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Physics Important Questions Chapter 2 Mathematical Methods
Maharashtra Board Class 11 Physics Important Questions Chapter 2 Mathematical Methods
Balbharti Maharashtra State Board 11th Physics Important Questions Chapter 2 Mathematical Methods Important Questions and Answers.
Maharashtra State Board 11th Physics Important Questions Chapter 2 Mathematical Methods
Question 1.
Explain the representation of a vector graphically and symbolically.
1. Graphical representation:
A vector is graphically represented by a directed line segment or an arrow.
eg.: displacement of a body from P to Q is represented as P → Q.
2. Symbolic representation:
Symbolically a vector is represented by a single letter with an arrow above it, such as \(\overrightarrow{\mathrm{A}}\). The magnitude of the vector \(\overrightarrow{\mathrm{A}}\) is denoted as
|A| or | \(\overrightarrow{\mathrm{A}}\) | or A.
Question 2.
A vector has both magnitude and direction. Does it mean that anything that has magnitude and direction is necessarily a vector?
1. For a physical quantity, only having magnitude and direction is not a sufficient condition to be a vector.
2. A physical quantity also has to obey vectors law of addition to be termed as vector.
3. Hence, anything that has magnitude and direction is not necessarily a vector.
Example: Though current has definite magnitude and direction, it is not a vector.
Question 3.
Define and explain the following terms:
i. Zero vector (Null vector)
ii. Resultant vector
iii. Negative vectors
iv. Equal vectors
v. Position vector
i. Zero vector (Null vector):
A vector having zero magnitude and arbitrary direction is called zero vector. It is denoted as \(\overrightarrow{0}\).
Example: Velocity vector of stationary particle, acceleration vector of a body moving with uniform velocity.
ii. Resultant vector:
The resultant of two or more vectors is defined as that single vector, which produces the same effect as produced by all the vectors together.
iii. Negative vectors:
A negative vector of a given vector is a vector of the same magnitude but opposite in direction to that of the given vector.
Negative vectors are antiparallel vectors.
In figure, \(\vec{b}\) = – \(\vec{a}\)
iv. Equal vectors:
Two vectors A and B representing same physical quantity are said to be equal if and only if they have the same magnitude and direction.
In the given figure |\(\overrightarrow{\mathrm{P}}\)| = |\(\overrightarrow{\mathrm{Q}}\)| = |\(\overrightarrow{\mathrm{R}}\)| = |\(\overrightarrow{\mathrm{S}}\)|
v. Position vector:
A vector which gives the position of a particle at a point with respect to the origin of chosen co-ordinate system is called position vector.
In the given figure \(\overrightarrow{\mathrm{OP}}\) represents position vector of \(\vec{P}\) with respect to O.
Question 4.
Whether the resultant of two vectors of unequal magnitude be zero?
The resultant of two vectors of different magnitude cannot give zero resultant.
Question 5.
Define unit vector and give its physical significance.
Unit vector: A vector having unit magnitude in a given direction is called a unit vector in that direction.
If \(\vec{p}\) is a non zero vector (P ≠ 0) then the unit vector \(\hat{\mathrm{u}}_{\mathrm{p}}\) in the direction of \(\overrightarrow{\mathrm{P}}\) is given by,
\(\hat{\mathrm{u}}_{\mathrm{p}}\) = \(\frac{\overrightarrow{\mathrm{P}}}{\mathrm{P}}\)
∴ \(\overrightarrow{\mathrm{P}}\) = \(\hat{u}_{p} P\)
Significance of unit vector:
i. The unit vector gives the direction of a given vector.
ii. Unit vector along X, Y and Z direction of a rectangular (three dimensional) coordinate is represented by \(\hat{\mathrm{i}}\), \(\hat{\mathrm{j}}\) and \(\hat{\mathrm{k}}\) respectively Such that
\(\hat{\mathbf{u}}_{x}\) = \(\hat{\mathrm{i}}\), \(\hat{\mathbf{u}}_{y}\) = \(\hat{\mathrm{j}}\) and \(\hat{\mathbf{u}}_{z}\) = \(\hat{\mathrm{k}}\)
This gives \(\hat{\mathrm{i}}\) = \(\frac{\overrightarrow{\mathrm{X}}}{\mathrm{X}}\), \(\hat{\mathrm{j}}\) = \(\frac{\overrightarrow{\mathrm{Y}}}{\mathrm{X}}\) and \(\hat{\mathrm{k}}\) = \(\frac{\
Question 6.
Explain multiplication of a vector by a scalar.
1. When a vector \(\overrightarrow{\mathrm{A}}\) is multiplied by a scalar ‘s’, it becomes ‘s\overrightarrow{\mathrm{A}}’ whose magnitude is s times the magnitude of \(\overrightarrow{\mathrm{A}}\).
2. The unit of \(\overrightarrow{\mathrm{A}}\) is different from the unit of ‘s \(\overrightarrow{\mathrm{A}}\)’.
For example,
If \(\overrightarrow{\mathrm{A}}\) = 10 newton and s = 5 second, then s\(\overrightarrow{\mathrm{A}}\) = 10 newton × 5 second = 50 Ns.
Question 7.
Explain addition of vectors.
1. The addition of two or more vectors of same type gives rise to a single vector such that the effect of this single vector is the same as the net effect of the original vectors.
2. It is important to note that only the vectors of the same type (physical quantity) can be added.
3. For example, if two vectors, \(\overrightarrow{\mathrm{P}}\) = 3 unit and \(\overrightarrow{\mathrm{Q}}\) = 4 unit are acting along the same line, then they can be added as, |\(\overrightarrow{\
mathrm{R}}\)| = |\(\overrightarrow{\mathrm{P}}\)| + |\(\overrightarrow{\mathrm{Q}}\)|
|\(\overrightarrow{\mathrm{R}}\)| = 3 + 4 = 7
[Note: When vectors are not in the same direction, then they can be added using triangle law of vector addition.]
Question 8.
State true or false. If false correct the statement and rewrite.
It is possible to add two vectors representing physical quantities having different dimensions.
It is not possible to add two vectors representing physical quantities having different dimensions.
Question 9.
Explain subtraction of vectors.
1. When two vectors are anti-parallel (in the opposite direction) to each other, the magnitude
2. It is important to note that only vectors of the same type (physical quantity) can be subtracted.
3. For example, if two vectors \(\overrightarrow{\mathrm{P}}\) = 3 unit and \(\overrightarrow{\mathrm{Q}}\) = 4 unit are acting in opposite direction, they are subtracted as, |\(\overrightarrow{\
mathrm{R}}\)| = ||\(\overrightarrow{\mathrm{P}}\)| – |\(\overrightarrow{\mathrm{Q}}\)||
= |3 – 4| = 1 unit, directed along \(\overrightarrow{\mathrm{Q}}\)
Question 10.
How can resultant of two vectors of a type inclined with each other be determined?
When two vectors of a type are inclined with each other, their resultant can be determined by using triangle law of vector addition.
Question 11.
What is triangle law of vector addition?
Triangle law of vector addition:
If two vectors describing the same physical quantity are represented in magnitude and direction, by the two sides of a triangle taken in order, then their resultant is represented in magnitude and
direction by the third side of the triangle drawn in the opposite sense, i.e., from the starting point (tail) of the first vector to the end point (head) of the second vector.
Let \(\overrightarrow{\mathrm{P}}\) and \(\overrightarrow{\mathrm{Q}}\) be the two vectors of same type taken in same order as shown in figure.
∴ Resultant vector will be given by third side taken in opposite order, i.e., \(\overline{\mathrm{OA}}\) + \(\overline{\mathrm{AB}}\) = \(\overline{\mathrm{OB}}\)
∴ \(\overrightarrow{\mathrm{P}}\) + \(\overrightarrow{\mathrm{Q}}\) = \(\overrightarrow{\mathrm{R}}\)
Question 12.
Using triangle law of vector addition, explain the process of adding two vectors which are not lying in a straight line.
i. Two vectors in magnitude and direction are drawn in a plane as shown in figure (a)
Let these vectors be \(\overrightarrow{\mathrm{P}}\) and \(\overrightarrow{\mathrm{Q}}\)
ii. Join the tail of \(\overrightarrow{\mathrm{Q}}\) to head of \(\overrightarrow{\mathrm{P}}\) in the given direction. The resultant vector will be the line which is obtained by joining tail of \(\
overrightarrow{\mathrm{P}}\) to head of \(\overrightarrow{\mathrm{Q}}\) as shown in figure (b).
iii. If \(\overrightarrow{\mathrm{R}}\) is the resultant vector of \(\overrightarrow{\mathrm{P}}\) and \(\overrightarrow{\mathrm{Q}}\) then using triangle law of vector addition, we have, \(\
overrightarrow{\mathrm{R}}\) = \(\overrightarrow{\mathrm{P}}\) + \(\overrightarrow{\mathrm{Q}}\)
Question 13.
Is it possible to add two velocities using triangle law?
Yes, it is possible to add two velocities using triangle law.
Question 14.
Explain, how two vectors are subtracted. Find their resultant by using triangle law of vector addition.
1. Let \(\overrightarrow{\mathrm{P}}\) and \(\overrightarrow{\mathrm{Q}}\) be the two vectors in a plane as shown in figure (a).
2. To subtract \(\overrightarrow{\mathrm{Q}}\) from \(\overrightarrow{\mathrm{P}}\), vector \(\overrightarrow{\mathrm{Q}}\) is reversed so that we get the vector –\(\overrightarrow{\mathrm{Q}}\) as
shown in figure (b).
3. The resultant vector is obtained by –\(\overrightarrow{\mathrm{R}}\) joining tail of \(\overrightarrow{\mathrm{P}}\) to head of – \(\overrightarrow{\mathrm{Q}}\) as shown in figure (c).
4. From triangle law of vector addition, \(\overrightarrow{\mathrm{R}}\) = \(\overrightarrow{\mathrm{P}}\) + (-\(\overrightarrow{\mathrm{Q}}\)) = \(\overrightarrow{\mathrm{P}}\) – \(\overrightarrow
Question 15.
Prove that: Vector addition is commutative.
Commutative property of vector addition:
According to commutative property, for two
vectors \(\overrightarrow{\mathrm{P}}\) and \(\overrightarrow{\mathrm{Q}}\), \(\overrightarrow{\mathrm{P}}\) + \(\overrightarrow{\mathrm{Q}}\) = \(\overrightarrow{\mathrm{Q}}\) + \(\overrightarrow{\
i. Let two vectors \(\overrightarrow{\mathrm{P}}\) and \(\overrightarrow{\mathrm{Q}}\) be represented in magnitude and direction by two sides \(\overrightarrow{\mathrm{OA}}\) and \(\overrightarrow{\
mathrm{AB}}\) respectively.
ii. Complete a parallelogramOABC such that
\(\overrightarrow{\mathrm{OA}}\) = \(\overrightarrow{\mathrm{CB}}\) = \(\overrightarrow{\mathrm{P}}\) and \(\overrightarrow{\mathrm{AB}}\) = \(\overrightarrow{\mathrm{OC}}\) = \(\overrightarrow{\
mathrm{Q}}\) then join OB.
iii. In △OAB, \(\overrightarrow{\mathrm{OA}}\) + \(\overrightarrow{\mathrm{AB}}\) = \(\overrightarrow{\mathrm{OB}}\)
(By triangle law of vector addition)
∴ \(\overrightarrow{\mathrm{P}}\) + \(\overrightarrow{\mathrm{Q}}\) = \(\overrightarrow{\mathrm{R}}\) … (1)
In △OCB, \(\overrightarrow{\mathrm{OC}}\) + \(\overrightarrow{\mathrm{CB}}\) = \(\overrightarrow{\mathrm{OB}}\)
(By triangle law of vector addition)
∴ \(\overrightarrow{\mathrm{Q}}\) + \(\overrightarrow{\mathrm{P}}\) = \(\overrightarrow{\mathrm{R}}\) … (2)
iv. From equation (1) and (2),
\(\overrightarrow{\mathrm{P}}\) + \(\overrightarrow{\mathrm{Q}}\) = \(\overrightarrow{\mathrm{Q}}\) + \(\overrightarrow{\mathrm{P}}\)
Hence, addition of two vectors obeys commutative law.
Question 16.
Prove that: Vector addition is associative.
Associative property of vector addition:
According to associative property, for three vectors \(\overrightarrow{\mathrm{P}}\), \(\overrightarrow{\mathrm{Q}}\) and \(\overrightarrow{\mathrm{R}}\),
(\(\overrightarrow{\mathrm{P}}\) + \(\overrightarrow{\mathrm{Q}}\)) + \(\overrightarrow{\mathrm{R}}\) = \(\overrightarrow{\mathrm{P}}\) + (\(\overrightarrow{\mathrm{Q}}\) + \(\overrightarrow{\mathrm
On comparing, equation (2) and (4), we get,
(\(\overrightarrow{\mathrm{P}}\) + \(\overrightarrow{\mathrm{Q}}\)) + \(\overrightarrow{\mathrm{R}}\) = \(\overrightarrow{\mathrm{P}}\) + (\(\overrightarrow{\mathrm{Q}}\) + \(\overrightarrow{\mathrm
Hence, associative law is proved.
Question 17.
State true or false. If false correct the statement and rewrite.
The subtraction of given vectors is neither commutative nor associative.
Question 18.
State and prove parallelogram law of vector addition and determine magnitude and direction of resultant vector.
i. Parallelogram law of vector add addition;
If two vectors of same type starting from the same point (tails cit the same point), are represented in magnitude and direction by the two adjacent sides of a parallelogram then, their resultant
vector is given in magnitude and direction, by the diagonal of the parallelogram starting from the same point.
ii. Proof:
a. Consider two vectors \(\overrightarrow{\mathrm{P}}\) and \(\overrightarrow{\mathrm{Q}}\) of the same type, with their tails at the point O’ and θ’ is the angle between \(\overrightarrow{\mathrm
{P}}\) and \(\overrightarrow{\mathrm{Q}}\) as shown in the figure below.
b. Join BC and AC to complete the parallelogram OACB, with \(\overline{\mathrm{OA}}\) = \(\overrightarrow{\mathrm{P}}\) and \(\overline{\mathrm{AC}}\) = \(\overrightarrow{\mathrm{Q}}\) as the
adjacent sides. We have to prove that diagonal \(\overline{\mathrm{OC}}\) = \(\overrightarrow{\mathrm{R}}\), the resultant of sum of the two given vectors.
c. By the triangle law of vector addition, we have,
\(\overrightarrow{\mathrm{OA}}\) + \(\overrightarrow{\mathrm{AC}}\) = \(\overrightarrow{\mathrm{OC}}\) … (1)
As \(\overrightarrow{\mathrm{AC}}\) is parallel to \(\overrightarrow{\mathrm{OB}}\),
\(\overrightarrow{\mathrm{AC}}\) = \(\overrightarrow{\mathrm{OB}}\) = \(\overrightarrow{\mathrm{Q}}\)
Substituting \(\overrightarrow{\mathrm{OA}}\) and \(\overrightarrow{\mathrm{OC}}\) in equation (1) we have,
\(\overrightarrow{\mathrm{P}}\) + \(\overrightarrow{\mathrm{Q}}\) = \(\overrightarrow{\mathrm{R}}\)
Hence proved.
iii. Magnitude of resultant vector:
a. To find the magnitude of resultant vector \(\overrightarrow{\mathrm{R}}\) = \(\overrightarrow{\mathrm{OC}}\), draw a perpendicular from C to meet OA extended at S.
c. Using Pythagoras theorem in right angled triangle, OSC
(OC)^2 = (OS)^2 + (SC)^2
= (OA + AS)^2 + (SC)^2
∴ (OC)^2 = (OA)^2 + 2(OA).(AS) + (AS^2) + (SC)^2 . . . .(4)
d. From right angle trianle ASC,
(AS)^2 + (SC)^2 = (AC)^2 …. (5)
e. From equation (4) and (5), we get
(OC)^2 = (OA)^2 + 2(OA) (AS) + (AC)^2
… .(6)
f. Using (2) and (6), we get
(OC)^2 = (OA)^2 + (AC)^2 + 2(OA)(AC) cos θ
∴ R^2 = P^2 + Q^2 + 2 PQ cos θ
∴ R = \(\sqrt{\mathrm{P}^{2}+\mathrm{Q}^{2}+2 \mathrm{PQ} \cos \theta}\) ….(7)
Equation (7) gives the magnitude of resultant vector \(\overrightarrow{\mathrm{R}}\).
iv. Direction of resultant vector:
To find the direction of resultant vector \(\overrightarrow{\mathrm{R}}\), let \(\overrightarrow{\mathrm{R}}\) make an angle α with \(\overrightarrow{\mathrm{P}}\).
Equation (9) represents direction of resultant vector.
[Note: If β is the angle between \(\overrightarrow{\mathrm{R}}\) and \(\overrightarrow{\mathrm{Q}}\), it can be similarly derived that
Question 19.
Complete the table for two vectors \(\overrightarrow{\mathrm{P}}\) and \(\overrightarrow{\mathrm{Q}}\) inclined at angle θ.
Question 20.
The diagonal of the parallelogram made by two vectors as adjacent sides is not passing through common point of two vectors. What does it represent?
The diagonal of the parallelogram made by two vectors as adjacent sides not passing through common point of two vectors represents triangle law of vector addition.
Question 21.
If | \(\overrightarrow{\mathbf{A}}\) + \(\overrightarrow{\mathbf{B}}\) | = | \(\overrightarrow{\mathbf{A}}\) – \(\overrightarrow{\mathbf{B}}\) | then what can be the
angle between \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) ?
Let θ be the angle between \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\), then
Thus, if |\(\overrightarrow{\mathrm{A}}\) + \(\overrightarrow{\mathrm{B}}\)| = |\(\overrightarrow{\mathrm{A}}\) – \(\overrightarrow{\mathrm{B}}\) |, then vectors \(\overrightarrow{\mathrm{A}}\) and \
(\overrightarrow{\mathrm{B}}\) must be at right angles to each other.
Question 22.
Express vector \(\overrightarrow{\mathbf{A C}}\) in terms of vectors \(\overrightarrow{\mathbf{A B}}\) and \(\overrightarrow{\mathbf{C B}}\) shown in the following figure.
Using the triangle law of addition of vectors,
\(\overrightarrow{\mathbf{A C}}\) + \(\overrightarrow{\mathbf{C B}}\) = \(\overrightarrow{\mathbf{A B}}\)
∴\(\overrightarrow{\mathbf{A C}}\) = \(\overrightarrow{\mathbf{A B}}\) – \(\overrightarrow{\mathbf{C B}}\)
Question 23.
From the following figure, determine the resultant of four forces \(\overrightarrow{\mathbf{A}}_{1}\), \(\overrightarrow{\mathbf{A}}_{2}\), \(\overrightarrow{\mathbf{A}}_{3}\), \(\overrightarrow{\
Join \(\overrightarrow{\mathrm{OB}}\) to complete ∆OAB as shown in figure below
Now, using triangle law of vector addition,
\(\overrightarrow{\mathrm{OB}}\) = \(\overrightarrow{\mathrm{OA}}\) + \(\overrightarrow{\mathrm{AB}}\) = \(\overrightarrow{\mathrm{A}}_{1}\) + \(\overrightarrow{\mathrm{A}}_{2}\)
Join \(\overrightarrow{\mathrm{OC}}\) to complete triangle OBC as shown figure below
Similarly, \(\overrightarrow{\mathrm{OC}}\) = \(\overrightarrow{\mathrm{OB}}\) + \(\overrightarrow{\mathrm{BC}}\) = \(\overrightarrow{\mathrm{A}}_{1}\) + \(\overrightarrow{\mathrm{A}}_{2}\) + \(\
\(\overrightarrow{O D}\) is the resultant of the four vectors.
Question 24.
Find the vector that should be added to the sum of (2\(\hat{\mathbf{i}}\) – 5\(\hat{\mathbf{j}}\) + 3\(\hat{\mathbf{k}}\)) and (4\(\hat{\mathbf{i}}\) + 7\(\hat{\mathbf{j}}\) – 4\(\hat{\mathbf{k}}\))
to give a unit vector along the X-axis.
Let vector \(\overrightarrow{\mathrm{p}}\) be added to get unit vector (\(\hat{\mathbf{i}}\)) along X-axis.
Sum of given vectors is given as,
(2\(\hat{\mathbf{i}}\) – 5\(\hat{\mathbf{j}}\) + 3\(\hat{\mathbf{k}}\) ) + (4\(\hat{\mathbf{i}}\) + 7\(\hat{\mathbf{j}}\) – 4\(\hat{\mathbf{k}}\)) = 6\(\hat{\mathbf{i}}\) + 2\(\hat{\mathbf{j}}\) – \
According to given condition, (6\(\hat{\mathbf{i}}\) + 2\(\hat{\mathbf{j}}\) – \(\hat{\mathbf{k}}\)) + \(\hat{\mathbf{P}}\) = \(\hat{\mathbf{i}}\)
∴ \(\overrightarrow{\mathrm{P}}\) = \(\hat{\mathbf{i}}\) – (6\(\hat{\mathbf{i}}\) + 2\(\hat{\mathbf{j}}\) – \(\hat{\mathbf{k}}\)) = \(\hat{\mathbf{i}}\) – 6\(\hat{\mathbf{i}}\) – 2\(\hat{\mathbf{j}}
\) + \(\hat{\mathbf{k}}\) = -5\(\hat{\mathbf{i}}\) – 2\(\hat{\mathbf{j}}\) + \(\hat{\mathbf{k}}\)
The required vector is -5\(\hat{\mathbf{i}}\) – 2\(\hat{\mathbf{j}}\) + \(\hat{\mathbf{k}}\).
Question 25.
If \(\overrightarrow{\mathbf{P}}\) = 2\(\hat{\mathbf{i}}\) + 3\(\hat{\mathbf{j}}\) – \(\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{Q}}\) = 2\(\hat{\mathbf{i}}\) – 5\(\hat{\mathbf{j}}\) + 2\(\hat
i. \(\overrightarrow{\mathbf{P}}\) + \(\overrightarrow{\mathbf{Q}}\)
ii. 3\(\overrightarrow{\mathbf{P}}\) – 2\(\overrightarrow{\mathbf{Q}}\)
Given \(\overrightarrow{\mathbf{P}}\) = 2\(\hat{\mathbf{i}}\) + 3\(\hat{\mathbf{j}}\) – \(\hat{\mathbf{k}}\), \(\overrightarrow{\mathbf{Q}}\) = 2\(\hat{\mathbf{i}}\) – 5\(\hat{\mathbf{j}}\) + 2\(\hat
To find:
i. \(\overrightarrow{\mathbf{P}}\) + \(\overrightarrow{\mathbf{Q}}\)
ii. 3\(\overrightarrow{\mathbf{P}}\) – 2\(\overrightarrow{\mathbf{Q}}\)
i. \(\overrightarrow{\mathbf{P}}\) + \(\overrightarrow{\mathbf{Q}}\) = (2\(\hat{\mathbf{i}}\) + 3\(\hat{\mathbf{j}}\) – k) + (2\(\hat{\mathbf{i}}\) – 5\(\hat{\mathbf{j}}\) + 2k)
= (2 + 2)\(\hat{\mathbf{i}}\) + (3 – 5)\(\hat{\mathbf{j}}\) + (-1 + 2)\(\hat{\mathbf{k}}\)
= 4\(\hat{\mathbf{i}}\) – 2\(\hat{\mathbf{j}}\) + \(\hat{\mathbf{k}}\)
ii. 3\(\overrightarrow{\mathbf{P}}\) = 3(2\(\hat{\mathbf{i}}\) + 3\(\hat{\mathbf{j}}\) – \(\hat{\mathbf{k}}\)) = 6\(\hat{\mathbf{i}}\) + 9\(\hat{\mathbf{j}}\) – 3\(\hat{\mathbf{k}}\)
2\(\overrightarrow{\mathbf{Q}}\) = 2(2\(\hat{\mathbf{i}}\) – 5\(\hat{\mathbf{j}}\) + 2\(\hat{\mathbf{k}}\)) = 4\(\hat{\mathbf{i}}\) – 10\(\hat{\mathbf{j}}\) + 4\(\hat{\mathbf{k}}\)
Question 26.
Find unit vector parallel to the resultant of the vectors \(\overrightarrow{\mathbf{A}}\) = \(\hat{\mathbf{i}}\) + 4\(\hat{\mathbf{j}}\) – 2\(\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{B}}\) =
3\(\hat{\mathbf{i}}\) – 5\(\hat{\mathbf{j}}\) + \(\hat{\mathbf{k}}\).
The resultant of \(\overrightarrow{\mathrm{A}}\) and \(\overrightarrow{\mathrm{B}}\) is,
The required unit vector is \(\frac{1}{3 \sqrt{2}}\)(4\(\hat{\mathbf{i}}\) – \(\hat{\mathbf{j}}\) – \(\hat{\mathbf{k}}\))
Question 27.
Two forces, F[1] and F[2], each of magnitude 5 N are inclined to each other at 60°. Find the magnitude and direction of their resultant force.
Given: F[1] = 5 N, F[2] = 5 N, θ = 60°
To find: Magnitude of resultant force (R),
Direction of resultant force (α)
i. The magnitude of resultant force is 8.662 N.
ii. The direction of resultant force is 30° w.r.t. \(\overrightarrow{\mathrm{F}_{1}}\).
Question 28.
Water is flowing in a stream with velocity 5 km/hr in an easterly direction relative to the shore. Speed of a boat relative to still water is 20 km/hr. If the boat enters the stream heading north,
with what velocity will the boat actually travel?
The resultant velocity \(\overrightarrow{\mathrm{R}}\) of the boat can be obtained by adding the two velocities using ∆ OAB shown in the figure.
The direction ot the resultant velocity is
Answer: The velocity of the boat is 20.616 km/hr in a direction 14.04° east of north. .
[Note: tan^-1 (0.25) ≈ 14.04° which equals 14°2]
Question 29.
Rain is falling vertically with a speed of 35 m/s. Wind starts blowing at a speed of 12 m/s in east to west direction. In which direction should a boy waiting at a bus stop hold his umbrella? (NCERT)
Let the velocity of rain and wind be \(\overrightarrow{\mathbf{V}_{\mathrm{R}}}\) and \(\overrightarrow{\mathbf{V}_{\mathrm{W}}}\), then resultant velocity \(\overrightarrow{\mathrm{v}}\) has
magnitude of
If \(\overrightarrow{\mathrm{v}}\) makes an angle θ with vertical then, from the figure
Answer: The boy should hold his umbrella in vertical plane at an angle of about 19° with vertical towards the east.
Question 30.
What are components of a vector?
1. The given vector can be written as sum of two or more vectors along certain fixed directions. The vectors into which the given single vector is splitted are called components of the vector.
2. Let \(\overrightarrow{\mathrm{A}}\) = \(\mathrm{A}_{1} \hat{\alpha}\) + \(\mathrm{A}_{2} \hat{\beta}\) + \(\mathrm{A}_{3} \hat{\gamma}\) where, \(\hat{\alpha}\), \(\hat{\beta}\) and \(\hat{\
gamma}\) are unit vectors along chosen directions. Then, A[1], A[2] and A[3] are known as components of \(\overrightarrow{\mathrm{A}}\) along three directions \(\hat{\alpha}\), \(\hat{\beta}\)
and \(\hat{\gamma}\).
3. It two vectors are equal then, their corresponding components are also equal and vice-versa.
[Note: The magnitude of a vector is a scalar while each component of a vector is always a vector.]
Question 31.
What is meant by resolution of vector?
1. The process of splitting a given vector into its components is called resolution of the vector.
2. Resolution of vector is equal to replacing the original vector with the sum of the component vectors.
Question 32.
That are rectangular components of vectors? Explain their uses.
i. Rectangular components of a vector:
If components of a given vector are mutually perpendicular to each other then they are called rectangular components of that vector.
ii. Consider a vector \(
\) = \(
\) originating from the origin O’ of a rectangular co-ordinate system as shown in figure.
iii. Draw CA ⊥ OX and CB ⊥ OY.
Let component of \(
\) along X-axis \(
\) and component of \(
\) along Y-axis = \(
By parallelogram law of vectors,
where, \(
\) and \(
\) are unit vectors along positive direction of X and Y axes respectively.
iv. If θ is angle made by \(
\) with X-axis, then
v. Squaring and adding equation (1) and (2) we get,
Equation (3) gives the magnitude of \(
vi. Direction of \(
\) can be found out by dividing equation (2) by (1),
Equation (4) gives direction of \(
vii. When vectors are noncoplanar, it becomes necessary to use the third dimension. If \(
\), \(
\) and \(
\) are three rectangular components of \(
\) along X, Y and Z axes of a three dimensional rectangular cartesian co-ordinate system then.
Question 33.
Find a unit vector in the direction of the vector 3\(
\) + 4\(
Question 34.
Given \(
\) = \(
\) + 2\(
\) and \(
\) = 2\(
\) + \(
\), what are the magnitudes of the two vectors? Are these two vectors equal?
The magnitudes of \(
\) and \(
\) are equal. However, their corresponding components are not equal, i.e., a[x] ≠ b[x] and a[y] ≠ b[y]. Hence, the two vectors are not equal.
Magnitudes of two vectors are equal, but vectors are unequal.
Question 35.
Find the vector drawn from the point (-4, 10, 7) to the point (3, -2, 1). Also find its magnitude.
If \(
\) is a vector drawn from the point (x[1], y[1], z[1]) to the point (x[2], y[2], z[2]), then
Question 36.
In a cartesian co-ordinate system, the co-ordinates of two points P and Q are (2, 4, 4) and (-2, -3, 7) respectively, find \(
\overrightarrow{\mathbf{P Q}}
\) and its magnitude.
Given: Position vector of P = (2,4,4)
∴ |\(
\)| = 8.6 units
Answer: Vector \(
\) is -4\(
\) – 7\(
\) + 3\(
\) and its magnitude is 8.6 units.
Question 37.
If \(
\) = 3\(
\) + 4[/latex] = 3\(
\) and \(
\) = 7\(
\) + 24\(
\), find a vector having the same magnitude as \(
\) and parallel to \(
The magnitude of vector \(
\) is | \(
\) |
Answer: The required vector is 15\(
\) + 20\(
Question 38.
Complete the table.
Question 39.
State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful.
i. Adding any two scalars,
ii. Adding a scalar to a vector of the same dimensions,
iii. Multiplying any vector by any scalar,
iv. Multiplying any two scalars,
v. Adding any two vectors. (NCERT)
1. Not any two scalars can be added. To add two scalars it is essential that they represent same physical quantity.
2. This operation is meaningless. Only a vector can be added to another vector.
3. This operation is possible. When a vector is multiplied with a dimensional scalar, the resultant vector will have different dimensions.
eg.: acceleration vector is multiplied with mass (a dimensional scalar), the resultant vector has the dimensions of force.
When a vector is multiplied with non – dimensional scalar, it will be a vector having dimensions as that of the given vector.
eg.: \(
\) × 3 = 3\(
4. This operation is possible. Multiplication of non-dimensional scalars is simply algebraic multiplication. Multiplication of non dimensional scalars will result into scalar with different
eg.: Volume × density = mass.
5. Not any two vectors can be added. To add two vectors it is essential that they represent same physical quantity.
Question 40.
Explain scalar product of two vectors with the help of suitable examples.
Scalar product of two vectors:
1. The scalar product of two non-zero vectors is defined as the product of the magnitude of the two vectors and cosine of the angle θ between the two vectors.
2. The dot sign is used between the two vectors to be multiplied therefore scalar product is also called dot product.
3. The scalar product of two vectors \(
\) and \(
\) is given by, \(
\) . \(
\) = PQ cos θ
where, p = magnitude of \(
\), Q = magnitude of \(
θ = angle between \(
\) and \(
4. Examples of scalar product:
1. Power (P) is a scalar product of force (\(
\)) and velocity (\(
∴ P = \(
\) . \(
2. Work is a scalar product of force (\(
\)) and displacement (\(
∴ W = \(
\overrightarrow{\mathrm{F}} \cdot \overrightarrow{\mathrm{s}}
Question 41.
Discuss characteristics of scalar product of two vectors.
Characteristics of the scalar product of two vectors:
i. The scalar product of two vectors is equivalent to the product of magnitude of one vector with component of the other in the direction of the first.
vi. Scalar product of two vectors is expressed in terms of rectangular components as
\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}}
\) = A[x] + B[x] + A[y]B[y] + A[z]B[z]
vii. For \(
\vec{a} \neq 0, \vec{a} \cdot \vec{b}=\vec{a} \cdot \vec{c}
\) does not necessarily mean \(
\) = \(
Question 42.
Complete the table vector given below:
Question 43.
Define and explain vector product of two vectors with suitable examples.
i. The vector product of two vectors is a third vector whose magnitude is equal to the product of magnitude of the two vectors and sine of the smaller angle θ between the two vectors.
ii. Vector product is also called cross product of vectors because cross sign is used to represent vector product.
iii. Explanation:
a. The vector product of two vectors \(
\) and \(
\), is a third vector \(
\) and is written as, \(
\) = \(
\) × \(
\) = AB sin θ \(
\) where, \(
\) is unit vector in direction of \(
\), i.e., perpendicular to plane containing two vectors. It is given by right handed screw rule.
c. Examples of vector product:
1. Force experienced by a charge q moving with velocity \(\overrightarrow{\mathrm{V}}\) in uniform magnetic field of induction (strength) \(\overrightarrow{\mathrm{B}}\) is given as \(\overrightarrow
{\mathrm{F}}\) = q\(\overrightarrow{\mathrm{V}}\) × \(\overrightarrow{\mathrm{B}}\)
2. Moment of a force or torque (\(\begin{aligned}
\end{aligned}\)) is the vector product of the position vector (\(\vec{r}\)) and the force (\(\overrightarrow{\mathrm{F}}\)).
i.e., \(\begin{aligned}
\end{aligned}\) = \(\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}\)
3. The instantaneous velocity (\(\overrightarrow{\mathrm{v}}\)) of a rotating particle is equal to the cross product of its angular velocity (\(\vec{\omega}\)) and its position (\(\overrightarrow{\
mathrm{r}}\)) from axis of rotation.
\(\overrightarrow{\mathrm{v}}\) = \(\overrightarrow{\mathrm{r}}\) × \(\vec{\omega}\)
Question 44.
State right handed screw rule.
Statement of Right handed screw rule: Hold a right handed screw with its axis perpendicular to the plane containing vectors and the screw rotated from first vector to second vector through a small
angle, the direction in which the screw tip would advance is the direction of the vector product of two vectors.
Question 45.
State the characteristics of the vector product (cross product) of two vectors.
Characteristics of the vector product (cross product):
i. The vector product of two vectors does not obey the commutative law of multiplication.
vi. The magnitude of cross product of two vectors is numerically equal to the area of a parallelogram whose adjacent sides represent the two vectors.
Question 46.
Derive an expression for cross product of two vectors and express it in determinant form.
Expression for cross product of two vectors:
i. Let two vectors \(\overrightarrow{\mathrm{R}}\) and \(\overrightarrow{\mathrm{Q}}\) be represented in magnitude and direction by,
iii. Determinant form of cross product of two vectors \(\overrightarrow{\mathrm{R}}\) and \(\overrightarrow{\mathrm{Q}}\) is given by,
Question 47.
Show that magnitude of vector product of two vectors is numerically equal to the area of a parallelogram formed by the two
Suppose OACB is a parallelogram of adjacent sides, \(\overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{P}}\) and \(\overrightarrow{\mathrm{OB}}=\overrightarrow{\mathrm{Q}}\).
Question 48.
Distinguish between scalar product (dot product) and vector product (cross product).
Question 49.
Given \(\overrightarrow{\mathbf{P}}\) = 4\(\hat{\mathbf{i}}\) – \(\hat{\mathbf{j}}\) + 8\(\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{Q}}\) = 2\(\hat{\mathbf{i}}\) – m\(\hat{\mathbf{j}}\) + 4\(\
hat{\mathbf{k}}\) find m if \(\overrightarrow{\mathbf{P}}\) and \(\overrightarrow{\mathbf{Q}}\) have the same direction. Solution:
Since \(\overrightarrow{\mathbf{P}}\) and \(\overrightarrow{\mathbf{Q}}\) have the same direction, their corresponding components must be in the same proportion, i.e.,
Question 50.
Find the scalar product of the two vectors \(\overrightarrow{\mathbf{v}}_{1}\) = \(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\mathbf{3} \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{v}}_{2}\) = \(3 \hat
{\mathbf{i}}+4 \hat{\mathbf{j}}-\mathbf{5} \hat{\mathbf{k}}\)
Answer: Scalar product of two given vectors is – 4.
Question 51.
A force \(\overrightarrow{\mathbf{F}}\) = \(4 \hat{\mathbf{i}}+6 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) acting on a particle produces a displacement of \(\overrightarrow{\mathbf{S}}\) = \(\
overrightarrow{\mathrm{s}}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\mathbf{5} \hat{\mathbf{k}}\) where F is expressed in newton and s in metre. Find the work done by the force.
Answer: The work done by the force is 41 J.
Question 52.
Find ‘a’ if \(\overrightarrow{\mathbf{A}}\) = \(3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{B}}\) = \(\mathbf{a} \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat
{\mathbf{k}}\) are perpendicular to one another.
Question 53.
If \(\overrightarrow{\mathbf{A}}\) = \(5 \hat{\mathbf{i}}+6 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{B}}\) = \(2 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\)
determine the angle between \(\) and \(\). Solution:
Question 54.
Find the angle between the vectors
\(\overrightarrow{\mathbf{A}}\) = \(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{B}}\) = \(-\hat{\mathbf{i}}+\hat{\mathbf{j}}-\mathbf{2} \hat{\mathbf{k}}\).
Let angle between the vectors be θ
Answer: The angle between the vectors is 60°.
Question 55.
If \(\overrightarrow{\mathbf{A}}\) = \(2 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) and \(\vec{B}\) = \(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\mathbf{5} \hat{\mathbf{k}}\), find the
component of \(\overrightarrow{\mathbf{A}}\) along \(\overrightarrow{\mathbf{B}}\).
Question 56.
\(\hat{\mathbf{i}}\) and \(\hat{\mathbf{j}}\) are unit vectors along X-axis and Y-axis respectively. What is the magnitude and direction of the vector \(\hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\hat
{\mathbf{i}}-\hat{\mathbf{j}}\)? What are the components of a vector
\(\overrightarrow{\mathbf{A}}=2 \hat{\mathbf{i}}+\mathbf{3} \hat{\mathbf{j}}\) along the directions of \((\hat{\mathbf{i}}+\hat{\mathbf{j}})\) and \((\hat{\mathbf{i}}-\hat{\mathbf{j}})\)? (NCERT)
Question 57.
The angular momentum \(\overrightarrow{\mathrm{L}}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{p}}\), where \(\overrightarrow{\mathbf{r}}\) is a position vector and \(\overrightarrow{\
mathrm{p}}\) is linear momentum of a body.
Question 58.
If \(\overrightarrow{\mathbf{A}}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{B}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) are two vectors, find
\(|\overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}}|\)
Question 59.
Find unit vectors perpendicular to the plane of the vectors, \(\overrightarrow{\mathbf{A}}\) = \(\) and
\(\overrightarrow{\mathbf{B}}\) = \(2 \hat{\mathbf{i}}-\hat{\mathbf{k}}\)
Let required unit vector be \(\hat{\mathrm{u}}\).
Question 60.
\(\overrightarrow{\mathbf{P}}\) = \(\hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{Q}}\) = \(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}\) are two vectors,
find the unit vector parallel to \(\overrightarrow{\mathbf{P}} \times \overrightarrow{\mathbf{Q}}\). Also find the vector perpendicular to P and Q of magnitude 6 units.
Question 61.
Find the area of a triangle formed by \(\overrightarrow{\mathbf{A}}\) = \(\hat{3} \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{B}}\) = \(\hat{\mathbf{i}}+\
hat{\mathbf{j}}-\boldsymbol{2} \hat{\mathbf{k}}\) as adjacent sides measure in metre. Solution:
Given: Two adjacent sides of triangle,
\(\overrightarrow{\mathrm{A}}\) = \(3 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}\), \(\overrightarrow{\mathrm{B}}\) = \(\hat{i}+\hat{j}-2 \hat{k}\)
To find: Area of triangle
Formula: Area of triangle =
Area of the triangle is 6.1 m^2.
Question 62.
Find the derivatives of the functions,
i. f(x) = x^8
ii. f(x) = x^3 + sin x
i. Using \(\frac{\mathrm{dx}^{\mathrm{n}}}{\mathrm{dx}}\) = nx^n-1,
\(\frac{d\left(x^{8}\right)}{d x}\) = 8x^7
Question 63.
Find derivatives of e^2x – tan x
Question 64.
Find the derivatives of the functions.
f(x) = x^3 sin x
Question 65.
Find derivatives of \(\frac{d}{d x}(x \times \ln x)\)
Question 66.
Evaluate the following integrals.
i. \(\int x^{8} d x\)
Using formula \(\int x^{n} d x\) = \(\frac{x^{n+1}}{n+1}\),
\(\int x^{8} d x\) = \(\frac{x^{9}}{9}\)
ii. \(\int_{2}^{5} x^{2} d x\)
iii) \(\int(x+\sin x) d x\)
iv) \(\int\left(\frac{10}{x}+e^{x}\right) d x\)
v) \(\int_{1}^{4}\left(x^{3}-x\right) d x\)
f[1](x) – f[2](x) = \(\int f_{1}(x)-\int f_{2}(x)\)
Question 67.
A man applies a force of 10 N on a garbage crate. If another man applies a force of 8 N on the same crate at an angle of 60° with respect to previous, then what will be the resultant force and
direction of the crate, if crate is stationary.
A resultant force of 15.62 N is applied on a crate at an angle of 26.56°.
Question 68.
A lady dropped her wallet in the parking lot of a super market. A boy picked the wallet up and ran towards the lady. He set off at 60° to the verge, heading towards the lady with a speed of 10 m s^
-1, as shown in the diagram.
Find the component of velocity of boy directly across the parking strip.
The angle between velocity vector and the direction of path is 60°.
∴ Component of velocity across the parking strip
= v × cos 60°
= 10^-1 × cos 60°
= 5 m s^-1
Question 69.
On an open ground, a biker follows a track that turns to his left by an angle of 60° after every 600 m. Starting from a given turn, specify the displacement of the biker at the third and sixth turn.
Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
The path followed by the biker will be a closed hexagonal path. Suppose the motorist starts his journey from the point O.
= 1200 m
= 1.2 km
∴ Total path length = \(|\overrightarrow{\mathrm{OA}}|+|\overrightarrow{\mathrm{AB}}|+|\overrightarrow{\mathrm{BC}}|\)
= 600 + 600 + 600
= 1800 m
= 1.8 km
The ratio of the magnitude of displacement to the total path-length = \(\frac{1.2}{1.8}\) = \(\frac{2}{3}\) = 0.67
ii. The motorist will take the sixth turn at O.
Displacement is zero.
path-length is = 3600 m or 3.6 km.
Ration of magnitude of displacement and path-length is zero.
Question 70.
What is the resultant of vectors shown in the figure below?
If number of vectors are represented by the various sides of a closed polygon taken in one order then, their resultant is always zero.
Question 71.
If \(\overrightarrow{\mathbf{P}}\) is moving away from a point and \(\overrightarrow{\mathbf{Q}}\) is moving towards a point then, can their resultant be found using parallelogram law of vector
No. Resultant cannot be found by parallelogram law of vector addition because to apply law of parallelogram of vectors the two vectors and should either act towards a point or away from a point.
Question 72.
Which of the throwing is a vector?
(A) speed
(B) displacement
(C) mass
(D) time
(B) displacement
Question 73.
The equation \(\vec{a}+\vec{a}=\vec{a}\) is
(A) meaningless
(B) always truc
(C) may he possible for limited values of a’
(D) true only when \(\overrightarrow{\mathrm{a}}=0\)
(D) true only when \(\overrightarrow{\mathrm{a}}=0\)
Question 74.
