temperatures (Supplementary Figs. 13). The resistivity data in Fig. 1a were taken on sample #7. The ARPES data in Fig. 2a, b were taken on sample #6. The ARPES data in Figs. 2d, f and 3 were taken on sample #1. The XRD data were taken on sample #13. The ARPES data taken at 370 K in Figs. 1 and 3 were taken on sample #14. Photon energy-dependent data in Fig. 4 was taken on sample #15.
## Data availability
The authors declare that the data supporting the findings of this study are available within the article and its Supplementary Information. All raw data are available from the corresponding author upon reasonable request.
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## Acknowledgements
We gratefully thank Y. Zhang and Z. Liu for stimulating discussions. We gratefully thank S.T. Cui for his help on the ARPES experiment at NSRL. This work is supported by the National Natural Science Foundation of China (NSFC) (Grant No. 11888101), by the National Key Research and Development Program of China (Grant Nos. 2018YFA0305602 and 2016YFA0301003), and by the NSFC (Grant Nos. 11574004 and 91421107).
## Author information
Authors
### Contributions
Y.Z. conceived and instructed the project. W.L.Y. synthesized the single crystals. Y.L. supported the sample synthetization. Y.D.W. took the ARPES and XRD measurements with the contribution of Z.M.X., T.T.H., Z.G.W., L.C. and C.C. Y.D.W. and Y.Z. analyzed the data and wrote the paper with the input from all authors.
### Corresponding author
Correspondence to Y. Zhang.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
Peer review information Nature Communications thanks Y. Liu, Luca Perfetti and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Wang, Y.D., Yao, W.L., Xin, Z.M. et al. Band insulator to Mott insulator transition in 1T-TaS2. Nat Commun 11, 4215 (2020). https://doi.org/10.1038/s41467-020-18040-4
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<meta http-equiv="refresh" content="1; url=/nojavascript/"> Nuclear Reactors ( Real World ) | Physics | CK-12 Foundation
# Nuclear Reactors
%
Progress
Progress
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Fusion Energy
### Fusion Energy
Credit: Robert Lopez
Source: CK-12 Foundation
When two light nuclei fuse into a heavier nucleus, energy is released. Long thought to be the next step in energy production, energy harvested from nuclear fusion has yet to be achieved due to several setbacks and the sheer difficulty inherent in fusion reactions.
#### Amazing But True
Credit: NASA
Source: commons.wikimedia.org/wiki/File:Sun_in_X-Ray.png
The sun is an example of a natural fusion reactor [Figure2]
• While fuel for nuclear fusion is virtually limitless, a viable method of efficiently harvesting energy for commercial use has yet to be found. One type of fusion reaction currently being researched is the following:
${^2 H} + {^3} H \rightarrow {^4 H}+n+17.6 \ MeV$
• Very large kinetic energies are needed to cause a fusion reaction between $^2 H$ and $^3 H$. One method to achieve these energies is by increasing the atom's kinetic energy by heating the system with temperatures around $kT=10 \ keV$.
• The temperature that corresponds to the above relationship is on the order of $10^8 \ K$. Temperature of this scale, while usually only seen in stars, has been recently achieved in laboratory experiments. Despite 60 years of nuclear fusion research, scientists have a long way to go before finding a viable implementation of nuclear fusion. Currently, commercial power production using fusion is not expected till after 2050.
#### Explore More
Using the information provided above, answer the following questions.
1. The temperature listed is actually an overestimate of the temperature that is needed to produce a reaction. Why? (Hint: Think of possible quantum effects)
2. Why must the plasma being using in fusion experiments be prevented from contacting the walls of the container used?
3. Why are runaway reactions not a major concern in fusion reactors?
1. [1]^ Credit: Robert Lopez; Source: CK-12 Foundation; License: CC BY-NC 3.0
2. [2]^ Credit: NASA; Source: commons.wikimedia.org/wiki/File:Sun_in_X-Ray.png; License: CC BY-NC 3.0
### Explore More
Sign in to explore more, including practice questions and solutions for Nuclear Equations.
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# Integration by parts
CellCoree
hi, i would like help on a problem i am currently stuck on.
$$\int(e^x)/(1+e^(2x))dx$$ <-- it's suppose to be $$\int$$ (e^x)/(1+e^(2x))dx
using integration by parts, here's what i done:
u=e^x
du=e^x
dv=(1+e^(2x))
v = (need to use anti-differentiation, which i don't remeber...)
can i use integration by parts with this? this is cal 2.
Last edited:
Crumbles
CellCoree said:
hi, i would like help on a problem i am currently stuck on.
dv=(1+e^(2x))
v = (need to use anti-differentiation, which i don't remeber...)
Yes, v would be the integral of (1+e^(2x))
Homework Helper
Erm, by-parts doesn't seem to make sense because actually:
$$u = e^x$$
$$dv = \frac{1}{1 + e^{2x}}$$
To me, it just looks like it is going to get nastier and nastier.
I would suggest using the substitution $t = e^x$ because $dt = e^xdx$ and if you look at the integral like this it becomes quite simple:
$$\int \frac{e^x dx}{1 + \left( e^x \right)^2}$$
Last edited:
Homework Helper
Gold Member
CellCoree said:
hi, i would like help on a problem i am currently stuck on.
$$\int(e^x)/(1+e^(2x))dx$$ <-- it's suppose to be $$\int$$ (e^x)/(1+e^(2x))dx
using integration by parts, here's what i done:
u=e^x
du=e^x
dv=(1+e^(2x))
v = (need to use anti-differentiation, which i don't remeber...)
can i use integration by parts with this? this is cal 2.
Hi,
I would not try an integration by parts. I would simply do a simple substitution u= e^x. Then you have the integral of du/(1+u^2) which is a basic one.
Pat
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# Puzzle: A coin rolls without slipping around another coin
If a coin rolls without slipping around another coin of the same or different size, how many times will it rotate while making one revolution?
The proof given is like this:
Cut the curve open at some point and uncoil it into a straight line segment. Rolling the circle along this segment it will rotate (length curve)/(length circle) times.
Keeping the circle attached to one end of the segment, we then recoil the segment back into thecurve, which contributes the final rotation.
I can understand the first part of the proof, but I couldn't understand the second part, any ideas? Also, I am also interested in different approaches for proving this one.
Problem/Solution source.
-
Both ways are the same thing,only difference is that in one of them the line segment is stationary and in other one the circle is stationary while the segment is wrapped around it.(by relative motion funda :) – Tomarinator May 24 '12 at 14:24
@Willie Wong:Hm, assuming anticlockwise movement and from what I can understand from that .gif file in wike page shouldn't the winding number in this case is 1? – Quixotic May 24 '12 at 14:25
Winding number explains the "extra" twist. In general if you "roll a coin" where the coin has circumference $C$ along a closed curve $\gamma$, the total number of turns the coin goes through is the winding number of $\gamma$ plus $|\gamma| / C$. The latter is the number of turns in the "co-moving" coordinate system, and the former is the number of turns the coordinate system itself made. – Willie Wong May 24 '12 at 14:29
I suggest that you first regard the case with "total slipping".
Let a coin slide around another coin while having always the same point attached to the central coin. Note that the outer coin makes exactly one rotation in this scenario.
Now, roll out the inner circle and let the outer coin slide along. Clearly, it does not rotate at all.
So, there is one rotation of the outer coin that comes just from going around the inner coin. The rest of the rotation can then be found by looking at the rotation along the line and combining the two.
-
You do it in three steps (where I just recall the first two steps, which you understood).
1. The two circles touch at one point. You cut one of the circles open at that point and unravel it to make it into a line. Let us assume we do this such that the line is horizontal and the circle sits at the left end.
2. You roll the second circle along that line to the right end.
3. You bend the line back to a circle.
For step three note that the circle is "attached" to the right end. You fix the left end of the line take the right end of the line and start to move it. The circle - which is still attached to that end - will make another whole turn.
-
Sorry, but I still don't understand why are we joining the right end after the the second step. – Quixotic May 24 '12 at 14:30
It is maybe easiest if you imagine an "axis" joining the two centers of the circles all the time (so one end of the axis doesn't advance at all an the other end turns a full circe), and a camera mounted on this axis filming the circles.
Since the camera makes a full tour in one direction (call it positive), it will see the fixed circle make a complete tour in the negative direction. If that fixed circle has radius $R$ and the other radius $r$, then at the point of contact this negative rotation produces a displacement of $2\pi R$ in the backwards direction, which will cause the other circle to make $R/r$ tours in the positive direction (so that it also produces a backward displacement of $(R/r)2\pi r=2\pi R$ at the point of contact, corresponding to "no slipping"). But since the camera also made a complete tour in the positive direction while filming, it is clear that the mobile circle made $(R/r)+1$ tours in reality (i.e., with repect to a fixed observer).
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## Fun with principal ideal domains
A commutative ring $R$ is called a principal ideal domain (PID) if every ideal of $R$ can be generated by a single element. If $R$ is a principal ideal domain, is every subring of $R$ a principal ideal domain? No, definitely not. That is because you can take any integral domain that is not a principal ideal domain, like $\Z[x]$, and take its fraction field. Its fraction field is a PID and the original ring sits inside it as a subring.
Another more interesting example is the ring $\Q[x]$ of polynomials with rational coefficients. It is a PID, yet the subring $\Q[x^2,x^3,x^4,\dots]$ is not. The ideal $(x^2,x^3)$ in this ring is not a principal ideal. By the way, is the ring $\Q[x^2,x^3,\dots]$ Noetherian? Does there exist an ideal in it that needs at least three generators?
…read the rest of this post!
## A quick intro to Galois descent for schemes
This is a very quick introduction to Galois descent for schemes defined over fields. It is a very special case of faithfully flat descent and other topos-descent theorems, which I won't go into at all. Typically, if you look up descent in an algebraic geometry text you will quickly run into all sorts of diagrams and descent data. In my opinion, that is a very counterintuitive way to present the basic idea.
## What is the descent theorem?
Here is the main topic of this post:
Theorem. Let $E/F$ be a finite Galois extension of fields with Galois group $G$. Then the functor
\begin{align*} \{\text{quasiproj. $F$-schemes}\}&\to \{\text{quasiproj. $E$ schemes with compatible $G$ action} \} \\ X&\mapsto X\otimes_F E\end{align*} where $X\otimes_F E$ is given an Galois action via the canonical action on $E$, is an equivalence of categories.
This is the basic theorem of Galois descent. What does it mean, and how does it work? First, I have to tell you what a compatible Galois action is. Well, if $X$ is an $E$-scheme, then there is a map $X\to{\rm Spec}(E)$, and there is the usual action of $G$ on ${\rm Spec}(E)$. Compatible just means that for each $\sigma\in$, the square
commutes.
…read the rest of this post!
## Dividing a square into triangles of equal area
Take a square and divide it down a diagonal, dividing the square into two triangles. Drawing the opposite diagonal now divides it into four triangles. In these two examples, we divided a square into an even number of triangles, all with equal area. Can we divide a square into an odd number of nonoverlapping triangles, all with equal area? In this question, we do not require that all the triangles be congruent, as in the above examples.
It turns out, you can't. Paul Monksy proved this in a 1970 American Mathematical Monthly paper [1], though John Thomas proved this earlier when the vertices of the triangles are restricted to having rational coordinates.
The proof progresses in several steps. I won't go through every detail, but try and convey the flavour of the proof. The reader is invited to read the proof in its entirety, which is something I just did and I recommend it.
…read the rest of this post!
## Finite-dimensional k[x]-modules: projective or not?
Let's suppose $M$ is a nonzero projective $\Z$-module. Can it be finite? Nope. I'm sure there are plenty ways to prove it, but one way is to observe that a projective $\Z$-module is free, and hence if $M$ is nonzero it must have at least one copy of $\Z$. So, $M$ is infinite.
What's the analogue for the ring $k[x]$ where $k$ is a field? If $M$ is a nonzero projective $k[x]$-module, can it be finite? It certainly can't if $k$ is infinite dimensional, since any nonzero $k[x]$-module (whether projective or not) is also a $k$-vector space. What about if $k$ is finite?
…read the rest of this post!
## Explicit example showing non-residual finiteness
This is mostly a continuation on the group I gave in the last post, which is given by the presentation
$$G = \langle a,t ~|~ t^{-1}a^2t = a^3\rangle.$$ At the risk of beating a dead horse, I proved that the homomorphism $f:G\to G$ given on generators by $f(t) = t$ and $f(a) = a^2$ is surjective but not injective. Groups for which surjective homomorphisms are isomorphisms are called Hopfian, and so our group $G$ is not Hopfian.
As I've been talking about frequently in the past little while, a group $G$ is called residually finite if for every nontrivial $x\in G$ there exists a homomorphism $\varphi:G\to F$ such that $F$ is finite and $\varphi(x)$ is not the identity of $F$. In the post on residually finite groups, I explained the classic proof that a finitely-generated, residually finite group is Hopfian.
Now the particular group $G$ here that is not Hopfian is finitely-generated, and so of course it can't be residually finite. I was wondering, can we find an explicit nontrivial element $x\in G$ such that for every homomorphism $\varphi:G\to F$ where $F$ is finite, the element $\varphi(x)\in F$ is the identity of $F$?
Yes, that's actually quite easy. It is because in the last post we already proved that the commutator $[t^{-1}at,a]$ is sent to the identity under the endomorphism $f:G\to G$ (recall, which was given by $f(t) = t$ and $f(a) = a^2$). But if we carefully examine the proof of the statement "every finitely-generated residually finite group is Hopfian", we see that the kernel of $f$ is actually contained in every finite-index normal subgroup of $G$. Therefore, in particular, the commutator $[t^{-1}at,a]$, which is nontrivial by Britton's lemma, is mapped to the identity under every homomorphism $G\to F$ where $F$ is a finite group!
## Yet another group that is not Hopfian
A few weeks ago I gave an example of a non-Hopfian finitely-presented group. Recall that a group $G$ is said to be Hopfian if every surjective group homomorphism $G\to G$ is actually an isomorphism. All finitely-generated, residually finite groups are Hopfian. So for example, the group of the integers $\Z$ is Hopfian.
Another example of a group that is not Hopfian was given by Gilbert Baumslag and Donald Solitar. Their group is the one-relator group
$$G = \langle a,t ~|~ t^{-1}a^2t = a^3\rangle.$$ …read the rest of this post!
## A zero-dimensional ring that is not von Neumann regular
An associative ring $R$ is called von Neumann regular if for each $x\in R$ there exists a $y\in R$ such that $x = xyx$.
Now let $R$ be a commutative ring. Its dimension is the supremum over lengths of chains of prime ideals in $R$. So for example, fields are zero dimensional because the only prime ideal in a field is the zero ideal.
Theorem. Let $R$ be a commutative ring. If $R$ is von Neumann regular, then it is zero dimensional.
The proof follows directly from the definition: suppose $P\subset R$ is a prime ideal of a von Neumann regular ring. If $x\not\in P$ and $y\in R$ is an element such that $x = xyx$, then $x(1 – yx) = 0$. Since $x\not\in P$, we must have $1 = yx$. Therefore, $P$ is maximal.
…read the rest of this post!
## A finitely generated flat module that is not projective
Let's see an example of a finitely-generated flat module that is not projective!
## What does this provide a counterexample to?
If $R$ is a ring that is either right Noetherian or a local ring (that is, has a unique maximal right ideal or equivalently, a unique maximal left ideal), then every finitely-generated flat right $R$-module is projective.
So what happens if we drop the Noetherian and local hypotheses?
## The Example
Let $R = \prod_{j=1}^\infty F_j$ be an infinite product of fields and let $I = \oplus_{i=1}^\infty F_j$ be the ideal that is the direct sum of all the fields. Then the module $R/I$ is finitely generated. It is also flat, because $R$ is von Neumann regular and in such rings, every module is flat. Why is it not projective?
To see that it is not projective, consider the exact sequence
$$0\to I\to R\to R/I\to 0.$$ If $R/I$ were projective, that would mean that the map $R\to R/I$ splits, which gives a direct sum decomposition $I\oplus R/I\xrightarrow{\sim} R$ where the composition of the map $I\to I\oplus R/I\to R$ is the inclusion $I\to R$. The image of $R/I$ then corresponds to a nonzero ideal in $R$. But any nonzero ideal intersects $I$, so such a splitting is impossible.
This example is part of my new counterexamples project.
## Kourkovka Notebook: Open problems in group theory
Every once in a while I spot a true gem on the arXiv. Unsolved Problems in Group Theory: The Kourkovka Notebook is such a gem: it is a huge collection of open problems in group theory. Started in 1965, this 19th volume contains hundreds of problems posed by mathematicians around the world. Additionally, problems solved from past volumes are also included with references.
For example, F.M. Markel proved that if $G$ is a finite supersolvable group with no two conjugacy classes having the same number of elements, then $G$ is actually isomorphic to the symmetric group $S_3$. Pretty cool right? Jiping Zhang extended this theorem by replacing 'supersolvable' by just 'solvable'. Problem 16.3 in the Kourkovka notebook asks the obvious: if $G$ is any finite group where no two conjugacy classes have the same size, is $G\cong S_3$? There are of course many more problems of varying technicality, but there should be something in here for any group theorist.
I've always thought that you can gauge the health of a discipline by the quality of open problems in it. If that's true, then the Kourkovka notebook shows that group theory is thriving very well.
## Britton's lemma and a non-Hopfian fp group
In a recent post on residually finite groups, I talked a bit about Hopfian groups. A group $G$ is Hopfian if every surjective group homomorphism $G\to G$ is an isomorphism. This concept connected back to residually finite groups because if a group $G$ is residually finite and finitely generated, then it is Hopfian. A free group on infinitely many generators is an example of a residually finite group that is not Hopfian.
Are there examples of finitely generated groups that are not Hopfian? Such an example would then of course give us an example of a group that is not residually finite.
In this post, we'll see an example of a group that is finitely presented and not Hopfian. Not only that, but I promise the construction is actually not even scary, unlike those finitely presented groups with unsolvable word problem.
…read the rest of this post!
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# Lecture 2: Standard Library: Vectors, Iterators, Sets, and Maps
## 1 Standard Template Library (STL)
STL is a misnomer. It was a library that informed the C++ standard library, so it refers to a library that is very similar to some parts of the C++ standard library. Using the name STL for the whole standard library is incorrect. It refers to some parts of the standard library implemented using templates, hence the name. Specifically, it has four components (the current standard library of C++ grew leaps and bounds beyond this):
Algorithms
It provides some standard algorithms optimized for general use cases so that you don't have to write up a good algorithms for sortingThe standard library comes with plenty flavors of sorting algorithms., binary search, running sum, find minimum, etc. every time you need it.
Containers
The standard library comes with very flexible containers (vector, deque, linked lists, maps, sets, priority queue) that support certain operations cheaply (keep & traverse elements in sorting order, find the smallest element, find a given element).
Iterators
Iterators allow traversing data structures or a range of items (the newest version of C++ also has ranges for a similar purpose, but we will not get into it). With iterators, we can generalize some algorithms: For example, a copying algorithm can use iterators, so the same implementation works if copying from a vector to a set, or vice versa.
Functions
The standard library comes with some abstractions to pass functions around like normal objects, so we can write a generic algorithm that uses an unknown function (for example, abstracting over the comparison function in a sorting algorithm). We will take a look into function objects in later lectures.
## 2 The Standard Library
It's really rare if a programming language provides all of the necessary tools to accomplish a task "out of the box". Programmers usually use provided building blocks to create something specific to fit their needs. Programmers are also able to build additional building blocks to build upon as well
These libraries usually come standard with the language.
• The libraries should be cross-platform compatible (you shouldn't have to code differently based on running it with Windows, Mac, or Linux)
Implementing and maintaining libraries come with a cost.
• Python has a dedicated organization called the Python Software Foundation.
• Java was developed and maintained by Sun Microsystems, which has been bought by Oracle. Its reference implementation is open-source: OpenJDK.
• Similarly, Rust was originally backed by Mozilla, but it is a community-developed open-source project now.
• C++ isn't "owned" by anyone really. The standard is maintained by a consortium.
Note that all of these languages, the language evolves through series of community improvement proposals that the core developers (or the standards committee members) discuss and refine.
Since C++ isn't a product of a large organization, and is organized like open-source.
• http://isocpp.org/, http://www.open-std.org/.
• Individual C++ compilers are then implemented based on the specifications.
• g++ / clang++ for Unix.
• MSVC for Microsoft.
• These compilers also ship with their implementation of the standard library, although you can mix-and-match the standard libraries and compilers to an extent.
• … there isn't any guarantee that these behave EXACTLY the same, but they do for the most part based on the specifications. For example, they may implement different sorting algorithms for std::sort but the algorithm being used is guaranteed to run in $$\bigO(n \log n)$$ time. In practice, the standard libraries implement roughly the same algorithm with slightly different performance tuning.
• In this class, we'll assume we're using the C++17 specification unless stated otherwise.
## 3 Standard Libarary Containers
There are many implementations of containers.
• Containers are data abstractions where you can store a sequence of elements.
• Iterators are a common part of these containers, which allow you to "iterate" through the components.
• They also act as handles to specific objects (rather than iterating over data), we will talk about this when we are discussing maps.
• Depending on the container, you can even read from/write to these elements using iterators.
### 3.1std::vector
• A vector is a sequence of objects that are conceptually stored one after the other
• Vectors are implemented with templates, so you can store one kind of object type in the vector container
# Makefile
CXX=g++
main: main.o
\${CXX} -o main -std=C++17 main.o
clean:
rm -f *.o main
// main.cpp
#include<vector>
int main() {
std::vector<int> v; // a vector containing int elements
return 0;
}
Under-the-hood, vectors are implemented using arrays and behave similar to arrays.
• Vectors can be indexed starting from 0 to $$size - 1$$ yet they're different than arrays.
• Vectors are dynamically-resizable
• Vectors have a size associated with it.
• Arrays do not know their size and the programmer must be aware of it.
### 3.2 Adding to a vector example
// main.cpp
#include<vector>
template <class T>
void printVector(std::vector<T> &v) {
for (int i = 0; i < v.size(); i++) {
std::cout << "v[" << i << "] = " << v[i] << std::endl;
}
// range-based for loop example
// for (const T& i : v) {
// std::cout << "v[" << index << "] = " << i << std::endl;
// }
}
int main() {
std::vector<int> v;
for (int i = 0; i < 5; i++) // it could be any reasonable size
v.push_back(i);
printVector(v);
return 0;
}
• Like arrays, if you index a vector element that is out of range, you will probably get junk data or make your program crash.
• You can also use the .at() function to access an element. Which is the better option especially for containers other than std::vector. We will talk about why later.
• Unlike operator [], if .at() references an element that the vector doesn't contain, an exception is thrown (more on exceptions later).
### 3.3 Example
std::cout << v.at(4) << std::endl;
std::cout << v.at(5) << std::endl; // EXCEPTION THROWN
std::cout << v1[5] << std::endl; // JUNK
Other supported operations are:
front()
returns the first element
back()
returns the last element
pop_back()
delete the last element
std::cout << "v.front() = " << v.front() << std::endl;
std::cout << "v.back() = " << v.back() << std::endl;
v.pop_back();
printVector(v);
### 3.4 Vector Initialization
push_back() is one way to create elements in a vector. Though it's not the only way
• You can declare a vector with a size initially
• You can also initialize a vector with a size and default values.
### 3.5 Example:
std::vector<int> v1(100); // initializes vector with 100 elements.
std::vector<int> v2(100, 1); //initializes vector with 100 elements = 1
### 3.6 Example creating a vector on the heap with a pointer reference to the vector contents on the heap
The following code snippet allocates both the vector itself and the contents on the heap. It is usually not what we want. std::vector already allocates the data on the heap so it uses a small space on the stack.
std::vector<int>* v = new vector<int>(10,1); // vector with 10 elements = 1
std::cout << v->size() << std::endl;
printVector(*v);
So, consider the following two vectors:
std::vector<int>* a = new vector<int>(1000,1); // vector with 1000 elements = 1
std::vector<int> b(1000,1); // vector with 1000 elements = 1
Here is where a, *a, and b are allocated on the memory:
Stack Heap
┌────────────────────────┐ ┌────────────────────────┐
│ │ │ │
│ ┌─┐ │ │ ┌───────────┐ │
│ │a├───────────────────┼───┼───►│*a │ │
│ └─┘ │ │ │ │ │
│ │ │ │size: 1000 │ │
│ ┌────────────────┐ │ │ │data: ────┼────┐ │
│ │b │ │ │ │ │ │ │
│ │ │ │ │ └───────────┘ │ │
│ │size: 1000 │ │ │ │ │
│ │data: ──────┐ │ │ │ │ │
│ │ │ │ │ │ ┌───────────────────▼┐ │
│ └─────────────┼──┘ │ │ │actual data of *a │ │
│ │ │ │ │(this is huge) │ │
│ │ │ │ └────────────────────┘ │
│ │ │ │ │
│ │ │ │ │
│ │ │ │ ┌──────────────────┐ │
│ └───────┼───┼─►│actual data of *b │ │
│ │ │ │(also huge) │ │
│ │ │ └──────────────────┘ │
│ │ │ │
└────────────────────────┘ └────────────────────────┘
b stores only a pointer to the actual array, and the size on the stack. The 1000-element array is stored on the heap in both cases.
## 4 Iterators
• An iterator is an abstraction for a position in a collection of objects.
• Container classes in the C++ standard library support iterators.
• It's common to think of an iterator as a pointer to an element's position
• Though technically it's not a pointer, but most likely uses a pointer in its implementation.
• Even though iterators are supported between different types of containers, an iterator can only be used with its own container type.
### 4.1 Example
vector<std::string> v2;
v2.push_back("Hello.");
v2.push_back("My");
v2.push_back("name");
v2.push_back("is");
v2.push_back("Batman");
for (vector<std::string>::iterator i = v2.begin(); i < v2.end(); i++) {
std::cout << *i << " "; // std::string value
std::cout << i->size() << std::endl; // prints the size of the strings
}
In the above example, we've seen vector functions that deal specifically with iterators.
begin()
returns an iterator that points to the first element
end()
returns an iterator that points to the last element
++
increments the iterator to the next element
<
compares positions of the iterator
*
dereferences an iterator to get the object
### 4.2 Example (Showing different ways to index elements using iterators):
vector<std::string>::iterator i = v2.begin();
std::cout << v2[4] << std::endl; // Batman
std::cout << i[4] << std::endl; // Batman
std::cout << *(i + 4) << std::endl; // Batman
In order to erase items in the vector, there is an erase method that requires iterators to do this
### 4.3 Example of erasing elements
// Removing 2nd index of the vector
v2.erase(v2.begin() + 2); // remove "name"
printVector(v2);
// -- separate example --
// Removing 1st and 2nd index - [1,3)
v2.erase(v2.begin() + 1, v2.begin() + 3);
printVector(v2);
## 5 Sets
• A set is a collection of unique values containing no duplicates.
• Sets support iterators
• Items in a set are in sorted order when iterating through them.
• std::set are implemented using a variant of binary search trees (BSTs) you learned in 24. They are self-balancing, so their .
### 5.1 Example
#include <set>
int main() {
std::set<std::string> s;
s.insert("Case");
s.insert("Molly");
s.insert("Armitage");
s.insert("Case"); // duplicate (only stored once)
s.insert("Wintermute");
// print out the contents
//
// question: why is there a const_ below?
for (set<std::string>::const_iterator i = s.begin(); i != s.end(); i++) {
std::cout << *i << std::endl;
}
// We can use auto to let the compiler infer the type of i. This
// is useful when dealing with long type names. This is equivalent to above.
for (auto i = s.begin(); i != s.end(); i++) {
std::cout << *i << std::endl;
}
}
Note: We will use std::set in the examples for now, we will talk about another option that usually performs better when we talk about hashing.
Moreover, the standard library sets' design is constrained because they support some operations used rarely. Other C++ libraries such as Abseil support faster, almost-always drop-in replacements.
## 6 Finding an element in a set
• find() returns an iterator to the item in a set if it exists.
• Otherwise, find() returns set.end(). This is a special iterator marking the end of the set (similar to other containers). Such special marker values are also called sentinel values.
• count() is another alternative. It returns number of times an element appears in a set. So, it is 1 if the element is in the set and it is 0 otherwise. C++ also supports multisets, when using a multiset you can put the same element more than once.
### 6.1 Example
if (s.find("Case") != s.end()) {
std::cout << "Case exists!" << std::endl; // prints this
} else {
std::cout << "Case does not exist" << std::endl;
}
if (s.find("Neuromancer") != s.end()) {
std::cout << "Neuromancer exists!" << std::endl;
} else {
std::cout << "Neuromancer does not exist" << std::endl; // prints this
}
## 7 Maps
• A map is an associated container containing a key / value mapping.
• Like a set, the keys are unique.
• Unlike a set, there is a value associated with each key.
• std::map is also implemented using a self-balancing BST (usually red-black trees). So, their elements are also sorted in order when you traverse them.
### 7.1 Example
#include <map>
std::map<int, std::string> students; // mapping studentIDs to studentNames
// Use bracket notation for creation
students[0] = "Richert";
students[1] = "John Doe";
students[2] = "Jane Doe";
std::cout << "students[1] = " << students[1] << std::endl;
### 7.2 Example using find()
• Similar to a set, find will look for a specific key and return map.end() if the key does not exist.
// Check if a student id exists
if (students.find(1) == students.end()) {
std::cout << "Can’t find id = 1" << std::endl;
} else {
std::cout << "Found student id = 1, Name = " << students[1] << std::endl;
}
### 7.3 Example using string and double types
map<std::string, double> stateTaxes;
stateTaxes["CA"] = 0.88;
stateTaxes["NY"] = 1.65;
if (stateTaxes.find("OR") == stateTaxes.end()) {
std::cout << "Can't find OR" << std::endl;
} else {
std::cout << "Found state OR" << std::endl;
}
### 7.4 Example between insert vs. []
• insert() will add a key / value pair to the map.
• If the key already exists, then .insert() will not replace the existing value.
• [] will map a key to a specific value.
• If the key already exists, then [] will replace the existing value.
• If the key does not exist, it first creates a default value.
• This may be expensive!
• insert_or_assign() is the better option over [] when you want to insert/update a value but don't know if it is in the map already.
• It is usually better to use .at() over [] when reading
• It will throw an exception rather than inserting a default value silently.
• It is also better to use .insert() or .insert_or_assign() when inserting rather than [].
• In short, avoid using [] with maps unless you know that the key is already in the map.
• Why are there loads of slightly different ways to do the same thing?
• C++ is an old language.
• The standards committee stuck with a behavior for [] and .at(). Rather than changing the behavior and breaking existing programs, they introduce new functions such as .insert_or_assign()
#include <utility> // for std::pair
// ...
students.insert(std::pair<int, std::string>(2, "Flatline")); // does not replace
// you can use curly braces with the pair
students.insert(std::pair<int, std::string>{2, "Flatline"});
// it is OK to drop the name as well, the compiler can figure it out
// most of the time
students.insert({2, "Chris Gaucho"});
students[2] = "Chris Gaucho"; // replaces
std::cout << students[2] << std::endl;
## 8 Erasing using iterators
• erase() can either erase an item in a map using an iterator location OR a specific key value.
### 8.1 Example
// Erasing by iterator
auto p = students.find(2); // p's type is std::map<int,
// std::string>::iterator. It is a
// mouthful.
students.erase(p); // erases "Jane Doe"
// Erasing by key
students.erase(0); // erases "Richert"
// print out the entire map.
for (auto i = students.begin(); i != students.end(); i++) {
std::cout << i->first << ": " << i->second << std::endl;
}
## Footnotes:
1
The standard library comes with plenty flavors of sorting algorithms.
2
C++ also supports multisets, when using a multiset you can put the same element more than once.
Author: Mehmet Emre
Created:
The material for this class is based on Prof. Richert Wang's material for CS 32
|
{}
|
Data structures and file formats¶
This page documents the internal data structures and storage mechanisms of Borg. It is partly based on mailing list discussion about internals and also on static code analysis.
Repository¶
Borg stores its data in a Repository, which is a file system based transactional key-value store. Thus the repository does not know about the concept of archives or items.
Each repository has the following file structure:
simple text file telling that this is a Borg repository
config
repository configuration
data/
directory where the actual data is stored
hints.%d
hints for repository compaction
index.%d
repository index
lock.roster and lock.exclusive/*
used by the locking system to manage shared and exclusive locks
Transactionality is achieved by using a log (aka journal) to record changes. The log is a series of numbered files called segments. Each segment is a series of log entries. The segment number together with the offset of each entry relative to its segment start establishes an ordering of the log entries. This is the “definition” of time for the purposes of the log.
Config file¶
Each repository has a config file which is a INI-style file and looks like this:
[repository]
version = 2
segments_per_dir = 1000
max_segment_size = 524288000
id = 57d6c1d52ce76a836b532b0e42e677dec6af9fca3673db511279358828a21ed6
This is where the repository.id is stored. It is a unique identifier for repositories. It will not change if you move the repository around so you can make a local transfer then decide to move the repository to another (even remote) location at a later time.
Keys¶
Repository keys are byte-strings of fixed length (32 bytes), they don’t have a particular meaning (except for the Manifest).
Normally the keys are computed like this:
key = id = id_hash(plaintext_data) # plain = not encrypted, not compressed, not obfuscated
The id_hash function depends on the encryption mode.
As the id / key is used for deduplication, id_hash must be a cryptographically strong hash or MAC.
Segments¶
Objects referenced by a key are stored inline in files (segments) of approx. 500 MB size in numbered subdirectories of repo/data. The number of segments per directory is controlled by the value of segments_per_dir. If you change this value in a non-empty repository, you may also need to relocate the segment files manually.
A segment starts with a magic number (BORG_SEG as an eight byte ASCII string), followed by a number of log entries. Each log entry consists of (in this order):
• crc32 checksum (uint32): - for PUT2: CRC32(size + tag + key + digest) - for PUT: CRC32(size + tag + key + payload) - for DELETE: CRC32(size + tag + key) - for COMMIT: CRC32(size + tag)
• size (uint32) of the entry (including the whole header)
• tag (uint8): PUT(0), DELETE(1), COMMIT(2) or PUT2(3)
• key (256 bit) - only for PUT/PUT2/DELETE
• payload (size - 41 bytes) - only for PUT
• xxh64 digest (64 bit) = XXH64(size + tag + key + payload) - only for PUT2
• payload (size - 41 - 8 bytes) - only for PUT2
PUT2 is new since repository version 2. For new log entries PUT2 is used. PUT is still supported to read version 1 repositories, but not generated any more. If we talk about PUT in general, it shall usually mean PUT2 for repository version 2+.
Those files are strictly append-only and modified only once.
When an object is written to the repository a PUT entry is written to the file containing the object id and payload. If an object is deleted a DELETE entry is appended with the object id.
A COMMIT tag is written when a repository transaction is committed. The segment number of the segment containing a commit is the transaction ID.
When a repository is opened any PUT or DELETE operations not followed by a COMMIT tag are discarded since they are part of a partial/uncommitted transaction.
The size of individual segments is limited to 4 GiB, since the offset of entries within segments is stored in a 32-bit unsigned integer in the repository index.
All data (the manifest, archives, archive item stream chunks and file data chunks) is compressed, optionally obfuscated and encrypted. This produces some additional metadata (size and compression information), which is separately serialized and also encrypted.
See Encryption for a graphic outlining the anatomy of the encryption in Borg. What you see at the bottom there is done twice: once for the data and once for the metadata.
An object (the payload part of a segment file log entry) must be like:
• length of encrypted metadata (16bit unsigned int)
• msgpacked dict with:
• ctype (compression type 0..255)
• clevel (compression level 0..255)
• csize (overall compressed (and maybe obfuscated) data size)
• psize (only when obfuscated: payload size without the obfuscation trailer)
• size (uncompressed size of the data)
• encrypted data (incl. encryption header), when decrypted:
• compressed data (with an optional all-zero-bytes obfuscation trailer)
This new, more complex repo v2 object format was implemented to be able to efficiently query the metadata without having to read, transfer and decrypt the (usually much bigger) data part.
The metadata is encrypted to not disclose potentially sensitive information that could be used for e.g. fingerprinting attacks.
The compression ctype and clevel is explained in Compression.
Index, hints and integrity¶
The repository index is stored in index.<TRANSACTION_ID> and is used to determine an object’s location in the repository. It is a HashIndex, a hash table using open addressing.
It maps object keys to:
• segment number (unit32)
• offset of the object’s entry within the segment (uint32)
• flags (uint32)
The hints file is a msgpacked file named hints.<TRANSACTION_ID>. It contains:
• version
• list of segments
• compact
• storage_quota_use
The integrity file is a msgpacked file named integrity.<TRANSACTION_ID>. It contains checksums of the index and hints files and is described in the Checksumming data structures section below.
If the index or hints are corrupted, they are re-generated automatically. If they are outdated, segments are replayed from the index state to the currently committed transaction.
Compaction¶
For a given key only the last entry regarding the key, which is called current (all other entries are called superseded), is relevant: If there is no entry or the last entry is a DELETE then the key does not exist. Otherwise the last PUT defines the value of the key.
By superseding a PUT (with either another PUT or a DELETE) the log entry becomes obsolete. A segment containing such obsolete entries is called sparse, while a segment containing no such entries is called compact.
Since writing a DELETE tag does not actually delete any data and thus does not free disk space any log-based data store will need a compaction strategy (somewhat analogous to a garbage collector).
Borg uses a simple forward compacting algorithm, which avoids modifying existing segments. Compaction runs when a commit is issued with compact=True parameter, e.g. by the borg compact command (unless the Append-only mode (forbid compaction) is active).
The compaction algorithm requires two inputs in addition to the segments themselves:
1. Which segments are sparse, to avoid scanning all segments (impractical). Further, Borg uses a conditional compaction strategy: Only those segments that exceed a threshold sparsity are compacted.
To implement the threshold condition efficiently, the sparsity has to be stored as well. Therefore, Borg stores a mapping (segment id,) -> (number of sparse bytes,).
2. Each segment’s reference count, which indicates how many live objects are in a segment. This is not strictly required to perform the algorithm. Rather, it is used to validate that a segment is unused before deleting it. If the algorithm is incorrect, or the reference count was not accounted correctly, then an assertion failure occurs.
These two pieces of information are stored in the hints file (hints.N) next to the index (index.N).
Compaction may take some time if a repository has been kept in append-only mode or borg compact has not been used for a longer time, which both has caused the number of sparse segments to grow.
Compaction processes sparse segments from oldest to newest; sparse segments which don’t contain enough deleted data to justify compaction are skipped. This avoids doing e.g. 500 MB of writing current data to a new segment when only a couple kB were deleted in a segment.
Segments that are compacted are read in entirety. Current entries are written to a new segment, while superseded entries are omitted. After each segment an intermediary commit is written to the new segment. Then, the old segment is deleted (asserting that the reference count diminished to zero), freeing disk space.
A simplified example (excluding conditional compaction and with simpler commit logic) showing the principal operation of compaction:
(The actual algorithm is more complex to avoid various consistency issues, refer to the borg.repository module for more comments and documentation on these issues.)
Storage quotas¶
Quotas are implemented at the Repository level. The active quota of a repository is determined by the storage_quota config entry or a run-time override (via borg serve). The currently used quota is stored in the hints file. Operations (PUT and DELETE) during a transaction modify the currently used quota:
• A PUT adds the size of the log entry to the quota, i.e. the length of the data plus the 41 byte header.
• A DELETE subtracts the size of the deleted log entry from the quota, which includes the header.
Thus, PUT and DELETE are symmetric and cancel each other out precisely.
The quota does not track on-disk size overheads (due to conditional compaction or append-only mode). In normal operation the inclusion of the log entry headers in the quota act as a faithful proxy for index and hints overheads.
By tracking effective content size, the client can always recover from a full quota by deleting archives. This would not be possible if the quota tracked on-disk size, since journaling DELETEs requires extra disk space before space is freed. Tracking effective size on the other hand accounts DELETEs immediately as freeing quota.
Enforcing the quota
The storage quota is meant as a robust mechanism for service providers, therefore borg serve has to enforce it without loopholes (e.g. modified clients). The following sections refer to using quotas on remotely accessed repositories. For local access, consider client and serve the same. Accordingly, quotas cannot be enforced with local access, since the quota can be changed in the repository config.
The quota is enforcible only if all borg serve versions accessible to clients support quotas (see next section). Further, quota is per repository. Therefore, ensure clients can only access a defined set of repositories with their quotas set, using --restrict-to-repository.
If the client exceeds the storage quota the StorageQuotaExceeded exception is raised. Normally a client could ignore such an exception and just send a commit() command anyway, circumventing the quota. However, when StorageQuotaExceeded is raised, it is stored in the transaction_doomed attribute of the repository. If the transaction is doomed, then commit will re-raise this exception, aborting the commit.
The transaction_doomed indicator is reset on a rollback (which erases the quota-exceeding state).
Compatibility with older servers and enabling quota after-the-fact
If no quota data is stored in the hints file, Borg assumes zero quota is used. Thus, if a repository with an enabled quota is written to with an older borg serve version that does not understand quotas, then the quota usage will be erased.
The client version is irrelevant to the storage quota and has no part in it. The form of error messages due to exceeding quota varies with client versions.
A similar situation arises when upgrading from a Borg release that did not have quotas. Borg will start tracking quota use from the time of the upgrade, starting at zero.
If the quota shall be enforced accurately in these cases, either
• delete the index.N and hints.N files, forcing Borg to rebuild both, re-acquiring quota data in the process, or
• edit the msgpacked hints.N file (not recommended and thus not documented further).
The object graph¶
On top of the simple key-value store offered by the Repository, Borg builds a much more sophisticated data structure that is essentially a completely encrypted object graph. Objects, such as archives, are referenced by their chunk ID, which is cryptographically derived from their contents. More on how this helps security in Structural Authentication.
The manifest¶
The manifest is the root of the object hierarchy. It references all archives in a repository, and thus all data in it. Since no object references it, it cannot be stored under its ID key. Instead, the manifest has a fixed all-zero key.
The manifest is rewritten each time an archive is created, deleted, or modified. It looks like this:
{
'version': 1,
'timestamp': '2017-05-05T12:42:23.042864',
'item_keys': ['acl_access', 'acl_default', ...],
'config': {},
'archives': {
'2017-05-05-system-backup': {
'id': b'<32 byte binary object ID>',
'time': '2017-05-05T12:42:22.942864',
},
},
'tam': ...,
}
The version field can be either 1 or 2. The versions differ in the way feature flags are handled, described below.
The timestamp field is used to avoid logical replay attacks where the server just resets the repository to a previous state.
item_keys is a list containing all Item keys that may be encountered in the repository. It is used by borg check, which verifies that all keys in all items are a subset of these keys. Thus, an older version of borg check supporting this mechanism can correctly detect keys introduced in later versions.
The tam key is part of the tertiary authentication mechanism (formerly known as “tertiary authentication for metadata”) and authenticates the manifest, since an ID check is not possible.
config is a general-purpose location for additional metadata. All versions of Borg preserve its contents (it may have been a better place for item_keys, which is not preserved by unaware Borg versions, releases predating 1.0.4).
Feature flags¶
Feature flags are used to add features to data structures without causing corruption if older versions are used to access or modify them. The main issues to consider for a feature flag oriented design are flag granularity, flag storage, and cache invalidation.
Feature flags are divided in approximately three categories, detailed below. Due to the nature of ID-based deduplication, write (i.e. creating archives) and read access are not symmetric; it is possible to create archives referencing chunks that are not readable with the current feature set. The third category are operations that require accurate reference counts, for example archive deletion and check.
As the manifest is always updated and always read, it is the ideal place to store feature flags, comparable to the super-block of a file system. The only problem is to recover from a lost manifest, i.e. how is it possible to detect which feature flags are enabled, if there is no manifest to tell. This issue is left open at this time, but is not expected to be a major hurdle; it doesn’t have to be handled efficiently, it just needs to be handled.
Lastly, cache invalidation is handled by noting which feature flags were and which were not understood while manipulating a cache. This allows borg to detect whether the cache needs to be invalidated, i.e. rebuilt from scratch. See Cache feature flags below.
The config key stores the feature flags enabled on a repository:
config = {
'feature_flags': {
'mandatory': ['some_feature'],
},
'check': {
'mandatory': ['other_feature'],
}
'write': ...,
'delete': ...
},
}
The top-level distinction for feature flags is the operation the client intends to perform,
the read operation includes extraction and listing of archives,
the write operation includes creating new archives,
the delete (archives) operation,
the check operation requires full understanding of everything in the repository.
These are weakly set-ordered; check will include everything required for delete, delete will likely include write and read. However, read may require more features than write (due to ID-based deduplication, write does not necessarily require reading/understanding repository contents).
Each operation can contain several sets of feature flags. Only one set, the mandatory set is currently defined.
Upon reading the manifest, the Borg client has already determined which operation should be performed. If feature flags are found in the manifest, the set of feature flags supported by the client is compared to the mandatory set found in the manifest. If any unsupported flags are found (i.e. the mandatory set is not a subset of the features supported by the Borg client used), the operation is aborted with a MandatoryFeatureUnsupported error:
Unsupported repository feature(s) {‘some_feature’}. A newer version of borg is required to access this repository.
Older Borg releases do not have this concept and do not perform feature flags checks. These can be locked out with manifest version 2. Thus, the only difference between manifest versions 1 and 2 is that the latter is only accepted by Borg releases implementing feature flags.
Therefore, as soon as any mandatory feature flag is enabled in a repository, the manifest version must be switched to version 2 in order to lock out all Borg releases unaware of feature flags.
Cache feature flags
The cache does not have its separate set of feature flags. Instead, Borg stores which flags were used to create or modify a cache.
All mandatory manifest features from all operations are gathered in one set. Then, two sets of features are computed;
• those features that are supported by the client and mandated by the manifest are added to the mandatory_features set,
• the ignored_features set comprised of those features mandated by the manifest, but not supported by the client.
Because the client previously checked compliance with the mandatory set of features required for the particular operation it is executing, the mandatory_features set will contain all necessary features required for using the cache safely.
Conversely, the ignored_features set contains only those features which were not relevant to operating the cache. Otherwise, the client would not pass the feature set test against the manifest.
When opening a cache and the mandatory_features set is not a subset of the features supported by the client, the cache is wiped out and rebuilt, since a client not supporting a mandatory feature that the cache was built with would be unable to update it correctly. The assumption behind this behaviour is that any of the unsupported features could have been reflected in the cache and there is no way for the client to discern whether that is the case. Meanwhile, it may not be practical for every feature to have clients using it track whether the feature had an impact on the cache. Therefore, the cache is wiped.
When opening a cache and the intersection of ignored_features and the features supported by the client contains any elements, i.e. the client possesses features that the previous client did not have and those new features are enabled in the repository, the cache is wiped out and rebuilt.
While the former condition likely requires no tweaks, the latter condition is formulated in an especially conservative way to play it safe. It seems likely that specific features might be exempted from the latter condition.
Defined feature flags
Currently no feature flags are defined.
From currently planned features, some examples follow, these may/may not be implemented and purely serve as examples.
• A mandatory read feature could be using a different encryption scheme (e.g. session keys). This may not be mandatory for the write operation - reading data is not strictly required for creating an archive.
• Any additions to the way chunks are referenced (e.g. to support larger archives) would become a mandatory delete and check feature; delete implies knowing correct reference counts, so all object references need to be understood. check must discover the entire object graph as well, otherwise the “orphan chunks check” could delete data still in use.
Archives¶
Each archive is an object referenced by the manifest. The archive object itself does not store any of the data contained in the archive it describes.
Instead, it contains a list of chunks which form a msgpacked stream of items. The archive object itself further contains some metadata:
• version
• name, which might differ from the name set in the manifest. When borg check rebuilds the manifest (e.g. if it was corrupted) and finds more than one archive object with the same name, it adds a counter to the name in the manifest, but leaves the name field of the archives as it was.
• item_ptrs, a list of “pointer chunk” IDs. Each “pointer chunk” contains a list of chunk IDs of item metadata.
• cmdline, the command line which was used to create the archive
• hostname
• time and time_end are the start and end timestamps, respectively
• comment, a user-specified archive comment
• chunker_params are the chunker-params used for creating the archive. This is used by borg recreate to determine whether a given archive needs rechunking.
• Some other pieces of information related to recreate.
Items¶
Each item represents a file, directory or other file system item and is stored as a dictionary created by the Item class that contains:
• path
• list of data chunks (size: count * ~40B)
• user
• group
• uid
• gid
• mode (item type + permissions)
• rdev (for device files)
• mtime, atime, ctime, birthtime in nanoseconds
• xattrs
• acl (various OS-dependent fields)
• flags
All items are serialized using msgpack and the resulting byte stream is fed into the same chunker algorithm as used for regular file data and turned into deduplicated chunks. The reference to these chunks is then added to the archive metadata. To achieve a finer granularity on this metadata stream, we use different chunker params for this chunker, which result in smaller chunks.
A chunk is stored as an object as well, of course.
Chunks¶
Borg has these chunkers:
• “fixed”: a simple, low cpu overhead, fixed blocksize chunker, optionally supporting a header block of different size.
• “buzhash”: variable, content-defined blocksize, uses a rolling hash computed by the Buzhash algorithm.
For some more general usage hints see also --chunker-params.
“fixed” chunker¶
The fixed chunker triggers (chunks) at even-spaced offsets, e.g. every 4MiB, producing chunks of same block size (the last chunk is not required to be full-size).
Optionally, it supports processing a differently sized “header” first, before it starts to cut chunks of the desired block size. The default is not to have a differently sized header.
borg create --chunker-params fixed,BLOCK_SIZE[,HEADER_SIZE]
• BLOCK_SIZE: no default value, multiple of the system page size (usually 4096 bytes) recommended. E.g.: 4194304 would cut 4MiB sized chunks.
The fixed chunker also supports processing sparse files (reading only the ranges with data and seeking over the empty hole ranges).
borg create --sparse --chunker-params fixed,BLOCK_SIZE[,HEADER_SIZE]
“buzhash” chunker¶
The buzhash chunker triggers (chunks) when the last HASH_MASK_BITS bits of the hash are zero, producing chunks with a target size of 2^HASH_MASK_BITS bytes.
Buzhash is only used for cutting the chunks at places defined by the content, the buzhash value is not used as the deduplication criteria (we use a cryptographically strong hash/MAC over the chunk contents for this, the id_hash).
The idea of content-defined chunking is assigning every byte where a cut could be placed a hash. The hash is based on some number of bytes (the window size) before the byte in question. Chunks are cut where the hash satisfies some condition (usually “n numbers of trailing/leading zeroes”). This causes chunks to be cut in the same location relative to the file’s contents, even if bytes are inserted or removed before/after a cut, as long as the bytes within the window stay the same. This results in a high chance that a single cluster of changes to a file will only result in 1-2 new chunks, aiding deduplication.
Using normal hash functions this would be extremely slow, requiring hashing approximately window size * file size bytes. A rolling hash is used instead, which allows to add a new input byte and compute a new hash as well as remove a previously added input byte from the computed hash. This makes the cost of computing a hash for each input byte largely independent of the window size.
Borg defines minimum and maximum chunk sizes (CHUNK_MIN_EXP and CHUNK_MAX_EXP, respectively) which narrows down where cuts may be made, greatly reducing the amount of data that is actually hashed for content-defined chunking.
borg create --chunker-params buzhash,CHUNK_MIN_EXP,CHUNK_MAX_EXP,HASH_MASK_BITS,HASH_WINDOW_SIZE can be used to tune the chunker parameters, the default is:
• CHUNK_MIN_EXP = 19 (minimum chunk size = 2^19 B = 512 kiB)
• CHUNK_MAX_EXP = 23 (maximum chunk size = 2^23 B = 8 MiB)
• HASH_MASK_BITS = 21 (target chunk size ~= 2^21 B = 2 MiB)
• HASH_WINDOW_SIZE = 4095 [B] (0xFFF)
The buzhash table is altered by XORing it with a seed randomly generated once for the repository, and stored encrypted in the keyfile. This is to prevent chunk size based fingerprinting attacks on your encrypted repo contents (to guess what files you have based on a specific set of chunk sizes).
The cache¶
The files cache is stored in cache/files and is used at backup time to quickly determine whether a given file is unchanged and we have all its chunks.
In memory, the files cache is a key -> value mapping (a Python dict) and contains:
• key: id_hash of the encoded, absolute file path
• value:
• file inode number
• file size
• file ctime_ns (or mtime_ns)
• age (0 [newest], 1, 2, 3, …, BORG_FILES_CACHE_TTL - 1)
• list of chunk ids representing the file’s contents
To determine whether a file has not changed, cached values are looked up via the key in the mapping and compared to the current file attribute values.
If the file’s size, timestamp and inode number is still the same, it is considered to not have changed. In that case, we check that all file content chunks are (still) present in the repository (we check that via the chunks cache).
If everything is matching and all chunks are present, the file is not read / chunked / hashed again (but still a file metadata item is written to the archive, made from fresh file metadata read from the filesystem). This is what makes borg so fast when processing unchanged files.
If there is a mismatch or a chunk is missing, the file is read / chunked / hashed. Chunks already present in repo won’t be transferred to repo again.
The inode number is stored and compared to make sure we distinguish between different files, as a single path may not be unique across different archives in different setups.
Not all filesystems have stable inode numbers. If that is the case, borg can be told to ignore the inode number in the check via --files-cache.
The age value is used for cache management. If a file is “seen” in a backup run, its age is reset to 0, otherwise its age is incremented by one. If a file was not seen in BORG_FILES_CACHE_TTL backups, its cache entry is removed. See also: It always chunks all my files, even unchanged ones! and I am seeing ‘A’ (added) status for an unchanged file!?
The files cache is a python dictionary, storing python objects, which generates a lot of overhead.
Borg can also work without using the files cache (saves memory if you have a lot of files or not much RAM free), then all files are assumed to have changed. This is usually much slower than with files cache.
The on-disk format of the files cache is a stream of msgpacked tuples (key, value). Loading the files cache involves reading the file, one msgpack object at a time, unpacking it, and msgpacking the value (in an effort to save memory).
The chunks cache is stored in cache/chunks and is used to determine whether we already have a specific chunk, to count references to it and also for statistics.
The chunks cache is a key -> value mapping and contains:
• key:
• chunk id_hash
• value:
• reference count
• size
The chunks cache is a HashIndex. Due to some restrictions of HashIndex, the reference count of each given chunk is limited to a constant, MAX_VALUE (introduced below in HashIndex), approximately 2**32. If a reference count hits MAX_VALUE, decrementing it yields MAX_VALUE again, i.e. the reference count is pinned to MAX_VALUE.
Indexes / Caches memory usage¶
Here is the estimated memory usage of Borg - it’s complicated:
chunk_size ~= 2 ^ HASH_MASK_BITS (for buzhash chunker, BLOCK_SIZE for fixed chunker)
chunk_count ~= total_file_size / chunk_size
repo_index_usage = chunk_count * 48
chunks_cache_usage = chunk_count * 40
files_cache_usage = total_file_count * 240 + chunk_count * 80
mem_usage ~= repo_index_usage + chunks_cache_usage + files_cache_usage
= chunk_count * 164 + total_file_count * 240
Due to the hashtables, the best/usual/worst cases for memory allocation can be estimated like that:
mem_allocation = mem_usage / load_factor # l_f = 0.25 .. 0.75
mem_allocation_peak = mem_allocation * (1 + growth_factor) # g_f = 1.1 .. 2
All units are Bytes.
It is assuming every chunk is referenced exactly once (if you have a lot of duplicate chunks, you will have fewer chunks than estimated above).
It is also assuming that typical chunk size is 2^HASH_MASK_BITS (if you have a lot of files smaller than this statistical medium chunk size, you will have more chunks than estimated above, because 1 file is at least 1 chunk).
If a remote repository is used the repo index will be allocated on the remote side.
The chunks cache, files cache and the repo index are all implemented as hash tables. A hash table must have a significant amount of unused entries to be fast - the so-called load factor gives the used/unused elements ratio.
When a hash table gets full (load factor getting too high), it needs to be grown (allocate new, bigger hash table, copy all elements over to it, free old hash table) - this will lead to short-time peaks in memory usage each time this happens. Usually does not happen for all hashtables at the same time, though. For small hash tables, we start with a growth factor of 2, which comes down to ~1.1x for big hash tables.
E.g. backing up a total count of 1 Mi (IEC binary prefix i.e. 2^20) files with a total size of 1TiB.
1. with create --chunker-params buzhash,10,23,16,4095 (custom, like borg < 1.0):
mem_usage = 2.8GiB
1. with create --chunker-params buzhash,19,23,21,4095 (default):
mem_usage = 0.31GiB
Note
There is also the --files-cache=disabled option to disable the files cache. You’ll save some memory, but it will need to read / chunk all the files as it can not skip unmodified files then.
HashIndex¶
The chunks cache and the repository index are stored as hash tables, with only one slot per bucket, spreading hash collisions to the following buckets. As a consequence the hash is just a start position for a linear search. If a key is looked up that is not in the table, then the hash table is searched from the start position (the hash) until the first empty bucket is reached.
This particular mode of operation is open addressing with linear probing.
When the hash table is filled to 75%, its size is grown. When it’s emptied to 25%, its size is shrinked. Operations on it have a variable complexity between constant and linear with low factor, and memory overhead varies between 33% and 300%.
If an element is deleted, and the slot behind the deleted element is not empty, then the element will leave a tombstone, a bucket marked as deleted. Tombstones are only removed by insertions using the tombstone’s bucket, or by resizing the table. They present the same load to the hash table as a real entry, but do not count towards the regular load factor.
Thus, if the number of empty slots becomes too low (recall that linear probing for an element not in the index stops at the first empty slot), the hash table is rebuilt. The maximum effective load factor, i.e. including tombstones, is 93%.
Data in a HashIndex is always stored in little-endian format, which increases efficiency for almost everyone, since basically no one uses big-endian processors any more.
HashIndex does not use a hashing function, because all keys (save manifest) are outputs of a cryptographic hash or MAC and thus already have excellent distribution. Thus, HashIndex simply uses the first 32 bits of the key as its “hash”.
The format is easy to read and write, because the buckets array has the same layout in memory and on disk. Only the header formats differ. The on-disk header is struct HashHeader:
• First, the HashIndex magic, the eight byte ASCII string “BORG_IDX”.
• Second, the signed 32-bit number of entries (i.e. buckets which are not deleted and not empty).
• Third, the signed 32-bit number of buckets, i.e. the length of the buckets array contained in the file, and the modulus for index calculation.
• Fourth, the signed 8-bit length of keys.
• Fifth, the signed 8-bit length of values. This has to be at least four bytes.
All fields are packed.
The HashIndex is not a general purpose data structure. The value size must be at least 4 bytes, and these first bytes are used for in-band signalling in the data structure itself.
The constant MAX_VALUE (defined as 2**32-1025 = 4294966271) defines the valid range for these 4 bytes when interpreted as an uint32_t from 0 to MAX_VALUE (inclusive). The following reserved values beyond MAX_VALUE are currently in use (byte order is LE):
• 0xffffffff marks empty buckets in the hash table
• 0xfffffffe marks deleted buckets in the hash table
HashIndex is implemented in C and wrapped with Cython in a class-based interface. The Cython wrapper checks every passed value against these reserved values and raises an AssertionError if they are used.
Encryption¶
The Cryptography in Borg section for an in-depth review.
For new repositories, borg only uses modern AEAD ciphers: AES-OCB or CHACHA20-POLY1305.
For each borg invocation, a new sessionkey is derived from the borg key material and the 48bit IV starts from 0 again (both ciphers internally add a 32bit counter to our IV, so we’ll just count up by 1 per chunk).
The encryption layout is best seen at the bottom of this diagram:
No special IV/counter management is needed here due to the use of session keys.
A 48 bit IV is way more than needed: If you only backed up 4kiB chunks (2^12B), the IV would “limit” the data encrypted in one session to 2^(12+48)B == 2.3 exabytes, meaning you would run against other limitations (RAM, storage, time) way before that. In practice, chunks are usually bigger, for big files even much bigger, giving an even higher limit.
Legacy modes¶
Old repositories (which used AES-CTR mode) are supported read-only to be able to borg transfer their archives to new repositories (which use AEAD modes).
AES-CTR mode is not supported for new repositories and the related code will be removed in a future release.
Both modes¶
Encryption keys (and other secrets) are kept either in a key file on the client (‘keyfile’ mode) or in the repository config on the server (‘repokey’ mode). In both cases, the secrets are generated from random and then encrypted by a key derived from your passphrase (this happens on the client before the key is stored into the keyfile or as repokey).
The passphrase is passed through the BORG_PASSPHRASE environment variable or prompted for interactive usage.
Key files¶
The Offline key security section for an in-depth review of the key encryption.
When initializing a repository with one of the “keyfile” encryption modes, Borg creates an associated key file in \$HOME/.config/borg/keys.
The same key is also used in the “repokey” modes, which store it in the repository in the configuration file.
The internal data structure is as follows:
version
currently always an integer, 2
repository_id
the id field in the config INI file of the repository.
crypt_key
the initial key material used for the AEAD crypto (512 bits)
id_key
the key used to MAC the plaintext chunk data to compute the chunk’s id
chunk_seed
the seed for the buzhash chunking table (signed 32 bit integer)
These fields are packed using msgpack. The utf-8 encoded passphrase is processed with argon2 to derive a 256 bit key encryption key (KEK).
Then the KEK is used to encrypt and authenticate the packed data using the chacha20-poly1305 AEAD cipher.
The result is stored in a another msgpack formatted as follows:
version
currently always an integer, 1
salt
random 256 bits salt used to process the passphrase
argon2_*
some parameters for the argon2 kdf
algorithm
the algorithms used to process the passphrase (currently the string argon2 chacha20-poly1305)
data
The encrypted, packed fields.
The resulting msgpack is then encoded using base64 and written to the key file, wrapped using the standard textwrap module with a header. The header is a single line with a MAGIC string, a space and a hexadecimal representation of the repository id.
Compression¶
Borg supports the following compression methods, each identified by a ctype value in the range between 0 and 255 (and augmented by a clevel 0..255 value for the compression level):
• none (no compression, pass through data 1:1), identified by 0x00
• lz4 (low compression, but super fast), identified by 0x01
• zstd (level 1-22 offering a wide range: level 1 is lower compression and high speed, level 22 is higher compression and lower speed) - identified by 0x03
• zlib (level 0-9, level 0 is no compression [but still adding zlib overhead], level 1 is low, level 9 is high compression), identified by 0x05
• lzma (level 0-9, level 0 is low, level 9 is high compression), identified by 0x02.
The type byte is followed by a byte indicating the compression level.
Speed: none > lz4 > zlib > lzma, lz4 > zstd Compression: lzma > zlib > lz4 > none, zstd > lz4
Be careful, higher compression levels might use a lot of resources (CPU/memory).
The overall speed of course also depends on the speed of your target storage. If that is slow, using a higher compression level might yield better overall performance. You need to experiment a bit. Maybe just watch your CPU load, if that is relatively low, increase compression until 1 core is 70-100% loaded.
Even if your target storage is rather fast, you might see interesting effects: while doing no compression at all (none) is a operation that takes no time, it likely will need to store more data to the storage compared to using lz4. The time needed to transfer and store the additional data might be much more than if you had used lz4 (which is super fast, but still might compress your data about 2:1). This is assuming your data is compressible (if you backup already compressed data, trying to compress them at backup time is usually pointless).
Compression is applied after deduplication, thus using different compression methods in one repo does not influence deduplication.
See borg create --help about how to specify the compression level and its default.
Lock files¶
Borg uses locks to get (exclusive or shared) access to the cache and the repository.
The locking system is based on renaming a temporary directory to lock.exclusive (for exclusive locks). Inside this directory, there is a file indicating hostname, process id and thread id of the lock holder.
There is also a json file lock.roster that keeps a directory of all shared and exclusive lockers.
If the process is able to rename a temporary directory (with the host/process/thread identifier prepared inside it) in the resource directory to lock.exclusive, it has the lock for it. If renaming fails (because this directory already exists and its host/process/thread identifier denotes a thread on the host which is still alive), lock acquisition fails.
The cache lock is usually in ~/.cache/borg/REPOID/lock.*. The repository lock is in repository/lock.*.
In case you run into troubles with the locks, you can use the borg break-lock command after you first have made sure that no Borg process is running on any machine that accesses this resource. Be very careful, the cache or repository might get damaged if multiple processes use it at the same time.
Checksumming data structures¶
As detailed in the previous sections, Borg generates and stores various files containing important meta data, such as the repository index, repository hints, chunks caches and files cache.
Data corruption in these files can damage the archive data in a repository, e.g. due to wrong reference counts in the chunks cache. Only some parts of Borg were designed to handle corrupted data structures, so a corrupted files cache may cause crashes or write incorrect archives.
Therefore, Borg calculates checksums when writing these files and tests checksums when reading them. Checksums are generally 64-bit XXH64 hashes. The canonical xxHash representation is used, i.e. big-endian. Checksums are stored as hexadecimal ASCII strings.
For compatibility, checksums are not required and absent checksums do not trigger errors. The mechanisms have been designed to avoid false-positives when various Borg versions are used alternately on the same repositories.
Checksums are a data safety mechanism. They are not a security mechanism.
Choice of algorithm
XXH64 has been chosen for its high speed on all platforms, which avoids performance degradation in CPU-limited parts (e.g. cache synchronization). Unlike CRC32, it neither requires hardware support (crc32c or CLMUL) nor vectorized code nor large, cache-unfriendly lookup tables to achieve good performance. This simplifies deployment of it considerably (cf. src/borg/algorithms/crc32…).
Further, XXH64 is a non-linear hash function and thus has a “more or less” good chance to detect larger burst errors, unlike linear CRCs where the probability of detection decreases with error size.
The 64-bit checksum length is considered sufficient for the file sizes typically checksummed (individual files up to a few GB, usually less). xxHash was expressly designed for data blocks of these sizes.
Lower layer — file_integrity¶
To accommodate the different transaction models used for the cache and repository, there is a lower layer (borg.crypto.file_integrity.IntegrityCheckedFile) wrapping a file-like object, performing streaming calculation and comparison of checksums. Checksum errors are signalled by raising an exception (borg.crypto.file_integrity.FileIntegrityError) at the earliest possible moment.
Calculating checksums
Before feeding the checksum algorithm any data, the file name (i.e. without any path) is mixed into the checksum, since the name encodes the context of the data for Borg.
The various indices used by Borg have separate header and main data parts. IntegrityCheckedFile allows borg to checksum them independently, which avoids even reading the data when the header is corrupted. When a part is signalled, the length of the part name is mixed into the checksum state first (encoded as an ASCII string via %10d printf format), then the name of the part is mixed in as an UTF-8 string. Lastly, the current position (length) in the file is mixed in as well.
The checksum state is not reset at part boundaries.
A final checksum is always calculated in the same way as the parts described above, after seeking to the end of the file. The final checksum cannot prevent code from processing corrupted data during reading, however, it prevents use of the corrupted data.
Serializing checksums
All checksums are compiled into a simple JSON structure called integrity data:
{
"algorithm": "XXH64",
"digests": {
"final": "e2a7f132fc2e8b24"
}
}
The algorithm key notes the used algorithm. When reading, integrity data containing an unknown algorithm is not inspected further.
The digests key contains a mapping of part names to their digests.
Integrity data is generally stored by the upper layers, introduced below. An exception is the DetachedIntegrityCheckedFile, which automatically writes and reads it from a “.integrity” file next to the data file. It is used for archive chunks indexes in chunks.archive.d.
Upper layer¶
Storage of integrity data depends on the component using it, since they have different transaction mechanisms, and integrity data needs to be transacted with the data it is supposed to protect.
Main cache files: chunks and files cache
The integrity data of the chunks and files caches is stored in the cache config, since all three are transacted together.
The [integrity] section is used:
[cache]
version = 1
repository = 3c4...e59
manifest = 10e...21c
timestamp = 2017-06-01T21:31:39.699514
key_type = 2
previous_location = /path/to/repo
[integrity]
manifest = 10e...21c
chunks = {"algorithm": "XXH64", "digests": {"HashHeader": "eab...39e3", "final": "e2a...b24"}}
The manifest ID is duplicated in the integrity section due to the way all Borg versions handle the config file. Instead of creating a “new” config file from an internal representation containing only the data understood by Borg, the config file is read in entirety (using the Python ConfigParser) and modified. This preserves all sections and values not understood by the Borg version modifying it.
Thus, if an older versions uses a cache with integrity data, it would preserve the integrity section and its contents. If a integrity-aware Borg version would read this cache, it would incorrectly report checksum errors, since the older version did not update the checksums.
However, by duplicating the manifest ID in the integrity section, it is easy to tell whether the checksums concern the current state of the cache.
Integrity errors are fatal in these files, terminating the program, and are not automatically corrected at this time.
chunks.archive.d
Indices in chunks.archive.d are not transacted and use DetachedIntegrityCheckedFile, which writes the integrity data to a separate “.integrity” file.
Integrity errors result in deleting the affected index and rebuilding it. This logs a warning and increases the exit code to WARNING (1).
Repository index and hints
The repository associates index and hints files with a transaction by including the transaction ID in the file names. Integrity data is stored in a third file (“integrity.<TRANSACTION_ID>”). Like the hints file, it is msgpacked:
{
'version': 2,
'hints': '{"algorithm": "XXH64", "digests": {"final": "411208db2aa13f1a"}}',
}
The version key started at 2, the same version used for the hints. Since Borg has many versioned file formats, this keeps the number of different versions in use a bit lower.
The other keys map an auxiliary file, like index or hints to their integrity data. Note that the JSON is stored as-is, and not as part of the msgpack structure.
Integrity errors result in deleting the affected file(s) (index/hints) and rebuilding the index, which is the same action taken when corruption is noticed in other ways (e.g. HashIndex can detect most corrupted headers, but not data corruption). A warning is logged as well. The exit code is not influenced, since remote repositories cannot perform that action. Raising the exit code would be possible for local repositories, but is not implemented.
Unlike the cache design this mechanism can have false positives whenever an older version rewrites the auxiliary files for a transaction created by a newer version, since that might result in a different index (due to hash-table resizing) or hints file (hash ordering, or the older version 1 format), while not invalidating the integrity file.
For example, using 1.1 on a repository, noticing corruption or similar issues and then running borg-1.0 check --repair, which rewrites the index and hints, results in this situation. Borg 1.1 would erroneously report checksum errors in the hints and/or index files and trigger an automatic rebuild of these files.
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# mom_interface_heights module reference¶
Functions for calculating interface heights, including free surface height.
More…
## Functions/Subroutines¶
find_eta_3d() Calculates the heights of all interfaces between layers, using the appropriate form for consistency with the calculation of the pressure gradient forces. find_eta_2d() Calculates the free surface height, using the appropriate form for consistency with the calculation of the pressure gradient forces.
## Detailed Description¶
Functions for calculating interface heights, including free surface height.
## Function/Subroutine Documentation¶
subroutine mom_interface_heights/find_eta_3d(h, tv, G, GV, US, eta, eta_bt, halo_size, eta_to_m, dZref)
Calculates the heights of all interfaces between layers, using the appropriate form for consistency with the calculation of the pressure gradient forces. Additionally, these height may be dilated for consistency with the corresponding time-average quantity from the barotropic calculation.
Parameters
• g :: [in] The ocean’s grid structure.
• gv :: [in] The ocean’s vertical grid structure.
• us :: [in] A dimensional unit scaling type
• h :: [in] Layer thicknesses [H ~> m or kg m-2]
• tv :: [in] A structure pointing to various thermodynamic variables.
• eta :: [out] layer interface heights [Z ~> m] or [1/eta_to_m m].
• eta_bt :: [in] optional barotropic variable that gives the “correct” free surface height (Boussinesq) or total water column mass per unit area (non-Boussinesq). This is used to dilate the layer thicknesses when calculating interface heights [H ~> m or kg m-2]. In Boussinesq mode, eta_bt and GbathyT use the same reference height.
• halo_size :: [in] width of halo points on which to calculate eta.
• eta_to_m :: [in] The conversion factor from the units of eta to m; by default this is USZ_to_m.
• dzref :: [in] The difference in the reference height between GbathyT and eta [Z ~> m]. The default is 0.
Call to
subroutine mom_interface_heights/find_eta_2d(h, tv, G, GV, US, eta, eta_bt, halo_size, eta_to_m, dZref)
Calculates the free surface height, using the appropriate form for consistency with the calculation of the pressure gradient forces. Additionally, the sea surface height may be adjusted for consistency with the corresponding time-average quantity from the barotropic calculation.
Parameters
• g :: [in] The ocean’s grid structure.
• gv :: [in] The ocean’s vertical grid structure.
• us :: [in] A dimensional unit scaling type
• h :: [in] Layer thicknesses [H ~> m or kg m-2]
• tv :: [in] A structure pointing to various thermodynamic variables.
• eta :: [out] free surface height relative to mean sea level (z=0) often [Z ~> m].
• eta_bt :: [in] optional barotropic variable that gives the “correct” free surface height (Boussinesq) or total water column mass per unit area (non-Boussinesq) [H ~> m or kg m-2]. In Boussinesq mode, eta_bt and GbathyT use the same reference height.
• halo_size :: [in] width of halo points on which to calculate eta.
• eta_to_m :: [in] The conversion factor from the units of eta to m; by default this is USZ_to_m.
• dzref :: [in] The difference in the reference height between GbathyT and eta [Z ~> m]. The default is 0.
Call to
mom_density_integrals::int_specific_vol_dp
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# Homework Help: Question about Sigma and Sum Notation
1. Jan 14, 2007
### Bucs44
Here is the problem I'm at currently. I'm not sure that I'm on the right track or not. Also I'm not sure what numbers I should be plugging into the equation. I think it would be 2 through 6 but...?
The sum of elements in the set {ti | i = 3 } #7 on top of sigma notation
tn = 2n - 1, n greater than or equal to 1
Here is my calculation so far:
i1 + i2 + i3 + i4 + i5 + i6 + i7 = (1-1)+(2-1)+(3-1)+(4-1)+(5-1)+(6-1)+(7-1)
Where do I go next?
2. Jan 14, 2007
### Tom1992
just use the formula sigma(i, i from 1 to n) = n(n+1)/2.
In this question, there are only 7 terms, so you can also add up by hand.
3. Jan 14, 2007
### Bucs44
I don't understand that - why would I be adding if ti = 2n - 1?
4. Jan 14, 2007
### cristo
Staff Emeritus
I don't really understand your question. Is it calculate $$\sum_{n=1}^7t_n$$, where tn=2n-1?
If not, please could you quote the exact question as written. (NB, click on the formula to get the code for the LaTex equation)
5. Jan 14, 2007
### Bucs44
Yes sorry - I'm not able to write it exactly - don't have math software - but how you have shown it is correct
6. Jan 14, 2007
### cristo
Staff Emeritus
Note that the software is preloaded into the forum, and so anyone can write in LaTex. See here for a tutorial.
If my sum above is correct, then you simply sum over n in the range 1 to 7. So, $$\sum_{n=1}^7(2n-1)= (2*1-1)+(2*2-1)+ \cdots$$ Do you see where I'm going here? Just plug in the remaining values of n, and then simplify the sum (to obtain a number) which will be the answer.
7. Jan 14, 2007
### Bucs44
Okay - so I do that up to 7 and then add them together or subtract?
I'd get 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49
8. Jan 14, 2007
### Bucs44
Correct me if I'm wrong, but in order to obtain the product, I simply multiply those numbers for my total?
9. Jan 14, 2007
### cristo
Staff Emeritus
Thats right, you add the terms.
Is this a different question now? Do you want to find $$\Pi_{n=1}^7(2n-1)$$? If so, yes, you would multiply the terms.
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# A ball with a mass of 7 kg moving at 6 m/s hits a still ball with a mass of 9 kg. If the first ball stops moving, how fast is the second ball moving?
$4.67 \frac{m}{s}$
$M o m e n t u m = M a s s \cdot V e l o c i t y$
$7 \cdot 6 = 9 \cdot v e l o c i t {y}_{b a l l 2}$
$v e l o c i t {y}_{b a l l 2} = \frac{7 \cdot 6}{9} = 4.67 \frac{m}{s}$
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# coral metrics sketch
As part of the Coral Project, we're trying to come up with some interesting and useful metrics about community members and discussion on news sites.
It's an interesting exercise to develop metrics which embody an organization's principles. For instance - perhaps we see our content as the catalyst for conversations, so we'd measure an article's success by how much discussion it generates.
Generally, there are two groups of metrics that I have been focusing on:
• Asset-level metrics, computed for individual articles or whatever else may be commented on
• User-level metrics, computed for individual users
For the past couple of weeks I've been sketching out a few ideas for these metrics:
• For assets, the principles that these metrics aspire to capture are around quantity and diversity of discussion.
• For users, I look at organizational approval, community approval, how much discussion this user tends to generate, and how likely they are to be moderated.
Here I'll walk through my thought process for these initial ideas.
## Asset-level metrics
For assets, I wanted to value not only the amount of discussion generated but also the diversity discussions. A good discussion is one in which there's a lot of high-quality exchange (something else to be measured, but not captured in this first iteration) from many different people.
There are two scores to start:
• a discussion score, which quantifies how much discussion an asset generated. This looks at how much people are talking to each other as opposed to just counting up the number of comments. For instance, a comments section in which all comments are top-level should not have a high discussion score. A comments section in which there are some really deep back-and-forths should have a higher discussion score.
• a diversity score, which quantifies how many different people are involved in the discussions. Again, we don't want to look at diversity in the comments section as a whole because we are looking for diversity within discussions, i.e. within threads.
The current sketch for computing the discussion score is via two values:
• maximum thread width: what is the highest number of replies for a comment?
These are pretty rough approximations of "how much discussion" there is. The idea is that for sites which only allow one level of replies, a lot of replies to a comment can signal a discussion, and that a very deep thread signals the same for sites which allow more nesting.
The discussion score of a top-level thread is the product of these two intermediary metrics:
$$\text{discussion score}_{\text{thread}} = \max(\text{thread}_{\text{depth}}) \max(\text{thread}_{\text{width}})$$
The discussion score for the entire asset is the value that answers this question: if a new thread were to start in this asset, what discussion score would it have?
The idea is that if a section is generating a lot of discussion, a new thread would likely also involve a lot of discussion.
The nice thing about this approach (which is similar to the one used throughout all these sketches) is that we can capture uncertainty. When a new article is posted, we have no idea how good of a discussion a new thread might be. When we have one or two threads - maybe one is long and one is short - we're still not too sure, so we still have a fairly conservative score. But as more and more people comment, we begin to narrow down on the "true" score for the article.
More concretely (skip ahead to be spared of the gory details), we assume that this discussion score is drawn from a Poisson distribution. This makes things a bit easier to model because we can use the gamma distribution as a conjugate prior.
By default, the gamma prior is parameterized with $k=1, \theta=2$ since it is a fairly conservative estimate to start. That is, we begin with the assumption that any new thread is unlikely to generate a lot of discussion, so it will take a lot of discussion to really convince us otherwise.
Since this gamma-Poisson model will be reused elsewhere, it is defined as its own function:
def gamma_poission_model(X, n, k, theta, quantile):
k = np.sum(X) + k
t = theta/(theta*n + 1)
return stats.gamma.ppf(quantile, k, scale=t)
Since the gamma is a conjugate prior here, the posterior is also a gamma distribution with easily-computed parameters based on the observed data (i.e. the "actual" top-level threads in the discussion).
We need an actual value to work with, however, so we need some point estimate of the expected discussion score. However, we don't want to go with the mean since that may be too optimistic a value, especially if we only have a couple top-level threads to look at. So instead, we look at the lower-bound of the 90% credible interval (the 0.05 quantile) to make a more conservative estimate.
So the final function for computing an asset's discussion score is:
def asset_discussion_score(threads, k=1, theta=2):
n = len(X)
k = np.sum(X) + k
t = theta/(theta*n + 1)
return {'discussion_score': gamma_poission_model(X, n, k, theta, 0.05)}
A similar approach is used for an asset's diversity score. Here we ask the question: if a new comment is posted, how likely is it to be a posted by someone new to the discussion?
We can model this with a beta-binomial model; again, the beta distribution is a conjugate prior for the binomial distribution, so we can compute the posterior's parameters very easily:
def beta_binomial_model(y, n, alpha, beta, quantile):
alpha_ = y + alpha
beta_ = n - y + beta
return stats.beta.ppf(quantile, alpha_, beta_)
Again we start with conservative parameters for the prior, $\alpha=2, \beta=2$, and then compute using threads as evidence:
def asset_diversity_score(threads, alpha=2, beta=2):
X = set()
n = 0
X = X | users
y = len(X)
return {'diversity_score': beta_binomial_model(y, n, alpha, beta, 0.05)}
Then averages for these scores are computed across the entire sample of assets in order to give some context as to what good and bad scores are.
## User-level metrics
User-level metrics are computed in a similar fashion. For each user, four metrics are computed:
• a community score, which quantifies how much the community approves of them. This is computed by trying to predict the number of likes a new post by this user will get.
• an organization score, which quantifies how much the organization approves of them. This is the probability that a post by this user will get "editor's pick" or some equivalent (in the case of Reddit, "gilded", which isn't "organizational" but holds a similar revered status).
• a discussion score, which quantifies how much discussion this user tends to generate. This answers the question: if this user starts a new thread, how many replies do we expect it to have?
• a moderation probability, which is the probability that a post by this user will be moderated.
The community score and discussion score are both modeled as gamma-Poission models using the same function as above. The organization score and moderation probability are both modeled as beta-binomial models using the same function as above.
## Time for more refinement
These metrics are just a few starting points to shape into more sophisticated and nuanced scoring systems. There are some desirable properties missing, and of course, every organization has different principles and values, and so the ideas presented here are not one-size-fits-all, by any means. The challenge is to create some more general framework that allows people to easily define these metrics according to what they value.
# Automatically identifying voicemails
Back in 2015, prosecutor Alberto Nisman was found dead under suspicious circumstances, just as he was about to bring a complaint accusing the Argentinian President Fernández over interfering with investigations into the AMIA bombing that took place in 1994 (This Guardian piece provides some good background).
There were some 40,000 phone calls related to the case that La Nación was interested in exploring further. Naturally, that is quite a big number and it's hard to gather the resources to comb through that many hours of audio.
La Nación crowdsourced the labeling of about 20,000 of these calls into those that were interesting and those that were not (e.g. voicemails or bits of idle chatter). For this process they used CrowData, a platform built by Manuel Aristarán and Gabriela Rodriguez, two former Knight-Mozilla Fellows at La Nación. This left about 20,000 unlabeled calls.
While Juan and I were in Buenos Aires for the Buenos Aires Media Party and our OpenNews fellows gathering, we took a shot at automatically labeling these calls.
## Data preprocessing
The original data we had was in the form of mp3s and png images produced from the mp3s. wav files are easier to work with so we used ffmpeg to convert the mp3s. With wav files, it is just a matter of using scipy to load them as numpy arrays.
For instance:
import scipy.io import wavfile
print(data)
# [15,2,5,6,170,162,551,8487,1247,15827,...]
In the end however, we used librosa, which normalizes the amplitudes and computes a sample rate for the wav file, making the data easier to work with.
import librosa
print(data)
# [0.1,0.3,0.46,0.89,...]
These arrays can be very large depending on the audio file's sample rate, and quite noisy too, especially when trying to identify silences. There may be short spikes in amplitude in an otherwise "silent" section, and in general, there is no true silence. Most silences are just low amplitude but not exactly 0.
In the example below you can see that what a person might consider silence has a few bits of very quiet sound scattered throughout.
There is also "noise" in the non-silent parts; that is, the signal can fluctuate quite suddenly, which can make analysis unwieldy.
To address these concerns, our preprocessing mostly consisted of:
• Reducing the sample rate a bit so the arrays weren't so large, since the features we looked at don't need the precision of a higher sample rate.
• Applying a smoothing function to deal with intermittent spikes in amplitude.
• Zeroing out any amplitudes below 0.015 (i.e. we considered any amplitude under 0.015 to be silence).
Since we had about 20,000 labeled examples to process, we used joblib to parallelize the process, which improved speeds considerably.
## Feature engineering
Typically, the main challenge in a machine learning problem is that of feature engineering - how do we take the raw audio data and represent it in a way that best suits the learning algorithm?
Audio files can be easily visualized, so our approach benefited from our own visual systems - we looked at a few examples from the voicemail and non-voicemail groups to see if any patterns jumped out immediately. Perhaps the clearest two patterns were the rings and the silence:
• A voicemail file will also have a greater proportion of silence than sound. For this, we looked at the images generated from the audio and calculated the percentage of white pixels (representing silence) in the image.
• A voicemail file will have several distinct rings, and the end of the file comes soon after the last ring. The intuition here is that no one picks up during a voicemail - hence many rings - and no one stays on the line much longer after the phone stops ringing. So we consider both the number of rings and the time from the last ring to the end of the file.
### Ring analysis
Identifying the rings is a challenge in itself - we developed a few heuristics which seem to work fairly well. You can see our complete analysis here, but the general idea is that we:
• Identify non-silent parts, separated by silences
• Check the length of the silence that precedes the non-silent part, if it is too short or too long, it is not a ring
• Check the difference between maximum and minimum amplitudes of the non-silent part; it should be small if it's a ring
The example below shows the original audio waveform in green and the smoothed one in red. You can see that the rings are preceded by silences of a roughly equivalent length and that they look more like plateaus (flat-ish on the top). Another way of saying this is that rings have low variance in their amplitude. In contrast, the non-ring signal towards the end has much sharper peaks and vary a lot more in amplitude.
### Other features
We also considered a few other features:
• Variance: voicemails have greater variance, since there is lots of silence punctuated by high-amplitude rings and not much in between.
• Length: voicemails tend to be shorter since people hang up after a few rings.
• Max amplitude: under the assumption that human speech is louder than the rings
• Mean silence length: under the assumption that when people talk, there are only short silences (if any)
However, after some experimentation, the proportion of silence and the ring-based features performed the best.
## Selecting, training, and evaluating the model
With the features in hand, the rest of the task is straightforward: it is a simple binary classification problem. An audio file is either a voicemail or not. We had several models to choose from; we tried logistic regression, random forest, and support vector machines since they are well-worn approaches that tend to perform well.
We first scaled the training data and then the testing data in the same way and computed cross validation scores for each model:
LogisticRegression
roc_auc: 0.96 (+/- 0.02)
average_precision: 0.94 (+/- 0.03)
recall: 0.90 (+/- 0.04)
f1: 0.88 (+/- 0.03)
RandomForestClassifier
roc_auc: 0.96 (+/- 0.02)
average_precision: 0.95 (+/- 0.02)
recall: 0.89 (+/- 0.04)
f1: 0.90 (+/- 0.03)
SVC
roc_auc: 0.96 (+/- 0.02)
average_precision: 0.94 (+/- 0.03)
recall: 0.91 (+/- 0.04)
f1: 0.90 (+/- 0.02)
We were curious what features were good predictors, so we looked at the relative importances of the features for both logistic regression:
[('length', -3.814302896584862),
('last_ring_to_end', 0.0056240364270560934),
('percent_silence', -0.67390678402142834),
('ring_count', 0.48483923341906693),
('white_proportion', 2.3131580570928114)]
And for the random forest classifier:
[('length', 0.30593363755717351),
('last_ring_to_end', 0.33353202776482688),
('percent_silence', 0.15206534339705702),
('ring_count', 0.0086084243372190443),
('white_proportion', 0.19986056694372359)]
Each of the models perform about the same, so we combined them all with a bagging approach (though in the notebook above we forgot to train each model on a different training subset, which may have helped performance), where we selected the label with the majority vote from the models.