The minimum number of numerically equal vectors whose vector sum can be zero is
(A) 4
(B) 3
(C) 2
(D) 1
(C) 2
Question 75.
If \(\vec{A}+\vec{B}=\vec{A}-\vec{B}\) then vector \(\overrightarrow{\mathrm{B}}\) must be
(A) zero vector
(B) unit vector
(C) Non zero vector
(D) equal to \(\overrightarrow{\mathrm{A}}\)
(A) zero vector
Question 76.
If \(\hat{\mathrm{n}}\) is the unit vector in the direction of \(\overrightarrow{\mathrm{A}}\), then,
(A) \(\hat{n}=\frac{\vec{A}}{|\vec{A}|}\)
(B) \(\hat{\mathrm{n}}=\overrightarrow{\mathrm{A}}|\overrightarrow{\mathrm{A}}|\)
(C) \(\hat{\mathrm{n}}=\frac{|\overrightarrow{\mathrm{A}}|}{\overrightarrow{\mathrm{A}}}\)
(D) \(\hat{\mathrm{n}}=\hat{\mathrm{n}} \times \overrightarrow{\mathrm{A}}\)
(A) \(\hat{n}=\frac{\vec{A}}{|\vec{A}|}\)
Question 77.
Two quantities of 5 and 12 unit when added gives a quantity 13 unit. This quantity is
(A) time
(B) mass
(C) linear momentum
(D) speed
(C) linear momentum
Question 78.
A force of 60 N acting perpendicular to a force of 80 N, magnitude of resultant force is
(A) 20N
(B) 70N
(C) 100 N
(D) 140 N
(C) 100 N
Question 79.
A river is flowing at the rate of 6 km h^-1. A man swims across it with a velocity of 9 km h^-1. The resultant velocity of the man will be
(A) \(\sqrt{15} \mathrm{~km} \mathrm{~h}^{-1}\)
(B) \(\sqrt{45} \mathrm{~km} \mathrm{~h}^{-1}\)
(C) \(\sqrt{117} \mathrm{~km} \mathrm{~h}^{-1}\)
(D) \(\sqrt{225} \mathrm{~km} \mathrm{~h}^{-1}\)
(C) \(\sqrt{117} \mathrm{~km} \mathrm{~h}^{-1}\)
Question 80.
If \(\overrightarrow{\mathrm{A}}=\overrightarrow{\mathrm{B}}+\overrightarrow{\mathrm{C}}\) and magnitudes of \(\overrightarrow{\mathrm{A}}\), \(\overrightarrow{\mathrm{B}}\) and \(\overrightarrow{\
mathrm{C}}\) are 5, 4 and 3 unit respectively, then angle between \(\overrightarrow{\mathrm{A}}\) and \(\overrightarrow{\mathrm{B}}\) is
(A) sin^-1 (3/4)
(B) cos^-1 (4/5)
(C) tan^-1 (5/3)
(D) cos^-1 (3/5)
(B) cos^-1 (4/5)
Question 81.
If \(\vec{A}=\hat{i}+2 \hat{j}+3 \hat{k}\) and \(\overrightarrow{\mathrm{B}}=3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\), then the area of parallelogram formed from these vectors as the
adjacent sides will be
(A) 2\(\sqrt{3}\) square units
(B) 4\(\sqrt{3}\) square units
(C) 6\(\sqrt{3}\) square units
(D) 8\(\sqrt{3}\) square units
(D) 8\(\sqrt{3}\) square units
Question 82.
A person moves from a point S and walks along the path which is a square of each side 50 m. He runs east, south, then west and finally north. Then the total displacement covered is
(A) 200m
(B) 100 m
(C) 50\(\sqrt{2}\) m
(D) zero
(D) zero
Question 83.
The maximum value of magnitude of \((\vec{A}-\vec{B})\) is
(A) A – B
(B) A
(C) A + B
(D) \(\sqrt{\left(A^{2}+B^{2}\right)}\)
(C) A + B
Question 84.
The magnitude of the X and Y components of \(\overrightarrow{\mathrm{A}}\) are 7 and 6. Also the magnitudes of the X and Y components of \(\vec{A}+\vec{B}\) are 11 and 9 respectively. What is the
magnitude of
(A) 5
(B) 6
(C) 8
(A) 5
Question 85.
What is the maximum n Limber of components into which a force can be resolved?
(A) Two
(B) Three
(C) Four
(D) Any number
(D) Any number
Question 86.
The resultant of two vectors of magnitude \(|\overrightarrow{\mathrm{P}}|\) is also \(|\overrightarrow{\mathrm{P}}|\). They act at an angle
(A) 60°
(B) 90°
(C) 120°
(D) 180°
(C) 120°
Question 87.
The vectors \(\overrightarrow{\mathrm{A}}\) and \(\overrightarrow{\mathrm{B}}\) are such that \(\overrightarrow{\mathrm{A}}\) + \(\overrightarrow{\mathrm{B}}\) = \(\overrightarrow{\mathrm{C}}\) and A
^2 + B^2 = C^2. Angle θ between positive directions of \(\overrightarrow{\mathrm{A}}\) and \(\overrightarrow{\mathrm{B}}\) is
(A) \(\frac{\pi}{2}\)
(B) 0
(C) π
(D) \(\frac{2 \pi}{3}\)
(A) \(\frac{\pi}{2}\)
Question 88.
The expression \(\frac{1}{\sqrt{2}}(\hat{\mathrm{i}}+\hat{\mathrm{j}})\) is a
(A) unit vector
(B) null vector
(C) vector of magnitude \(\sqrt{2}\)
(D) scalar
(A) unit vector
Question 89.
What is the angle between \(\hat{i}+\hat{j}+\hat{k}\) and \(\hat{\mathrm{i}}\)?
(A) 0°
(B) \(\frac{\pi}{6}\)
(C) \(\frac{\pi}{3}\)
(D) None of the above
(D) None of the above
Question 90.
\((\overrightarrow{\mathrm{P}}+\overrightarrow{\mathrm{Q}})\) is a unit vector along X-axis. If \(\overrightarrow{\mathrm{P}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}\) then \(\
overrightarrow{\mathrm{Q}}\) is
(A) \(\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}\)
(B) \(\hat{\mathrm{j}}-\hat{\mathrm{k}}\)
(C) \(\hat{i}+\hat{j}+\hat{k}\)
(D) \(\hat{\mathrm{j}}+\hat{\mathrm{k}}\)
(B) \(\hat{\mathrm{j}}-\hat{\mathrm{k}}\)
Question 91.
The magnitude of scalar product of the vectors \(\overrightarrow{\mathrm{A}}=2 \hat{\mathrm{i}}+5 \hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{B}}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}}\) is
(A) 20
(B) 22
(C) 26
(D) 29
(C) 26
Question 92.
Three vectors \(\overrightarrow{\mathrm{A}}\), \(\overrightarrow{\mathrm{B}}\) and \(\overrightarrow{\mathrm{C}}\) satisfy the relation \(\overrightarrow{\mathrm{A}}\) . \(\overrightarrow{\mathrm{B}}
\) = 0 and \(\overrightarrow{\mathrm{A}}\). \(\overrightarrow{\mathrm{C}}\) = 0, then \(\overrightarrow{\mathrm{A}}\) is parallel to
(A) \(\overrightarrow{\mathrm{B}}\)
(B) \(\overrightarrow{\mathrm{C}}\)
(C) \(\overrightarrow{\mathrm{B}}\) × \(\overrightarrow{\mathrm{C}}\)
(D) \(\overrightarrow{\mathrm{A}}\) . \(\overrightarrow{\mathrm{C}}\)
(C) \(\overrightarrow{\mathrm{B}}\) × \(\overrightarrow{\mathrm{C}}\)
Question 93.
What vector must be added to the sum of two vectors \(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}}\) and \(3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}\) so that the resultant
is a unit vector along Z axis?
(A) \(5 \hat{\hat{i}}+\hat{\mathrm{k}}\)
(B) \(-5 \hat{i}+3 \hat{j}\)
(C) \(3 \hat{j}+5 \hat{k}\)
(D) \(-3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}\)
(B) \(-5 \hat{i}+3 \hat{j}\)
Question 94.
\(\overrightarrow{\mathrm{A}}=5 \overrightarrow{\mathrm{i}}-2 \overrightarrow{\mathrm{j}}+3 \overrightarrow{\mathrm{k}}\) and \(\overrightarrow{\mathrm{B}}=2 \overrightarrow{\mathrm{i}}+\
overrightarrow{\mathrm{j}}+2 \overrightarrow{\mathrm{k}}\), then component of \(\overrightarrow{\mathrm{B}}\) along \(\overrightarrow{\mathrm{A}}\) is
(A) \(\frac{\sqrt{28}}{38}\)
(B) \(\frac{28}{\sqrt{38}}\)
(C) \(\frac{\sqrt{28}}{48}\)
(D) \(\frac{14}{\sqrt{38}}\)
(D) \(\frac{14}{\sqrt{38}}\)
Question 95.
Choose the WRONG statement
(A) The division of vector by scalar is valid.
(B) The multiplication of vector by scalar is valid.
(C) The multiplication of vector by another vector is valid by using vector algebra.
(D) The division of a vector by another vector is valid by using vector algebra.
(D) The division of a vector by another vector is valid by using vector algebra.
Question 96.
The resultant of two forces of 3 N and 4 N is 5 N, the angle between the forces is
(A) 30°
(B) 60°
(C) 90°
(D) 120°
(C) 90°
Question 97.
The unit vector along \(\hat{\mathrm{i}}+\hat{\mathrm{j}}\) is
(A) \(\hat{\mathrm{k}}\)
(B) \(\hat{\mathrm{i}}+\hat{\mathrm{j}}\)
(C) \(\frac{\hat{\mathrm{i}}+\hat{\mathrm{j}}}{\sqrt{2}}\)
(D) \(\frac{\hat{\mathrm{i}}+\hat{\mathrm{j}}}{2}\)
(C) \(\frac{\hat{\mathrm{i}}+\hat{\mathrm{j}}}{\sqrt{2}}\) | {"url":"https://maharashtraboardsolutions.guru/maharashtra-board-class-11-physics-important-questions-chapter-2/","timestamp":"2024-11-05T15:24:16Z","content_type":"text/html","content_length":"144176","record_id":"<urn:uuid:b2849a42-9b67-473e-9c97-5c9cae0ded3c>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00334.warc.gz"} |
urlongs to Meters
Furlongs to Meters Converter
β Switch toMeters to Furlongs Converter
How to use this Furlongs to Meters Converter π €
Follow these steps to convert given length from the units of Furlongs to the units of Meters.
1. Enter the input Furlongs value in the text field.
2. The calculator converts the given Furlongs into Meters in realtime β using the conversion formula, and displays under the Meters label. You do not need to click any button. If the input
changes, Meters value is re-calculated, just like that.
3. You may copy the resulting Meters value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Furlongs to Meters?
The formula to convert given length from Furlongs to Meters is:
Length[(Meters)] = Length[(Furlongs)] / 0.0049709695
Substitute the given value of length in furlongs, i.e., Length[(Furlongs)] in the above formula and simplify the right-hand side value. The resulting value is the length in meters, i.e., Length
Calculation will be done after you enter a valid input.
Consider that a horse race is 8 furlongs long.
Convert this distance from furlongs to Meters.
The length in furlongs is:
Length[(Furlongs)] = 8
The formula to convert length from furlongs to meters is:
Length[(Meters)] = Length[(Furlongs)] / 0.0049709695
Substitute given weight Length[(Furlongs)] = 8 in the above formula.
Length[(Meters)] = 8 / 0.0049709695
Length[(Meters)] = 1609.344
Final Answer:
Therefore, 8 fur is equal to 1609.344 m.
The length is 1609.344 m, in meters.
Consider that a traditional country road stretches for 12 furlongs.
Convert this distance from furlongs to Meters.
The length in furlongs is:
Length[(Furlongs)] = 12
The formula to convert length from furlongs to meters is:
Length[(Meters)] = Length[(Furlongs)] / 0.0049709695
Substitute given weight Length[(Furlongs)] = 12 in the above formula.
Length[(Meters)] = 12 / 0.0049709695
Length[(Meters)] = 2414.016
Final Answer:
Therefore, 12 fur is equal to 2414.016 m.
The length is 2414.016 m, in meters.
Furlongs to Meters Conversion Table
The following table gives some of the most used conversions from Furlongs to Meters.
Furlongs (fur) Meters (m)
0 fur 0 m
1 fur 201.168 m
2 fur 402.336 m
3 fur 603.504 m
4 fur 804.672 m
5 fur 1005.84 m
6 fur 1207.008 m
7 fur 1408.176 m
8 fur 1609.344 m
9 fur 1810.512 m
10 fur 2011.68 m
20 fur 4023.36 m
50 fur 10058.4001 m
100 fur 20116.8002 m
1000 fur 201168.0015 m
10000 fur 2011680.0153 m
100000 fur 20116800.1534 m
A furlong is a unit of length used primarily in horse racing and agriculture. One furlong is equivalent to 220 yards or approximately 201.168 meters.
The furlong is defined as one-eighth of a mile, making it a useful measurement for shorter distances, especially in contexts like racetracks and land measurement.
Furlongs are commonly used in horse racing to describe the length of a race and in agriculture for measuring field lengths. The unit is less frequently used in modern contexts but remains important
in specific areas where its historical relevance endures.
A meter (m) is a unit of length in the International System of Units (SI). One meter is equivalent to approximately 3.2808 feet.
The meter is defined by the distance light travels in a vacuum in 1/299,792,458 seconds.
Meters are used worldwide to measure length and distance in various fields, including science, engineering, and everyday life. Most countries have adopted the meter as the standard unit of
measurement for length.
Frequently Asked Questions (FAQs)
1. How do I convert furlongs to meters?
To convert furlongs to meters, multiply the number of furlongs by 201.168, since one furlong is approximately 201.168 meters. For example, 5 furlongs Γ 201.168 β 1,005.84 meters.
2. What is the formula for converting furlongs to meters?
The formula is: \( \text{meters} = \text{furlongs} \times 201.168 \).
3. How many meters are in a furlong?
There are approximately 201.168 meters in 1 furlong.
4. Is 8 furlongs equal to 1,609.344 meters?
Yes, 8 furlongs Γ 201.168 meters per furlong equals 1,609.344 meters, which is exactly one mile.
5. How do I convert meters to furlongs?
To convert meters to furlongs, divide the number of meters by 201.168. For example, 1,005.84 meters Γ· 201.168 β 5 furlongs.
6. Why do we multiply by 201.168 to convert furlongs to meters?
Because one furlong is defined as 220 yards, and since one yard is 0.9144 meters, multiplying 220 yards Γ 0.9144 meters per yard gives 201.168 meters in a furlong.
7. How many meters are there in half a furlong?
Half a furlong equals approximately 100.584 meters because 0.5 Γ 201.168 = 100.584 meters.
8. Are furlongs shorter than kilometers?
Yes, furlongs are shorter than kilometers. One furlong is approximately 0.201168 kilometers. | {"url":"https://convertonline.org/unit/?convert=furlongs-meters","timestamp":"2024-11-13T05:01:12Z","content_type":"text/html","content_length":"98818","record_id":"<urn:uuid:4734589b-f87d-49bf-a8f7-0c914ca07f1a>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00098.warc.gz"} |
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I loved loved loved loved loved reading this book. I have never felt such a renewed connection and understanding of mathematics than I have from this book and I offer my sincere thanks to Professor
Hart for sharing this book with the world.
As someone with a math degree and a library degree, this book felt custom written for me. Although I'm not well-versed in the classics, I did appreciate the connections that Hart made between books
like Moby Dick and math. I most enjoyed the parts where she discussed books that I've read, such as Life of Pi, Alice in Wonderland, and Arcadia. I learned some knew trivia, thought about some math
topics differently, and overall enjoyed it. Now I am off to read Half a Yellow Sun - I love Adichie so I don't know how I missed that one and I'm excited that it has a math connection!
A book bridging mathematics and literature has a few options: 1) It can be dull and dry; 2) It can be unobtainable; 3) or it can be Sarah Hart's book, Once Upon a Prime.
Funny, rich, approachable, and just the right amount of quirkiness, Hart dispels any myths about the separation between mathematics and literature. Relying on examples in-depth, from well-known to
more obscure texts, Hart pulls mathematical concepts from the pages of story deftly and with an academic personableness. Many of the texts I have read or am familiar with, and Hart pointed out
mathematical elements that I had simply glossed over with the previous reads.
I've often wondered about the layers and structuring of literature. Metaphor and simile offer hooks to hang many elements, including texture, emotion, color, and movement, but they are not elements
of structure. What is the architecture of literature?
We are not left stranded as writers and readers, for mathematics steps in to offer structures, equations, and models. In Once Upon a Prime, Sarah Hart rifles through classic and contemporary
literature from Moby Dick to A Gentleman in Moscow, from beloved novels to outlier texts, to share the plethora of mathematical referneces, and not only an arrow pointing to mathematics, but also a
snapshot into the ways mathematics provides a platform for story of strength, movement, and centenial wisdom. Mathematics, and the many solved and unsolved mysteries of mathematics, have been
harnessed into literature for centuries. Puzzles, secret codes, as in Sherlock Holmes, and inaccurate mathematics, such as in Gulliver's Travels, keep the literary landscape rich and intriguing.
Also, the book touches on mathematicians who were also obsessed with ones, ones I was familiar with such as Lewis Carroll and his charming Alice In Wonderland, and ones I was unaware of such as
George Eliot, who weaved her daily, voluntary lessons in mathematics into her beloved literature.
It's a causal assumption that literature and mathematics are two very distant points on the map of disciplinary space. In other words, we think the two don't mix much and when they do they don't mix
well. Here, enters Sarah Hart, professor of mathematics at Birkbeck, University of London to try and disabuse us of this notion in her book Once Upon a Prime.
Once Upon a Prime is divided into three parts, roughly three chapters per section. The first discusses the "fundamental structures of literary text, from plot in novels to rhyme scheme in verse." The
second is focused on the use of mathematical metaphors in literature. And the final section illustrates how mathematics can be deployed creatively in literature and takes a critical eye at some
well-known uses: The Life of Pi, Sherlock Holmes, Alice in Wonderland, and Flatland. There is a lot of interesting, entertaining, and edifying information distributed throughout the text, but the
middle section drags and feels a bit like random trivia. I also wish the first section, especially "The Geometry of Narrative" chapter, was expanded significantly. For instance, Hart mentions
Vonnegut's "Shape of Story" lecture but doesn't engage comprehensively or deeply enough with these ideas. I was disappointed by this as the introduction led me to believe this content figure
prominently in the book. Fortunately, Hart does provide a detailed analysis of the structure of verse though the analysis is primarily focused on rhyme schema. There is a discussion of meter, but I
found it somewhat confusing relative to more traditional discussions of how meter functions. It would have also benefitted to book to explore if there are certain poetic structures that are more
inherently pleasing than others to humans. Don't the ostensibly recurring patterns in visual, auditory, and literary arts suggest a universal structure to beauty? Or are the structures variable
enough across time and culture to suggest otherwise? Some of these topics may be outside the scope of the book, but I think exploring them would have made the book more resonant with a broader
readership and would have provided a deep link between math and art for readers.
After finishing, Once Upon a Prime I confess I am not entirely persuaded by the premise. I grant that literature has structure and that this can be described mathematically or statistically. But this
seems like a reading a bit much into a trivial observation. If we accept the Chomskyite theory of universal grammar (probably our best theory of language), can't we make claims about structure for
all spoken or written communication? Maybe Hart would concede this and argue that there are certain math structures that elevate language aesthetically. Unfortunately, claims to this effect aren't
made in the book. Instead the treatment of math in literature is mostly as a playful and experimental exercise. Hence, we are blessed with lots of discussion about Oulipo or the "workshop of
potential literature." This was a group of French intellectuals who essentially tried mathematical experiments in literature, such as writing whole novels without using certain letters (i.e. a
lipogram). I enjoyed learning about Oulipo and their members, but this seemed content tangential to the purported central claim of Once Upon a Prime.
Despite the grab bagginess of the book, Hart communicates complex ideas clearly and accessibly. There is a lot to amuse readers within the book, and there are a few instances where Hart belabors or
indulges concepts beyond what would be tolerable to general readers. She also always shows her work, sketching out the equation and computations that accompany the described math. Still, the reading
experiences can feel a bit like being inside a pinball machine. Hart bounces from topic to topic rapidly and sometimes wanders various tangents down too far before returning.
Although my criticisms may seem strong, I really appreciated the attempt made by Hart. In fact, the effort was probably a bit too ambitious. Each section (or even some of the chapters) could have
probably been given book-length treatments. Plus, these concepts and topics aren't necessarily the most general-audience-friendly. It certainly took considerable creativity and economy from Hart to
even acceptably assemble this work. Considering all this, I recommend this one. It's unique, making its weaknesses quite tolerable if not useful. And honestly, I did enjoy many portions immensely:
the section on cryptography, the miscellaneous history of mathematical and literary figures, and various esoterica one could only find in a book like this.
Kenneth S, Media/Journalist
This is a clever look at the intersection of math and literature. It will appeal to a cross-section of some people who have a foot in two worlds that the author convincingly argues are tightly bound.
This book was jam packed with fascinating connections between the worlds of mathematics and literature. In spite of the dense subject matter, Hart's prose was light and entertaining throughout. It
was fun to look through such an unusual lens on a beloved topic. I've already recommended this to some of my math-minded friends and family.
My thanks to both NetGalley and the publisher Flatiron Books for an advanced copy of this book on mathematics, literature, how they compliment each other, and the hidden use of numbers inside of so
many classic books.
Every math teacher I ever had made it quite clear that all the geometry, algebra, calculus and every little bit of math I learned in school, not well, was going to be necessary in life out of school.
Some of the math yes, I need to understand some or the hard science fiction I was reading. And yes, after graduating math did make an appearance in balancing what I had in a band account, and all the
things I wanted to buy. However it was working in a bookstore that made math important and necessary. In the day before phones we would have to be able to figure out the price of books with various
discount stickers. Institutional orders would take a certain amount of math, and geometry always came in when designing our tables for display. A briefly foray in chain stores taught me metrics of
customers per sale, and membership conversions, something like calculus I have rid my mind of. So books did teach me math appreciation in more ways than one. And now after reading Professor Sarah
Hart's Once Upon a Prime: The Wondrous Connections Between Mathematics and Literature I see the influence of math on many books that I have read and can appreciate the connections even more.
The book begins with a brief description of the author, a love of both books and numbers, rising to be one of the first woman to hold England's oldest mathematical chairs, and rediscovering
literature through a happy incident. While rocking her baby sleep at night, Professor Hart found it best to use an e-reader to read, which brought her back to literature and the numbers that she
never really thought about that appeared in many books. There is a discussion about cycloids, a beautiful mathematical arc the appears in both Moby Dick and other writings, that engaged the interests
of Blaise Pascal and Galileo. A chapter on poetry, looks at rhyme schemes, and how one poet with skill could make two poems with the same rhymes a large groups of poems, just by changing lines
around. Actually the section on poetry was one I enjoyed the most, learning about the numbers that make up a haiku or a sonnet, and all the possibilities that are hidden among the words. Sherlock
Holmes is of course a featured player as his arch-nemesis the "Napoleon of Crime" was of course a mathematician who was known for a book about asteroids, not for trying to kill Holmes numerous time.
Dan Brown makes an appearance in a part about bad numbers in literature, which is both funny and revealing how anything can be tweaked to make it sound good.
A very enjoyable book, with a different view of both math and literature. Not stogey, but very well written with a lot of humor, and great examples of math and literature. The writing is very good,
funny and informative, without much lecturing. Professor Hart is very good at explaining what principles Hart is discussing, and why it is important. Also readers can tell Hart enjoys reading and
math, and wants to share that joy, with others. Math is a part of life, honestly we are counting the moments till we are gone everyday. To know math is important to even what we do for fun, is a
great lesson, and I can appreciate more the beauty and mystery among some classics of literature.
Recommended for people interested in math, mainly to remind them to read a book and look at the world once in awhile. Also for people studying or interested in literature, especially poetry. Readers
learn a new way of looking. Very good and a lot of fun.
It is said that there are two kinds of people: numbers people and letters people. Sarah Hart challenges this notion, being a mathematician who loves to read. In this fascinating book, she explains
how many authors integrated math into literary masterpieces. From novels with mathematical structures to characters obsessed with numbers, from Moby Dick to Sherlock Holmes, it is amazing to learn
that so much math can be found there. Hart has a great sense of humor and explains some complicated concepts in an approachable way, without talking down at readers. I particularly enjoyed the
examples of “bad math” that can be found in Dan Brown or Stieg Larsson. She also clearly shows how the image that we have of math geeks as being ultra-logical machines with no feelings is wrong. As a
book lover who knows nothing about math, I enjoyed and understood it.
I chose to read this book and all opinions in this review are my own and completely unbiased. Thank you, #NetGalley/#Flatiron Books!
⭐️⭐️⭐️⭐️ I loved once upon a prime. Though at first I found it a little disorienting. I’m a mathematician who is currently struggling through War and Peace, and Moby Dick on my ereader. And I’m the
beginning the author talks about struggling through war and peace and Moby dick on, wait for it, her ereader. I sort of was wondering if I had had a bout of amnesia and forgotten I wrote a whole
book. This book is full of interesting Mathematical tidbits and facts about classic works of literature. If your interested in one of those things and can tolerate the other—I think you’d like this
book. It’s not a book that’s trying to hide from you or play games. If you hear “mathematical discussions of literature” and think “hey I’d like to read 250 pages about that” you’ll like this book.
It’s exactly what it says on the cover. I’m just not sure how wide the audience is for this book. I’d love to be wrong and for there to be throngs of people queueing up to buy this but I just think
there’s a small population of people interested in fractals and a small population interested in Moby Dick so how big can the intersection of those populations be? I did very much enjoy how little
time the author has for Dan Brown’s nonsense. I was provided an ARC in exchange for this honest review. #bookstagram #bookreview #mathematician #primenumbers #memoir
Will be posted on Goodreads/instagram/blog March 14 (PI DAY!)
This book instantly caught my attention with its punny title. Both math and literature are wide ranging subjects, and this book does a good job in demonstrating its scope. Explanations are clearly
written and easy to follow. The author's love for teaching definitely comes through and adds to the overall enjoyment of reading this book. The organization is logical and has good progression. I
would recommend this book to anyone interested in literature, math, or history, and anyone who likes learning more about patterns, structure, logic, and organization. | {"url":"https://www.netgalley.com/book/270164/reviews","timestamp":"2024-11-04T12:02:30Z","content_type":"text/html","content_length":"79003","record_id":"<urn:uuid:1ac3d5b8-2850-4504-9a91-2f92c1e95860>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00450.warc.gz"} |
Data science Archives - Dibyendu Deb
The logical functions in Power BI are essential while writing DAX expressions. Logical functions help us in decision making to check if any condition is true or false.
Once the data has been extracted through Power Query, these DAX expressions help us to fetch important information from the data. Here is an article explaining the difference between Power Query and
DAX, which you may be interested in.
The logical functions in Power BI I will discuss here are IF, AND, OR, NOT, IN and IFERROR. They are all true to their names and do the task exactly as they are used in English.
I will discuss them along with their application on a data set containing the area and production of different crops of different Indian states. Below is a glimpse of the dataset.
Dataset with crop production
I have collected the data from web with the data scraping feature of Power BI. Here is the article where I have explained how you can take advantage of this nifty feature of loading data from the web
in Power BI.
“IF” logical function
The IF function accepts three arguments. The expression of this logical function is as below. We can see that it has the same English conditional context and very easy to understand.
IF (expression, True_Info, False_Info)
The first argument of this function is a Boolean expression. If this expression has some positive value the IF function returns the second argument otherwise the third argument.
Let’s see a practical example of its use on the India_statewise_crop_production dataset. I have created a new column Production_category using the IF function. If the production is less than 10, then
it is under LOW production_category; otherwise HIGH production_category.
Creating new column using IF function
“Nested IF” function
We can use IF within another IF function, which is called the nested IF function. It helps us to check more than one condition at a time.
For example, I have placed two conditions here. One is the earlier one I used in the IF function and added another that if Production is greater than 500 then the production is HIGH else MEDIUM.
See the result below, how the Production_category column has the new values according to the NESTED IF condition.
Nested IF function
“AND” logical function
It can take two arguments. If both the arguments are correct, it returns TRUE else FALSE. Its syntax is as below:
AND (Logical_condition1, Logical_condition2)
I have applied the AND function to find out if the productivity is high or low. I have used AND to check if the conditions Area is less than 10 and production is higher than 200. If both the
conditions are TRUE then it returns “High Productivity” else “Low Productivity”.
Use of AND function
“OR” logical function
Unlike AND logical function, in the case of OR function if anyone condition holds true, the function returns TRUE. It returns FALSE only if both of the conditions are FALSE.
For the crop production data set, I have applied the OR function to check if both the conditions that are Area<10 and Production<20 are true then it should return “Low production” else “High
Use of OR function
“NOT” function
The NOT logical function simply changes FALSE to TRUE and TRUE to FALSE. It is very simple to use. See the below example.
I have used NOT with the IF function. If the IF checks the condition Season=” Kharif”, if it is true, IF returns True, again the NOT function turns it to False. See the output column “Kharif_check”,
it has False corresponding to Kharif and for other entries it has True.
Use of NOT function
“IN” logical function
The IN function lets us check the specific entries under a column and calculate corresponding values for other columns.
In this example, I wanted to calculate the total production for only three states “Assam”, “Bihar” and “Uttar Pradesh”. In order to do that, I have created one measure using the SUM and IN function
nested under the CALCULATE function. And see the result on a card.
Use of IN Function
“IF ERROR” logical function
The IF ERROR is another very useful logical function that checks for any error and returns values accordingly. This function is very useful while checking arithmetic overflow or any other kind of
The syntax for this function is as below:
IFERROR (Value, ValueIfError )
You can get the syntax guide when you will select the function in the Power BI editor, see in the below image. As soon as I have started to type the function name, Power BI IntelliSense guided me
with the autocomplete and the syntax for the function.
Use of IFERROR function
In my example, I have checked if there is an error in the Crop column. In case of any error found it should return “Error”. As there was no such error in the column so the IFERROR column has the
exact values as in the Crop column.
How to use the “COUNT” function in Power BI?
The COUNT() is an important function in writing the DAX formula in Power BI used. It is one of the time intelligence functions of DAX, which means it can manipulate data using time periods like days,
weeks, months, quarters etc. and then use them in analytics.
We apply DAX to slice and dice the data to extract valuable information. To import data from different data sources and perform required transformations we need to know the use of Power Query. If you
are curious to know the difference between Power Query and DAX, Here is an article you may be interested in.
Use of COUNT() in Power BI
The syntax for count function is very simple, we have to pass only the column name as argument like below
Measure = COUNT (Table_name [Column_name])
Count function when applied on any column, it returns the count of cells containing numbers. So it returns only whole numbers and skips the blank cells. If any cell of a column does not contain
anything (string, date or numerical) then the function returns blank.
Here is an example of the application of COUNT() on the data set I have on the rainfall of different Indian states. The dataset has three columns “SUBDIVISION” containing different ecological zones
of the country, “YEAR” from 1901 to 2019 and “ANNUAL” containing rainfall in mm of the corresponding year.
Application of COUNT() in Power BI
I data I have collected from the web using the data scraping feature of Power BI desktop. Here is a glimpse of the dataset.
Glimpse of the rainfall data
First, I have created a new measure using DAX (see here how can you create a new measure in Power BI). A measure has a default name “Measure” which I have changed to “Measure_count“.
Using COUNT() function in Power BI
Here you can see COUNT() is used to get the count of ANNUAL column cells having numbers. To see the result of COUNT() I have used a “Card”. The number “4090” in the card shows the cell count of the
ANNUAL column having a number.
If we change the column and replace ANNUAL with SUBDIVISION, then the count function returns “4116”. This is because rainfall of all the subdivisions are not present in the ANNUAL column. We can
check the difference and know how many subdivisions and year combinations do not have rainfall data.
The COUNTA() function
If a column consists of binary values like True and False, COUNT() fails to count them. To count such values COUNT() has another version which is COUNTA(). COUNTA() is for counting any logical value
or text and also the empty cells of the column.
In this data set we dont have any logical values. If COUNTA() function is applied on the same columns i.e. ANNUAL and SUBDIVISION, the results are same as COUNT() gave.
The COUNTAX() function
For those columns which have values other than strings, digits, logical values, date like formulae then there is another useful variation of COUNT() which is COUNTAX(). It returns the count of
non-blank rows evaluating the result of an expression on a table.
The DAX formula for COUNTAX() is:
COUNTAX ( <table>, <expression>)
It also returns whole number and unlike COUNTA() function, it iterates through the cells of that column, evaluates the expression and returns count of nonblank rows.
Here is an example of the application of COUNTAX() on the same table. I have used this function to calculate the count of row number of ANNUAL column for a particular YEAR in the rainfall table. I
have used the FILTER() function nested under COUNTAX() to filter the particular rows corresponding to the YEAR=1910 and 2010.
Application of COUNTAX()
From the above figure we can see that the COUNTAX() function has returned two different whole numbers for two different years 1910 and 2010. This is because not all the SUBDIVISION has the record of
annual rainfall for the year1910.
An overview of DAX in Power BI
As the name suggests Data Analysis eXpressions or DAX in Power BI is nothing but collection of operators, functions and constants which we use in writing formula or expressions to return value/
values. It is a native language for data analytics tools of Microsoft. DAX is also a highly versatile and functional language with the capacity to work with a relational database.
DAX helps us to dig into the data we already have in our hand to explore new information. It helps us to perform dynamic aggressions, slice and dice the data. It is different from Power Query with M
language at its core. Power Query performs the data extraction from different sources. Whereas DAX is applied to the extracted data source for analysis purpose.
It is very common to confuse between DAX and Power Query. You can refer this article to know a detailed comparison between Power Query and DAX.
Excel formula is similar to the DAX formula. Anyone with experience in writing Excel formula finds it easy to write DAX formula. However, DAX is far advanced than the Excel worksheet formula.
DAX is mainly used to create “Measures” and “Calculated Columns”. Below is an example of creating a measure using DAX.
Example of DAX formula
Writing effective DAX formula is the key. An effective DAX formula helps us to get the most out of the data. Writing the DAX formula in Power BI is easy. Power BI DAX editor has a smart complete
feature, which automatically prompts us with probable options.
Now let’s try writing a DAX formula to perform a simple calculation. I already have a data set in the Power BI desktop on the rainfall of different Indian subdivisions. The data was scraped from the
web using the data scraping tool of Power BI. You can get the details of how to do it in this article.
Below is an example of how a DAX measure has been created on the Power BI desktop. The screenshots from my Power BI desktop shows the steps of creating a measure. The purpose of the measure is to
create total annual rainfall.
First of all to create a new measure, right-click on the “Fields” pane of the Power BI desktop report/data window and then choose “New measure“.
Creating new measure
The default name of the measure is “Measure“. I have changed it to “Rainfall“. As you start writing the function name Power BI starts suggesting with relevant functions name. Here I have selected
“CALCULATE“. It is a very popular and frequently used function of DAX.
Steps for creating a measure using DAX
As we enter into the “CALCULATE” function, it starts to prompt us to show that it will accept an expression followed by filters. I have selected the “SUM” function and the “ANNUAL” column of the
“rainfall_india” table inside it as we want to calculate the total annual rainfall.
With this, the measure has been created. We can check the “Rainfall” measure in the “Fields” pane under the “rainfall_india” table.
Nested function in DAX
Inside the “CALCULATE” function again I have chosen the “SUM” function. This is an example of a nested function, which is a function within another function. Nested functions help us to narrow down
the query to achieve the desired result.
DAX can have up to 64 nested functions. Although using this many numbers of nested functions is very uncommon as debugging of such complex functions is very tough and the execution time of such
functions is also high.
Using a measure in another measure
Another useful feature of the DAX formula is it allows using a measure already created within another measure. For example, if want to further narrow down the result to calculate the total annual
rainfall of any particular subdivision, we can use the “Rainfall” measure we already created. Let’s see how to do it.
For example, we want to know the total annual rainfall of the state “Kerala“. The measure “Rainfall” calculates the total annual rainfall. So, we need to provide a filter within the calculate
function along with the “Rainfall” measure.
Using a measure within a measure
See the above image where I have nested one measure within another. A table and a bar chart are also created to compare the total annual rainfall and Kerala_rainfall just show how the measures are
Row context and filter context of DAX
These two concepts of context are very important for the effective use of DAX. Context refers to the dynamic analysis of the data.
Row context is related to functions while applying filters to identify a single row from the table. In most of the cases, we even dont realize that we are applying the concept of row context.
Filter context is a more complex concept than row context. It applies to narrow down the data. For example, here you can see how the column “SUBDIVISION” of “rainfall_india” has filtered the context
and helped us to get the annual rainfall of a particular subdivision.
An overview of Power Query in Power BI
Power Query in Power BI plays the role of a data connection technology. It does the data mashup i.e. connect, combine and refine data from many sources to meet the need of our data analysis.
Power Query is available in Excel 2016 or later version of Excel. It can also be added in Excel 2010 as an add-in. It is mainly used for data Extraction-Transformation and Load (ETL) in Excel
worksheet or Power BI model.
ETL is something which takes the major portion of time of a data analyst. To ease this task Power Query takes raw data from the source and convert to something more workable form. This form of data
is easy to analyze and to draw insights.
Data sources for Power Query
Power Query in Power BI and Excel allows us to extract data from almost any external sources and Excel itself. Here are some examples of the external sources we can bring data from. And there are
many more…
Some examples of external sources power query in Power BI can bring data from
After the data has been extracted from the desired source, Power Query helps us clean and prepare the data.
Using Power Query, we can easily append or stack different data tables. We can create relationships by merging different data tables, group and summarize using Pivot feature provided by Power Query.
The beauty of Power Query in Power BI lies in the fact that all this data transformation does not affect the original data set. The data transformation happens in the Power BI memory and we can
anytime get back our old data just by removing any particular data transformation step.
Applied Steps can be managed from Query Settings
Once we have summarized the data extracted from diverse sources, the report can be refreshed with one click. Every time new data added in the source data folder, Power Query helps us to update the
report accordingly with this refresh feature.
Flow of data processing by Power Query in Power BI
The M language and structure of Power Query
The M language is at the core of Power Query. It is the same as the F# language, case sensitive and contains code blocks starting with "let" and "in" as shown below.
<em> variable </em> = <em> expression </em> [,....]
<em> variable </em>
These blocks consists prcocedural steps of declaring and defining variables. Power Query is very flexible with physical position of these logical steps. That means we can declare a variable at the
begining of coding and then can define at the last.
But such a type of coding with a different logical and physical structure is very tough to debug. So, unless absolutely necessary, we should maintain the same logical and physical structure of Power
Editing the Power Query
Luckily we don’t need to write the Power Query in Power BI from scratch. It is already written in the background when we perform the data transformation steps. If it is needed we can tweak the Power
Query to make desired changes.
First of all, we need to open the data transformation window by clicking the “Transform data” option in Power BI. Then the Power Query can be edited using either the “Advanced Editor” or editing the
code for each “Applied Steps” of “Query Settings“.
Editing the Power Query in Power BI
The image below consists of an example of Power Query where the data is stored in a variable called “source“. Some other variables are also declared here to store the data with different
transformation steps.
The programming blocks of M language
The variables can be of any supported type with a unique name. Only if the variable name contains spaces, then the variable must contain a hashtag in the beginning and enclosed with quotes. It is the
protocol of declaring Power Query variables.
Power query Vs DAX: what is the difference?
This article presents a discussion on Power query Vs DAX. As they can perform many similar tasks so it is very normal to get confused when to use which one. Except for some features in common these
two are completely different in use and syntax.
Power Query performs the query from the source, through ETL process format and store the physical data tables in Excel or Power BI. DAX comes into the picture once the data is already queried from
the source to calculate tables and different analysis.
Here we will briefly go through the concepts of these two with uses and example.
Power Query
It is a powerful query language and helps us build queries to mashup data. The name “M” of the M language which is actually behind the Power Query has come from “Mashup”. Power Query is available
with Microsoft Excel 2016 Get & Transform and Power Query in Excel.
It is the M language syntax used in both Excel and Power BI Desktop. Mainly used for data Extraction, Transform and Load (ETL). ETL is an important step and allows us to start with our analysis
• Power Query is used in both Excel worksheets as well as Power BI models.
• It has lots of similarities in syntax with F# multi paradigm language which encompasses imperative, functional and object oriented programming methods.
• Case sensitive and contains programming blocks with “Let" & “In“
• The user may need a little programming experience in order to create advanced data mashup queries.
Power Query is used for query-time transformations in order to shape the data while extracting it.
Uses of Power Query
As mentioned, Power Query is basically for data extraction from the source. So, if we need any kind of data transformation operation to perform, then we will do it using Power Query before loading it
into Power BI.
Particularly in Power BI desktop, if we are clicking “Transform” instead of “Load” we are using Power Query or M to make a required transformation in the source data.
Example of Power Query
In the above image, we have an example of Power Query, used for calculating the length of the target text.
Suppose the source data has two columns with First Name and Last Name. And we want to concatenate them to create one single name column, then we should use M or Power Query to do that.
It can also be done using DAX but it will require several lines of codes. So better to use M for creating a new column, Pivot or Unpivot in Excel etc during ETL itself.
DAX (Data Analysis Expressions)
DAX has few functions identical to Excel but many other functions too and way powerful than Excel functions. It is used for Power Pivot, summarize, slice and dice complex data. Unlike Power Query,
DAX performs In-Memory transformation to analyze the extracted data.
DAX is a common language which can be found in SQL server analysis services Tabular, in Power BI and Power Pivot in Excel.
Use of DAX
As the name suggests, DAX is typically for data analysis tasks. After the data is already extracted, it is used for data modelling and reporting. Data analysis, slice and dice can be easily done with
It is very similar to Excel functions, does not have any programming blocks like M. So, any person has experience in Excel, can easily use DAX.
While creating Power BI report, DAX helps us to calculate Year to Date, calculate aggregates, ranges and several other analytical tasks easily with built in functions.
Use of DAX
Here is an example of DAX in the above image. It is used here to create a new column for reporting. The new column calculated the total value of another column. Here you can notice the difference
between the syntax of Power Query and DAX.
Final words on Power Query Vs DAX
Power Query and DAX are different. They are built with different purposes, they have different syntax and used in different stages.
Power Query is available both in Excel and Power BI. Users with Excel and the above version has this feature. Whereas DAX is exclusively available with Power BI.
The Data Analysis eXpression language is the far more advanced version of Excel worksheet functions. Still, DAX has many similarities with Excel functions hence easy to use as almost every one of us
have experience of using Excel.
Both of them are required to use the Power BI platform to its full potential. You can not chose any one of them as they are specialized for different stages of a complete analytical tasks.
How to do forecasting in power bi desktop?
Forecasting is predicting the future with the help of present and past data. It uses the concept of Exponential Smoothing to predict the future. The Power BI desktop has a very nifty feature of
forecasting. This article will describe the process with practical data.
The data has been collected from Wikipedia using the data scraping feature provided in Power BI. I have described here how you can load the data from the web with this feature.
The data I have collected has several years of information on the monthly and annual rainfall of different regions of India. This data can be used to predict the rainfall of those particular regions
for the coming years.
The data
I have selected only a single region including West Bengal and Sikkim with rainfall data from 1901 to 2016. The rainfall has been recorded in mm. On the basis of these many years data, lets try to
predict what will be the rainfall of next 5 years.
Here is a glimpse of the data.
Creating the line chart
In order to apply the Forecasting feature in the Power BI desktop, we need to create a line chart first. The line chart option is available in the “Visualizations” pane of the application.
Select the “Line chart” option from visualizations and then select appropriate variables from “Fields“.
The line chart option and variable selection
The next step is selecting the “Year” as the Axis variable and “Annual” rainfall in the Values. Consequently, the line chart will be created.