## Classification
We tried two variations on classifying the audio files, differing in where we set the probability cutoff for classifying a file as uninteresting or not.
in the balanced classification, we set the probability threshold to 0.5, so any audio file that has ≥ 0.5 of being uninteresting is classified as uninteresting. This approach labeled 8,069 files as discardable.
in the unbalanced classification, we set the threshold to the much stricter 0.9, so an audio file must have ≥ 0.9 chance of being uninteresting to be discarded. This approach labeled 5,785 files as discardable.
## Validation
We have also created a validation Jupyter notebook where we can cherry pick random results from our classified test set and verify the correctness ourselves by listening to the audio file and viewing its image.
The validation code is available here.
## Summary
Even though using machine learning to classify audio is noisy and far from perfect, it can be useful making a problem more manageable. In our case, our solution narrowed the pool of audio files to only those that seem to be more interesting, reducing the time and resources needed to find the good stuff. We could always double check some of the discarded ones if there’s time to do that.
# broca
At this year's OpenNews Code Convening, Alex Spangher of the New York Times and I worked on broca, which is a Python library for rapidly experimenting with new NLP approaches.
Conventional NLP methods - bag-of-words vector space representations of documents, for example - generally work well, but sometimes not well enough, or worse yet, not well at all. At that point, you might want to try out a lot of different methods that aren't available in popular NLP libraries.
Prior to the Code Convening, broca was little more than a hodgepodge of algorithms I'd implemented for various projects. During the Convening, we restructured the library, added some examples and tests, and implemented in the key piece of broca: pipelines.
## Pipelines
The core of broca is organized around pipes, which take some input and produce some output, which are then chained into pipelines.
Pipes represent different stages of an NLP process - for instance, your first stage may involve preprocessing or cleaning up the document, the next may be vectorizing it, and so on.
In broca, this would look like:
from broca.pipeline import Pipeline
from broca.preprocess import Cleaner
from broca.vectorize import BoW
docs = [
# ...
# some string documents
# ...
]
pipeline = Pipeline(
Cleaner(),
BoW()
)
vectors = pipeline(docs)
Since a key part of broca is rapid prototyping, it makes it very easy to simultaneously try different pipelines which may vary in only a few components:
from broca.vectorize import DCS
pipeline = Pipeline(
Cleaner(),
[BoW(), DCS()]
)
This would produce a multi-pipeline consisting of two pipelines: one which vectorizes using BoW, the other using DCS.
Multi-pipelines often have shared components. In the example above, Cleaner() is in both pipelines. To avoid redundant processing, a key part of broca's pipelines is that the output for each pipe is "frozen" to disk.
These frozen outputs are identified by a hash derived from the input data and other factors. If frozen output exists for a pipe and its input, that frozen output is "defrosted" and returned, saving unnecessary processing time.
This way, you can tweak different components of the pipeline without worrying about needing to re-compute a lot of data. Only the parts that have changed will be re-computed.
## Included pipes
broca includes a few pipes:
• broca.tokenize includes various tokenization methods, using lemmas and a few different keyword extractors.
• broca.vectorize includes a traditional bag-of-words vectorizer, an implementation of "dismabiguated core semantics", and Doc2Vec.
• broca.preprocess includes common preprocessors - cleaning punctuation, HTML, and a few others.
## Other tools
Not everything in broca is a pipe. Also included are:
• broca.similarity includes similarity methods for terms and documents.
• broca.distance includes string distance methods (this may be renamed later).
• broca.knowledge includes some tools for dealing with external knowledge sources (e.g. other corpora or Wikipedia).
Though at some point these may also become pipes.
We made it really easy to implement your own pipes. Just inherit from the Pipe class, specify the class's input and output types, and implement the __call__ method (that's what's called for each pipe).
For example:
from broca.pipeline import Pipe
class MyPipe(Pipe):
input = Pipe.type.docs
output = Pipe.type.vecs
def __init__(self, some_param):
self.some_param = some_param
def __call__(self, docs):
# do something with docs to get vectors
vecs = make_vecs_func(docs, self.some_param)
return vecs
We hope that others will implement their own pipes and submit them as pull requests - it would be great if broca becomes a repository of sundry NLP methods which makes it super easy to quickly try a battery of techniques on a problem.
broca is available on GitHub and also via pip:
pip install broca
# Fellowship Status Update
I've long been fascinated with the potential for technology to liberate people from things people would rather not be doing. This relationship between human and machine almost invariably manifests in the context of production processes - making this procedure a bit more efficient here, streamlining this process a bit there.
But what about social processes? A hallmark of the internet today is the utter ugliness that's possible of people; a seemingly inescapable blemish on the grand visions of the internet's potential for social transformation. And the internet's opening of the floodgates has had the expected effect of information everywhere, though perhaps in volumes greater than anyone anticipated.
Here we are, trying to engage little tyrants at considerable emotional expense. Here we are, futilely chipping away at the info-deluge we're suspended in. Here we are, both these things gradually chipping away at us. Things people would rather not be doing.
Prior to my fellowship, these kinds of inquiries had to be relegated to off-hours skunkworks. The fellowship has given me the rare privilege of autonomy, both financial and temporal, and the resources, especially of the human kind, with which I can actually explore these questions as my job.
With the Coral Project, I'm researching what makes digital communities tick, surveying the problems with which they grapple, and learning about how different groups are approaching them - from video games to journalism to social networks both in the mainstream and the fringes (you can read my notes here). Soon we'll be building software to address these issues.
For my own projects I'm working on automatic summarization of comments sections, a service that keeps up with news when you can't, a reputation system for new social network, and all the auxiliary tools these kinds of projects tend to spawn, laying the groundwork for work I hope to continue long after the fellowship. I've been toying with the idea of simulating social networks to provide testing grounds for new automated community management tools. The best part is that it's up to me whether or not I pursue it.
A huge part of the fellowship is learning, which is something I love but have had to carve out my own time for. Here, it's part of the package. I've had ample opportunity to really dig into the sprawling landscape of machine learning and AI (my in-progress notes are here), something I've long wanted to do but never had the space for.
The applications for the 2016 fellowship are open, and I encourage you to apply. Rub shoulders with fantastic and talented folks from a variety of backgrounds. Pursue the questions that conventional employment prohibits you from. Explore topics and skills you've never had time for. It's really what you make of it. At the very least, it's a unique opportunity to be deliberate about where your work takes you.
The halfway mark of my OpenNews fellowship has just about passed. I knew from the start the time would pass by quickly, but I hadn't considered how much could happen in this short a time. There are only about 5 months left - the fellowship does end, but, fortunately, the work it inaugurates doesn't have to.
# Geiger (Intro/Update)
A couple months ago I thought it would be interesting to see if a summary could be generated for a comment section. As a comment section grows, the comments become more repetitive as more people pile into make the same point. It also seems that some natural clustering forms as some commenters focus on particular aspects of an article or topic.
When there are hundreds to thousands of comments, there is little to be gained by reading all of them. However, it may be useful to quantify how much support certain opinions have, or what is most salient about a particular topic. What if there we had some automated means of presenting us such insight? For example, for an article about a new coal industry regulation: 27 comments are focused on how this regulation affects jobs, 39 are arguing about the environmental impacts, 6 are mentioning the meaning of this regulation in an international context, etc.
Having such insight can serve a number of purposes:
• Provide a quick understanding of the salient points for readers of an article
• Direct focus for future articles on the topic
• Give a quick view into how people are responding to an article
• Provide fodder for follow-up pieces on how people are responding
• Surface entry points for other readers into the conversation
Geiger is still very much a work in progress and has led to a lot of experimentation, some of which worked ok, some of which didn't work at all, but so far nothing has worked as well as I'd like.
Below is a screenshot from an early prototype of Geiger which allowed me to try a battery of common techniques (TF-IDF bag of words with DBSCAN, K-Means, HAC, and LDA) and compare their results on any New York Times article with comments.
None of those led to particularly promising results, but a few alternatives were available.
## Aspect summarization
This problem of clustering-to-summarize comments is similar to aspect summarization, which is more closely associated with ratings and reviews. For instance, you may have seen how Yelp's business pages have a few sentences selected at the top, with some term (the "aspect") highlighted, and then the number of reviewers that mentioned this term. That's aspect summarization - the aggregate reviews are being summarized by highlighting aspects which are mentioned the most.
Sometimes aspect summarization includes an additional layer of sentiment analysis, so that instead of just quantifying the number of people talking about an aspect, whether they are talking positively or negatively can also be surfaced (Yelp isn't doing this, however).
The process of aspect summarization can be broken down into three steps:
1. Identify aspects
2. Group documents by aspect
3. Rank aspect groups
To identify aspects I used a few keyword extraction approaches (PoS tagging for noun phrases, named entity recognition, and other methods like Rapid Automatic Keyword Extraction) and then learned phrases by looking at keyword co-occurrences. If two keywords are adjacent (or separated by only a conjunction or hyphen) in at least 80% in the documents where they are present, we consider it a key phrase.
This simple co-occurrence approach is surprisingly effective. Here are some phrases learned on a set of comments for the coal industry regulation article:
'carbon tax', 'green energy', 'sun and wind', 'clean coal', 'air and water', 'high level', 'slow climate', 'middle class', 'signature environmental', 'mitch mcconnell', 'poor people', 'coal industry', 'true cost', 'clerical error', 'coal miner', 'representative democracy', 'co2 emission', 'power source', 'clean air', 'future generation', 'blah blah', 'ice age', 'planet earth', 'climate change', 'energy industry', 'critical thinking', 'particulate matter', 'coal mining', 'corporate interest', 'solar and wind', 'air act', 'acid rain', 'carbon dioxide', 'heavy metal', 'obama administration', 'monied interest', 'greenhouse gas', 'human specie', 'president obama', 'long term', 'political decision', 'big coal', 'coal and natural', 'al gore', 'bottom line', 'power generation', 'wind and solar', 'nuclear plant', 'global warming', 'human race', 'supreme court', 'environmental achievement', 'renewable source', 'coal ash', 'legal battle', 'united state', 'wind power', 'epa regulation', 'economic cost', 'federal government', 'state government', 'natural gas', 'west virginia', 'nuclear power', 'radioactive waste', 'battle begin', 'coal fire', 'energy source', 'common good', 'renewable energy', 'coal burning', 'nuclear energy', 'big tobacco', 'carbon footprint', 'red state', 'sea ice', 'peabody coal', 'tobacco industry', 'american citizen', 'fossil fuel', 'fuel industry', 'climate scientist', 'carbon credit', 'power plant', 'republican president', 'electricity cost'
Some additional processing steps were performed, such as removing keywords that were totally subsumed by key phrases; that is, keywords which only ever appear as part of a key phrase. Keywords were also stemmed and merged, e.g. "polluter", "pollute", "pollutant", "pollution" are grouped as a single aspect.
Grouping documents by aspects is straightforward (just look at overlaps). For this task I treated individual sentences as the documents, much like Yelp does.
Ranking them is a bit trickier. I used a combination of token length (assuming that phrases are more interesting than single keywords), support (number of sentences which mention the aspect), and IDF weighting of the aspect. The latter is useful because, for instance, we expect many comments will mention the "coal industry" if the article is about the coal industry, rendering it uninformative.
Although you get a bit of insight into what commenters are discussing, the results of this approach aren't very interesting. We don't really get any summary of what people are saying about an aspect. This is problematic when commenters are talking about an aspect in different ways. For instance, many commenters are talking about "climate change", but some ask whether or not the proposed regulation would be effective in mitigating it, whereas others debate whether or not climate change is a legitimate concern.
Finally, one problem here, which is consistent across all methods, is that this method is ignorant of synonymy - it cannot recognize when two words which look different mean essentially the same thing. For instance, colloquially people use "climate change" and "global warming" interchangeably, but here they are treated as two different aspects. This is a consequence of text similarity approaches which rely on matching the surface form of words - that is, which only look at exact term overlap.
This is especially challenging when dealing with short text documents, which I explain in greater length here.
## Word2Vec word embeddings
There has been a lot of excitement around neural networks, and rightly so - their ability to learn representations is very useful. Word2Vec is capable of learning vector representations of words ("word embeddings") which allow us to capture some degree of semantic quality in vector space. For example, we could say that two words are semantically similar if their word embeddings are close to each other in vector space.
I loaded up Google's pre-trained Word2Vec model (trained on 100 billion words from a Google News dataset) and tested it out a bit. It seemed promising:
w2v.similarity('climate_change', 'global_warming')
>>> 0.88960381786226284
I made some attempts at approaches which leaned on this Word2Vec similarity of terms rather than their exact overlap - when comparing two documents A and B, each term from A is matched to is maximally-similar term in B and vice versa. Then the documents' similarity score is computed from these pairs' similarity values, weighted by their average IDF.
A problem with using Word2Vec word embeddings is that they are not really meant to quantify synonymy. Words that have embeddings close together do not necessarily mean similar things, all that it means is that they are exchangeable in some way. For example:
w2v.similarity('good', 'bad')
>>> 0.71900512146338569
The terms "good" and "bad" are definitely not synonyms, but they serve the same function (indicating quality or moral judgement) and so we expect to find them in similar contexts.
Because of this, Word2Vec ends up introducing more noise on occasion.
## Measuring salience
Another angle I spent some time on was coming up with some better way of computing term "salience" - how interesting a term is. IDF is a good place to start, since a reasonable definition of a salient term is one that doesn't appear in every document, nor does it only appear in one or two documents. We want something more in the middle, since that indicates a term that commenters are congregating around.
Thus middle IDF values should be weighted higher than those closer to 0 or 1 (assuming these values are normalized to $[0,1]$). To capture this, I put terms' IDF values through a Gaussian function with its peak at $x=0.5$ and called the resulting value the term's "salience". Then, using the Word2Vec approach above, maximal pairs' similarity scores are weighted by their average salience instead of their average IDF.
The results of this technique look less noisy than before, but there is still ample room for improvement.
## Challenges
Working through a variety of approaches has helped clarify what the main difficulties of the problem are:
• Short texts lack a lot of helpful context
• Recognizing synonymy is tricky
• Noise - some terms aren't very interesting given the article or what other people are saying
## What's next
More recently I have been trying a new clustering approach (hscluster) and exploring ways of better measuring short text similarity. I'm also going to take a deeper look into topic modeling methods, which I don't have a good grasp on yet but seem promising.
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# Efficient Partial Dependence Plots with decision trees
Partial Dependence Plots (PDPs) are a standard inspection technique for machine learning models. There are two ways to compute PDPs:
1. The slow and generic way that works for any model
2. The fast way that only works for models that are based on (regression) decision trees. As far as I know, the fast method was originally presented in Friedman’s Gradient Boosting paper (pdf link).
This post will describe both techniques, and explain why the fast way is… well, faster. We will also see that they are not always equivalent.
## Partial Dependence: definition
We will briefly describe partial dependence functions. For a more thorough introduction to PDPs, you can refer to the Bible, or to the Interpretable Machine Learning book.
In what follows, we will keep things simple and only consider a dataset with two features. The first feature $X_0$ is the one we want to get a PDP for, and the second feature $X_1$ is the one that will get averaged out. Following the notation from the references above, $X_0$ corresponds to $X_S$ and $X_1$ corresponds to $X_C$
The partial dependence on $X_0$ is:
$pd_{X_0}(x) \overset{def}{=} \mathbb{E}_{X_1}\left[ f(x, X_1) \right],$
where $f(\cdot, \cdot)$ is our model, taking both features as input. This is also called the average dependence, which IMHO is a better name because it’s clear that the rest of the features get averaged out.
## The slow way in general
For a given value $x$ of $X_0$, the slow method approximates $pd_{X_0}(x)$ with an average over some data. This is usually the training set, but that could be also be some validation data:
$pd_{X_0}(x) \approx \frac{1}{n} \sum_{i=1}^n f(x, x_1^{(i)})$
where $n$ is the number of samples and $x_1^{(i)}$ is the value of feature $X_1$ for the $i$th sample. To compute the above approximation, we need to:
• create $n$ fake samples, namely $(x, x_1^{(1)}), (x, x_1^{(2)}), … (x, x_1^{(n)})$
• compute the prediction of the model for each of these fake samples
• compute the average of the predictions
Repeating that procedure for each value $x$ of $X_0$, we end up with a PDP like the one below.
The issue here is that for each value $x$, we need a full pass on the dataset. That’s slow, but that’s a very generic way that will work for any model.
## The slow way on decision trees
Let’s consider the following tree. Each non-leaf node indicates whether its split is on $X_0$ or on $X_1$:
That tree will partition the input space as in the following plot (thresholds are arbitrary). Each line corresponds to a split, i.e. a non-leaf node, and each rectangle corresponds to a leaf, with its own value.
Let’s apply the slow method to compute $pd_{X_0}(5)$. We need to create fake samples (whose values for $X_1$ depend on our dataset at hand), and then compute the prediction of the tree for each of these fake samples. Our fake samples are represented as dots in the following plot (notice that they naturally all have $X_0=5$). The prediction for a given sample corresponds to the value of the leaf where the sample lands on.
To approximate $pd_{X_0}(5)$, we now average these predictions following the formula above, and get:
$pd_{X_0}(5) \approx \frac{1}{6} (3 v_H + 1 v_I + 2 v_G) = \frac{1}{2} v_H + \frac{1}{6} v_I + \frac{1}{3} v_G$
Repeat this for all values of $X_0$, and you get a PDP.
## The fast way on decision trees
In blue are the paths that the fake samples have followed:
We then just computed the proportion of fake samples landing in each leaf (H, I and G), and averaged the predictions. The key to the fast way is that we can compute the proportions ($\frac{1}{2}, \frac{1}{6}, \frac{1}{3}$) without having to create the fake samples. We’ll use the fact that each node of the tree remembers how many training samples went through it during the training stage.
For a given value $x$, the fast way simulates the $n$ tree traversals of the fake samples in a single tree traversal. During the traversal, we keep track of the proportions of training samples that would have followed each path (the “would” part is important here):
# input: x, a value of X0
# output: pd, an approximation of the expectation pd_X0(x)
pd = 0
def dfs(node, prop):
if node.is_leaf:
pd += prop * node.value
return
if node.split_feature is X0:
# follow normal path: either left or right child
child = node.left if x <= node.threshold else node.right
dfs(child, prop)
else:
# follow both left and right, with appropriate proportions
prop_samples_left = node.left.n_train_samples / node.n_train_samples
dfs(node.left, prop * prop_samples_left)
dfs(node.right, prop * (1 - prop_samples_left))
dfs(root, prop=1)
This is almost a regular tree dfs traversal. The main twist is that when a node splits on $X_1$, we follow both children instead of just one of them. You can think of the fast way as doing exactly what the slow way would do with the training data, that is, creating fake samples from the training data and passing them through the tree. Except that, in the fast version, we don’t explicitly create the fake samples, we just simulate them by keeping track of the proportions.
The nodes that are visited by the fast method are the same nodes that are visited by the fake samples, had they been created. In particular, notice in the tree plot above that when the split is on $X_0$, only one of the children is followed. When the split is on $X_1$, both children are followed, by a given proportions of samples.
We only needed one traversal instead of $n$ traversals, and we got the same result!
All this generalizes very naturally to more than one feature in $X_S$ or in $X_C$. If you’re curious about how we implemented this in scikit-learn, here are the slow and the fast method implementations.
## Differences beween the fast and the slow method
The slow and the fast methods are almost equivalent, but they might differ in two ways.
First, they can only be equivalent if the slow method is using the training data. Indeed, both methods try to approximate the expectation $\mathbb{E}_{X_1}\left[ f(x, X_1) \right]$. The difference lies in the values of $X_1$ that are used to approximate this expectation with an average: the fast method always uses the values of the training data, the slow one uses whatever you feed it. As long as the data used by the slow method follows the same distribution as the training data, the differences should be in general minimal.
The second reason is trickier to grasp. Consider the degenerated dataset below, where all points have a target value of 0 except for one of them (in yellow) whose target value is 1000.
The corresponding tree is:
As we’ll see, the fast and the slow method will strongly disagree on the value of $pd_{X_0}(4)$.
The fast way will (virtually) map all the samples to the root’s right child, and then give a weight of $\frac{1}{2}$ to its left leaf, and a weight of $\frac{1}{2}$ to its right leaf. This results in $pd_{X_0}(4) = \frac{1}{2} \times 1000 + \frac{1}{2} \times 0 = 500$
The slow way will create $n$ fake samples who all fall on the vertical line $X_0 = 4$, and whose value for $X_1$ correspond to those present in the training set. All these fake samples will be mapped to the root’s right child, just like in the fast method. But then, the vast majority of these fake samples have $X_1 < 9.5$. So most of them will have a predicted value of 1000 while only a handful will have a predicted value of 0. Taking the average of the predictions, the partial dependence will be estimated at $pd_{X_0}(4) = 950$, which is very far from the 500 as predicted by the fast method.
Which one is correct? Probably none of them.
The slow method is creating fake samples that have unrealistic characteristics. It’s clear from the training data that samples with $X_0 > 0$ are very unlikely, and yet the slow method relies on the assumption that such samples exist.
On the other hand, the fast method considers that $\frac{1}{2}$ / $\frac{1}{2}$ is a reasonable approximation of the distribution of samples at that node. It might be true, but clearly the proportions are computed on the basis of very few samples (only 2), so they are not necessarily reliable.
As always with interpretation methods, one needs to be careful and always check the assumptions of method that is being used.
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## Photoshop 2020 Crack [Latest] 2022
Note
On the Mac, this keyboard shortcut is Control+drag.
You can drag the picture between the different locations on your hard drive to make a folder full of pictures. Or you can drag it to another program on your computer (Figure 4-6. You may want to copy the picture to another folder instead so that it’s easier to find later. If your clipboard already has the picture, just paste it wherever you like using File→Paste.”), left).
The following options appear at the bottom of the box where you drag:
* **Drag and drop** means that you can drop the picture into its new destination by pressing . This option is the equivalent of pressing Enter on the Mac.
* **Copy** means that you can copy the picture onto your Windows Clipboard (Ctrl+C on the Mac) and paste it at any time later.
* **Move to new location** means that you can drag the picture out of its current location to another location on your computer hard drive.
* **Paste into** specifies the folder, disk, or program
## Photoshop 2020 Crack+ Patch With Serial Key [Updated-2022]
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Q:
Building a graph from $\sum\limits_{i=1}^n \frac{1}{i}$ and $\sum\limits_{i=1}^\infty \frac{1}{i^2}$
I am trying to build a graph from the following two sums:
$$\sum_{i=1}^n \frac{1}{i} \qquad\text{and}\qquad \sum_{i=1}^\infty \frac{1}{i^2}$$
The first sum diverges and the second converges. However, I am unable to represent their convergence in a graph.
The only way I could represent the first sum is by graphing $x\sum\limits_{i=1}^n \frac{1}{i}=\frac{1}{n}-\frac{1}{n+1}$ and using the fact that $x\in\left(-1,1\right)$ leads to this graph:
Does this graph represent the divergence of the first sum?
If not, what can I use as a graphing function of the second sum to represent its convergence?
A:
First it’s $x\sum_{i=1}^n\frac1i\leq \frac1n$ so that $x$ must be between $0$ and $1$.
When $x=1$ the left part is $\frac1n-\frac{n-1}n$, that gives $\frac1n$ when $n\to\infty$.
When $x\in(0,1)$
$$\frac1n-\frac1{n+x}\leq \frac1n-\frac1{2n}$$ so
$$\frac1{n+x}\leq\frac{1-x}2$$ and the limit is $\frac{(1-x)}2$ when $x\to 0$.
$$\frac1n-\frac1{n+x}\geq \frac1n-\frac{n-1}n-\frac1{n+1}-\frac1{2n}\geq\frac1n-\frac1{2n}$$
and the limit is $\frac12$.
So the limit is $1$ for \$x\ge
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Serine proteinase inhibitor, secretory leukocyte proteinase inhibitor (SLPI) is an important anti-microbial protein in cervicovaginal secretions. SLPI and SLPI2 are two identical proteins and exist in cervicovaginal secretions. These two proteins, however, have different functions. It is unclear whether SLPI and SLPI2 contribute to different functions. As the defense mechanism by which epithelial cells control their local environment, some cervical epithelial cells have a capability to produce and secrete high levels of SLPI. However, the regulation of these SLPI proteins by bacterial exposure has not been well elucidated. Therefore, we investigated the temporal regulation of SLPI and SLPI2 in an immortalized human cervical epithelial cell line (MeT-5A). In this study, we showed that IL-8 induced SLPI and SLPI2 expression in MeT-5A cells as early as 1 h after stimulation. Furthermore, we showed that LPS stimulated at 1 h also enhanced these SLPI and SLPI2 expressions. Interestingly, treatment with LPS down-regulated their expressions, although IL-8 up-regulated them. Finally, we investigated the effect of SLPI and SLPI2 overexpression on LPS-induced nitric oxide production. Overexpression of SLPI but not SLPI2 suppressed LPS-induced production of nitric oxide. Our study demonstrated that the temporal expression of SLPI and SLPI2 after LPS exposure was dependent on IL-8, suggesting that IL-8 is an important regulator of SLPI and SLPI2 expression.A blockchain network enables distributed coordination and engagement of computing processes and the shared storage of information in a publicly viewable and distributed ledger. The distributed database is also referred to as a distributed ledger because each user is able to access the recorded information in the database with the permission of any other user. Some blockchain networks are permissionless (i.e. there are no restrictions on a user’s ability to read or modify the database) while other networks may have a permissioned database, where the entities are regulated by the network, such as the set of nodes that may communicate and share information within the network.
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Mac systems released since Mid-2007, that is, MacBook Pro/MacBook Air/MacBook Pro Retina/
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# Center of Mass double integral
A lamina occupies the region which is the intersection of $x^2+y^2-2y \leq 0$ and the first quadrant of the $xy$-plane. Find the center of mass if the density at a point of the lamina is twice the point's distance from the origin.
Does the setup of this look like this:$$\int_0^{\frac{\pi}{2}}\int_0^{2\sin\theta}2r^2drd\theta?$$
-
I'm assuming $(x, y) \in \mathbb{R}^2$. If so, then $\{(x, y) \in \mathbb{R}^2 \mid x^2 + y^2 \leq 0\} = \{(0, 0)\}$. – Michael Albanese Apr 18 '13 at 16:15
Sorry, forgot the (-2y), you have to complete the square so the radius becomes 1. – user73064 Apr 18 '13 at 16:18
Actually, the factor of $2$ is unimportant. The center of mass in the $y$ direction is given by
$$\bar{y} = \frac{\displaystyle \int_0^{\pi/2} d\theta \: \int_0^{2 \sin{\theta}} dr \, r (2 r)(r \sin{\theta})}{\displaystyle \int_0^{\pi/2} d\theta \: \int_0^{2 \sin{\theta}} dr \, r (2 r)}$$
Note that the weight of the $y$ coordinate is expressed in the $r \sin{\theta}$ term in the numerator. Note also the boundary of the lamina is expressed in its polar form, $r=\sin{\theta}$. Evaluating the radial integrals, we get
$$\bar{y} = \frac{\frac{16}{4} \displaystyle \int_0^{\pi/2} d\theta \: \sin^5{\theta}}{\frac{8}{3} \displaystyle \int_0^{\pi/2} d\theta \: \sin^3{\theta}}$$
or
$$\bar{y} = \frac{6}{5}$$
For $\bar{x}$, we do a similar calculation:
\begin{align}\bar{x} &= \frac{\displaystyle \int_0^{\pi/2} d\theta \: \int_0^{2 \sin{\theta}} dr \, r (2 r)(r \cos{\theta})}{\displaystyle \int_0^{\pi/2} d\theta \: \int_0^{2 \sin{\theta}} dr \, r (2 r)}\\ &= \frac{\frac{16}{4} \displaystyle \int_0^{\pi/2} d\theta \: \cos{\theta} \sin^4{\theta}}{\frac{8}{3} \displaystyle \int_0^{\pi/2} d\theta \: \sin^3{\theta}}\\ &= \frac{9}{20}\end{align}
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I originally meant the equation for the mass because once you have that it's very easy to go from mass to center of mass...I don't understand why you went from (0 to pi) and(0 to sinθ)...if the lamina is restricted to the first quadrant, shouldn't it be (0 to pi/2)? – user73064 Apr 18 '13 at 17:23
I didn't catch the first quadrant, let me fix. That will not affect the $y$ result, but will affect the $x$. – Ron Gordon Apr 18 '13 at 17:23
Okay, I still don't get how you can just get rid of the 2 in r=2sinθ? It would result in a different answer so I don't see how it's unimportant. – user73064 Apr 18 '13 at 17:38
Because the center of mass calculation involves a ratio - the denominator is the total mass. In doing that calculation, any constant factor will cancel with the numerator. The important thing is that the density is proportional to $r$. – Ron Gordon Apr 18 '13 at 17:39
But it wouldn't be wrong to put 0 to 2sinθ as a bound? – user73064 Apr 18 '13 at 17:59
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## Electronic Journal of Probability
### A Williams decomposition for spatially dependent superprocesses
#### Abstract
We present a genealogy for superprocesses with a non-homogeneous quadratic branching mechanism, relying on a weighted version of the superprocess introduced by Engländer and Pinsky and a Girsanov theorem. We then decompose this genealogy with respect to the last individual alive (Williams' decomposition). Letting the extinction time tend to infinity, we get the Q-process by looking at the superprocess from the root, and define another process by looking from the top. Examples including the multitype Feller diffusion (investigated by Champagnat and Roelly) and the superdiffusion are provided.
#### Article information
Source
Electron. J. Probab., Volume 18 (2013), paper no. 37, 43 pp.
Dates
Accepted: 12 March 2013
First available in Project Euclid: 4 June 2016
https://projecteuclid.org/euclid.ejp/1465064262
Digital Object Identifier
doi:10.1214/EJP.v18-1801
Mathematical Reviews number (MathSciNet)
MR3035765
Zentralblatt MATH identifier
1294.60104
Rights
#### Citation
Delmas, Jean-François; Hénard, Olivier. A Williams decomposition for spatially dependent superprocesses. Electron. J. Probab. 18 (2013), paper no. 37, 43 pp. doi:10.1214/EJP.v18-1801. https://projecteuclid.org/euclid.ejp/1465064262
#### References
• R. Abraham and J.-F. Delmas. A continuum-tree-valued Markov process. To appear in The Annals of Probability, 2012.
• Abraham, Romain; Delmas, Jean-François. Williams' decomposition of the Lévy continuum random tree and simultaneous extinction probability for populations with neutral mutations. Stochastic Process. Appl. 119 (2009), no. 4, 1124–1143.
• Aldous, David. The continuum random tree. I. Ann. Probab. 19 (1991), no. 1, 1–28.
• Bertoin, Jean; Fontbona, Joaquin; Martínez, Servet. On prolific individuals in a supercritical continuous-state branching process. J. Appl. Probab. 45 (2008), no. 3, 714–726.
• Bertoin, Jean; Le Gall, Jean-François. The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes. Probab. Theory Related Fields 117 (2000), no. 2, 249–266.
• Champagnat, Nicolas; Roelly, Sylvie. Limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions. Electron. J. Probab. 13 (2008), no. 25, 777–810.
• Y.-T. Chen and J.-F. Delmas. Smaller population size at the CA time for stationary branching processes. To appear in The Annals of Probability, 2012.
• Cranston, M.; Koralov, L.; Molchanov, S.; Vainberg, B. A solvable model for homopolymers and self-similarity near the critical point. Random Oper. Stoch. Equ. 18 (2010), no. 1, 73–95.
• Dhersin, Jean-Stéphane; Serlet, Laurent. A stochastic calculus approach for the Brownian snake. Canad. J. Math. 52 (2000), no. 1, 92–118.
• Donnelly, Peter; Kurtz, Thomas G. Particle representations for measure-valued population models. Ann. Probab. 27 (1999), no. 1, 166–205.
• Duquesne, Thomas; Le Gall, Jean-François. Random trees, Lévy processes and spatial branching processes. Astérisque No. 281 (2002), vi+147 pp.
• Duquesne, Thomas; Winkel, Matthias. Growth of Lévy trees. Probab. Theory Related Fields 139 (2007), no. 3-4, 313–371.
• Dynkin, Eugene B. An introduction to branching measure-valued processes. CRM Monograph Series, 6. American Mathematical Society, Providence, RI, 1994. x+134 pp. ISBN: 0-8218-0269-0
• Engländer, János; Harris, Simon C.; Kyprianou, Andreas E. Strong law of large numbers for branching diffusions. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 1, 279–298.
• Engländer, János; Kyprianou, Andreas E. Local extinction versus local exponential growth for spatial branching processes. Ann. Probab. 32 (2004), no. 1A, 78–99.
• Engländer, János; Pinsky, Ross G. On the construction and support properties of measure-valued diffusions on $D\subseteq{\bf R}^ d$ with spatially dependent branching. Ann. Probab. 27 (1999), no. 2, 684–730.
• Klebaner, F. C.; Rösler, U.; Sagitov, S. Transformations of Galton-Watson processes and linear fractional reproduction. Adv. in Appl. Probab. 39 (2007), no. 4, 1036–1053.
• Kurtz, Thomas G.; Rodrigues, Eliane R. Poisson representations of branching Markov and measure-valued branching processes. Ann. Probab. 39 (2011), no. 3, 939–984.
• Le Gall, J.-F. A class of path-valued Markov processes and its applications to superprocesses. Probab. Theory Related Fields 95 (1993), no. 1, 25–46.
• Li, Zeng Hu. A note on the multitype measure branching process. Adv. in Appl. Probab. 24 (1992), no. 2, 496–498.
• Perkins, Edwin. Dawson-Watanabe superprocesses and measure-valued diffusions. Lectures on probability theory and statistics (Saint-Flour, 1999), 125–324, Lecture Notes in Math., 1781, Springer, Berlin, 2002.
• Pinsky, Ross G. Positive harmonic functions and diffusion. Cambridge Studies in Advanced Mathematics, 45. Cambridge University Press, Cambridge, 1995. xvi+474 pp. ISBN: 0-521-47014-5
• Roynette, Bernard; Yor, Marc. Penalising Brownian paths. Lecture Notes in Mathematics, 1969. Springer-Verlag, Berlin, 2009. xiv+275 pp. ISBN: 978-3-540-89698-2
• Seneta, E. Non-negative matrices and Markov chains. Revised reprint of the second (1981) edition [Springer-Verlag, New York; ]. Springer Series in Statistics. Springer, New York, 2006. xvi+287 pp. ISBN: 978-0387-29765-1; 0-387-29765-0
• Serlet, Laurent. The occupation measure of super-Brownian motion conditioned to nonextinction. J. Theoret. Probab. 9 (1996), no. 3, 561–578.
• Williams, David. Path decomposition and continuity of local time for one-dimensional diffusions. I. Proc. London Math. Soc. (3) 28 (1974), 738–768.
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<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
You are reading an older version of this FlexBook® textbook: CK-12 Basic Algebra Concepts Go to the latest version.
# 11.11: Box-and-Whisker Plots
Difficulty Level: Basic Created by: CK-12
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Practice Box-and-Whisker Plots
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Suppose that in the first 20 screenings of a movie, the number of paying customers at a theater were as follows: 16, 34, 22, 19, 59, 33, 60, 45, 50, 27, 75, 38, 49, 52, 20, 40, 13, 15, 26, 21. Could you analyze this data with a box-and-whisker plot? If so, what would you need to do first? In this Concept, you'll learn how to use box-and-whisker plots to analyze data sets like this one.
### Guidance
A box-and-whisker plot is another type of graph used to display data. It shows how the data are dispersed around a median, but it does not show specific values in the data. It does not show a distribution in as much detail as does a stem-and-leaf plot or a histogram.
A box-and-whisker plot is a graph based upon medians. It shows the minimum value, the lower median, the median, the upper median, and the maximum value of a data set. It is also known as a box plot .
This type of graph is often used when the number of data values is large or when two or more data sets are being compared.
#### Example A
You have a summer job working at Paddy’s Pond. Your job is to measure as many salmon as possible and record the results. Here are the lengths (in inches) of the first 15 fish you found: 13, 14, 6, 9, 10, 21, 17, 15, 15, 7, 10, 13, 13, 8, 11
Create a box-and-whisker plot.
Solution:
Since a box-and-whisker plot is based on medians, the first step is to organize the data in order from smallest to largest.
$6, \ 7, \ 8, \ 9, \ 10, \ 10, \ 11, \ \boxed{13}, \ 13, \ 13, \ 14, \ 15, \ 15, \ 17, \ 21$
Step 1: Find the median: $median=13$ .
Step 2: Find the lower median.
The lower median is the median of the lower half of the data. It is also called the lower quartile or $Q_1$ .
$6, \ 7, \ 8, \ & \boxed{9}, \ 10, \ 10, \ 11\\Q_1& =9$
Step 3: Find the upper median.
The upper median is the median of the upper half of the data. It is also called the upper quartile or $Q_2$ .
$13, \ 13, \ 14, \ \boxed{15}, \ 15, \ 17, \ 21$
$Q_3=15$
Step 4: Draw the box plot. The numbers needed to construct a box-and-whisker plot are called the five-number summary .
The five-number summary are: the minimum value, $Q_1$ , the median, $Q_2$ , and the maximum value.
$Minimum=6; \ Q_1=9;\ median=13; \ Q_3=15; \ maximum=21$
The three medians divide the data into four equal parts. In other words:
• One-quarter of the data values are located between 6 and 9.
• One-quarter of the data values are located between 9 and 13.
• One-quarter of the data values are located between 13 and 15.
• One-quarter of the data values are located between 15 and 21.
From its whiskers, any outliers (unusual data values that can be either low or high) can be easily seen on a box-and-whisker plot. An outlier would create a whisker that would be very long.
Each whisker contains 25% of the data and the remaining 50% of the data is contained within the box. It is easy to see the range of the values as well as how these values are distributed around the middle value. The smaller the box, the more consistent the data values are with the median of the data.
#### Example B
After one month of growing, the heights of 24 parsley seed plants were measured and recorded. The measurements (in inches) are given here: 6, 22, 11, 25, 16, 26, 28, 37, 37, 38, 33, 40, 34, 39, 23, 11, 48, 49, 8, 26, 18, 17, 27, 14.
Construct a box-and-whisker plot to represent the data.
Solution:
To begin, organize your data in ascending order. There is an even number of data values so the median will be the mean of the two middle values. $Med=\frac{26+26}{2}=26$ . The median of the lower quartile is the number between the 6th and 7th positions, which is the average of 16 and 17, or 16.5. The median of the upper quartile is also the number between the 6th and 7th positions, which is the average of 37 and 37, or 37. The smallest number is 6, and the largest number is 49.
Creating Box-and-Whisker Plots Using a Graphing Calculator
The TI-83 can also be used to create a box-and-whisker plot. The five-number summary values can be determined by using the trace function of the calculator.
#### Example C
'Make a histogram of the data from the previous example on your calculator.
Solution:
Enter the data into $[L_1]$ .
Change the [STATPLOT] to a box plot instead of a histogram.
Box-and-whisker plots are useful when comparing multiple sets of data. The graphs are plotted, one above the other, to visualize the median comparisons.
### Guided Practice
Using the data from the previous Concept, determine whether the additive improved the gas mileage.
540 550 555 570 570
580 585 587 588 590
591 610 615 640 660
500 589 618 619 629
633 635 637 638 639
659 664 689 694 709
Solution:
Smallest # 540 500
$Q_1$ 570 619
Median 587 637
$Q_3$ 610 664
Largest # 660 709
From the above box-and-whisker plots, where the blue one represents the regular gasoline and the yellow one the premium gasoline, it is safe to say that the additive in the premium gasoline definitely increases the mileage. However, the value of 500 seems to be an outlier.
### Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Box-and-Whisker Plots (13:14)
1. Describe a five-number summary.
2. What is the purpose of a box-and-whisker plot? When it is useful?
3. What are some disadvantages to representing data with a box-and-whisker plot?
4. The following is the data that represents the amount of money that males spent on prom night. $&25 && 60 && 120 && 64 && 65 && 28 && 110 && 60\\&70 && 34 && 35 && 70 && 58 && 100 && 55 && 95\\&55 && 95 && 93 && 50 && 75 && 35 && 40 && 75\\&90 && 40 && 50 && 80 && 85 && 50 && 80 && 47\\&50 && 80 && 90 && 42 && 49 && 84 && 35 && 70$ Construct a box-and-whisker graph to represent the data.
5. Forty students took a college algebra entrance test and the results are summarized in the box-and-whisker plot below. How many students would be allowed to enroll in the class if the pass mark were set at:
1. 65 %
2. 60 %
1. Harika is rolling three dice and adding the scores together. She records the total score for 50 rolls, and the scores she gets are shown below. Display the data in a box-and-whisker plot, and find both the range and the inter-quartile range. 9, 10, 12, 13, 10, 14, 8, 10, 12, 6, 8, 11, 12, 12, 9, 11, 10, 15, 10, 8, 8, 12, 10, 14, 10, 9, 7, 5, 11, 15, 8, 9, 17, 12, 12, 13, 7, 14, 6, 17, 11, 15, 10, 13, 9, 7, 12, 13, 10, 12
2. The box-and-whisker plots below represent the times taken by a school class to complete a 150-yard obstacle course. The times have been separated into boys and girls. The boys and the girls both think that they did best. Determine the five-number summary for both the boys and the girls and give a convincing argument for each of them.
3. Draw a box-and-whisker plot for the following unordered data. 49, 57, 53, 54, 49, 67, 51, 57, 56, 59, 57, 50, 49, 52, 53, 50, 58
4. A simulation of a large number of runs of rolling three dice and adding the numbers results in the following five-number summary: 3, 8, 10.5, 13, 18. Make a box-and-whisker plot for the data.
5. The box-and-whisker plots below represent the percentage of people living below the poverty line by county in both Texas and California. Determine the five-number summary for each state, and comment on the spread of each distribution.
6. The five-number summary for the average daily temperature in Atlantic City, NJ (given in Fahrenheit) is 31, 39, 52, 68, 76. Draw the box-and-whisker plot for this data and use it to determine which of the following would be considered an outlier if it were included in the data.
1. January’s record-high temperature of $78^\circ$
2. January’s record-low temperature of $-8^\circ$
3. April’s record-high temperature of $94^\circ$
4. The all-time record high of $106^\circ$
1. In 1887, Albert Michelson and Edward Morley conducted an experiment to determine the speed of light. The data for the first ten runs (five results in each run) is given below. Each value represents how many kilometers per second over 299,000 km / sec were measured. Create a box-and-whisker plot of the data. Be sure to identify outliers and plot them as such. 900, 840, 880, 880, 800, 860, 720, 720, 620, 860, 970, 950, 890, 810, 810, 820, 800, 770, 850, 740, 900, 1070, 930, 850, 950, 980, 980, 880, 960, 940, 960, 940, 880, 800, 850, 880, 760, 740, 750, 760, 890, 840, 780, 810, 760, 810, 790, 810, 820, 850
2. Using the following box-and-whisker plot, list three pieces of information you can determine from the graph.
3. In a recent survey done at a high school cafeteria, a random selection of males and females were asked how much money they spent each month on school lunches. The following box-and-whisker plots compare the responses of males to those of females. The lower one is the response by males.
1. How much money did the middle 50% of each gender spend on school lunches each month?
2. What is the significance of the value of $42 for females and$46 for males?
3. What conclusions can be drawn from the above plots? Explain.
1. Multiple Choice. The following box-and-whisker plot shows final grades last semester. How would you best describe a typical grade in that course? A. Students typically got between 82 and 88. B. Students typically got between 41 and 82. C. Students typically got around 62. D. Students typically got between 58 and 82.
Mixed Review
1. Find the mean, median, mode, and range for the following salaries in an office building: 63,450; 45,502; 63,450; 51,769; 63,450; 35,120; 45,502; 63,450; 31,100; 42,216; 49,108; 63,450; 37,904
2. Graph $g(x)=2\sqrt{x-1}-3$ .
3. Translate into an algebraic sentence: The square root of a number plus six is less than 18 .
4. Solve for $y$ : $6(y-11)+9=\frac{1}{3} (27+3y)-16$ .
5. A fundraiser is selling two types of items: pizzas and cookie dough. The club earns $5 for each pizza sold and$4 for each container of cookie dough. They want to earn more than \$550.