Creating the Line chart
Forecasting in the Power BI desktop
Now as the line chart is ready we need to create a Forecast for future time points. In “Visualizations” pane, under “Analytics” you can get the option for “Forecast“. But unless your data has at
least 60-time points the option will not be available.
Forecast tool in Power BI
Go with the default values of Forecast and click apply. Now forecast for 10 future time points is produced. As in my case each year is individual time point, so forecast for next 10 years will be
The confidence interval is 95% by default. In layman language, out of 100 times the experiment conducted, 95 times the forecast will lie within the interval shown around the forecast values.
Producing forecast with default options
But you can see the forecast produced, does not appear to be very realistic. It has no similarity with the historical trend. So, something is wrong here. The seasonality is left to be selected
automatically which is not working in the present case.
Seasonality in Forecasting
We need to provide appropriate value to the “Seasonality“. This parameter is the most important in the case of forecasting. So let’s try to adjust these value to get the most accurate result.
Seasonality in time series forecasting refers to a time period during which the data shows some regular and predictable changes. This period may be weeks, months or years with a cyclic pattern.
Identifying the “Seasonality” in forecasting
This cycle we can identify from the line chart we created. If we closely analyze the line chart and zoom it a little, we can notice the line repeats a pattern every 5-6 years period.
So, I will try to create the forecast with seasonality values close to 5 time points.
Checking accuracy of the forecast
To check the performance of the forecast, the forecast tool of the Power BI desktop has one feature “Ignore last”. It simply help us to produce the forecast leaving the last few points as mentioned
in this field.
Which means, for this many time period we have both the observed as well as forecast values. Thus we can compare how precise the forecast is.
If we take seasonality of 4 and 6 time points, the forecast has big differences with the observed ones. See the below images. For example, for the year 2011, the actual rainfall is 2418.70mm and the
forecast is 2733.56mm.
Forecast with seasonality 4
Again if we set seasonality as 6, again the forecast is very different from the original value.
Forecast with seasonality 6
But if we provide seasonality as 5 we achieve the best forecast with the closest values to the original rainfall. If we again take the example of the year 2011, with seasonality as 6 time points, the
rain forecast is 2337.89mm.
The “Format” option allows us to change the style of the forecast report generated. We can change the confidence interval pattern, line pattern and colour etc.
How to use Goal Seek and solver in Excel 2016?
Goal seek and solver in Microsoft Excel 2016 are two very important functions. These two help us to perform some back calculations. Among these two, Goal Seek is the simpler one. So let’s start with
the Goal Seek function.
I will demonstrate the use of Goal Seek with a very practical example. Every one of us wants to know the future value of their investment amount. And also how much they should invest to reach their
Here are some useful articles on the use of Power BI to create map visualization, data modelling, web scraping to collect data.
There are lots of online tools available to calculate this. Here we will use the formula for compound interest. We know that our investment either in Bank deposits or the market earns compound
Compound interest means annual interest gets accumulated with the principal amount and next time the interest is calculated on this increased amount.
For example, if we invest Rs 100.00 and gets a 10% interest in the first year, in the next year the principal amount becomes Rs 110.00. So the interest in the next year is 10% of Rs 110.00 that is Rs
11.00, it also gets added with the principal amount (Rs 110.00+11=Rs 121.00) and the process goes on.
See the below screenshot from my excel spreadsheet. It contains the formula for calculating the return from compound interest. Let’s take an example where we assumed an annual interest of 7% and want
to know the future value of Rs 10000.00 invested for a period of 10 years.
Use of Goal Seek in Excel 2016
Calculating return on investment
Now what if we want to know how much we need to invest in order to get a return of Rs 25000.00 keeping other conditions same. Here we need to back calculate the investment amount using Goal Seek.
You can find the Goal Seek option in the “What-If Analysis” under the Data tab of Microsoft Excel. It has three fields which we need to fill. See the below figure to understand the process.
We need the Future Value of cell B5 to be 25000. So the “Set Cell” is B5, “To Value” is 25000 and we want to change the value of cell B2.
Using Goal Seek in Excel
Now click “OK” to know the investment amount. Now we know that we need to invest Rs 12708.73 (the cell was not set to 2 decimal places) to become Rs 25000.00 in 10 years with an interest rate of 7%.
Result of Goal Seek
Again if we want to know how many years it takes to make the same amount Rs 10000.00 to Rs 25000.00 with 7% interest. We need to use Goal Seek and will change the cell B4. See the below image, now we
know that we have to keep the amount invested for 14 years.
Another example of Goal Seek
But the problem with Goal Seek is that it can not be used for changing more than one variables. It is for simple calculations. If we have a more complex situation and needs several variables to be
changed to at the same time, we need to use Solver.
So, lets know how Solver functions with an example.
Use of Solver in Excel 2016
“Solver” is not a default application in Excel and comes as an Add-in. You have to add it to make the option appear under the “Data” tab. Follow the steps as shown in the screenshots below to add
this Add-in.
Go to “File” and then click “Options” to open Excel Options page.
Opening Excel options
In Excel Options, go to Add-ins and click on “Go...” to open the window containing list of available Add-ins. Now select “Solver Add-in” and click OK.
Activating Solver Add-in
Now check if the Solver Add-in has been added under the Data tab.
Solver Add-in under Data tab
Application of Solver Add-in
Now to see the application of Solver, let’s take another simple yet practical example. Below I have shown a small stock portfolio created in Excel. Here I have calculated the total invested amount of
some stocks with some hypothetical prices. As shown the invested amount has been calculated by multiplying the cost with the stocks quantity.
The total amount stands as Rs. 202660.00. But I want to invest Rs 40000.00. So my goal is to calculate the quantity of some stocks lying within a specified range. And the constraints for this
calculation are also mentioned in the below image.
The example data set and constrain
Like Goal Seek, Solver also needs to input the “Objective cell” and “Variable cells” where we want to change the values. See the screenshot below where I have shown how to mention the cells as per
our requirement. The value field has 40000 as we want to invest Rs 40000.00 in total.
Application of Solver
In the “Add Constraint” field, you need to provide the cell reference and their specific “Constraint“. They need to be provided one by one and finally they are added to the “Subject to the
constraints” field. See the image below to understand the process.
Adding constrains in Solver
Now click “Solve” and then if Solver is able to find a solution for your problem, the next window appears where you need to confirm the change by clicking OK.
Now you have the number of stocks and their costs which you can buy within your budget of Rs 40000.00.
Changed values with Solver
I hope the article will help you understand how to use both Goal Seek and Solver in Excel 2016. Please comment below if you have any questions or doubts regarding the article.
How to create new column from existing column Power BI
Create new column from existing column in Power BI is very useful when we want to perform a particular analysis. Many times the new column requires clipping a string part from an existing column.
Such a situation I faced a few days back and was looking for a solution. And this is when I again realised the power of “Power Query“. This article is to share the same trick with you.
If you are new to Power BI, then I would suggest going through this article for a quick idea of its free version, Power BI desktop. It has numerous feature for data manipulation, transformation,
visualization, report preparation etc.
Here are some popular application of Power BI as a Business Intelligence tool.
This article covers another super useful feature of Power BI. Adding a new column derived from the existing columns are very common in data analytics. It may be required as an intermediate step of
data analysis or fetching some specific information.
For example, we may need only the month or day from the date column, or only the user-id from the email-id list etc.
In this article I will demonstrate the process using the data sets related to India’s state-wise crop production and rainfall data.
Lets start the process step by step and see how I have done this.
Use of “Add Column” and “Transform” options
Power BI desktop offers two options for extracting information from an existing column. Both the options namely “Add column” and “Transform” has different purposes altogether.
Create new column from existing column Power BI
Add column option adds a new column extracting the desired information from the selected column. Whereas the Transform option replaces the existing information with the extracted text.
Here our purpose is to create new column from existing column in Power BI. So let’s explore the “Add Column” feature and the options it offers.
Create new column from existing column Power BI with “Add column” option
First of all, you need to open the “Power Query Editor” by clicking “Transform data" from the Power BI desktop. Here you will get the option “Extract” under the “Add column” tab as shown in the below
Extracting the “Length“
This option is to fetch the length of any particular string and populate the new column. See in the below example, I have created a new column with the length of the state names from “State_Name”
Extracting length
The power query associated with this feature is as given below. The M language is very easy to understand. You can make necessary changes here itself if you want.
= Table.AddColumn(#"Changed Type", "Length", each Text.Length([State_Name]), Int64.Type)
Extracting the “First Characters“
If we select the “First Character” option, then as the name suggests, it extracts as many characters as we want from the start. As shown in the below image, upon clicking the option, a new window
appears asking the number of characters you want to keep from the first.
As a result, the new column named “First Characters” contains the first few characters of our choice.
Extracting first characters
Extracting “Last Characters“
In the same way, we can extract the last characters as we need. See the below image, as we select the option "Last Characters", again it will prompt for the number of characters we wish to fetch from
the last of the selected column.
Last characters
As we have provided 7 in the window asking the number of characters, it has extracted a matching number of characters and populated the column “Last Characters”.
Extract “Text Range“
This option offers to select strings from exact location you want. You can select starting index and number of characters from the starting index. See the bellow example where I wished to extract
only the string “Nicobar”.
Keeping this in mind, I have provided 12 as the starting index and 7 as the number of characters from the 12th character. The result is the column “Text Range” has been populated with the string
“Nicobar” as we wanted.
Extracting text range
Extracting using delimiters
Another very useful feature is using delimiters for extracting necessary information. It has again three options for using delimiters, which are:
• Text Before Delimiter
• Text After Delimiter
• Text Between Delimiter
The image below demonstrates the use of the first two options. As the column “State_Name” has only one delimiter i.e. the blank space between the words, so in both the cases, I have used this
delimiter only.
You can clearly observe differences between the outputs here.
Use of text before delimiters
The script for executing this operation is given below.
= Table.AddColumn(#"Removed Columns", "Text After Delimiter", each Text.AfterDelimiter([State_Name], " "), type text)
Below is the example where we need to extract text between the delimiters. The process is same as before.
Text between delimiters
The code is as below.
= Table.AddColumn(#"Inserted Text After Delimiter", "Text Between Delimiters", each Text.BetweenDelimiters([State_Name], " ", " "), type text)
Another example with different delimiters
Below is another example where you have some delimiter other than the blank space. For example one of the columns in the data table has a range of years. And the years separated with a “-“.
Now if we use this in both cases of text before and text after delimiter, the results are as in the below image.
Use of text before delimiters
Use of “Conditional Column“
This is another powerful feature of the Power BI desktop. Where we create a new column applying some conditions to another column.
For example, in the case of the agricultural data, the crop cover of different states have different percentages. And I wish to create 6 classes using these percentage values.
First, open the Power Query Editor and click the “Conditional Column” option under the “Add column” tab. You will see the window as in the below image.
The classes will be as below
• Class I: crop cover with <1%
• Class II: crop cover 1-2%
• Class III: crop cover 2-3%
• Class IV: crop cover 3-4%
• Class V: crop cover 4-5% and
• Class VI: crop cover >5%
See the resultant column created with the class information as we desired.
Use of conditional column
Using DAX to create new column from existing column Power BI
We can also use DAX for creating columns with substrings. The option is available under the “Table tools” in Power BI desktop. See the image below.
New column option in Power BI Desktop
Now we need to write the DAX expressions for the exact substring you want to populate the column with. See the image below, I have demonstrated few DAX for creating substring from a column values.
Comparing to the original values you can easily understand how the expressions work to extract the required information.
Creating column with substrings
Combining values of different columns
This is the last function which we will discuss in this article. Unlike fetching a part of the column values, we may need to combine the values of two columns. Such situation may arise in order to
create an unique id for creating keys too.
Likewise here my purpose was to combine the state name and corresponding districts to get a unique column. I have used the COMBINEDVALUES() function of DAX to achieve the goal.
See the below image. I have demonstrated the whole process taking screenshots from my Power BI desktop.
Use of COMBINEDVALUES function
I have tried to cover the steps in a single image. The original two column values as well as the combined value columns are shown side by side so that you can compare the result.
Final words
In this blog I have covered several options available in Power BI desktop for creating new column extracting values from other columns. We frequently face such situation where we need to create such
columns in order to get desired analysis results or visualization.
I hope the theory explained along with the detailed screenshots will help you understand all the steps easily. In case of any doubt please mention them in the comment below. I would like to answer
How to add data from website to Power BI desktop
Different webpages are rich source of data. Either structured or unstructured, these data are very useful and can provide good insights.
Power BI has recently added and enhanced the existing feature of data extraction from the web. This feature was already compelling, with the recent enhancement it has become even more powerful.
In this article I will discuss this feature in detail with a practical example.
A practical example of web scraping with Power BI desktop
I have a data set with information on state-wise crop production in India. The data was collected from data.world. I have discussed this data and how I have analyzed it with the Power BI desktop in
this article.
Now my purpose was to analyze this crop production data in context with India’s economic growth. As we know that any country’s GDP (Gross Domestic Product) and GSDP (Gross State Domestic Product) are
very good indicators of its economic growth.
So we need to collect this data in order to correlate the state wise crop production with GDP and GSDP of corresponding state.
But the problem is such data is not readily available. So…..
Is web scraping an alternative option?
In this scenario, web scraping is generally the only solution.
You can read this article to know how to write web scrapers with python to collect the necessary information.
But writing a web scraper with Python needs coding knowledge. It is not possible for a person with zero knowledge in software development or at least any single programming language (preferably
Add data from website to Power BI desktop
Here comes Power BI desktop with its immense powerful feature to add data directly from the website. It is really a boon for the data analysts/scientists who want this data import process smooth real
It does not require any software development background. Anyone with no idea about any programming language at all (really!!) can use it.
So without any further ado, lets jump to see how you can also do the same.
The data source
I will use an authentic website like Wikipedia for open-source data. A data source that has unquestionable authority. If you simply search Google with the search query “Indian states and union
territories with their GDP” the first result you will get is from Wikipedia.
Google search result for Indian GDP
Importing the data from website to Power BI desktop
Power BI allows importing data from numerous sources as I have mentioned in the introductory article on Power BI. Among several sources, one is from “Web“.
As you can see in the below figure, you need to select the “Web” from “Get data” option under “Home” tab. Consequently, a new window will open where the URL is to be provided.
Importing data using URL of particular website
If you have already imported the data once from a website, then the address gets stored in “Recent sources“. It will help you to quickly import the data in case you need the data again to import.
“Recent source” of previously used website URL
As you provide the URL, it will first establish a connection to fetch the data. Next, a “Navigator” will open to show you a preview of the data.
See the below screenshot of the Power BI app on my computer; on the left side, all the tables from the web page will be listed. You can click the particular table you are interested and the table
preview will be displayed on the right-hand side.
Navigator and table view
Similarly, if you want to get a glimpse of the web view, just click the “Web view“. A web view of the page will be displayed as shown below.
Web view of the page
Transform/load data to Power BI desktop
Now if you are satisfied and found the particular information you are looking for, proceed with the data by clicking “Load” or “Transform". I would suggest going for the transform option as it will
enable you to make the necessary changes in the data.
As I have shown the data below before loading it in Power BI. With the help of Power Query, I have made minor changes like changing the column name, replacing the blank rows or replacing the “Null”
values, adding necessary columns etc.
I have already described all these operations in data transformation steps here.
Window for data transformation
Once you are satisfied with the table you created, you can load it in Power BI for further processing.
Add table using examples
Another very useful feature is to "Add table using examples". As you can see this option at the bottom left corner of the window in the below screenshot.
Add table using example
This option is very helpful when the tables Power BI automatically shows do not cater to your purpose.
Suppose for the above web page you can see almost all the structured data in table form on the left pane. But you are looking for some information which is scattered on the page and not in a table
In that case, if you click the option “Add table using example” you will be provided with a blank table along with the web view as shown in the above image.
Upon clicking the row of the table Power BI provides several options which you can choose from to fill the table. As shown in the above image, some information with no table structure are there to
populate the table.
You can also add several columns by clicking on the column header with “*”. Also change column name or later at the data transformation stage.
Final words
So, I hope this article will help you to collect required information from any web page using this feature of Power BI. It is very simple to use only you need to be aware of its existence.
My purpose was to provide you with a practical example of real world data which will make you familiar to this feature. And also to document all the steps for my future reference.
For data analysts and those who just want to get their desired data, writing web scrapers is pure time waste. I myself have written several web scrapers. It obviously has some benefits and can get
you some very specific data from several webpages.
But if your data is not scattered in multiple webpages and can be fetched from specific URL, Power BI is your best friend. It will save your lots of time of collecting data and you can straight way
jump to the main task of data analysis.
If you find the article helpful, please let me know commenting below. Also if you have any question regarding the topic I would love to discuss with you.
How to join tables in power bi desktop: a practical example
Joining tables is an important feature to combine information from several tables. We can join tables in Power BI desktop with a very nifty merging feature provided with it..
In this article, I am going to demonstrate how to join tables in Power BI desktop with some practical data. The data are all open source. You can collect them from the links provided and use them to
Combining data
When it comes to combining data in tables, it can be done in two ways. One is you may need to increase the rows of a table with new data. This type of data combination is known as “Appending“.
Whereas when you add columns with new information with an existing table, it is called “Merging“.
Power BI provides both of these features under the “Home ” tab (as in the below figure). You need to use them according to your requirement.
Merge and Append queries
This particular article is on joining tables. So we are going to discuss the Merge queries option here only with suitable example.
Lets first discuss different kinds of joins. This will be an overview so that you can make the right decision while selecting join types while merging queries. Also will suggest going through the
Microsoft documentation page for details.
The data set
The data set I have used for demonstration purpose is on India’s state-wise crop production collected from data. world. And another data set with India’s state-wise rainfall from different years.
Both the data set presents a real-world experience. The data is collected in raw form and refined using data transformation feature of Power BI. You can go through all the data transformation steps
The measures used for different calculations are described in this article.
The first table that is the crop production table contains the area under different crops in hectare (ha) respective to different districts of different Indian states. Whereas the second table i.e.
the rainfall table has the rainfall record in mm respective to different districts of Indian states.
Replacing/removing errors
While importing the data from the CSV file or from the web itself, you may face some missing values. When it gets imported in Power BI, the missing values are shown as “Error“.
Now you can not proceed without handling these errors. One way is to replace these errors with proper values. It may be mean or median of rest of the values of particular variable or you can simple
remove the rows with missing values.
In the below screenshot you can see how I have replaced the error with a suitable descriptive value. The “Remove errors” option will simply remove the corresponding rows.
Now suppose we want information on both the cropped area as well as rainfall of particular state and districts. Then we need to join the tables with proper conditions.
Different kinds of join tables in power bi desktop
Now Power BI will ask you about the particular join type you are interested to apply. And in order to use the correct join, we should have the idea about different joins. So here is a vivid
description with examples of different joins.
For demonstration purpose I have picked few rows from both the tables and created two tables.
Left outer join
It is a prevalent joining process and the default one in Power BI, where the left or 1st table (as in the figure below) retains all of its rows and matching rows from the right or 2nd table. As the
text from “Merge queries” option of Power BI displays “All from first, matching from second“.
Suppose we want to know the rainfall of some particular districts with a definite amount of cropped area. So how will you join the two tables to fetch the information? Here is left outer join to help
you. See the below figure
Left outer join
In the above figure the particular information we are interested are colored as green cells. The corresponding information has the same color code in the second table.
So, as per the rule of the left outer join, all the rows (yellow coloured) of the left table and the green-coloured rows of the right table has been joined in the new table.
The Venn diagram at the bottom right corner describes the joining process in colour codes. According to this diagram, the matching rows are called the interaction of the two sets. So here the new
table consists of the complete left set and the interaction of the sets.
Right outer join
Now suppose we are interested to know the information exactly opposite to what we have fetched earlier. This time we want to know the cropped area of districts having particular rainfall. So, here
the right outer join we need to perform.
Right outer join of tables in Power b
BI desktop
Full outer join
When we are in need of information of all the states and districts with their cropped area and rainfall, we should go for a full outer join. As the name suggests, this join will return all the
records including the matching ones.
Full outer join of tables in Power BI desktop
Here is the final output from the full outer join. The rows contain both the rainfall and cropped area information including the matching rows (in green colour).
Inner join
Again if we need information on the rainfall of those districts with cropped area data as we have nothing to do with rainfall data for those districts not having any cropped area information. In this
situation, the inner join produces the desired result.
See the below figure where we have applied inner join on both the tables.
Inner join of tables in Power BI desktop
See in the above image, only the matching rows are kept and all other rows have been excluded.
Left anti join
Suppose for the sake of data analysis, we need only those states and respective districts for which we don’t have any rainfall data, how can we fetch the required information?
Not to worry here the particular join type we need to apply is left anti join. Which will keep all the rows from the left table removing all those which have a match in the right table. See the below
Left anti join of tables in Power BI
Here in the above image we just have the required information from the left table of state wise cropped area and everything else have been excluded.
Right anti join
Now suppose we need information exactly opposite the earlier one i.e. we need only the rainfall data of all those districts for which we don’t have any information on the cropped area. The join we
will apply here is right anti join.
See the below demonstration with the two tables.
Right anti join of tables in Power BI
In the above figure, the crop production tables have been joined with the rainfall table using the right anti join. You can see that all the rows of rainfall table have been retained excluding the
matched rows from the crop production table.
Join tables in Power BI desktop
Now lets see a practical application of joining tables in Power BI desktop.
When you click the merge query option of Power BI desktop, you will see the first table as the active table (the selected able while clicking the merge query option).
You need to select a column from the table with unique values and can act as a key. Then a drop-down will allow to chose from the available tables. Again you have to select another column from the
second table using which both the tables can be joined.
Then you need to select the particular join option from the drop-down as shown below. All the join options are available in the same sequence as we discussed above.
Join tables options in Power BI desktop
For example here in the above image, you can see I have selected column “State_name” from the table “India_statewise_crop_production” and the column “SUBDIVISION” from the second table
“rainfall_India” as the key columns for joining.
Now our purpose was to keep all the crop production information with matching rainfall data of the corresponding districts. So, I opted for “Left outer join” which is also the default option. You can
choose any of them as per your requirement.
Now simply click and see the operation in process.
Selecting rows from the joined table
Now the newly created column will appear in the table. You can see in the below image that the whole table is displayed as a column. So you need to select the particular column and deselect the
option “Use original column as prefix“.
See in the above image, by default the complete table appears as row element in the first table. You need to select the particular columns of the table you want here.
Use of advanced editor
Use of advanced editor of Power BI allows more flexibility to change the M script itself yo make desired changes in the output. The M language is at the core of every Power BI application.
See in the below image, the option “Advanced Editor” under the “Home” tab shows the M script behind the particular application. You can tweak them to change the join type or column names etc.
Editing M script for Joining tables in Power BI desktop
Final words
So, here is all about on join tables in Power BI desktop. I have tried to cover all the basics of each kind of join with examples. So that the you can understand the logic behind the joins. And apply
the right kind for your need.
Joining tables and fetching the exact information is the core of data modelling. In order to obtain a good visual representation of the Power BI report, a good data modelling is must.
I hope this article will help you to get a good grip on this very fundamental operation of Power BI. If you have queries or doubt, please comment below. I would like to answer them. | {"url":"https://dibyendudeb.com/category/data-science/","timestamp":"2024-11-14T21:57:46Z","content_type":"text/html","content_length":"245898","record_id":"<urn:uuid:ee1fc7f1-280f-49fd-a8d2-c8292627acb9>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00662.warc.gz"} |
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Nature Communications volume 14, Article number: 3105 (2023)
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Epidemiological models are commonly fit to case and pathogen sequence data to estimate parameters and to infer unobserved disease dynamics. Here, we present an inference approach based on sequence
data that is well suited for model fitting early on during the expansion of a viral lineage. Our approach relies on a trajectory of segregating sites to infer epidemiological parameters within a
Sequential Monte Carlo framework. Using simulated data, we first show that our approach accurately recovers key epidemiological quantities under a single-introduction scenario. We then apply our
approach to SARS-CoV-2 sequence data from France, estimating a basic reproduction number of approximately 2.3-2.7 under an epidemiological model that allows for multiple introductions. Our approach
presented here indicates that inference approaches that rely on simple population genetic summary statistics can be informative of epidemiological parameters and can be used for reconstructing
infectious disease dynamics during the early expansion of a viral lineage.
Phylodynamic inference methods use pathogen sequence data to estimate epidemiological quantities such as the basic reproduction number and to reconstruct epidemiological patterns of incidence and
prevalence. These inference methods have been applied to sequence data across a broad range of RNA viruses, including HIV^1,2,3,4, Ebola virus^5,6,7, dengue viruses^8, influenza viruses^9, and most
recently severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2)^10,11,12. Most commonly, phylodynamic inference methods rely on underlying coalescent models or birth-death models.
Coalescent-based approaches have been generalized to accommodate time-varying population sizes and structured epidemiological models, for example, susceptible-exposed-infected-recovered (SEIR) models
and models with spatial subdivision^6, 13. Birth-death approaches^14,15, where a birth in the context of infectious diseases corresponds to a new infection and death corresponds to a recovery from
infection, carry advantages such as capturing the role of demographic stochasticity in disease dynamics, which may be particularly important in emerging diseases that start with low infection numbers
^16. Birth-death approaches have also been expanded to incorporate the complex nature of infectious disease dynamics including structured populations^17. Both coalescent-based and birth-death
phylodynamic inference approaches rely on time-resolved phylogenies and have been incorporated into the phylogenetics software packages BEAST1^18 and BEAST2^19 to allow for joint estimation of
epidemiological parameters and dynamics while integrating over phylogenetic uncertainty^6,20. Integrating over phylogenetic uncertainty is crucial when applying these methods to viral sequence data
that are sampled over a short period of time and contain only low levels of genetic diversity. However, integrating over phylogenetic uncertainty can be computationally intensive. Moreover,
phylodynamic approaches that use reconstructed trees for inference require estimation of parameters associated with models of sequence evolution, along with parameters that are of more immediate
epidemiological interest.
Here, we present an alternative sequence-based statistical inference method that may be particularly useful when viral sequences are sampled over short time periods and when phylogenetic uncertainty
present in time-resolved viral phylogenies is considerable. Instead of relying on viral phylogenies to infer epidemiological parameters or to reconstruct patterns of viral spread, the “tree-free”
method we propose here fits epidemiological models to time series of the number of segregating sites (that is, the number of polymorphic sites) present in a sampled viral population. The approach we
propose here allows for structured infectious disease models to be considered in a straightforward “plug-and-play” manner. It also incorporates the effect that demographic noise has on
epidemiological dynamics. Below, we first describe how segregating site trajectories are calculated using sequence data and how they are impacted by sampling effort, rates of viral spread, and
transmission heterogeneity. We then describe our proposed statistical inference method and apply it to simulated data to demonstrate the ability of this method to infer epidemiological parameters and
to reconstruct unobserved epidemiological dynamics. Finally, we apply our segregating sites method to SARS-CoV-2 sequence data from France, arriving at quantitatively similar parameter estimates to
those arrived at using epidemiological data.
The number of segregating sites present in a set of sampled viral sequences is defined as the number of nucleotide sites at which genetic variation is present in the sample set. To determine whether
the number of segregating sites that are observed over time in a viral population may be informative of underlying epidemiological dynamics, we forward-simulated a classic
susceptible-exposed-infected-recovered (SEIR) epidemiological model, augmented with viral evolution, under various sampling efforts and parameterizations (Fig. 1; Methods). Simulations of this
augmented SEIR model, initialized with a single infected individual, first indicate that segregating site trajectories are sensitive to sampling effort, as expected (Fig. 1a, b). More specifically,
we considered three different sampling strategies, each with sequences binned in consecutive, nonoverlapping 4-day time windows to calculate segregating site trajectories. These three sampling
strategies consisted of a strategy with full sampling effort (all sequences per 4-day time window), one with dense sampling effort (40 sequences per 4-day time window) and one with sparse sampling
effort (20 sequences per 4-day time window). With all three of these sampling efforts, the number of segregating sites first increases as the epidemic grows, with mutations accumulating in the virus
population. Following the peak of the epidemic, the number of segregating sites starts to decline as viral sublineages die out, reducing the amount of genetic variation present in the viral
population. A comparison between full, dense, and sparse sampling efforts indicates that lowering sampling effort results in a lower number of observed segregating sites during any time window. This
is because at lower sampling effort, less of the genetic variation present in a viral population over a given time window is likely to be sampled. The patterns shown here across sampling strategies
are robust to the time window length used for the calculation of segregating site trajectories (Figure S1).
a Dynamics of infected individuals (I) under an SEIR model simulated with an R[0] of 1.6. b Segregating site trajectories under full (black dashed line), dense (black lines), and sparse (gray lines)
sampling efforts. Dense and sparse sampling correspond to 40 and 20 sequences sampled per time window, respectively. c Simulated infected dynamics under the SEIR model with an R[0] of 2.0 (blue line)
compared to those of the R[0]=1.6 simulation (black line). d Segregating site trajectories for the two simulations shown in panel c. e Simulated infected dynamics under the SEIR model with
transmission heterogeneity (green, dashed line) compared to those of the R[0]=1.6 simulation (black line) without transmission heterogeneity. Transmission heterogeneity was included by setting the
parameter p[h] to 0.06. For ease of comparing segregating site trajectories, the transmission heterogeneity simulation was shifted later in time (green, solid line). f Segregating site trajectories
for the shifted transmission heterogeneity simulation (green lines) and the original simulation (black lines). g Simulated infected dynamics under the SEIR model with changing R[0]. In the
simulations shown in red and yellow, when the number of infected individuals reached 400, R[0] was decreased to 1.1 and 0.75, respectively. The simulation in black has R[0] remaining at 1.6. h
Segregating site trajectories for the three simulations shown in panel g. Dense sampling effort was used to generate all segregating site trajectories shown in panels d, f, and h. 30 randomly-sampled
segregating site trajectories are shown for each sampling effort in panel b and for each epidemiological scenario in panels d, f, and h. In all model simulations, ({gamma }_{E}=1/2) days^−1, ({gamma
}_{I}=1/3) days^−1, population size N=10^5, and the per genome, per transmission mutation rate μ=0.2. Initial conditions are S(t[0])=N-1, E(t[0])=0, I(t[0])=1, and R(t[0])=0. For the
transmission heterogeneity simulation (panel e), I[h](t[0])=1 and I[l](t[0])=0 was used instead of I(t[0])=1. A time step of τ=0.1 days was used in the Gillespie τ -leap algorithm.
To assess whether segregating site trajectories could be used for statistical inference, we first considered whether these trajectories differed between epidemics governed by different basic
reproduction numbers (R[0] values). Figure 1c shows simulations of the SEIR model under two parameterizations of the basic reproduction number: an R[0] of 1.6, corresponding to the simulation shown
in Fig. 1a, and a higher R[0] of 2.0 (implemented via a higher transmission rate β). The epidemic with the higher R[0] expanded more rapidly (Fig. 1c) and, under the same sampling effort, resulted in
a more rapid increase in the number of segregating sites (Fig. 1d). This indicates that segregating site trajectories can be informative of R[0] early on in an epidemic.
We next considered the effect of transmission heterogeneity on segregating site trajectories. Many viral pathogens are characterized by ‘superspreading’ dynamics, where a relatively small proportion
of infected individuals are responsible for a large proportion of secondary infections^21. The extent of transmission heterogeneity is often gauged relative to the 20/80 rule (where the most
infectious 20% of infected individuals are responsible for 80% of the secondary cases^22). Some pathogens like SARS-CoV-2 exhibit extreme levels of superspreading, with as low as 10-15% of infected
individuals responsible for 80% of secondary cases^10,23,24,25. Because transmission heterogeneity is known to impact patterns of viral genetic diversity^26, we simulated the above SEIR model with
transmission heterogeneity to ascertain its effects on segregating site trajectories (Methods). Because transmission heterogeneity has a negligible impact on epidemiological dynamics once the number
of infected individuals is large^27, epidemiological dynamics with and without transmission heterogeneity should be quantitatively similar to one another, with transmission heterogeneity simply
expected to shorten the timing of epidemic onset in simulations with successful invasion^21. Our simulations, parameterized with extreme transmission heterogeneity of 6/80, confirm this pattern
(Fig. 1e). To compare segregating site trajectories between these simulations, we therefore shifted the simulation with transmission heterogeneity later in time such that the two simulated epidemics
peaked at similar times (Fig. 1e). Comparisons of segregating site trajectories between these simulations indicated that transmission heterogeneity decreased the number of segregating sites during
every time window (Fig. 1f). As expected, lower levels of transmission heterogeneity result in less substantial decreases in the number of segregating sites (Figure S2). Together, these results
indicate that transmission heterogeneity needs to be taken into consideration when estimating epidemiological parameters using segregating site trajectories.
Finally, we wanted to assess whether changes in R[0] over the course of an epidemic would leave signatures in segregating site trajectories. We considered this scenario because phylodynamic inference
has often been used to quantify the effect of public health interventions on R[0], most recently in the context of SARS-CoV-2^10,11. We thus implemented simulations with R[0] starting at 1.6 and then
either remaining at 1.6 or reduced to either 1.1 or 0.75 when the number of infected individuals reached 400 (Fig. 1g). The segregating site trajectories for these three simulations indicate that
reductions in R[0] over the course of an epidemic leave signatures in this summary statistic of viral diversity (Fig. 1h). The signatures left in the segregating site trajectories reflect the
epidemiological dynamics that result from the reductions in R[0]. Reducing R[0] to 1.1 results in a slower increase in the number of cases and a delayed, as well as broader, epidemic peak; as such,
the number of segregating sites increases more slowly and the decline in the number of segregating sites is not apparent over the time period shown. Reducing R[0] to 0.75 results in an immediate
decline in cases, with an observed drop in the number of segregating sites due to the stochastic loss of viral sublineages. Similar magnitude reductions in R[0] that were implemented later on in the
simulated epidemic yielded fainter signatures of this effect in the segregating site trajectories (Figure S3).
To examine the extent to which inference based on segregating sites can be used for epidemiological parameter estimation, we generated a mock segregating site trajectory by forward simulating an SEIR
model with an R[0] of 1.6. From this simulation, we randomly sampled 500 viral sequences (corresponding to approximately 0.78% of infections being sampled) and binned these sequences into 4-day time
windows based on their sampling times (Fig. 2a). Figure 2b shows the segregating site trajectory from these binned sequences. From this trajectory, we first attempted to estimate only R[0] under the
assumption that the timing of the index case t[0] is known (Methods). We estimated an R[0] value of 1.58 (95% confidence interval of 1.37 to 1.81; Fig. 2c), demonstrating that our segregating sites
inference approach applied to this simulated dataset is able to recover the true R[0] value of 1.6. Lower levels of sampling effort (100 viral sequences) resulted in an R[0] estimate to 1.65 and a
broader 95% confidence interval (1.30 to 2.06; Figure S4). Instead of random sampling of sequences, adopting a more uniformly distributed sampling strategy acted to reduce the uncertainty in the R[0]
estimate (Figure S5). In Figure S6, we present results for the same set of sequences as those used in Fig. 2, with the sequence data binned instead in time windows of 1 day, 2 days, 6 days, and 10
days, rather than in a time window of 4 days. These results show that R[0] estimates are not biased by the use of different time window lengths.
a, top The number of sampled sequences over time, binned by 4-day time windows. Sampling was done in proportion to the number of individuals recovering in a time window. In all, 500 sequences were
sampled over the course of the simulated epidemic. a, bottom The proportion of sampled individuals in each time window, obtained by dividing the number of sampled individuals by the number of
individuals who recovered during a time window. b Simulated segregating site trajectory from the sampled sequences, by time window. c Estimation of R[0] using Sequential Monte Carlo (SMC). Points
show log-likelihood values from different SMC simulations. R[0] values between 1.0 and 1.25 and between 2.0 and 2.5 were considered with a step size of 0.1. R[0] values between 1.25 and 2.0 were
considered with a step size of 0.01. Solid black curve shows the mean of 20 data points for each R[0] value. The vertical red dashed line shows the maximum likelihood estimate (MLE) of R[0]. The red
band shows the 95% confidence interval of R[0]. The vertical blue line shows the true value of R[0]=1.6. The MLE and 95% CI were obtained using the mean log-likelihood values. The 95% CI band
included the set of R[0] values with log-likelihoods that fell within 1.92 units of the highest mean log-likelihood value, based on a chi-squared distribution with 1 degree of freedom. Model
parameters for the simulated data set are: ({R}_{0}) =1.6, ({gamma }_{E})=^1/2 days^−1, ({gamma }_{I})=1/3 days^−1, population size N=10^5, t[0]=0, and the per genome, per transmission
mutation rate μ=0.2. Initial conditions are S(t[0])=N-1, E(t[0])=0, I(t[0])=1, and R(t[0])=0. A time step of τ=0.1 days was used in the Gillespie τ -leap algorithm.
Because the timing of the index case t[0] (in cases with a single introduction) is almost certainly not known for an emerging epidemic, we further attempted to estimate both R[0] and t[0] using the
segregating site trajectory shown in Fig. 2b. We considered a range of R[0] values between 1.0 to 2.5 and a broad range of t[0] starting 50 days prior to the true start date of 0 and ending at the
date of the first sampled sequence. We divided this parameter space into fine-resolution parameter combinations (R[0] intervals of 0.1 and t[0] intervals of 2 days) and ran 20 SMC simulations for
every parameter combination. In Fig. 3a, we plot the mean value of the 20 SMC log-likelihoods for every parameter combination in the considered parameter space. Examination of this plot indicates
that there is a log-likelihood ridge that runs between early t[0]/low R[0] parameter sides, indicating that inference using segregating site trajectories can in principle estimate both t[0] and R[0].
The parameter combination with the highest mean log-likelihood was R[0]=1.7 and t[0]=16 days, with the true parameter combination of R[0]=1.6 and t[0]=0 days falling within the 95% confidence
region of the estimated parameters. Our results therefore indicate that joint estimation of these parameters is thus possible in cases where a single introduction is responsible for igniting local
circulation. Using our estimates of R[0] and t[0], we reconstructed the dynamics of the segregating sites (Fig. 4a) and unobserved state variables: the number of susceptible, exposed, and infected
individuals over time (Fig. 4b-d). These reconstructed state variables captured the true epidemiological dynamics, demonstrating that our segregating sites approach can be used to infer
epidemiological variables that generally go unobserved.
a The log-likelihood surface based on the segregating site trajectory shown in Fig. 2b is shown over a range of R[0] and t[0] parameter combinations. The log-likelihood value shown in each cell is
the mean log-likelihood value calculated from 20 SMC simulations. Blank cells yielded mean log-likelihood values of negative infinity. The red boundary shows the set of (R[0], t[0]) values that fall
within the 95% confidence region. Parameter combinations within the red boundary have mean log-likelihood values that fall within 2.996 units of the highest mean log-likelihood value, based on a
chi-squared distribution with 2 degrees of freedom. b Joint density plot for R[0] and the time of the most recent common ancestor (tMRCA), as estimated using PhyDyn^6 on the same set of 500 sampled
sequences. Dashed red line in the joint density plot shows the 95% HPD interval of the joint density.
a Simulated trajectory of the number of segregating sites (dashed red), alongside reconstructed trajectories of the number of segregating sites (gray). b Simulated dynamics of susceptible individuals
(dashed red), alongside reconstructed dynamics of susceptible individuals (gray). c Simulated dynamics of exposed individuals (dashed red), alongside reconstructed dynamics of exposed individuals
(gray). d Simulated dynamics of infected individuals (dashed red), alongside reconstructed dynamics of infected individuals (gray). Reconstructed state variables were obtained by running the particle
filter using R[0] and t[0] parameter values randomly sampled from within the 95% CI region, with a further condition that the log-likelihood from the run exceeded the 95% CI region log-likelihood
cutoff shown in Fig. 3a. To show that resampling of particles during the SMC performs effectively, we show in Figure S7 the dynamics of these unobserved state variables in particles that are sampled
at different time points during the SMC procedure that may be lost by the end of the simulation as a result of resampling.
As mentioned in the Introduction, there are existing phylodynamic inference approaches available that can estimate epidemiological model parameters using viral phylogenies that have been
reconstructed from sequence data. Of particular note is the coalescent-based inference approach developed by Volz^13 that has been implemented as PhyDyn^6 in BEAST2. To compare our results using the
segregating sites approach to results using PhyDyn, we generated mock viral nucleotide sequences from our set of 500 sampled sequences (Methods) and used these nucleotide sequences as input into
PhyDyn. Assuming the same epidemiological model structure and using uninformative priors, PhyDyn was similarly able to recover the true R[0] value of 1.6 used in the forward simulation (Fig. 3b; 95%
credible interval = 1.44 to 1.61). Because PhyDyn infers epidemiological parameters using a tree-based method, the program does not estimate the time of the index case t[0]. Instead, it estimates the
time of the most recent common ancestor (tMRCA) of the viral phylogeny. The credible interval of PhyDyn’s tMRCA estimate spanned from −26.89 to 1.87 days post the true time of the index case (t[0]=
0). Times of a most recent common ancestor, however, are generally later (and never earlier) than the time of the index case. This is because some viral lineages likely go unsampled and the pruning
of these unsampled lineages results in a tMRCA that can be considerably later than the time of the index case t[0]^28. As such, interpretation of the PhyDyn results would almost certainly result in
timing the index case t[0] as less than 0 (too early), given 1.87 days as the top end of the tMRCA credible interval. This potentially early estimate of t[0] may be due to the “push-of-the-past”
effect^29, which results from the assumption of deterministic dynamics in the inference process when the underlying population dynamics are stochastic (and conditioned on the persistence of a
lineage). This “push-of-the-past” effect is usually reflected in an overestimate of the growth rate (or an overestimate in R[0]) in coalescent-based inference approaches that are applied to datasets
with small population sizes during their exponential growth phase^16. Here, because R[0] controls not only the rate of increase in the number of infected individuals at the start of the simulated
epidemic but also the time at which the simulated epidemic starts to decline, the “push-of-the-past” effect may instead be reflected in a tMRCA estimate that likely occurs too early. Because our
inference approach implements stochastic population dynamics, it appropriately accounts for the push-of-the-past effect, as do phylodynamic inference approaches that incorporate stochastic population
dynamics (e.g., birth-death models).
Because the impetus for developing the segregating sites inference approach was based on the extent of phylogenetic uncertainty present early on in an epidemic, we re-applied the inference approach
to sequences sampled early on during the simulated epidemic, with time window bins ending on days 36, 40, 44, 48, and 52 (Fig. 5a). During each of these five-time windows, we sampled 10 sequences,
resulting in a total of 50 sampled sequences. Our results on this subset of simulated data indicate that R[0] and t[0] could again be jointly estimated, although the confidence intervals for R[0] and
t[0] were both considerably broader, as expected with a much shorter time series (Fig. 5b). Similarly, on this same subset of data, PhyDyn’s 95% credible intervals were considerably broader (95%
credible interval for R[0]=1.48 to 10.80). For this particular time series, both the segregating sites approach and PhyDyn tended to overestimate the true value of R[0]=1.6 (Figs. 5b, 5c). For
PhyDyn, the “push-of-the-past” effect^29 may have contributed to the overestimation of R[0].
a Simulated trajectory of the number of segregating sites using early sequences. Sequences were binned into 4-day windows, with 10 individuals sampled from each time window. b The log-likelihood
surface based on a segregating site trajectory shown in panel (a). As in Fig. 3a, the log-likelihood value shown in each cell is the mean log-likelihood value calculated from 20 SMC simulations and
the 95% CI boundary shown in red contains sets of parameter combinations that fall within 2.966 log-likelihood units of the maximum log-likelihood. Blank cells had mean log-likelihood values of
negative infinity. (c) Joint density plot for R[0] and the time of the most recent common ancestor (tMRCA), as estimated using PhyDyn^6 on the same set of 50 sampled sequences. Dashed red line in the
joint density plot shows the 95% HPD interval of the joint density. For R[0], only the lower bound of the 95% HPD is shown as the upper bound is above 6. In panels a through c, simulations were
parameterized with a per genome, per transmission mutation rate of μ=0.2.