1. Write this situation as an inequality.
2. Give four combinations that will make this sentence true.
1. Find the equation for a line parallel to $x+2y=10$ containing the point (2, 1).
### Vocabulary Language: English
Extremes
Extremes
The extremes are the maximum and minimum values in a data set.
five point summary
five point summary
The numbers needed to construct a box-and-whisker plot are called the five-point-summary. The five points are the minimum, the lower median (Q1), the median, the upper median (Q3), and the maximum.
line of fit
line of fit
A line of fit is a straight or continuously curved line representing the trend of changes in the comparison of two data sets (or one set of bivariate data).
Median
Median
The median of a data set is the middle value of an organized data set.
observed data
observed data
Observed data are the values that result from computations performed on the input variable.
Outlier
Outlier
In statistics, an outlier is a data value that is far from other data values.
Quartile
Quartile
A quartile is each of four equal groups that a data set can be divided into.
skewed
skewed
As with the horizontal skewing of a histogram, stem plots with a obvious skew toward one end or the other tend to indicate an increased number of outliers either lesser than or greater than the mode.
statistical correlation
statistical correlation
Statistical correlation is a representation of possible related changes in values between the two sets of data.
trends
trends
Trends in data sets or samples are indicators found by reviewing the data from a general or overall standpoint
uniform
uniform
A uniform shaped histogram indicates data that is very consistent; the frequency of each class is very similar to that of the others.
Basic
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Oct 26, 2012
Aug 20, 2015
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# How do you graph x=8 by plotting points?
Feb 11, 2017
Refer Explanation section
#### Explanation:
$x = 8$ This line is parallel to Y-axis
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# Exponents - Numbers with Variables
#### cgr4
##### New member
(9x)^-1/2
It's got a negative exponent, so I assume its 1 / something.
My guess for the answer would be:
1 / 3x
[(25xy)^3/2] / x2y
For this question, would I compute 253/2 and then x3/2 y3/2 and then divide by x2y
#### Reckoner
##### Member
(9x)^-1/2
It's got a negative exponent, so I assume its 1 / something.
My guess for the answer would be:
1 / 3x
Close. Remember that in addition to taking the square root of the $9,$ you also need to take the square root of the $x.$
\begin{align*}
(9x)^{-1/2} &= \frac1{(9x)^{1/2}}\\
&= \frac1{9^{1/2}x^{1/2}}\\
&= \frac1{3\sqrt x}.
\end{align*}
[(25xy)^3/2] / x2y
For this question, would I compute 253/2 and then x3/2 y3/2 and then divide by x2y
Yes. What do you get after you do that?
#### cgr4
##### New member
For the second question. It would look something like this?
125x3/2y3/2[HR][/HR]x2y
then I would subtract x2 and y
which would equal 125x1/2y ?
#### MarkFL
Staff member
No, you want to apply the property of exponents:
$$\displaystyle \frac{a^b}{a^c}=a^{b-c}$$
In other words, for each like base, you want to subtract the exponent in the denominator from the exponent in the numerator.
#### cgr4
##### New member
Still a little confused about the 2nd problem.
So
125x(3/2-2)y(3/2-2)
correct?
#### cgr4
##### New member
resulting in 125x-1y
125y[HR]x[/HR]
125y over x
#### earboth
##### Active member
Still a little confused about the 2nd problem.
So
125x(3/2-2)y(3/2-2)
correct?
Very close!
You have made a small typo.
The final result should be:
$$\displaystyle 125 \cdot x^{-\frac12} \cdot y^{\frac12}$$
Re-write this term without the fractional exponents.
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## Corona Virus Outbreak from Turd
Discussion of the recent unfolding of history.
### Re: Corona Virus Outbreak from Turd
promethean75 wrote:No that's not a marketable or even practical idea. If you need a ventilator, you'll be in a hospital... not hanging out at the mall with a portable ventilator.
Let's run it by Mr. Reasonable first.
He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles
Start here: viewtopic.php?f=1&t=176529
Then here: viewtopic.php?f=15&t=185296
And here: viewtopic.php?f=1&t=194382
iambiguous
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### Re: Corona Virus Outbreak from Turd
He did once say that he could play blues guitar like nobody's business. Those weren't his exact words but that was the gist of it. So I've prepared a simple three chord progression in a blues scale for him to lead over. This might draw him out.
https://vocaroo.com/nhteJ7swIvE
promethean75
Philosopher
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### Re: Corona Virus Outbreak from Turd
Irony on steroids:
"Britain’s prime minister, Boris Johnson, who was hospitalized with persistent coronavirus-related symptoms, was moved to intensive care on Monday after his condition worsened, his office said." NYT
If you know what I mean.
He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles
Start here: viewtopic.php?f=1&t=176529
Then here: viewtopic.php?f=15&t=185296
And here: viewtopic.php?f=1&t=194382
iambiguous
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Joined: Tue Nov 16, 2010 8:03 pm
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### Re: Corona Virus Outbreak from Turd
MagsJ wrote:
Zero_Sum wrote:Boris Johnson currently is in quarantine, I'm really worried he is going to die from the virus. So very sad.
I wonder what the reaction from the U.K. would be afterwards.
Boris is alive and well and recovering at No.10, where he posts regular updates to his Instagram, on the need to continue to self-isolate and social-distance, and his latest piece of advice.. to stay local this weekend, i.e. don’t go to the beach.
Update: Boris is on oxygen, but not on a ventilator.. his condition remains unchanged.
Dominic Raab is now Acting Prime-minister
Breaking news: Michael Gove: the Chancellor of the Duchy of Lancaster, is now in self isolation, due to a family member displaying symptoms.
An end to self-isolation is very unlikely in the foreseeable months.
The possibility of anything we can imagine existing is endless and infinite.. - MagsJ
I haven't got the time to spend the time reading something that is telling me nothing, as I will never be able to get back that time, and I may need it for something at some point in time.. Huh! - MagsJ
You’re suggestions and I, just simply don’t mix.. like oil on water, or a really bad DJ - MagsJ
MagsJ
The Londonist: a chic geek
Posts: 21457
Joined: Wed Nov 01, 2006 2:59 pm
Location: Suryaloka / LDN Town
### Re: Corona Virus Outbreak from Turd
England's Donald Trump.
Back then as it were...
https://youtu.be/n3NAx3tsy-k
He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles
Start here: viewtopic.php?f=1&t=176529
Then here: viewtopic.php?f=15&t=185296
And here: viewtopic.php?f=1&t=194382
iambiguous
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Joined: Tue Nov 16, 2010 8:03 pm
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### Re: Corona Virus Outbreak from Turd
14 mins ago - Boris Johnson is "responding to treatment" for coronavirus as he spends his third day in hospital. The prime minister was being kept in St Thomas' Hospital in London "for close monitoring" and remained clinically stable, Downing Street said. Downing Street said he was not working ...
The possibility of anything we can imagine existing is endless and infinite.. - MagsJ
I haven't got the time to spend the time reading something that is telling me nothing, as I will never be able to get back that time, and I may need it for something at some point in time.. Huh! - MagsJ
You’re suggestions and I, just simply don’t mix.. like oil on water, or a really bad DJ - MagsJ
MagsJ
The Londonist: a chic geek
Posts: 21457
Joined: Wed Nov 01, 2006 2:59 pm
Location: Suryaloka / LDN Town
### Re: Corona Virus Outbreak from Turd
Virapolitical contest:
Joe Biden holds a 4-point lead over Trump in a new national poll, which comes a day after Bernie Sanders suspended his presidential campaign.
According to the Monmouth University poll, Biden attracts the support of 48% of registered voters, while Trump stands at 44%. The result is similar to Biden’s 3-point lead over Trump in a poll released last month.
With Sanders dropping our of the race, such head-to-head polls between Trump and Biden have taken on a heightened level of significance as America looks ahead to the general election in November.
However, national polls do not reflect the state-by-state nature of presidential elections. Hillary Clinton famously won the popular vote in 2016 but lost the electoral college, handing Trump a victory.
From guardian
Meno_
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### Re: Corona Virus Outbreak from Turd
Was 310, now 1,373
Spain 3,261
Italy 2,375
France 1,804
Germany 1,379
Portugal 1,369
U.K. 895
Canada 548
World 203
He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles
Start here: viewtopic.php?f=1&t=176529
Then here: viewtopic.php?f=15&t=185296
And here: viewtopic.php?f=1&t=194382
iambiguous
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Joined: Tue Nov 16, 2010 8:03 pm
Location: baltimore maryland
### Re: Corona Virus Outbreak from Turd
Here we go again!
"Trump seeks to reopen much of U.S. next month" WP headline
'Experts fear a possible covid-19 resurgence if Americans return to their normal lives before the virus is truly stamped out. But President Trump wants a strategy for resuming business activity by May 1, if not sooner.
So...
1] will the health experts manage to talk him back as they did with the Easter Sunday gambit?
2] if not, what actual power does he have to bring this reopening about?
3] if he does succeed in bringing it about, what will the consequences be?
Many in Trumpworld will jump on an assessment like this one: https://www.washingtonpost.com/opinions ... story.html
In other words, that dire predictions are just not being matched by the reality of the pandemic itself.
Yet another rendition of this:
There are known knowns. These are things we know that we know. There are known unknowns. That is to say, there are things that we know we don't know. But there are also unknown unknowns. There are things we don't know we don't know.
He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles
Start here: viewtopic.php?f=1&t=176529
Then here: viewtopic.php?f=15&t=185296
And here: viewtopic.php?f=1&t=194382
iambiguous
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### Re: Corona Virus Outbreak from Turd
Here we go again, continued...
'New federal projections show a spike in infections if shelter in place orders are lifted at 30 days.
Stay-at-home orders, school closures and social distancing greatly reduce infections of the coronavirus, but lifting those restrictions after just 30 days will lead to a dramatic infection spike this summer and death tolls that would rival doing nothing, government projections indicate.
The projections obtained by The New York Times come from the departments of Homeland Security and Health and Human Services. The models use three scenarios. The first has policymakers doing nothing to mitigate the spread of the coronavirus. The second, labeled “steady state,” assumes schools remain closed until summer, 25 percent of Americans telework from home, and some social distancing continues. The third scenario includes a 30-day shelter in place, on top of those “steady state” restrictions.'
NYT
That's the gamble that Trumpworld is considering. Which scenario?
And then the "unknowns" embedded in the "wave" theory. What happens if, in the Fall and Winter, the next big wave does hit?
For example, just in time for the November elections.
He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles
Start here: viewtopic.php?f=1&t=176529
Then here: viewtopic.php?f=15&t=185296
And here: viewtopic.php?f=1&t=194382
iambiguous
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Joined: Tue Nov 16, 2010 8:03 pm
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### Re: Corona Virus Outbreak from Turd
What to believe? Who to believe?
Then what and who to trust?
For example:
https://www.washingtonpost.com/politics ... us-deaths/
'To advocate for a quick or immediate return to America as normal, one must figure out how to rationalize the fact that this week alone, tens of thousands of people have died of covid-19, the disease caused by the coronavirus that emerged in China last year. You can’t simply say, “open up the economy and let the bodies fall where they may.” Instead, we get two different arguments: “open up the economy because the number of deaths is comparable to other causes,” or “open up the economy because the number of deaths isn’t a high as suggested.
'The second argument has been fairly common recently, advocated by people such as former Fox News host Bill O’Reilly and current Fox News personality Brit Hume. Hume in particular has repeatedly suggested that the number of covid-19 deaths being recorded is inflated because people who have preexisting conditions such as cancer are dying and having their deaths attributed to the virus.'
That's always the predicament for most of us. Those folks who do not have access to "deep state" data, and who are not privy to the "behind the curtain" politics that play out as those in power and/or with power jockey to game the system to their own advantage.
Instead, we are all embedded in our own particular "set of circumstances", hoping that when the chips do fall, they don't fall squarely on us.
This part basically: https://www.nytimes.com/2020/04/09/opin ... e=Homepage
He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles
Start here: viewtopic.php?f=1&t=176529
Then here: viewtopic.php?f=15&t=185296
And here: viewtopic.php?f=1&t=194382
iambiguous
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Posts: 39669
Joined: Tue Nov 16, 2010 8:03 pm
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### Re: Corona Virus Outbreak from Turd
He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles
Start here: viewtopic.php?f=1&t=176529
Then here: viewtopic.php?f=15&t=185296
And here: viewtopic.php?f=1&t=194382
iambiguous
ILP Legend
Posts: 39669
Joined: Tue Nov 16, 2010 8:03 pm
Location: baltimore maryland
### Re: Corona Virus Outbreak from Turd
Update number.... I can’t recall.
For the forum hobos and anarchists, the page on the IRS site that you can post your direct deposit to the American IRS to get your check in, if you didn’t make enough to file the last few years.
https://www.irs.gov/coronavirus/non-fil ... -info-here
I’ve done a search yesterday for Tab, noticed he hasn’t posted in a while.
Current updates regarding the treatment of coronavirus:
Micro Embolisms are a concern in the lungs, when pneumonia is present. Anti Coagulants are often used, but the oxygen reader they clip to your finger only reads the oxygen in your blood, not the ability of your cells to absorb oxygen. They have a separate way of testing for this, but may have to nudge the doctor with over reliance on the finger readings.
The Chinese attempt to end the Corona Virus has failed completely. I’m sure many of you saw the riots in Wuhan, flipping police vehicles. The Communist Party has banned reporting of new infections except in foreigners. It doesn’t matter color of skin now, if you are foreign you are considered a source of infection, but they are especially vicious against Africans. There has been mass buying of food across China, most cities only keep a two week supply of rice and grains on hand, and in many cases it is gone now. People are increasingly found dead on the streets, and many people can’t find their supposedly cured family members. They aren’t online or anything. Just vanished.
I’m seeing the first signs of deflation in the stores here for products not related to food. The midget Zero Sum keeps screaming about inflation but right now the manufacturing base is facing the opposite, a lot of people don’t have a need for large purchases. You see it on commercials too, going out of the way to guarantee risk for up to a year on buying a new car in delaying payments.
Corona Virus continues to get whittled down as predicted in western countries. US and Italy seems to be flattening the curve at about the same time. Expect for the next half year a game of wackamo as Corona Virus flairs up in micro hotspots. I’m expecting eventually meds prior to a vaccine will be given out as a ration by zip code to create a sort of herd immunity in such situations, first via places of employment then along bus routes and other tracing analytics, until finally they call fuck it and just pass it out to 10,000 people at once when they realize they can’t control it in emerging hotspots.
The underlining reason we will have hotspots is less because of internal US spread- that’s largely gonna be hammered away, but rather because of China which lacks a reliable rest for Corona Virus (their test is absolute shit) and they are increasingly traveling around the world again. We will get people from random nations infecting over and over. I recognize that looks precisely like the Chinese propaganda against foreigners, but point out the US has a very aggressive testing program that is much more accurate now, as well as a wider array of medications, and save for forum nitwits here, most trust the US medical system and FEMA to disperse medications. Currently the only way I see controlling the foreign influx is employment testing and area blanketing with antivirals till a vaccine goes mainstream. Way better than Chinese picking on Africans and beating the shit out of them on security footage. Africans tend to take antimalarial drugs so are unlikely to get it, it is usually the chines reporting then who are the more likely ones infected.
Japan is paying businesses to move factories out of China. US considering similar. WHO is likely to be defunded, starting next week. The nations the did best in the plague tended to be countries most critical and least trusting of China- Taiwan for example completely crushed the virus. South Korea moved early. US was later but still early. Outbreaks in San Francisco was quickly crushed, we have a lot of illegal Wuhan tailors working in fashion houses I used to guard, but most Chinese in San Francisco are pro Taiwan and the city locked down hard on everything. New York did not, most Chinese there are later immigrants from PRC era. The fashion industry in Italy is mostly ran by illegal Chinese immigrants from Wuhan. Iran absolutely and completely trusted in China and kept running flights a week after official lockdown to China, and Pakistan has open borders to China and Iran, and when they do test they get high numbers of infected.
Right now I’m guessing the US will have to test people wanting to fly to the US at airports overseas, using our 15 minute tests before international flights are resumed but this won’t stop private flights (I tried convincing Erik last year to fly with me to Mexico on a private jet I was getting a free ride on to go to a Donkey Show but he backed out. He likes Animal Porn, ask him- was gonna abandon him there if he said yes).
So that’s the current state of things. I’ve been doing a lot of hiking. Most of my face mask orders failed during the corona lockdown but one got through from Shenzhen of all places so I’ve had some good masks, wear in combination with safety glasses (it can get into your eyes).
Oh, it is quite common for people to have scar tissue from this virus, reduces air supple with apparently permanent shortness of breath even in athletes.
And work has been increasingly done on tracking down the Chinese expats who bought out all the PPE in the US and Australia back in early January for resale at 10x to 50x costs. It isn’t all tongs, most are black market sellers who focus on avoiding Chinese Taxes- they used to hit San Francisco hard and IRS would scout my store. That was for luxury goods them (over $10,000), but now it is mass buying of emergency supplies and removing them from the shelves of one nation for resale in China. I’m aware France and UK had similar issues but not certain what is being done in those cases. Expect China to lash out this year militarily, likely against Taiwan (unlikely to succeed) or increased naval actions, it needs more confrontation to take criticism away from Xi. Everyone expects this so it will likely be a few sterile grandstanding hits. I’m not seeing a scenario in the next year where China recovers. It has no way of tracking its corona infected, swears it conquered the disease and hunts foreigners down as scapegoats. That’s not a good sign for foreign investment. It will have warehouses for lease in exchange for foreign capital but I guarantee you it is bleeding wealth like crazy, and corruption is likely to skyrocket in this time of fast decline. It is like Japan prior to WW2 with the oil embargo thinking of hitting Pearl Harbor, but there is no one obvious place to attack to make all its woes go away. Unless of course China plans on attacking Beijing. This is unlikely. Traditionally when China enters a period of political fragmentation that leads to civil war, they go through a prolonged initial period of pretend unity when growing regional autonomy emerges. They rarely have outright violence in the beginning, and don’t think we are close to having the cliques that took control of China 100 years ago re-emerge for a few more years in a warring capacity- but even then they claimed unity. Expect the CCP to be around for quite some time, with a occasional Coup. It won’t be obvious at first which regions will be autonomous for a while. You’ll know it when Beijing insists on taxes already paid, and the US giving regional visas to residents of parts of China and threatening to sanction party leaders of other areas not getting along. That’s when it will first occur to the media something is up. The fighting will only be sporadic and will stop all together for periods of time and you will think it is over. It won’t be. They have a few thousand years of this behavior. It is very predictable. I would strongly recommend to people viewing this website to avoid most people posting on this site, they don’t research shit and half are convicts with their heads up their ass switching between Communism, Anarchism and Fascist in a game of musical chairs, and their beliefs are mostly fake, delusional and designed to pass the time. The rest are on rather shaky ground for grasping the ideologies the claim to follow. Look at YouTube sites like MedCram. Avoid the dipshits here. They are a step away from alien autopsy stuff. Maia Philosopher Posts: 2792 Joined: Thu Jun 07, 2012 2:22 am Location: UK ### Re: Corona Virus Outbreak from Turd For the forum hobos and anarchists And we thank you for that, T. G. Turd. I actually did end up filing for 2019 real quick through one of those free online deals. I was in a hurry and on a smart phone so I didn't itemize my deductions and shit, which I now regret. I ended up owing about four grand for 2019. I coulda got that down to around two-five if I had the patience. But that wasn't the point. What I needed was to get a direct routing number and an account into the IRS's system so those niggas don't have to track me down with a paper check. Incidentally to do this, I had to file for 2019. We thank you nonetheless for this information, but it's a little late.... like everything else the trumpf administration decides to do. Ohhhhhhhhh promethean75 Philosopher Posts: 4022 Joined: Thu Jan 31, 2019 7:10 pm ### Re: Corona Virus Outbreak from Turd Consider... 'Some organizers and demonstrators had affiliations with the Tea Party and displayed the “Don’t Tread on Me” logo that was an unofficial slogan for the movement. Others waved flags and banners in support of President Trump, who has pushed to reopen the economy. But the size of the protests in places like Michigan suggested that anger over the no-end-in-sight nature of the lockdowns is not limited to the far right, and that the public’s patience has a limit. As anxiety, uncertainty and joblessness grow, the next few weeks will pose a test for governors and local leaders who are likely to face increased pressure to loosen some of the restrictions. In Michigan alone, more than 1 million people — roughly a quarter of the state’s work force — have filed for unemployment benefits.' NYT Will it come down to a choice between opening the economy back up and just accepting that thousands upon thousands more will die until we reach that crucial "herd immunity" point? Or create a vaccine? On the other hand, even that is problematic: https://www.nytimes.com/interactive/202 ... e=Homepage In other words, a kind of "the old vs. the young world". A world in which jobs become more important to those who, even if they are infected, are likely to experience mild or no symptoms at all. While the old and already infirm folks bite the dust. They "take a bullet" for the rest of us. The Texas Lt. Governor Dan Patrick approach to it all. Yo, Turd, your own insights here please. He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles Start here: viewtopic.php?f=1&t=176529 Then here: viewtopic.php?f=15&t=185296 And here: viewtopic.php?f=1&t=194382 iambiguous ILP Legend Posts: 39669 Joined: Tue Nov 16, 2010 8:03 pm Location: baltimore maryland ### Re: Corona Virus Outbreak from Turd "President Trump on Friday began openly fomenting right-wing protests of social distancing restrictions in states where groups of his conservative supporters have been violating stay-at-home orders, less than a day after announcing guidelines for how governors could decide on an orderly reopening of their communities. NYT Forget Trump's typically calculating/lying stance here. His fanatic base would support him even if he did take a gun and start shooting down liberals on Pennsylvania Avenue. I think basically the 2020 presidential campaign will revolve around 1] Democrats hell bent on reminding voters of Trump's catastrophic refusal to take the coronavirus seriously early on, and 2] Republicans hell bent on reminding voters that it was the Democrats that cost them their jobs by going too far in shutting down the economy. Of course the coronavirus itself is still largely in command here. To the extent that Trumpworld is successful in reopening the economy, everything comes down to whether there either is or is not a new resurgence in infections/deaths. That and the extent to which there either is or is not an even more virulent "second wave" in the fall, no matter what anyone does. He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles Start here: viewtopic.php?f=1&t=176529 Then here: viewtopic.php?f=15&t=185296 And here: viewtopic.php?f=1&t=194382 iambiguous ILP Legend Posts: 39669 Joined: Tue Nov 16, 2010 8:03 pm Location: baltimore maryland ### Re: Corona Virus Outbreak from Turd https://www.nytimes.com/2020/04/18/opin ... e=Homepage 'What Trump was saying with those tweets was: Everybody just go back to work. From now on, each of us individually, and our society collectively, is going to play Russian roulette. We’re going to bet that we can spin through our daily lives — work, shopping, school, travel — without the coronavirus landing on us. And if it does, we’ll also bet that it won’t kill us.' Yes, but as with everything else of this sort, there is encompassing it in a "general description intellectual contraption", and there is reacting to it in terms of what your own individual circumstances are. Me, I'm one of the "lucky" ones. For two reasons. One, I hardly ever left my apartment before the pandemic struck. So, now, instead of going out into the world and playing Russian Roulette two hours a week, I go out for just one hour every other week. That is simply what my situation is. Two, I made a good living when I was younger. I have enough moolah stashed away in the coolah to take me all the way to the grave. Here my main concern is what Joker has prophesized: the pandemic leading to a bout of hyperinflation that brings my stash crashing down. Of course, it not likely that Thomas L. Friedman is desparately waiting for his own$1,200 check to keep him and his loved ones heads above water.
So I can definitely sympathize with those who are fuming that perhaps the economic lockdown has gone way too far.
When you are literally living from paycheck to paycheck and, now, by the tens of millions, you find yourself without one for weeks or months on end what the fuck difference will $1,200 really make?! I'm especially drawn and quartered on this one. All those talking heads making their big bucks in the media industrial complex yapping about how we are all in this together when they don't have a fucking clue as to how those in the working class are struggling just to subsist from week to week. There are definitely two sides [at least] to this tragedy. He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles Start here: viewtopic.php?f=1&t=176529 Then here: viewtopic.php?f=15&t=185296 And here: viewtopic.php?f=1&t=194382 iambiguous ILP Legend Posts: 39669 Joined: Tue Nov 16, 2010 8:03 pm Location: baltimore maryland ### Re: Corona Virus Outbreak from Turd https://www.washingtonpost.com/nation/2 ... ronavirus/ Headline in WP: #FloridaMorons trends after people flock to reopened Florida beaches 'Aerial snapshots of people flocking to a reopened beach in Jacksonville, Fla., made waves on the Internet on Saturday. Local news aired photos and videos of Florida’s shoreline dotted with people, closer than six feet apart, spurring #FloridaMorons to trend on Twitter after Gov. Ron DeSantis (R) gave the go-ahead for local beachfront governments to decide whether to reopen their beaches during a news briefing Friday. Duval and St. Johns counties have reopened their beaches, while Miami-Dade County officials said they are considering following suit.' If nothing else, this is an excellent opportunity to test to see which side is more rational in their approach to the pandemic. In other words, someone [healthcare officials in particular] should be on the beach attempting to track the fate of those who ventured out. Will it prompt a surge in infections down there in this community? Right now there are 25,492 cases in Florida with 748 deaths. Today Florida reported 58 deaths from the virus. The highest one day total to date. If the government down there proceeds to loosen restrictions all the more, how will that impact the stats? He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles Start here: viewtopic.php?f=1&t=176529 Then here: viewtopic.php?f=15&t=185296 And here: viewtopic.php?f=1&t=194382 iambiguous ILP Legend Posts: 39669 Joined: Tue Nov 16, 2010 8:03 pm Location: baltimore maryland ### Re: Corona Virus Outbreak from Turd https://www.nytimes.com/2020/04/18/heal ... e=Homepage And then -- inevitably? eventually? maybe? not likely? -- this part: 'Imagine an America divided into two classes: those who have recovered from infection with the coronavirus and presumably have some immunity to it; and those who are still vulnerable. “It will be a frightening schism,” Dr. David Nabarro, a World Health Organization special envoy on Covid-19, predicted. “Those with antibodies will be able to travel and work, and the rest will be discriminated against.” Already, people with presumed immunity are very much in demand, asked to donate their blood for antibodies and doing risky medical jobs fearlessly.' NYT So, get tested. The new Ubermen! 'Soon the government will have to invent a way to certify who is truly immune. A test for IgG antibodies, which are produced once immunity is established, would make sense, said Dr. Daniel R. Lucey, an expert on pandemics at Georgetown Law School. Many companies are working on them. Dr. Fauci has said the White House was discussing certificates like those proposed in Germany. China uses cellphone QR codes linked to the owner’s personal details so others cannot borrow them. The California adult-film industry pioneered a similar idea a decade ago. Actors use a cellphone app to prove they have tested H.I.V. negative in the last 14 days, and producers can verify the information on a password-protected website.' He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles Start here: viewtopic.php?f=1&t=176529 Then here: viewtopic.php?f=15&t=185296 And here: viewtopic.php?f=1&t=194382 iambiguous ILP Legend Posts: 39669 Joined: Tue Nov 16, 2010 8:03 pm Location: baltimore maryland ### Re: Corona Virus Outbreak from Turd promethean75 wrote:The LL Bean guy who's making those masks is actually considering making fashionable masks with different styles and colors and shit. Hey why not, right? If this is gonna be the new thing, might as well make it a little more exciting with fully customizable face masks. Say a little about yourself with your face mask. I mean you do that with your shoes and hats and scarves and you don't think that's silly. Mowk my words; if stylish face masks start trending and people start wearing them, you will too. Eventually. So start thinking about how to customize your face mask so it has some personality and says something about you. Is it weird that I'm kinda thinking that if this thing keeps going that I really want companies like Arcteryx and Patagonia to make some bad ass lightweight high end masks? You see...a pimp's love is very different from that of a square. Dating a stripper is like eating a noisy bag of chips in church. Everyone looks at you in disgust, but deep down they want some too. What exactly is logic? -Magnus Anderson Support the innocence project on AmazonSmile instead of Turd's African savior biker dude. http://www.innocenceproject.org/ Mr Reasonable resident contrarian Posts: 28300 Joined: Sat Mar 17, 2007 8:54 am Location: pimping a hole straight through the stratosphere itself ### Re: Corona Virus Outbreak from Turd One possible "worst case scenario": https://www.nytimes.com/2020/04/20/worl ... e=Homepage 'The spread of the coronavirus in this tidy city-state suggests that it might be difficult for the United States, Europe and the rest of the world to return to the way they were anytime soon, even when viral curves appear to have flattened. Although countries can closely track contacts to try to keep an outbreak at bay as Singapore did, the coronavirus is sickening, killing and spreading with each passing day, leaving scientists and political leaders racing to catch up with its relentless pace and new dangers. 'If anything, the trials of this intensely urban, hyper-international country hint at a global future in which travel is taboo, borders are shut, quarantines endure and industries like tourism and entertainment are battered. Weddings, funerals and graduation parties will have to wait. Vulnerable populations, such as migrants, cannot be ignored.' He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles Start here: viewtopic.php?f=1&t=176529 Then here: viewtopic.php?f=15&t=185296 And here: viewtopic.php?f=1&t=194382 iambiguous ILP Legend Posts: 39669 Joined: Tue Nov 16, 2010 8:03 pm Location: baltimore maryland ### Re: Corona Virus Outbreak from Turd There may come a far worse coronavirus infection in the next wave: "Even as states move ahead with plans to reopen their economies, the director of the Centers for Disease Control and Prevention warned Tuesday that a second wave of the novel coronavirus will be far more dire because it is likely to coincide with the start of flu season. “There’s a possibility that the assault of the virus on our nation next winter will actually be even more difficult than the one we just went through,” CDC Director Robert Redfield said in an interview with The Washington Post. “And when I’ve said this to others, they kind of put their head back, they don’t understand what I mean.” “We’re going to have the flu epidemic and the coronavirus epidemic at the same time,” he said. Having two simultaneous respiratory outbreaks would put unimaginable strain on the health-care system, he said. The first wave of covid-19, the disease caused by the coronavirus, has already killed more than 42,000 people across the country. It has overwhelmed hospitals and revealed gaping shortages in test kits, ventilators and protective equipment for health-care workers. In a wide-ranging interview, Redfield said federal and state officials need to use the coming months to prepare for what lies ahead. As stay-at-home orders are lifted, officials need to stress the continued importance of social distancing, he said. They also need to massively scale up their ability to identify the infected through testing and find everyone they interact with through contact tracing. Doing so prevents new cases from becoming larger outbreaks. Asked about the appropriateness of protests against stay-at-home orders and calls on states to be “liberated” from restrictions, Redfield said: “It’s not helpful.” Put out by CDC director Robert Redfield today. It is noteworthy to comment that the first wave of the 1918 Spanish flu was followed by a second, far worse then the first, and then the third. The final tally came in at approx. 50 million deaths. Meno_ breathless Posts: 8042 Joined: Tue Dec 08, 2015 2:39 am Location: Mysterium Tremendum ### Re: Corona Virus Outbreak from Turd Yet another "worst case scenario" account: https://www.washingtonpost.com/business ... its-scary/ 'There’s growing consensus among economists and epidemiologists that the recovery period from the deadly coronavirus is going to be long — and bumpy. 'Hopes of a quick bounce back for the economy — dubbed a V-shaped recovery — have faded. Even as parts of the nation reopen, many Americans will be afraid to venture out, and it looks increasingly likely that restaurants, stadiums and yoga classes are going to be operating at partial capacity, at best, for a while. 'What isn’t getting as much attention is the possibility of a W-shaped recovery, the scary scenario when the economy starts looking better and then there’s a second downturn later this year or next. The “W” could be triggered by reopening the economy too quickly and seeing a second spike in deaths from covid-19, the disease caused by the coronavirus. Businesses would have to shutter again, and people would be even more afraid to venture out until a vaccine is found. 'Something else could also cause a “W” pattern: A wave of bankruptcies and defaults later this year. As companies go belly up, a domino effect ensues: Workers aren’t rehired, suppliers aren’t paid, and fear rises about who will be next to fall. '“Pretending the world will return to normal in three months or six months is just wrong,” said Diane Swonk, chief economist at Grant Thornton. “The economy went into an ice age overnight. We’re in a deep freeze. As the economy thaws, we’ll see the damage done as well. Flooding will occur.” 'Early warning signs are here. Major retailers like Macy’s and Neiman Marcus face significant financial duress, and analysts anticipate bankruptcies ahead in the retail sector. Oil prices plunging below$20 a barrel is another blow to America’s fragile energy sector that will reverberate for months. Rystad Energy predicts more than 500 U.S. companies will go bankrupt by the end of 2021.
'Big law firms like Hogan Lovells are urging their lawyers to brush up on bankruptcy and restructuring law “in anticipation of a wave of bankruptcy filings in the coming months,” and the major banks spent a lot of time on their earnings calls last week predicting a surge in credit card, auto loan and business loan defaults. Both firms and consumers are teetering on the financial edge.'
Precipitating this:
'As people lose jobs, they stop paying their rent or mortgage, which can lead to eviction and a bad credit rating that drags them down for years. They lose health insurance and possibly their car. Often, they lose hope. This is the scenario the nation needs to avoid, and policymakers could be doing a much better job trying to prevent this, economists say.'
Imagine the political repercussions of this in a Presidential election year. Fear of the unknown on steroids if this scenario plays out.
He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles
Start here: viewtopic.php?f=1&t=176529
Then here: viewtopic.php?f=15&t=185296
And here: viewtopic.php?f=1&t=194382
iambiguous
ILP Legend
Posts: 39669
Joined: Tue Nov 16, 2010 8:03 pm
Location: baltimore maryland
### Re: Corona Virus Outbreak from Turd
it weird that I'm kinda thinking that if this thing keeps going that I really want companies like Arcteryx and Patagonia to make some bad ass lightweight high end masks?
Breathable waterproof coronacore nanofiber technology in our highly customizable face masks. Choose from four colors: matte orange, onyx black, real tree camo, electric blue. Removable inner liner for winter season and machine washable.
promethean75
Philosopher
Posts: 4022
Joined: Thu Jan 31, 2019 7:10 pm
### Re: Corona Virus Outbreak from Turd
Gore-tex pro for maximum rain resistance while maintaining breathability and a special treatment to prevent it from picking up the smell of my breath. Like those shirts that you can camp in for days that don't get stinky. Good stuff really.
You see...a pimp's love is very different from that of a square.
Dating a stripper is like eating a noisy bag of chips in church. Everyone looks at you in disgust, but deep down they want some too.
What exactly is logic? -Magnus Anderson
Support the innocence project on AmazonSmile instead of Turd's African savior biker dude.
http://www.innocenceproject.org/
Mr Reasonable
resident contrarian
Posts: 28300
Joined: Sat Mar 17, 2007 8:54 am
Location: pimping a hole straight through the stratosphere itself
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# Just how do you locate what procedure is holding a documents open in Windows?
One point that frustrates me no end concerning Windows is the old sharing offense mistake. Usually you can not recognize what's holding it open. Generally it's simply an editor or traveler simply indicating a pertinent directory site yet occasionally I've needed to consider restarting my equipment.
Any kind of pointers on just how to locate the wrongdoer?
0
2019-05-04 09:11:03
Source Share
Just be really mindful with shutting takes care of ; it is a lot more unsafe than you would certainly assume, as a result of take care of reusing - if you close the documents take care of, and also the program opens up another thing, that initial documents manage you shut might be recycled for that "something else." And currently presume what takes place if the program proceeds, assuming it is working with the documents (whose manage you shut), when actually that documents take care of is currently indicating another thing.
see Raymond Chen's post on this subject
Suppose a search index solution has a documents open for indexing yet has actually obtained stuck momentarily and also you intend to delete the documents, so you (unwisely) compel the take care of shut. The search index solution opens its log documents in order to videotape some details, and also the take care of to the removed documents is reused as the take care of to the log documents. The stuck procedure ultimately finishes, and also the search index solution ultimately navigates to shutting that manage it had open, yet it winds up unintentionally shutting the log documents take care of.
The search index solution opens up an additional documents, claim an arrangement apply for creating so it can upgrade some relentless state. The take care of for the log documents obtains reused as the take care of for the arrangement documents. The search index solution intends to log some details, so it contacts its log documents. However, the log documents take care of was shut and also the take care of recycled for its arrangement documents. The logged details enters into the arrangement documents, damaging it.
At the same time, an additional manage you compelled shut was recycled as a mutex take care of, which is made use of to aid protect against information from being damaged. When the initial documents take care of is shut, the mutex take care of is shut and also the defenses versus information corruption are shed. The longer the solution runs, the even more damaged its indexes come to be. At some point, someone notifications the index is returning wrong outcomes. And also when you attempt to reactivate the solution, it falls short due to the fact that its arrangement documents have actually been damaged.
You report the trouble to the firm that makes the search index solution and also they establish that the index has actually been damaged, the log documents has actually strangely quit logging, and also the arrangement documents was overwritten with waste. Some inadequate professional is appointed the helpless job of identifying why the solution damages its indexes and also arrangement documents, not aware that the resource of the corruption is that you compelled a take care of shut.
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2019-12-04 05:12:15
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I obtained transformed on to the Exteneded Task Manager a while earlier by Jeremy Zawodny is blog site, and also it is wonderful for locating better details on procedures also.+1 for Process Explorer as above, also, specifically for eliminating procedures that the typical Task Manager will not end
0
2019-05-31 13:02:31
Source
Who Lock Me functions well and also maintains individuals entertained with the name!
0
2019-05-17 01:53:17
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There is NirSoft's Opened Files View too.
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2019-05-11 23:00:42
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I've made use of Handle with success to locate such procedures in the past.
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2019-05-08 01:44:04
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On a remote web server, when you're examining a network share, something as straightforward as the Computer Management console can present this details and also close the documents.
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2019-05-08 01:19:54
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Try the openfiles command.
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2019-05-08 00:48:30
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I've had success with Sysinternals Process Explorer. With this, you can search to locate what procedure (es ) have a documents open, and also you can utilize it to close the take care of (s ) if you desire. Certainly, it is more secure to close the entire procedure. Workout care and also reasoning.
To locate a details documents, make use of the food selection alternative Find->Find Handle or DLL... Type in component of the path to the documents. The checklist of procedures will certainly show up listed below.
If you favor command line, Sysinternals collection consists of command line device Handle, that details open takes care of. A couple of instances on just how to utilize it :
• c:\Program Files\SysinternalsSuite>handle.exe |findstr /i e:\ - locate all documents opened up from drive E :
• c:\Program Files\SysinternalsSuite>handle.exe |findstr /i file-or-path-in-question
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2019-05-07 22:28:00
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Mathematics » Equations and Inequalities » Solving Linear Equations
# Solving Linear Equations
## Solving Linear Equations
The simplest equation to solve is a linear equation. A linear equation is an equation where the highest exponent of the variable is $$\text{1}$$. The following are examples of linear equations:
\begin{align*} 2x + 2 & = 1 \\ \cfrac{2 – x}{3x + 1} & = 2 \\ 4(2x – 9) – 4x & = 4 – 6x \\ \cfrac{2a – 3}{3} – 3a & = \cfrac{a}{3} \end{align*}
Solving an equation means finding the value of the variable that makes the equation true. For example, to solve the simple equation $$x + 1 = 1$$, we need to determine the value of $$x$$ that will make the left hand side equal to the right hand side. The solution is $$x = 0$$.
The solution, also called the root of an equation, is the value of the variable that satisfies the equation. For linear equations, there is at most one solution for the equation.
To solve equations we use algebraic methods that include expanding expressions, grouping terms, and factorising.
For example:
\begin{align*} 2x + 2 & = 1 \\ 2x & =1 – 2 \quad \text{ (rearrange)} \\ 2x & = -1 \quad \text{ (simplify)} \\ x & = -\cfrac{1}{2} \quad \text{(divide both sides by } 2\text{)} \end{align*}
Check the answer by substituting $$x=-\cfrac{1}{2}$$.
\begin{align*} \text{LHS } & = 2x + 2 \\ & = 2(-\cfrac{1}{2}) + 2 \\ & = -1 + 2 \\ & = 1 \\ \text{RHS } & =1 \end{align*}
Therefore $$x=-\cfrac{1}{2}$$
The following video gives an introduction to solving linear equations.
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EN FR
• Legal notice
• Accessibility - non conforme
## Section: New Software and Platforms
### Jorek-Django
Functional Description
Jorek-Django is a new version of the JOREK software, for MHD modelling of plasma dynamic in tokamaks geometries. The numerical approximation is derived in the context of finite elements where 3D basic functions are tensor products of 2D basis functions in the poloidal plane by 1D basis functions in the toroidal direction. More specifically, Jorek uses curved bicubic isoparametric elements in 2D and a spectral decomposition (sine, cosine) in the toroidal axis. Continuity of derivatives and mesh alignment to equilibrium surface fluxes are enforced. Resulting linear systems are solved by the PASTIX software developed at Inria-Bordeaux.
• Participants: Boniface Nkonga, Hervé Guillard, Emmanuel Franck, Ayoub Iaagoubi, Ahmed Ratnani
• Contact: Hervé Guillard
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# The intricate link between galaxy dynamics and intrinsic shape (or why so-called prolate rotation is a misnomer)
@article{Foster2019TheIL,
title={The intricate link between galaxy dynamics and intrinsic shape (or why so-called prolate rotation is a misnomer)},
author={Caroline Foster and R. Bassett},
journal={Proceedings of the International Astronomical Union},
year={2019},
volume={14},
pages={222 - 225}
}
• Published 1 June 2019
• Physics
• Proceedings of the International Astronomical Union
Abstract Many recent integral field spectroscopy (IFS) survey teams have used stellar kinematic maps combined with imaging to statistically infer the underlying distributions of galaxy intrinsic shapes. With now several IFS samples at our disposal, the method, which was originally proposed by M. Franx and collaborators in 1991, is gaining in popularity, having been so far applied to ATLAS3D, SAMI, MANGA and MASSIVE. We present results showing that a commonly assumed relationship between…
## References
SHOWING 1-10 OF 20 REFERENCES
Prospects for recovering galaxy intrinsic shapes from projected quantities
• Physics
Monthly Notices of the Royal Astronomical Society
• 2019
The distribution of three dimensional intrinsic galaxy shapes has been a longstanding open question. The difficulty stems from projection effects meaning one must rely on statistical methods applied
The SAMI Galaxy Survey: the intrinsic shape of kinematically selected galaxies
• Physics
• 2017
Using the stellar kinematic maps and ancillary imaging data from the Sydney AAO Multi Integral field (SAMI) Galaxy Survey, the intrinsic shape of kinematically selected samples of galaxies is
The ATLAS 3D project - XXIV. The intrinsic shape distribution of early-type galaxies
• Physics
• 2014
We use the ATLAS^(3D) sample to perform a study of the intrinsic shapes of early-type galaxies, taking advantage of the available combined photometric and kinematic data. Based on our ellipticity
The SLUGGS Survey: stellar kinematics, kinemetry and trends at large radii in 25 early-type galaxies
• Physics
• 2016
Due to longer dynamical time-scales, the outskirts of early-type galaxies retain the footprint of their formation and assembly. Under the popular two-phase galaxy formation scenario, an initial in
The intrinsic shape of galaxies in SDSS/Galaxy Zoo
• Physics
• 2013
By modelling the axis ratio distribution of SDSS DR8 galaxies we find the intrinsic 3D shapes of spirals and ellipticals. We use morphological information from the Galaxy Zoo project and assume a
The ordered nature of elliptical galaxies - Implications for their intrinsic angular momenta and shapes
• Physics
• 1991
The observations of rotation along the major and minor axes of 38 elliptical galaxies are analyzed to determine their intrinsic structure. Rotation along the minor axis occurs (i) as a result of
SDSS-IV MaNGA: The Intrinsic Shape of Slow Rotator Early-type Galaxies
• Physics
The Astrophysical Journal
• 2018
By inverting the distributions of galaxies' apparent ellipticities and misalignment angles (measured around the projected half-light radius $R_{\rm e}$) between their photometric and kinematic axes,
Introducing the Illustris Project: the evolution of galaxy populations across cosmic time
• Physics
• 2014
We present an overview of galaxy evolution across cosmic time in the Illustris Simulation. Illustris is an N-body/hydrodynamical simulation that evolves 2*1820^3 resolution elements in a (106.5Mpc)^3
Specific angular momentum of disc merger remnants and the λR‐parameter
• Physics
• 2009
We use two-dimensional kinematic maps of simulated binary disc mergers to investigate the λ R -parameter, which is a luminosity-weighted measure of projected angular momentum per unit mass. This
Triaxial galaxy models with thin tube orbits
• Physics, Geology
• 1992
It is shown how to construct self-consistent distribution functions for triaxial galaxy models with Stackel potentials. The Stackel potentials cause the dynamics to be simple, with all the stellar
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Question: is it possible to find sample batch # in CEL files?
0
6.9 years ago by
Brian Tsai40
Brian Tsai40 wrote:
Hi, I've been downloading raw CEL files from the gene expression omnibus, and have been trying to process them -- i'd like to account for batch effect when computing differential expression, but the authors didn't provide the information explicitly in their annotations. Is this information stored/retrievable through the CEL files through Bioconductor? [[alternative HTML version deleted]]
process • 948 views
modified 4.6 years ago by suprun.maria0 • written 6.9 years ago by Brian Tsai40
Answer: is it possible to find sample batch # in CEL files?
0
6.9 years ago by
Sean Davis21k
United States
Sean Davis21k wrote:
Not a direct answer, but you might look at the sva package which does not rely on externally-defined batch effects. Sean On Thu, Jan 24, 2013 at 8:23 AM, Brian Tsai <btsai00 at="" gmail.com=""> wrote: > Hi, > > I've been downloading raw CEL files from the gene expression omnibus, and > have been trying to process them -- i'd like to account for batch effect > when computing differential expression, but the authors didn't provide the > information explicitly in their annotations. Is this information > stored/retrievable through the CEL files through Bioconductor? > > [[alternative HTML version deleted]] > > _______________________________________________ > Bioconductor mailing list > Bioconductor at r-project.org > https://stat.ethz.ch/mailman/listinfo/bioconductor > Search the archives: http://news.gmane.org/gmane.science.biology.informatics.conductor
Answer: is it possible to find sample batch # in CEL files?
0
6.9 years ago by
Guido Hooiveld2.5k
Wageningen University, Wageningen, the Netherlands
Guido Hooiveld2.5k wrote:
Hi, Some time ago I came across these lines of code, that could be of help: http://bios.ucdenver.edu/images/a/a1/Affy_headerinfo.txt Never used it myself, though. HTH, Guido -----Original Message----- From: bioconductor-bounces@r-project.org [mailto:bioconductor- bounces@r-project.org] On Behalf Of Brian Tsai Sent: Thursday, January 24, 2013 14:23 To: bioconductor at r-project.org Subject: [BioC] is it possible to find sample batch # in CEL files? Hi, I've been downloading raw CEL files from the gene expression omnibus, and have been trying to process them -- i'd like to account for batch effect when computing differential expression, but the authors didn't provide the information explicitly in their annotations. Is this information stored/retrievable through the CEL files through Bioconductor? [[alternative HTML version deleted]] _______________________________________________ Bioconductor mailing list Bioconductor at r-project.org https://stat.ethz.ch/mailman/listinfo/bioconductor Search the archives: http://news.gmane.org/gmane.science.biology.informatics.conductor
Answer: is it possible to find sample batch # in CEL files?
0
6.9 years ago by
James F. Reid120
James F. Reid120 wrote:
Hi Brian, On 24/01/13 13:23, Brian Tsai wrote: > Hi, > > I've been downloading raw CEL files from the gene expression omnibus, and > have been trying to process them -- i'd like to account for batch effect > when computing differential expression, but the authors didn't provide the > information explicitly in their annotations. Is this information > stored/retrievable through the CEL files through Bioconductor? you should be able to access the date the chip was scanned using the readCelHeader function provided in the affxparser package. Look for the 'datheader' entry. James.
Hi I haven't tried this in a while, but afaIcs the 'readAffy' function in the 'affy' package automatically populates the 'ScanDate' field in the resulting AffyBatch object, which you can access with syntax like protocolData(a)$ScanDate where I have assumed that 'a' is an AffyBatch. Best wishes Wolfgang Il giorno Jan 24, 2013, alle ore 2:43 PM, James F. Reid <reidjf at="" gmail.com=""> ha scritto: > Hi Brian, > > On 24/01/13 13:23, Brian Tsai wrote: >> Hi, >> >> I've been downloading raw CEL files from the gene expression omnibus, and >> have been trying to process them -- i'd like to account for batch effect >> when computing differential expression, but the authors didn't provide the >> information explicitly in their annotations. Is this information >> stored/retrievable through the CEL files through Bioconductor? > you should be able to access the date the chip was scanned using the readCelHeader function provided in the affxparser package. Look for the 'datheader' entry. > > James. > > _______________________________________________ > Bioconductor mailing list > Bioconductor at r-project.org > https://stat.ethz.ch/mailman/listinfo/bioconductor > Search the archives: http://news.gmane.org/gmane.science.biology.informatics.conductor ADD REPLYlink written 6.9 years ago by Wolfgang Huber13k Answer: is it possible to find sample batch # in CEL files? 0 6.9 years ago by Rob Dunne230 Rob Dunne230 wrote: Hi Brian, affxparser has a function called readCelHeader. library(affxparser) dates<-rep(0,length(files)) for (i in 1:length(files)){ datheader<-readCelHeader(ff[i])$datheader dd<-gsub(".*([0-9]{2,2}/[0-9]{2,2}/[0-9]{2,2}).*","\\1", datheader) dates[i]<-dd } Bye Rob ________________________________________ From: bioconductor-bounces@r-project.org [bioconductor- bounces@r-project.org] On Behalf Of Brian Tsai [btsai00@gmail.com] Sent: Friday, January 25, 2013 12:23 AM To: bioconductor at r-project.org Subject: [BioC] is it possible to find sample batch # in CEL files? Hi, I've been downloading raw CEL files from the gene expression omnibus, and have been trying to process them -- i'd like to account for batch effect when computing differential expression, but the authors didn't provide the information explicitly in their annotations. Is this information stored/retrievable through the CEL files through Bioconductor? [[alternative HTML version deleted]] _______________________________________________ Bioconductor mailing list Bioconductor at r-project.org https://stat.ethz.ch/mailman/listinfo/bioconductor Search the archives: http://news.gmane.org/gmane.science.biology.informatics.conductor
Answer: is it possible to find sample batch # in CEL files?
0
4.6 years ago by
United States
suprun.maria0 wrote:
We are getting batch date using the following code:
pData(protocolData(a)[sampleNames(a),])$ScanDate Or this code to process all the samples: a$Batch <- sapply(pData(protocolData(a)[sampleNames(a),])$ScanDate, function(x){substr(x,1,10)}) If you only use protocolData(a)$ScanDate it might depend on the sorting and will assign some dates incorrectly.
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## anonymous 4 years ago 3x^2=7
1. Lukecrayonz
3x=sqrt(7) x=sqrt(7)/3
2. anonymous
hi, i got the answer. not sure if anyone done math labs online, how do you input this
3. anonymous
|dw:1349872076838:dw|
4. Lukecrayonz
034 what in the world is wrong with your x's lol
5. anonymous
yes, i work this the same, however when inputing it online, coming back incorrect maybe the symbol
6. anonymous
@Lukecrayonz hahaha why???
7. Lukecrayonz
Looks like a 2pi or 2r or something :P
8. anonymous
what ever it looks like but its my habit
9. anonymous
help, how to write the correct solution set (4-3x)^2=39?
10. Lukecrayonz
Expand (4-3x)^2
11. anonymous
expand?
12. anonymous
If you expand it you get: (4-3x)*(4-3x)=9x^2 -24x +16
13. anonymous
and then = 9x^2 -24x +16 -39 =0=9x^2 -24x -23
14. anonymous
and use the abc formula
15. anonymous
16. anonymous
I'm dutch, sorry
17. anonymous
@ Frank 1991yes I got what you where saying
18. anonymous
i see what you have here, and i have similar, I guess I need to simplify this because it does not show in my options in labs
19. anonymous
@Frank1991, i meant have to simplify all radicals. heck its multiply choice
20. anonymous
use the square root property to solve for the equation. (x-9)^2=64
21. anonymous
in the last case you could say $\sqrt{(x-9)^{2}}=\sqrt{64}=8=x-9$
22. anonymous
so x = 8+9
23. anonymous
@Frank1991, my answers are as yours, but when i type them in them in the computer labs they come back wrong.
24. anonymous
oh that is strange you could check what wolframalpha is saying
25. anonymous
@Frank1991 sure thing
26. anonymous
Yes wolfram alpha is also saying something different
27. anonymous
I forgot about that a sqaure could also be negative
28. anonymous
29. anonymous
yes
30. anonymous
@Frank1991 .. thanks
31. anonymous
The formula A=P (1+r)^2 gives the amount A in dollars that P dollars will grow to in 2 years interst rate r ( where r is given as a decimal), using compound intreset. What interest rate will cause $3000 to grow to$3370.80 in 2 years..
32. anonymous
The formula S=16t^2 is used to approximate the distance S, in feet, that an object falls freely from the rest in t seconds. The height of a building is 1048 feet. How long would it take for an object to fall from the top?
33. anonymous
To the first question:3000=3370.8(1+r)^2 => 3000=3370.8 (r^2 +2r +1) => r^2 +2r +1 = 3000/3370.8 => r^2 +2r +(1-3000/3370.8) = 0 and solve this for r. Probably one of the answers is negative and that makes no sense
34. mayankdevnani
|dw:1349875404939:dw|
35. mayankdevnani
@johnson32 got it
36. anonymous
second one: 1048=16 *t ^2 => t = squarroot( 1048/16) = ...
37. anonymous
I hope it helped @johnson32
38. anonymous
@Frank1991 , so you got 0% interest rate,
39. anonymous
@mayankdevnani , yes thank you got that one solve already
40. mayankdevnani
ok
41. anonymous
no r is the intrest rate
42. anonymous
@Frank1991 , I got 65.5 seconds for the object falling at
43. anonymous
so use the quadratic formula for: r^2 +2r +(1-3000/3370.8) = 0 and r would be your answer
44. anonymous
got it
45. anonymous
oh the 3000 and 3370.8 should switch
46. anonymous
You obtain r = -2.06 or r=0.06
47. anonymous
r=-2.06 makes no sense because there is an increase not a decrease, so your final answer is 0.06
48. anonymous
got it? @johnson32
49. anonymous
yes I got it, thank you
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# method of sections example problems with solutions pdf
## method of sections example problems with solutions pdf
(Use the ratio of initial rates to get the orders). 800 lb. Like most static structural analysis, we must first start by locating and solving the reactions at supports. 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A B CD E G F Method of Sections Example: Determine the forces BC, CG, and GF in the following truss. Problem 003-ms The truss in Fig. 0000004808 00000 n Problems and Solutions in Real and Complex Analysis, Integration, Functional Equations and Inequalities by Willi-Hans Steeb International School for Scienti c Computing Preface The purpose of this book is to supply a collection of P-428. In this unit, you will again use some of 0000004117 00000 n The Simplex Method A-5 The Simplex Method Finally, consider an example wheres 1 0 and s 2 0. Taking the sum of moments about the left support gets us: ∑MA=0(15m)(−10kN)+(25m)(−15kN)+(30m)(RB)=0RB=17.5kN So the re… Simplifying the structure to just include the loads and supports: Without spending too much time on calculating the reactions, you generally start by taking the sum of moments about a point. In the early stage, approximate modelling establishes whether the … Graphical Educational content for Mathematics, Science, Computer Science. T-04 is pinned to the wall at point F, and supported by a roller at point C. Calculate the force (tension or compression) in members BC, BE, and DE. 3 5 Method of Sections Monday, October 22, 2012 Method of Sections " The method of sections utilizes both force and moment equilibrium. " H�bf�ba�bd@ AV da�8� L��e����.��Rq٭r���3�(��7��S��3z7$K�MY�qkS� Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Example problem using method of sections for truss analysis - statics and structural analysis. 0000000751 00000 n 0000003887 00000 n This engineering statics tutorial goes over a method of sections example problem for truss analysis. The truss in Fig. Each method of expressing concentration of the solutions has its … method of sections. Truss Problem 428 - Howe Truss by Method of Sections Problem 428 Use the method of sections to determine the force in members DF, FG, and GI of the triangular Howe truss shown in Fig. Consider the table of initial rates for the reaction: 2ClO 2 + 2OH 1- … Chapter 6 : Surface Integrals Here are a set of practice problems for the Surface Integrals chapter of the Calculus III notes. 246 0 obj << /Linearized 1 /O 249 /H [ 1370 543 ] /L 424218 /E 51307 /N 17 /T 419179 >> endobj xref 246 20 0000000016 00000 n Principles of Statics Equilibrium of Force System Demand engt’s utility function is U(x1, x2)= x1 + ln x 2 x 1 - stamps x 2 - beer Bengts budget p1 x 1 Truss – Example Problem Draw the free-body diagram. z�7��S/�M�U�6]U@W�X���,���,:�����H��8L�.�f�a���]��z]s�l�o��I�,�/1�Kd(^��� �B�Y@�L����X�XYa$I�� a�` �������=|Ξ���)����R��\!XDB��1� �q�#�Rp�2/����:�yZ�h8�jtz L�E�52�P� C++ Solved programs, problems/Examples with solutions. Figure 3.10: Method of Sections Example - Free Body Diagram for Cut Section to the Right of b-b Use a free body diagram to solve for the unknown internal axial forces For a simpler problem, only one cut would be needed if the section had only three members crossing the cut. This page contains List of all programs from Basic to Advance with source code and output to differ.. C++ Program to Remove Characters in String Except Alphabets C++ The sections are obtained by cutting through some of the members of the truss to For example, 1.00 mol kg–1 (or 1.00 m) solution of KCl means that 1 mol (74.5 g) of KCl is dissolved in 1 kg of water. 0000020469 00000 n Here is a list of all the sections for which practice problems have been written as well as a brief description of the material covered in the notes for that particular section. 0000003147 00000 n 0000003125 00000 n T-04 is pinned to the wall at point F, and supported by a roller at point C. Calculate the force (tension or compression) in members BC, BE, and DE. Q����w��������$ʌ�I��\y��U��*3'1���zAh�ؾ���RA�����,�Zwr�����:���u�Ԁ��Z;=v^��O��D�I. Problem 003-ms 0000002348 00000 n 0000001116 00000 n 3.5 The Method of Joints 3.6 The Method of Sections 3.7 Practice Problems 3.7a Selected Problem Answers Chapter 4: Analysis of Determinate Beams and Frames Chapter 5: Deflections of Determinate Structures Chapter 6 Mathematical Economics Practice Problems and Solutions – Second Edition – G. Stolyarov II 1 MatheMatical econoMics Practice ProbleMs and solutions Second Edition G. Stolyarov II, ASA, ACAS, MAAA, CPCU, ARe, ARC 0000003393 00000 n Unit 1 Lesson 3: Graphical method for solving LPP. 0000004592 00000 n Problems with solutions, Intermediate microeconomics, part 1 Niklas Jakobsson, nja@nova.no Katarina.Katz@kau.se Problem 1. The Method of Sections The method of sections is a process used to solve for the unknown forces acting on members of a truss.The method involves breaking the truss down into individual sections and analyzing each section as a separate rigid body. 0000002307 00000 n 2. %PDF-1.3 %���� 0000023147 00000 n These values result in the follow-ing set of equations. H��T�n�0��+��̧����� i�[��"S�Kr%����K�n�DN[ia93;�Kzљ��gg4�o5��|U5����1y��Kx�Y�:o�ik�_ui��U���G:����A*+�oGD(��[��痋�.�=p�@_4� Open Digital Education.Data for CBSE, GCSE, ICSE and Indian state boards. DIFFERENTIAL EQUATION PROBLEMS 12 Example 1.6 We shall here concentrate on the scalar case n = m =1,inr =1to4 dimensions and with orders L = 1 or 2, i.e. Useful solutions to standard problems in Introduction and synopsis Modelling is a key part of design. minimization problems with any combination of ≤, ≥, = constraints 2 Example Maximize P = 2x 1 + x 2 subject to x 1 + x 2 < 10 –x 1 + x 2 > 2 0 3 x 1, x 2 > To form an equation out of the first inequality, we introduce a s 1x 1 + MethodofSections The Method of Sections involves analytically cutting the truss into sections and solving for static equilibrium for each section. CHAPTER 1. Visualizations are in the form of Java applets and HTML5 visuals. They are used to span greater distances and to carry larger loads than Read moreabout 500 lb. For example, find the force in member EF: Read Examples 6.2 and 6.3 from the book. 500 lb.$F_{BE} = 150.78 \, \text{ kN tension}$answer,$F_{BC} = 120 \, \text{ kN compression}$answer,$F_{DE} = 64 \, \text{ kN tension}\$ answer, Method of Sections | Analysis of Simple Trusses, Method of Joints | Analysis of Simple Trusses, Problem 417 - Roof Truss by Method of Sections, Problem 418 - Warren Truss by Method of Sections, Problem 419 - Warren Truss by Method of Sections, Problem 420 - Howe Truss by Method of Sections, Problem 421 - Cantilever Truss by Method of Sections, Problem 422 - Right-triangular Truss by Method of Sections, Problem 423 - Howe Roof Truss by Method of Sections, Problem 424 - Method of Joints Checked by Method of Sections, Problem 425 - Fink Truss by Method of Sections, Problem 426 - Fink Truss by Method of Sections, Problem 427 - Interior Members of Nacelle Truss by Method of Sections, Problem 428 - Howe Truss by Method of Sections, Problem 429 - Cantilever Truss by Method of Sections, Problem 430 - Parker Truss by Method of Sections, Problem 431 - Members in the Third Panel of a Parker Truss, Problem 432 - Force in Members of a Truss by Method of Sections, Problem 433 - Scissors Truss by Method of Sections, Problem 434 - Scissors Truss by Method of Sections, Problem 435 - Transmission Tower by Method of Sections, Problem 436 - Howe Truss With Counter Braces, Problem 437 - Truss With Counter Diagonals, Problem 438 - Truss With Redundant Members, Method of Members | Frames Containing Three-Force Members. Since truss members are subjected to only tensile or compressive 4.2 Maximization Problems (Continued) Example 4: Solve using the Simplex Method Kool T-Dogg is ready to hit the road and go on tour. on scalar ordinary and partial we focus onlinearrL 0000001891 00000 n Unit 18 Trusses: Method of Joints Frame 18-1 *Introduction A truss is a structure composed of several members joined at their ends so as to form a rigid body. This will give us the boundary conditions we need to progress in solving the structure. In situations where we need to find the internal forces only in a few specific members of a truss , the method of sections is more appropriate. 0000001370 00000 n 0000001913 00000 n Useful solutions for standard problems Preface Modelling is a key part of design. 0000002277 00000 n 0000020262 00000 n THE METHOD OF SECTIONS In the method of sections, a truss is divided into two parts by taking an imaginary “cut” (shown here as a-a) through the truss. Unit 19 Trusses: Method of Sections Frame 19-1 *Introduction In the preceding unit you learned some general facts about trusses as well as a method of solution called the "Method of Joints." The method of sections is often utilized when we want to know the forces in just a few CS Topics covered : Greedy … Learning outcome 1.Finding the graphical solution to the linear programming model Graphical Method of solving Linear Programming Problems Introduction Dear students, during the
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# On the proof of the hamiltonian flow box theorem
The hamiltonian flow box theorem, as stated in Abraham and Marsden's Foundations of Mechanics, says that:
Given an hamiltonian system $(M,\omega,h)$ with $dh(x_0)\neq 0$ for some $x_0$ in $M$, there is a symplectic chart $(U,\phi)$ on $M$ centered at $x_0$ such that $\phi_{\ast}h(x)=h(x_0)+\omega_0(\phi_{\ast}X_h(x_0),x)$, where $\omega_0$ is the canonical symplectic form.
*Question:*I know some different proofs of this theorem, but I would know if, at your knowledge, in the literature there is a proof which uses the Moser's trick as in the proof of the Darboux' theorem.
In Abraham and Marsden there is a proof using the contact structure associated to the symplectic one and its canonical transormations. I know even that it has an extension in a theorem of Cartan which says: Given a $2n$-dimensional symplectic manifold $(M,\omega)$, it is possible to extend to a system of symplectic coordinates on $(M,\omega)$ any set of local functions $f_1,\ldots,f_k,g_1,\ldots,g_l$ on $M$ such that $f_1,\ldots,f_k$ are independent and in involution, $g_1,\ldots,g_l$ are independent and in involution, and $\{f_i,g_j\}=\delta_{ij}$ for any $i,j$.
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My guess would be that you can find such a proof in the literature, since the Moser trick is such a powerful tool, though I don't know where.
Instead let me sketch a proof of the fact that any two Hamiltonian systems $(M_i,\omega_i,h_i)$ are locally isomorphic around non deg. points $x_i\in M_i, i=0,1$ using the Moser trick. That's the statement I see behind the flow box theorem.
First as usal one proves the linearized fact (i.e. any two $2n$ dim vector spaces $V_i$ supplied with non-degenerate two-forms $\omega_i$ and non degenerate one forms $v_i, i=0,1$ are isomorphic). Using this one constructs a local diffeomorphism between $(M_0,\omega_0,h_0)$ and $(M_1,\omega_1,h_1)$ satisfying:
1. point $x_0$ goes to $x_1$
2. the symplectic forms coincide on the above points and
3. the Hamiltonian function $h_0$ goes to $h_1$.
Now we have a manifold $M$ with two symplectic forms $\omega_0,\omega_1$ coinciding at $x\in M$ and one function $h$ which is nondegenerate at $x$. Next you try the usual Moser trick to morph $\omega_1$ to $\omega_2$ with the flow of a time dependent vector field $X_t$, imposing the additional requirement that $X_t$ preserve the function $h$. I.e. $L_{X_t}h=0$. Hence $X_t$ should lie in the $2n-1$ dimensional distribution $\ker dh$.
At some point in the Moser trick one chooses a one-form $\alpha$ such that $d\alpha=\omega_1-\omega_0$, and here we have the freedom to fullfill the additional restriction since we can add any closed one form $df$ to $\alpha$. We want the result $\alpha+df$ to lie in the $2n-1$ dimensional subspace of one-forms satisfyinig $i_{Y_t}\alpha'=0$, where $Y_t$ denotes the Hamiltonian vector field associated to $h$ w.r.t. the symplectic structure $\omega_t$. This can always be achieved since $Y_t$ is non deg and you are solving the equation $Y_t(f)=g_t$ where $g_t=-i_{Y_t}\alpha$.
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Dear Michael, Thank you very much. Your answer is very enlightening and properly what I was searching for. I have tried to read it careful. In order to express you my appreciation I have posted an answer in which I write what I have understood with just a slight modification about your condition 3. – Giuseppe Tortorella Apr 14 '11 at 20:59
You're welcome! I'll try to explain how i though of point 3 as soon as I have a moment. – Michael Bächtold Apr 15 '11 at 6:19
Warning: I have posted this as an answer and not as comment, not to gain in reputation, but just in order to have enough space to write to Michael what I had understood of his answer that has been very much useful to me.
Let $(M_0,\omega_1)$ and $(M_1,\omega_1)$ be symplectic manifolds of the same dimension. If $h_i$ is a smooth function on $M_i$ with $dh_i(x_i)\neq 0$ for some $x_i\in M_i$, and $i=0,1$, then there exists a local diffeomorphism $\phi$ from $M_0$ to $M_1$ such that $\phi(x_0)=x_1, \phi_{\ast}\omega_0=\omega_1$, and $d\phi_{\ast}h_0=dh_1$.
For the result from linear algebra reported in Michael's answer, there a local diffeomorphism $\psi$ from $M_0$ to $M_1$ such that $\psi(x_0)=x_1, \psi_{\ast}\omega_0(x_1)=\omega_1(x_1)$, and $d\psi_{\ast}h_0(x_1)=dh_1(x_1)$.
So we have now a manifold $M$ with symplectic forms $\Omega_0$ and $\Omega_1$ coinciding at a point $x_0$, and smooth regular functions $H_0$ and $H_1$ with $dH_0(x_0)=dH_1(x_1)$. With no loss of generality we can assume also $H_0(x_0)=H_1(x_0)$.
Let us introduce the following time dependent forms on $M$:
$H_t=H_0+t\tilde{H}=H_0+t(H_1-H_0)$ and $\Omega_t=\Omega_0+t\tilde{\Omega}=\Omega_0+t(\Omega_1-\Omega_0)$.
In order to construct the required local diffeomorphism using the Moser's trick, we need a time dependent local vector field $X_t$ around $x_0$ satisfying: $di_{X_t}\Omega_t+\tilde{\Omega}=0$, $i_{X_t}dH_t+\tilde{H}=0$, $X_t(x_0)=0$, for $t\in$0,1$$. Really the third condition follows from the second one because $\tilde{H}(x_0)=0$ and $H_t$ is regular.
Let $\alpha$ be a local primitive of $\tilde{\Omega}$ vanishing at $x_0$. The first condition becomes $i_{X_t}\Omega_t=-\alpha+df_t$, and determines a unique $X_t$ for each smooth function $f=\{f_t\}_t$ on a neighborhood of $M\times$0,1$$.
Finally the second condition becomes the following one only on $f=\{f_t\}_t$: $\mathcal{L}(Y_t).(f_t)=g_t\equiv\tilde{H}-i_{Y_t}\alpha$. Where $Y_t$ is the Hamiltonian vector field corresponding to $H_t$ w.r.t. $\Omega_t$.
A solution for this equation $\mathcal{L}(Y_t+0\frac{\partial}{\partial t}).f=g$ could be constructed using the method of characterics, considering that $Y\equiv Y_t+0\frac{\partial}{\partial t}$ is non singular because such is $dH_t$.
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That seems absolutely right. – Michael Bächtold Apr 15 '11 at 6:14
@Michael: If the final step invoking the method of characteristic is correct, then you could also manage the case when $H$ is replaced by a set of functions that are independent and in involution. So you take in account an hamiltonian systems with a set of its first integral. – Giuseppe Tortorella Apr 15 '11 at 6:25
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# Mixed-effects regression
Martijn Wieling
University of Groningen
## Introduction
• Consider the following situation (taken from Clark, 1973):
• Mr. A and Mrs. B study reading latencies of verbs and nouns
• Each randomly selects 20 words and tests 50 subjects
• Mr. A finds (using a sign test) verbs to have faster responses
• Mrs. B finds nouns to have faster responses
• How is this possible?
## The language-as-fixed-effect fallacy
• The problem is that Mr. A and Mrs. B disregard the (huge) variability in the words
• Mr. A included a difficult noun, but Mrs. B included a difficult verb
• Their set of words does not constitute the complete population of nouns and verbs, therefore their results are limited to their words
• This is known as the language-as-fixed-effect fallacy (LAFEF)
• Fixed-effect factors have repeatable and a small number of levels
• Word is a random-effect factor (a non-repeatable random sample from a larger population)
## Why linguists are not always good statisticians
• LAFEF occurs frequently in linguistic research until the 1970's
• Many reported significant results are wrong (the method is anti-conservative)!
• Clark (1973) combined a by-subject ($$F_1$$) analysis and by-item ($$F_2$$) analysis in measure min F'
• Results are significant and generalizable across subjects and items when min F' is significant
• Unfortunately many researchers (>50%!) incorrectly interpreted this study and may report wrong results (Raaijmakers et al., 1999)
• E.g., they only use $$F_1$$ and $$F_2$$ and not min F' or they use $$F_2$$ while unneccesary (e.g., counterbalanced design)
## Our problems solved...
• Apparently, analyzing this type of data is difficult...
• Fortunately, using mixed-effects regression models solves these problems!
• The method is easier than using the approach of Clark (1973)
• Results can be generalized across subjects and items
• Mixed-effects models are robust to missing data (Baayen, 2008, p. 266)
• We can easily test if it is necessary to treat item as a random effect
• No balanced design necessary (as in repeated-measures ANOVA)!
• But first some words about regression...
## Recap: multiple regression (1)
• Multiple regression: predict one numerical variable on the basis of other independent variables (numerical or categorical)
• We can write a regression formula as $$y = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \epsilon$$
• $$y$$: (value of the) dependent variable
• $$x_i$$: (value of the) predictor
• $$\beta_0$$: intercept, value of $$y$$ when all $$x_i = 0$$
• $$\beta_i$$: slope, change in $$y$$ when the value of $$x_i$$ increases with 1
• $$\epsilon$$: residuals, difference between observed values and predicted (fitted) values
## Recap: multiple regression (2)
• Factorial predictors are (automatically) represented by binary-valued predictors: $$x_i = 0$$ (reference level) or $$x_i = 1$$ (alternative level)
• Factor with $$n$$ levels: $$n-1$$ binary predictors
• Interpretation of factorial $$\beta_i$$: change from reference to alternative level
• Example of regression formula:
• Predict the reaction time of a subject on the basis of word frequency, word length, and subject age: RT = 200 - 5WF + 3WL + 10SA
## Mixed-effects regression modeling: introduction
• Mixed-effects regression distinguishes fixed effects and random-effect factors
• Fixed effects:
• All numerical predictors
• Factorial predictors with a repeatable and small number of levels (e.g., Gender)
• Random-effect factors:
• Only factorial predictors!
• Levels are a non-repeatable random sample from a larger population
• Generally a large number of levels (e.g., Subject, Item)
## What are random-effects factors?
• Random-effect factors are factors which are likely to introduce systematic variation (here: subject and item)
• Some subjects have a slow response (RT), while others are fast
= Random Intercept for Subject (i.e. $$\beta_0$$ varies per subject)
• Some items are easy to recognize, others hard
= Random Intercept for Item (i.e. $$\beta_0$$ varies per item)
• The effect of item frequency on RT might be higher for one subject than another: e.g., non-native participants might benefit more from frequent words than native participants
= Random Slope for Item Frequency per Subject (i.e. $$\beta_{\textrm{WF}}$$ varies per subject)
• The effect of subject age on RT might be different for one item than another: e.g., modern words might be recognized faster by younger participants
= Random Slope for Subject Age per Item (i.e. $$\beta_{\textrm{SA}}$$ varies per item)
• Note that it is essential to test for random slopes!
## Random slopes are necessary!
Estimate Std. Error t value Pr(>|t|)
Linear regression DistOrigin -6.418e-05 1.808e-06 -35.49 <2e-16
+ Random intercepts DistOrigin -2.224e-05 6.863e-06 -3.240 <0.001
+ Random slopes DistOrigin -1.478e-05 1.519e-05 -0.973 n.s.
(This example is explained at the HLP/Jaeger lab blog)
## Modeling the variance structure
• Mixed-effects regression allow us to use random intercepts and slopes (i.e. adjustments to the population intercept and slopes) to include the variance structure in our data
• Parsimony: a single parameter (standard deviation) models this variation for every random slope or intercept (a normal distribution with mean 0 is assumed)
• The slope and intercept adjustments are Best Linear Unbiased Predictors
• Model comparison determines the inclusion of random intercepts and slopes
• Mixed-effects regression is only required when each level of the random-effect factor has multiple observations (e.g., participants respond to multiple items)
## Specific models for every observation
• RT = 200 - 5WF + 3WL + 10SA (general model)
• The intercepts and slopes may vary (according to the estimated variation for each parameter) and this influences the word- and subject-specific values
• RT = 400 - 5WF + 3WL - 2SA (word: scythe)
• RT = 300 - 5WF + 3WL + 15SA (word: twitter)
• RT = 300 - 7WF + 3WL + 10SA (subject: non-native)
• RT = 150 - 5WF + 3WL + 10SA (subject: fast)
• And it is not hard to use!
• lmer( RT ~ WF + WL + SA + (1+SA|Wrd) + (1+WF|Subj) )
• (lmer automatically discovers random-effects structure: nested/crossed)
## Random slopes and intercepts may be (cor)related
• For example:
Subject Intercept WF slope
S1 525 -2
S2 400 -1
S3 500 -2
S4 550 -3
S5 450 -2
S6 600 -4
S7 300 0
## BLUPs of lmer do not suffer from shrinkage
• The BLUPS (i.e. adjustment to the model estimates per item/participant) are close to the real adjustments, as lmer takes into account regression towards the mean (fast subjects will be slower next time, and slow subjects will be faster) thereby avoiding overfitting and improving prediction - see Efron & Morris (1977)
## Center your variables (i.e. subtract the mean)!
• Otherwise random slopes and intercepts may show a spurious correlation
• Also helps the interpretation of interactions (see this lecture)
## Mixed-effects regression assumptions
• Independent observations within each level of the random-effect factor
• Relation between dependent and independent variables linear
• No strong multicollinearity
• Residuals are not autocorrelated
• Homoscedasticity of variance in residuals
• Residuals are normally distributed
• (Similar assumptions as for regression)
## Model criticism
• Check the distribution of residuals: if not normally distributed and/or heteroscedastic residuals then transform dependent variable or use generalized linear mixed-effects regression modeling
• Check outlier characteristics and refit the model when large outliers are excluded to verify that your effects are not 'carried' by these outliers
## Model selection II
• My stepwise variable-selection procedure (for exploratory analysis):
• Include random intercepts
• Add other potential explanatory variables one-by-one
• Insignificant predictors are dropped
• Test predictors for inclusion which were excluded at an earlier stage
• Test possible interactions (don't make it too complex)
• Try to break the model by including significant predictors as random slopes
• Only choose a more complex model if supported by model comparison
## Model selection III
• For a hypothesis-driven analysis, stepwise selection is problematic
• Solutions:
• Careful specification of potential a priori models lining up with the hypotheses (including optimal random-effects structure) and evaluating only these models
• This may be followed by an exploratory procedure
• Validating a stepwise procedure via cross validation (e.g., bootstrap analysis)
## Case study: influence of alcohol on L1 and L2
• Reported in Wieling et al. (2019) and Offrede et al. (2020)
• Assess influence of alcohol (BAC) on L1 (clarity) vs. L2 (nativelikeness) ratings
• Prediction: higher BAC has negative effect on L1, but positive effect on L2 nativelikeness (based on Renner et al., 2018)
• ~80 recordings from native Dutch speakers included (all BACs < 0.8, no drugs)
• Dutch ratings were given by >100 native (sober) Dutch speakers (at Lowlands)
• English ratings were given by >100 native American English speakers (online)
• Dependent variable is $$z$$-transformed rating (5-point Likert scale)
• Numerical variables are centered (not $$z$$-transformed)
## Dataset
load("lls.rda")
# SID BAC Gender BirthYear L2cnt SelfEN LivedEN L2anxiety Edu LID Lang Rating
# 1 S0045188-17 0.392 F -5.66 -1.04 0.463 Y 0.357 0.817 L0637009 NL 0.946
# 2 S0045188-17 0.392 F -5.66 -1.04 0.463 Y 0.357 0.817 L196 EN 0.330
# 3 S0045188-17 0.392 F -5.66 -1.04 0.463 Y 0.357 0.817 L86 EN -0.298
# 4 S0045188-17 0.392 F -5.66 -1.04 0.463 Y 0.357 0.817 L0614758 NL 0.325
# 5 S0045188-17 0.392 F -5.66 -1.04 0.463 Y 0.357 0.817 L220 EN -0.435
# 6 S0045188-17 0.392 F -5.66 -1.04 0.463 Y 0.357 0.817 L225 EN -0.239
## Fitting our first model
#### (fitted in R version 4.0.5 (2021-03-31), lme4 version 1.1.27)
library(lme4)
m <- lmer(Rating ~ BAC + (1 | SID) + (1 | LID), data = lls) # does not converge
# boundary (singular) fit: see ?isSingular
summary(m)$coef # show fixed effects # Estimate Std. Error t value # (Intercept) 0.144 0.0638 2.252 # BAC -0.221 0.3262 -0.677 summary(m)$varcor # show random-effects part only: no variability per listener (due to z-scaling)
# Groups Name Std.Dev.
# LID (Intercept) 0.000
# SID (Intercept) 0.560
# Residual 0.738
## Evaluating our hypothesis (1)
#### (note: random intercept for LID excluded)
m1 <- lmer(Rating ~ Lang * BAC + (1 | SID), data = lls)
• This model represents our hypothesis test (but likely without the correct random-effects structure)
## Results (likely overestimating significance)
summary(m1, cor = F) # suppress expected correlation of regression coefficients; Note: |t| > 2 => p < .05 (N > 100)
# Linear mixed model fit by REML ['lmerMod']
# Formula: Rating ~ Lang * BAC + (1 | SID)
# Data: lls
#
# REML criterion at convergence: 5853
#
# Scaled residuals:
# Min 1Q Median 3Q Max
# -3.668 -0.647 0.032 0.707 3.187
#
# Random effects:
# Groups Name Variance Std.Dev.
# SID (Intercept) 0.305 0.552
# Residual 0.533 0.730
# Number of obs: 2541, groups: SID, 82
#
# Fixed effects:
# Estimate Std. Error t value
# (Intercept) 0.0883 0.0635 1.39
# LangNL 0.2430 0.0358 6.78
# BAC -0.0592 0.3255 -0.18
# LangNL:BAC -0.7286 0.1938 -3.76
## Visualization helps interpretation
library(visreg)
visreg(m1, "BAC", by = "Lang", overlay = T, xlab = "BAC (centered)", ylab = "Rating")
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## In a parallelogram opposite sides are 2x+3 and 5x-6.So, Find x?
Question
In a parallelogram opposite sides are 2x+3 and 5x-6.So, Find x?
in progress 0
3 weeks 2021-09-09T18:02:31+00:00 2 Answers 0 views 0
x = 3
Step-by-step explanation:
We know that opposite sides of a parallelogram are parallel and equal.
Thus, 2x + 3 = 5x – 6
5x – 2x = 6 + 3
3x = 9
Sides are – 2x +3 and 5x-6
We know that
Opposite sides of a parallelogram are equal.
So, According to the condition,
2x+3=5x-6
or, 5x-2x=6+3
or, 3x = 9
Or, x=3.
Therefore, the value of x is 3 unit.
Step-by-step explanation:
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# Problem while decrypting Hill cipher
I have a plaintext "monday" and ciphertext "IKTIWM" and $$m=2$$. I want to find the key of the Hill cipher.
I made a matrix $$\begin{bmatrix} a_1 & a_2 \\ a_3 & a_4 \end{bmatrix}\begin{bmatrix} m \\ o \end{bmatrix} = \begin{bmatrix} I \\ K \end{bmatrix} \pmod{26}$$
$$X=\{\{m,o\},\{n,d\}\}$$, $$Y=\{\{I,K\},\{T,I\} \}$$, I want to find $$X \times K=Y$$.
I will multiply this equation with inverse($$X$$).
But for the modulo inverse you need $$gcd$$(determinant($$X), 26) =1$$ . Which is not happening here.
• I am making a matrix X={ {m,o}, {nd} },Y={ {I,K} ,{T,I} },I want to find X*K=Y; – Manoharsinh Rana Feb 1 at 11:20
• I edited it.I don't know how to write a matrix here. – Manoharsinh Rana Feb 1 at 11:23
• Hint: not all systems of 6 equations with 4 unknowns have a unique solution. Find them all. – fgrieu Feb 1 at 11:52
• these are the equations. 12a + 14b = 8 , 12c + 14d = 10 ,13a + 3d = 19 ,13c + 3d = 8 , 24b=22 , 24d= 12. I have replaced a1 with a , a2 with b , a3 with c , a4 with d. Can we solve them? – Manoharsinh Rana Feb 1 at 12:33
• you are right. But can you help me solve it? – Manoharsinh Rana Feb 1 at 12:55
$$\begin{bmatrix}7&2\\ 10& 20\end{bmatrix}, \begin{bmatrix}7&2\\ 23& 7\end{bmatrix}, \begin{bmatrix}20&15\\ 10& 20\end{bmatrix}, \begin{bmatrix}20&15\\ 23& 7\end{bmatrix}$$
are all the $$2 \times 2$$ matrices over $$\mathbb{Z}_{26}$$ would transform 'monday' to IKTIWM, the first and third have even determinant so are not invertible so the second or the fourth candidate encryption matrix is the correct one: invert them and check the rest of the text which is one is actually correct.
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Relational rather than hierarchical OO packaging?
Nothing stunning here, just a basic/simplistic question -
Assuming relational better than hierarchical, for "visibility" of things in languages, are there / could there be a successful approach to visibility of things which is relational, not hierarchical?
C++ has the "friend" visibility keyword, Java and C# don't seem to have a cross-cutting visibility tool. Does C++'s "friend" suffice, or does it have too much of a preference of hierarchical over relational, and/or does it have some land-mine drawbacks (I mean besides the standard chances of abuse)? I guess I was always just queasy with it e.g.
If encountering friend functions for the first time, you might feel slightly uneasy since they seem to violate encapsulation. This feeling may stem from the fact that a friend function is not strictly a member of the class.
By thinking of a friend function as part of the class’s public interface, you can get a better understanding of how friends work. From a design perspective, friends can be treated in a similar way to public member functions. The concept of a class interface can be extended from public members to include friend functions and friend classes.
Comment viewing options
I guess it depends how you use it
Another quote from the article:
Another reason for using a friend function is one of efficiency: directly accessing data members saves the overhead of using get/set members, if the compiler has not inlined these
Now if that does not invite violations of encapsulation...
Note that C++ has private and protected, so that is not as if hierarchical visibility would not allow for some encapsulation itself.
Classes are not always proper abstraction boundaries...
...often times, you want abstraction boundaries to not be congruent with (natural) classes.
There are many good reasons, in C++, to define free functions which are friends of a particular class--to assume that doing so violates modularity is only true if a) you assume that the class is the only proper abstraction boundary, and/or b) do something silly like letting somebody else specify the implementation of a friend. But if the class and the friend function are part of the same "module" (here in quotes as C++ doesn't have modules or packages), and developed and deployed in tandem, there is no abstraction violation.
The same is true in the opposite direction--there are many times that I want to grant access to a particular attribute in a class *only* to a few select methods; and want to prevent access to the attribute by any other class member (let alone an external client).
IMO, "protected" is often a bigger violation of abstraction than "friend" is.
b) do something silly like
b) do something silly like letting somebody else specify the implementation of a friend.
There is something to be said about keeping related code in the same source file ;)
Who knows who will revisit your code after you wrote it?
Or not use files at all
It always seemed to me that sorting code into files is a little odd. Especially since the ways that we need to access definitions are much more like searches than reading a book.
Visual Age
I think Visual Age IDEs used to show you a more fragment (by method) view of the source code. It was a little annoying because sometimes I really do just like to scroll through the whole file to try to get the gestalt of things, and because I'm relatively good at maps+directions so once I know the layout of a file I can find what I want by direct navigation rather than search.
Basically the idea of One Way To Do It when it comes to viewing code is probably wrong; it would be nice to enable many different views of it. Like, sometimes I want to see the subclass as if it had all the parent class code inlined in it.
Eiffel does that sort of thing...
IIRC, the Eiffel tools from Betrand Meyer's company allow four views of a class--the product of two orthogonal parameters:
* Whether all features (including private one) were shown, or the public interface only
and
* Whether or not the "resulting" class (containing all base class features not overridden or renamed), or just the class definition itself (showing only changes from the base classes)
Of course, such a feature is not technically limited to Eiffel.