To determine whether there might be an upwards bias in the estimation of R[0] using the segregating sites approach, we simulated an additional short dataset under the same epidemiological model
structure and model parameterization, with the exception of the mutation rate μ, which we increased from 0.2 to 0.4. To calculate the segregating sites trajectory, we sampled from this simulation as
we did for Fig. 5a–c, with 10 sequences sampled in each of the five time windows (Figure S8a). The maximum likelihood estimates of R[0] using our segregating sites approach did not overestimate the
true R[0] of 1.6 in this dataset, although the time of the index case was again estimated to be slightly later than the true value of t[0]=0 (Figure S8b). Compared to the results on the μ=0.2
short dataset (Fig. 5b), the 95% confidence region spanned over a similar extent of parameter space. PhyDyn also did not overestimate R[0] on this μ=0.4 short dataset (Figure S8c). Moreover, its
95% credible interval was considerably smaller than on the μ=0.2 short dataset. This result makes sense: at higher mutation rates, phylogenetic uncertainty is reduced and tree-based inference
approaches are expected to improve. In contrast, a low-dimensional summary statistic, such as the number of segregating sites cannot take advantage of the higher-dimensional structure present in the
sequence data.
We applied the segregating sites inference approach to a set of SARS-CoV-2 sequences sampled from France between January 23, 2020, and March 17, 2020 (the date on which a country-wide lockdown
began). We decided to apply our approach to this set of sequences for several reasons. First, many of the 479 available full-genome sequences from France over this time period appear to be
genetically very similar to one another^30, indicating that one major lineage took off in France (or at least, that most sampled sequences derived from one major lineage). This lineage would be the
focus of our analysis. Second, an in-depth epidemiological analysis previously inferred R[0] for France prior to the March 17 lockdown measures that were implemented^31. That analysis fit a
compartmental infectious disease model to epidemiological data that included case, hospitalization, and death data. Because our segregating sites inference approach can accommodate epidemiological
model structures of arbitrary complexity, we could adopt the same model structure as in this previous analysis. We could also set the epidemiological parameters that were assumed fixed in this
previous analysis to their same values. By controlling for model structure and the set of model parameters assumed as given, we could ask to what extent sequence data corroborate the R[0] estimates
arrived at from detailed fits to epidemiological data.
To apply our segregating sites approach to the viral sequences from France, we first identified the subset of the 479 sequences that constituted a single, large lineage. To keep with the “tree-free”
emphasis of our approach, we identified this subset of sequences (n=432) without inferring a phylogeny (Methods). Using phylogenetic inference, however, we confirmed that our subset of sequences
constituted a single clade, with sequences from France falling outside of this clade being excluded (Figure S9). To generate a segregating site trajectory from these sequences, we defined 4-day time
windows such that the last time window ended on March 17, 2020. Figure 6a shows the number of sequences falling into each time window. Figure 6b shows the segregating site trajectory calculated from
these sequences.
a The number of sequences sampled over time, calculated using a 4-day time window. b The segregating site trajectory calculated from the binned sequences shown in panel (a). c Estimation of the
per-genome, per-transmission mutation rate μ. The histogram shows the fraction of 87 analyzed transmission pairs with consensus sequences that differ from one another by the number of mutations shown
on the x-axis. The mean number of mutations per transmission is μ=0.33 (95% CI=0.22–0.48). Black dots represent the probability of observing 0, 1, 2, and 3 mutations assuming a Poisson
distribution with a mean of 0.33. Vertical black error bars span the probability of observing 0, 1, 2, and 3 mutations assuming Poisson distributions with mean values of 0.22 and 0.48.
We parameterized the model with a per genome, per transmission mutation rate μ using consensus sequence data from established SARS-CoV-2 transmission pairs that were available in the literature^32,33
,34,35 (Methods). Specifically, for each of the 87 transmission pairs we had access to, we calculated the nucleotide distance between the consensus sequence of the donor sample and that of the
recipient sample and fit a Poisson distribution to these data (Fig. 6c). Using this approach, we estimated a μ value of 0.33, corresponding approximately to one mutation occurring every 3
transmission events.
Similar to the approach we undertook with our simulated data, we first attempted to jointly estimate R[0] and the timing of the index case t[0] for this segregating site trajectory. We considered a
broad parameter space over which to calculate log-likelihood values. Specifically, we considered R[0] values between 1.0 and 4.5 and t[0] values of between December 1st, 2019 and February 14th, 2020.
We ran 10 SMC simulations and calculated the mean log-likelihood for each parameter combination (Fig. 7). We estimated R[0] to be 3.0 (95% confidence interval = 1.6 to 4.2), consistent with the R[0]
estimate of 2.9 (95% confidence interval = 2.81 to 3.01) arrived at through epidemiological time series analysis^31. We estimated t[0] to be February 8th, 2020 (95% confidence interval = December 25,
2019, to February 14, 2020).
The joint log-likelihood surface based on the estimated segregating site trajectory for the France data. Each cell shows the mean log-likelihood value based on 10 SMC simulations. Blank cells
indicate mean log-likelihood values of negative infinity. Gray cells indicate where log-likelihood values were not evaluated. The red lines denote the set of parameter values that fall within the 95%
confidence interval. A few ‘islands’ of parameter combinations that fall either outside or inside the 95% CI are apparent and are due to the variation in the log-likelihood values obtained from the
SMC simulations.
We decided to further consider an alternative model that allowed for multiple introductions of the focal lineage into France (Methods). This decision was based on evolutionary analyses that have
shown that regional SARS-CoV-2 epidemics in Europe (as well as in the United States) were initiated through multiple introductions rather than only a single one^36. Instead of attempting to jointly
estimate R[0] and t[0], we attempted to jointly estimate R[0] and a parameter η using the segregating site trajectory. The parameter η quantifies the extent to which transmission between France and
regions outside of France is reduced relative to transmission occurring within France. This model further required specification of the time at which the basal genotype evolved outside of France,
which we refer to as t[e]. We considered a broad parameter space over which to calculate log-likelihood values (R[0] values between 1.0 and 4.0 and η values between 10^−8 and 10^−1) and three
different t[e] values: December 24, 2019, January 1, 2020, and January 8, 2020 (Methods). At each of these t[e] values, we ran 10 SMC simulations and calculated the mean log-likelihood for each
parameter combination (Fig. 8a–c). We estimated R[0] to be 2.6 (95% CI=2.0 to 4.0), 2.7 (95% CI=2.0 to 4.0), and 2.3 (95% CI=2.1 to 4.0), respectively, under t[e] = December 24, 2019, January
1, 2020, and January 8, 2020. These results indicate that the inferred R[0] values are relatively insensitive to the assumed emergence time of the basal genotype outside of France. At later assumed
values of t[e], our estimates for η were higher, indicating that later emergence times were compensated for by a higher transmission rate between infected individuals outside of France and
susceptible individuals within France.
The joint log-likelihood surface based on the estimated segregating site trajectory for the France data is shown under three different basal genotype emergence times: t[e] = December 24, 2019 (a),
January 1, 2020 (b), and January 8, 2020 (c). Each cell shows the mean log-likelihood value based on 10 SMC simulations. Blank cells indicate mean log-likelihood values of negative infinity. Gray
cells indicate where log-likelihood values were not evaluated due to extended simulation time. The red lines in each panel denote the set of parameter combinations that fall within the 95% confidence
interval. As in Fig. 7, a few ‘islands’ of parameter combinations are apparent due to the variation in the log-likelihood values obtained from the SMC simulations.
We reconstructed the unobserved state variables for the multiple-introductions model using SMC simulations parameterized with R[0] and η values that were sampled from the parameter spaces shown in
Fig. 8, using the same approach we used for reconstructing state variables on the mock segregating sites trajectory. These reconstructed variables are shown in Fig. 9. As expected for an epidemic
with an R[0]>1, the total number of infected individuals increased exponentially over the time period considered (Fig. 9d–f). In Fig. 9g–i, we plot the reconstructed cumulative number of recovered
individuals over time. These cumulative trajectories indicate that by mid-March 2020, approximately 0.009% to 2.044% of individuals in France had recovered from infection from this SARS-CoV-2
lineage. These cumulative predictions can be roughly compared against findings from a serological study that was conducted over this time period in France^37. Based on a survey of 3221 individuals,
this study found that 0.41% of individuals (95% confidence interval = 0.05% to 0.88%) had gotten infected with SARS-CoV-2 by March 9 to 15, 2020 (Fig. 9g–i). Our estimates fall in line with these
independent estimates. Of note, our estimates should fall on the low side of these independent estimates because other, smaller clades were also circulating in France during the time period studied
and infections with viruses from these other clades would also contribute to seropositivity levels. We also emphasize that this is necessarily a rough comparison because seroconversion does not occur
exactly at the point of recovery. It can occur over a broader range of times, ranging from prior to recovery to many days following symptom onset^38. Finally, in Fig. 9j–l, we plotted the
reconstructed cumulative number of infections that resulted directly from contact with individuals outside of France. By the first sampled time window (ending on February 22, 2022), our SMC results
indicate that there were very likely repeated introductions of this lineage into France, with the majority of sampled particles pointing towards hundreds of introductions of this lineage into France
by this time point.
State variables are reconstructed for the multiple-introductions model with three different values assumed for the emergence time of the basal genotype: t[e] = December 24, 2019 (first column),
January 1, 2020 (second column), and January 8, 2020 (third column). a–c Segregating site trajectory for the France SARS-CoV-2 data (red), alongside reconstructed segregating site trajectories
(gray). d–f Reconstructed dynamics of the number of infected individuals (E[1]+E[2]+I) over time, shown in percent of France’s population. g–i Reconstructed dynamics of the cumulative number of
recovered individuals over time, shown in percent of France’s population. Independent estimates of the fraction of the population that has been infected with SARS-CoV-2 by mid-March are shown in
black. Estimates are from a serological study conducted during the time window March 9-15, 2020^37. j–l Reconstructed dynamics of the cumulative number of infections in France that resulted from
contact with infected individuals outside of France. Reconstructed state variables shown in panels (a–l) were obtained by running the particle filter using R[0] and t[0] parameter values randomly
sampled from within the 95% CI region, with a further condition that the log-likelihood from the run exceeded the 95% CI region log-likelihood cutoff shown in Fig. 8a–c, respectively.
Here, we developed a statistical inference approach to estimate epidemiological parameters from virus sequence data. Our inference approach is a “tree-free” approach in that it does not rely on the
reconstruction of viral phylogenies to estimate model parameters. One benefit of using such an approach for parameter estimation of emerging viral pathogens is that, early on in an epidemic,
phylogenetic uncertainty present in time-resolved viral phylogenies is significant, and tree-based phylodynamic inference approaches would need to integrate over this uncertainty. This is oftentimes
computationally intensive, especially when many sequences have been sampled. The computational complexity of our “tree-free” approach, in contrast, does not scale with the number of sampled
sequences. Instead, the runtime required for parameter inference depends on the number of genotypes that evolve over the course of the model simulations. This number in turn is affected by the
proposed basic reproduction number, the proposed time of the index case in the single introduction model, and the magnitude of the per genome, per transmission mutation rate μ. A second benefit to
our tree-free approach is that it can estimate the time of the index case (in a single-introduction scenario), whereas tree-based inference methods estimate the time of the most recent common
ancestor. This is a benefit when the question of interest focuses on when a viral lineage emerges and starts to spread. Instead of viral phylogenies being the data that statistically interface with
the epidemiological models, our approach uses a population genetic summary statistic of the sequence data, namely the number of segregating sites present in time-binned sets of viral sequences. Our
inference approach benefits from being plug-and-play in that it can easily accommodate different epidemiological model structures.
Based on fits to a simulated data set, we have shown that segregating site trajectories can be used to estimate the basic reproduction number R[0] and the timing of the index case t[0] in cases where
a single introduction can be assumed. We further fit a multiple-introductions epidemiological model to a segregating site trajectory that was calculated from SARS-CoV-2 sequence data from France,
estimating a basic reproduction number R[0] of approximately 2.3-2.7. These results are consistent with previous estimates from an epidemiological analysis and consistent with a serological study
conducted in mid-March 2020.
Our inference approach relies on several assumptions that are shared by existing phylodynamic inference methods. Most notably, it relies on an assumption that all mutations are phenotypically
neutral. However, a recent analysis of SARS-CoV-2 sequences has shown evidence for purifying selection, even early on during the pandemic^39. Indeed, within the set of SARS-CoV-2 sequences from
France, we observe 170 nonsynonymous mutations and 138 synonymous mutations (a ratio of 1.23:1). Given the number of nonsynonymous sites (n=68,540) and the number of synonymous sites (n=19,255)
in the SARS-CoV-2 genome, we would expect, under neutrality, a ratio of 3.56:1. This underrepresentation of nonsynonymous genetic variation points towards purifying selection in our analyzed dataset.
A more recent analysis also raises the possibility of adaptive evolution occurring during early 2020^40. Incorporating non-neutral genetic variation into inference approaches such as ours and
existing phylodynamic ones is complicated, although some statistical approaches have started to tackle this goal^9. In the context of our segregating sites inference approach, directly incorporating
non-neutral evolution will increase model complexity considerably, and assumptions would need to be made about the distribution of mutational fitness effects. Rather than incorporating non-neutral
evolution within our approach, we can for now consider how the occurrence of non-neutral evolution would impact our parameter estimates. With purifying selection at play, we would expect to see less
genetic variation than in its absence. As such, the number of segregating sites in any time window would be lower than it would be under neutrality. Our inference approach, assuming neutrality, would
therefore bias R[0] estimates to be low and, in single-introduction models, the timing of the index case t[0] to be late. In multi-introduction models, our estimate of η would be biased high.
Our approach also assumes infinite sites and the absence of homoplasies. While these assumptions are limiting over longer periods of sequence evolution, our approach is intended to be used for
emerging viral pathogens, sampled over shorter periods of time, when levels of genetic diversity are still low. As such, these assumptions will likely not be violated in cases where this approach
will come in useful. We would also like to note that the infinite sites assumption could in principle be relaxed, but this would make the simulations in the inference approach substantially more
costly. Furthermore, as time goes on, not only do chances of repeated mutations at sites increase, but genetic diversity increases. As such, phylogenetic uncertainty will decrease, such that existing
tree-based phylodynamic inference approaches will become increasingly informative and segregating site trajectories less informative.
While our inference approach does adopt assumptions of phenotypic neutrality and infinite sites, it does not assume a constant sampling rate or a specific sampling process throughout the time period
over which sequences are collected. As we have shown in Fig. 1b, sampling effort does impact the segregating sites trajectory: the greater the sampling effort, the larger the number of segregating
sites. For our inference approach to perform effectively, sampling effort therefore needs to be matched between the simulations and the empirical data. This matching of sampling effort is implemented
in the particle filter. However, the number of samples sequenced per time window is not particularly informative of model parameters (except in the case of extremely high sampling effort when certain
low R[0] model parameterizations cannot appropriately evaluate the expected number of segregating sites in a time window because the number of sampled sequences exceed the number of simulated
recoveries). The reason why the number of samples is not particularly informative of model parameters is because, under our approach, sampling of individuals does not impact the underlying
epidemiological dynamics: individuals are sampled upon recovery, once they are no longer infectious. That the number of observed samples is not highly informative of model parameters we see as a
benefit of our approach because sampling effort and testing rates can change dramatically over the course of an emerging pandemic or over the early period of an emerging viral lineage as surveillance
efforts ramp up. In contrast, sampling times of sequences have been shown to be highly informative of model parameters in the case of birth-death models, with sampling process misspecification
resulting in the possibility of arriving at biased parameter estimates^41.
While the number of sampled sequences is largely uninformative of model parameters, our approach does have to make an assumption of when individuals are sampled. In our simulated dataset and in our
application to SARS-CoV-2, we assumed that individuals were sampled as they recovered. This sampling scheme decision was based on our understanding that the time of symptom onset often follows peak
viral load for many emerging viral pathogens^42 and an assumption that most testing early on in a pandemic involves individuals who develop symptoms. It is important to note that if the assumed
sampling scheme is mismatched with the empirical sampling scheme, parameter estimates may be biased. For example, if individuals were instead sampled as they transitioned from the exposed class to
the infectious class, rather than upon recovery, and we assumed in our model that individuals were sampled upon recovery, then our R[0] estimates would be biased high.
Finally, we would like to note that setting the per genome, per transmission mutation rate to a constant value does not correspond to an assumption of a constant molecular clock. A constant molecular
clock requires that the number of substitutions per unit time remains the same. Our assumption is that the mean number of nucleotide changes that occur during a transmission event between a donor and
a recipient (at the consensus level) stays constant over time. This would almost certainly be the case unless the fidelity of the viral polymerase was evolving over the period considered. Changes in
the substitution rate could come about if the generation interval between transmission events changes due, for example, to the implementation of non-pharmaceutical interventions or increased symptom
awareness. A shortening of the generation interval (defined as the time between infection and onward transmission) would increase the number of transmission events that occur per unit time and
thereby result in an increase in the substitution rate. In contrast, a lengthening of the generation interval would result in fewer transmission events occurring per unit time, thereby decreasing the
population-level substitution rate. Changes in the generation interval can emerge from an underlying epidemiological model, such that our assumption of a constant per genome, per transmission event
mutation rate does not preclude or conflict with the observation of changes in the substitution rate over time.
The analysis we presented here focuses on statistical inference using sequence data alone. In recent years, there has also been a growing interest in combining multiple data sources – for example,
sequence data and epidemiological data or serological data – to more effectively estimate model parameters. The few existing studies that have incorporated additional data while performing
phylodynamic inference have shown the value in pursuing this goal^7,43,44. As a next step, we aim to extend the segregating sites approach developed here to incorporate epidemiological data and/or
serological data more explicitly. Straightforward extension is possible due to the state-space model structure that is at the core of the particle filtering routine we use.
Our analysis focused on phylodynamic inference based on sequence data belonging to a single viral lineage, with either a single index case or multiple introductions from an outside reservoir. Our
approach, however, can be expanded in a straightforward manner to multiple viral lineages. This is especially useful in cases like SARS-CoV-2, where many regions have witnessed the introduction of
multiple clades^10,45. In this case, a single segregating sites trajectory could be calculated for each clade, such that multiple segregating site trajectories could be simultaneously fit to under
specified constraints such as the basic reproduction number being the same across all clades. Different clades could also be allowed to differ in their reproductive numbers, such that questions
relating to the selective advantage of some clades over others could be addressed. As such, this inference method, designed for emerging pathogens with low levels of genetic diversity, may continue
to be useful for endemic pathogens to address questions related to the emergence of new viral lineages.
Mutations occur during viral replication within infected individuals and these have the potential to be transmitted. During the epidemiological spread of an emerging virus or viral lineage, the virus
population (distributed across infected individuals) thus accrues mutations and diversifies genetically. This joint process of viral spread and evolution can be simulated forward in time using
compartmental models, with patterns of epidemiological spread leaving signatures in the evolutionary trajectory of the virus population. Parameters of these compartmental models that govern patterns
of epidemiological spread can thus in principle be estimated using viral sequence data. Here, similar in spirit to existing inference approaches based on summary statistics^46,47,48,49,50, we develop
a statistical inference approach that fits compartmental epidemiological models to time series of a low-dimensional summary statistic calculated from sequence data. Specifically, we use trajectories
of the number of segregating sites from samples of the viral population taken over time for statistical inference. Because we propose the use of our method early on in an epidemic (or during the
early expansion of a viral lineage), we focus primarily on estimating the basic reproduction number R[0] using this inference approach.
To simulate mock data of segregating site trajectories, we specify a compartmental epidemiological model and simulate the model under demographic stochasticity using Gillespie’s τ-leap algorithm.
Here, we use a susceptible-exposed-infected-recovered (SEIR) model whose stochastic dynamics are governed by the following equations:
where β is the transmission rate, N is the host population size, γ[E] is the rate of transitioning from the exposed to the infected class, γ[I] is the rate of recovering from infection, and Δt is the
τ-leap time step used. R[0] is given by β /γ[I]. The epidemiological dynamics of this model can be simulated from the above equations alone. Additional complexity is needed to incorporate virus
evolution throughout the course of the simulation. To incorporate virus evolution, we partition exposed individuals and infected individuals into genotype classes, with genotype 0 being the reference
genotype present at the start of the simulation. Mutations to the virus occur at the time of transmission, with the number of mutations that occur in a single transmission event given by a Poisson
random variable with mean μ, the per-genome, per-transmission event mutation rate. We assume infinite sites such that new mutations necessarily result in new genotypes. New mutations and new
genotypes are both assigned integer indices in order of their appearance. When new mutations are generated at a transmission event, the new genotype harbors the same mutation(s) as its parent
genotype plus any new mutations. We use a sparse matrix approach to store genotypes and their associated mutations to save on memory.
There are three types of events that occur in the SEIR model simulations: transitions from exposed to infected; transitions from infected to recovered; and transmission. To simulate transitions from
exposed to infected, during a time step Δt, ({n}_{Eto I}) individuals are drawn at random from the set of individuals who are currently reside in the exposed class. These individuals will transition
to the infected class during this time step, while retaining their current genotype statuses. To simulate transitions from infected to recovered, during a time step Δt, ({n}_{Ito R}) individuals are
drawn at random from the set of individuals who are currently residing in the infected class. These individuals will transition to the recovered class during this time step. To simulate transmission,
during a time step Δt, we add ({n}_{Sto E}) new individuals to the set of exposed individuals. For each newly exposed individual, we randomly choose (with replacement) a currently infected individual
as its ‘parent’. If no mutations occur during transmission, then this newly exposed individual enters the same genotype class as its parent. If one or more mutations occur during transmission, this
newly exposed individual enters a new genotype class, and the sparse matrix is extended to document the new genotype and its associated mutations (given as integers, without a bitstring or explicit
genome structure).
We start the simulation with one infected individual carrying a viral genotype that we consider as the reference genotype (genotype 0). To calculate a time series of segregating sites, we define a
time window length T (T>Δt) of a certain number of days and partition the simulation time course into discrete, non-overlapping time windows. During simulation, we keep track of the individuals
that recover (transition from I to R) within a time window. For each time window i, we then sample n[i] of these individuals at random, where n[i] is the number of sequences sampled in a given time
window based on the sampling scheme chosen. Because we have the genotypes of the sampled individuals from the sparse matrix, we can calculate the number of segregating sites s[i] in any time window i
. Since s[i] is the number of polymorphic sites across the sampled individuals in time window i, it is simply calculated from the set of mutations harbored by the sequences of the sampled
individuals. While in our simulations, we sample individuals as they recover, alternative sampling schemes can instead be assumed. For example, individuals could be sampled as they transition from
the exposed to the infected class, or while they are in the infected class. We chose to sample upon recovery based on symptom development (and thereby testing) often occurring following peak viral
We implement transmission heterogeneity in the epidemiological model by splitting the infected classes into a high-transmission and a low-transmission class, as has been done elsewhere^6,10. For an
SEIR model, the model extended to incorporate transmission heterogeneity becomes:
The parameter p[H] quantifies the proportion of exposed individuals who transition to the high-transmission I[h] class. Parameters β[h] and β[l] quantify the transmission rates of the infectious
classes that have high and low transmissibility, respectively. We set the values of β[h] and β[l] based on a given parameterization of overall R[0] and the parameter p[H]. To do this, we first
define, as in previous work^6,10, the relative transmissibility of infected individuals in the I[h] and I[l] classes as (c=frac{{beta }_{h}}{{beta }_{l}}). We further define a parameter P as the
fraction of secondary infections that result from a fraction p[H] of the most transmissible infected individuals. Based on given values of p[H] and P, we set c, as in previous work^10, to (frac{[frac
{1-{p}_{H}}{{p}_{H}}]}{[frac{1}{P}-1]}). With c defined in this way, p[H] can be interpreted as the proportion of most infectious individuals that result in P of secondary infections. We set P to
0.80, to make p[H] easily interpretable relative to the “20/80” rule in disease ecology^22. Recognizing that ({R}_{0}=frac{{p}_{H}{beta }_{h}+({1-p}_{H}){beta }_{l}}{{gamma }_{I}}) in this model, we
can then solve for β[l]:(frac{{R}_{0}{gamma }_{I}}{{p}_{H}c+({1-p}_{H})}), and set ({beta }_{h}=c{beta }_{l}.) Note that the interpretation of p[H] in the context of the disease ecology rule is an
approximation since this calculation does not take into consideration variation in individual R[0] that results from differences in the duration of infection or variation in individual R[0] that
results from differences in the number of secondary infections that are due to stochastic effects.
Our inference approach relies on particle filtering, also known as Sequential Monte Carlo (SMC), to estimate model parameters and reconstruct unobserved (latent) state variables. Particle filtering
calculates the likelihood of a parameterized model (more precisely, the probability of observing the time-series data marginalized over the unobserved state variables) by recurrently updating a set
of particles (Figure S10). In our case, each of these particles holds a state-space model, which includes a process model component that simulates underlying epidemiological and evolutionary dynamics
and an observation model that relates these latent state variables to the observed segregating sites data (Figure S11). The process model includes the unobserved epidemiological variables (e.g., S, E
, I, and R) and the evolutionary components of the model (viral genotypes and mutations). From one observed segregating sites data point to the next one, the model is simulated using Gillespie’s τ
-leap algorithm, as described in the section above.
At the end of each time window, when the simulation reaches the next observed segregating sites data point, the observation model is used to calculate the probability of observing the observed data
point given the underlying process model. This probability is calculated as follows. We calculate the expected number of segregating sites from the model simulation by performing k ‘grabs’ of sampled
individuals, with each grab consisting of the following steps:
pick (without replacement) n[i] individuals from the set of individuals who recovered during time window i, where n[i] is the number of samples present in the empirical dataset in window i. This step
mimics the process of sample collection at the same effort as in the observed data. We control for sampling effort because the extent of sampling impacts the number of segregating sites.
calculate the simulated number of segregating sites s[i]^sim, based on the genotypes of the sampled n[i] individuals.
Between grabs, the replacement of previously sampled individuals occurs. We then calculate the mean number of segregating sites for window i by taking the average of all k s[i]^sim values. Finally,
we calculate the probability of observing s[i] segregating sites in window i, given the model-simulated mean number of segregating sites, using a Poisson probability mass function parameterized with
the mean s[i]^sim value and evaluated at s[i]. As a special case in the calculation of this probability, if the number of individuals who recovered during a given time window i is less than the
number that needs to be sampled (n[i]), then the particle’s probability of observing the number of segregating sites s[i] is set to 0. The calculated probabilities serve as the weights for the
Particle weights obtained at the end of each window are used 1) to resample particles for the next time window according to their assigned weights and 2) to calculate the likelihood of a
parameterized model. In the particle filtering algorithm, the likelihood is obtained by averaging particle weights within each window and then multiplying these average particle weights across all
time windows with observations. For time windows without observations (n[i]=0), particle weights are assigned a value of 0 if the virus has died out stochastically and 1 if the virus continues to
persist in the population. These weights are used for resampling, but do not contribute to the calculation of the likelihood. We adopt this approach to filter out particles during early time windows
that have undergone stochastic extinction.
Latent state variables are reconstructed by randomly sampling a particle at the end of an SMC simulation and plotting the values of its simulated latent state variables over time. All of our SMC
simulations were performed with 200 particles and k=50 grabs. Note that the complexity of this inference method is largely independent of the number of input sequences. This stands in contrast to
phylodynamic inference approaches that frequently down-sample sequences to reduce runtime.
Simulated sparse matrices were converted to nucleotide alignments by first generating a reference sequence with the same length as the maximum number of mutations in the sparse matrix and choosing an
A, C, G, or T nucleotide at each site with equal probability. A mutated sequence was generated for each genotype represented in the sparse matrix by replacing the reference allele at that position
with another nucleotide chosen with equal probability. The final FASTA alignment was generated by identifying the simulated sequence associated with each sampled individual. Generation of the
simulated FASTA file was done using Python v3.9.4 with Numpy v1.19.4.
The simulated FASTA alignment was used to generate a BEAST2 XML file from a template XML which was generated in part using Beauti v2.6.6. This template used a JC69 nucleotide substitution model with
no invariant sites. We assumed an uncorrelated log-normally distributed relaxed clock with a uniform [0.0, 1E-2] prior on the mean and a uniform [0.0,2.0] prior on the standard deviation.
A single-deme structured coalescent prior as defined by the following equations was implemented using PhyDyn v1.3.8:
where (beta={R}_{0}{gamma }_{I}). A population size of 10^5 with a single initially infected individual was used. We assume infected individuals remain exposed for an average of 2 days (1/γ[E]) and
infectious (1/γ[I]) for an average of 3 days. R[0] was estimated using a uniform [1.0, 10.0] prior. All sampled sequences were assigned to the infected (“I”) class.
Sampled parameters and trees were logged every 1000 states and all MCMC chains were run for at least 209M (Fig. 3b), 64 million (Fig. 5c), 150 million (Figure S8c) iterations. The first 10% of MCMC
chains were discarded as burn-in and the ESS values of all parameters were >200, as identified by Tracer v1.7.1 (10.1093/sysbio/syy032).
The process model we use in our application to SARS-CoV-2 sequence data from France is based on a previously published epidemiological model^31. We base our process model on this published model to
allow for a direct comparison of inferred R[0] values between our sequence-based analysis and their analysis that focuses on SARS-CoV-2 spread in France over a similar time period. Their analysis was
based on fitting an epidemiological model to a combination of case, hospitalization, and death data. Their model structure, once implemented using Gillespie’s τ-leap algorithm, is given by:
The parameters are the transmission rate β, the rate of transitioning from the E[1] class to the E[2] class ({gamma }_{E1}), the rate of transitioning from the E[2] class to the I class ({gamma }_
{E2}), and the rate of transition from the I class to the R class ({gamma }_{I}). The average duration of time spent in the E[1] class given by ({1/gamma }_{E1})=4 days, the average duration of
time spent in the E[2] class given by ({1/gamma }_{E2})=1 day, and the average duration of time spent in the infected class given by ({1/gamma }_{I})=3 days. Their model assumes that the
transmission efficiency β of exposed class 2 (E[2]) and that of the infected class I are the same; their model considers E[2] and I as distinct classes to interface with the case data, where symptoms
are assumed to not appear before an individual has transitioned to class I. We maintain the model structure with E[1], E[2], and I rather than reducing it to a model structure with just a single E
and a single I class to keep the same overall distribution of infection times as in their model.
Because SARS-CoV-2 dynamics are characterized by substantial levels of transmission heterogeneity^10,23,51 and we have shown in Fig. 1 that transmission heterogeneity impacts segregating site
trajectories, we expanded the compartmental epidemiological model for SARS-CoV-2 described above to include transmission heterogeneity in a manner similar to the one we used in Fig. 1. Based
specifically on the analysis by Paireau and colleagues^52, we set p[H] to 0.10, such that 10% of infections are responsible for 80% of secondary infections. Analogous to the approach we undertook for
the simulated data, we jointly estimated R[0] and t[0] using the segregating site trajectory shown in Fig. 6b.
Based on phylogenetic analyses that have indicated that early introductions of SARS-CoV-2 into focal regions likely resulted from multiple introductions rather than a single one, we considered a
modified version of the epidemiological model that would allow for multiple introductions. The modification relied on the incorporation of infections within France that resulted from direct contact
with infected individuals outside of France, termed the viral “reservoir”. Similar to the approach adopted by some existing phylodynamic analyses^12, the viral population dynamics in this reservoir
are simplified to exponential growth. This infected population from outside of France acts as another source of infection for susceptible individuals within France, allowing for multiple
introductions of SARS-CoV-2 into France.
As in the focal region, new genotypes are expected to emerge in the outside reservoir. As we assume an infinite sites model, the genotypes that emerge in the outside reservoir and in the focal region
will not overlap except in the basal genotype that is first introduced to the focal region. For this reason, and because the basal genotype is expected to be considerably more common than any of the
viral genotypes that stem from it, we consider only the repeated introduction of the basal genotype into France. Starting at the time of emergence of the basal genotype in the outside reservoir (t
[e]), we let the number of individuals infected with this basal genotype Y[t] grow exponentially:
where r is the intrinsic growth rate of the basal genotype. Based on empirical estimates^53,54, we set the intrinsic growth rate to 0.22 day^−1. To set t[e], we first identified the genotype sampled
in France that is genetically closest to the reference strain Wuhan/Hu-1 (MN908947.3). This basal genotype differs from Wuhan/Hu-1 by 4 nucleotides: C241T, C3037T, C14408T, and A23403G. Using GISAID
data, we then identified sequences with collection locations outside of France that carried all four of these mutations that define the basal genotype. The earliest of these sequences including the
four basal genotype-defining mutations was collected on January 25, 2020, in Australia, suggesting that the basal genotype had been circulating prior to January 25, 2020. Considering the potential
delay between emergence and the time of first detection, we considered three distinct ({t}_{e}) values: December 24th, 2019, January 1st, 2020, and January 8th, 2020.
Individuals infected in this outside reservoir can transmit their infection to susceptible individuals within France. The rate at which they transmit the infection is reduced relative to the rate at
which infected individuals within France transmit the infection to susceptible individuals within France. We let the factor by which transmission is reduced be given by the factor η. During a τ-leap
timestep, the number of individuals within France who become infected from contact with an infected individual outside of France is therefore given by:
As we are considering only the transmission of the basal genotype from infected individuals in the outside reservoir to susceptible individuals in France, all of these newly infected individuals will
carry the basal genotype unless mutation occurs during the transmission process. Our simplifying assumption that only the basal genotype can be introduced into France from the outside reservoir
ignores the possibility that genotypes that are derived from the basal genotype enter France from the outside reservoir. Strictly speaking, we think this assumption is unlikely to be met. However, at
very early time points in France’s epidemic, most of the genotypes outside of France should still be the basal genotype, and only at later time points should the frequencies of derived genotypes
increase outside of France. Introduction of these derived genotypes at these later time points could result in the establishment of viral sublineages in France. However, because autochthonous
infections would be high at this point, these viral sublineages would very likely go unsampled. As such, we do not think that our assumption of only the basal genotype being introduced into France
would have a dramatic effect on our results. We can consider, however, the effects that violation of this assumption would have on our parameter estimates: if derived genotypes were introduced into
France and sampled (or their descendants sampled), then the number of segregating sites that would have evolved within France would be lower than we are currently taking it to be. As such, our
current estimate of R[0] would be biased high.
We set the per-genome, per-transmission mutation rate parameter μ to 0.33. This is based on the fit of a Poisson distribution to the number of de novo substitutions that were observed in 87
transmission pairs of SARS-CoV-2 from four studies^32,33,34,35. Accession numbers for 78/87 of these transmission pairs are available in Table S1. Accession numbers for the remaining pairs were
provided by the corresponding authors of the relevant publication^34. Sequence data were aligned to Wuhan/Hu-1 (MN908947.3)^55 using MAFFT v.7.464^56. Insertions relative to Wuhan/Hu-1 were removed
and the first 55 and last 100 nucleotides of the genome were masked. De novo substitutions for each pair were identified in Python v.3.9.4 (http://www.python.org) using NumPy v.1.19.4^57. Ambiguous
nucleotides were permissively included in the identification of de novo substitutions (e.g., an R nucleotide was assumed to match both an A and a G). The mean number of substitutions between
transmission pairs is the maximum likelihood estimate for the rate parameter of the Poisson distribution. The 95% confidence interval was calculated using the exact method using SciPy v.1.5.4^58.
The value for μ=0.33 is consistent with population-level substitution rate estimates for SARS-CoV-2, which range from 7.9 ×10−^4 to 1.1 ×10−^3 substitutions per site per year^28,59. With a genome
length of SARS-CoV-2 of approximately 30,000 nucleotides and a generation interval of approximately 4.5 days^60, these population-level substitution rates would correspond to per genome, per
transmission mutation rates of between 0.29 and 0.41, respectively.
We downloaded all complete and high-coverage SARS-CoV-2 sequences with complete sampling dates sampled through March 17th, 2020 (https://doi.org/10.55876/gis8.230123mt) in France and uploaded through
April 29th, 2021 from GISAID^61. Sequences were aligned to Wuhan/Hu-1 using MAFFT v.7.464. Insertions relative to Wuhan/Hu-1 were removed. Any sequences with fewer than 28000A, C, T, or G characters
were removed. Following this filtering protocol, our dataset included 479 sequences. We masked the first 55 and last 100 nucleotides in the genome as well as positions marked as “highly homoplasic”
in early SARS-CoV-2 sequencing data (https://github.com/W-L/ProblematicSites_SARS-CoV2/blob/master/archived_vcf/problematic_sites_sarsCov2.2020-05-27.vcf). Pairwise SNP distances were calculated in a
manner that accounted for IUPAC ambiguous nucleotides in Python using NumPy. To subset these data to a single clade circulating within France, we identified the connected components of this pairwise
distance matrix with a cutoff of 1 SNP in Python using SciPy and identified the shared SNPs relative to Wuhan/Hu-1 between all sequences in each connected component. The largest connected component
contained 308 sequences which shared the substitutions C241T, C3037T, C14408T, and A23403G. Our final dataset included these 308 as well as 124 sequences from connected components that shared these
four substitutions relative to Wuhan/Hu-1. We included connected components in which all sequences had an N at any of the four clade-defining sites of the largest connected component. Two sequences
were excluded as they differed from all other sequences in the dataset by > 7 SNPs. This dataset includes 112 of the 186 sequences analyzed in Danesh et al.^11. Sequences were binned into four-day
windows, aligned such that the last window ended on the latest sampling date. The number of segregating sites in each window was calculated in Python using NumPy. Ambiguous nucleotides were
permissively considered in the calculation of segregating sites, e.g., an N nucleotide was assumed to match all four nucleotides, whereas an R nucleotide was assumed to match only A and G
nucleotides. This matching assumption results in a lower bound estimate for the number of segregating sites in any time window. If we instead count an N nucleotide at a site as a mutation, the number
of segregating sites in each time window is much larger (Figure S12a). However, it is unlikely that an N nucleotide indicates a mutation; it is much more likely that an N indicates an inability to
call a nucleotide based on low read depth or poor quality scores at a site. If we count N nucleotides as matching observed variation but count other ambiguous nucleotides (e.g., R) as mutations, the
segregating site trajectory is barely affected (Figure S12b). This is because there are very few non-N ambiguous nucleotides in the dataset. As such, our parameter estimates on the France dataset are
unlikely to be impacted by our assumption of ambiguous nucleotides matching observed genetic variation at their respective sites.
To confirm that the subset of sequences from France obtained from finding connected components formed an evolutionary lineage/clade, we first combined the 479 sequences sampled from France with 100
randomly-selected complete, high-coverage sequences sampled from outside France through March 17th, 2020 and uploaded to GISAID through April 29th, 2021. These sequences were aligned to Wuhan/Hu-1
using MAFFT, insertions were removed, and the sites described above were masked. This alignment was concatenated with the aligned sequences from France. IQ-Tree v. 2.0.7^62 was used to construct a
maximum likelihood phylogeny, and ModelFinder^63 was used to find the best fit nucleotide substitution model (GTR+F+I). Small branches were collapsed. TreeTime v. 0.8.0^64 was used to remove any
sequences with more than four interquartile distances from the expected evolutionary rate, rooting at Wuhan/Hu-1. Treetime was also used to generate a time-aligned phylogeny assuming a clock rate of
1 ×10−^3 substitutions per site per year with a standard deviation of 5 ×10−^4 substitutions per site per year, a skyline coalescent model, marginal time reconstruction, accounting for covariation,
and resolving polytomies. Maximum likelihood phylogenies were visualized in Python using Matplotlib v. 3.3.3^65 and Baltic (https://github.com/evogytis/baltic).
Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.
The simulated data generated in this study are available at https://github.com/koellelab/segregating-sites. The transmission pair data used to estimate the per-genome, per-transmission event mutation
rate is provided in Table S1. The SARS-CoV-2 viral genome sequences used in the France analysis are available from GISAID (Supplementary information; https://doi.org/10.55876/gis8.230123mt). Due to
the size of datasets, source data (excluding genome sequences downloaded from GISAID) are available at https://github.com/koellelab/segregating-sites.
Python code used for generation of all figures is available on GitHub: https://github.com/koellelab/segregating-sites.
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The research reported in this paper was supported by the National Institute of General Medical Sciences grant NIH/NIGMS R01 GM124280 R01 GM 12480 and R01 GM124280-03S1 (K.K.), the National Institute
of Allergy and Infectious Diseases Centers of Excellence for Influenza Research and Response (CEIRR) contract # 75N93021C00017 (K.K.), and NIH NIAID F31AI154738 (M.A.M.). We thank the Koelle lab,
Aaron King, Sally Otto, and Ailene MacPherson for feedback, as well as the BIRS Mathematics and Statistics of Genomic Epidemiology workshop for the opportunity to discuss this work.
Michael A. Martin
Present address: Department of Pathology, Johns Hopkins University School of Medicine, Baltimore, MD, USA
Graduate Program in Population Biology, Ecology, and Evolution, Emory University, Atlanta, GA, 30322, USA
Yeongseon Park & Michael A. Martin
Department of Biology, Emory University, Atlanta, GA, 30322, USA
Katia Koelle
Emory Center of Excellence for Influenza Research and Response (CEIRR), Atlanta, GA, USA
Katia Koelle
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You can also search for this author in PubMed Google Scholar
You can also search for this author in PubMed Google Scholar
Y.P.: Conceptualization, Methodology, Software, Validation, Formal Analysis, Investigation, Writing, Visualization. M.A.M.: Conceptualization, Methodology, Software, Formal Analysis, Investigation,
Writing, Visualization. K.K.: Conceptualization, Methodology, Investigation, Writing, Supervision.
Correspondence to Katia Koelle.
The authors declare no competing interests.
Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.
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NCERT Solutions for Class 11 Economics Chapter 2 – Collection of DataNCERT Solutions for Class 11 Economics Chapter 2 – Collection of Data
NCERT Solutions for Class 11 Economics Chapter 2 – Collection of Data
Page No 20:
Question 1(i):
Which of the following is the most important when you buy a new dress?
Which of the following is the most important when you buy a new dress?
a. Colour
b. Brand
c. Price
d. Size
Page No 21:
Question 1(ii):
How often do you use computers?
How often do you use computers?
a. Less than 1 hour a day
b. 1 to 3 hours a day
c. 3 to 5 hours a day
d. More than 5 hours
Question 1(iii):
Which of the following newspaper/s do you read regularly?
Which of the following newspaper/s do you read regularly?
a. The Times of India
b. The Hindustan Times
c. Indian Express
d. The Tribune
Question 1(iv):
Rise in the price of petrol is justified.
Rise in the price of petrol is justified.
a. Because of rise in demand of petrol
b. Because of rise in supply of petrol
c. Because of rise in petrol duty
d. Because of no close substitute for petrol
Question 1(v):
What is the monthly income of your family?
What is the monthly income of your family?
a. Less than Rs 10,000
b. Rs 10,000 to Rs 20,000
c. Rs 20,000 to Rs 30,000
d. More than Rs 30,000
Question 2:
Frame five two-way questions (with 'Yes' or 'No').
a. Do you smoke?
b. Do you use cosmetics daily?
c. Have you ever being to any foreign country?
d. Are you satisfied with your present income?
e. Do you have a two-wheeler?
Question 3(i):
There are many sources of data (true/false).
There are many sources of data. False
There are only two sources of data- primary and secondary data sources. The collection of data from its original source is termed as primary source of data. On the other hand, the collection of data
that has already been collected by somebody else in the past is termed as secondary source of data.
Question 3(ii):
Telephone survey is the most suitable method of collecting data, when the population is literate and spread over a large area (true/false).
Telephone survey is the most suitable method of collecting data when the population is literate and spread over a large area. False
In a Telephone Survey, the investigator asks question over the phone or mobile. So, there is no need for the person to read the survey form. Hence, this method of collecting information is the most
suitable for the illiterate population spread over a large area. Moreover, this method is less time consuming as the investigator need not to visit each person individually from door to door.