Thanks for the notes
It seems to me that, even with things like "friend", C++/C#/Java start off with and mainly stick to the hierarchical meme. Are there approaches to code visibility which start with something else instead? (e.g. relational, network.)
Hierarchies...
In many OO languages, there arenot one but two hierarchies present: The inheritance hierarchy (which is often not a hierarchy but a DAG, in OO languages with multiple inheritance), and the scoping hierarchy (which remains tree-structured FTMP; though languages with delegation may also exhibit DAG-structure here as well).
I'm not aware of any languages in which visibility is based on the relational model. I'm sufficiently unfamiliar with "network databases" (other than RDBMS dogma that they are bad) to comment there; though capabilities (you can see things that you hold references to; there is no global ambient authority available through queries over a namespace or a set of relations) might suffice for an example of a "network model".
I'm not sure that a full-blown relational model (or a Prolog-style inference model, etc.) would be all that beneficial--such a thing might well be harder to specify and/or make secure rather than easier. What is the use case for such a thing?
Use case?
Partly just a thought experiment. But, there are times when I want some part of something exposed to some part of something else, and in Java/C# I guess it would generally require either opening up everything, or making various adapter classes. Neither is an attractive option.
On the other hand, I do wonder if a relational visiblity system would simply lead to a rat's nest of completely non-un-tangle-able code.
Friend in C++
lets you expose all of something to part of something else; though the "something else" is limited to either an entire class (which is often too coarse), or a list of specific methods within that class.
Friend does not, however, let you open up only part of a class's implementation to the designated friend--any class/method declared a friend has access to the entire stinking class
You can do that, albeit
You can do that, albeit clumsily, in Java by resorting to the (default) "package" visibility.
Packages
I always stressed to my students that Ada uses the package construct for visibility, and tagged types for inheritance thus cutting the Gordian knot tying these two aspects of OOP in class based OOPLs.
In Ada, for example, two tagged types (classes) can be defined in the same package, and thus visibility exists between the implementations. If you don't want that (as is usually the case), simply define the types in separate packages.
Traits
There are many known problems with using classes as the unit of code reuse and abstraction. Many such problems are solved by traits, and I believe they solve this "friend functions" problem as well.
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# The latest developments in energy storage technology
1. Jan 28, 2017
### Trainee Engineering
Hi all,
I'm interested in the latest development in battery technology. as of now, from what I understand, the most advanced tech in energy storage (battery) is created by Tesla, the PowerWall. but, even that, can only store 14kWh per unit. my house consumption is about 10kWh per day. my question is, what are the type of batteries out there available to the public? I'm only concerned of these things:
1. capacity --> I need to store 2-3 weeks worth of energy (in case of blackouts), so, it's somewhere around 140kWh - 210 kWh. if possible, I need it to be less than 10 units (so each unit is around 21kWh)
2. durability --> must be able to store huge amount of energy for a long period of time without dissipating. Now, it wont be charged and discharging frequently. charged only after a blackout, and wont be discharging until the next blackout, so storing for long time
3. warranty --> if possible, above 10 years warranty that energy leakage is less than 5%
price is not an issue. what are my options?
what's the latest type of battery suited for this type of usage?
thanks
2. Jan 28, 2017
### oz93666
One 14 kWh Powerwall battery $5,500 ... that means you need to spend around$80,000 every ten years ! and there are many loopholes in the 10 year warranty.
There's nothing special in these tesla packs ... just Lithium ion batteries .... someone with moderate technical knowledge could make an equivalent for a quarter of the cost .
You could buy 10kw of solar panels for $5000 ...this would generate on average 30kwHrs /day three times your consumption ... to cover you for cloudy days , and night time consumption a modest sized battery ....the whole setup much less than$10,000...
Li -ion are the best batteries available for what you want (long life compactness) ...lead acid are still the most cost effective.
Last edited: Jan 29, 2017
3. Jan 29, 2017
### Trainee Engineering
ok, when you put it that way...
for solar panel? what's the price per wp now?
4. Jan 29, 2017
### oz93666
Alibaba $0.4 /W or less delivered to a port of your choosing ... Solar industry is a big rip off , companies make a big mark up for re selling these panels and putting them on your roof or land ... I bought 5KW about 4 years ago at$0.5 /W , alibaba ... I had to pick them up at the port , no taxes where I am ... prices have dropped a lot since then ... Philippines government just bought a GW @ $0.2 /W including inverters !!! ... panels are just sand and a bit of silver ...dirt cheap to make .... It depends if you want to do it all yourself , if you do , 10kw with off grid inverter and batteries less than$10K ...
If you get a company to install everything ...no idea ... will depend on country , but if they give you a high price you can tell them the price of panels and inverters on alibaba don't let companies tell you their panels are better quality , they're all the same , the panels will still deliver 75% even after 25 years , inverters, many different types , best to buy good quality one .... Any schoolboy could wire it up for you , ideally put the panels on open land , easier to clean than on a roff.
Last edited: Jan 29, 2017
5. Jan 29, 2017
Staff Emeritus
If you regularly have blackouts lasting weeks, I wouldn't be thinking about an $80,000 storage device. I'd be thinking about a$5000 gas generator.
6. Jan 29, 2017
I found a 700 W generator for less than $150, and 2 kW for less than$1000. An additional battery for peak loads (cooking) and smoothing out the gas generator load should be around $1000. If you want to avoid CO2 emissions, you can buy a gas generator plus a small-scale photovoltaic/battery system that covers most of your electricity consumption but does not have to cover it all. 7. Jan 29, 2017 ### Trainee Engineering interesting, first time hearing about gas generator. so, how much kwH per kg of gas? anybody has experience with these gas generators? or perhaps can point me to some links? I dont mind CO2 emissions since this is not gonna be used frequently, only during blackouts, which usually lasts less than 3 weeks in a year (in total) 8. Jan 29, 2017 ### Trainee Engineering after some researching, both diesel and gas generators will cost more to generate 1 kWh compared to electricity price from the grid (I guess that's to be expected). now, between gas, gasoline and diesel generator, what's the most recommended? my requirements: 1. be able to run in long period of time (24 hours non stop if possible) 2. for household use (around 2kW) 3. not too noisy (residential area) 4. efficiency (most kwH generated per unit of$$resource needed) not too worried about air pollution since I'll be putting the generator on open space on 3rd floor Thanks 9. Jan 29, 2017 ### mfb ### Staff: Mentor ~2 kWh/liter look realistic. A bit more per kilogram as it is lighter than water. At ~$1/liter that makes 50 cents/kWh (more expensive than from the grid - not surprising). 10kWh/day over 3 weeks is 210 kWh, or ~$100 in fuel costs for a single 3-week blackout. If you combine a 2 kW generator with a small battery, you can run it at 50% load for 10 hours a day - during the night the battery is sufficient to power the fridge and so on. Just for longer blackouts, the gas generator is probably the cheapest option. Photovoltaics would also deliver energy the rest of the year, of course, that should be taken into account. 10. Jan 29, 2017 ### Vanadium 50 Staff Emeritus Of course. But like you say, it's 3 weeks out of the year. And during those 3 weeks you don't have a grid option. You need to think about the load, and the source of fuel. If you have natural gas to your house already, and if it is still delivered during blackouts. that's a much simpler option than storing fuel. 11. Jan 29, 2017 ### mheslep Without subsidies, all alternatives cost more than grid power in the mainland US. 12. Jan 29, 2017 ### mheslep Installation (connection to the home electrical service box with a switchover) and a natural gas utility connection is at least$1500.
13. Jan 29, 2017
### rbelli1
One of the new quiet Honda inverter units is a good option as they are surprisingly (shockingly?) quiet. They are pricey but worth the money. There are less expensive (by about half) alternatives with equivalent specs but I have no experience or other knowledge about them.
Is that for an auto-switch-over or manual?
BoB
14. Jan 30, 2017
### mheslep
Auto. I think the equipment (transfer switch) cost is relatively inexpensive compared to the labor. Most of the cost is labor, i.e. licensed electrician.
15. Jan 30, 2017
### sophiecentaur
There's the rub. In the UK the regs are very tight and you need to be part of 'the system' and be part of the feed in tariff arrangement. The approved installers charge a fortune and the government incentives for green energy are reducing.
Any alternative approach to a 230V standby supply in UK homes is really not on the cards, I reckon.
16. Jan 30, 2017
### Staff: Mentor
Three weeks of grid outage per year in the UK? I doubt that. We are probably talking about a developing country.
17. Jan 30, 2017
### mheslep
Or, six weeks of half power from a solar array cloaked in a UK would require the three weeks of battery backup.
18. Jan 31, 2017
### CWatters
https://blogs.scientificamerican.co...me-can-increase-energy-consumption-emissions/
It seems to be suggesting that if you send excess solar PV to the grid then all of the electricity generated is used to reducing C02 emissions. Whereas if you store it in a battery and use it yourself later then some is lost in the charging efficiency. So overall it's greener NOT to use a solar PV backup battery.
19. Jan 31, 2017
### Staff: Mentor
Depends on where you are. As an example, on a very sunny day in Germany photovoltaics produces so much that there are no additional CO2 savings - no fossil fuel power plant gets shut down if you dump even more power from photovoltaics into the grid.
20. Jan 31, 2017
### Staff: Mentor
Sure, but that is a different question than what you were asking in the OP. In the OP you asked about emergency energy storage, not full time grid replacement (which would be illegal with a generator and only semi-legal with solar...and, by the way, solar is generally illegal to use for emergency power).
The comparison of a generator to battery backup is a no-brainer: your requirement is for about 30 gallons of gas, costing about $80 (or$800 over 10 years if you use it once a year) versus a battery pack costing \$80,000.
Another option is natural gas/methane if you have a connection to that already. It is much cheaper than gasoline (roughly 1/3 the cost) and is not subject to outages like electricity is -- it would pretty much only go down during a zombie apocalypse or giant meteor impact.
Last edited: Jan 31, 2017
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#### Sentence Examples
• Similarly if we have F more unknowns than we have equations to determine them, we must fix arbitrarily F coordinates before we fix the state of the whole system.
• The number F is called the number of degrees of freedom of the system, and is measured by the excess of the number of unknowns over the number of variables.
• But questions remained—the big three unknowns of who, why, and when.
• Sometimes his x has to do duty twice, for different unknowns, in one problem.
• He first divides by the factor x -x', reducing it to the degree m - I in both x and x' where m>n; he then forms m equations by equating to zero the coefficients of the various powers of x'; these equations involve the m powers xo, x, - of x, and regarding these as the unknowns of a system of linear equations the resultant is reached in the form of a determinant of order m.
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## Fearnley, John and Savani, Rahul - The Complexity of All-switches Strategy Improvement
lmcs:3794 - Logical Methods in Computer Science, October 31, 2018, Volume 14, Issue 4
The Complexity of All-switches Strategy Improvement
Authors: Fearnley, John and Savani, Rahul
Strategy improvement is a widely-used and well-studied class of algorithms for solving graph-based infinite games. These algorithms are parameterized by a switching rule, and one of the most natural rules is "all switches" which switches as many edges as possible in each iteration. Continuing a recent line of work, we study all-switches strategy improvement from the perspective of computational complexity. We consider two natural decision problems, both of which have as input a game $G$, a starting strategy $s$, and an edge $e$. The problems are: 1.) The edge switch problem, namely, is the edge $e$ ever switched by all-switches strategy improvement when it is started from $s$ on game $G$? 2.) The optimal strategy problem, namely, is the edge $e$ used in the final strategy that is found by strategy improvement when it is started from $s$ on game $G$? We show $\mathtt{PSPACE}$-completeness of the edge switch problem and optimal strategy problem for the following settings: Parity games with the discrete strategy improvement algorithm of V\"oge and Jurdzi\'nski; mean-payoff games with the gain-bias algorithm [14,37]; and discounted-payoff games and simple stochastic games with their standard strategy improvement algorithms. We also show $\mathtt{PSPACE}$-completeness of an analogous problem to edge switch for the bottom-antipodal algorithm for finding the sink of an Acyclic Unique Sink Orientation on a cube.
Source : oai:arXiv.org:1507.04500
DOI : 10.23638/LMCS-14(4:9)2018
Volume: Volume 14, Issue 4
Published on: October 31, 2018
Submitted on: July 18, 2017
Keywords: Computer Science - Data Structures and Algorithms,Computer Science - Computational Complexity,Computer Science - Computer Science and Game Theory,Computer Science - Logic in Computer Science
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# Iterate a user defined function to load Table or ParametricPlot function
Posted 26 days ago
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2 Replies
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1 Total Likes
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I would like to plot series of points on a graph starting at a given point and continuing to the point of greatest magnitude. This could be accomplished by either creating a Table[] of points and then plotting the points, or embedding ParametricPlot[] in the Show[] function. However using either Table[]or ParametricPlot [] functions would require iterating a user defined gradient function and then using the output of the previous iteration as the input to the subsequent iterations and I am not sure how to do that in Mathematica and would appreciate guidance in performing the iterations (see attached notebook).Additional if my whole approach is incorrect, I would be open to any and all suggestions because I am always interested it trying a better method to visualize my calculations. Thank you so much for your assistance. Attachments:
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Posted 26 days ago
Hi Mitchell, iterating a user defined gradient function and then using the output of the previous iteration as the input to the subsequent iterations Nest and related functions can do this e.g. NestList[Normalize@df[Sequence @@ #] &, {3, 4}, 3] (* {{3, 4}, {7/25, -(24/25)}, {527/625, 336/625}, {-(164833/390625), -(354144/390625)}} *) In the attached notebook P1, P2, P3, P4 ignore the sign of the previous term. If that is what you want, just add an Abs.
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home Forums # Technical Support qtFuzzyLite and rules
This topic contains 1 reply, has 2 voices, and was last updated by Juan Rada-Vilela 3 years, 1 month ago.
Viewing 2 posts - 1 through 2 (of 2 total)
• Author
Posts
• #825
jcredberry
Participant
Hi Juan,
Before anything else, I would like to Thank for your efforts, they seem in the right direction.
I was trying to use qtFuzzyLite and couldn’t find where rules could be defined. I even loaded one of the examples, and while at first hand the rules were loaded, after pressing the “Process Rules” button, they disappeared. Then, the following message was presented for all the rules:
if Ambient is DARK then Power is ?
if Ambient is MEDIUM then Power is ?
if Ambient is BRIGHT then Power is ?
#---------------------------------------------
# Total rules: 3. Good Rules: 0. Bad Rules: 3.
Any idea on what is happening here? Thanks in advance.
Julio Rojas-Mora
#826
Keymaster
Hi Julio,
The problem is that you did not press the process rules button, but rather the generate all possible rules.
The rules are defined in text, so you can edit the text in the window where the rules are, that is, where you found:
if Ambient is DARK then Power is ?
if Ambient is MEDIUM then Power is ?
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# Specific heat of selected metals7 0.6 0.5 1 0.3 0.20.4510.3850.2370.1310.2220.128100 150 Atomic mass (amu)200250
###### Question:
Specific heat of selected metals 7 0.6 0.5 1 0.3 0.2 0.451 0.385 0.237 0.131 0.222 0.128 100 150 Atomic mass (amu) 200 250
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Your desired sample size of 751, you find out that about 70% of PTs are female and about 40% of PTs are over age 50. You decide that you should take a stratified random sampling approach based on these two variables. Then determine the sample size of the 4 subgroups: (12 pts) a) Female PTs over...
##### E 15 of 39 Questions Assignment Score: Resources Give Up Feedback Try Again 1207/3900 O 12...
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##### If the general solution of = e y(2x _ 4) is dx =ey = 22 4x + c and y(6)=0,then y(8):bsJl siys | a_ In(21) b. In(36) C_ In(16)d. In(6)
If the general solution of = e y(2x _ 4) is dx =ey = 22 4x + c and y(6)=0,then y(8) :bsJl siys | a_ In(21) b. In(36) C_ In(16) d. In(6)...
##### [~/6 Points]DETAILSFind sin(a) and cos(8), tan(a) and cot(8), and sec(a) and csc(B).(a)sin(c) and cos(B)(b)tan(a) and cot(8)sec(a) and csc(B)Type here t0 search
[~/6 Points] DETAILS Find sin(a) and cos(8), tan(a) and cot(8), and sec(a) and csc(B). (a) sin(c) and cos(B) (b) tan(a) and cot(8) sec(a) and csc(B) Type here t0 search...
##### A bacterial isolate from a urine specimen was grown in culture, gram stained, and then tested...
A bacterial isolate from a urine specimen was grown in culture, gram stained, and then tested for its ability to ferment sugars and hydrolyze various subtrates. What approach to bacterial identification is this an example of? a.phenotypic b. antibiotic c. genetic...
##### Question 190.25 PointsTwc coins are tossed and one dice rolled_ Answer the following:What is the probability of [ having number 5 on the dice and atmost head?Note: Draw tree diagram to show all tne possible outcomes and write the sampl space in a sneet of paperto nelp you answering the question:0.0830.1250.1670.58Question 200.25 PointsIfP(A) 0.6 P(B) 0.4 and P(A and B) = 0.1, find P(A or B)Answer: Blank 1BlankAdd your answer
Question 19 0.25 Points Twc coins are tossed and one dice rolled_ Answer the following: What is the probability of [ having number 5 on the dice and atmost head? Note: Draw tree diagram to show all tne possible outcomes and write the sampl space in a sneet of paperto nelp you answering the question:...
##### Harrelson Company manufactures pizza sauce through two production departments: Cooking and Canning. In each process, materials...
Harrelson Company manufactures pizza sauce through two production departments: Cooking and Canning. In each process, materials and conversion costs are incurred evenly throughout the process. For the month of April, the work in process accounts show the following increases. Direct Materials Factory ...
##### Find two positive numbers with product 625 and whose sum is minimum:Enter your answers in increasing orderFirst number: NumberSecond Number: NumberIn a few sentences, please explain how you got _ your answers_
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##### Find the area of the surface generated when the given curve is revolved about the X-axis_ x3 3) y = on [1,.2] 4x
Find the area of the surface generated when the given curve is revolved about the X-axis_ x3 3) y = on [1,.2] 4x...
##### Laplace problem find solution lp need he
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##### An amusement park studied methods for decreasing the waiting time (in minutes) for its rides by...
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##### Suppose motor connected to a 140 V source draws 21.5 A when it first starts_(a) What is its resistance?What current does it draw at its normal operating speed when it develops a 100 V back emf?
Suppose motor connected to a 140 V source draws 21.5 A when it first starts_ (a) What is its resistance? What current does it draw at its normal operating speed when it develops a 100 V back emf?...
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##### Lab NotebookOnly complete the table for your successful titrations: Titration 1 Titration 2 Titration 3KHP (g)1.6521.6771802Buret before (mL)0.90.71.7Buret after (mL)26.628.230.7Volume used (mL) Concentration of NaOH (M)25.727.529Averageconcentration of NaOH (M)Calculation example
Lab Notebook Only complete the table for your successful titrations: Titration 1 Titration 2 Titration 3 KHP (g) 1.652 1.677 1802 Buret before (mL) 0.9 0.7 1.7 Buret after (mL) 26.6 28.2 30.7 Volume used (mL) Concentration of NaOH (M) 25.7 27.5 29 Average concentration of NaOH (M) Calculation exampl...
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# MLE for normal distribution with restrictive parameters
Suppose that $$X_1, . . . , X_n$$, $$n\geq 2$$, is a sample from a $$N(\mu,\sigma^2)$$ distribution. Suppose $$\mu$$ and $$\sigma^2$$ are both known to be nonnegative but otherwise unspecified. Now, I want to find the MLE of $$\mu$$ and $$\sigma^2$$. I have drawn the MLE for non-restrictive parameters but I am stuck on this one.
# Solution
Let $$\bar{x}$$ denote the sample mean:
$$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$$
The constrained maximum likelihood mean $$\hat{\mu}$$ and variance $$\hat{\sigma}^2$$ are:
$$\hat{\mu} = \left\{ \begin{array}{cl} \bar{x} & \bar{x} \ge 0 \\ 0 & \text{Otherwise} \\ \end{array} \right.$$
$$\hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^n (x_i – \hat{\mu})^2$$
That is, we simply take the sample mean and clip it to zero if it’s negative. Then, plug it into the usual expression for the (uncorrected) sample variance. I obtained these expressions by setting up the constrained optimization problem, then solving for the parameters that satisfy the KKT conditions, as described below.
# Derivation
### Objective function
Maximizing the likelihood is equivalent to minimizing the negative log likelihood $$L(\mu, \sigma^2)$$, which will be more convenient to work with:
$$L(\mu, \sigma^2) = -\sum_{i=1}^n \log \mathcal{N}(x_i \mid \mu, \sigma^2)$$
$$= \frac{n}{2} \log(2 \pi) + \frac{n}{2} \log(\sigma^2) + \frac{1}{2 \sigma^2} \sum_{i=1}^n (x_i-\mu)^2$$
We’ll also need its partial derivatives w.r.t. $$\mu$$ and $$\sigma^2$$:
$$\frac{\partial}{\partial \mu} L(\mu, \sigma^2) = \frac{n \mu}{\sigma^2} – \frac{1}{\sigma^2} \sum_{i=1}^n x_i$$
$$\frac{\partial}{\partial \sigma^2} L(\mu, \sigma^2) = \frac{n}{2 \sigma^2} – \frac{1}{2 \sigma^4} \sum_{i=1}^n (x_i-\mu)^2$$
### Optimization problem
The goal is to find the parameters $$\hat{\mu}$$ and $$\hat{\sigma}^2$$ that minimize the negative log likelihood, subject to a non-negativity constraint on the mean. The variance is non-negative by definition and the solution below turns out to automatically respect this constraint, so we don’t need to impose it explicitly. The optimization problem can be written as:
$$\hat{\mu}, \hat{\sigma}^2 = \arg \min_{\mu, \sigma^2} \ L(\mu, \sigma^2) \quad \text{s.t. } g(\mu, \sigma^2) \le 0$$
$$\text{where } \ g(\mu, \sigma^2) = -\mu$$
I’ve written the constraint this way to follow convention, which should hopefully make it easier to match this up with other discussions about constrained optimization. In our problem, this just amounts to the constraint $$\mu \ge 0$$.
### KKT conditions
If $$(\hat{\mu}, \hat{\sigma}^2)$$ is an optimal solution, there must exist a constant $$\lambda$$ such that the KKT conditions hold: 1) stationarity, 2) primal feasibility, 3) dual feasibility, and 4) complementary slackness. Furthermore, we have a convex loss function with a convex, continuously differentiable constraint. This implies that the KKT conditions are sufficient for optimality, so we can find the solution by solving for the parameters that satisfy these conditions.
Stationarity:
$$\frac{\partial}{\partial \mu} L(\hat{\mu}, \hat{\sigma}^2) + \lambda \frac{\partial}{\partial \mu} g(\hat{\mu}, \hat{\sigma}^2) = 0$$
$$\frac{\partial}{\partial \sigma^2} L(\hat{\mu}, \hat{\sigma}^2) + \lambda \frac{\partial}{\partial \sigma^2} g(\hat{\mu}, \hat{\sigma}^2) = 0$$
Plug in expressions for the derivatives and solve for the parameters:
$$\hat{\mu} = \frac{1}{n} \hat{\sigma}^2 \lambda + \frac{1}{n} \sum_{i=1}^n x_i \tag{1}$$
$$\hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^n (x_i-\hat{\mu})^2 \tag{2}$$
Primal feasibility:
$$g(\hat{\mu}, \hat{\sigma}^2) \le 0 \implies \hat{\mu} \ge 0$$
This just says the parameters must respect the constraints
Dual feasibility:
$$\lambda \ge 0$$
Complementary slackness:
$$\lambda g(\hat{\mu}, \hat{\sigma}^2) = 0 \implies \lambda \hat{\mu} = 0$$
This says that either $$\lambda$$ or $$\hat{\mu}$$ (or both) must be zero.
### Solving
Note that the RHS of equation $$(1)$$ is a multiple of $$\lambda$$ plus the sample mean $$\frac{1}{n} \sum_{i=1}^n x_i$$. If the sample mean is non-negative, set $$\lambda$$ to zero (satisfying the dual feasibility and complementary slackness conditions). It then follows from equation $$(1)$$ (the stationarity condition) that $$\hat{\mu}$$ is equal to the sample mean. This also satisfies the primal feasibility condition, since it’s non-negative.
Otherwise, if the sample mean is negative, set $$\hat{\mu}$$ to zero (satisfying the primal feasibility and complementary slackness conditions). To satisfy equation $$(1)$$ (the stationarity condition), set $$\lambda = -\hat{\sigma}^{-2} \sum_{i=1}^n x_i$$. Since the sample mean is negative and the variance is positive, $$\lambda$$ takes a positive value, satisfying the dual feasibility conditionn.
In both cases, we can plug $$\hat{\mu}$$ into equation $$(2)$$ to obtain $$\hat{\sigma}^2$$.
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# If the velocity-time graph has the shape AMB,
Question:
If the velocity-time graph has the shape $\mathrm{AMB}$, what would be the shape of the corresponding acceleration-time graph?
1. (1)
2. (2)
3. (3)
4. (4)
Correct Option: 1
Solution:
$a=\frac{d v}{d t}=$ slope of $(v-t)$ curve
If $m=+$ ve, then equation of straight line isc
$y=m x+c \Rightarrow v=m t+c$
(for MB)
If $m=-$ ve, then equation of straight line is
$y=-m x+c \Rightarrow v=-m t+c($ for $A M)$
If we differentiate equation (1) and (2), we get
$\mathrm{a}_{\mathrm{MB}}=+\mathrm{ve}=\mathrm{m}$
$a_{A M}=-v e=-m$, so graph of $(a-t)$ will be
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SEO – How many links per page can we maximally have in an HTML sitemap?
There is a limit of 50,000 per Sitemap in the Sitemap. Is there a limit on links that may be present in the HTML-based sitemap?
If I have more than 100,000 pages or posts, should I use them as a pagination method for HTML Sitemaps?
PS: XML sitemap is different from HTML based sitemap.
What are the best merchant category codes for a maximally successful transaction?
Hello everyone,
I see a much greater decline than successful debit card capture.
Properties of the collection of maximally independent sets of a graph
To let $$G$$ be a graph and define
$$mathscr {I} (G) = {S subset V (G) | S$$ is a maximum independent group of $$G }$$
1. What is known? $$mathscr {I} (G)$$?
2. What are some of the properties of $$mathscr {I} (G)$$?
3. How does it work $$mathscr {I} (G)$$ refers to other properties of $$G$$ for example chromatic number?
4. Is it possible to decide if there is a collection? $$mathscr {A}$$ corresponds to $$mathscr {I} (H)$$ for a diagram $$H$$?
java – How can I maximally work against any group in Android Studio?
I have a question about a program I'm doing right now. In principle, I create a configuration so that the user can choose how many groups he wants to create and how many players should be created per group. When it's created, I can give it the ability to insert players and select which group I want you to go. My question is, how can I form a limiter by groups, ie if I target players in group 1, and that is the maximum, in which I do not get more players in group 1, and that happens the same applies to the number of groups , that I have. I've tried creating an array list to put several array lists into others and try to get them to work, but I can not get them. Thank you
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# if some he gas is introduced into equilibrium pcl5=pcl3+cl2 at const pressure and temp then kc will increase decrease unchanged none
Dear Student. Kc will be unchanged because it depends on temperature only. I am giving you some more detail. Consider the dissociation equilibrium of PCl5 : PCl5 (g) ⇌ PCl3 (g) + Cl2 (g) According to law of chemical equilibrium , equilibrium constant (Kc) : Kc = [PCl3] [Cl2][PCl5] a) If an inert gas is added at constant volume then there is no effect on the dissociation of a compound as only the total pressure of the system increases and there is no change in the partial pressures of the gases. Hence , there will be not effect on the state of equilibrium. b) However, if an inert gas is added at constant pressure and the total volume (number of moles) increases, then in order to maintain total pressure and Kp same, the partial pressures have to adjust (i.e. decrease as volume and pressure are inversely proportional). Hence , the molar concentration of both reactants and products will decrease at equilibrium. There are two terms of concentration in the numerator and only one term of concentration in the denominator , therefore Kc should decrease. Kc is equilibrium constant at constant temperature. Hence, if we want to keep the Kc constant , either we have to decrease the [PCl5] or [PCl3] and we have to increase [Cl2] . This can only be achieved when more of PCl5 dissociates to give PCl3 and Cl2 . Hence, the dissociation increases with the addition of an inert gas. Number of moles increase on dissociation. According to Le Chatelier's principle, a decrease in pressure due to increasing volume causes the reaction to shift to the side with more moles of gas. The equilibrium will move in such a way that the pressure increases again. It can do that by producing more molecules. Thus , the equilibrium would shift to the side with more gas molecules which is the right side and hence dissociation will increase. Regards
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# Simplify Your Code with %>%
Removing duplication is an important principle to keep in mind with your code; however, equally important is to keep your code efficient and readable. Efficiency is often accomplished by leveraging functions and control statements in your code. However, efficiency also includes eliminating the creation and saving of unnecessary objects that often result when you are trying to make your code more readable, clear, and explicit. Consequently, writing code that is simple, readable, and efficient is often considered contradictory. For this reason, the magrittr package is a powerful tool to have in your data wrangling toolkit.
The magrittr package was created by Stefan Milton Bache and, in Stefan’s words, has two primary aims: “to decrease development time and to improve readability and maintainability of code.” Hence, it aims to increase efficiency and improve readability; and in the process it greatly simplifies your code. The following covers the basics of the magrittr toolkit.
## Pipe (%>%) Operator
The principal function provided by the magrittr package is %>%, or what’s called the “pipe” operator. This operator will forward a value, or the result of an expression, into the next function call/expression. For instance a function to filter data can be written as:
filter(data, variable == numeric_value)
or
data %>% filter(variable == numeric_value)
Both functions complete the same task and the benefit of using %>% may not be immediately evident; however, when you desire to perform multiple functions its advantage becomes obvious. For instance, if we want to filter some data, group it by categories, summarize it, and then order the summarized results we could write it out three different ways. Don’t worry, you’ll learn how to operate these specific functions in the next section.
Nested Option:
arrange(
summarize(
group_by(
filter(mtcars, carb > 1),
cyl
),
Avg_mpg = mean(mpg)
),
desc(Avg_mpg)
)
## Source: local data frame [3 x 2]
##
## cyl Avg_mpg
## (dbl) (dbl)
## 1 4 25.90
## 2 6 19.74
## 3 8 15.10
This first option is considered a “nested” option such that functions are nested within one another. Historically, this has been the traditional way of integrating code; however, it becomes extremely difficult to read what exactly the code is doing and it also becomes easier to make mistakes when making updates to your code. Although not in violation of the DRY principle1, it definitely violates the basic principle of readability and clarity, which makes communication of your analysis more difficult. To make things more readable, people often move to the following approach…
Multiple Object Option:
a <- filter(mtcars, carb > 1)
b <- group_by(a, cyl)
c <- summarise(b, Avg_mpg = mean(mpg))
d <- arrange(c, desc(Avg_mpg))
print(d)
## Source: local data frame [3 x 2]
##
## cyl Avg_mpg
## (dbl) (dbl)
## 1 4 25.90
## 2 6 19.74
## 3 8 15.10
This second option helps in making the data wrangling steps more explicit and obvious but definitely violates the DRY principle. By sequencing multiple functions in this way you are likely saving multiple outputs that are not very informative to you or others; rather, the only reason you save them is to insert them into the next function to eventually get the final output you desire. This inevitably creates unnecessary copies and wrecks havoc on properly managing your objects…basically it results in a global environment charlie foxtrot! To provide the same readability (or even better), we can use %>% to string these arguments together without unnecessary object creation…
%>% Option:
library(magrittr)
library(dplyr)
mtcars %>%
filter(carb > 1) %>%
group_by(cyl) %>%
summarise(Avg_mpg = mean(mpg)) %>%
arrange(desc(Avg_mpg))
## Source: local data frame [3 x 2]
##
## cyl Avg_mpg
## (dbl) (dbl)
## 1 4 25.90
## 2 6 19.74
## 3 8 15.10
This final option which integrates %>% operators makes for more efficient and legible code. Its efficient in that it doesn’t save unncessary objects (as in option 2) and performs as effectively (as both option 1 & 2) but makes your code more readable in the process. Its legible in that you can read this as you would read normal prose (we read the %>% as “and then”): “take mtcars and then filter and then group by and then summarize and then arrange.”
And since R is a functional programming language, meaning that everything you do is basically built on functions, you can use the pipe operator to feed into just about any argument call. For example, we can pipe into a linear regression function and then get the summary of the regression parameters. Note in this case I insert “data = .” into the lm() function. When using the %>% operator the default is the argument that you are forwarding will go in as the first argument of the function that follows the %>%. However, in some functions the argument you are forwarding does not go into the default first position. In these cases, you place “.” to signal which argument you want the forwarded expression to go to.
mtcars %>%
filter(carb > 1) %>%
lm(mpg ~ cyl + hp, data = .) %>%
summary()
##
## Call:
## lm(formula = mpg ~ cyl + hp, data = .)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.6163 -1.4162 -0.1506 1.6181 5.2021
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 35.67647 2.28382 15.621 2.16e-13 ***
## cyl -2.22014 0.52619 -4.219 0.000353 ***
## hp -0.01414 0.01323 -1.069 0.296633
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.689 on 22 degrees of freedom
## Multiple R-squared: 0.7601, Adjusted R-squared: 0.7383
## F-statistic: 34.85 on 2 and 22 DF, p-value: 1.516e-07
You can also use %>% to feed into plots:
library(ggplot2)
mtcars %>%
filter(carb > 1) %>%
qplot(x = wt, y = mpg, data = .)
You will also find that the %>% operator is now being built into packages to make programming much easier. For instance, in the tutorials where I illustrate how to reshape and transform your data with the dplyr and tidyr packages, you will see that the %>% operator is already built into these packages. It is also built into the ggvis and dygraphs packages (visualization packages), the httr package (which I covered in the data scraping tutorials), and a growing number of newer packages.
In addition to the %>% operator, magrittr provides several additional functions which make operations such as addition, multiplication, logical operators, re-naming, etc. more pleasant when composing chains using the %>% operator. Some examples follow but you can see the current list of the available aliased functions by typing ?magrittr::add in your console.
# subset with extract
mtcars %>%
extract(, 1:4) %>%
## mpg cyl disp hp
## Mazda RX4 21.0 6 160 110
## Mazda RX4 Wag 21.0 6 160 110
## Datsun 710 22.8 4 108 93
## Hornet 4 Drive 21.4 6 258 110
## Hornet Sportabout 18.7 8 360 175
## Valiant 18.1 6 225 105
# add, subtract, multiply, divide and other operations are available
mtcars %>%
extract(, "mpg") %>%
multiply_by(5)
## [1] 105.0 105.0 114.0 107.0 93.5 90.5 71.5 122.0 114.0 96.0 89.0
## [12] 82.0 86.5 76.0 52.0 52.0 73.5 162.0 152.0 169.5 107.5 77.5
## [23] 76.0 66.5 96.0 136.5 130.0 152.0 79.0 98.5 75.0 107.0
# logical assessments and filters are available
mtcars %>%
extract(, "cyl") %>%
equals(4)
## [1] FALSE FALSE TRUE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE
## [12] FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE TRUE TRUE FALSE
## [23] FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE TRUE
# renaming columns and rows is available
mtcars %>%
set_colnames(paste("Col", 1:11, sep = ""))
## Col1 Col2 Col3 Col4 Col5 Col6 Col7 Col8 Col9 Col10 Col11
## Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4
## Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4
## Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1
## Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1
## Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2
## Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1
magrittr also offers some alternative pipe operators. Some functions, such as plotting functions, will cause the string of piped arguments to terminate. The tee (%T>%) operator allows you to continue piping functions that normally cause termination.
# normal piping terminates with the plot() function resulting in
# NULL results for the summary() function
mtcars %>%
filter(carb > 1) %>%
extract(, 1:4) %>%
plot() %>%
summary()
## Length Class Mode
## 0 NULL NULL
# inserting %T>% allows you to plot and perform the functions that
mtcars %>%
filter(carb > 1) %>%
extract(, 1:4) %T>%
plot() %>%
summary()
## mpg cyl disp hp
## Min. :10.40 Min. :4.00 Min. : 75.7 Min. : 52.0
## 1st Qu.:15.20 1st Qu.:6.00 1st Qu.:146.7 1st Qu.:110.0
## Median :17.80 Median :8.00 Median :275.8 Median :175.0
## Mean :18.62 Mean :6.64 Mean :257.7 Mean :163.7
## 3rd Qu.:21.00 3rd Qu.:8.00 3rd Qu.:351.0 3rd Qu.:205.0
## Max. :30.40 Max. :8.00 Max. :472.0 Max. :335.0
The compound assignment %<>% operator is used to update a value by first piping it into one or more expressions, and then assigning the result. For instance, let’s say you want to transform the mpg variable in the mtcars data frame to a square root measurement. Using %<>% will perform the functions to the right of %<>% and save the changes these functions perform to the variable or data frame called to the left of %<>%.
# note that mpg is in its typical measurement
## mpg cyl disp hp drat wt qsec vs am gear carb
## Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4
## Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4
## Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1
## Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1
## Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2
## Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1
# we can square root mpg and save this change using %<>%
mtcars$mpg %<>% sqrt head(mtcars) ## mpg cyl disp hp drat wt qsec vs am gear carb ## Mazda RX4 4.582576 6 160 110 3.90 2.620 16.46 0 1 4 4 ## Mazda RX4 Wag 4.582576 6 160 110 3.90 2.875 17.02 0 1 4 4 ## Datsun 710 4.774935 4 108 93 3.85 2.320 18.61 1 1 4 1 ## Hornet 4 Drive 4.626013 6 258 110 3.08 3.215 19.44 1 0 3 1 ## Hornet Sportabout 4.324350 8 360 175 3.15 3.440 17.02 0 0 3 2 ## Valiant 4.254409 6 225 105 2.76 3.460 20.22 1 0 3 1 Some functions (e.g. lm, aggregate, cor) have a data argument, which allows the direct use of names inside the data as part of the call. The exposition (%$%) operator is useful when you want to pipe a dataframe, which may contain many columns, into a function that is only applied to some of the columns. For example, the correlation (cor) function only requires an x and y argument so if you pipe the mtcars data into the cor function using %>% you will get an error because cor doesn’t know how to handle mtcars. However, using %$% allows you to say “take this dataframe and then perform cor() on these specified columns within mtcars.” # regular piping results in an error mtcars %>% subset(vs == 0) %>% cor(mpg, wt) ## Error in pmatch(use, c("all.obs", "complete.obs", "pairwise.complete.obs", : object 'wt' not found # using %$% allows you to specify variables of interest
mtcars %>%
subset(vs == 0) %\$%
cor(mpg, wt)
## [1] -0.830671
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Integral using cauchy theorem
1. Feb 9, 2014
Dassinia
Hello,
I don't get why using the fact that ∫dz/z = 2*pi*i accros the circle
This integral gives:
1/3∫(1/(z-2)-1/(z-1/2))dz = 1/3(-2*pi*i) accross the circle
Thanks !
2. Feb 9, 2014
Dick
Why do you think 1/z has anything to do with (1/3)*(1/(z-2)-1/(z-1/2))??
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## Local Explanation Methods for Deep Neural Networks Lack Sensitivity to Parameter Values
Feb 12, 2018 Submission readers: everyone Show Bibtex
• Abstract: Explaining the output of a complicated machine learning model like a deep neural network (DNN) is a central challenge in machine learning. Several proposed local explanation methods address this issue by identifying what dimensions of a single input are most responsible for a DNN's output. The goal of this work is to assess the sensitivity of local explanations to DNN parameter values. Somewhat surprisingly, we find that DNNs with randomly-initialized weights produce explanations that are both visually and quantitatively similar to those produced by DNNs with learned weights. Our conjecture is that this phenomenon occurs because these explanations are dominated by the lower level features of a DNN, and that a DNN's architecture provides a strong prior which significantly affects the representations learned at these lower layers.
• Keywords: Interpretability, Saliency, DNNs, Random, Weights
• TL;DR: Local explanations for DNNs remain visually and quantitatively similar to DNNs with learned weights and those with randomized weights.
0 Replies
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# Chapter 7 - Section 7.5 - Solving Equations Containing Rational Expressions - Exercise Set: 38
This equation has no solution
#### Work Step by Step
$\dfrac{1}{x+2}=\dfrac{4}{x^{2}-4}-\dfrac{1}{x-2}$ Factor the denominator of the second fraction: $\dfrac{1}{x+2}=\dfrac{4}{(x-2)(x+2)}-\dfrac{1}{x-2}$ Multiply the whole equation by $(x-2)(x+2)$: $(x-2)(x+2)\Big[\dfrac{1}{x+2}=\dfrac{4}{(x-2)(x+2)}-\dfrac{1}{x-2}\Big]$ $x-2=4-x-2$ Take all terms to the left side of the equation and simplify it by combining like terms: $x-2-4+x+2=0$ $2x-4=0$ Solve for $x$: $2x=4$ $x=2$ Substituting $x=2$ in the original equation makes both denominator on the right side $0$, this means that this equation has no solution.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.