Question 3(iii):
Data collected by investigator is called the secondary data (true/false).
Data collected by an investigator is called the secondary data. False
The data collected by an investigator is called the primary data, whereas, the data that is already in existence being collected by any other investigator is known as secondary data.
Question 3(iv):
There is a certain bias involved in the non-random selection of samples (true/false).
There is a certain bias involved in the non-random selection of samples. True
Non-random selection of items obstructs the equal chance of every item of the Universe of getting selected. Hence, there may be a high probability of involvement of personal biasness of the
investigator in the non-random sample selected.
Question 3(v):
Non-sampling errors can be minimised by taking large samples (true/ false).
Non-sampling errors can be minimised by taking large samples. False
The errors that are related to the collection of data are termed as Non-sampling Errors. If the field of investigation or the population size is large, then the possibility of Non-sampling Errors
also increases. So, Non-sampling Errors maximises by taking large samples.
Question 4(iv):
(a) Do you agree with the use of chemical fertilisers?
(b) Do you use fertilisers in your fields?
(c) What is the yield per hectare in your field?
The chronological order of the questions asked is incorrect. The order lacks the direction of causation. It should move from general to specific that makes the respondents easy and comfortable
(i) The correct order of the above questions should be: What is the yield per hectare in your field?
(ii) Do you use fertilisers in your fields?
(iii) Do you agree with the use of chemical fertilisers?
Question 4(iii):
Wouldn't you be opposed to increase in price of petrol?
Although the answer to this question needs a wider view and knowledge of the economic condition and far reaching effect of rise in petrol price, but the majority of people will argue against such
rise. Thus, more effective question could be:
Do you think the hike of prices of petrol is justified?
a) Yes
b) No
Question 4(ii):
If plastic bags are only 5 percent of our garbage, should it be banned?
The particular question, ‘If plastic bags are only 5 percent of our garbage, should it be banned?’ is too long which discourages people to complete the questionnaire.
The correct question should be:
Do you think use of plastic bags should be banned?
a) Yes
b) No
Question 4(i):
How far do you live from the closest market?
This question is ambiguous. The respondents will not be able to answer the question correctly. The correct question should be:
How many kilometers is your home away from the closest market?
a) Less than 1 km
b) Between 1 km to 2 km
c) More than 2 km
Question 5:
You want to research on the popularity of Vegetable Atta Noodles among children. Design a suitable questionnaire for collecting this information.
│ QUESTIONNAIRE │
│ │
│ Popularity of Vegetable Atta Noodles │
│ │
│Name ¦ Age │
│ │
│Address ¦ Sex: Male Female │
│ │
│1. Do you like Vegetable Atta Noodles? │
│ │
│(a) Yes (b) No │
│ │
│2. Do you find this reasonable? │
│ │
│(a) Yes (b) No │
│ │
│3. How many packets do you consume in a month? │
│ │
│(a) 1“ 2 Packets │
│ │
│(b) 2“ 3 Packets │
│ │
│(c) 3“ 6 Packets │
│ │
│(d) More than 6 packets │
│ │
│4. Do you prefer Atta noodles over Maida noodles? │
│ │
│(a) Yes (b) No │
│ │
│5. Which Vegetables according to you should be added in present Atta noodles?│
│ │
│. │
│ │
│6. Do you think it should be spicier? │
│ │
│(a) Yes (b) No │
│ │
│7. At what time of the day do you prefer the Atta noodles the most? │
│ │
│(a) Day │
│ │
│(b) Afternoon │
│ │
│(c) Evening │
│ │
│(d) Night │
│ │
│8. Do your parents accompany you while having noodles? │
│ │
│(a) Yes (b) No │
Question 6:
In a village of 200 farms, a study was conducted to find the cropping pattern. Out of the 50 farms surveyed, 50% grew only wheat. Identify the population and the sample here.
Population refers to the aggregate or the total items to be studied for an investigation. So, the population here is 200 farms.
Sample is the subset of the population. In other words, a small set selected from the population for statistical study is referred as sample population.
Out of 200 farms, only 50 farms are selected for survey; therefore, the sample population is 50 farms.
Question 7:
Give two examples each of sample, population and variable.
Sample is the subset of the population. In other words, a small set selected from the population for statistical study is referred as sample population. For example, in order to study the growth
pattern of students, the heights of 50 students (sample) are recorded from a school of 500 students (population). Similarly, in order to record the level of sugar in the blood, blood sample of 2000
people (sample) was taken from 20,000 people (population).
Population refers to the aggregate or the total items to be studied for an investigation. In the above examples, 500 students and 20,000 people constitute the population.
Variables are the characteristics of a sample or population that can be expressed in numbers such as, height, income, age, etc.
Question 8:
Which of the following methods give better results and why?
Sample Method gives better results than the Census Method due to the following reasons.
1. Accuracy– Although census method provides more accurate and reliable results as compared to the sample method but in the sample method the errors can be easily located and rectified in the
sampling methods due to the smaller number of items. Therefore, despite sample method providing lesser reliable results (as not all units are studied) yet the sample method is efficient in a sense
that errors committed can be easily located.
2. Less Time and Energy Consuming- The Sample Method involves study of fewer items of the universe. So, this method not only saves time but also energy of the investigator.
3. Cost Efficient- The cost of approaching each individual unit for interrogation and collection of data is comparatively lower due to small size of sample.
4. Lesser Non-sampling Errors- The probability of Non-sampling Errors is also low as the sample size is smaller as compared to that of the Census Method.
5. More Efficient- As the sample size is smalle, so the small teams of enumerators can be formed. These small teams can easily be well trained and supervised. Consequently, the team can work more
efficiently than the team engaged in the Census Method.
Question 9:
Which of the following errors is more serious and why?
│(a)│Sampling error │(b)│Non-Sampling error │
Non-sampling Errors are more serious than the Sampling Errors because the latter can be minimised by taking a larger sample. Non-sampling Errors emerge due to the use of faulty means of collection of
data, whereas, the Sampling Errors emerge due to the divergence between the estimated and the actual value of a parameter of a small sized sample population. For example, errors due to personal
biasness, misinterpretation of results, miscalculations, etc.
The Sampling Errors can be minimised by increasing the size of a small sample, so that the difference between the actual and the estimated value is reduced. But the Non-sampling Errors are difficult
to rectify as it would require selection of a new sample and conducting a fresh survey. Thus, Non-sampling Errors are more serious than the Sampling Errors.
Question 10:
Suppose there are 10 students in your class. You want to select three out of them. How many samples are possible?
Population = 10
Number of possible samples=Crn=C310 =10!3! 10-3! =10!3! 7! = 120Thus, 120 samples are possible in the above scenario.
Question 11:
Discuss how you would use the lottery method to select 3 students out of 10 in your class?
The following method can be used while selecting 3 students out of 10 of the class.
(i) Make ten paper slips of equal size.
(ii) Write the name of each student on each slip.
(iii) Make sure that no two slips contain the same names.
(iv) Now, put all the slips in a box and mix them well.
(v) Draw three slips at random without replacement (i.e. one by one).
(vi) The names of the three students that are written on the three slips drawn are considered as selected.
Question 12:
Does the lottery method always give you a random sample? Explain.
Yes, the lottery method always gives a random sample’s outcome. In a random sample, each individual unit has an equal chance of getting selected. Similarly, in a lottery method, each individual unit
is selected at random from the population and thereby has equal opportunity of getting selected. For example, in order to select a student as monitor, the slips containing the names of all the
students are mixed well, and then a slip is drawn out at random. In this case, all students of the class have equal chance of getting selected. The probability of a student getting selected through
the lottery method is exactly same as the probability of any one student randomly selected.
Question 13:
Explain the procedure of selecting a random sample of 3 students out of 10 in your class, by using random number tables.
The procedure of selecting random sample of 3 students out of 10 in a class is as follows:
1. Assign a particular number between 1 to 10 to all the 10 students like, 01, 02, 03, 04, 05, 06, 07, 08, 09, 10.
2. Select a number randomly. Let us assume that the number selected is 05.
3. Consult the Two-Digit Random Number Table, two numbers successive to the selected random number (i.e. 05) either horizontally or vertically are the remaining two students (i.e. 06 and 07).
Question 14:
Do samples provide better results than surveys? Give reasons for your answer.
Samples provide better results than surveys. The advantages of samples over survey are as follows:
1. Provides Reliable and Accurate Results- The Sample Method provides reliable and accurate results. This is because of the fact that errors can be easily located and rectified in the sampling
methods due to the smaller number of items.
2. Less Time and Energy Consuming- The Sample Method involves study of fewer items of the universe. So, this method not only saves time but also energy of the investigator.
3. Cost Efficient- The cost of approaching each individual unit for interrogation and collection of data is comparatively lower due to the small size of sample.
4. Lesser Non-sampling Errors- The probability of Non-sampling Errors is also low as the sample size is smaller.
5. More Efficient- As the sample size is smaller, so small teams of enumerators can be formed. These small teams can easily be well trained and supervised. Consequently, the team is more efficient
than that of the team engaged in the surveys.
6. Large Investigations- When the size of the population is very large and inapproachable, the method of sample is the most feasible. | {"url":"https://www.studyguide360.com/2019/03/ncert-solutions-for-class-11-economics-chapter-2-collection-of-data.html","timestamp":"2024-11-09T22:04:24Z","content_type":"application/xhtml+xml","content_length":"271704","record_id":"<urn:uuid:75e2269a-9662-4875-855a-9cef08bc4fcb>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00340.warc.gz"} |
Excel Bubble Chart Multiple Series Example 2024 - Multiplication Chart Printable
Excel Bubble Chart Multiple Series Example
Excel Bubble Chart Multiple Series Example – You could make a multiplication graph in Shine through a web template. You will find many samples of templates and figure out how to formatting your
multiplication graph or chart using them. Here are several tips and tricks to make a multiplication graph or chart. When you have a format, all you need to do is backup the formulation and mixture it
in the new mobile. You may then take advantage of this method to flourish several numbers by an additional set. Excel Bubble Chart Multiple Series Example.
Multiplication table template
You may want to learn how to write a simple formula if you are in the need to create a multiplication table. Very first, you have to fasten row among the header line, then grow the amount on row A by
mobile B. A different way to build a multiplication desk is by using blended referrals. In such a case, you would probably key in $A2 into line A and B$1 into row B. The outcome is really a
multiplication desk using a formulation that works for columns and rows.
You can use the multiplication table template to create your table if you are using an Excel program. Just open up the spreadsheet together with your multiplication dinner table change and template
the name on the student’s name. You can also adjust the sheet to fit your individual needs. It comes with an option to modify the colour of the cells to modify the appearance of the multiplication
dinner table, as well. Then, you may modify all the different multiples suitable for you.
Making a multiplication graph in Shine
When you’re using multiplication dinner table application, it is simple to develop a straightforward multiplication table in Excel. Just create a sheet with rows and columns numbered from a single to
30. Where the columns and rows intersect will be the respond to. For example, if a row has a digit of three, and a column has a digit of five, then the answer is three times five. The same goes for
the opposite.
Initially, you may enter in the phone numbers that you need to increase. If you need to multiply two digits by three, you can type a formula for each number in cell A1, for example. To help make the
phone numbers larger sized, find the cellular material at A1 and A8, and then select the correct arrow to select an array of cells. Then you can type the multiplication solution within the tissue in
the other rows and columns.
Gallery of Excel Bubble Chart Multiple Series Example
How To Make Bubble Chart In Excel Excelchat Excelchat
Bubble Chart
Creating Multi series Bubble Charts In Excel Bubble Chart Bubbles Chart
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5.8 Item information | Child Development with the D-score: Tuning instruments to Unity
5.8 Item information
Item information is a psychometric measure that quantifies the sensitivity of the item to changes in the person’s ability. An item is most sensitive around the D-score value where the PASS
probability equals the FAIL probability, which corresponds to the item difficulty (\(\delta_i\)). One unit change around \(\delta_i\) has a large effect on the probability of endorsing, while one
unit change far away from \(\delta_i\) has negligible impact. Suppose person A had passing probability \(0.7\) for some item. The information delivered by that item for person A is the product \(0.7
\times (1.0 - 0.7) = 0.21\). Suppose person B has a D-score that coincides with the difficulty level of the item. In that case, the information for B equals \(0.5 \times (1 - 0.5) = 0.25\), the
maximum. Likewise, for a person C with high ability, the information could be \(0.98 * 0.02 = 0.02\), so that item carries almost no information for person C.
The information is inversely related to the error of measurement. More information amounts to less measurement error. For each response in the data, we can compute the amount of information it
contributed to the model D-score. By summing the information over persons, we obtain a measure of certainty about the difficulty estimate of the item. This sum of information incorporates both the
number of administrations and the quality of the match between person abilities and item difficulty.
Figure 5.7 displays the summed information for each item, divided into four grades: A(best) to D (worst). The information grade measures the stability of the difficulty estimate. Most items receive
grades higher than C. In total, 30 milestones have grade D. Adding these items to future studies may yield important additional information.
The red circles indicate active equate groups. Most have grade A, so we have a lot of information about the items that form the active equate groups. Table 5.2 displays more detailed information for
the active equate groups. The sample sizes are reasonably large. Many information statistics are well is above 100; the criterion for Grade A. The interpretation of this criterion is as follows.
Suppose that we obtain a sample of 400 persons who are all perfectly calibrated to the item of interest. In that case, the information for that item will be equal to 100.
Table 5.2: Equate group information in the final
Equate group Difficulty Sample Size Information Grade
EXP2 11.44 3608 162.32846 A
REC6 30.90 5428 95.39963 B
GM25 36.43 6380 470.62671 A
FM26 42.93 4155 296.78327 A
GM35 44.01 5522 356.04417 A
COG36 44.53 7912 230.02909 A
GM42 49.86 5953 327.74297 A
FM31 53.17 10991 731.65850 A
COG55 54.08 5647 420.34928 A
FM72 57.07 5430 253.63655 A
EXP26 59.15 9119 578.79355 A
SA1 60.08 3363 172.10653 A
FM38 60.87 10236 491.68110 A
FM52 67.80 13487 1159.93870 A
FM43 69.66 15765 1563.88651 A
GM60 70.09 9519 1070.60909 A
REC40 71.04 10393 1182.90580 A
FM61 72.56 10612 945.86689 A | {"url":"https://d-score.org/dbook2/sec-iteminformation.html","timestamp":"2024-11-10T05:15:50Z","content_type":"text/html","content_length":"26969","record_id":"<urn:uuid:eb7bdaa6-9a40-45d2-8eb0-ac01011436a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00702.warc.gz"} |
state transition diagram LTL model checking
+ General Questions (11)
It is not wise to ask for a standard procedure that may exist, but may not be efficient if applied to any problem. In general, LTL model checking is done as follows to compute the states that satisfy
A phi:
1. translate the negation of the LTL formula phi to an equivalent existential nondeterministic omega-automaton
2. compute the product of the Kripke structure and the omega-automaton
3. determine the states that have a path in the product structure that starts in an initial state of the automaton and that satisfies the acceptance condition
How these steps are done in detail, may be different for the particular example that is considered. Sometimes, a big simplification can be done by observing something, mostly by omitting unreachable
states, and deadend states. That can be done in many ways, and neither is there a unique solution nor is there a general recipe; just keep your eyes open when looking at such examples. | {"url":"https://q2a.cs.uni-kl.de/3140/state-transition-diagram-ltl-model-checking?show=3141","timestamp":"2024-11-07T19:02:55Z","content_type":"text/html","content_length":"51303","record_id":"<urn:uuid:8bacb59e-b934-4bc2-a5a3-cfb5a34777bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00010.warc.gz"} |
SK Enrichment Centre|Tuition Services|Testimonials|tuition agency
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Developing Creativity of Schoolchildren through the Course “Developmental Mathematics”
The relevance of the present study is due to the importance of developing creativity which can be achieved through a variety of school subjects including mathematics. In the article the potential of
extended (supplementary) mathematical education (in primary and secondary schools) is highlighted. The main objective of this study is to examine and evaluate the contents, practices and methods that
are currently employed in extended education. The main empirical method of this study is modeling of the modular system of lessons (the course) that offers a variety of assignments including
non-standard tasks, puzzles and problems; tasks and topics from academic Olympiads and other mathematical competitions; creative tasks, practical assignments and experiments with mathematical
materials ("empirical" mathematics); team and individual competitions and organization of home readings on a specific subject. The article describes the author's methodology. The main feature of the
developed course is the inclusion of various organizational forms and diverse materials aimed at sustaining schoolchildren’s interest towards mathematics, enabling them to deal with advanced level
mathematical problems and developing their curiosity and creativity. The course "Developmental Mathematics" that is presented in this article has been empirically tested since 2008.
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
Article Type: Research Article
EURASIA J Math Sci Tech Ed, Volume 13, Issue 6, June 2017, 1799-1815
Publication date: 18 Apr 2017
Article Views: 2458
Article Downloads: 1215
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Commit 2020-06-30 08:06 a143c386
View on Github →
refactor(linear_algebra/affine_space): allow empty affine subspaces (#3219) When definitions of affine spaces and subspaces were added in #2816, there was some discussion of whether an affine
subspace should be allowed to be empty. After further consideration of what lemmas need to be added to fill out the interface for affine subspaces, I've concluded that it does indeed make sense to
allow empty affine subspaces, with nonempty hypotheses then added for those results that need them, to avoid artificial nonempty hypotheses for other results on affine spans and intersections of
affine subspaces that don't depend on any way on affine subspaces being nonempty and can be more cleanly stated if they can be empty. Thus, change the definition to allow affine subspaces to be empty
and remove the bundled direction. The new definition of direction (as the vector_span of the points in the subspace, i.e. the submodule.span of the vsub_set of the points) is the zero submodule for
both empty affine subspaces and for those consisting of a single point (and it's proved that in the nonempty case it contains exactly the pairwise subtractions of points in the affine subspace). This
doesn't generally add new lemmas beyond those used in reproving existing results (including what were previously the add and sub axioms for affine subspaces) with the new definitions. It also doesn't
add the complete lattice structure for affine subspaces, just helps enable adding it later.
Estimated changes | {"url":"https://mathlib-changelog.org/v3/commit/a143c386","timestamp":"2024-11-04T01:53:56Z","content_type":"text/html","content_length":"24541","record_id":"<urn:uuid:fc137e8b-1d5c-4828-b6e0-fb943802c1fa>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00588.warc.gz"} |
From Encyclopedia of Mathematics
Hi! Under "Generalized Zeta Function" the formula should have "n=0" and NOT "n=1". If a=1, the first component=1/2 which is not how the zeta function begins. Google "Hurwitz Zeta Function" to
confirm. -robinlrandall@gmail.com
Corrected, thanks. TBloom 10:01, 26 May 2012 (CEST)
How to Cite This Entry:
Zeta-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zeta-function&oldid=26859 | {"url":"https://encyclopediaofmath.org/wiki/Talk:Zeta-function","timestamp":"2024-11-09T01:16:01Z","content_type":"text/html","content_length":"12903","record_id":"<urn:uuid:a5bb539c-6731-48fc-b50f-d073debf8f77>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00276.warc.gz"} |
(Get Answer) - Auric Enterprises conducts exploratory gold mining operations in...| Transtutors
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1. Auric Enterprises conducts exploratory gold mining...
Auric Enterprises conducts exploratory gold mining operations in western North...
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Auric Enterprises conducts exploratory gold mining operations in western North Carolina. In order to fund the operation, investors form partnerships, which provide the financial support necessary to
explore a fixed number of locations. Each location explored is then classified as either "potential" or "worthless". Past experience shows that this type of exploratory operation discovers
"potential" mines 25 percent of the time. Suppose a newly formed partnership has provided the financial support for exploring at 19 locations. Assume the 19 explorations are independent of each
a) What is the expected value of the number of "potential" mines that will be discovered?
1) 4
2) 4.75
3) 5
4) 5.25
5) None of the above
b) What is the probability that exactly 2 "potential" mines will be discovered?
1) 0.0803 =BINOM.DIST(2,19,0.25,FALSE)
2) 0.1113 =BINOM.DIST(2,19,0.25,TRUE)
3) 0.8887 =1-BINOM.DIST(2,19,0.25,TRUE)
4) 0.9197 =1-BINOM.DIST(2,19,0.25,FALSE)
5) None of the above
c) What is the probability that at least 4 "potential" mines will be discovered?
1) 0.2631 =BINOM.DIST(3,19,0.25,TRUE)
2) 0.4654 =BINOM.DIST(4,19,0.25,TRUE)
3) 0.5346 =1-BINOM.DIST(4,19,0.25,TRUE)
4) 0.7369 =1-BINOM.DIST(3,19,0.25,TRUE)
5) None of the above
d) What is the probability that at least 3 but no more than 7 "potential" mines will be discovered?
1) 0.6594 =BINOM.DIST(7,19,0.25,TRUE)-BINOM.DIST(3,19,0.25,TRUE)
2) 0.7082 =BINOM.DIST(8,19,0.25,TRUE)-BINOM.DIST(3,19,0.25,TRUE)
3) 0.8112 =BINOM.DIST(7,19,0.25,TRUE)-BINOM.DIST(2,19,0.25,TRUE)
4) 0.8599 =BINOM.DIST(8,19,0.25,TRUE)-BINOM.DIST(2,19,0.25,TRUE)
5) None of the above
Recent Questions in Statistics - Others
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MHT CET 2023 9th May Morning Shift | Alternating Current Question 37 | Physics | MHT CET - ExamSIDE.com
MHT CET 2023 9th May Morning Shift
MCQ (Single Correct Answer)
A capacitor, an inductor and an electric bulb are connected in series to an a.c. supply of variable frequency. As the frequency of the supply is increased gradually, then the electric bulb is found
MHT CET 2023 9th May Morning Shift
MCQ (Single Correct Answer)
In an $$\mathrm{AC}$$ circuit, the current is $$\mathrm{i}=5 \sin \left(100 \mathrm{t}-\frac{\pi}{2}\right) \mathrm{A}$$ and voltage is $$\mathrm{e}=200 \sin (100 \mathrm{t})$$ volt. Power
consumption in the circuit is $$\left(\cos 90^{\circ}=0\right)$$
MHT CET 2022 11th August Evening Shift
MCQ (Single Correct Answer)
A capacitor of capacitance $$50 \mu \mathrm{F}$$ is connected to a.c. source $$\mathrm{e}=220 \sin 50 \mathrm{t}$$ ($$\mathrm{e}$$ in volt, $$\mathrm{t}$$ in second). The value of peak current is
MHT CET 2022 11th August Evening Shift
MCQ (Single Correct Answer)
The resistance offered by an inductor $$\left(X_L\right)$$ in an a.c. circuit is | {"url":"https://questions.examside.com/past-years/jee/question/pa-capacitor-an-inductor-and-an-electric-bulb-are-connect-mht-cet-physics-motion-dpyprm2pclymjest","timestamp":"2024-11-10T07:25:11Z","content_type":"text/html","content_length":"196154","record_id":"<urn:uuid:97091c0e-3b89-4098-982e-fc2f80c57acc>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00650.warc.gz"} |
From Fluid Dynamics to Gravity and Back
How the Movement of Water Molecules Corresponds to Ripples in Spacetime
Shiraz Minwalla has uncovered an unexpected connection between the equations of fluid and superfluid dynamics and Einstein’s equations of general relativity.
There is an interesting connection between two of the best-studied nonlinear partial differential equations in physics: the equations of hydrodynamics and the field equations of gravity.
Let’s start with a brief review of hydrodynamics. At the microscopic level a tank of water is a collection of, say, 10^25 molecules that constantly collide with one another. The methods of physics
may be used to model this collection of water molecules as follows: we set up equations that track the position and momentum of each of the water molecules and predict their time evolution. These
conceptually complete equations have of order 10^25 variables and so are clearly too difficult to handle in practice.
Does it then follow that tanks of water cannot be usefully studied using the methods of physics? As every plumber knows, this conclusion is false: a useful description of water is obtained by keeping
track of average properties of water molecules, rather than each individual molecule.
Think of a tank of water as a union of non-overlapping lumps of water. Each lump is big enough to contain a large number of molecules but small enough so that gross macroscopic properties of the
water (energy density, number density, momentum density) are approximately uniform. The fundamental assumption of hydrodynamics is that under appropriate conditions, all the “average” properties of
any lump are completely determined by its conserved charge densities (in the case of water, molecule number density, energy density, and momentum density). In particular, the conserved current for
molecule number j^µ and the conserved current for energy and momentum T^µν are themselves dynamically determined functionals of local thermodynamical densities in a locally equilibrated system
(fluctuations away from these dynamically determined values are suppressed by a factor proportional to the square root of the number of molecules in each lump). The equations that express conserved
currents as functionals of conserved densities are difficult to compute theoretically but are easily measured experimentally and are known as constitutive relations.
When supplemented with constitutive relations, the conservation equations ∂[µ] j^µ =0, and ∂[µ] T^µν=0(2) turn into a well-posed initial value problem for the dynamic of conserved densities. They are
the equations of hydrodynamics. Let me reemphasize that the effect of the ignored degrees on the evolution of conserved densities is inversely proportional to the square root of the number of
molecules in a lump, and so is negligible in an appropriate thermodynamic limit, allowing the formulation of a closed dynamical system for conserved densities.
My research concerns how the equations of hydrodynamics pop up in an apparently completely unrelated setting: in the study of the long wavelength dynamics of black holes governed by Einstein’s
equations with a negative cosmological constant.
Einstein’s gravitational equations describe the dynamics of the geometry of spacetime. The ripples of spacetime (gravitational waves) have interesting dynamics even in the absence of any matter. For
most of this article, I will be referring to Einstein’s equations in the absence of matter.
The simplest solution of the most familiar Einstein equation G[µν]=0 is simply flat Minkowskian spacetime. However, the usual Einstein equations can be deformed to admit the so-called cosmological
constant term G[µν]=λg[µν]. This deformation, which was first suggested and later rejected by Einstein himself, appears to be needed to model the cosmological expansion of our universe. The observed
accelerated expansion of our universe is plausibly explained by the existence of a positive cosmological constant (the equation above with a positive value of λ).
Recent theoretical investigations within string theory have focused attention on Einstein’s equations with a negative cosmological constant (negative value of λ). This equation does not have flat
space as a solution. Its simplest solution is a highly symmetric spacetime called anti-de Sitter (AdS) space. In this article, I will explore asymptotically AdS solutions of Einstein’s equations with
a negative cosmological constant in five spacetime dimensions.
Einstein’s equations, with or without a gravitational constant, admit a huge variety of black hole solutions. The equations with a negative cosmological constant also admit rather unusual related
solutions called black brane. These solutions have finite energy and momentum density rather than a finite energy and momentum. Stationary black brane solutions are analytically well known, and
appear in a four-parameter set, labeled by a uniform energy and momentum density.
It is often the case in physics that if a uniform distribution of “something” (a field, the orientation of a spin, etc.) leads to a time independent solution of the equations of motion, then slowly
varying configurations of that thing result in slow dynamics. It turns out that this general expectation applies to the distribution of energy and momentum density on black branes. The exact
four-parameter set of time-independent black brane solutions may be generalized to an infinite number of approximate solutions of Einstein’s equations. These solutions are characterized by varying
(rather than uniform) energy and momentum density fields. The fields are functions of spatial position as well as time, but are constrained to obey dynamical equations. It has been demonstrated that
these equations take the form of conservation equations (conservation of the stress tensor in the case of the vacuum Einstein equations), with all components of the stress tensor determined to be a
particular functional of the local energy density by an effective constitutive relation.^1 In other words, the long wavelength fluctuations of black branes are governed by the equations of
hydrodynamics, with gravitationally determined constitutive relations. This fact is the so-called fluid-gravity correspondence.
The fluid-gravity correspondence was established constructively; there is an explicit construction of an approximate solution to Einstein’s equations dual to any fluid flow. The construction of these
solutions proceeds in an expansion in derivatives. The procedure that determines the solutions to Einstein’s equations also simultaneously determines the constitutive relations to the given order in
It turns out that the gravitational solutions that participate in the fluid-gravity correspondence all have regular event horizons; moreover, Hawking’s area theorem, which states that event horizons
can only stay the same or increase but never decrease, may be used to determine a positive divergence entropy current for fluid flows. This establishes that the fluid flows generated by gravity obey
a basic physical requirement; even locally, entropy in such flows never decreases.
Why does the equation that describes the average motion of water molecules in a water tank also govern the ripples of the event horizon of asymptotically AdS black brane? At least in some
circumstances, we believe this is because Einstein’s equations in the presence of black branes actually describe the averaged dynamics of a large number of underlying microscopic variables. In
particular, the AdS/conformal field theory correspondence of string theory proposes that the uniform black brane of negative cosmological constant gravity is dual, in a particular context, to a gas
of gluons of a large U(N) gauge theory at large N. The hydrodynamical solutions of fluid gravity are presumably the duals to the fluid flows of this collection of gluons. Fluctuations about these
hydrodynamical flows are suppressed by ^1⁄[N] and may be thought of as quantum gravity fluctuations from the gravitational point of view.
At the conceptual level, the fluid-gravity correspondence suggests a novel view of the role of Einstein’s equations in the presence of event horizons. At a more practical level, this correspondence
has had a completely unanticipated application in the reverse direction: to the theory of the equations of relativistic hydrodynamics, a subject that was thought to have been closed in the 1930s.
Analyses by Lev Landau and Evgeny Lifshitz in the 1930s claimed to have determined the most general form of the constitutive relations of relativistic fluid at first order in the derivative
expansion. Motivated by the fluid-gravity correspondence, it has been discovered that the Landau-Lifshitz constitutive relations must be generalized in the case of certain parity violating charged
fluids.^2 In particular, if the fluid charge has a U(1)^3 triangle anomaly, then there are new terms in the constitutive relations of this fluid––roughly in terms of the fluid current proportional to
the vorticity––that are completely determined by the anomaly coefficient plus the thermodynamics of the fluid. This discovery may turn out to have experimental consequences (in the study of fluid
flows in Brookhaven National Laboratory’s Relativistic Heavy Ion Collider experiment, for example, which aims to study the first few moments after the universe’s creation), surely a surprising
application for the esoteric study of black hole physics in higher-dimensional gravity.^3
1. “Nonlinear Fluid Dynamics from Gravity,” Sayantani Bhattacharyya, Veronika E. Hubeny, Shiraz Minwalla, and Mukund Rangamani, Journal of High Energy Physics 0802:045 (2008).
2. “Hydrodynamics with Triangle Anomalies,” Dam T. Son and Piotr Surowka, Physical Review Letters 103, 191601 (2009).
3. “The Fluid/Gravity Correspondence,” Veronika E. Hubeny, Shiraz Minwalla, and Mukund Rangamani, arXiv:1107.5780. | {"url":"https://www.ias.edu/ideas/2014/minwalla-dynamics","timestamp":"2024-11-14T10:48:54Z","content_type":"text/html","content_length":"72224","record_id":"<urn:uuid:bc08a7b7-2d72-4e5b-bf7c-f900a5472e00>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00644.warc.gz"} |
Algebra Tiles Solving Equations Worksheet - Equations Worksheets
Algebra Tiles Solving Equations Worksheet
Algebra Tiles Solving Equations Worksheet – The purpose of Expressions and Equations Worksheets is for your child to be able to learn more effectively and efficiently. The worksheets contain
interactive exercises as well as problems that are based on sequence of operations. These worksheets make it simple for children to grasp complicated concepts and concepts fast. These PDF resources
are completely free to download and may be used by your child in order to learn math concepts. These resources are helpful to students in the 5th-8th grades.
Download Free Algebra Tiles Solving Equations Worksheet
The worksheets are suitable for use by students from the 5th-8th grades. These two-step word problems are constructed using decimals or fractions. Each worksheet contains ten problems. They are
available at any print or online resource. These worksheets can be used to practice rearranging equations. These worksheets can be used for practicing rearranging equations. They assist students with
understanding equality and inverse operations.
These worksheets can be used by fifth- and eighth grade students. They are great for students who struggle to compute percentages. There are three kinds of questions you can choose from. There is the
option to either work on single-step problems that include whole numbers or decimal numbers or to use words-based techniques for fractions and decimals. Each page will contain 10 equations. These
Equations Worksheets can be used by students from 5th through 8th grades.
These worksheets are a great way for practicing fraction calculations and other concepts in algebra. Some of the worksheets let you to choose from three different kinds of problems. You can choose
the one that is word-based, numerical or a combination of both. The type of the problem is vital, as each will have a distinct problem type. Each page has ten questions, making them a great source
for students in the 5th-8th grade.
These worksheets are designed to teach students about the relationship between variables and numbers. They allow students to work on solving polynomial problems and to learn how to apply equations in
daily life. If you’re in search of an excellent educational tool to discover the basics of equations and expressions begin by exploring these worksheets. These worksheets will help you learn about
various kinds of mathematical problems as well as the different symbols that are used to express them.
These worksheets can be very beneficial for students in their first grades. These worksheets will teach students how to solve equations as well as graph. They are great to practice polynomial
variables. These worksheets will help you simplify and factor these variables. There are plenty of worksheets that can be used to teach children about equations. The best method to learn about
equations is to complete the work yourself.
There are a variety of worksheets to teach quadratic equations. Each level has its own worksheet. The worksheets were designed for you to help you solve problems of the fourth degree. After you’ve
solved a step, you’ll be able proceed to solve other kinds of equations. Then, you can tackle the same problems. For example, you might solve a problem using the same axis in the form of an elongated
Gallery of Algebra Tiles Solving Equations Worksheet
Solving Two step Equations With Algebra Tiles Worksheet
Free Printable Algebra Tiles Worksheets PDF Number Dyslexia
Free Printable Algebra Tiles Worksheets PDF Number Dyslexia
Leave a Comment | {"url":"https://www.equationsworksheets.net/algebra-tiles-solving-equations-worksheet/","timestamp":"2024-11-05T17:12:48Z","content_type":"text/html","content_length":"63376","record_id":"<urn:uuid:cb363bb1-05c0-4ecc-9d8c-4eb1389c56c4>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00004.warc.gz"} |
Reasoning and Proving
A flaming arrow is
shot into the air to
mark the beginning
of a festival. Its
height, in metres,
after t seconds is modelled by
the function h(t) = -4.9t² +24.5t+2.
Selecting Tools
Problem Solving
a) Determine the height of the arrow at 2 s.
b) Determine the rate of change of the height
of the arrow at 1, 2, 4, and 5 s.
c) What happens at 5 s?
d) How long does it take the arrow to return
to the ground?
e) How fast is the arrow travelling when it hits
the ground? Explain how you arrived at
your answer.
f) Graph the function. Use the graph to
confirm your answers in parts a) to e).
Fig: 1 | {"url":"https://tutorbin.com/questions-and-answers/3-reasoning-and-proving-a-flaming-arrow-is-shot-into-the-air-to-mark-the-beginning-of-a-festival-its-height-in-metres","timestamp":"2024-11-06T01:17:55Z","content_type":"text/html","content_length":"63489","record_id":"<urn:uuid:338d4b5a-d680-4bf3-b7dc-9e4e4f2760e8>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00277.warc.gz"} |
Mathematics Semester-1 Answer Key ICSE 2021Class 10 - ICSEHELP
Mathematics Semester-1 Answer Key ICSE 2021Class 10
Mathematics Semester-1 Answer Key ICSE 2021 Class 10. Step by step Answer Key of ICSE Class-10 Paper Solution. During Answer Key of semester-1 Mathematics examination paper we explain with figure ,
graph, table whenever necessary so that student can achieve their goal in next upcoming exam of council. Visit official website CISCE for detail information about ICSE Board Class-10.
Mathematics Sem-1 Answer Key ICSE 2021 Class 10
Board ICSE
Class 10th (X)
Subject Mathematics
Topic Semester-1 ICSE Paper Answer Key
Syllabus on bifurcated syllabus (after reduction)
session 2021-22
Topic Answer Key of Mathematics
Total Total-25 with all parts
Max 40
Mathematics Sem-1 ICSE Answer Key
Answer key of Question 1 to Question 24 with all parts step by step.
Council of Indian School Certificate Examination (CISCE) successfully conducted the ICSE Class 10 Sem-1 Mathematics exam 2021-22, December 06, 2021. The Class 10 exam was held in the offline format,
covering a total of 25 questions with all parts. Maximum marks of Mathematics question paper is 40.
Answer key of ICSE Mathematics
-: End of Mathematics Sem-1 ICSE Answer Key class 10 :-
Return to – CISCE Semester-1 Answer Key for ICSE and ISC
6 thoughts on “Mathematics Semester-1 Answer Key ICSE 2021Class 10”
1. Answer key Is not correct
□ may be . it is not officially
2. Many of the answers are wrong… Please recheck
□ ok we will look into matter
3. These answers are wrong .u can check another website for correct answers
□ update soon
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This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://icsehelp.com/mathematics-semester-1-answer-key-icse-2021class-10/","timestamp":"2024-11-08T09:07:40Z","content_type":"text/html","content_length":"79909","record_id":"<urn:uuid:d0f41aab-abaf-4b34-a486-d514196f3a35>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00051.warc.gz"} |
Paper on decisional second-preimage resistance @ASIACRYPT 2019
In a recent paper with Daniel J. Bernstein, we introduce a new security notion for cryptographic hash functions called decisional second-preimage resistance (DSPR) that asks to decide if a given
domain element has a colliding value within the domain of the hash function. It turns out that DSPR and conventional second-preimage resistance (SPR) together imply one-wayness (PRE). This fills a
gap in the theory of hash functions showing when SPR implies PRE. Moreover, it has applications in achieving tight proofs for hash-based signatures (which was the reason for us to start looking into
this). The paper will appear at ASIACRYPT 2019. | {"url":"https://huelsing.net/wordpress/?p=828","timestamp":"2024-11-14T18:09:53Z","content_type":"text/html","content_length":"26925","record_id":"<urn:uuid:57eedf8d-2291-435f-9a2b-ccf220dcd958>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00305.warc.gz"} |
Mat4 | 8th Wall
Interface representing a 4x4 matrix. A 4x4 matrix is represented by a 16 dimensional array of data, with elements stored in column major order. A special kind of matrix, known as a TRS matrix (for
Translation, Rotation, and Scale) is common in 3D geometry for representing the position, orientation, and size of points in a 3D scene. Many special types of matrices have easily specified inverses.
By specifying these ahead of time, Mat4 allows for matrix inverse to be a very fast O(1) operation. Mat4 objects are created with the ecs.math.mat4 Mat4Factory, or through operations on other Mat4
Code Example
const {mat4, quat, vec3} = ecs.math
const [targetX, targetY, targetZ] = [1, 2, 3]
const [pitch, yaw, distance] = [30, 90, 5]
// Compute orbit controls position based on target position, pitch, yaw, and distance.
const orbitPos = mat4.t(vec3.xyz(targetX, targetY, targetZ))
.times(mat4.r(quat.pitchYawRollDegrees(vec3.xyz(pitch, yaw, 0))))
.times(mat4.t(vec3.xyz(0, 0, distance)))
Mat4 objects are created with the ecs.math.mat4 MatFactory, with the following methods:
i: () => Mat4
Identity matrix.
of: (data: number[], inverseData?: number[]) => Mat4
Create the matrix with directly specified data, in column major order. An optional inverse can be specified. If the inverse is not specified, it will be computed if the matrix is invertible. If the
matrix is not invertible, calling inv() will throw an error.
r: (q: QuatSource) => Mat4
Create a rotation matrix from a quaternion.
rows: (dataRows: number[][], inverseDataRows?: number[][]) => Mat4
Create a matrix with specified row data, and optionally specified inverse row data. dataRows and inverseDataRows should be four arrays, each with four numbers. If the inverse is not specified, it
will be computed if the matrix is invertible. If the matrix is not invertible, calling inv() will throw an error.
s: (v: Vec3Source) => Mat4
Create a scale matrix. No scale element should be zero.
t: (v: Vec3Source) => Mat4
Create a translation matrix.
tr: (t: Vec3Source, r: QuatSource) => Mat4
Create a translation and rotation matrix.
trs: (t: Vec3Source, r: QuatSource, s: Vec3Source) => Mat4
Create a translation, rotation and scale matrix.
Mat4 has no enumerable properties, but its underlying 16 element data array and inverseData array can be accessed with data() and inverseData() respectively.
Immutable API
The following methods perform computations based on the current value of a Mat4, but do not modify its contents. Methods that return Mat4 types return new objects. Immutable APIs are typically safer,
more readable, and less error-prone than mutable APIs, but may be inefficient in situations where thousands of objects are allocated each frame. In cases where object garbage collection is a
performance concern, prefer the Mutable API (below).
clone: () => Mat4
Create a new matrix with the same components as this matrix.
data: () => number[]
Get the raw data of the matrix, in column-major order.
decomposeTrs: (target?: {t: Vec3, r: Quat, s: Vec3}) => {t: Vec3, r: Quat, s: Vec3}
Decompose the matrix into its translation, rotation, and scale components, assuming it was formed by a translation, rotation, and scale in that order. If ‘target’ is supplied, the result will be
stored in ‘target’ and ‘target’ will be returned. Otherwise, a new {t, r, s} object will be created and returned.
determinant: () => number
Compute the determinant of the matrix.
equals: (m: Mat4, tolerance: number) => boolean
Check whether two matrices are equal, with a specified floating point tolerance. | {"url":"https://www.8thwall.com/docs/ja/studio/api/ecs/math/mat4/","timestamp":"2024-11-11T04:04:19Z","content_type":"text/html","content_length":"77868","record_id":"<urn:uuid:40e6c4ac-3318-4ffd-91ac-1e3309f09f15>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00599.warc.gz"} |
Fraction calculator
This calculator divides fractions. The first step makes the reciprocal value of the second fraction - exchange numerator and denominator of 2nd fraction. Then multiply both numerators and place the
result over the product of both denominators. Then simplify the result to the lowest terms or a mixed number.
The result:
1/2 : 1/4 = 2/1 = 2
The spelled result in words is two.
How do we solve fractions step by step?
1. Divide: 1/2 : 1/4 = 1/2 · 4/1 = 1 · 4/2 · 1 = 4/2 = 2 · 2 /2 · 1 = 2
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and
denominator, reciprocal of 1/4 is 4/1) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, cancel by a common
factor of 2 gives 2/1.
In other words - one half divided by one quarter is two.
Rules for expressions with fractions:
- use a forward slash to divide the numerator by the denominator, i.e., for five-hundredths, enter
. If you use mixed numbers, leave a space between the whole and fraction parts.
Mixed numerals
(mixed numbers or fractions) keep one space between the integer and
fraction and use a forward slash to input fractions i.e.,
1 2/3
. An example of a negative mixed fraction:
-5 1/2
Because slash is both sign for fraction line and division, use a colon (:) as the operator of division fractions i.e.,
1/2 : 1/3
Decimals (decimal numbers) enter with a decimal point
and they are automatically converted to fractions - i.e.
The calculator follows well-known rules for the order of operations. The most common mnemonics for remembering this order of operations are:
PEMDAS - Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
BEDMAS - Brackets, Exponents, Division, Multiplication, Addition, Subtraction
BODMAS - Brackets, Of or Order, Division, Multiplication, Addition, Subtraction.
GEMDAS - Grouping Symbols - brackets (){}, Exponents, Multiplication, Division, Addition, Subtraction.
MDAS - Multiplication and Division have the same precedence over Addition and Subtraction. The MDAS rule is the order of operations part of the PEMDAS rule.
Be careful; always do multiplication and division before addition and subtraction. Some operators (+ and -) and (* and /) have the same priority and must be evaluated from left to right.
Last Modified: October 9, 2024 | {"url":"https://www.hackmath.net/en/calculator/fraction?input=1%2F2+%3A+1%2F4","timestamp":"2024-11-12T22:04:25Z","content_type":"text/html","content_length":"31429","record_id":"<urn:uuid:a4beb383-1003-446f-9205-bb4bc9a8b74c>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00750.warc.gz"} |
Objects of class "fv" are returned by Fest, Gest,Jest, and Kest along with many other functions.