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byShow For the definitions below, assume A, B and C are all mXn matrices. Let matrices defined property) According to this law, the order in which two quantities are multiplied does not affect the final product. Matrix addition is associative, that and show that matrix addition is commutative that is show that if A and B are both m*n matrices, then A+B=B+A? . Once the matrices are in a nice order, you can pick whichever "+" you want to do first. This means that (a + b) + c = a + (b + c). -th Properties of matrix scalar multiplication. You should be happy with the following rules of matrix addition. This lecture introduces matrix addition, one of the basic algebraic operations the assertion is true. This is an immediate consequence of the fact Addition and multiplication are both commutative. Let and Non-commutative rings are not models of RT+Ind where Ind is first order induction. is a matrix such that its columns are equal to the rows of matrix:Define the If $$A$$ is an $$m\times p$$ matrix, $$B$$ is a $$p \times q$$ matrix, and $$C$$ is a $$q \times n$$ matrix, then $A(BC) = (AB)C.$ This important property makes simplification of many matrix expressions possible. To solve a problem like the one described for the soccer teams, we can use a matrix, which is a rectangular array of numbers. element of and Truong-Son N. Dec 27, 2016 No, but it is not too difficult to show that it is anticommutative. sum: Let and (Warning!! is. -th Two matrices are equal if and only if 1. any matrices Let Matrix subtraction is not commutative because you have to subtract term by term your two matrices and the order in the subtraction counts. more familiar addition of real numbers. The multiplication of matrix A by the scalar k yields a matrix B of the same shape as A, according to (4.32)B = kA, with bij = k aij for all i and j. element is equal to the sum of the that can be performed on matrices. Subtraction and division are not commutative. Matrix addition enjoys properties that are similar to those enjoyed by the A + (B + C) = (A + B) + C (iii) Existence of additive identity : Null or zero matrix is the additive identity for matrix addition. Thus, we have shown that matrices are commutative. Adding matrices is easier than you might think! The rules for matrix addition and multiplication by a scalar give unambiguous meaning to linear forms involving matrices of conforming dimensions. The Commutative Property of Matrix Addition is just like the Commutative Property of Addition! have the same dimension, we can compute their Rules for Matrix Addition. be two Proposition (commutative I'm aware there are many possible binary operations and not all of them are commutative, but I'm specifically looking for examples which are conventionally spelled "+" and called addition. element-by-element sums that are performed when carrying out matrix addition. The transpose is symmetric if it is equal to its transpose. The only sure examples I can think of where it is commutative is multiplying by the identity matrix, in which case … property) Addition is commutative. Proposition (commutative property) Matrix addition is commutative, that is, for any matrices and and such that the above additions are meaningfully defined. eureka-math.org -M2 TE 1.3.0 08.2015 This work is licensed under a Creative … Just find the corresponding positions in each matrix and add the elements in them! The transpose of The product of two block matrices is given by multiplying each block. This is the currently selected item. Two matrices can be added together if and only if they have the same For example, three matrices named A,B,A,B, and CCare shown below. for all :Now, Another similar law is the commutative law of multiplication. matrix defined The addition of vectors is commutative, because. What are the Commutative Properties of Addition and Multiplication. isThus, Show that matrix addition is commutative: + = + NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M2 PRECALCULUS AND ADVANCED TOPICS Lesson 11: Matrix Multiplication Is Commutative This file derived from PreCal S.81 This work is derived from Eureka Math ™ and licensed by Great Minds. sum If moving the numbers in a calculation by switching their places does not affect the answer, then the calculation is commutative. and dimension. Two well-known examples of commutative binary operations: The addition of real numbers is commutative, since. y … the Of course you're correct that non-abelian groups, by definition, are non-commutative, but all of the examples I've found don't call the operator "addition" or spell it "+". In order to compute the sum of So you have those equations: Solution Step 1:Let A be an matrix and let 0 be an matrix has all entries equal to zero then we have to show that Step 2:consider matrices A and B So adding this two matrices we get Hence matrix Abo gives an example of a phi(x) we can prove using induction that is false in matrix arithmetic. Each of these operations has a precise definition. Proof This is an immediate consequence of the fact that the commutative property applies to sums of scalars, and therefore to the element-by-element sums that are performed when carrying out matrix addition. and , "Matrix addition", Lectures on matrix algebra. Connect number words and numerals to the quantities they represent, using various physical models and representations. that the associative property applies to sums of scalars, and therefore to the follows:Computewhere Most of the learning materials found on this website are now available in a traditional textbook format. Matrices (plural) are enclosed in [ ] or ( ) and are usually named with capital letters. Their sum is obtained by summing each element of one matrix to the $\endgroup$ – Russell Easterly Feb 19 '13 at 4:07. add a comment | 3 Answers Active Oldest Votes. satisfying be two Matrix addition is commutative, that Commutative Property Of Addition: There are three basic properties of numbers, and your textbook will probably have just a little section on these properties, somewhere near the beginning of the course, and then you’ll probably never see them again (until the beginning of the next course). youtube.com. The corresponding elements of the matrices are the same Since matrices form an Abelian group under addition, matrices form a ring . Mathematics. their sum. This tutorial can show you the entire process step-by-step. We can remember that the word ‘commute’ means to move. A + B = B + A; A + 0 = 0 + A = A; 0 + 0 = 0; These look the same as some rules for addition of real numbers. Matrix addition is associative. Covers the following skills: Develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers. However, matrix multiplication is not, in general, commutative (although it is commutative if and are diagonal and of the same dimension). The latter Taboga, Marco (2017). Properties of matrix addition & scalar multiplication. Let A column in a matrix is a set of numbers that are aligned vertically. This tutorial uses the Commutative Property of Addition and an example to explain the Commutative Property of Matrix Addition. that the sum of #Properties of addition of matrices commutative associative existence of identity additive inverse. (19) {\displaystyle {\vec {a}}+ {\vec {b}}= {\vec {b}}+ {\vec {a}}} . In this video you will learn about Properties of Matrix for Addition - Commutative, Associative and Additive Inverse - Matrices - Maths - Class 12/XII - ISCE,CBSE - NCERT. matrices. is the transpose of is,for Example Matrix addition and subtraction, where defined (that is, where the matrices are the same size so addition and subtraction make sense), can be turned into homework problems. such that the above additions are meaningfully defined. (i) Matrix addition is commutative : If A and B are any two matrices of same order, then. https://www.statlect.com/matrix-algebra/matrix-addition. In math, the associative and commutative properties are laws applied to addition and multiplication that always exist. In each rule, the matrices are assumed to all have the same dimensions. Below you can find some exercises with explained solutions. For example, 3 + 5 = 8 and 5 + 3 = 8. ©2015 Great Minds. Subtraction is not Commutative. isThe Why is it that multiplication is not commutative and addition is commutative? such that the above additions are meaningfully defined. corresponding element of the other matrix. and Example This operation is commutative, with kA = Ak. . be a matrix such that its Show that matrix addition is both commutative and associative. Learn about the properties of matrix addition (like the commutative property) and how they relate to real number addition. These techniques are used frequently in machine learning and deep learning so it is worth familiarising yourself with them. , Matrix multiplication is NOT commutative. Commutative Law of Multiplication . Remember that column vectors and row vectors are also matrices. Email. Commutative operations in mathematics. and we need to sum each element of This preview shows page 15 - 18 out of 35 pages.. 15 Solution: 9.5.2 PROPERTIES OF MATRIX ADDITION/SUBTRACTION i) Matrix addition is commutative A B B A ii) Matrix subtraction is NOT commutative A B B Solution: 9.5.2 PROPERTIES OF MATRIX ADDITION/SUBTRACTION i) Matrix addition is commutative A B B A ii) Matrix subtraction is NOT commutative A B B sum of This tutorial defines the commutative property and provides examples of how to use it. be the following with the corresponding element of If A is a matrix of order m x n, then , Even though matrix multiplication is not commutative, it is associative in the following sense. and is. When R is a commutative ring, the matrix ring M n (R) is an associative algebra, and may be called a matrix algebra. Any subring of a matrix ring is a matrix ring. be two A=[1234],B=[1270−… matrix The order of the matrices are the same 2. This is an immediate consequence of the fact any matrices Some students spoil my fun by realizing that (since matrix addition is commutative) the matrices can be rearranged into a more favorable order. Next lesson. more. matricesTheir Why "rings with non-commutative addition" are a somewhat side story and commutativity of addition is the usual assumption? So: #A-B!=B-A#. The commutative property is a fundamental building block of math, but it only works for addition and multiplication. This tutorial uses the Commutative Property of Addition and an example to explain the Commutative Property of Matrix Addition. If you've ever wondered what variables are, then this tutorial is for you! and element-by-element sums that are performed when carrying out matrix addition. Not all rules for matrix math look the same as for real number math.) and Show that matrix addition is commutative; that is, show that if A and B are both m × n matrices, then A + B = B + A. When A+B=B+A, we say that the commutative property is satisfied. What does it mean to add two matrices together? For example, consider: Answer link. and Properties of matrix addition. Simply because the basic and main examples of these rings, those which primarily occur doing mathematics, do have this property. a → + b → = b → + a →. As a This video demonstrates how addition of two matrices satisfies the commutative property. 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Moving the numbers in a nice order, you can find some exercises with explained solutions the positions! With them learning materials found on this website are now available in a ring! On matrix algebra obtained by summing each element of the basic and main examples of commutative binary operations: addition! Not affect the final product they can be added together if and only if they have the same.... An operation and hopefully see that it is worth familiarising yourself with them subring of matrix! Ca n't do algebra without working with variables, but it is equal to its.... What does it mean to add two matrices are the commutative Property of!... Always exist matrices is given by multiplying each block the matrices are assumed to all have the same.!, Lectures on matrix algebra, b and c are all mXn matrices an entry, called... Added to scalars, vectors and row vectors are also matrices rule, the associative and commutative Properties of!! Find the corresponding positions in each rule, the associative and commutative Properties laws. Its transpose quantities are multiplied does not affect the Answer, then this tutorial defines the commutative law of.. Under addition, one of many basic laws that are aligned horizontally their sum is by...
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# Lesson 4 - NLP, Tabular, and Collaborative Filtering
These are my personal notes from fast.ai Live (the new International Fellowship programme) course and will continue to be updated and improved if I find anything useful and relevant while I continue to review the course to study much more in-depth. Thanks for reading and happy learning!
Live date: 14 Nov 2018, GMT+8
## Topics
• Natural Language Processing (NLP)
• Language modelling
• Deeper dive into NLP transfer learning
• Text classification
• Tabular data
• Continuous vs. categorical variable
• Collaborative filtering
• MovieLens dataset
• Cold start problem
• Embeddings
• Neural network
• What's happening mathematically
## Lesson Resources
• Video
• Jupyter Notebook and code
## Assignments
• Run lesson 4 notebooks.
• Replicate lesson 4 notebooks with your own dataset.
# My Notes
We are going to finish our journey through these key applications. We've already looked at a range of vision applications.
We've looked a classification, localization, image regression. We briefly touched on NLP. We're going to do a deeper dive into NLP transfer learning today. We're going to then look at tabular data and collaborative filtering which are both super useful applications.
Then we're going to take a complete u-turn. We're going to take that collaborative filtering example and dive deeply into it to understand exactly what's happening mathematically—exactly what's happening in the computer. And we're going to use that to gradually go back in reverse order through the applications again in order to understand exactly what's going on behind the scenes of all of those applications.
Correction on CamVid result
Before we do, somebody on the forum is kind enough to point out that when we compared ourselves to what we think might be the state of the art or was recently the state of the art for CamVid, there wasn't a fair comparison because the paper actually used a small subset of the classes, and we used all of the classes. So Jason in our study group was kind enough to rerun the experiments with the correct subset of classes from the paper, and our accuracy went up to 94% compared to 91.5% of the paper. So I think that's a really cool result and a great example of how pretty much just using the defaults nowadays can get you far beyond what was the best of a year or 2 ago. It was certainly the best last year when we were doing this course because we started it quite intensely. So that's really exciting.
## Natural Language Processing (NLP) [2:00]
What I wanted to start with is going back over NLP a little bit to understand really what was going on there.
### A quick review
So first of all, a quick review. Remember NLP is Natural Language Processing. It's about taking text and doing something with it. Text classification is particularly useful﹣practically useful applications. It's what we're going to start off focusing on. Because classifying a text or classifying a document can be used for anything from:
• Spam prevention
• Identifying fake news
• Finding a diagnosis from medical reports
• Finding mentions of your product in Twitter
So it's pretty interesting. And actually there was a great example during the week from one of our students @howkhang who is a lawyer and he mentioned on the forum that he had a really great results from classifying legal texts using this NLP approach. And I thought this was a great example. This is the post that they presented at an academic conference this week describing the approach:
This series of three steps that you see here (and I'm sure you recognize this classification matrix) is what we're going to start by digging into.
We're going to start out with a movie review like this one and decide whether it's positive or negative sentiment about the movie. But, here's the the problem. We have, in the training set, 25,000 movie reviews and for each one we have like one bit of information: they liked it, or they didn't like it. That's what we're going to look into a lot more detail today and in the current lessons. Our neural networks (remember, they're just a bunch of matrix multiplies and simple nonlinearities﹣particularly replacing negatives with zeros), those weight matrices start out random. So if you start out with with some random parameters and try to train those parameters to learn how to recognize positive vs. negative movie reviews, you literally have 25,000 ones and zeros to actually tell you I like this one I don't like that one. That's clearly not enough information to learn, basically, how to speak English﹣how to speak English well enough to recognize they liked this or they didn't like this. Sometimes that can be pretty nuanced. Particularly with movie reviews because these are like online movie reviews on IMDB, people can often use sarcasm. It could be really quite tricky.
So, for a long time, until very recently like this year, neural nets didn't do a good job at all of this kind of classification problem. And that was why﹣there's not enough information available. So the trick, hopefully you can all guess, is to use transfer learning. It's always the trick.
Last year in this course I tried something crazy which was I thought what if I try transform learning to demonstrate that it can work for NLP as well. I tried it out and it worked extraordinarily well. So here we are, a year later, and transfer learning in NLP is absolutely the hit thing. And I'm going to describe to you what happens.
### Transfer learning in NLP [6:04]
The key thing is we're going to start with the same kind of thing that we used for computer vision﹣a pre-trained model that's been trained to do something different to what we're doing with it. For ImageNet, that was originally built as a model to predict which of a thousand categories each photo falls into. And people then fine-tune that for all kinds of different things as you've seen. So we're going to start with a pre-trained model that's going to do something else, not movie review classification. We're going to start with a pre-trained model which is called a language model.
A language model has a very specific meaning in NLP and it's this. A language model is a model that learns to predict the next word of a sentence. To predict the next word of a sentence, you actually have to know quite a lot about English (assuming you're doing it in English) and quite a lot of world knowledge. By world knowledge, I'll give you an example.
Here's your language model and it has read:
• "I'd like to eat a hot _": Obviously, "dog", right?
• "It was a hot _": Probably "day"
Now previous approaches to NLP use something called n-grams largely which is basically saying how often do these pairs or triplets of words tend to appear next to each other. And n-grams are terrible at this kind of thing. As you can see, there's not enough information here to decide what the next word probably is. But with a neural net, you absolutely can.
So here's the nice thing. If you train a neural net to predict the next word of a sentence then you actually have a lot of information. Rather than having a single bit to every 2,000 word movie review: "liked it" or "didn't like it", every single word, you can try and predict the next word. So in a 2,000 word movie review, there are 1,999 opportunities to predict the next word. Better still, you don't just have to look at movie reviews. Because really the hard thing isn't so much as "does this person like the movie or not?" but "how do you speak English?". So you can learn "how do you speak English?" (roughly) from some much bigger set of documents. So what we did was we started with Wikipedia.
WikiText-103 [8:30]
Stephen Merity and some of his colleagues built something called WikiText-103 dataset which is simply a subset of most of the largest articles from Wikipedia with a little bit of pre-processing that's available for download.
So you're basically grabbing Wikipedia and then I built a language model on all of Wikipedia. So I just built a neural net which would predict the next word in every significantly sized Wikipedia article. That's a lot of information. If I remember correctly, it's something like a billion tokens. So we've got a billion separate things to predict. Every time we make a mistake on one of those predictions, we get the loss, we get gradients from that, and we can update our weights, and they can better and better until we can get pretty good at predicting the next word of Wikipedia.
Why is that useful? Because at that point, I've got a model that knows probably how to complete sentences like this, so it knows quite a lot about English and quite a lot about how the world works﹣what kinds of things tend to be hot in different situations, for instance. Ideally, it would learn things like "in 1996 in a speech to the United Nations, United States president _ said "... Now that would be a really good language model, because it would actually have to know who is this United States president in that year. So getting really good at training language models is a great way to teach a neural net a lot about what is our world, what's in our world, how do things work in our world. It's a really fascinating topic, and it's actually one that philosophers have been studying for hundreds of years now. There's actually a whole theory of philosophy which is about what can be learned from studying language alone. So it turns out, apparently, quite a lot.
So here's the interesting thing. You can start by training a language model on all of Wikipedia, and then we can make that available to all of you. Just like a pre-trained ImageNet model for vision, we've now made available a pre-trained WikiText model for NLP not because it's particularly useful of itself (predicting the next word of sentences is somewhat useful, but not normally what we want to do), but it's a model that understands a lot about language and a lot about what language describes. So then, we can take that and we can do transfer learning to create a new language model that's specifically good at predicting the next word of movie reviews.
### Fine-tuning WikiText to create a new language model [11:10]
If we can build a language model that's good at predicting the next word of movie reviews pre-trained with the WikiText model, then that's going to understand a lot about "my favorite actor is Tom . " Or "I thought the photography was fantastic but I wasn't really so happy about the _ (director)." It's going to learn a lot about specifically how movie reviews are written. It'll even learn things like what are the names of some popular movies.
That would then mean we can still use a huge corpus of lots of movie reviews even if we don't know whether they're positive or negative to learn a lot about how movie reviews are written. So for all of this pre-training and all of this language model fine-tuning, we don't need any labels at all. It is what the researcher Yann LeCun calls self supervised learning. In other words, it's a classic supervised model﹣we have labels, but the labels are not things that somebody else have created. They're built into the dataset itself. So this is really really neat. Because at this point, we've now got something that's good at understanding movie reviews and we can fine-tune that with transfer learning to do the thing we want to do which in this case is to classify movie reviews to be positive or negative. So my hope was (when I tried this last year) that at that point, 25,000 ones and zeros would be enough feedback to fine-tune that model and it turned out it absolutely was.
Question: Does the language model approach works for text in forums that are informal English, misspelled words or slangs or shortforms like s6 instead of Samsung S6? [12:47]
Yes, absolutely it does. Particularly if you start with your WikiText model and then fine-tune it with your "target" corpus. Corpus is just a bunch of documents (emails, tweets, medical reports, or whatever). You could fine-tune it so it can learn a bit about the specifics of the slang, abbreviations, or whatever that didn't appear in the full corpus. So interestingly, this is one of the big things that people were surprised about when we did this research last year. People thought that learning from something like Wikipedia wouldn't be that helpful because it's not that representative of how people tend to write. But it turns out it's extremely helpful because there's a much a difference between Wikipedia and random words than there is between like Wikipedia and Reddit. So it kind of gets you 99% of the way there.
So language models themselves can be quite powerful. For example there was a blog post from SwiftKey (the folks that do the mobile-phone predictive text keyboard) and they describe how they kind of rewrote their underlying model to use neural nets. This was a year or two ago. Now most phone keyboards seem to do this. You'll be typing away on your mobile phone, and in the prediction there will be something telling you what word you might want next. So that's a language model in your phone.
Another example was the researcher Andrej Karpathy who now runs all this stuff at Tesla, back when he was a PhD student, he created a language model of text in LaTeX documents and created these automatic generation of LaTeX documents that then became these automatically generated papers. That's pretty cute.
We're not really that interested in the output of the language model ourselves. We're just interested in it because it's helpful with this process.
Review of the basic process [15:14]
We briefly looked at the process last week. The basic process is, we're going to start with the data in some format. So for example, we've prepared a little IMDB sample that you can use which is in CSV file. You can read it in with Pandas and there's negative or positive, the text of each movie review, and boolean of is it in the validation set or the training set.
path = untar_data(URLs.IMDB_SAMPLE)path.ls()
[PosixPath('/home/jhoward/.fastai/data/imdb_sample/texts.csv'), PosixPath('/home/jhoward/.fastai/data/imdb_sample/models')]
df = pd.read_csv(path/'texts.csv')df.head()
label text is_valid 0 negative Un-bleeping-believable! Meg Ryan doesn't even ... False 1 positive This is a extremely well-made film. The acting... False 2 negative Every once in a long while a movie will come a... False 3 positive Name just says it all. I watched this movie wi... False 4 negative This movie succeeds at being one of the most u... False
So there's an example of a movie review:
df['text'][1]
'This is a extremely well-made film. The acting, script and camera-work are all first-rate. The music is good, too, though it is mostly early in the film, when things are still relatively cheery. There are no really superstars in the cast, though several faces will be familiar. The entire cast does an excellent job with the script.<br /><br />But it is hard to watch, because there is no good end to a situation like the one presented. It is now fashionable to blame the British for setting Hindus and Muslims against each other, and then cruelly separating them into two countries. There is some merit in this view, but it\'s also true that no one forced Hindus and Muslims in the region to mistreat each other as they did around the time of partition. It seems more likely that the British simply saw the tensions between the religions and were clever enough to exploit them to their own ends.<br /><br />The result is that there is much cruelty and inhumanity in the situation and this is very unpleasant to remember and to see on the screen. But it is never painted as a black-and-white case. There is baseness and nobility on both sides, and also the hope for change in the younger generation.<br /><br />There is redemption of a sort, in the end, when Puro has to make a hard choice between a man who has ruined her life, but also truly loved her, and her family which has disowned her, then later come looking for her. But by that point, she has no option that is without great pain for her.<br /><br />This film carries the message that both Muslims and Hindus have their grave faults, and also that both can be dignified and caring people. The reality of partition makes that realisation all the more wrenching, since there can never be real reconciliation across the India/Pakistan border. In that sense, it is similar to "Mr & Mrs Iyer".<br /><br />In the end, we were glad to have seen the film, even though the resolution was heartbreaking. If the UK and US could deal with their own histories of racism with this kind of frankness, they would certainly be better off.'
So you can just go TextDataBunch.from_csv to grab a language model specific data bunch:
data_lm = TextDataBunch.from_csv(path, 'texts.csv')
And then you can create a learner from that in the usual way and fit it.
data_lm.save()
You can save the data bunch which means that the pre-processing that is done, you don't have to do it again. You can just load it.
data = TextDataBunch.load(path)
What happens behind the scenes if we now load it as a classification data bunch (that's going to allow us to see the labels as well)?
data = TextClasDataBunch.load(path)data.show_batch()
text label xxbos xxfld 1 raising victor vargas : a review \n\n you know , raising victor vargas is like sticking your hands into a big , xxunk bowl of xxunk . it 's warm and gooey , but you 're not sure if it feels right . try as i might negative xxbos xxfld 1 now that che(2008 ) has finished its relatively short australian cinema run ( extremely limited xxunk screen in xxunk , after xxunk ) , i can xxunk join both xxunk of " at the movies " in taking steven soderbergh to task . \n\n it 's usually negative xxbos xxfld 1 many xxunk that this is n't just a classic due to the fact that it 's the first xxup 3d game , or even the first xxunk - up . it 's also one of the first xxunk games , one of the xxunk definitely the first positive xxbos xxfld 1 i really wanted to love this show . i truly , honestly did . \n\n for the first time , gay viewers get their own version of the " the bachelor " . with the help of his obligatory " hag " xxunk , james , a negative xxbos xxfld 1 this film sat on my xxunk for weeks before i watched it . i xxunk a self - indulgent xxunk flick about relationships gone bad . i was wrong ; this was an xxunk xxunk into the xxunk - up xxunk of new xxunk . \n\n the positive
As we described, it basically creates a separate unit (i.e. a "token") for each separate part of a word. So most of them are just for words, but sometimes if it's like an 's from it's, it will get its own token. Every bit of punctuation tends to get its own token (a comma, a full stop, and so forth).
Then the next thing that we do is a numericalization which is where we find what are all of the unique tokens that appear here, and we create a big list of them. Here's the first ten in order of frequency:
data.vocab.itos[:10]
['xxunk', 'xxpad', 'the', ',', '.', 'and', 'a', 'of', 'to', 'is']
And that big list of unique possible tokens is called the vocabulary which we just call it a "vocab". So what we then do is we replace the tokens with the ID of where is that token in the vocab:
data.train_ds[0][0]
Text xxbos xxfld 1 he now has a name , an identity , some memories and a a lost girlfriend . all he wanted was to disappear , but still , they xxunk him and destroyed the world he hardly built . now he wants some explanation , and to get ride of the people how made him what he is . yeah , jason bourne is back , and this time , he 's here with a vengeance .
data.train_ds[0][0].data[:10]
array([ 43, 44, 40, 34, 171, 62, 6, 352, 3, 47])
That's numericalization. Here's the thing though. As you'll learn, every word in our vocab is going to require a separate row in a weight matrix in our neural net. So to avoid that weight matrix getting too huge, we restrict the vocab to no more than (by default) 60,000 words. And if a word doesn't appear more than two times, we don't put it in the vocab either. So we keep the vocab to a reasonable size in that way. When you see these xxunk, that's an unknown token. It just means this was something that was not a common enough word to appear in our vocab.
We also have a couple of other special tokens like (see fastai.text.transform.py for up-to-date info):
• xxfld: This is a special thing where if you've got like title, summary, abstract, body, (i. e. separate parts of a document), each one will get a separate field and so they will get numbered (e.g. xxfld 2).
• xxup: If there's something in all caps, it gets lower cased and a token called xxup will get added to it.
With the Data Block API [18:31]
Personally, I more often use the data block API because there's less to remember about exactly what data bunch to use, and what parameters and so forth, and it can be a bit more flexible.
data = (TextList.from_csv(path, 'texts.csv', cols='text') .split_from_df(col=2) .label_from_df(cols=0) .databunch())
So another approach to doing this is to just decide:
• What kind of list you're creating (i.e. what's your independent variable)? So in this case, my independent variable is text.
• What is it coming from? A CSV.
• How do you want to split it into validation versus training? So in this case, column number two was the is_valid flag.
• How do you want to label it? With positive or negative sentiment, for example. So column zero had that.
• Then turn that into a data bunch.
That's going to do the same thing.
path = untar_data(URLs.IMDB)path.ls()
[PosixPath('/home/jhoward/.fastai/data/imdb/imdb.vocab'), PosixPath('/home/jhoward/.fastai/data/imdb/models'), PosixPath('/home/jhoward/.fastai/data/imdb/tmp_lm'), PosixPath('/home/jhoward/.fastai/data/imdb/train'), PosixPath('/home/jhoward/.fastai/data/imdb/test'), PosixPath('/home/jhoward/.fastai/data/imdb/README'), PosixPath('/home/jhoward/.fastai/data/imdb/tmp_clas')]
(path/'train').ls()
[PosixPath('/home/jhoward/.fastai/data/imdb/train/pos'), PosixPath('/home/jhoward/.fastai/data/imdb/train/unsup'), PosixPath('/home/jhoward/.fastai/data/imdb/train/unsupBow.feat'), PosixPath('/home/jhoward/.fastai/data/imdb/train/labeledBow.feat'), PosixPath('/home/jhoward/.fastai/data/imdb/train/neg')]
Now let's grab the whole data set which has:
• 25,000 reviews in training
• 25,000 interviews in validation
• 50,000 unsupervised movie reviews (50,000 movie reviews that haven't been scored at all)
### Language Model [19:44]
We're going to start with the language model. Now the good news is, we don't have to train the WikiText 103 language model. Not that it's difficult—you can just download the WikiText 103 corpus, and run the same code. But it takes two or three days on a decent GPU, so not much point in you doing it. You may as well start with ours. Even if you've got a big corpus of like medical documents or legal documents, you should still start with WikiText 103. There's just no reason to start with random weights. It's always good to use transfer learning if you can.
So we're gonna start fine-tuning our IMDB language model.
bs=48
data_lm = (TextList.from_folder(path) #Inputs: all the text files in path .filter_by_folder(include=['train', 'test']) #We may have other temp folders that contain text files so we only keep what's in train and test .random_split_by_pct(0.1) #We randomly split and keep 10% (10,000 reviews) for validation .label_for_lm() #We want to do a language model so we label accordingly .databunch(bs=bs))data_lm.save('tmp_lm')
We can say:
• It's a list of text files﹣the full IMDB actually is not in a CSV. Each document is a separate text file.
• Say where it is﹣in this case we have to make sure we just to include the train and test folders.
• We randomly split it by 0.1.
A little good trick
Now this is interesting﹣10%. Why are we randomly splitting it by 10% rather than using the predefined train and test they gave us? This is one of the cool things about transfer learning. Even though our validation set has to be held aside, it's actually only the labels that we have to keep aside. So we're not allowed to use the labels in the test set. If you think about in a Kaggle competition, you certainly can't use the labels because they don't even give them to you. But you can certainly use the independent variables. So in this case, you could absolutely use the text that is in the test set to train your language model. This is a good trick﹣when you do the language model, concatenate the training and test set together, and then just split out a smaller validation set so you've got more data to train your language model. So that's a little trick.
So if you're doing NLP stuff on Kaggle, for example, or you've just got a smaller subset of labeled data, make sure that you use all of the text you have to train in your language model, because there's no reason not to.
• How are we going to label it? Remember, a language model kind of has its own labels. The text itself is labels so label for a language model (label_for_lm) does that for us.
• And create a data bunch and save it. That takes a few minutes to tokenize and numericalize.
Since it takes some few minutes, we save it. Later on you can just load it. No need to run it again.
data_lm = TextLMDataBunch.load(path, 'tmp_lm', bs=bs)
data_lm.show_batch()
idx text 0 xxbos after seeing the truman show , i wanted to see the other films by weir . i would say this is a good one to start with . the plot : \n\n the wife of a doctor ( who is trying to impress his bosses , so he can get xxunk trying to finish her written course , while he s at work . but one day a strange man , who says that he s a plumber , tells her he s been called out to repair some pipes in there flat . 1 and turn to the wisdom of homeless people & ghosts . that 's a good plan . i would never recommend this movie ; partly because the sexual content is unnecessarily graphic , but also because it really does n't offer any valuable insight . check out " yentl " if you want to see a much more useful treatment of jewish tradition at odds with society . xxbos creep is the story of kate ( potente ) , an intensely unlikeable bourgeois bitch that finds herself somehow sleeping through the noise of the last 2 been done before but there is something about the way its done here that lifts it up from the rest of the pack . \n\n 8 out of 10 for dinosaur / monster lovers . xxbos i rented this movie to see how the sony xxunk camera shoots , ( i recently purchased the same camera ) and was blown away by the story and the acting . the directing , acting , editing was all above what i expected from what appeared at first glance to be a " low budget " type of 3 troubles . nigel and xxunk are the perfect team , i 'd watch their show any day ! i was so crushed when they removed it , and anytime they had it on xxup tv after that i was over the moon ! they put it on on demand one summer ( only the first eight episodes or so ) and i 'd spend whole afternoons watching them one after the other ... but the worst part ? it is now back on a channel called xxunk - and xxup it xxup 's xxup on 4 movie ! ) the movie is about edward , a obsessive - compulsive , nice guy , who happens to be a film editor . he is then lent to another department in the building , and he is sent to the posh yet violent world of sam campbell , the splatter and gore department . sam campbell , eddy 's new boss , is telling eddy about the big break on his movies , the gruesome loose limbs series , and he needs eddy to make the movie somewhat less violent so they can
Training [22:29]
At this point things are going to look very familiar. We create a learner:
learn = language_model_learner(data_lm, pretrained_model=URLs.WT103, drop_mult=0.3)
But instead of creating a CNN learner, we're going to create a language model learner. So behind the scenes, this is actually not going to create a CNN (a convolutional neural network), it's going to create an RNN (a recurrent neural network). We're going to be learning exactly how they're built over the coming lessons, but in short they're the same basic structure. The input goes into a weight matrix (i.e. a matrix multiply), that then you replace the negatives with zeros, and it goes into another matrix multiply, and so forth a bunch of times. So it's the same basic structure.
As usual, when we create a learner, you have to pass in two things:
• data_lm: language model data
• pretrained_model: what pre-trained model we want to use—here, the pre-trained model is the WikiText 103 model that will be downloaded for you from fastai if you haven't used it before just like ImageNet pre-trained models are downloaded for you.
This here (drop_mult=0.3) sets the amount of dropout. We haven't talked about that yet. We've talked briefly about this idea that there is something called regularization and you can reduce the regularization to avoid underfitting. So for now, just know that by using a number lower than one is because when I first tried to run this, I was under fitting. So if you reduced that number, then it will avoid under fitting.
Okay. so we've got a learner, we can lr_find and looks pretty standard:
learn.lr_find()
learn.recorder.plot(skip_end=15)
Then we can fit one cycle.
learn.fit_one_cycle(1, 1e-2, moms=(0.8,0.7))
Total time: 12:42epoch train_loss valid_loss accuracy1 4.591534 4.429290 0.251909 (12:42)
What's happening here is we are just fine-tuning the last layers. Normally after we fine-tune the last layers, the next thing we do is we go unfreeze and train the whole thing. So here it is:
learn.unfreeze()learn.fit_one_cycle(10, 1e-3, moms=(0.8,0.7))
Total time: 2:22:17epoch train_loss valid_loss accuracy1 4.307920 4.245430 0.271067 (14:14)2 4.253745 4.162714 0.281017 (14:13)3 4.166390 4.114120 0.287092 (14:14)4 4.099329 4.068735 0.292060 (14:10)5 4.048801 4.035339 0.295645 (14:12)6 3.980410 4.009860 0.298551 (14:12)7 3.947437 3.991286 0.300850 (14:14)8 3.897383 3.977569 0.302463 (14:15)9 3.866736 3.972447 0.303147 (14:14)10 3.847952 3.972852 0.303105 (14:15)
As you can see, even on a pretty beefy GPU that takes two or three hours. In fact, I'm still under fitting. So probably tonight, I might train it overnight and try and do a little bit better. I'm guessing I could probably train this a bit longer because you can see the accuracy hasn't started going down again. So I wouldn't mind try to train that a bit longer. But the accuracy, it's interesting. 0.3 means we're guessing the next word of the movie review correctly about a third of the time. That sounds like a pretty high number﹣the idea that you can actually guess the next word that often. So it's a good sign that my language model is doing pretty well. For more limited domain documents (like medical transcripts and legal transcripts), you'll often find this accuracy gets a lot higher. So sometimes this can be even 50% or more. But 0.3 or more is pretty good.
Predicting with Language Model [25:43]
You can now run learn.predict and pass in the start of a sentence, and it will try and finish off that sentence for you.
learn.predict('I liked this movie because ', 100, temperature=1.1, min_p=0.001)
Total time: 00:10
'I liked this movie because of course after yeah funny later that the world reason settings - the movie that perfect the kill of the same plot - a mention of the most of course . do xxup diamonds and the " xxup disappeared kill of course and the movie niece , from the care more the story of the let character , " i was a lot \'s the little performance is not only . the excellent for the most of course , with the minutes night on the into movies ( ! , in the movie its the first ever ! \n\n a'
Now I should mention, this is not designed to be a good text generation system. This is really more designed to check that it seems to be creating something that's vaguely sensible. There's a lot lot of tricks that you can use to generate much higher quality text﹣none of which we're using here. But you can kind of see that it's certainly not random words that it's generating. It sounds vaguely English like even though it doesn't make any sense.
At this point, we have a movie review model. So now we're going to save that in order to load it into our classifier (i.e. to be a pre-trained model for the classifier). But I actually don't want to save the whole thing. A lot of the second half of the language model is all about predicting the next word rather than about understanding the sentence so far. So the bit which is specifically about understanding the sentence so far is called the encoder, so I just save that (i.e. the bit that understands the sentence rather than the bit that generates the word).
learn.save_encoder('fine_tuned_enc')
### Classifier [27:18]
Now we're ready to create our classifier. Step one, as per usual, is to create a data bunch, and we're going to do basically exactly the same thing:
data_clas = (TextList.from_folder(path, vocab=data_lm.vocab) #grab all the text files in path .split_by_folder(valid='test') #split by train and valid folder (that only keeps 'train' and 'test' so no need to filter) .label_from_folder(classes=['neg', 'pos']) #remove docs with labels not in above list (i.e. 'unsup') .filter_missing_y() #label them all with their folders .databunch(bs=50))data_clas.save('tmp_clas')
But we want to make sure that it uses exactly the same vocab that are used for the language model. If word number 10 was the in the language model, we need to make sure that word number 10 is the in the classifier. Because otherwise, the pre-trained model is going to be totally meaningless. So that's why we pass in the vocab from the language model to make sure that this data bunch is going to have exactly the same vocab. That's an important step.
split_by_folder﹣remember, the last time we had split randomly, but this time we need to make sure that the labels of the test set are not touched. So we split by folder.
And then this time we label it not for a language model but we label these classes (['neg', 'pos']). Then finally create a data bunch.
Sometimes you'll find that you ran out of GPU memory. I was running this in an 11G machine, so you should make sure this number (bs) is a bit lower if you run out of memory. You may also want to make sure you restart the notebook and kind of start it just from here (classifier section). Batch size 50 is as high as I could get on an 11G card. If you're using a p2 or p3 on Amazon or the K80 on Google, for example, I think you'll get 16G so you might be able to make this bit higher, get it up to 64. So you can find whatever batch size fits on your card.
So here is our data bunch:
data_clas = TextClasDataBunch.load(path, 'tmp_clas', bs=bs)data_clas.show_batch()
text label xxfld 1 match 1 : tag team table match bubba ray and spike dudley vs eddie guerrero and chris benoit bubba ray and spike dudley started things off with a tag team table match against eddie guerrero and chris benoit . according to the rules of the match , both pos xxfld 1 i have never seen any of spike lee 's prior films , as their trailers never caught my interest . i have seen , and admire denzel washington , and jodie foster 's work , and have several of their dvds . i was , however , entirely neg xxfld 1 pier paolo pasolini , or pee - pee - pee as i prefer to call him ( due to his love of showing male genitals ) , is perhaps xxup the most overrated european marxist director - and they are thick on the ground . how anyone can neg xxfld 1 chris rock deserves better than he gives himself in " down to earth . " as directed by brothers chris & paul weitz of " american pie " fame , this uninspired remake of warren beatty 's 1978 fantasy " heaven can wait , " itself a rehash neg xxfld 1 yesterday , i went to the monthly antique flea market that comes to town . i really have no interest in such things , but i went for the fellowship of friends who do have such an interest . looking over the hundreds of vendor , passing many pos
learn = text_classifier_learner(data_clas, drop_mult=0.5)learn.load_encoder('fine_tuned_enc')learn.freeze()
This time, rather than creating a language model learner, we're creating a text classifier learner. But again, same thing﹣pass in the data that we want, figure out how much regularization we need. If you're overfitting then you can increase this number (drop_mult). If you're underfitting, you can decrease the number. And most importantly, load in our pre train model. Remember, specifically it's this half of the model called the encoder which is the bit that we want to load in.
Then freeze, lr_find, find the learning rate and fit for a little bit.
learn.lr_find()
learn.recorder.plot()
learn.fit_one_cycle(1, 2e-2, moms=(0.8,0.7))
Total time: 02:46epoch train_loss valid_loss accuracy1 0.294225 0.210385 0.918960 (02:46)
We're already up nearly to 92% accuracy after less than three minutes of training. So this is a nice thing. In your particular domain (whether it be law, medicine, journalism, government, or whatever), you probably only need to train your domain's language model once. And that might take overnight to train well. But once you've got it, you can now very quickly create all kinds of different classifiers and models with that. In this case, already a pretty good model after three minutes. So when you first start doing this. you might find it a bit annoying that your first models take four hours or more to create that language model. But the key thing to remember is you only have to do that once for your entire domain of stuff that you're interested in. And then you can build lots of different classifiers and other models on top of that in a few minutes.
learn.save('first')
learn.load('first');
learn.freeze_to(-2)learn.fit_one_cycle(1, slice(1e-2/(2.6**4),1e-2), moms=(0.8,0.7))
Total time: 03:03epoch train_loss valid_loss accuracy1 0.268781 0.180993 0.930760 (03:03)
We can save that to make sure we don't have to run it again.
And then, here's something interesting. I'm not going to say unfreeze. Instead, I'm going to say freeze_to. What that says is unfreeze the last two layers, don't unfreeze the whole thing. We've just found it really helps with these text classification not to unfreeze the whole thing, but to unfreeze one layer at a time.
• unfreeze the last two layers
• train it a little bit more
• unfreeze the next layer again
• train it a little bit more
• unfreeze the whole thing
• train it a little bit more
learn.save('second')
learn.load('second');
learn.freeze_to(-3)learn.fit_one_cycle(1, slice(5e-3/(2.6**4),5e-3), moms=(0.8,0.7))
Total time: 04:06epoch train_loss valid_loss accuracy1 0.211133 0.161494 0.941280 (04:06)
learn.save('third')
learn.load('third');
learn.unfreeze()learn.fit_one_cycle(2, slice(1e-3/(2.6**4),1e-3), moms=(0.8,0.7))
Total time: 10:01epoch train_loss valid_loss accuracy1 0.188145 0.155038 0.942480 (05:00)2 0.159475 0.153531 0.944040 (05:01)
You also see I'm passing in this thing (moms=(0.8,0.7))﹣momentums equals 0.8,0.7. We are going to learn exactly what that means probably next week. We may even automate it. So maybe by the time you watch the video of this, this won't even be necessary anymore. Basically we found for training recurrent neural networks (RNNs), it really helps to decrease the momentum a little bit. So that's what that is.
That gets us a 94.4% accuracy after about half an hour or less of training. There's quite a lot less of training the actual classifier. We can actually get this quite a bit better with a few tricks. I don't know if we'll learn all the tricks this part. It might be next part. But even this very simple standard approach is pretty great.
If we compare it to last year's state of the art on IMDb, this is from The CoVe paper from McCann et al. at Salesforce Research. Their paper was 91.8% accurate. And the best paper they could find, they found a fairly domain-specific sentiment analysis paper from 2017, they've got 94.1%. And here, we've got 94.4%. And the best models I've been able to build aince have been about 95.1%. So if you're looking to do text classification, this really standardized transfer learning approach works super well.
## Tabular [33:10]
So that was NLP. We'll be learning more about NLP later in this course. But now, I wanted to switch over and look at tabular. Now tabular data is pretty interesting because it's the stuff that, for a lot of you, is actually what you use day-to-day at work in spreadsheets, in relational databases, etc.
Question: Where does the magic number of $2.6^{4}$ in the learning rate come from? [33:38]
learn.fit_one_cycle(2, slice(1e-3/(2.6**4),1e-3), moms=(0.8,0.7))
Good question. So the learning rate is various things divided by 2.6 to the fourth. The reason it's to the fourth, you will learn about at the end of today. So let's focus on the 2.6. Why 2.6? Basically, as we're going to see in more detail later today, this number, the difference between the bottom of the slice and the top of the slice is basically what's the difference between how quickly the lowest layer of the model learns versus the highest layer of the model learns. So this is called discriminative learning rates. So really the question is as you go from layer to layer, how much do I decrease the learning rate by? And we found out that for NLP RNNs, the answer is 2.6.
How do we find out that it's 2.6? I ran lots and lots of different models using lots of different sets of hyper parameters of various types (dropout, learning rates, and discriminative learning rate and so forth), and then I created something called a random forest which is a kind of model where I attempted to predict how accurate my NLP classifier would be based on the hyper parameters. And then I used random forest interpretation methods to basically figure out what the optimal parameter settings were, and I found out that the answer for this number was 2.6. So that's actually not something I've published or I don't think I've even talked about it before, so there's a new piece of information. Actually, a few months after I did this, Stephen Merity and somebody else did publish a paper describing a similar approach, so the basic idea may be out there already.
Some of that idea comes from a researcher named Frank Hutter and one of his collaborators. They did some interesting work showing how you can use random forests to actually find optimal hyperparameters. So it's kind of a neat trick. A lot of people are very interested in this thing called Auto ML which is this idea of like building models to figure out how to train your model. We're not big fans of it on the whole. But we do find that building models to better understand how your hyper parameters work, and then finding those rules of thumb like oh basically it can always be 2.6 quite helpful. So there's just something we've kind of been playing with.
Back to Tabular [36:41]
Let's talk about tabular data. Tabular data such as you might see in a spreadsheet, a relational database, or financial report, it can contain all kinds of different things. I tried to make a little list of some of the kinds of things that I've seen tabular data analysis used for:
Using neural nets for analyzing tabular data﹣when we first presented this, people were deeply skeptical. They thought it was a terrible idea to use neural nets to analyze tabular data, because everybody knows that you should use logistic regression, random forests, or gradient boosting machines (all of which have their place for certain types of things). But since that time, it's become clear that the commonly held wisdom is wrong. It's not true that neural nets are not useful for tabular data, in fact they are extremely useful. We've shown this in quite a few of our courses, but what's really helped is that some really effective organizations have started publishing papers and posts describing how they've been using neural nets for analyzing tabular data.
One of the key things that comes up again and again is that although feature engineering doesn't go away, it certainly becomes simpler. So Pinterest, for example, replaced the gradient boosting machines that they were using to decide how to put stuff on their homepage with neural nets. And they presented at a conference this approach, and they described how it really made engineering a lot easier because a lot of the hand created features weren't necessary anymore. You still need some, but it was just simpler. So they ended up with something that was more accurate, but perhaps even more importantly, it required less maintenance. So I wouldn't say you it's the only tool that you need in your toolbox for analyzing tabular data. But where else, I used to use random forests 99% of the time when I was doing machine learning with tabular data, I now use neural nets 90% of the time. It's my standard first go-to approach now, and it tends to be pretty reliable and effective.
One of the things that's made it difficult is that until now there hasn't been an easy way to create and train tabular neural nets. Nobody has really made it available in a library. So we've actually just created fastai.tabular and I think this is pretty much the first time that's become really easy to use neural nets with tabular data. So let me show you how easy it is.
### Tabular examples [39:51]
This is actually coming directly from the examples folder in the fastai repo. I haven't changed it at all. As per usual, as well as importing fastai, import your application﹣so in this case, it's tabular.
from fastai import *from fastai.tabular import *
We assume that your data is in a Pandas DataFram. Pandas DataFrame is the standard format for tabular data in Python. There are lots of ways to get it in there, but probably the most common might be pd.read_csv. But whatever your data is in, you can probably get it into a Pandas data frame easily enough.
path = untar_data(URLs.ADULT_SAMPLE)df = pd.read_csv(path/'adult.csv')
Question: What are the 10% of cases where you would not default to neural nets? [40:41]
Good question. I guess I still tend to give them a try. But yeah, I don't know. It's kind of like as you do things for a while, you start to get a sense of the areas where things don't quite work as well. I have to think about that during the week. I don't think I have a rule of thumb. But I would say, you may as well try both. I would say try a random forest and try a neural net. They're both pretty quick and easy to run, and see how it looks. If they're roughly similar, I might dig into each and see if I can make them better. But if the random forest is doing way better, I'd probably just stick with that. Use whatever works.