See plot.fv for plotting an "fv" object.
See as.function.fv to convert an "fv" object to an R function.
Use cbind.fv to combine several "fv" objects. Use bind.fv to glue additional columns onto an existing "fv" object.
Undocumented functions for modifying an "fv" object include fvnames, fvnames<-, tweak.fv.entry and rebadge.fv. | {"url":"https://www.rdocumentation.org/packages/spatstat.explore/versions/3.0-6/topics/fv.object","timestamp":"2024-11-03T22:19:47Z","content_type":"text/html","content_length":"65441","record_id":"<urn:uuid:edad9b9c-8b46-4965-b26d-6670ac090d85>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00429.warc.gz"} |
7 Search Results
CoCoA is a system for Computations in Commutative Algebra. It is able to perform simple and sophisticated operations on multivaraiate polynomials and on various data related to them (ideals, modules,
matrices, rational functions). For example, it can readily compute Grobner bases, syzygies and minimal free resolution, intersection, division, the radical of an ideal, the ideal of zero-dimensional
schemes, Poincare' series and Hilbert functions, factorization of polynomials, toric ideals. The capabilities of CoCoA and the flexibility of its use are further enhanced by the dedicated high-level
programming language. For convenience, the system offers a textual interface, an Emacs mode, and a graphical user interface common to most platforms.
GAP is a system for computational discrete algebra, with particular emphasis on Computational Group Theory. GAP provides a programming language, a library of thousands of functions implementing
algebraic algorithms written in the GAP language as well as large data libraries of algebraic objects. GAP is used in research and teaching for studying groups and their representations, rings,
vector spaces, algebras, combinatorial structures, and more. GAP is developed by international cooperation. The system, including source, is distributed freely under the terms of the GNU General
Public License. You can study and easily modify or extend GAP for your special use. The current version is GAP 4, the older version GAP 3 is still available.
LiE is the name of a software package that enables mathematicians and physicists to perform computations of a Lie group theoretic nature. It focuses on the representation theory of complex semisimple
(reductive) Lie groups and algebras, and on the structure of their Weyl groups and root systems. LiE does not compute directly with elements of the Lie groups and algebras themselves; it rather
computes with weights, roots, characters and similar objects.
LinBox is a C++ template library for exact, high-performance linear algebra computation with dense, sparse, and structured matrices over the integers and over finite fields. LinBox has the following
top-level functions: solve linear system, matrix rank, determinant, minimal polynomial, characteristic polynomial, Smith normal form and trace. A good collection of finite field and ring
implementations is provided, for use with numerous black box matrix storage schemes.
polymake is an object-oriented system for experimental discrete mathematics. The typical working cycle of a polymake user starts with the construction of an object of interest, auch as a convex
polytope, a finite simplicial complex, a graph, etc. It is then possible to ask the system for some of the object's properties or for some form of visualization. Further steps might include more
elaborate constructions based on previously defined objects. Each class of polymake objects comes with a set of rules which describe how a new property of an object can be derived from previously
known ones. It is a key feature that the user can extend or modify the set of rules, add further properties or even new classes of objects (with entirely new rule bases). The functions provided
include: several convex hull algorithms, face lattices of convex polytopes, Voronoi diagrams and Delaunay decompositions (in arbitrary dimensions), simplicial homology (with integer coefficients),
simplicial cup and cap products, intersection forms of triangulated 4-manifolds. Several forms of (interactive) visualization via interfaces to Geomview, JavaView and other programs.
Risa/Asir is a general computer algebra system and also a tool for various computation in mathematics and engineering. The development of Risa/Asir started in 1989 at FUJITSU. Binaries have been
freely available since 1994 and now the source code is also free. Currently Kobe distribution is the most active branch of its development. We characterize Risa/Asir as follows: (1) An environment
for large scale and efficient polynomial computation. (2) A platform for parallel and distributed computation based on OpenXM protocols.
Maple program for computing the sum of values of a polynomial function over the set of integral points of a polygon and the corresponding weighted Ehrhart quasi-polynomial. | {"url":"https://orms.mfo.de/search@terms=small+groups.html","timestamp":"2024-11-11T06:52:30Z","content_type":"application/xhtml+xml","content_length":"8685","record_id":"<urn:uuid:29e5b472-41d2-4a40-9091-2c0e4fc1ee79>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00780.warc.gz"} |
LARGE - Excel docs, syntax and examples
The LARGE function in Excel returns the nth largest value from a range of data. It is useful for quickly identifying the top values within a dataset or array.
=LARGE(array, n)
array The range or array of data from which you want to find the nth largest value.
n The position (rank) of the largest value you want to return. It must be greater than or equal to 1.
Remark n^th^ largest isn't n^th^ value
About LARGE 🔗
When you need to pinpoint the highest values within a set of data in Excel, the LARGE function steps in to simplify the process. It allows you to swiftly extract the nth largest value from a
specified array or range, which proves valuable in various analytical scenarios where identifying top results is crucial. Whether you're analyzing sales figures, test scores, or any other numerical
data, LARGE facilitates the quick identification of significant data points within your dataset. Simply provide the array containing your data and specify the position of the desired value, and LARGE
does the rest by returning the nth largest value to aid your analysis.
Examples 🔗
Suppose you have a dataset of exam scores in cells A1:A10 and you want to find the 3rd largest score. The formula would be: =LARGE(A1:A10, 3)
Suppose you have a range of sales revenue in cells B2:B20 and you need to determine the 5th largest revenue amount. The formula to use: =LARGE(B2:B20, 5)
Ensure that the 'array' argument contains only numerical values. The 'n' argument specifies the position of the desired largest value and must be a positive integer. If 'n' is greater than the number
of values in the array, the function returns a #NUM! error.
Questions 🔗
What happens if the 'n' value in the LARGE function is less than 1?
If the 'n' value in the LARGE function is less than 1, Excel returns a #NUM! error indicating an invalid argument.
Can the LARGE function be used with non-numeric data?
No, the LARGE function is designed to work with arrays or ranges containing numerical values. If non-numeric data is present in the specified array, Excel will return a #NUM! error.
How does the LARGE function handle ties or duplicate values in the dataset?
If there are ties or duplicate values in the dataset, the LARGE function will return the highest occurrence of the nth largest value. Subsequent identical values are not considered when determining
the nth largest value.
Related functions 🔗
Leave a Comment | {"url":"https://spreadsheetcenter.com/excel-functions/large/","timestamp":"2024-11-15T02:58:40Z","content_type":"text/html","content_length":"28521","record_id":"<urn:uuid:7e8fefd9-b5fd-4375-9ce3-448dfea27ed7>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00461.warc.gz"} |
Ryder Cup Opportunity Lost, Rory McIlroy & Europe Weep Over Relationships, New Memories & Time Passed By - Mountain View Golf Club
Relationships in team golf are often elusive, which makes weeks like this one of the best in sport
SHEBOYGAN, Wis. — In 2009, a 19-year-old wunderkind from Northern Ireland, with the kingship of an entire sport laid out in front of him, had some things to say about the biennial Ryder Cup.
“The Ryder Cup is a great spectacle but an exhibition at the end of the day, and it should be there to be enjoyed,” said a baby-faced, underaged Rory McIlroy. “In the big scheme of things, it’s not
that important to me.”
McIlroy eventually went back on those words and proceeded to win three straight Ryder Cups (plus one in Paris in 2018) before losing at Hazeltine in 2016 and then again, this year at Whistling
Straits. He played every session of every event from the moment he said those words until Saturday morning foursomes this year. In the process, he evolved into not only a convert of what he now says
is the best event in golf but perhaps its foremost ambassador.
You have certainly seen the two interviews by now. A broken 32-year-old prince of sport, shortly after dusting Xander Schauffele in Sunday singles, unable to choke out the words he wanted to use to
describe these magical weeks. I was standing right in front of him when it happened, and the entire thing was jarring.
The aroma of victory had begun to waft as the Americans closed in on their historic win, but Rory’s presence quelled whatever elation was felt by folks in those small circles.
The irony of the Ryder Cup is that nearly everyone involved is making money besides the players, and yet, the players — for the most part — wouldn’t trade these experiences for any amount of money
you could feasibly offer.
Why is that? How could that possibly be true? What is it about this week and this event?
There are plenty of reasons you could select, but one stands above the rest: Failing together can be far more meaningful than succeeding alone.
That’s it. That’s the whole thing. For 103 weeks every two years, they are singular. Then for seven days, they are not. Some players might tell you they would rather win on their own than fail with a
group, but buddy, I saw some emotions this week that paint a different picture.
“I can say those two days, those matches with Sergio [Garcia], what it means, the history of the game, an admirer of what Seve [Ballesteros] and Ollie [Jose Maria Olazabal] were able to do, to tee it
up with Sergio; he’s living Ryder Cup history,” said world No. 1, Jon Rahm. “To be able to win those matches with them the way we did it, that is undoubtedly the most fun I’ve had on a golf course by
Rahm, you may remember, just won the U.S. Open three months ago.
This was not an illusory presentation for the sake of appearances in a press conference. We spoke with Rahm more intimately on the 17th hole on Sunday as the week wound down, and he said the exact
same thing. He told us that the feeling of rolling home against the best in the world at Torrey Pines couldn’t hold a candle to what took place this week with Garcia.
And this was at the tail-end of an absolute ass-kicking!
Succeeding alone is vapid. That’s hardwired into our souls. There are too many existential quotes from too many successful, famous folks wondering whether there’s another peak beyond the summit they
just climbed to think anything other than this.
In golf, one of the famous examples is the David Duval story from 2001. He’s flying back from having won the Open Championship when he asks, “Is that all there is?”
Contrast that to what was said in the European team presser after losing in the most lopsided Ryder Cup in the modern era.
“It means a lot to be part of these teams,” said Ian Poulter. “We play a selfish sport week-in, week-out, and when we have this team spirit that we have … We have a good group. These things don’t
come around very often. It’s special to put the shirt on. It’s special. It’s special to get around all these guys in a way that you would not imagine. It means a lot in Europe to represent Europe in
the Ryder Cup, and that’s why it hurts and that’s why you see all the emotion that you see.”
“I spent years trying to make a Ryder Cup team, and I got here this week and didn’t know what to expect,” added rookie Shane Lowry. “I have probably done something that I only could have dreamed of,
like, I won The Open by six shots in my home country, and this week … has been by far the best week of my golfing career. I said to the lads last night, ‘I’m having the time of my life, and we’re six
points behind. What’s it going to be like when we’re leading?'”
Again, this is an Open Championship winner who looked at himself in the mirror before his final round at Portrush in 2019 and asked if he had what it took to win. Then he went out and did it at that
Open on the island where he grew up with his parents and his wife and kids and all his friends looking on. And this guy is calling a 1-2-0 record in a field next to a lake in rural Wisconsin the best
week of his life.
This is not normal! Lowry, too, wept as McIlroy embraced him on Sunday. Two kids who grew up dreaming about Claret Jugs embracing over a collective 2-5-0 mark at Whistling Straits. That’s incredible.
Tears were a theme this week, far more so in this Ryder Cup than others that I’ve covered. Perhaps because this week felt special as the event was delayed because of the pandemic.
U.S. captain Steve Stricker shed them at the opening ceremony, and he bookended the week by breaking down at the closing ceremony when he said, “I never won a major, but this is my major right here.”
Nearly every Euro player cried, too, but let’s get back to the most famous one of them all.
There’s a lesson about the human condition in Rory’s story. When you’re 19, you think jackets and trophies and having enough money to buy a big jet, or a small island are going to fill up your heart.
That’s the story of all of us. As we get older, we learn that the thing we wanted all along is the thing nobody could ever have enough money to purchase: relationship.
Being Rory McIlroy is probably not as fun as it seems. I’m sure (I know) a lot of it is grand, but so much of your time is spent constructing sandcastles by yourself. They might be the best
sandcastles ever built, but they’re still made of sand.
“I don’t think there’s any greater privilege to be part of one of these teams,” McIlroy said Sunday. “It’s an absolute privilege. They’ve always been the greatest experiences of my career. I’ve never
really cried or gotten emotional over what I’ve done as an individual. I couldn’t give a (beep). But this team and what it feels like to be a part of … it’s phenomenal, and I’m so happy to be a part
of it.”
I always think of the Pringles story. I think about it all the time. The loneliness of professional golf at the highest level. That’s not his existence these days, but it’s a microcosm of the world
of pro golf, and specifically his world of pro golf.
Being Rory McIlroy for 12 straight years with hardly a day where you don’t have to be on for somebody or something is probably even more exhausting than it sounds.
Psychoanalyzing McIlroy is one of the most fun parts of this job. Some of us feel as if we’ve built half our career around doing just that.
Not wanting to get this one wrong, I spoke with McIlroy on Monday to ask why he wept so openly and so vulnerably on Sunday afternoon. It couldn’t have been as simple as going 1-3-0 or having one of
the worst birdie percentages of all 24 players or feeling like he let his team down. It seemed like there was something more there, something deeper.
McIlroy said it meant the world to him to be sent out first on Sunday. That’s a big deal to the European team, and though he was initially slated to go 11th, a collection of those on the Euro side
said, Rory McIlroy doesn’t go 11th on Sunday at the Ryder Cup.
As a result, he was desperate not to let them down, especially after such a terrible start to the week and especially after losing to Patrick Reed in 2016 and Justin Thomas in 2018 from that same
leadoff spot.
“I didn’t really sleep much on Saturday night after they put me in that spot,” McIlroy told CBS Sports on Monday.
We also talked about how, when you become a dad, you start thinking more about your own mortality. We admitted crying more since becoming dads than the rest of our lives combined. McIlroy also said
he started thinking more about his own golf mortality (and the golf mortality of his teammates as well).
“I’ve played in six, which means I’ve probably played in the majority of these that I’m going to play in my career,” he said. “The end is not near, but you start thinking about that a little bit
more. The other thing is that this year was meaningful for our team because we knew it was probably the end for some of our older players who have been so great at Ryder Cups.”
McIlroy explained how outsiders don’t necessarily understand just how special these weeks are for not just the men who play in them but for their families as well. How players’ wives revel in getting
to spend that time together in ways they don’t get to in most other weeks. Add it all up, and you get what you saw on Sunday.
“I don’t think anybody believes I don’t care anymore,” he said.
What you probably didn’t see with McIlroy on Sunday was what happened with his own wife shortly after he completed his interviews. He meandered through a small army that stood inside the ropes, still
shook up from what they had just witnessed. Then he found Erica. As he walked toward her, she mouthed the words, ‘I love you,’ and he completely fell apart.
What McIlroy said and how he said it make both him and this event more likable, if that is even possible. After that scene was over, I saw him go to every player and captain on both teams, shake
their hands and say something meaningful and important about the week, something they would remember.
This truly is, like he said, the very best event.
I love the Ryder Cup. That’s no big secret. I love talking about it, thinking about it and experiencing it. I ran around for most of this one with Chris Solomon of No Laying Up, trying to see as many
shots as possible, trying to drink from the three-day firehose of this event, and in-between all of that, hollering at each other about how it should be played every year.
At one point, Rahm told us, “Think about the feelings you guys have in a week like this, now imagine how we feel.”
I believe the Ryder Cup reigns because it reminds us that we were created for something outside of ourselves. Sandcastles will not suffice. There’s a vulnerability in admitting that, and so often
going at it alone for these professionals is the safer, less exposed option, but it is not the better one. That’s why these European teams are so immensely likable. They understand that, and they are
willing to openly weep, not over losses but over the loss of that wisp of time.
The Ryder Cup, like the Masters for me, also represents a passage of time. Two years until the next one. Four years until the next one in the United States. So many different things will happen in
the lives of these players and those of us covering them in that span of time. Marriages will begin, babies will be born, family members will pass away.
Then we will gather again in Rome and then at Bethpage and do this whole thing over again. Insane moments will happen. A complete theater of the absurd. There’s a through line, though. And that
through line is that we will gather again.
C.S. Lewis once said art and culture (and I will add sport) are all mortal entities. They will pass away. The grass withers, the flowers fade. What lasts is relationship.
That’s too rarely highlighted in golf, and then it gets magnified and clarified during this week we call the Ryder Cup, and it is beautiful.
This event, too, will not last forever but damn, after weeks like this one, I certainly wish it could.
By Kyle Porter | CBS Sports
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Fixed-Point Theorem - (Advanced Matrix Computations) - Vocab, Definition, Explanations | Fiveable
Fixed-Point Theorem
from class:
Advanced Matrix Computations
The fixed-point theorem states that under certain conditions, a function will have at least one point at which the value of the function is equal to the value of that point. This concept is crucial
in iterative methods, where an approximate solution to a problem is refined through repeated applications of a function, particularly for sparse linear systems. The theorem ensures that these
iterative processes converge to a solution, making it foundational in numerical analysis and computational mathematics.
congrats on reading the definition of Fixed-Point Theorem. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Fixed-point theorems are essential in proving the existence of solutions in various mathematical problems, particularly in solving equations iteratively.
2. The most common application of the fixed-point theorem in sparse linear systems is in the context of methods like Jacobi and Gauss-Seidel iterations.
3. For the fixed-point theorem to guarantee convergence, the function must satisfy specific properties such as continuity and contractiveness within a certain domain.
4. The fixed-point theorem is closely linked to the concept of contraction mappings, where the distance between successive iterations decreases.
5. Understanding fixed-point theorems helps in analyzing the stability and efficiency of iterative methods when solving large sparse systems.
Review Questions
• How does the fixed-point theorem apply to iterative methods in solving sparse linear systems?
□ The fixed-point theorem provides a theoretical foundation for iterative methods by ensuring that these methods will converge to a solution under certain conditions. In the context of sparse
linear systems, techniques like Jacobi and Gauss-Seidel rely on applying functions repeatedly until the approximations stabilize at a fixed point. This convergence is crucial for ensuring
that these methods yield accurate solutions efficiently, particularly when dealing with large and sparse matrices.
• What conditions must be satisfied for the fixed-point theorem to guarantee convergence in iterative methods?
□ For the fixed-point theorem to ensure convergence in iterative methods, the function used must be continuous and must often be a contraction mapping. This means that there exists a constant
less than one such that the distance between the function's outputs decreases with each iteration. Additionally, the initial guess must be chosen appropriately within a certain range so that
the sequence remains bounded and approaches the fixed point. These conditions are vital for reliable and efficient computations.
• Evaluate how understanding fixed-point theorems enhances the effectiveness of numerical methods for large sparse systems.
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iterative method converges to a solution helps practitioners select appropriate algorithms and initial conditions. Furthermore, this knowledge allows for optimizations in computational
resources by focusing on methods guaranteed to yield results effectively. Ultimately, leveraging fixed-point theorems enables more accurate and faster solutions in real-world applications
where large matrices are prevalent.
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Central limit theorem - codefinance.trainingCentral limit theorem - codefinance.training
Central limit theorem
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal
distribution. This holds even if the original variables themselves are not normally distributed. There are several versions of the CLT, each applying in the context of different conditions.
The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other
types of distributions.
This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern form it was only precisely stated as late
as 1920.^[1]
In statistics, the CLT can be stated as: let denote a statistical sample of size from a population with expected value (average) and finite positive variance , and let denote the sample mean (which
is itself a random variable). Then the limit as of the distribution of is a normal distribution with mean and variance .^[2]
In other words, suppose that a large sample of observations is obtained, each observation being randomly produced in a way that does not depend on the values of the other observations, and that the
average (arithmetic mean) of the observed values is computed. If this procedure is performed many times, resulting in a collection of observed averages, the central limit theorem says that if the
sample size was large enough, the probability distribution of these averages will closely approximate a normal distribution.
The central limit theorem has several variants. In its common form, the random variables must be independent and identically distributed (i.i.d.). This requirement can be weakened; convergence of the
mean to the normal distribution also occurs for non-identical distributions or for non-independent observations if they comply with certain conditions.
The earliest version of this theorem, that the normal distribution may be used as an approximation to the binomial distribution, is the de Moivre–Laplace theorem.
Independent sequences
Whatever the form of the population distribution, the sampling distribution tends to a Gaussian, and its dispersion is given by the central limit theorem.^[3]
Classical CLT
Let be a sequence of i.i.d. random variables having a distribution with expected value given by and finite variance given by Suppose we are interested in the sample average
By the law of large numbers, the sample average converges almost surely (and therefore also converges in probability) to the expected value as
The classical central limit theorem describes the size and the distributional form of the stochastic fluctuations around the deterministic number during this convergence. More precisely, it states
that as gets larger, the distribution of the difference between the sample average and its limit when multiplied by the factor — that is, — approaches the normal distribution with mean and variance
For large enough the distribution of gets arbitrarily close to the normal distribution with mean and variance
The usefulness of the theorem is that the distribution of approaches normality regardless of the shape of the distribution of the individual Formally, the theorem can be stated as follows:
In the case convergence in distribution means that the cumulative distribution functions of converge pointwise to the cdf of the distribution: for every real number where is the standard normal cdf
evaluated at The convergence is uniform in in the sense that where denotes the least upper bound (or supremum) of the set.^[5]
Lyapunov CLT
In this variant of the central limit theorem the random variables have to be independent, but not necessarily identically distributed. The theorem also requires that random variables have moments of
some order , and that the rate of growth of these moments is limited by the Lyapunov condition given below.
Lyapunov CLT^[6] — Suppose is a sequence of independent random variables, each with finite expected value and variance . Define
If for some , Lyapunov’s condition is satisfied, then a sum of converges in distribution to a standard normal random variable, as goes to infinity:
In practice it is usually easiest to check Lyapunov's condition for .
If a sequence of random variables satisfies Lyapunov's condition, then it also satisfies Lindeberg's condition. The converse implication, however, does not hold.
Lindeberg (-Feller) CLT
In the same setting and with the same notation as above, the Lyapunov condition can be replaced with the following weaker one (from Lindeberg in 1920).
Suppose that for every where is the indicator function. Then the distribution of the standardized sums converges towards the standard normal distribution .
Multidimensional CLT
Proofs that use characteristic functions can be extended to cases where each individual is a random vector in , with mean vector and covariance matrix (among the components of the vector), and these
random vectors are independent and identically distributed. The multidimensional central limit theorem states that when scaled, sums converge to a multivariate normal distribution.^[7] Summation of
these vectors is done component-wise.
For let be independent random vectors. The sum of the random vectors is and their average is Therefore,
The multivariate central limit theorem states that where the covariance matrix is equal to
The multivariate central limit theorem can be proved using the Cramér–Wold theorem.^[7]
The rate of convergence is given by the following Berry–Esseen type result:
Theorem^[8] — Let be independent -valued random vectors, each having mean zero. Write and assume is invertible. Let be a -dimensional Gaussian with the same mean and same covariance matrix as . Then
for all convex sets , where is a universal constant, , and denotes the Euclidean norm on .
It is unknown whether the factor is necessary.^[9]
The Generalized Central Limit Theorem
The Generalized Central Limit Theorem (GCLT) was an effort of multiple mathematicians (Bernstein, Lindeberg, Lévy, Feller, Kolmogorov, and others) over the period from 1920 to 1937.^[10] The first
published complete proof of the GCLT was in 1937 by Paul Lévy in French.^[11] An English language version of the complete proof of the GCLT is available in the translation of Gnedenko and Kolmogorov
's 1954 book.^[12]
The statement of the GCLT is as follows:^[13]
A non-degenerate random variable Z is α-stable for some 0 < α ≤ 2 if and only if there is an independent, identically distributed sequence of random variables X[1], X[2], X[3], ... and constants
a[n] > 0, b[n] ∈ ℝ with
a[n] (X[1] + ... + X[n]) − b[n] → Z.
Here → means the sequence of random variable sums converges in distribution; i.e., the corresponding distributions satisfy F[n](y) → F(y) at all continuity points of F.
In other words, if sums of independent, identically distributed random variables converge in distribution to some Z, then Z must be a stable distribution.
Dependent processes
CLT under weak dependence
A useful generalization of a sequence of independent, identically distributed random variables is a mixing random process in discrete time; "mixing" means, roughly, that random variables temporally
far apart from one another are nearly independent. Several kinds of mixing are used in ergodic theory and probability theory. See especially strong mixing (also called α-mixing) defined by where is
so-called strong mixing coefficient.
A simplified formulation of the central limit theorem under strong mixing is:^[14]
Theorem — Suppose that is stationary and -mixing with and that and . Denote , then the limit exists, and if then converges in distribution to .
In fact, where the series converges absolutely.
The assumption cannot be omitted, since the asymptotic normality fails for where are another stationary sequence.
There is a stronger version of the theorem:^[15] the assumption is replaced with , and the assumption is replaced with
Existence of such ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see (Bradley 2007).
Martingale difference CLT
Theorem — Let a martingale satisfy
• in probability as n → ∞,
• for every ε > 0, as n → ∞,
then converges in distribution to as .^[16]^[17]
Proof of classical CLT
The central limit theorem has a proof using characteristic functions.^[18] It is similar to the proof of the (weak) law of large numbers.
Assume are independent and identically distributed random variables, each with mean and finite variance . The sum has mean and variance . Consider the random variable where in the last step we
defined the new random variables , each with zero mean and unit variance (). The characteristic function of is given by where in the last step we used the fact that all of the are identically
distributed. The characteristic function of is, by Taylor's theorem, where is "little o notation" for some function of that goes to zero more rapidly than . By the limit of the exponential function (
), the characteristic function of equals
All of the higher order terms vanish in the limit . The right hand side equals the characteristic function of a standard normal distribution , which implies through Lévy's continuity theorem that the
distribution of will approach as . Therefore, the sample average is such that converges to the normal distribution , from which the central limit theorem follows.
Convergence to the limit
The central limit theorem gives only an asymptotic distribution. As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the
normal distribution; it requires a very large number of observations to stretch into the tails.
The convergence in the central limit theorem is uniform because the limiting cumulative distribution function is continuous. If the third central moment exists and is finite, then the speed of
convergence is at least on the order of (see Berry–Esseen theorem). Stein's method^[19] can be used not only to prove the central limit theorem, but also to provide bounds on the rates of convergence
for selected metrics.^[20]
The convergence to the normal distribution is monotonic, in the sense that the entropy of increases monotonically to that of the normal distribution.^[21]
The central limit theorem applies in particular to sums of independent and identically distributed discrete random variables. A sum of discrete random variables is still a discrete random variable,
so that we are confronted with a sequence of discrete random variables whose cumulative probability distribution function converges towards a cumulative probability distribution function
corresponding to a continuous variable (namely that of the normal distribution). This means that if we build a histogram of the realizations of the sum of n independent identical discrete variables,
the piecewise-linear curve that joins the centers of the upper faces of the rectangles forming the histogram converges toward a Gaussian curve as n approaches infinity; this relation is known as de
Moivre–Laplace theorem. The binomial distribution article details such an application of the central limit theorem in the simple case of a discrete variable taking only two possible values.
Common misconceptions
Studies have shown that the central limit theorem is subject to several common but serious misconceptions, some of which appear in widely used textbooks.^[22]^[23]^[24] These include:
• The misconceived belief that the theorem applies to random sampling of any variable, rather than to the mean values (or sums) of iid random variables extracted from a population by repeated
sampling. That is, the theorem assumes the random sampling produces a sampling distribution formed from different values of means (or sums) of such random variables.
• The misconceived belief that the theorem ensures that random sampling leads to the emergence of a normal distribution for sufficiently large samples of any random variable, regardless of the
population distribution. In reality, such sampling asymptotically reproduces the properties of the population, an intuitive result underpinned by the Glivenko-Cantelli theorem.
• The misconceived belief that the theorem leads to a good approximation of a normal distribution for sample sizes greater than around 30,^[25] allowing reliable inferences regardless of the nature
of the population. In reality, this empirical rule of thumb has no valid justification, and can lead to seriously flawed inferences. See Z-test for where the approximation holds.
Relation to the law of large numbers
The law of large numbers as well as the central limit theorem are partial solutions to a general problem: "What is the limiting behavior of S[n] as n approaches infinity?" In mathematical analysis,
asymptotic series are one of the most popular tools employed to approach such questions.
Suppose we have an asymptotic expansion of :
Dividing both parts by φ[1](n) and taking the limit will produce a[1], the coefficient of the highest-order term in the expansion, which represents the rate at which f(n) changes in its leading term.
Informally, one can say: "f(n) grows approximately as a[1]φ[1](n)". Taking the difference between f(n) and its approximation and then dividing by the next term in the expansion, we arrive at a more
refined statement about f(n):
Here one can say that the difference between the function and its approximation grows approximately as a[2]φ[2](n). The idea is that dividing the function by appropriate normalizing functions, and
looking at the limiting behavior of the result, can tell us much about the limiting behavior of the original function itself.
Informally, something along these lines happens when the sum, S[n], of independent identically distributed random variables, X[1], ..., X[n], is studied in classical probability theory. If each X[i]
has finite mean μ, then by the law of large numbers, S[n]/n → μ.^[26] If in addition each X[i] has finite variance σ^2, then by the central limit theorem, where ξ is distributed as N(0,σ^2). This
provides values of the first two constants in the informal expansion
In the case where the X[i] do not have finite mean or variance, convergence of the shifted and rescaled sum can also occur with different centering and scaling factors: or informally
Distributions Ξ which can arise in this way are called stable.^[27] Clearly, the normal distribution is stable, but there are also other stable distributions, such as the Cauchy distribution, for
which the mean or variance are not defined. The scaling factor b[n] may be proportional to n^c, for any c ≥ 1/2; it may also be multiplied by a slowly varying function of n.^[28]^[29]
The law of the iterated logarithm specifies what is happening "in between" the law of large numbers and the central limit theorem. Specifically it says that the normalizing function √n log log n,
intermediate in size between n of the law of large numbers and √n of the central limit theorem, provides a non-trivial limiting behavior.
Alternative statements of the theorem
Density functions
The density of the sum of two or more independent variables is the convolution of their densities (if these densities exist). Thus the central limit theorem can be interpreted as a statement about
the properties of density functions under convolution: the convolution of a number of density functions tends to the normal density as the number of density functions increases without bound. These
theorems require stronger hypotheses than the forms of the central limit theorem given above. Theorems of this type are often called local limit theorems. See Petrov^[30] for a particular local limit
theorem for sums of independent and identically distributed random variables.
Characteristic functions
Since the characteristic function of a convolution is the product of the characteristic functions of the densities involved, the central limit theorem has yet another restatement: the product of the
characteristic functions of a number of density functions becomes close to the characteristic function of the normal density as the number of density functions increases without bound, under the
conditions stated above. Specifically, an appropriate scaling factor needs to be applied to the argument of the characteristic function.
An equivalent statement can be made about Fourier transforms, since the characteristic function is essentially a Fourier transform.
Calculating the variance
Let S[n] be the sum of n random variables. Many central limit theorems provide conditions such that S[n]/√Var(S[n]) converges in distribution to N(0,1) (the normal distribution with mean 0, variance
1) as n → ∞. In some cases, it is possible to find a constant σ^2 and function f(n) such that S[n]/(σ√n⋅f(n)) converges in distribution to N(0,1) as n→ ∞.
Lemma^[31] — Suppose is a sequence of real-valued and strictly stationary random variables with for all , , and . Construct
1. If is absolutely convergent, , and then as where .
2. If in addition and converges in distribution to as then also converges in distribution to as .
Products of positive random variables
The logarithm of a product is simply the sum of the logarithms of the factors. Therefore, when the logarithm of a product of random variables that take only positive values approaches a normal
distribution, the product itself approaches a log-normal distribution. Many physical quantities (especially mass or length, which are a matter of scale and cannot be negative) are the products of
different random factors, so they follow a log-normal distribution. This multiplicative version of the central limit theorem is sometimes called Gibrat's law.
Whereas the central limit theorem for sums of random variables requires the condition of finite variance, the corresponding theorem for products requires the corresponding condition that the density
function be square-integrable.^[32]
Beyond the classical framework
Asymptotic normality, that is, convergence to the normal distribution after appropriate shift and rescaling, is a phenomenon much more general than the classical framework treated above, namely, sums
of independent random variables (or vectors). New frameworks are revealed from time to time; no single unifying framework is available for now.
Convex body
Theorem — There exists a sequence ε[n] ↓ 0 for which the following holds. Let n ≥ 1, and let random variables X[1], ..., X[n] have a log-concave joint density f such that f(x[1], ..., x[n]) = f(|x[1]
|, ..., |x[n]|) for all x[1], ..., x[n], and E(X^2
[k]) = 1 for all k = 1, ..., n. Then the distribution of is ε[n]-close to in the total variation distance.^[33]
These two ε[n]-close distributions have densities (in fact, log-concave densities), thus, the total variance distance between them is the integral of the absolute value of the difference between the
densities. Convergence in total variation is stronger than weak convergence.
An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains
the term "central limit theorem for convex bodies".
Another example: f(x[1], ..., x[n]) = const · exp(−(|x[1]|^α + ⋯ + |x[n]|^α)^β) where α > 1 and αβ > 1. If β = 1 then f(x[1], ..., x[n]) factorizes into const · exp (−|x[1]|^α) … exp(−|x[n]|^α),
which means X[1], ..., X[n] are independent. In general, however, they are dependent.
The condition f(x[1], ..., x[n]) = f(|x[1]|, ..., |x[n]|) ensures that X[1], ..., X[n] are of zero mean and uncorrelated; still, they need not be independent, nor even pairwise independent. By the
way, pairwise independence cannot replace independence in the classical central limit theorem.^[34]
Here is a Berry–Esseen type result.
Theorem — Let X[1], ..., X[n] satisfy the assumptions of the previous theorem, then ^[35] for all a < b; here C is a universal (absolute) constant. Moreover, for every c[1], ..., c[n] ∈ R such that c
[1] + ⋯ + c^2
[n] = 1,
The distribution of X[1] + ⋯ + X[n]/√n need not be approximately normal (in fact, it can be uniform).^[36] However, the distribution of c[1]X[1] + ⋯ + c[n]X[n] is close to (in the total variation
distance) for most vectors (c[1], ..., c[n]) according to the uniform distribution on the sphere c^2
[1] + ⋯ + c^2
[n] = 1.
Lacunary trigonometric series
Theorem (Salem–Zygmund) — Let U be a random variable distributed uniformly on (0,2π), and X[k] = r[k] cos(n[k]U + a[k]), where
• n[k] satisfy the lacunarity condition: there exists q > 1 such that n[k + 1] ≥ qn[k] for all k,
• r[k] are such that
• 0 ≤ a[k] < 2π.
Then^[37]^[38] converges in distribution to .
Gaussian polytopes
Theorem — Let A[1], ..., A[n] be independent random points on the plane R^2 each having the two-dimensional standard normal distribution. Let K[n] be the convex hull of these points, and X[n] the
area of K[n] Then^[39] converges in distribution to as n tends to infinity.
The same also holds in all dimensions greater than 2.
The polytope K[n] is called a Gaussian random polytope.
A similar result holds for the number of vertices (of the Gaussian polytope), the number of edges, and in fact, faces of all dimensions.^[40]
Linear functions of orthogonal matrices
A linear function of a matrix M is a linear combination of its elements (with given coefficients), M ↦ tr(AM) where A is the matrix of the coefficients; see Trace (linear algebra)#Inner product.
A random orthogonal matrix is said to be distributed uniformly, if its distribution is the normalized Haar measure on the orthogonal group O(n,R); see Rotation matrix#Uniform random rotation matrices
Theorem — Let M be a random orthogonal n × n matrix distributed uniformly, and A a fixed n × n matrix such that tr(AA*) = n, and let X = tr(AM). Then^[41] the distribution of X is close to in the
total variation metric up to2√3/n − 1.
Theorem — Let random variables X[1], X[2], ... ∈ L[2](Ω) be such that X[n] → 0 weakly in L[2](Ω) and X^
[n] → 1 weakly in L[1](Ω). Then there exist integers n[1] < n[2] < ⋯ such that converges in distribution to as k tends to infinity.^[42]
Random walk on a crystal lattice
The central limit theorem may be established for the simple random walk on a crystal lattice (an infinite-fold abelian covering graph over a finite graph), and is used for design of crystal
Applications and examples
A simple example of the central limit theorem is rolling many identical, unbiased dice. The distribution of the sum (or average) of the rolled numbers will be well approximated by a normal
distribution. Since real-world quantities are often the balanced sum of many unobserved random events, the central limit theorem also provides a partial explanation for the prevalence of the normal
probability distribution. It also justifies the approximation of large-sample statistics to the normal distribution in controlled experiments.
Another simulation using the binomial distribution. Random 0s and 1s were generated, and then their means calculated for sample sizes ranging from 1 to 2048. Note that as the sample size increases
the tails become thinner and the distribution becomes more concentrated around the mean.
Regression analysis, and in particular ordinary least squares, specifies that a dependent variable depends according to some function upon one or more independent variables, with an additive error
term. Various types of statistical inference on the regression assume that the error term is normally distributed. This assumption can be justified by assuming that the error term is actually the sum
of many independent error terms; even if the individual error terms are not normally distributed, by the central limit theorem their sum can be well approximated by a normal distribution.
Other illustrations
Given its importance to statistics, a number of papers and computer packages are available that demonstrate the convergence involved in the central limit theorem.^[45]
Dutch mathematician Henk Tijms writes:^[46]
The central limit theorem has an interesting history. The first version of this theorem was postulated by the French-born mathematician Abraham de Moivre who, in a remarkable article published in
1733, used the normal distribution to approximate the distribution of the number of heads resulting from many tosses of a fair coin. This finding was far ahead of its time, and was nearly
forgotten until the famous French mathematician Pierre-Simon Laplace rescued it from obscurity in his monumental work Théorie analytique des probabilités, which was published in 1812. Laplace
expanded De Moivre's finding by approximating the binomial distribution with the normal distribution. But as with De Moivre, Laplace's finding received little attention in his own time. It was
not until the nineteenth century was at an end that the importance of the central limit theorem was discerned, when, in 1901, Russian mathematician Aleksandr Lyapunov defined it in general terms
and proved precisely how it worked mathematically. Nowadays, the central limit theorem is considered to be the unofficial sovereign of probability theory.
Sir Francis Galton described the Central Limit Theorem in this way:^[47]
I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the "Law of Frequency of Error". The law would have been personified by the Greeks
and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more
perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most
beautiful form of regularity proves to have been latent all along.
The actual term "central limit theorem" (in German: "zentraler Grenzwertsatz") was first used by George Pólya in 1920 in the title of a paper.^[48]^[49] Pólya referred to the theorem as "central" due
to its importance in probability theory. According to Le Cam, the French school of probability interprets the word central in the sense that "it describes the behaviour of the centre of the
distribution as opposed to its tails".^[49] The abstract of the paper On the central limit theorem of calculus of probability and the problem of moments by Pólya^[48] in 1920 translates as follows.
The occurrence of the Gaussian probability density 1 = e^−x^2 in repeated experiments, in errors of measurements, which result in the combination of very many and very small elementary errors, in
diffusion processes etc., can be explained, as is well-known, by the very same limit theorem, which plays a central role in the calculus of probability. The actual discoverer of this limit
theorem is to be named Laplace; it is likely that its rigorous proof was first given by Tschebyscheff and its sharpest formulation can be found, as far as I am aware of, in an article by
Liapounoff. ...
A thorough account of the theorem's history, detailing Laplace's foundational work, as well as Cauchy's, Bessel's and Poisson's contributions, is provided by Hald.^[50] Two historical accounts, one
covering the development from Laplace to Cauchy, the second the contributions by von Mises, Pólya, Lindeberg, Lévy, and Cramér during the 1920s, are given by Hans Fischer.^[51] Le Cam describes a
period around 1935.^[49] Bernstein^[52] presents a historical discussion focusing on the work of Pafnuty Chebyshev and his students Andrey Markov and Aleksandr Lyapunov that led to the first proofs
of the CLT in a general setting.
A curious footnote to the history of the Central Limit Theorem is that a proof of a result similar to the 1922 Lindeberg CLT was the subject of Alan Turing's 1934 Fellowship Dissertation for King's
College at the University of Cambridge. Only after submitting the work did Turing learn it had already been proved. Consequently, Turing's dissertation was not published.^[53]
See also
1. ^ Fischer (2011), p. .
2. ^ Montgomery, Douglas C.; Runger, George C. (2014). Applied Statistics and Probability for Engineers (6th ed.). Wiley. p. 241. ISBN 9781118539712.
3. ^ Rouaud, Mathieu (2013). Probability, Statistics and Estimation (PDF). p. 10. Archived (PDF) from the original on 2022-10-09.
4. ^ Billingsley (1995), p. 357.
5. ^ Bauer (2001), p. 199, Theorem 30.13.
6. ^ Billingsley (1995), p. 362.
7. ^ ^a ^b van der Vaart, A.W. (1998). Asymptotic statistics. New York, NY: Cambridge University Press. ISBN 978-0-521-49603-2. LCCN 98015176.
8. ^ O’Donnell, Ryan (2014). "Theorem 5.38". Archived from the original on 2019-04-08. Retrieved 2017-10-18.
9. ^ Bentkus, V. (2005). "A Lyapunov-type bound in ". Theory Probab. Appl. 49 (2): 311–323. doi:10.1137/S0040585X97981123.
10. ^ Le Cam, L. (February 1986). "The Central Limit Theorem around 1935". Statistical Science. 1 (1): 78–91. JSTOR 2245503.
11. ^ Lévy, Paul (1937). Theorie de l'addition des variables aleatoires [Combination theory of unpredictable variables]. Paris: Gauthier-Villars.
12. ^ Gnedenko, Boris Vladimirovich; Kologorov, Andreĭ Nikolaevich; Doob, Joseph L.; Hsu, Pao-Lu (1968). Limit distributions for sums of independent random variables. Reading, MA: Addison-wesley.
13. ^ Nolan, John P. (2020). Univariate stable distributions, Models for Heavy Tailed Data. Springer Series in Operations Research and Financial Engineering. Switzerland: Springer. doi:10.1007/
978-3-030-52915-4. ISBN 978-3-030-52914-7. S2CID 226648987.
14. ^ Billingsley (1995), Theorem 27.4.
15. ^ Durrett (2004), Sect. 7.7(c), Theorem 7.8.
16. ^ Durrett (2004), Sect. 7.7, Theorem 7.4.
17. ^ Billingsley (1995), Theorem 35.12.
18. ^ Lemons, Don (2003). An Introduction to Stochastic Processes in Physics. Johns Hopkins University Press. doi:10.56021/9780801868665. ISBN 9780801876387. Retrieved 2016-08-11.
19. ^ Stein, C. (1972). "A bound for the error in the normal approximation to the distribution of a sum of dependent random variables". Proceedings of the Sixth Berkeley Symposium on Mathematical
Statistics and Probability. 6 (2): 583–602. MR 0402873. Zbl 0278.60026.
20. ^ Chen, L. H. Y.; Goldstein, L.; Shao, Q. M. (2011). Normal approximation by Stein's method. Springer. ISBN 978-3-642-15006-7.
21. ^ Artstein, S.; Ball, K.; Barthe, F.; Naor, A. (2004). "Solution of Shannon's Problem on the Monotonicity of Entropy". Journal of the American Mathematical Society. 17 (4): 975–982. doi:10.1090/
22. ^ Brewer, J.K. (1985). "Behavioral statistics textbooks: Source of myths and misconceptions?". Journal of Educational Statistics. 10 (3): 252–268. doi:10.3102/10769986010003252. S2CID 119611584.
23. ^ Yu, C.; Behrens, J.; Spencer, A. Identification of Misconception in the Central Limit Theorem and Related Concepts, American Educational Research Association lecture 19 April 1995
24. ^ Sotos, A.E.C.; Vanhoof, S.; Van den Noortgate, W.; Onghena, P. (2007). "Students' misconceptions of statistical inference: A review of the empirical evidence from research on statistics
education". Educational Research Review. 2 (2): 98–113. doi:10.1016/j.edurev.2007.04.001.
25. ^ "Sampling distribution of the sample mean (video) | Khan Academy". 2 June 2023. Archived from the original on 2023-06-02. Retrieved 2023-10-08.
26. ^ Rosenthal, Jeffrey Seth (2000). A First Look at Rigorous Probability Theory. World Scientific. Theorem 5.3.4, p. 47. ISBN 981-02-4322-7.
27. ^ Johnson, Oliver Thomas (2004). Information Theory and the Central Limit Theorem. Imperial College Press. p. 88. ISBN 1-86094-473-6.
28. ^ Uchaikin, Vladimir V.; Zolotarev, V.M. (1999). Chance and Stability: Stable distributions and their applications. VSP. pp. 61–62. ISBN 90-6764-301-7.
29. ^ Borodin, A. N.; Ibragimov, I. A.; Sudakov, V. N. (1995). Limit Theorems for Functionals of Random Walks. AMS Bookstore. Theorem 1.1, p. 8. ISBN 0-8218-0438-3.
30. ^ Petrov, V. V. (1976). Sums of Independent Random Variables. New York-Heidelberg: Springer-Verlag. ch. 7. ISBN 9783642658099.
31. ^ Hew, Patrick Chisan (2017). "Asymptotic distribution of rewards accumulated by alternating renewal processes". Statistics and Probability Letters. 129: 355–359. doi:10.1016/j.spl.2017.06.027.
32. ^ Rempala, G.; Wesolowski, J. (2002). "Asymptotics of products of sums and U-statistics" (PDF). Electronic Communications in Probability. 7: 47–54. doi:10.1214/ecp.v7-1046.
33. ^ Klartag (2007), Theorem 1.2.
34. ^ Durrett (2004), Section 2.4, Example 4.5.
35. ^ Klartag (2008), Theorem 1.
36. ^ Klartag (2007), Theorem 1.1.
37. ^ Zygmund, Antoni (2003) [1959]. Trigonometric Series. Cambridge University Press. vol. II, sect. XVI.5, Theorem 5-5. ISBN 0-521-89053-5.
38. ^ Gaposhkin (1966), Theorem 2.1.13.
39. ^ Bárány & Vu (2007), Theorem 1.1.
40. ^ Bárány & Vu (2007), Theorem 1.2.
41. ^ Meckes, Elizabeth (2008). "Linear functions on the classical matrix groups". Transactions of the American Mathematical Society. 360 (10): 5355–5366. arXiv:math/0509441. doi:10.1090/
S0002-9947-08-04444-9. S2CID 11981408.
42. ^ Gaposhkin (1966), Sect. 1.5.
43. ^ Kotani, M.; Sunada, Toshikazu (2003). Spectral geometry of crystal lattices. Vol. 338. Contemporary Math. pp. 271–305. ISBN 978-0-8218-4269-0.
44. ^ Sunada, Toshikazu (2012). Topological Crystallography – With a View Towards Discrete Geometric Analysis. Surveys and Tutorials in the Applied Mathematical Sciences. Vol. 6. Springer. ISBN
45. ^ Marasinghe, M.; Meeker, W.; Cook, D.; Shin, T. S. (August 1994). Using graphics and simulation to teach statistical concepts. Annual meeting of the American Statistician Association, Toronto,
46. ^ Henk, Tijms (2004). Understanding Probability: Chance Rules in Everyday Life. Cambridge: Cambridge University Press. p. 169. ISBN 0-521-54036-4.
47. ^ Galton, F. (1889). Natural Inheritance. p. 66.
48. ^ ^a ^b Pólya, George (1920). "Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Momentenproblem" [On the central limit theorem of probability calculation and the problem
of moments]. Mathematische Zeitschrift (in German). 8 (3–4): 171–181. doi:10.1007/BF01206525. S2CID 123063388.
49. ^ ^a ^b ^c Le Cam, Lucien (1986). "The central limit theorem around 1935". Statistical Science. 1 (1): 78–91. doi:10.1214/ss/1177013818.
50. ^ Hald, Andreas (22 April 1998). A History of Mathematical Statistics from 1750 to 1930 (PDF). Wiley. chapter 17. ISBN 978-0471179122. Archived (PDF) from the original on 2022-10-09.
51. ^ Fischer (2011), Chapter 2; Chapter 5.2.
52. ^ Bernstein, S. N. (1945). "On the work of P. L. Chebyshev in Probability Theory". In Bernstein., S. N. (ed.). Nauchnoe Nasledie P. L. Chebysheva. Vypusk Pervyi: Matematika [The Scientific Legacy
of P. L. Chebyshev. Part I: Mathematics] (in Russian). Moscow & Leningrad: Academiya Nauk SSSR. p. 174.
53. ^ Zabell, S. L. (1995). "Alan Turing and the Central Limit Theorem". American Mathematical Monthly. 102 (6): 483–494. doi:10.1080/00029890.1995.12004608.
54. ^ Jørgensen, Bent (1997). The Theory of Dispersion Models. Chapman & Hall. ISBN 978-0412997112.
External links | {"url":"https://codefinance.training/programming-topic/mathematical-methods/central-limit-theorem/","timestamp":"2024-11-04T17:44:31Z","content_type":"text/html","content_length":"446167","record_id":"<urn:uuid:ee13a637-896f-4711-84d1-d916cc24beba>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00811.warc.gz"} |
166667 Hours to Minutes
166667 hr to min conversion result above is displayed in three different forms: as a decimal (which could be rounded), in scientific notation (scientific form, standard index form or standard form in
the United Kingdom) and as a fraction (exact result). Every display form has its own advantages and in different situations particular form is more convenient than another. For example usage of
scientific notation when working with big numbers is recommended due to easier reading and comprehension. Usage of fractions is recommended when more precision is needed.
If we want to calculate how many Minutes are 166667 Hours we have to multiply 166667 by 60 and divide the product by 1. So for 166667 we have: (166667 × 60) ÷ 1 = 10000020 ÷ 1 = 10000020 Minutes
So finally 166667 hr = 10000020 min | {"url":"https://unitchefs.com/hours/minutes/166667/","timestamp":"2024-11-07T06:36:52Z","content_type":"text/html","content_length":"22357","record_id":"<urn:uuid:dc4a1f3e-6ee2-4fab-beea-3f2960b0c0de>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00502.warc.gz"} |
e P
IEP Math Goals for Operations in the Primary Grades
How to make a calculator in MIT App Inventor 2 - YouTube
Also learn the facts to easily understand math glossary with fun math worksheet Example. 5+4 = 9. 5-4 =1. 6-3 = 3. 6+3 =9. 2 × 3 =6.
Or, does this just seem to be the historical standard or traditional way to perform equations based on history or You will also explore some of the laws that govern these operations and use
mathematical models to reinforce the algorithms being discussed. See Note 1 below. Dec 31, 2019 Here's a quick recap. The four basic Mathematical operations are Addition, Subtraction, Multiplication
and Division. This video reacquaints us to Operations are things like addition, subtraction, multiplication, and division. When you The order of operations is like grammar rules for the language of
math. This free poster showing the four operations (addition, subtraction, multiplication and division) and their associated vocabula… More Worksheets to practice order of operations with solutions,
how to perform mixed operations with addition, subtraction, multiplication 4 × 3 + 6 ÷ 2 = Try the free Mathway calculator and problem solver below to practice various mat Grade 4.
means (a/b)/cand nota/(b/c). (a+b).
SOM 306 syll - CALIFORNIA STATE UNIVERSITY
308–354). Reston, VA: National Council of Some of the worksheets for this concept are Operations in scienti c notation, Writing scienti c notation, Mscc7 ws 10a, Big ideas math answers grade 8
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Huvudsida - math.chalmers.se
Do you know the formula for the volume of a sphere? för 3 dagar sedan — 4) Ta vinst på övre bandet. ekmanin posath wenna thamai dagalanne. dan salli walata huruwela ewarai. kove dammath makala
hadanna ba.
4. D ivide-- 2/3 5. A ddition--- 2+3.
Rotary performance sverige
2) Write both original fractions as equivalent fractions with the least common denominator.
Mixing word problems encourages students to read and think about the questions, rather than simply recognizing a pattern to the solutions. Below are six versions of our grade 4 math worksheet with
word problems involving the 4 basic operations: addition, subtraction, multiplication and division. Some questions will have more than one step. 2014-03-21 · math page.
Vad finns det för likheter och skillnader mellan de olika perspektiven
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These key words are helpful for students to use as a reference when completing story/word problems for math. Younger grades will appreciat. Math operators take the values to their left and right and
perform a math operation on them. The four most common operators add, subtract, multiply, or divide values. The addition and subtraction operators look just like they do in most elementary school
math textbooks. Se hela listan på mathsteacher.com.au 2021-03-10 · According to order of operations, you solve whatever is in the parentheses first. | {"url":"https://jobbxcis.web.app/3717/60489.html","timestamp":"2024-11-14T18:06:25Z","content_type":"text/html","content_length":"8784","record_id":"<urn:uuid:51b7c87e-9103-433a-9090-ef66970decfa>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00431.warc.gz"} |
ISLR Chapter 3 - Linear Regression | Bijen Patel
Simple Linear Regression
Simple linear regression assumes a linear relationship between the predictor (\( X \)) and the response (\( Y \)). A simple linear regression model takes the following form:
\[ \hat{y} = \beta_{0}+\beta_{1}(X) \]
• \( \hat{y} \) - represents the predicted value
• \( \beta_{0} \) - represents a coefficient known as the intercept
• \( \beta_{1} \) - represents a coefficient known as the slope
• \( X \) - represents the value of the predictor
For example, we could build a simple linear regression model from the following statistician salary dataset:
Years of Experience (X) Salary (Y)
0.5 70,000
1.0 74,000
1.5 75,000
2.0 75,000
2.5 77,000
3.0 80,000
3.5 78,000
4.0 79,000
4.5 82,000
5.0 85,000
The simple linear regression model could be written as follows:
\[ Predicted\: Salary = \beta_{0}+\beta_{1}(Years\: of\: Experience) \]
Least Squares Criteria
The best estimates for the coefficients (\( \beta_{0}, \beta_{1} \)) are obtained by finding the regression line that fits the training dataset points as closely as possible. This line can be
obtained by minimizing the least squares criteria.
What does it mean to minimize the least squares criteria? Let's use the example of the regression model for the statistician salaries.
\[ Predicted\: Salary = \beta_{0}+\beta_{1}(Years\: of\: Experience) \]
The difference between the actual salary value in the dataset (\( y \)) and the predicted salary (\( \hat{y} \)) is known as the residual (\( e \)). The residual sum of squares (\( RSS \)) is defined
\[ RSS = e_{1}^2 + e_{2}^2 + ... \]
The least squares criteria chooses the \( \beta \) coefficient values that minimize the \( RSS \).
For our statistician salary dataset, the linear regression model determined through the least squares criteria is as follows:
\[ Predicted\: Salary = \beta_{0}+\beta_{1}(Years\: of\: Experience) \]
• \( \beta_{0} \) is represented by $70,545
• \( \beta_{1} \) is represented by $2,576
This final regression model can be visualized by the orange line below:
Interpretation of \( \beta \) Coefficients
How do we interpret the coefficients of a simple linear regression model in plain English? In general, we say that:
• If the predictor (\( X \)) were 0, the prediction (\( Y \)) would be \( \beta_{0} \), on average.
• For every one increase in the predictor, the prediction changes by \( \beta_{1} \), on average.
Using the example of the final statistician salary regression model, we would conclude that:
• If a statistician had 0 years of experience, he/she would have an entry-level salary of $70,545, on average.
• For every one additional year of experience that a statistician has, his/her salary increases by $2,576, on average.
True Population Regression vs Least Squares Regression
The true population regression line represents the "true" relationship between \( X \) and \( Y \). However, we never know the true relationship, so we use least squares regression to estimate it
with the data that we have available.
For example, assume that the true population regression line for statistician salaries was represented by the black line below. The least squares regression line, represented by the orange line, is
close to the true population regression line, but not exactly the same.
So, how do we estimate how accurate the least squares regression line is as an estimate of the true population regression line?
We compute the standard error of the coefficients and determine the confidence interval.
The standard error is a measure of the accuracy of an estimate. Knowing how to mathematically calculate the standard error is not important, as programs like R will determine them easily.
Standard errors are used to compute confidence intervals, which provide an estimate of how accurate the least squares regression line is.
The most commonly used confidence interval is the 95% confidence interval. The 95% confidence intervals for the coefficients are calculated as follows:
• 95% Confidence Interval for \( \beta_{0} \) = \( \beta_{0}\pm2*SE(\beta_{0}) \)
• 95% Confidence Interval for \( \beta_{1} \) = \( \beta_{1}\pm2*SE(\beta_{1}) \)
The confidence interval is generally interpreted as follows:
• There is a 95% probability that the interval contains the true population value of the coefficient.
For example, for the statistician salary regression model, the confidence intervals are as follows:
• \( \beta_{0} = [67852, 72281] \)
• \( \beta_{1} = [1989, 3417] \)
In the context of the statistician salaries, these confidence intervals are interpreted as follows:
• In the absence of any years of experience, the salary of an entry-level statistician will fall between $67,852 and $72,281.
• For each one additional year of experience, a statistician's salary will increase between $1,989 and $3,417.
Hypothesis Testing
So, how do we determine whether or not there truly is a relationship between \( X \) and \( Y \)? In other words, how do we know that \( X \) is actually a good predictor for \( Y \)?
We use the standard errors to perform hypothesis tests on the coefficients.
The most common hypothesis test involves testing the null hypothesis versus the alternative hypothesis:
• Null Hypothesis \( H_{0} \): No relationship between \( X \) and \( Y \)
• Alternative Hypothesis \( H_{1} \): Some relationship between \( X \) and \( Y \)
These can also be written mathematically as:
• \( H_{0} \): \( \beta_{1}=0 \)
• \( H_{1} \): \( \beta_{1} \cancel= 0 \)
\( t \)-statistic and \( p \)-value
So, how do we determine if \( \beta_{1} \) is non-zero? We use the estimated value of the coefficient and its standard error to determine the \( t \)-statistic:
\[ t=\frac{\beta_{1}-0}{SE(\beta_{1})} \]
The \( t \)-statistic measures the number of standard deviations that \( \beta_{1} \) is away from 0.
The \( t \)-statistic allows us to determine something known as the \( p \)-value, which ultimately helps determine whether or not the coefficient is non-zero.
The \( p \)-value indicates how likely it is to observe a meaningful association between \( X \) and \( Y \) by some bizarre random error or chance, as opposed to there being a true relationship
between \( X \) and \( Y \).
Typically, we want \( p \)-values less than 5% or 1% to reject the null hypothesis. In other words, rejecting the null hypothesis means that we are declaring that some relationship exists between \(
X \) and \( Y \).
Assessing Model Accuracy
There are 2 main assessments for how well a model fits the data: RSE and \( R^2 \).
Residual Standard Error (RSE)
The RSE is a measure of the standard deviation of the random error term (\( \epsilon \)).
In other words, it is the average amount that the actual response will deviate from the true regression line. It is a measure of the lack of fit of a model.
The value of RSE and whether or not it is acceptable will depend on the context of the problem.
R-Squared \( R^2 \)
\( R^2 \) measures the proportion of variability in \( Y \) that can be explained by using \( X \). It is a proportion that is calculated as follows:
\[ R^2 = \frac{TSS-RSS}{TSS} \]
• Total Sum of Squares = \( TSS = \sum(y_{i}-\bar{y})^2 \)
• Residual Sum of Squares = \( RSS = e_{1}^2 + e_{2}^2 + ... \)
TSS is a measure of variability that is already inherent in the response variable before regression is performed.
RSS is a measure of variability in the response variable after regression is performed.
The final statistician salary regression model has an \( R^2 \) of 0.90, meaning that 90% of the variability in the salaries of statisticians is explained by using years of experience as a predictor.
Multiple Linear Regression
Simple linear regression is useful for prediction if there is only one predictor ( \( X \) ). But what if we had multiple predictors ( \( X_{1} , X_{2}, X_{3}, \) etc.)? Multiple linear regression
allows for multiple predictors, and takes the following form:
\[ Y = \beta_{0} + \beta_{1}(X_{1}) + \beta_{2}(X_{2}) + ... \]
For example, let's take the statistician salary dataset, add a new predictor for college GPA, and add 10 new data points.
Years of Experience (X1) GPA (X2) Salary (Y)
0.00 3.46 70,000
1.00 3.47 74,000
1.50 3.50 75,000
2.00 3.33 75,000
2.50 3.57 77,000
3.00 3.52 80,000
3.50 3.48 78,000
4.00 3.32 79,000
4.50 3.52 82,000
5.00 3.41 85,000
0.00 3.59 73,000
1.00 3.53 76,000
1.50 3.58 76,000
2.00 3.71 78,000
2.50 3.60 81,000
3.00 3.78 85,000
3.50 3.76 85,000
4.00 3.57 84,000
4.50 3.72 85,000
5.00 3.81 87,000
The multiple linear regression model for the dataset would take the form:
\[ Y = \beta_{0} + \beta_{1}(Years\: of\: Experience) + \beta_{2}(GPA) \]
The multiple linear regression model would fit a plane to the dataset. The dataset is represented below as a 3D scatter plot with an X, Y, and Z axis.
In multiple linear regression, we're interested in a few specific questions:
1. Is at least one of the predictors useful in predicting the response?
2. Do all of the predictors help explain \( Y \), or only a few of them?
3. How well does the model fit the data?
4. How accurate is our prediction?
Least Squares Criteria
Similar to simple linear regression, the coefficient estimates in multiple linear regression are chosen based on the same least squares approach that minimizes RSS.
Interpretation of \( \beta \) Coefficients
The interpretation of the coefficients is also very similar to the simple linear regression setting, with one key difference (indicated in bold). In general, we say that:
• If all of the predictors were 0, the prediction ( \( Y \) ) would be \( \beta_{0} \), on average.
• For every one increase in some predictor \( X_{j} \), the prediction changes by \( \beta_{j} \), on average, holding all of the other predictors constant.
Hypothesis Testing
So, how do we determine whether or not there truly is a relationship between the \( X \)s and \( Y \)? In other words, how do we know that the \( X \)s are actually good predictors for \( Y \)?
Similar to the simple linear regression setting, we perform a hypothesis test. The null and alternative hypotheses are slightly different:
• Null Hypothesis \( H_{0} \): No relationship between the \( X \)s and \( Y \)
• Alternative Hypothesis \( H_{1} \): At least one predictor has a relationship to the response
These can also be written mathematically as:
• \( H_{0} \): \( \beta_{1}=\beta_{2}=\beta_{3}=...=0 \)
• \( H_{1} \): At least one \( \beta_{j} \cancel= 0 \)
\( F \)-statistic and \( p \)-value
So, how do we determine if at least one \( \beta_{j} \) is non-zero? In simple regression, we determined the \( t \)-statistic. In multiple regression, we determine the \( F \)-statistic instead.
\[ F=\frac{(TSS-RSS)/p}{RSS/(n-p-1)} \]
When there is no relationship between the response and predictors, we generally expect the \( F \)-statistic to be close to 1.
If there is a relationship, we generally expect the \( F \)-statistic to be greater than 1.
Similar to the \( t \)-statistic, the \( F \)-statistic also allows us to determine the \( p \)-value, which ultimately helps decide whether or not a relationship exists.
The \( p \)-value is essentially interpreted in the same way that it is interpreted in simple regression.
Typically, we want \( p \)-values less than 5% or 1% to reject the null hypothesis. In other words, rejecting the null hypothesis means that we are declaring that some relationship exists between the
\( X \)s and \( Y \).
\( t \)-statistics in Multiple Linear Regression
In multiple linear regression, we will receive outputs that indicate the \( t \)-statistic and \( p \)-values for each of the different coefficients.
However, we have to use the overall \( F \)-statistic instead of the individual coefficient \( p \)-values. This is because when the number of predictors is large (e.g. \( p=100 \)), about 5% of the
coefficients will have low \( p \)-values less than 5% just by chance. Therefore, in this scenario, choosing whether or not to reject the null hypothesis based on the individual \( p \)-values would
be flawed.
Determining Variable Significance
So, after concluding that at least one predictor is related to the response, how do we determine which specific predictors are significant?
This process is called variable selection, and there are three approaches: forward selection, backward selection, and mixed selection.
Forward Selection
Assume that we had a dataset of credit card balance and 10 predictors.
Income Limit Rating Cards Age Education Gender Student Married Ethnicity Balance
106025 6645 483 3 82 15 Female No Yes Asian 903
... ... ... ... ... ... ... ... ... ... ...
Forward selection begins with a null model with no predictors:
\[ Balance = \beta_{0} \]
Then, 10 different simple linear regression models are built for each of the predictors:
\[ Balance = \beta_{0} + \beta_{1}(Income) \]
\[ Balance = \beta_{0} + \beta_{2}(Limit) \]
\[ Balance = \beta_{0} + \beta_{3}(Rating) \]
\[ ... \]
The predictor that results in the lowest RSS is then added to the initial null model. Assume that the Limit variable is the variable that results in the lowest RSS. The forward selection model would
then become:
\[ Balance = \beta_{0} + \beta_{2}(Limit) \]
Then, a second predictor is added to this new model, which will result in building 9 different multiple linear regression models for the remaining predictors:
\[ Balance = \beta_{0} + \beta_{2}(Limit) + \beta_{1}(Income) \]
\[ Balance = \beta_{0} + \beta_{2}(Limit) + \beta_{3}(Rating) \]
\[ Balance = \beta_{0} + \beta_{2}(Limit) + \beta_{4}(Cards) \]
\[ ... \]
The second predictor that results in the lowest RSS is then added to the model. Assume that the model with the Income variable resulted in the lowest RSS. The forward selection model would then
\[ Balance = \beta_{0} + \beta_{2}(Limit) + \beta_{1}(Income) \]
This process of adding predictors is continued until some statistical stopping rule is satisfied.
Backward Selection
Backward selection begins with a model with all predictors:
\[ Balance = \beta_{0} + \beta_{1}(Income) + \beta_{2}(Limit) \\ + \beta_{3}(Rating) + \beta_{4}(Cards) + \beta_{5}(Age) + \beta_{6}(Education) \\ + \beta_{7}(Gender) + \beta_{8}(Student) + \beta_{9}
(Married) + \beta_{10}(Ethnicity) \]
Then, the variable with the largest \( p \)-value is removed, and the new model is fit.
Again, the variable with the largest \( p \)-value is removed, and the new model is fit.
This process is continued until some statistical stopping rule is satisfied, such as all variables in the model having low \( p \)-values less than 5%.
Mixed Selection
Mixed selection is a combination of forward and backward selection. We begin with a null model that contains no predictors. Then, variables are added one by one, exactly as done in forward selection.
However, if at any point the \( p \)-value for some variable rises above a chosen threshold, then it is removed from the model, as done in backward selection.
Assessing Model Accuracy
Similar to the simple regression setting, RSE and \( R^2 \) are used to determine how well the model fits the data.
In multiple linear regression, RSE is calculated as follows:
\[ RSE=\sqrt{\frac{1}{n-p-1}RSS} \]
R-Squared \( R^2 \)
\( R^2 \) is interpreted in the same manner that it is interpreted in simple regression.
However, in multiple linear regression, adding more predictors to the model will always result in an increase in \( R^2 \).
Therefore, it is important to look at the magnitude at which \( R^2 \) changes when adding or removing a variable. A small change will generally indicate an insignificant variable, whereas a large
change will generally indicate a significant variable.
Assessing Prediction Accuracy
How do we estimate how accurate the actual predictions are? Confidence intervals and prediction intervals can help assess prediction accuracy.
Confidence Intervals
Confidence intervals are determined through the \( \beta \) coefficient estimates and their inaccuracy through the standard errors.
This means that confidence intervals only account for reducible error.
Prediction Intervals
Reducible error isn't the only type of error that is present in regression modeling.
Even if we knew the true values of the \( \beta \) coefficients, we would not be able to predict the response variable perfectly because of random error \( \epsilon \) in the model, which is an
irreducible error.
Prediction intervals go a step further than confidence intervals by accounting for both reducible and irreducible error. This means that prediction intervals will always be wider than confidence
Other Considerations in Regression Modeling
Qualitative Predictors
It is possible to have qualitative predictors in regression models. For example, assume that we had a predictor that indicated gender.
For qualitative variables with only two levels, we could simply create a "dummy" variable that takes on two values:
\[ X_{i} =\begin{cases}0 & \text{if person is male}\\1 & \text{if person is female}\end{cases} \]
We'd simply use that logic to create a new column for the dummy variable in the data, and use that for regression purposes:
Gender Gender Dummy
Male 0
Female 1
Male 0
Female 1
But what if the qualitative variable had more than two levels? For example, assume we had a predictor that indicated ethnicity:
African American
In this case, a single dummy variable cannot represent all of the possible values. In this situation, we create multiple dummy variables:
\[ X_{i1} =\begin{cases}0 & \text{if person is not Caucasian}\\1 & \text{if person is Caucasian}\end{cases} \]
\[ X_{i2} =\begin{cases}0 & \text{if person is not Asian}\\1 & \text{if person is Asian}\end{cases} \]
For qualitative variables with multiple levels, there will always be one fewer dummy variable than the total number of levels. In this example, we have three ethnicity levels, so we create two dummy
The new variables for regression purposes would be represented as follows:
Ethnicity Caucasian Asian
Caucasian 1 0
African American 0 0
Asian 0 1
Extensions of the Linear Model
Regression models provide interpretable results and work well, but make highly restrictive assumptions that are often violated in practice.
Two of the most important assumptions state that the relationship between the predictors and response are additive and linear.
The additive assumption means that the effect of changes in some predictor \( X_{j}\) on the response is independent of the values of the other predictors.
The linear assumption states that the change in the response \( Y \) due to a one-unit change in some predictor \( X_{j} \) is constant, regardless of the value of \( X_{j}\).
Removing the Additive Assumption
Assume that we had an advertising dataset of money spent on TV ads, money spent on radio ads, and product sales.
TV Spend Radio Spend Product Sales
230,000 37,800 22,100
44,500 39,300 10,400
17,200 45,900 9,300
151,500 41,300 18,500
180,800 10,800 12,900
8,700 48,900 7,200
57,500 32,800 11,800
120,200 19,600 13,200
8,600 2,100 4,800
199,800 2,600 10,600
The multiple linear regression model for the data would have the form:
\[ Sales = \beta_{0} + \beta_{1}(TV) + \beta_{2}(Radio) \]
This model states that the average effect on sales for a $1 increase in TV advertising spend is \( \beta_{1} \), on average, regardless of the amount of money spent on radio ads.
However, it is possible that spending money on radio ads actually increases the effectiveness of TV ads, thus increasing sales further. This is known as an interaction effect or a synergy effect. The
interaction effect can be taken into account by including an interaction term:
\[ Sales = \beta_{0} + \beta_{1}(TV) + \beta_{2}(Radio) + \beta_{3}(TV*Radio) \]
The interaction term relaxes the additive assumption. Now, every $1 increase in TV ad spend increases sales by \( \beta_{1} + \beta_{3}(Radio) \).
Sometimes it is possible for the interaction term to have a low p-value, yet the main terms no longer have a low \( p \)-value. The hierarchical principle states that if an interaction term is
included, then the main terms should also be included, even if the \( p \)-values for the main terms are not significant.
Interaction terms are also possible for qualitative variables, as well as a combination of qualitative and quantitative variables.
Non-Linear Relationships
In some cases, the true relationship between the predictors and response may be non-linear. A simple way to extend the linear model is through polynomial regression.
For example, for automobiles, there is a curved relationship between miles per gallon and horsepower.
A quadratic model of the following form would be a great fit to the data:
\[ MPG = \beta_{0} + \beta_{1}(Horsepower) + \beta_{2}(Horsepower)^2 \]
Potential Problems with Linear Regression
When fitting linear models, there are six potential problems that may occur: non-linearity of data, correlation of error terms, nonconstant variance of error terms, outliers, high leverage data
points, and collinearity.
Non-Linearity of Data
Residual plots are a useful graphical tool for the identification of non-linearity.
In simple linear regression, the residuals (\( y_{i}-\hat{y}_{i} \)) are plotted against the predictor \( X \).
In multiple linear regression, the residuals are plotted against the predicted values (\( \hat{y}_{i} \)).
If there is some kind of pattern in the residual plot, then it is an indication of potential non-linearity.
Non-linear transformations, such as log-transformation, of the predictors could be a simple method to solving the issue.
For example, take a look at the below residual graphs, which represent different types of fits for the automobile data mentioned previously. The graph on the left represents what the residuals look
like if a simple linear model is fit to the data. Clearly, there is a curved pattern in the residual plot, indicating non-linearity. The graph on the right represents what the residuals look like if
a quadratic model is fit to the data. Fitting a quadratic model seems to resolve the issue, as a pattern doesn't exist in the plot.
Correlation of Error Terms
Proper linear models should have residual terms that are uncorrelated. This means that the sign (positive or negative) of some residual \( e_{i} \) should provide no information about the sign of the
next residual \( e_{i+1} \).
If the error terms are correlated, we may have an unwarranted sense of confidence in the linear model.
To determine if correlated errors exist, we plot residuals in order of observation number. If the errors are uncorrelated, there should not be a pattern. If the errors are correlated, then we may see
tracking in the graph. Tracking is where adjacent residuals have similar signs.
For example, take a look at the below graphs. The graph on the left represents a scenario in which residuals are not correlated. In other words, just because one residual is positive doesn't seem to
indicate that the next residual will be positive. The graph on the right represents a scenario in which the residuals are correlated. There are 15 residuals in a row that are all positive.
Nonconstant Variance of Error Terms
Proper linear models should also have residual terms that have a constant variance. The standard errors, confidence intervals, and hypothesis tests associated with the model rely on this assumption.
Nonconstant variance in errors is known as heteroscedasticity. It is identified as the presence of a funnel shape in the residual plot.
One solution to nonconstant variance is to transform the response using a concave function, such as \( log(Y) \) or \( \sqrt{Y} \). Another solution is to use weighted least squares instead of
ordinary least squares.
The graphs below represent the difference between constant and nonconstant variance. The residual plot on the right has a funnel shape, indicating nonconstant variance.
Depending on how many outliers are present and their magnitude, they could either have a minor or major impact on the fit of the linear model. However, even if the impact is small, they could cause
other issues, such as impacting the confidence intervals, \( p \)-values, and \( R^2 \).
Outliers are identified through various methods. The most common is studentized residuals, where each residual is divided by its estimated standard error. Studentized residuals greater than 3 in
absolute value are possible outliers.
These outliers can be removed from the data to come up with a better linear model. However, it is also possible that outliers indicate some kind of model deficiency, so caution should be taken before
removing the outliers.
The red data point below represents an example of an outlier that would greatly impact the slope of a linear regression model.
High Leverage Data Points
Observations with high leverage have an unusual value for \( x_{i} \) compared to the other observation values. For example, you might have a dataset of \( X \) values between 0 and 10, and just one
other data point with a value of 20. The value of 20 is a high leverage data point.
High leverage is determined through the leverage statistic. The leverage statistic is always between \( \frac{1}{n} \) and 1.
The average leverage is defined as:
\[ Average\: Leverage = \frac{p+1}{n} \]
If the leverage statistic of a data point is greatly higher than the average leverage, then we have reason to suspect high leverage.
The red data point below represents an example of a high leverage data point that would impact the linear regression fit.
Collinearity refers to the situation in which 2 or more predictor variables are closely related. Collinearity makes it difficult to separate out the individual effects of collinear variables on the
response. It also reduces the accuracy of the estimates of the regression coefficients by causing the coefficient standard errors to grow, thus reducing the credibility of hypothesis testing.
A simple way to detect collinearity is to look at the correlation matrix of the predictors.
However, not all collinearity can be detected through the correlation matrix. It is possible for collinearity to exist between multiple variables instead of pairs of variables, which is known as
The better way to assess collinearity is through the Variance Inflation Factor (VIF). VIF is the ratio of the variance of a coefficient when fitting the full model, divided by the variance of the
coefficient when fitting a model only on its own. The smallest possible VIF is 1. In general, a VIF that exceeds 5 or 10 may indicate a collinearity problem.
One way to solve the issue of collinearity is to simply drop one of the predictors from the linear model. Another solution is to combine collinear variables into one variable.
The chart below demonstrates an example of collinearity. As we know, an individual's credit limit is directly related to their credit rating. A dataset that includes both of these predictor should
only include one of them for regression purposes, to avoid the issue of collinearity.
ISLR Chapter 3 - R Code
Simple Linear Regression
library(MASS) # For model functions
library(ISLR) # For datasets
library(ggplot2) # For plotting
# Working with the Boston dataset to predict median house values
# Fit a simple linear regression model
# Median house value is the response (Y)
# Percentage of low income households in the neighborhood is the predictor (X)
Boston_lm = lm(medv ~ lstat, data=Boston)
# View the fitted simple linear regression model
# View all of the objects stored in the model, and get one of them, such as the coefficients
# 95% confidence interval of the coefficients
confint(Boston_lm, level=0.95)
# Use the model to predict house values for specific lstat values
lstat_predict = c(5, 10, 15) # The lstat values we want to predict with
lstat_predict = data.frame(lstat=lstat_predict) # Convert the values to a dataframe
predict(Boston_lm, lstat_predict) # Predictions
predict(Boston_lm, lstat_predict, interval="confidence", level=0.95) # Confidence interval predictions
predict(Boston_lm, lstat_predict, interval="prediction", level=0.95) # Prediction interval predictions
# Use ggplot to create a residual plot of the model
Boston_lm_pred_resid = data.frame(Prediction=Boston_lm$fitted.values, Residual=Boston_lm$residuals)
# Create a separate dataframe of the predicted values and residuals
Boston_resid_plot = ggplot(Boston_lm_pred_resid, aes(x=Prediction, y=Residual)) +
geom_point() +
labs(title="Boston Residual Plot", x="House Value Prediction", y="Residual")
# Use ggplot to create a studentized residual plot of the model (for outlier detection)
Boston_lm_pred_Rstudent = data.frame(Prediction=Boston_lm$fitted.values, Rstudent=rstudent(Boston_lm))
# Create a separate dataframe of the predicted values and Rstudent values
Boston_Rstudent_plot = ggplot(Boston_lm_pred_Rstudent, aes(x=Prediction, y=Rstudent)) +
geom_point() +
labs(title="Boston Rstudent Plot", x="House Value Prediction", y="Rstudent")
# Determine leverage statistics for the lstat values (for high leverage detection)
Boston_leverage = hatvalues(Boston_lm) # Function for determining leverage statistics
head(order(-Boston_leverage), 10) # See top 10 lstat values with the highest leverage statistics
Multiple Linear Regression
# Multiple linear regression model with two predictors
# Median house value is the response (Y)
# Percentage of low income households in the neighborhood is the first predictor (X1)
# Percentage of houses in the neighborhood built before 1940 is the second predictor (X2)
Boston_lm_mult_1 = lm(medv ~ lstat + age, data=Boston)
## Coefficients:
## (Intercept) lstat age
## 33.22276 -1.03207 0.03454
# Multiple linear regression model with all predictors
# Median house value is the response (Y)
# Every variable in the Boston dataset is a predictor (X)
Boston_lm_mult_2 = lm(medv ~ ., data=Boston) # The period "." represents all predictors
# Multiple linear regression model with all predictors except specified (age)
Boston_lm_mult_3 = lm(medv ~ . -age, data=Boston) # The minus "-" represents exclusion
# Multiple linear regression model with an interaction term
Boston_lm_mult_4 = lm(medv ~ crim + lstat:age, data=Boston)
# The colon ":" will include an (lstat)(age) interaction term
Boston_lm_mult_5 = lm(medv ~ crim + lstat*age, data=Boston)
# The asterisk "*" will include an (lstat)(age) interaction term
# It will also include the terms by themselves, without having to specify separately
# Multiple linear regression model with nonlinear transformation
Boston_lm_mult_6 = lm(medv ~ lstat + I(lstat^2), data=Boston)
# Multiple linear regression model with nonlinear transformation (easier method)
Boston_lm_mult_7 = lm(medv ~ poly(lstat, 5), data=Boston)
# Multiple linear regression model with log transformation
Boston_lm_mult_8 = lm(medv ~ log(rm), data=Boston)
# ANOVA test to compare two different regression models
anova(Boston_lm_mult, Boston_lm_mult_6)
# Null Hypothesis: Both models fit the data equally well
# Alternative Hypothesis: The second model is superior to the first
# 95% confidence interval for the coefficients
confint(Boston_lm_mult_1, level=0.95)
Linear Regression with Qualitative Variables
# The Carseats data has a qualitative variable for the quality of shelf location
# It takes on one of three values: Bad, Medium, Good
# R automatically generates dummy variables ShelveLocGood and ShelveLocMedium
# Multiple linear regression model to predict carseat sales
Carseats_lm_mult = lm(Sales ~ ., data=Carseats)
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ...
## ShelveLocGood 4.8501827 0.1531100 31.678 < 2e-16 ***
## ShelveLocMedium 1.9567148 0.1261056 15.516 < 2e-16 ***
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LegoLoom Part 1: Background
The other weekend I experience the pleasure and fascination of visiting the Antique Gas and Steam Engine Museum. While there, my friend and I perused the [DEL:loom room:DEL] Weaver’s Barn. At first,
I’m shocked that people spend time weaving, as it looks like such a task should be fully mechanized. We’ll get to that in the history section!
The Weaver’s Barn at the Antique Gas and Steam Engine Museum
Being of a technical inclination, I went around the shop floor looking at the different loom setups. They all share some common features: foot bars connected to shafts that move groups of threads up
and down to make the pattern of the weave, a shuttle to pass thread across, a big comb to press the horizontal threads into the cloth, a roller of threads on the back and another roller of threads on
the front. Plenty of the loom used thick threads, like yarn, and some even had ribbon for the horizontal thread, weaving what looked like a rug or mat.
All but one of the looms controlled the weave pattern via a sequence of shafts that group the heddles. Those designs produce a recurrent pattern, typically a repeating one like plain weave, twill, or
houndstooth. But I see a weave — an over or under — as a pixel, and a machine with 4, 6, or even 12 shafts simply can’t produce arbitrary patterns like this woven portrait of Jacquard — woven from a
program of 24,000 punched cards in 1839.
A woven silk portrait of Jacquard, made on a Jacquard loom in 1839 with 24000 punchcards.
By Michel Marie Carquillat (tisseur) d’après Claude Bonnefond – Link
I simply can’t resist injecting here that the most awesome part about that picture is the meta-referential inclusion of the loom in the background and cards in the foreground!
What makes Jacquard and his loom so special?
From a wonderful TV series, Connections, presented by James Burke, that I recall watching an episode about the Jacquard loom. Mention of the machine also occasionally occurs in the history of
Computer Science, because we attribute credit for it being the first programmable machine, weaving from a program of punch cards.
If you can program the weave as you would a grid of pixels, then you can make a pattern that doesn’t repeat! Very early on, this ability got used for Jacquard’s portrait and other imagery as part of
bespoke fabric design. Although I remain quite impressed by the ability of the loom, specifically its importance as a stepping stone in the history of computing, I can’t help but notice that the
programmed sequence is just… longer. It still loops!
In the case of Jacquard’s portrait, keep using the card stack and we’d cycle through and producing an endless sequence of the same portrait. A very large for-loop over a fixed pattern. In 1800’s
England they didn’t have a formalized notion of a non-repeating pattern.
A new shape dropped this year!
Earlier this year, a few mathematical tinkerers discovered the Einstein Tile! Actually, they found a whole parameterized family of aperiodic monotiles! Shortly afterward, they followed up with the
Spectre, fixing an itch to avoid reflections.
Shoot. I got excited. Let’s back up to 1961, when Wang worked on the Domino Problem — Whether a set of square tiles with a color on each side can cover the plane without rotation or reflection when
placed side by side with matching colors.
An aperiodic tiling of the plane with a set of 11 Wang tiles discovered in 2015
By Claudio Rocchini – Own work, CC BY-SA 3.0, Link
In 1966, Berger solved the problem by showing how translate any Turning Machine into a set of Wang tiles that tiles the plane if and only if the Turning Machine does not halt, thus demonstrating that
no algorithm can decide an answer for all tile sets. So much more info is at Wikipedia.
The original work needed 20426 tiles and obviously weren’t drawn but only described. In 1971, Robinson worked the number down to 52 tiles. Work in 2015 reduced it further to a set of 11.
By Parcly Taxel – Own work, FAL, Link
But the original solution led to a yet more interesting observation: There must exist a finite set of Wang tiles that covers the plane, but only does so without periodicity. This launched inquiry
into finding such tile sets, and more generally tile shapes that tile the plane, but without periodicity.
In the List of aperiodic sets of tiles you can see how most of the approaches have tiles with rules — dots, colors, etc — that restrict the matching of edges. By 1971, Penrose famously discovered 2
shapes that would tile the plane aperiodically. And there we stood for 50 years.
Roger Penrose standing in the foyer of the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University on a floor decorated by a Penrose tiling
By Solarflare100 – Own work, CC BY 3.0, Link
Again! The meta-referentiality here is delicious. That’s Roger Penrose in the foyer of the Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, standing on a floor with a
Penrose tiling.
Earlier this year, we got a new shape! An Einstein Tile! One Stone that tiles the plane aperiodically! They called it The Hat. It’s beautiful in simplicity and the proof is by fractal construction.
The Hat, an aperiodic monotile. Strauss (CC BY-SA 4.0) Image from Scientific American Newfound Mathematical ‘Einstein’ Shape Creates a Never-Repeating Pattern
I was really excited by this work! Seriously I told everyone I met for about 2 weeks after I heard about it. And I went and read the proof! I mean, how often is it that we get a large breakthrough
that occurs on a kitchen table? Or with a shape that comes from combining 4 darts found in a common toy tile set for children?
But, I had an issue with this tiling, because it sometimes uses a reflection. While technically not against the rules, it’s an itch. The discover’s agreed, because within a couple months they found a
chiral aperiodic monotile called The Spectre! With straight edges, it also admits a periodic tiling, but curving or notching the edges forces only aperiodic tilings.
The Spectre, a chiral aperiodic monotile. Image from Now that’s what I call an aperiodic monotile
A return to the loom
Given that the aperiodic tiling is so simple, yet deviously complex in its coverage of the plane, it makes for a beautiful challenge of a Jacquard loom! But this time, we can’t have a loop of
punchcards. Instead, we must compute each scanline of the tiling.
So I went back and looked at the Jacquard loom in the Weaver’s barn. It could be done by creating a box that presents a set of holes, dynamically acting as a new card for each line of fabric.
We can definitely create such a box with solenoids and an arduino. Even though I’m not strong in the computer graphics department, I’m pretty sure I could learn how to program a tiling and then
raster scan it for producing lines of fabric. But that approach uses modern tech attached to an old device.
If they had known about such a thing, could someone in 1800 have produced an aperiodic pattern by purely mechanical means? I think so, but we’d have to pick an even simpler pattern.
Cellular Automata, Rule 30
Back in the early 2000s, Wolfram published A New Kind of Science, a book about trying to find the laws of physics in very simple rule structures like cellular automata. I read that book (but not all
the footnotes, which were even larger in word count than the main text).
Current pattern 111 110 101 100 011 010 001 000
New state for center cell 0 0 0 1 1 1 1 0
A very simple rule. Computable with a few gears! They had those in 1800. I did a K-map that works out to (!L and R) or (!L and C) or (L and !C and !R) for left, center, and right. Not very many logic
operations at all. Completely achievable with 1800s technology!
Turns out this rule is Turing Complete and produces an aperiodic pattern.
Many generations of Rule 30
The Jacquard Project
Use 1800s tech to compute an aperiodic pattern, and weave it on an old Jacquard just as they would have done.
OK, so it won’t be the prettiest of fabrics. And the arbitrarily long runs of the same pixel setting means instability in the weave. But the more troublesome part: I’ve got a problem with the fabric
being of fixed width on the loom, so I’ll have to choose — anchor to 0, anchor to 1, or do a computation. I’ll write up a program to exercise the visuals, but I’m tempted to use a mod 2 checksum or
some other simple trick that’ll introduce a random element into the fabric. Maybe hold a register with one lookback and just “bounce off the wall”.