So we start with the data in a data frame, and so we've got an adult sample﹣it's a classic old dataset. It's a pretty small simple old dataset that's good for experimenting with. And it's a CSV file, so you can read it into a data frame with Pandas read CSV (pd.read_csv). If your data is in a relational database, Pandas can read from that. If it's in spark or Hadoop, Pandas can read from that. Pandas can read from most stuff that you can throw at it. So that's why we use it as a default starting point.
dep_var = '>=50k'cat_names = ['workclass', 'education', 'marital-status', 'occupation', 'relationship', 'race']cont_names = ['age', 'fnlwgt', 'education-num']procs = [FillMissing, Categorify, Normalize]
test = TabularList.from_df(df.iloc[800:1000].copy(), path=path, cat_names=cat_names, cont_names=cont_names)
data = (TabularList.from_df(df, path=path, cat_names=cat_names, cont_names=cont_names, procs=procs) .split_by_idx(list(range(800,1000))) .label_from_df(cols=dep_var) .add_test(test, label=0) .databunch())
As per usual, I think it's nice to use the data block API. So in this case, the list that we're trying to create is a tabular list and we're going to create it from a data frame. So you can tell it:
• What the data frame is.
• What the path that you're going to use to save models and intermediate steps is.
• Then you need to tell it what are your categorical variables and what are your continuous variables.
### Continuous vs. Categorical [43:07]
We're going to be learning a lot more about what that means to the neural net next week, but for now the quick summary is this. Your independent variables are the things that you're using to make predictions with. So things like education, marital status, age, and so forth. Some of those variables like age are basically numbers. They could be any number. You could be 13.36 years old or 19.4 years old or whatever. Where else, things like marital status are options that can be selected from a discrete group: married, single, divorced, whatever. Sometimes those options might be quite a lot more, like occupation. There's a lot of possible occupations. And sometimes, they might be binary (i.e. true or false). But anything which you can select the answer from a small group of possibilities is called a categorical variable. So we're going to need to use a different approach in the neural net to modeling categorical variables to what we use for continuous variables. For categorical variables, we're going to be using something called embeddings which we'll be learning about later today. For continuous variables, they could just be sent into the neural net just like pixels in a neural net can. Because pixels in a neural net are already numbers; these continuous things are already numbers as well. So that's easy.
So that's why you have to tell the tabular list from data frame which ones are which. There are some other ways to do that by pre-processing them in Pandas to make things categorical variables, but it's kind of nice to have one API for doing everything; you don't have to think too much about it.
### Processor [45:04]
Then we've got something which is a lot like transforms in computer vision. Transforms in computer vision do things like flip a photo on its axis, turn it a bit, brighten it, or normalize it. But for tabular data, instead of having transforms, we have things called processes. And they're nearly identical but the key difference, which is quite important, is that a processor is something that happens ahead of time. So we basically pre-process the data frame rather than doing it as we go. So transformations are really for data augmentation﹣we want to randomize it and do it differently each time. Or else, processes are the things that you want to do once, ahead of time.
procs = [FillMissing, Categorify, Normalize]
We have a number of processes in the fastai library. And the ones we're going to use this time are:
• FillMissing: Look for missing values and deal with them some way.
• Categorify: Find categorical variables and turn them into Pandas categories
• Normalize : Do a normalization ahead of time which is to take continuous variables and subtract their mean and divide by their standard deviation so they are zero-one variables.
The way we deal with missing data, we'll talk more about next week, but in short, we replace it with the median and add a new column which is a binary column of saying whether that was missing or not.
For all of these things, whatever you do to the training set, you need to do exactly the same thing to the validation set and the test set. So whatever you replaced your missing values with, you need to replace them with exactly the same thing in the validation set. So fastai handles all these details for you. They are the kinds of things that if you have to do it manually, if you like me, you'll screw it up lots of times until you finally get it right. So that's what these processes are here.
Then we're going to split into training versus validation sets. And in this case, we do it by providing a list of indexes so the indexes from 800 to a thousand. It's very common. I don't quite remember the details of this dataset, but it's very common for wanting to keep your validation sets to be contiguous groups of things. If they're map tiles, they should be the map tiles that are next to each other, if their time periods, they should be days that are next to each other, if they are video frames, they should be video frames next to each other. Because otherwise you're kind of cheating. So it's often a good idea to use split_by_idx and to grab a range that's next to each other if your data has some kind of structure like that or find some other way to structure it in that way.
All right, so that's now given us a training and a validation set. We now need to add labels. In this case, the labels can come straight from the data frame we grabbed earlier, so we just have to tell it which column it is. So the dependent variable is whether they're making over \$50,000 salary. That's the thing we're trying to predict.
We'll talk about test sets later, but in this case we can add a test set. And finally get our data bunch. At that point, we have something that looks like this:
data.show_batch(rows=10)
workclass education marital-status occupation relationship race education-num_na age fnlwgt education-num target Private Prof-school Married-civ-spouse Prof-specialty Husband White False 0.1036 0.9224 1.9245 1 Self-emp-inc Bachelors Married-civ-spouse Farming-fishing Husband White False 1.7161 -1.2654 1.1422 1 Private HS-grad Never-married Adm-clerical Other-relative Black False -0.7760 1.1905 -0.4224 0 Private 10th Married-civ-spouse Sales Own-child White False -1.5823 -0.0268 -1.5958 0 Private Some-college Never-married Handlers-cleaners Own-child White False -1.3624 0.0284 -0.0312 0 Private Some-college Married-civ-spouse Prof-specialty Husband White False 0.3968 0.4367 -0.0312 1 ? Some-college Never-married ? Own-child White False -1.4357 -0.7295 -0.0312 0 Self-emp-not-inc 5th-6th Married-civ-spouse Sales Husband White False 0.6166 -0.6503 -2.7692 1 Private Some-college Married-civ-spouse Sales Husband White False 1.5695 -0.8876 -0.0312 1 Local-gov Some-college Never-married Handlers-cleaners Own-child White False -0.6294 -1.5422 -0.0312 0
There is our data. Then to use it, it looks very familiar. You get a learner, in this case it's a tabular learner, passing in the data, some information about your architecture, and some metrics. And you then call fit.
learn = tabular_learner(data, layers=[200,100], metrics=accuracy)
learn.fit(1, 1e-2)
Total time: 00:03epoch train_loss valid_loss accuracy1 0.362837 0.413169 0.785000 (00:03)
Question: How to combine NLP (tokenized) data with meta data (tabular data) with Fastai? For instance, for IMDb classification, how to use information like who the actors are, year made, genre, etc. [49:14]
Yeah, we're not quite up to that yet. So we need to learn a little bit more about how neural net architectures work as well. But conceptually, it's kind of the same as the way we combine categorical variables and continuous variables. Basically in the neural network, you can have two different sets of inputs merging together into some layer. It could go into an early layer or into a later layer, it kind of depends. If it's like text and an image and some metadata, you probably want the text going into an RNN, the image going into a CNN, the metadata going into some kind of tabular model like this. And then you'd have them basically all concatenated together, and then go through some fully connected layers and train them end to end. We will probably largely get into that in part two. In fact we might entirely get into that in part two. I'm not sure if we have time to cover it in part one. But conceptually, it's a fairly simple extension of what we'll be learning in the next three weeks.
Question: Do you think that things like scikit-learn and xgboost will eventually become outdated? Will everyone will use deep learning tools in the future? Except for maybe small datasets? [50:36]
I have no idea. I'm not good at making predictions. I'm not a machine learning model. I mean xgboost is a really nice piece of software. There's quite a few really nice pieces of software for gradient boosting in particular. Actually, random forests in particular has some really nice features for interpretation which I'm sure we'll find similar versions for neural nets, but they don't necessarily exist yet. So I don't know. For now, they're both useful tools. scikit-learn is a library that's often used for pre-processing and running models. Again, it's hard to predict where things will end up. In some ways, it's more focused on some older approaches to modeling, but I don't know. They keep on adding new things, so we'll see. I keep trying to incorporate more scikit-learn stuff into fastai and then I keep finding ways I think I can do it better and I throw it away again, so that's why there's still no scikit-learn dependencies in fastai. I keep finding other ways to do stuff.
[52:12]
We're gonna learn what layers= means either towards the end of class today or the start of class next week, but this is where we're basically defining our architecture just like when we chose ResNet34 or whatever for conv nets. We'll look at more about metrics in a moment, but just to remind you, metrics are just the things that get printed out. They don't change our model at all. So in this case, we're saying I want you to print out the accuracy to see how we're doing.
So that's how to do tabular. This is going to work really well because we're gonna hit our break soon. And the idea was that after three and a half lessons, we're going to hit the end of all of the quick overview of applications, and then I'm going to go down on the other side. I think we're going to be to the minute, we're going to hit it. Because the next one is collaborative filtering.
## Collaborative Filtering [53:08]
Collaborative filtering is where you have information about who bought what, or who liked what—it's basically something where you have something like a user, a reviewer, or whatever and information about what they've bought, what they've written about, or what they reviewed. So in the most basic version of collaborative filtering, you just have two columns: something like user ID and movie ID and that just says this user bought that movie. So for example, Amazon has a really big list of user IDs and product IDs like what did you buy. Then you can add additional information to that table such as oh, they left a review, what review did they give it? So it's now like user ID, movie ID, number of stars. You could add a timecode so this user bought this product at this time and gave it this review. But they are all basically the same structure.
There are two ways you could draw that collaborative filtering structure. One is a two-column approach where you've got user and movie. And you've got user ID, movie ID﹣each pair basically describes that user watch that movie, possibly also number of stars (3, 4, etc). The other way you could write it would be you could have like all the users down here and all the movies along here. And then, you can look and find a particular cell in there to find out what could be the rating of that user for that movie, or there's just a 1 there if that user watched that movie, or whatever.
So there are two different ways of representing the same information. Conceptually, it's often easier to think of it this way (the matrix on the right), but most of the time you won't store it that way. Explicitly because most of the time, you'll have what's called a very sparse matrix which is to say most users haven't watched most movies or most customers haven't purchased most products. So if you store it as a matrix where every combination of customer and product is a separate cell in that matrix, it's going to be enormous. So you tend to store it like the left or you can store it as a matrix using some kind of special sparse matrix format. If that sounds interesting, you should check out Rachel's computational linear algebra course on fastai where we have lots and lots of information about sparse matrix storage approaches. For now though, we're just going to kind of keep it in this format on left hand side.
[56:38]
For collaborative filtering, there's a really nice dataset called MovieLens created by GroupLens group and you can download various different sizes (20 million ratings, 100,000 ratings). We've actually created an extra small version for playing around with which is what we'll start with today. And then probably next week, we'll use the bigger version.
from fastai import *from fastai.collab import *from fastai.tabular import *
You can grab the small version using URLs.ML_SAMPLE:
user,item,title = 'userId','movieId','title'
path = untar_data(URLs.ML_SAMPLE)path
PosixPath('/home/jhoward/.fastai/data/movie_lens_sample')
ratings = pd.read_csv(path/'ratings.csv')ratings.head()
userId movieId rating timestamp 0 73 1097 4.0 1255504951 1 561 924 3.5 1172695223 2 157 260 3.5 1291598691 3 358 1210 5.0 957481884 4 130 316 2.0 1138999234
It's a CSV so you can read it with Pandas and here it is. It's basically a list of user IDs﹣we don't actually know anything about who these users are. There's some movie IDs. There is some information about what the movies are, but we won't look at that until next week. Then there's the rating and the timestamp. We're going to ignore the timestamp for now. So that's a subset of our data. head in Pandas is just the first rows.
So now that we've got a data frame, the nice thing about collaborative filtering is it's incredibly simple.
data = CollabDataBunch.from_df(ratings, seed=42)
y_range = [0,5.5]
learn = collab_learner(data, n_factors=50, y_range=y_range)
That's all the data that we need. So you can now go ahead and say get collab_learner and you can pass in the data bunch. The architecture, you have to tell it how many factors you want to use, and we're going to learn what that means after the break. And then something that could be helpful is to tell it what the range of scores are. We're going to see how that helps after the break as well. So in this case, the minimum score is 0, the maximum score is 5.
learn.fit_one_cycle(3, 5e-3)
Total time: 00:04epoch train_loss valid_loss1 1.600185 0.962681 (00:01)2 0.851333 0.678732 (00:01)3 0.660136 0.666290 (00:01)
Now that you've got a learner, you can go ahead and call fit_one_cycle and trains for a few epochs, and there it is. So at the end of it, you now have something where you can pick a user ID and a movie ID, and guess whether or not that user will like that movie.
### Cold start problem [58:55]
This is obviously a super useful application that a lot of you are probably going to try during the week. In past classes, a lot of people have taken this collaborative filtering approach back to their workplaces and discovered that using it in practice is much more tricky than this. Because in practice, you have something called the cold start problem. So the cold start problem is that the time you particularly want to be good at recommending movies is when you have a new user, and the time you particularly care about recommending a movie is when it's a new movie. But at that point, you don't have any data in your collaborative filtering system and it's really hard.
As I say this, we don't currently have anything built into fastai to handle the cold start problem and that's really because the cold start problem, the only way I know of to solve it (in fact, the only way I think that conceptually can solve it) is to have a second model which is not a collaborative filtering model but a metadata driven model for new users or new movies.
I don't know if Netflix still does this, but certainly what they used to do when I signed up to Netflix was they started showing me lots of movies and saying "have you seen this?" "did you like it?" ﹣so they fixed the cold start problem through the UX, so there was no cold stat problem. They found like 20 really common movies and asked me if I liked them, they used my replies to those 20 to show me 20 more that I might have seen, and by the time I had gone through 60, there was no cold start problem anymore.
For new movies, it's not really a problem because like the first hundred users who haven't seen the movie go in and say whether they liked it, and then the next hundred thousand, the next million, it's not a cold start problem anymore.
The other thing you can do if you, for whatever reason, can't go through that UX of asking people did you like those things (for example if you're selling products and you don't really want to show them a big selection of your products and say did you like this because you just want them to buy), you can instead try and use a metadata based tabular model what geography did they come from maybe you know their age and sex, you can try and make some guesses about the initial recommendations.
So collaborative filtering is specifically for once you have a bit of information about your users and movies or customers and products or whatever.
[1:01:37]
Question: How does the language model trained in this manner perform on code switched data (Hindi written in English words), or text with a lot of emojis?
Text with emojis, it'll be fine. There's not many emojis in Wikipedia and where they are at Wikipedia it's more like a Wikipedia page about the emoji rather than the emoji being used in a sensible place. But you can (and should) do this language model fine-tuning where you take a corpus of text where people are using emojis in usual ways, and so you fine-tune the WikiText language model to your reddit or Twitter or whatever language model. And there aren't that many emojis if you think about it. There are hundreds of thousands of possible words that people can be using, but a small number of possible emojis. So it'll very quickly learn how those emojis are being used. So that's a piece of cake.
I'm not really familiar with Hindi, but I'll take an example I'm very familiar with which is Mandarin. In Mandarin, you could have a model that's trained with Chinese characters. There are about five or six thousand Chinese characters in common use, but there's also a romanization of those characters called pinyin. It's a bit tricky because although there's a nearly direct mapping from the character to the pinyin (I mean there is a direct mapping but that pronunciations are not exactly direct), there isn't direct mapping from the pinyin to the character because one pinyin corresponds to multiple characters.
So the first thing to note is that if you're going to use this approach for Chinese, you would need to start with a Chinese language model.
Actually fastai has something called Language Model Zoo where we're adding more and more language models for different languages, and also increasingly for different domain areas like English medical texts or even language models for things other than NLP like genome sequences, molecular data, musical MIDI notes, and so forth. So you would you obviously start there.
To then convert that (in either simplified or traditional Chinese) into pinyin, you could either map the vocab directly, or as you'll learn, these multi-layer models﹣it's only the first layer that basically converts the tokens into a set of vectors, you can actually throw that away and fine-tune just the first layer of the model. So that second part is going to require a few more weeks of learning before you exactly understand how to do that and so forth, but if this is something you're interested in doing, we can talk about it on the forum because it's a nice test of understanding.
Question: What about time series on tabular data? is there any RNN model involved in tabular.models? [1:05:09]
We're going to look at time series tabular data next week, but the short answer is generally speaking you don't use a RNN for time series tabular data but instead, you extract a bunch of columns for things like day of week, is it a weekend, is it a holiday, was the store open, stuff like that. It turns out that adding those extra columns which you can do somewhat automatically basically gives you state-of-the-art results. There are some good uses of RNNs for time series, but not really for these kind of tabular style time series (like retail store logistics databases, etc).
Question: Is there a source to learn more about the cold start problem? [1:06:14]
I'm gonna have to look that up. If you know a good resource, please mention it on the forums.
The half way point [1:06:34]
That is both the break in the middle of lesson 4, it's the halfway point of the course, and it's the point at which we have now seen an example of all the key applications. So the rest of this course is going to be digging deeper into how they actually work behind the scenes, more of the theory, more of how the source code is written, and so forth. So it's a good time to have a nice break. Furthermore, it's my birthday today, so it's a really special moment.
### Collaborative filter with Microsoft Excel [1:07:25]
Microsoft Excel is one of my favorite ways to explore data and understand models. I'll make sure I put this in the repo, and actually this one, we can probably largely do in Google sheets. I've tried to move as much as I can over the last few weeks into Google sheets, but I just keep finding this is such a terrible product, so please try to find a copy of Microsoft Excel because there's nothing close, I've tried everything. Anyway, spreadsheets get a bad rap from people that basically don't know how to use them. Just like people who spend their life on Excel and then they start using Python, and they're like what the heck is this stupid thing. It takes thousands of hours to get really good at spreadsheets, but a few dozen hours to get confident at them. Once you're confident at them, you can see everything in front of you. It's all laid out, it's really great.
### Jeremy's spreadsheet tip of the day! [1:08:37]
I'll give you one spreadsheet tip today which is if you hold down the ctrl key or command key on your keyboard and press the arrow keys, here's ctrl+➜, it takes you to the end of a block of a table that you're in. And it's by far the best way to move around the place, so there you go.
In this case, I want to skip around through this table, so I can hit ctrl+ ⬇︎ ➜ to get to the bottom right, ctrl+ ⬅︎ ⬆︎ to get to the top left. Skip around and see what's going on.
So here's some data, and as we talked about, one way to look at collaborative filtering data is like this:
What we did was we grabbed from the MovieLens data the people that watched the most movies and the movies that were the most watched, and just limited the dataset down to those 15. As you can see, when you do it that way, it's not sparse anymore. There's just a small number of gaps.
This is something that we can now build a model with. How can we build a model? What we want to do is we want to create something which can predict for user 293, will they like movie 49, for example. So we've got to come up with some function that can represent that decision.
Here's a simple possible approach. We're going to take this idea of doing some matrix multiplications. So I've created here a random matrix. So here's one matrix of random numbers (the left). And I've created here another matrix of random numbers (the top). More specifically, for each movie, I've created five random numbers, and for each user, I've created five random numbers.
So we could say, then, that user 14, movie 27; did they like it or not? Well, the rating, what we could do would be to multiply together this vector (red) and that vector (purple). We could do a dot product, and here's the dot product. Then we can basically do that for every possible thing in here. And thanks to spreadsheets, we can just do that in one place and copy it over, and it fills in the whole thing for us. Why would we do it this way? Well, this is the basic starting point of a neural net, isn't it? A basic starting point of a neural net is that you take the matrix multiplication of two matrices, and that's what your first layer always is. So we just have to come up with some way of saying what are two matrices that we can multiply. Clearly, you need a vector for a user (a matrix for all the users) and a vector for a movie (a matrix for all the movies) and multiply them together, and you get some numbers. So they don't mean anything yet. They're just random. But we can now use gradient descent to try to make these numbers (top) and these numbers (left) give us results that are closer to what we wanted.
So how do we do that? Well, we set this up now as a linear model, so the next thing we need is a loss function. We can calculate our loss function by saying well okay movie 27 for user ID 14 should have been a rating of 3. With this random matrices, it's actually a rating of 0.91, so we can find the sum of squared errors would be $(3-0.91)^{2}$ and then we can add them up. So there's actually a sum squared in Excel already sum X minus y squared (SUMXMY2), so we can use just sum X minus y squared function, passing in those two ranges and then divide by the count to get the mean.
Here is a number that is the square root of the mean squared error. You sometimes you'll see people talk about MSE so that's the Mean Squared Error, sometimes you'll see RMSE that's the Root Mean Squared Error. Since I've got a square root at the front, this is the square root mean square error.
Excel Solver [1:14:30]
We have a loss, so now all we need to do is use gradient descent to try to modify our weight matrices to make that loss smaller. Excel will do that for me.
If you don't have solver, go to Excel Options → Add-ins, and enable "Solver Add-in".
The gradient descent solver in Excel is called "Solver" and it just does normal gradient descent. You just go Data → Solver (you need to make sure that in your settings that you've enabled the solver extension which comes with Excel) and all you need to do is say which cell represents my loss function. So there it is, cell V41. Which cells contain your variables, and so you can see here, I've got H19 to V23 which is up here, and B25 to F39 which is over there, then you can just say "okay, set your loss function to a minimum by changing those cells" and click on Solve:
You'll see the starts a 2.81, and you can see the numbers going down. And all that's doing is using gradient descent exactly the same way that we did when we did it manually in the notebook the other day. But it's rather than solving the mean squared error for [email protected] in Python, instead it is solving the loss function here which is the mean squared error of the dot product of each of those vectors by each of these vectors.
We'll let that run for a little while and see what happens. But basically in micro, here is a simple way of creating a neural network which is really in this case, it's like just a single linear layer with gradient descent to solve a collaborative filtering problem.
Back to the notebook [1:17:02]
Let's go back and see what we do over here.
data = CollabDataBunch.from_df(ratings, seed=42)
y_range = [0,5.5]
learn = collab_learner(data, n_factors=50, y_range=y_range)
learn.fit_one_cycle(3, 5e-3)
Total time: 00:04epoch train_loss valid_loss1 1.600185 0.962681 (00:01)2 0.851333 0.678732 (00:01)3 0.660136 0.666290 (00:01)
So over here we used collab_learner to get a model. So the the function that was called in the notebook was collab_learner and as you dig deeper into deep learning, one of the really good ways to dig deeper into deep learning is to dig into the fastai source code and see what's going on. So if you're going to be able to do that, you need to know how to use your editor well enough to dig through the source code. Basically there are two main things you need to know how to do: 1. Jump to a particular "symbol", like a particular class or function by its name 2. When you're looking at a particular symbol, to be able to jump to its implementation
For example in this case, I want to find def collab_learner. In most editors including the one I use, vim, you can set it up so that you can hit tab or something and it jumps through all the possible completions, and you can hit enter and it jumps straight to the definition for you. So here is the definition of collab_learner. As you can see, it's pretty small as these things tend to be, and the key thing it does is to create model of a particular kind which is an EmbeddingDotBias model passing in the various things you asked for. So you want to find out in your editor how you jump to the definition of that, which in vim you just hit Ctrl+] and here is the definition of EmbeddingDotBias.
Now we have everything on screen at once, and as you can see there's not much going on. The models that are being created for you by fastai are actually PyTorch models. And a PyTorch model is called an nn.Module that's the name in PyTorch of their models. It's a little more nuanced than that, but that's a good starting point for now. When a PyTorch nn.Module is run (when you calculate the result of that layer, neural net, etc), specifically, it always calls a method for you called forward. So it's in here that you get to find out how this thing is actually calculated.
When the model is built at the start, it calls this thing called __init__ as we've briefly mentioned before in Python people tend to call this "dunder init". So dunder init is how we create the model, and forward is how we run the model.
One thing if you're watching carefully, you might notice is there's nothing here saying how to calculate the gradients of the model, and that's because PyTorch does it for us. So you only have to tell it how to calculate the output of your model, and PyTorch will go ahead and calculate the gradients for you.
So in this case, the model contains:
• a set of weights for a user
• a set of weights for an item
• a set of biases for a user
• a set of biases for an item
And each one of those is coming from this thing called embedding. Here is the definition of embedding:
All it does is it calls this PyTorch thing called nn.Embedding. In PyTorch, they have a lot of standard neural network layers set up for you. So it creates an embedding. And then this thing here (trunc_normal_) is it just randomizes it. This is something which creates normal random numbers for the embedding.
## Embedding [1:21:41]
So what's an embedding? An embedding, not surprisingly, is a matrix of weights. Specifically, an embedding is a matrix of weights that looks something like this:
It's a matrix of weights which you can basically look up into, and grab one item out of it. So basically an embedding matrix is just a weight matrix that is designed to be something that you index into it as an array, and grab one vector out of it. That's what an embedding matrix is. In our case, we have an embedding matrix for a user and an embedding matrix for a movie. And here, we have been taking the dot product of them:
But if you think about it, that's not quite enough. Because we're missing this idea that maybe there are certain movies that everybody likes more. Maybe there are some users that just tend to like movies more. So I don't really just want to multiply these two vectors together, but I really want to add a single number of like how popular is this movie, and add a single number of like how much does this user like movies in general. So those are called "bias" terms. Remember how I said there's this idea of bias and the way we dealt with that in our gradient descent notebook was we added a column of 1's. But what we tend to do in practice is we actually explicitly say I want to add a bias term. So we don't just want to have prediction equals dot product of these two things, we want to say it's the dot product of those two things plus a bias term for a movie plus a bias term for user ID.
Back to code [1:23:55]
So that's basically what happens. We when we set up the model, we set up the embedding matrix for the users and the embedding matrix for the items. And then we also set up the bias vector for the users and the bias vector for the items.
Then when we calculate the model, we literally just multiply the two together. Just like we did. We just take that product, we call it dot. Then we add the bias, and (putting aside y_range for a moment) that's what we return. So you can see that our model is literally doing what we did in the spreadsheet with the tweak that we're also adding the bias. So it's an incredibly simple linear model. For these kinds of collaborative filtering problems, this kind of simple linear model actually tends to work pretty well.
Then there's one tweak that we do at the end which is that in our case we said that there's y range of between 0 and 5.5. So here's something to point out. So you do that dot product and you add on the two biases and that could give you any possible number along the number line from very negative through to very positive numbers. But we know that we always want to end up with a number between zero and five. What if we mapped that number line like so, to this function. The shape of that function is called a sigmoid. And so, it's gonna asymptote to five and it's gonna asymptote to zero.
That way, whatever number comes out of our dot product and adding the biases, if we then stick it through this function, it's never going to be higher than 5 and never going to be smaller than 0. Now strictly speaking, that's not necessary. Because our parameters could learn a set of weights that gives about the right number. So why would we do this extra thing if it's not necessary? The reason is, we want to make its life as easy for our model as possible. If we actually set it up so it's impossible for it to ever predict too much or too little, then it can spend more of its weights predicting the thing we care about which is deciding who's going to like what movie. So this is an idea we're going to keep coming back to when it comes to like making neural network's work better. It's about all these little decisions that we make to basically make it easier for the network to learn the right thing. So that's the last tweak here:
return torch.sigmoid(res) * (self.y_range[1]-self.y_range[0]) + self.y_range[0]
We take the result of this dot product plus biases, we put it through a sigmoid. A sigmoid is just a function which is basically$\frac{1}{1+e^{-x}}$, but the definition doesn't much matter. But it just has the shape that I just mentioned, and that goes between 0 and 1. If you then multiply that by y_range[1] minus y_range[0] plus y_range[0], then that's going to give you something that's between y_range[0] and y_range[1].
So that means that this tiny little neural network, I mean it's a push to call it a neural network. But it is a neural network with one weight matrix and no nonlinearities. So it's kind of the world's most boring neural network with a sigmoid at the end. I guess it does have a non-linearity. The sigmoid at the end is the non-linearity, it only has one layer of weights. That actually turns out to give close to state-of-the-art performance. I've looked up online to find out like what are the best results people have on this MovieLens 100k database, and the results I get from this little thing is better than any of the results I can find from the standard commercial products that you can download that are specialized for this. And the trick seems to be that adding this little sigmoid makes a big difference.
[1:29:09]
Question: There was a question about how you set up your vim, and I've already linked to your .vimrc but I wanted know if you had more to say about. They really like your setup 🙂
You like my setup? There's almost nothing in my setup. It's pretty bare honestly. I mean whatever you're doing with your editor, you probably want it to look like this which is when you've got a class that you're not currently working on it should be this is called folded/folding﹣it should be closed up so you can't see it. So you basically want something where it's easy to close and open folds, so vim already does all this for you. Then as I mentioned, you also want something where you can jump to the definition of things which in vim called using tags (e.g. to jump to the definition of Learner, position the cursor over Learner and hit Ctrl+]). Basically vim already does all this for you. You just have to read instructions. My .vimrc is minimal. I basically hardly use any extensions or anything. Another great editor to use is a Visual Studio Code. It's free and it's awesome and it has all the same features that you're seeing that vim does, basically VS Sode does all of those things as well. I quite like using vim because I can use it on the remote machine and play around, but you can of course just clone the git repo into your local computer and open it up with VS Code to play around with. Just don't try and look through the code just on github or something. That's going to drive you crazy. You need to be able to open it and close it and jump and jump back. Maybe people can create some threads on the forum for vim tips, VS Code tips, Sublime tips, whatever. For me, I would if you're gonna pick an editor, if you want to use something on your local, I would go with the VS Code today. I think it's the best. If you want to use something on the terminal side, I would go with VIM or Emacs, to me they're clear winners.
## Neural Network - overview of important terminology [1:31:24]
So what I wanted to close with today is, to take this collaborative filtering example and describe how we're going to build on top of it for the next three lessons to create the more complex neural networks we've been seeing. Roughly speaking, this is the bunch of concepts that we need to learn about:
• Inputs
• Weights/parameters
• Random
• Activations
• Activation functions / nonlinearities
• Output
• Loss
• Metric
• Cross-entropy
• Softmax
• Fine tuning
• Layer deletion and random weights
• Freezing & unfreezing
Let's think about what happens when you're using a neural network to do image recognition. Let's take a single pixel. You've got lots of pixels, but let's take a single pixel. So you've got a red a green and a blue pixel. Each one of those is some number between 0 and 255, or we normalize them so they have the mean of zero and standard deviation of one. But let's just do 0 to 255 version. So red: 10, green: 20, blue 30. So what do we do with these? Well, what we do is we basically treat that as a vector, and we multiply it by a matrix. So this matrix (depending on how you think of the rows and the columns), let's treat the matrix is having three rows and then how many columns? You get to pick. Just like with the collaborative filtering version, I decided to pick a vector of size five for each of my embedding vectors. So that would mean that's an embedding of size 5. You get to pick how big your weight matrix is. So let's make it size 5. This is 3 by 5.
Initially, this weight matrix contains random numbers. Remember we looked at embedding weight matrix just now?
There were two lines; the first line was create the matrix, and the second was fill it with random numbers? That's all we do. I mean it all gets hidden behind the scenes by fastai and PyTorch, but that's all it's doing. So it's creating a matrix of random numbers when you set it up. The number of rows has to be 3 to match the input, and the number of columns can be as big as you like. So after you multiply the input vector by that weight matrix, you're going to end up with a vector of size 5.
People often ask how much linear algebra do I need to know to be able to do deep learning. This is the amount you need. And if you're not familiar with this, that's fine. You need to know about matrix products. You don't need to know a lot about them, you just need to know like computationally what are they and what do they do. You've got to be very comfortable with if a matrix of size blah times a matrix of size blah gives a matrix or size blah (i.e. how do the dimensions match up). So if you have 3, and they remember in NumPy and PyTorch, we use @ times 3 by 5 gives a vector of size 5.
Then what happens next; it goes through an activation function such as ReLU which is just max(0,x) and spits out a new vector which is, of course, going to be exactly the same size because no activation function changes the size﹣it only changes the contents. So that's still of size 5.
What happens next? We multiply by another matrix. Again, it can be any number of columns, but the number of rows has to map nicely. So it's going to be 5 by whatever. Maybe this one has 5, let's say, by 10. That's going to give some output﹣it should be size 10 and again we put that through ReLU, and again that gives us something of the same size.
Then we can put that through another matrix. Actually, just to make this a bit clearer (you'll see why in a moment), I'm going to use 8 not 10.
Let's say we're doing digit recognition. There are ten possible digits, so my last weight matrix has to be 10 in size. Because then that's going to mean my final output is a vector of 10 in size. Remember if you're doing that digit recognition, we take our actuals which is 10 in size. And if the number we're trying to predict was the number 3, then that means that there is a 1 in the third position ([0,0,0,1,0,...]).
So what happens is our neural net runs along starting with our input, and going weight matrix→ReLU→ weight matrix→ReLU→ weight matrix→ final output. Then we compare these two together to see how close they are (i.e. how close they match) using some loss function and we'll learn about all the loss functions that we use next week. For now, the only one we've learned is mean squared error. And we compare the output (you can think of them as probabilities for each of the 10) to the actual each of the 10 to get a loss, and then we find the gradients of every one of the weight matrices with respect to that, and we update the weight matrices.
The main thing I wanted to show right now is the terminology we use because it's really important.
These things (yellow) contain numbers. Specifically they initially are matrices containing random numbers. And we can refer to these yellow things, in PyTouch, they're called parameters. Sometimes we'll refer to them as weights, although weights is slightly less accurate because they can also be biases. But we kind of use the terms a little bit interchangeably. Strictly speaking, we should call them parameters.
Then after each of those matrix products, that calculates a vector of numbers. Here are some numbers (blue) that are calculated by a weight matrix multiply. And then there's some other set of numbers (purple) that are calculated as a result of a ReLU as well as the activation function. Either one is called activations.
Activations and parameters, both refer to numbers. They are numbers. But Parameters are numbers that are stored, they are used to make a calculation. Activations are the result of a calculation﹣the numbers that are calculated. So they're the two key things you need to remember.
So use these terms, and use them correctly and accurately. And if you read these terms, they mean these very specific things. So don't mix them up in your head. And remember, they're nothing weird and magical﹣they are very simple things.
• An activation is the result of either a matrix multiply or an activation function.
• Parameters are the numbers inside the matrices that we multiply by.
That's it. Then there are some special layers. Every one of these things that does a calculation, all of these things that does a calculation (red arrow), are all called layers. They're the layers of our neural net. So every layer results in a set of activations because there's a calculation that results in a set of results.
There's a special layer at the start which is called the input layer, and then at the end you just have a set of activations and we can refer to those special numbers (I mean they're not special mathematically but they're semantically special); we can call those the outputs. The important point to realize here is the outputs of a neural net are not actually mathematically special, they're just the activations of a layer.
So what we did in our collaborative filtering example, we did something interesting. We actually added an additional activation function right at the very end. We added an extra activation function which was sigmoid, specifically it was a scaled sigmoid which goes between 0 and 5. It's very common to have an activation function as your last layer, and it's almost never going to be a ReLU because it's very unlikely that what you actually want is something that truncates at zero. It's very often going to be a sigmoid or something similar because it's very likely that actually what you want is something that's between two values and kind of scaled in that way.
So that's nearly it. Inputs, weights, activations, activation functions (which we sometimes call nonlinearities), output, and then the function that compares those two things together is called the loss function, which so far we've used MSE.
That's enough for today. So what we're going to do next week is we're going to kind of add in a few more extra bits which is we're going to learn the loss function that's used for classification called cross-entropy, we're going to use the activation function that's used for single label classification called softmax, and we're also going to learn exactly what happens when we do fine-tuning in terms of how these layers actually, what happens with unfreeze, and what happens when we create transfer learning. Thanks everybody! Looking forward to seeing you next week.
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Physics INTRODUCTION, AC VOLTAGE APPLIED TO A RESISTOR, REPRESENTATION OF AC CURRENT AND VOLTAGE BY ROTATING VECTORS — PHASORS FOR CBSE-NCERT
### Topic covered
color{blue}{star} INTRODUCTION
color{blue}{star} AC VOLTAGE APPLIED TO A RESISTOR
color{blue}{star} REPRESENTATION OF AC CURRENT AND VOLTAGE BY ROTATING VECTORS — PHASORS
### INTRODUCTION
color{blue} ✍️ We have so far considered direct current (dc) sources and circuits with dc sources. These currents do not change direction with time.
color{blue} ✍️The electric mains supply in our homes and offices is a voltage that varies like a sine function with time. Such a voltage is called alternating voltage (ac voltage) and the current driven by it in a circuit is called the alternating current (ac current)*.
color{blue} ✍️ Today, most of the electrical devices we use require ac voltage. This is mainly because most of the electrical energy sold by power companies is transmitted and distributed as alternating current. The main reason for preferring use of ac voltage over dc voltage is that ac voltages can be easily and efficiently converted from one voltage to the other by means of transformers.
### AC VOLTAGE APPLIED TO A RESISTOR
color{blue} ✍️ Figure 7.1 shows a resistor connected to a source e of ac voltage. The symbol for an ac source in a circuit diagram is . We consider a source which produces sinusoidally varying potential difference across its terminals. Let this potential difference, also called ac voltage, be given by
color {blue}{v = v_m sin omega t}
...........(7.1)
color {blue}{➢➢}where v_m is the amplitude of the oscillating potential difference and omega is its angular frequency.
color {blue}{➢➢}To find the value of current through the resistor, we apply Kirchhoff’s loop rule sum epsilon(t ) = 0 , to the circuit shown in Fig. 7.1 to get
sin = v_m omega t R
or t = (v_m)/R sin omegat
color {blue}{➢➢}Since R is a constant, we can write this equation as
color {blue}{t = t_m sin omega t}
.............(7.2)
color {blue}{➢➢}where the current amplitude i_m is given by
color {blue}{i_m = (v_m)/R}
............(7.3)
color {blue}{➢➢}Equation (7.3) is just Ohm’s law which for resistors works equally well for both ac and dc voltages.
color {blue}{➢➢}The voltage across a pure resistor and the current through it, given by Eqs. (7.1) and (7.2) are plotted as a function of time in Fig. 7.2.
color {brown} bbul{"Note"}, in particular that both v and i reach zero, minimum and maximum values at the same time.
color {blue}{➢➢}Clearly, the voltage and current are in phase with each other. We see that, like the applied voltage, the current varies sinusoidally and has corresponding positive and negative values during each cycle.
color {blue}{➢➢}Thus, the sum of the instantaneous current values over one complete cycle is zero, and the average current is zero. The fact that the average current is zero, however, does not mean that the average power consumed is zero and that there is no dissipation of electrical energy.
color {blue}{➢➢}As you know, Joule heating is given by i^2 R and depends on i^2 (which is always positive whether i is positive or negative) and not on i.
color {blue}{➢➢}Thus, there is Joule heating and dissipation of electrical energy when an ac current passes through a resistor. The instantaneous power dissipated in the resistor is
color {blue}{p =i 2R =i_(m)^(R) sin^2omega t}
........(7.4)
color {blue}{➢➢}The average value of p over a cycle is*
color {blue}{R = , i^2 R > = < i_(m)^(2) sin^2 omega t > }
.........[7.5(a)]
color {blue}{➢➢}where the bar over a letter(here, p) denotes its average value and <......> denotes taking average of the quantity inside the bracket. Since, i_(m)^(2) and R are constants,
color {blue}{bar P = i_(m)^(2) R< sin^2 omega t > }
.........[7.5(b)]
color {blue}{➢➢}Using the trigonometric identity, sin2 wt = 1/2 (1– cos 2omegat ), we have < sin^2 omegat > = (1/2) (1– < cos 2omegat >)
and since < cos2omegat > = 0**, we have,
< sin^2 omegat> = 1/2
color {blue}{➢➢}Thus color {blue}{bar P= 1/2i_(m)^(2)R}
...........[7.5(c)]
color {blue}{➢➢}To express ac power in the same form as dc power (P = I^2R), a special value of current is defined and used.
It is called, root mean square (rms) or effective current (Fig. 7.3) and is denoted by I_(rms) or I.
color {blue}{➢➢}It is defined by
I = sqrt(i^2)= sqrt((1/2)i_(m)^(2)) = (i_m)/sqrt2
color {blue}{= 0.707 i_m}
.............(7.6)
color {blue}{➢➢}In terms of I, the average power, denoted by P is
color {blue}{P = bar P= 1/2 i_(m)^(2) R = I^2R}
.............(7.7)
color {blue}{➢➢}Similarly, we define the rms voltage or effective voltage by
color {blue}{V= (v_m)/(sqrt2)n = 0.707 v_m}
.............(7.8)
From Eq. (7.3), we have
v_m = i_mR
or , (v_m)/(sqrt2) = (i_m)/(sqrt2) R
or,
color {blue}{V= IR}
.............(7.9)
color {blue}{➢➢}Equation (7.9) gives the relation between ac current and ac voltage and is similar to that in the dc case.
color {blue}{➢➢}This shows the advantage of introducing the concept of rms values. In terms of rms values, the equation for power [Eq. (7.7)] and relation between current and voltage in ac circuits are essentially the same as those for the dc case. It is customary to measure and specify rms values for ac quantities. For example, the household line voltage of 220 V is an rms value with a peak voltage of
color {blue}{V_M = sqrt2 V = (1.414)(220V) = 311V}
color {blue}{➢➢}In fact, the I or rms current is the equivalent dc current that would produce the same average power loss as the alternating current. Equation
color {blue}{P = V^2 / R = I V " " ("since "V = I R)}
(7.7) can also be written as
Q 3118845700
A light bulb is rated at 100W for a 220 V supply. Find
(a) the resistance of the bulb; (b) the peak voltage of the source; and
(c) the rms current through the bulb.
Class 12 Chapter 7 Example 1
Solution:
a) We are given P = 100 W and V = 220 V. The resistance of the bulb is
R = (V^2)/(P) = (220V)^2/(100W) = 482Omega
(b) The peak voltage of the source is
V_m = sqrt2 V = 311V
(c) Since, P = I V
I = P/V (100W)/(220V) = 0.450A
### REPRESENTATION OF AC CURRENT AND VOLTAGE BY ROTATING VECTORS — PHASORS
color{blue} ✍️ As we learnt that the current through a resistor is in phase with the ac voltage. But this is not so in the case of an inductor, a capacitor or a combination of these circuit elements.
In order to show phase relationship between voltage and current in an ac circuit, we use the notion of phasors. The analysis of an ac circuit is facilitated by the use of a phasor diagram.
color{blue} ✍️ A "phasor" is a vector which rotates about the origin with angular speed w, as shown in Fig. 7.4.
color{blue} ✍️ The vertical components of phasors V and I represent the sinusoidally varying quantities v and i. The magnitudes of phasors V and I represent the amplitudes or the peak values v_m and i_m of these oscillating quantities.
color {blue}{➢➢}Figure 7.4(a) shows the voltage and current phasors and their relationship at time t1 for the case of an ac source connected to a resistor i.e., corresponding to the circuit shown in Fig. 7.1.
color{blue} ✍️ The projection of voltage and current phasors on vertical axis, i.e., v_m sinomegat and im sinomegat, respectively represent the value of voltage and current at that instant. As they rotate with frequency w, curves in Fig. 7.4(b) are generated.
color {blue}{➢➢}From Fig. 7.4(a) we see that phasors V and I for the case of a resistor are in the same direction. This is so for all times. This means that the phase angle between the voltage and the current is zero.
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## fourier transforms. mental block
I have a several calculations involving similar things to the one below. This is just a simple one so I can work out how to do it because I only have a vague idea.
How does one write the following in momentum space?
$f(x)=\int d^{3}y A(x)\omega(y)A(y)$
$\omega$ is an operator whose fourier transform I know. x and y are three vectors. I need to express it in terms of momentum space for x and y if that makes sense. Im very confused.
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