OK, so caveat. Even with a modern loom, you do eventually run out of warp threads. Although I could image a world with 10k spinners behind a machine so that doesn’t happen for as long as you can feed
it raw wool. But hypothetically solving that, you’d still cut off the fabric for sale. So you might as well just run a loop on a pre-computed image that’s only as long as the warp lasts, or even
shorter, as long as the reams being sold.
So technically, in 1800 they would have pre-computed a reasonable size of Rule 30 and run a sequence of cards. But I’m gonna use Lego gears and logic gates.
This is something they could have done in 1800 but for:
1. They didn’t have knowledge of Rule 30, by circumstance of not having developed that branch of mathematics (contingent on computers for quick experimentation).
2. They hadn’t yet developed the perspective of seeing problems from a computational perspective — Modulo Charles Babbage, who was very clearly inspired by Jacquard looms and owned one of those
woven portraits.
Modern Looms
Alright. I dove into loom technology for a couple of days after the museum visit. Modern mechanized looms are awesome. Fully programmable, fully automated, extremely fast. Did you know they got rid
of the shuttle!?! Now they just blast the string through the air!
I still really want this bedspread. | {"url":"http://www.cogitolingua.net/blog/2023/11/14/legoloom-part-1-background/","timestamp":"2024-11-04T13:18:28Z","content_type":"text/html","content_length":"39604","record_id":"<urn:uuid:860e69c0-fb1e-4a14-82a7-70f0bc7c3eee>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00665.warc.gz"} |
Mathematics (from Ancient Greek μάθημα; máthēma: ‘knowledge, study, learning’) is an area of knowledge that includes such topics as numbers (arithmetic, number theory), formulas and related
structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).
Most mathematical activity involves the use of pure reason to discover or prove the properties of abstract objects, which consist of either abstractions from nature or—in modern mathematics—entities
that are stipulated with certain properties, called axioms. A mathematical proof consists of a succession of applications of some deductive rules to already known results, including previously proved
theorems, axioms and (in case of abstraction from nature) some basic properties that are considered as true starting points of the theory under consideration.
Mathematics is used in science for modeling phenomena, which then allows predictions to be made from experimental laws. The independence of mathematical truth from any experimentation implies that
the accuracy of such predictions depends only on the adequacy of the model. Inaccurate predictions, rather than being caused by incorrect mathematics, imply the need to change the mathematical model
used. For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein’s general relativity, which replaced Newton’s law of gravitation as a better
mathematical model.
But for more post and free books from our site please make sure you subscribe to our site and if you need a copy of our notes as how it is in our site contact us any time we sell them in low cost in
form of PDF or WORD.
mathematics, the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and
quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter. Since the 17th century, mathematics has been an indispensable
adjunct to the physical sciences and technology, and in more recent times it has assumed a similar role in the quantitative aspects of the life sciences.
In many cultures—under the stimulus of the needs of practical pursuits, such as commerce and agriculture—mathematics has developed far beyond basic counting. This growth has been greatest in
societies complex enough to sustain these activities and to provide leisure for contemplation and the opportunity to build on the achievements of earlier mathematicians.
All mathematical systems (for example, Euclidean geometry) are combinations of sets of axioms and of theorems that can be logically deduced from the axioms. Inquiries into the logical and
philosophical basis of mathematics reduce to questions of whether the axioms of a given system ensure its completeness and its consistency. For full treatment of this aspect, see mathematics,
foundations of.
This article offers a history of mathematics from ancient times to the present. As a consequence of the exponential growth of science, most mathematics has developed since the 15th century ce, and it
is a historical fact that, from the 15th century to the late 20th century, new developments in mathematics were largely concentrated in Europe and North America. For these reasons, the bulk of this
article is devoted to European developments since 1500.
This does not mean, however, that developments elsewhere have been unimportant. Indeed, to understand the history of mathematics in Europe, it is necessary to know its history at least in ancient
Mesopotamia and Egypt, in ancient Greece, and in Islamic civilization from the 9th to the 15th century. The way in which these civilizations influenced one another and the important direct
contributions Greece and Islam made to later developments are discussed in the first parts of this article.
India’s contributions to the development of contemporary mathematics were made through the considerable influence of Indian achievements on Islamic mathematics during its formative years. A separate
article, South Asian mathematics, focuses on the early history of mathematics in the Indian subcontinent and the development there of the modern decimal place-value numeral system. The article East
Asian mathematics covers the mostly independent development of mathematics in China, Japan, Korea, and Vietnam.
Ancient mathematical sources
It is important to be aware of the character of the sources for the study of the history of mathematics. The history of Mesopotamian and Egyptian mathematics is based on the extant original documents
written by scribes. Although in the case of Egypt these documents are few, they are all of a type and leave little doubt that Egyptian mathematics was, on the whole, elementary and profoundly
practical in its orientation. For Mesopotamian mathematics, on the other hand, there are a large number of clay tablets, which reveal mathematical achievements of a much higher order than those of
the Egyptians. The tablets indicate that the Mesopotamians had a great deal of remarkable mathematical knowledge, although they offer no evidence that this knowledge was organized into a deductive
system. Future research may reveal more about the early development of mathematics in Mesopotamia or about its influence on Greek mathematics, but it seems likely that this picture of Mesopotamian
mathematics will stand.
From the period before Alexander the Great, no Greek mathematical documents have been preserved except for fragmentary paraphrases, and, even for the subsequent period, it is well to remember that
the oldest copies of Euclid’s Elements are in Byzantine manuscripts dating from the 10th century ce. This stands in complete contrast to the situation described above for Egyptian and Babylonian
documents. Although, in general outline, the present account of Greek mathematics is secure, in such important matters as the origin of the axiomatic method, the pre-Euclidean theory of ratios, and
the discovery of the conic sections, historians have given competing accounts based on fragmentary texts, quotations of early writings culled from nonmathematical sources, and a considerable amount
of conjecture.
Many important treatises from the early period of Islamic mathematics have not survived or have survived only in Latin translations, so that there are still many unanswered questions about the
relationship between early Islamic mathematics and the mathematics of Greece and India. In addition, the amount of surviving material from later centuries is so large in comparison with that which
has been studied that it is not yet possible to offer any sure judgment of what later Islamic mathematics did not contain, and therefore it is not yet possible to evaluate with any assurance what was
original in European mathematics from the 11th to the 15th century.
In modern times the invention of printing has largely solved the problem of obtaining secure texts and has allowed historians of mathematics to concentrate their editorial efforts on the
correspondence or the unpublished works of mathematicians. However, the exponential growth of mathematics means that, for the period from the 19th century on, historians are able to treat only the
major figures in any detail. In addition, there is, as the period gets nearer the present, the problem of perspective. Mathematics, like any other human activity, has its fashions, and the nearer one
is to a given period, the more likely these fashions will look like the wave of the future. For this reason, the present article makes no attempt to assess the most recent developments in the
Unlike the Egyptians, the mathematicians of the Old Babylonian period went far beyond the immediate challenges of their official accounting duties. For example, they introduced a versatile numeral
system, which, like the modern system, exploited the notion of place value, and they developed computational methods that took advantage of this means of expressing numbers; they solved linear and
quadratic problems by methods much like those now used in algebra; their success with the study of what are now called Pythagorean number triples was a remarkable feat in number theory. The scribes
who made such discoveries must have believed mathematics to be worthy of study in its own right, not just as a practical tool.
The four arithmetic operations were performed in the same way as in the modern decimal system, except that carrying occurred whenever a sum reached 60 rather than 10. Multiplication was facilitated
by means of tables; one typical tablet lists the multiples of a number by 1, 2, 3,…, 19, 20, 30, 40, and 50. To multiply two numbers several places long, the scribe first broke the problem down into
several multiplications, each by a one-place number, and then looked up the value of each product in the appropriate tables. He found the answer to the problem by adding up these intermediate
results. These tables also assisted in division, for the values that head them were all reciprocals of regular numbers.
Regular numbers are those whose prime factors divide the base; the reciprocals of such numbers thus have only a finite number of places (by contrast, the reciprocals of nonregular numbers produce an
infinitely repeating numeral). In base 10, for example, only numbers with factors of 2 and 5 (e.g., 8 or 50) are regular, and the reciprocals (1/8 = 0.125, 1/50 = 0.02) have finite expressions; but
the reciprocals of other numbers (such as 3 and 7) repeat infinitely and , respectively, where the bar indicates the digits that continually repeat). In base 60, only numbers with factors of 2, 3,
and 5 are regular; for example, 6 and 54 are regular, so that their reciprocals (10 and 1 6 40) are finite. The entries in the multiplication table for 1 6 40 are thus simultaneously multiples of its
reciprocal 1/54. To divide a number by any regular number, then, one can consult the table of multiples for its reciprocal.
Geometric and algebraic problems
In a Babylonian tablet now in Berlin, the diagonal of a rectangle of sides 40 and 10 is solved as 40 + 10^2/(2 × 40). Here a very effective approximating rule is being used (that the square root of
the sum of a^2 + b^2 can be estimated as a + b^2/2a), the same rule found frequently in later Greek geometric writings. Both these examples for roots illustrate the Babylonians’ arithmetic approach
in geometry. They also show that the Babylonians were aware of the relation between the hypotenuse and the two legs of a right triangle (now commonly known as the Pythagorean theorem) more than a
thousand years before the Greeks used it.
A type of problem that occurs frequently in the Babylonian tablets seeks the base and height of a rectangle, where their product and sum have specified values. From the given information the scribe
worked out the difference, since (b − h)^2 = (b + h)^2 − 4bh. In the same way, if the product and difference were given, the sum could be found. And, once both the sum and difference were known, each
side could be determined, for 2b = (b + h) + (b − h) and 2h = (b + h) − (b − h). This procedure is equivalent to a solution of the general quadratic in one unknown. In some places, however, the
Babylonian scribes solved quadratic problems in terms of a single unknown, just as would now be done by means of the quadratic formula.
Although these Babylonian quadratic procedures have often been described as the earliest appearance of algebra, there are important distinctions. The scribes lacked an algebraic symbolism; although
they must certainly have understood that their solution procedures were general, they always presented them in terms of particular cases, rather than as the working through of general formulas and
identities. They thus lacked the means for presenting general derivations and proofs of their solution procedures. Their use of sequential procedures rather than formulas, however, is less likely to
detract from an evaluation of their effort now that algorithmic methods much like theirs have become commonplace through the development of computers.
Mathematical astronomy
The sexagesimal method developed by the Babylonians has a far greater computational potential than what was actually needed for the older problem texts. With the development of mathematical astronomy
in the Seleucid period, however, it became indispensable. Astronomers sought to predict future occurrences of important phenomena, such as lunar eclipses and critical points in planetary cycles
(conjunctions, oppositions, stationary points, and first and last visibility). They devised a technique for computing these positions (expressed in terms of degrees of latitude and longitude,
measured relative to the path of the Sun’s apparent annual motion) by successively adding appropriate terms in arithmetic progression. The results were then organized into a table listing positions
as far ahead as the scribe chose. (Although the method is purely arithmetic, one can interpret it graphically: the tabulated values form a linear “zigzag” approximation to what is actually a
sinusoidal variation.) While observations extending over centuries are required for finding the necessary parameters (e.g., periods, angular range between maximum and minimum values, and the like),
only the computational apparatus at their disposal made the astronomers’ forecasting effort possible.
Within a relatively short time (perhaps a century or less), the elements of this system came into the hands of the Greeks. Although Hipparchus (2nd century bce) favoured the geometric approach of his
Greek predecessors, he took over parameters from the Mesopotamians and adopted their sexagesimal style of computation. Through the Greeks it passed to Arab scientists during the Middle Ages and
thence to Europe, where it remained prominent in mathematical astronomy during the Renaissance and the early modern period. To this day it persists in the use of minutes and seconds to measure time
and angles.
Aspects of the Old Babylonian mathematics may have come to the Greeks even earlier, perhaps in the 5th century bce, the formative period of Greek geometry. There are a number of parallels that
scholars have noted. For example, the Greek technique of “application of area” (see below Greek mathematics) corresponded to the Babylonian quadratic methods (although in a geometric, not arithmetic,
form). Further, the Babylonian rule for estimating square roots was widely used in Greek geometric computations, and there may also have been some shared nuances of technical terminology. Although
details of the timing and manner of such a transmission are obscure because of the absence of explicit documentation, it seems that Western mathematics, while stemming largely from the Greeks, is
considerably indebted to the older Mesopotamians.
Mathematics in ancient Egypt
The introduction of writing in Egypt in the predynastic period (c. 3000 bce) brought with it the formation of a special class of literate professionals, the scribes. By virtue of their writing
skills, the scribes took on all the duties of a civil service: record keeping, tax accounting, the management of public works (building projects and the like), even the prosecution of war through
overseeing military supplies and payrolls. Young men enrolled in scribal schools to learn the essentials of the trade, which included not only reading and writing but also the basics of mathematics.
One of the texts popular as a copy exercise in the schools of the New Kingdom (13th century bce) was a satiric letter in which one scribe, Hori, taunts his rival, Amen-em-opet, for his incompetence
as an adviser and manager. “You are the clever scribe at the head of the troops,” Hori chides at one point,
What is known of Egyptian mathematics tallies well with the tests posed by the scribe Hori. The information comes primarily from two long papyrus documents that once served as textbooks within
scribal schools. The Rhind papyrus (in the British Museum) is a copy made in the 17th century bce of a text two centuries older still. In it is found a long table of fractional parts to help with
division, followed by the solutions of 84 specific problems in arithmetic and geometry. The Golenishchev papyrus (in the Moscow Museum of Fine Arts), dating from the 19th century bce, presents 25
problems of a similar type. These problems reflect well the functions the scribes would perform, for they deal with how to distribute beer and bread as wages, for example, and how to measure the
areas of fields as well as the volumes of pyramids and other solids.
For larger numbers this procedure can be improved by considering multiples of one of the factors by 10, 20,…or even by higher orders of magnitude (100, 1,000,…), as necessary (in the Egyptian decimal
notation, these multiples are easy to work out). Thus, one can find the product of 28 by 27 by setting out the multiples of 28 by 1, 2, 4, 8, 10, and 20. Since the entries 1, 2, 4, and 20 add up to
27, one has only to add up the corresponding multiples to find the answer.
Computations involving fractions are carried out under the restriction to unit parts (that is, fractions that in modern notation are written with 1 as the numerator). To express the result of
dividing 4 by 7, for instance, which in modern notation is simply 4/7, the scribe wrote 1/2 + 1/14. The procedure for finding quotients in this form merely extends the usual method for the division
of integers, where one now inspects the entries for 2/3, 1/3, 1/6, etc., and 1/2, 1/4, 1/8, etc., until the corresponding multiples of the divisor sum to the dividend. (The scribes included 2/3, one
may observe, even though it is not a unit fraction.) In practice the procedure can sometimes become quite complicated (for example, the value for 2/29 is given in the Rhind papyrus as 1/24 + 1/58 + 1
/174 + 1/232) and can be worked out in different ways (for example, the same 2/29 might be found as 1/15 + 1/435 or as 1/16 + 1/232 + 1/464, etc.). A considerable portion of the papyrus texts is
devoted to tables to facilitate the finding of such unit-fraction values.
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Standard Model - Cosmos
The Standard Model of particle physics has been well measured from almost every angle. Here’s a scorecard of the results. The currently accepted and experimentally well-tested theory of
electromagnetic and weak interactions is called the Standard Model. The Standard Model is based on relativistic quantum gauge field theory. When physicists in the 1920s tried to […]
Relativistic quantum gauge field theory- Gauge bosons Read More »
Classical Physics vs Quantum Physics- The major Difference
What is the difference between classical physics and quantum physics? Physics that studies and explains the phenomena that occur in the domain of atoms, their nuclei, and elementary particles are
called quantum; and the basic mathematical theory that explains the movements and relationships in this field is called quantum mechanics. However, it should not be
Classical Physics vs Quantum Physics- The major Difference Read More »
Particle Physics Standard Model
What is the Standard Model of Particle Physics? A theory explaining the relationships between known fundamental interactions between elementary particles that make up all matter is the standard model
of particle physics. It is a quantum field theory that is consistent with quantum mechanics and special relativity that was developed between 1970 and 1973. Nearly | {"url":"https://cosmos.theinsightanalysis.com/tag/standard-model/","timestamp":"2024-11-14T04:59:42Z","content_type":"text/html","content_length":"140456","record_id":"<urn:uuid:6d740bb5-2306-4704-929d-163972f3fc5e>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00876.warc.gz"} |
Inference of the two dimensional GPR velocity field using collocated cokriging of Direct Push permittivity and conductivity logs and GPR profiles.
Gloaguen, Erwan; Lefebvre, René; Ballard, Jean-Marc; Paradis, Daniel; Tremblay, Laurie et Michaud, Yves (2012). Inference of the two dimensional GPR velocity field using collocated cokriging of
Direct Push permittivity and conductivity logs and GPR profiles. Journal of Applied Geophysics , vol. 78 . pp. 94-101. DOI: 10.1016/j.jappgeo.2011.10.015.
Ce document n'est pas hébergé sur EspaceINRS.
One of the main difficulties in processing GPR surface data is to infer the ground velocity models in order to convert the radargram time scale into depth. Common conversion techniques usually fail
if the ground lithology is complex. Consequently, the ground velocity has to be guessed and generally assumed laterally invariant. In this paper, we present a new geostatistical approach to the
velocity analysis of sandy aquifer based on the interpolation of the Direct Push relative permittivity and conductivity logs. The first step of the method consists in scaling the radargram to
maximize the correlation with the logs. Then, collocated cokriging of electrical relative permittivity and conductivity and the scaled radargram information is computed. The cokriged fields are
converted in velocity field following EM hypothesis and the time to depth conversion of the original radargram is then applied using the inferred velocity field. Results on real data analysis show
that the method works well in fairly homogeneous sandy aquifer with intercalated dipping thin silt layers.
Type de document: Article
Mots-clés libres: CPT; geostatistics; GPR; multivariate analysis
Centre: Centre Eau Terre Environnement
Date de dépôt: 19 oct. 2018 14:15
Dernière modification: 19 oct. 2018 14:15
URI: https://espace.inrs.ca/id/eprint/7245
Gestion Actions (Identification requise) | {"url":"https://espace.inrs.ca/id/eprint/7245/","timestamp":"2024-11-10T07:27:09Z","content_type":"application/xhtml+xml","content_length":"21098","record_id":"<urn:uuid:ac269858-e76d-4c56-bc89-cf1cf40545c1>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00014.warc.gz"} |
Catrice Nymphelia LE (review, swatches)
Catrice Nymphelia limited edition is out! At first inspection of the stand I bought 3 nail polishes. Brown one wasn’t that special and I already have Rambo No. 5, so those two I left in the store. I
was sooo sad that all the blushes were sold out … and I wouldn’t even want it that much if Maestra didn’t say how lovely the tester looked on me. But my darling came to the rescue and sacrificed part
of his lunch hour to go to the store where it was still available.
Catrice Nymphelia LE
Catrice Be Pool was the first shade I tried out and first time I used new brush. You can see how that went in Catrice Nymphelia video. :D It was kind of stressful moment for me – can you imagine how
disappointing would it be if one of your favorite brands switched brush to something unusable? Thankfully Catrice didn’t go overboard with width, so the brush is still OK. Maybe those of you with
smaller nails will have to switch techniques, but as far as I can see majority of “non-polish-addict” users prefer one (2 max) stroke application anyway.
Be Pool is limited edition color, so if you love this gorgeous blue-green shimmery nail polish, now is your chance to get it. I used only one coat of base coat but there was no staining whatsoever.
Will be reaching for this shade again.
It seems to me that Catrice changed their formula – polishes are not as thick as they used to be … at least those nail polishes that I tried so far. First five nails were little bit awkward to apply
(I’m sure camera in front of me had nothing to do with it … yeah right), but then application went smoothly. I used 2 coats + top coat.
Catrice Nymphelia LE Be Pool swatch
Catrice Fred Said Red has even thinner formula than Be Pool. This pretty warmer medium red will be part of regular line, so no rush to get it in this limited edition. VNL is still visible after two
coats, but if VNL does not bother you – 1 coat can be enough for nice application. I flooded my cuticles twice as I kept expecting thicker nail polish. Shame on me. 2 coats + top coat.
Catrice Nymphelia LE Fred Said Red swatch
I apologize in advance for disastrous swatch of Catrice Salmon&Garfunkel shade. This nail polish looks to me like nude apricot shade. Do keep in mind that it looks slightly darker on nails than it
does in the bottle. One moment this shade is a little more on pinkish side and other on orangey one. Of course my camera saw it as heavy duty orange and when I tried making photo with camcorder it
came out completely pink. fail. This polish look a lot prettier IRL. Most realistic color is shown in video at the end of the post. Color as shown on the photo is the one I see under artificial
lighting (but less on the orange side). Application was pretty nice for almost pastel like nail polish. 2 coats + top coat.
Catrice Nymphelia LE Salmon & Garfunkel swatch
Catrice Nymphelia LE Dancing Nymph marbled baked blush is also available only in this limited edition. I’m not a huge fan of shimmery blushes, bronzers and highlighters, but I like this blush. I
can’t really see any shimmer on my face, more of a glow … but then again I have no idea how will Dancing Nymph look like in sunlight … hopefully not to shimmery. Brownish-orange color is perfect for
me (I have pale, yellow to neutral skin-tone). I like the fact that blush is appropriate for those of us who are blush shy, but at the same time buildable for those of you who want more color.
Catrice Nymphelia LE Dancing Nymph baked blush swatch
The second time I stood in front of Nymphelia stand, I bought Have You Seen Alice? Intensif’ Eye baked eyeshadow. I liked both eyeshadows, but as they’re not that cheap I decided to get greenish-blue
shade with pretty dark base. I applied it over Rival de Loop esb and it turned out as dark as you can see on the swatch. On the eyes I applied it over Essence It’s A Snow-Woman’s World liquid
eyeshadow and it came out a lot lighter.
Catrice Nymphelia LE Have You Seen Alice? baked eyeshadow swatch
• 10ml (0.33 fl.oz.) of nail polish – 2,59 EUR
• 6.3g (0.22 oz.) of blush – 4,69 EUR
• 0.8g (0.028 oz) of eyeshadow – 3,99 EUR
Video review:
Have you tried new Catrice nail polish brush? Do you like it more or less than old one?
This post is also available in SLOVENŠČINA.
24 thoughts on “Review and swatches: Catrice Nymphelia LE”
1. Catrice has pretty stuff!
2. I haven’t seen the LE here, but the new products that are permanent are already in the stores. I bought Salmon & garfunkel. It’s such a pretty polish!
3. I’m so jealous, everything looks so pretty! I want Catrice in Greece NOW! :D x
4. Mislim da mi bo novi čopič še bolj všeč, ampak odtenki me nekako niso pritegnili tako da bom počakala na redno linijo, ki je sicer že v Mullerju!!!!
Opazila sem tvoj nenavaden način nanašanja laka od krivini, jaz imam vedno težave tam in vedno porabim veliko časa, pri tebi je pa to en, dva tri in si končala
Naslednjič bom tudi jaz preizkusila Gejba tehniko :yes:
5. joooj, meni pa nov čopič sploh ne odgovarja. Sumim, da imam zanj prekratke nohte in ga zelo težko obvladam pri nanosu. :) ampak, ker so laki tako noro lepi, pač malo potrpim in nanosu posvetim
več časa. :yes:
6. I love Be Pool, it’s gorgeous! :wub: I hope we get this collection here as well.
7. Be Pool is gorgeous! I love the shimmer.
8. Salmon and Garfunkel, I love the colour and the name too!! :haha:
9. I hope I can find Be Pool over here! I can’t resist greens, that’s it.
Is Salmon&Garfunkel similar to I Scream Peach by Essence? I love this colour but I know the Essence one just makes me look like a shrimp.
□ Hm – I only have I Scream Peach from Catrice and that one is not similar to Salmon&Garfunkel. Its a lot darker, coral/pinkish shade.
10. Joj, senčka zgleda super, zdaj mi je pa žal, da je nisem porajtala. Saj v bistvu ne vem, če je sploh še bila na stojalcu … Zdaj je verjetno nikjer več ne dobim, v Ljubljani imamo nesramno malo LE
od Catrice na voljo.
□ Senčke bodo v redni prodaji, tako da bo Have You Seen Alice? kmalu na vseh stojalih. :yes:
Že tako imamo na voljo premalo Catrice stojal, potem pa se ne odločijo niti za vsaj 2 LE stojali. :sigh:
11. nemam zelju za nicim iz ove kolekcije, a nadam se samo da mi nove cetkice nece bit prevelike, bojim se da bi mogle bit i ove stare su mi vec skoro prevelike :ermm:
□ Nadam se, da će ti biti OK, pošto su boje i formula super. :nails:
12. Hmmm..potem praviš, da je Be Pool bolj modro-zelen? Ker, jap, na promo slikcah mi je bolj zelen zgledal, na nekaj swatchih skoraj samo moder – kar je pa čist preveč modre zame :))
Salmon&Garfunkel bo pa treba v živo vidt, se mi zdi, da ima dober potencial.
□ Manj je zelen kot v filmčku in bolj kot na swatchu … vsaj v trenutni svetlobi. :silly:
13. Be Pool in pa Salmon & Garfunkel. :wub:
□ Mhm. :nails:
14. Kako to da je ova kolekcija već stigla.. nadam se da će onda uskoro i kod nas u Hrvatskoj stići, posebno mi se sviđa rumenilo..
□ Novi sortiment je nekud već postavljen, tako da ova LE već malo kasni. :silly:
15. Novi čopiči izgledajo vrhunsko! :w00t: Že zaradi tega bom morala nabaviti kakšen lak, hihi.
Zelo uporaben video. :thumb:
□ Mislim, da niso tako gromozanski kot tvoji najljubši, ampak bi rekla da ti bodo všeč. :biggrin:
16. I like Be Pool and Salmon&Garfunkel, they look really good :)
□ They’re even prettier IRL. :wub: | {"url":"https://www.parokeets.com/en/2012/02/review-and-swatches-catrice-nymphelia-le/","timestamp":"2024-11-06T04:52:31Z","content_type":"text/html","content_length":"144288","record_id":"<urn:uuid:ddb7d4fc-852e-40e9-8d1a-090f49079b41>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00304.warc.gz"} |
Yuchong Zhang
I am an Assistant Professor in the Department of Statistical Sciences at University of Toronto. I received my PhD degree from the University of Michigan in 2015, under the supervision of Prof. Erhan
Bayraktar. Before joining U of T, I worked at Columbia University as a Term Assistant Professor.
I am on leave in 2023.
Here is my CV.
Research Interests
Mathematical finance, stochastic control, game theory, applied probability.
• ACT240 Mathematics of Investment and Credit, Fall 2022.
• ACT460/STA2502 Stochastic Methods for Finance and Acturial Science, Fall 2018, Fall 2020, Fall 2021.
• STA4526 Stochastic Control and Applications in Finance, Fall 2019, Fall 2021.
• STA2570 Numerical Methods for Finance and Insurance Winter 2020 (1st half), Winter 2021, Winter 2022.
• STA4246 Research Topics in Mathematical Finance, Winter 2019 (1st half), Winter 2020, Winter 2021, Winter 2022.
Publications and Preprints
Selected Awards
• NSERC Discovery Grant RGPIN-2020-06290, 2020-2025.
• NSERC Discovery Launch Supplement DGECR-2020-00373, 2020-2021.
• NSF Grant in Applied Mathematics, DMS-1714607, 2017-2020 (terminated in 2018 due to moving to a non-US institution).
• SIAG/FME Conference Paper Prize, 2016.
• Rackham International Student Fellowship, University of Michigan, Spring/Summer 2012.
• Hong Kong Jockey Club Scholar, 2008.
• Admission scholarship covering four-year tuition and living expenses, CUHK, 2006. | {"url":"https://utstat.toronto.edu/yuchong/","timestamp":"2024-11-13T17:34:46Z","content_type":"text/html","content_length":"10151","record_id":"<urn:uuid:82b02ffc-8625-4c8f-93fb-cf84e33ffb3e>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00405.warc.gz"} |
Office Pro - Teachers Lounge March 25, 2021: Helping Students Excel on the MOS Excel Certification Exam
Watch Video
Our star trainers Stevie George and Allan Escobar are at it again! After a very successful first Teachers Lounge focusing on Word, we are setting up another Teachers Lounge to focus on how to help
your students pass the MOS Excel Certification Exam using TestOut Office Pro.
Join Stevie and Allan along with some amazing instructors as they discuss what has been effective in their classroom when teaching Excel charts, tables, formulas, and other questions you've had about
preparing students for the MOS Excel Certification Exam, including the following:
1. How to get students more experience when all they have is a Chromebook? Since our students use Chromebooks. Is there a shortcut, other than F4, for doing the absolute cell references?
Michelle: I have never actually had a Chromebook, so I was a little bit taken aback by this specific question, but my students have them all the time. One of the things that I did to find out
some of those answers was I used CertMaster . CertMaster , has a support window that says, “Hey, how can we help?” And as you scroll down the support window, you'll see that ‘Tips for
Chromebook users’ is the second article. It talks about how to deal with slowness or unresponsiveness and various things about the Chromebook and different things that you can do. And then
this one says “Other common issues and workarounds like if you're unable to use the function keys and like I said, I can't do this on my computer because I don't have a Chromebook.
Also, you were asking about how to get more experience out of a Chromebook. When students have a Chromebook, a lot of times they can't do every single thing that's out there. They can't just
open, for example, Excel and do the harder stuff that's Excel Expert related. There are a lot of things you just can't do in a Chromebook because Chromebooks aren't strong enough. We were
lucky enough in in our county that they purchased a program called Cameo, which is basically just online Microsoft Office. CertMaster has some actual Excel activities. And you [can] to save
them in your computer and upload them into Cameo and they'll work even on the Chromebook.
Kim: We have gone this year as one-to-one. So, all of our students do have Chromebooks. Fortunately, I still teach in a PC lab. So, when students are on campus, they can get into Excel and
get real application experience. But yes, CertMaster is great to have the feature to be in a simulated environment so that [the students] can get that practice. The Excel online through
Office 365 is a stripped-down version, but they can still get some practice because you can still do your functions and you can still do some formatting and, some things like that. So, [for]
coming up with some of your projects that you may do on a PC, we would have to kind of water them down a little bit. But, if you had to, students could get extra practice with projects using
Excel online also.
Gwen: We also are using Chromebooks, and I'm also very fortunate to have a PC Lab. At first, we were not one-to-one prior to COVID. And we were told in our county, “We’ll give you Office 365.
You can do everything you need with Office 365, with your Microsoft classes.” And the first three weeks were a nightmare. And then wonderful CertMaster , came into my life. In the Word
meeting somebody mentioned it just being like a blessing. And it truly was, it was a game changer for us and our whole county. It was several months before everybody had Chromebooks but, what
CertMaster did was allowed it for students on MacBooks, PCs, [and] the ones that did have Chromebooks, we were all able to do the same work. For that period of time, I was strictly using
CertMaster . I could not do anything interactive with actual Microsoft Office that was live, because we just had far too many scenarios. But I asked for feedback from my students about their
views on CertMaster , and they all loved it. Because, for the first time, they were actually able to learn and practice, and get immediate feedback. Even for the ones that did have Excel
installed on their computers, they would have to share their screen [and] I'd be trying to analyze it to see if it was correct. And it was just not very effective.
About the F4 key, where it's going to make a cell absolute. I've taught all of mine how to manually do dollar signs if they want to, as well. And so that way if that's the main thing, [that]
they need the function keys for [in] Excel, then they can just do the dollar signs in the formula bar. And it’s just a nice little quick way to get to it.
2. I am using Office Pro with 9th graders. I asked them what they found most confusing. To quote: "Why do they tell us what to do first as the last sentence of the directions?".
Kim:This is a very valid question, because [the course developers] don't want to come run out and give you the answer or give you too many clues. I have found that a strong vocabulary is very
important, and that's why [I use] all of the CertMaster lessons. And I have quizzes in different resources that I use to help reinforce that vocabulary so that they have an understanding of
the concepts and they aren't just memorizing steps. And so that's very important. I've referred back to the Word Teachers Lounge, and Kelly on there had discussed how you needed to break down
the task, read the entire task, find the location on the document, find location on the ribbon, and then the action. And it works the same way with Excel and even more so. Break it down,
first look and see what worksheet you're going to be on because that is common, even in CertMaster . I'm so glad that CertMaster works that way. It doesn't automatically bring up the
worksheet that you're going to do the task on. Neither does the Microsoft Office test. So, you have to first verify you're on the correct worksheet. Then, look at the cell or the range that
you're working with, and then what command are you looking at. As an example of how a task may be written, be sure that you read the entire task. And I always tell my students, if you're
unsure about it, don't waste time because that's a big factor: You don't want to run out of time. You can pass the test without answering every task. You want to make sure you get to the task
that you know how to do. If you're unsure about it, mark it for review and move on. But if you do want to tackle it, then be sure again that you look at what worksheet you're going to work
on. Are you on that worksheet? If not, switch to the worksheet. Then what cell or range are you going to work on? And then, once you get that selected, you can find out what the task is. Then
find the ribbon or command that you need in order to complete that. That's what I advise my students to do to just be sure to read the entire task. Don't get too bogged down on all the extra
words that there might be. Just pick out the worksheet, the cell range, and the task, and you can probably figure out that you know how to do it.
Michelle: I saw the Word presentation a couple of weeks ago and one of the comments that they made was about it being difficult to understand the questions. I did not feel that way, because I
take the students, and I'll highlight the words in the questions, and I'll point at the screen, and match up those words in the questions with the thing in the ribbon [that] they're supposed
to look for. And if they can just take and read the whole question and I really think that's so important for all students and adults too. Reading the whole question just makes a huge
difference in whether or not they will understand it. Read it all the way to the end of the period. Sometimes I even make them read it out loud.
Gwen: We are [on an] A day/B day [schedule]. On [the student’s] off days, they’re 100% [on] CertMaster . And on their on day, we are working together in the lab using Excel. I always try to
use information that they understand, like payroll, or car payments, which is one thing I'm going to show in a minute. Things that they're like, “Okay, yeah, I have a job, I get a paycheck.”
And I try to make it where they then can understand things like net pay. I think when they're working in CertMaster , or doing the certification test, they need to first look at their data,
figure out what type of information it is. So, then when they're telling you to do a function or format it, in a way it makes a little more sense to you because you understand the data
itself. But I do think it's very important to read the whole thing and then, of course, breaking down that question. The very first thing I do is say, “What worksheets should you be on? Get
there first. Now, what range were they talking about? What's in that range? Look at your data, now what are they telling you to do?” And if you break it down, it really does become a totally
different type of task. And one other thing that I'll say is that I frequently tell my students, “Pretend you're telling someone that knows nothing about Excel, how to do it.” You would first
have to tell them to go to that area and select it. And we have to tell the computer, exactly what to do, as if it was a person that had no clue. I always try to get them to think about it
that way instead of thinking [that] the computer can figure out [that] they meant a cell they didn't have selected. I do have ninth through twelfth graders, but we do require at our school
that you take Word before you take Excel. So, I don't have freshmen in my Excel class for that reason.
3. How to help students with What-if/lookup functions. Anything specific you do to help students remember?
Gwen: Before I started teaching, I used both of these religiously. I frequently tell [the students that] this might not seem like a big deal, but you will love it if you can use it. I did
make up a little example here. I have two different things here. The first thing was, we were talking about car payments and how much of a car you can afford. When it comes down to it, it's
really how much of a monthly payment can you afford? So, we were doing this for two different reasons: one, using the payment function, and the other, to say, “how much do you think you can
afford in a car payment?” Most people are like, “Oh, I'm going to buy this car, it’s 35,000, and then they calculate the payment they’re I like “Oh!”. So, we actually have a lot of fun with
this because I first ask about how much money they would like to spend on a car when they're 30 years old. And then we run the monthly payment and it's very high. I think it’s like $600 or
more. Then what we do is we talk about how much a month could you afford? And my students figured out they can afford about $250. That would be about where they were looking. So we already
had it set up for our payment calculator under Columns A and B, and so then we use the What-If [analysis], to see, “If I can only afford $200 a month, how much can I borrow?” I try to make it
first, where we start trying to figure it out ourselves. So, we get in here, and we start playing around with the price. We play around with it and they're trying to hit the right mark. And
so, then I'll tell them how the What-If will tell you straightforward what that amount is. I don't know if they want me to show it work but we always get something very real for them first.
And I have just set this up so that we can play with formatting, borders, and shading. When I go to my Data tab then, to do my What-If. And I'm going to use Goal Seek for this one, because my
goal is to get a payment of a certain amount. And I really break this down.
To zoom in on my screen I’m going to show the Windows and Plus key shortcut. When you do the Window and Plus [key], it'll zoom in on anything. When we are doing functions, I always make them
open up the dialog box, and my students in Zoom, and my students in my class, can't see it well on my board. And so, if you do the Window and Plus [key], it'll zoom in on the whole window,
and your ribbons, everything. It's really nice. It's been a game changer for us.
So then I’ll use the What-If, then I use the goal seek. This seems kind of confusing unless you really just read it like a sentence, or like a question. I always have them read the whole
thing first, before we begin. It says, set a certain cell to a certain value by changing another cell. When we do this as a class if I tell them to do it by themselves to begin with, they
don't know where to go. If I speak it to them, they can answer the questions like that. And so, I try to point that out to them. Pretend you're talking to a person. I'll say, “Help them to
tell me which number we are looking for?” We're looking for our monthly payment to be a certain amount. And so, what cell is that in? That’s in cell A6, and what value do I want it? If I want
it to be a $200 a month payment, then I'm going to get 200 and what cell would need to change? And so, we talked about a scenario where the only thing that really is going to vary here is our
price, not our down payment. So, that's what we have saved. Then we just click on the price and then click OK. And then, it lets us know at $17,000 with a 5000 dollar down payment for five
years, that would be $200 a month. And that's with zero percent interest. When we did this project, I had a student that said, “I know I will spend the money to have Excel once I get out of
school.” Because we do things that they can relate to.
And one thing I was just going to say with Lookup functions. What I do with my students live is to try to give them practice. And also, for those that are on Chromebooks, we can still do a
tremendous amount with formulas and functions with the Chromebook. We can't do things like naming ranges the same way that we can in Excel. But it works out pretty well. I'll give them a
spreadsheet, and they've got all the information they need. What they don't have is the unit cost. Well, that's where a Lookup function comes in the hand because I have another tab with all
the unit costs. That's where they have to do the Lookup function to pull up the unit cost. And one thing that I always try to explain to them is how important it is to get a spreadsheet set
up to be useful in the future. So, if this was my business and then the price of baby food went up. I want to change it in one place, not in 10,000 rows of information. We did the same thing
with the payroll. We had the different payroll taxes on a separate sheet, and I make them do external references. And that's a great way for us to practice absolute cell references. With this
one, this is the type of project that I'll do closer to the end the week. I'll tell them to complete my form and they're like, “Well, I don't know the unit cost. I say, “You have to do a
Lookup function. And how would you calculate total revenue? And how would you do a total cost and total profit?” And it's a great way for them to see the relevance of Excel. And I always tell
them there's not one correct solution. There’s one correct answer, but there's lots of solutions. I just strongly encourage them to look at my data and look at my column headings. You know
sometimes they won't even look at the headings because they're kind of used to learning material that's memorization. But when they make you do word problems in math, or when they make you do
a math problem and say, “How did you know how to find the answer?” I tell them, “Excel is totally right there. You have to tell the computer exactly how to do it.” We have a lot of fun with
Lookup functions and with What-Ifs.
Michelle: When I do Lookup functions, one of the things that I remind them is that after they type that equals If or equals Vlookup or whatever function it is that they're doing, if they'll
just look a little bit below where they're typing, Excel will tell them exactly what to type in next. It'll tell you to equals If and then what to look for. Then, to put the comma, and you
got to remember to not put the space, and it demonstrates after the comma. Then, it tells you what to put if it's Yes, if it's true. It says it right there, underneath what you're typing.
What do you want it to say if it's true? Then, comma and value if false. What do you want to say if it’s false? Then, close the parentheses and move on. Excel is very, very user friendly in
that, it seems like it's so complicated to write some of these equations, but what Gwen said about reading them out loud as you go, it will really tell you what to put next a lot of times.
Kim: Yes, I agree with trying to make it real for them. And I've had a few students who have went on to Excel Expert and have achieved that which I'm very proud of. For those [students], I
have some projects that I add an extension to. Maybe assigning a letter grade to the gradebook that we have using V lookup. It’s something that they can grasp and understand. And so, that
just kind of helps them understand the concepts and that they can hopefully use to achieve that Excel Expert. Or even if they're not going for that, as they get into college or work, that
they can see the value of those functions.
Gwen: One thing I'd like to add onto that, is that in our district, my classes are Excel honors, so, it’s Excel and Excel Expert. And before COVID, I would spend the first six weeks doing
Excel and then the remainder of it doing Excel Expert because it’s so extensive. Because of COVID, I only have my students two days a week on site where I can work with them instead of five
days. What I've done this semester, and it's worked out fantastically, is I incorporate Excel Expert at the same time I'm doing regular Excel. And an example of that is when we did some
functions and then I took it to some IFs and then SUMIFs. Then I made them do if statements based on the SUMIFs. And they were going seamlessly. They didn't know they were getting to an
expert level. And one day after we had been doing some IFs, extensively, I asked, “Do you find this very difficult now?” They said, “No.” And I said, “Well, this is expert.” And they said,
“Really? That's awesome.” I replied, “Yeah, you’re just that smart.”
4. A lot of my colleague’s students have already told her that they do not wish to take the MOS Excel exam because they fear the formulas. How do you help students understand this?
Gwen: Formulas are what's fun to me and what I try to do for my students is tell them to think about it like a video game or a new smartphone. And a lot of these students are like, “Well, we
haven't learned that.” My goal is to teach them that Excel can do unbelievable things and Excel has a lot of ways to help you when you're doing a function. We can pull up the dialog box, or
the arguments, and it will give us descriptions of what to do. What I want to teach them is not how to do every function in Excel. I don't have time to do that. What I want to teach them is
how to use the resources in Excel. And I explain it to them in the concept of a video game or smartphone, and I ask how many of them went to the Verizon store the first time they got a
smartphone and took the classes. Because you can take classes on how to use a smartphone. And they say, “Oh, Ms. Barnes, we didn't do that.” And I say, “Well did your parents teach you? Who
taught you how to use that? Because it's very complicated.” They say, “I figured it out myself. And same thing with video games.” And I'm like, “You know when you get a new video game, do you
just sit down for a little bit and read the manual to learn how to do it?” And they're like, “No, I just start playing and figuring it out.” I tell them to think about Excel like a video
game. And it's a challenge.
With the data, for example this is about selling these different items, like baby food and it’s to different countries. And what I do is I ask them questions like “How much baby food do we
sell to Europe?” And that's my question. And they have to figure out how to get an answer for me. I like for them to tackle it like a video game. I think it's in Mario Kart where you can
bounce up and get little coins. Well, I don't know how you figure that out. I figured it out because my kids told me. And so, I teach them that there are all kinds of little tricks like that
in Excel. And one is for example, go into the Formulas ribbon. And if I'm having to do something with text, then go look at my options in Text.
And one thing that I just want to mention, and I will use the V lookup as an example, but if I know it's Vlookup and I start my formula, once I do the open parentheses, I can see [hints] at
the bottom. That's great and very, very helpful if you're familiar with it. And [for] some functions, it's very easy. But for every formula like this, I go ahead and do Ctrl A. And what
that'll do is go ahead and open up the dialog box where I can see all the arguments. You can also click on the little FX to the left of the formula bar. And the key is going to open up your
parentheses. It’ll go straight into the arguments for that function. I make all of my students read out loud to me what a lookup value is. And I point out to them that this information is
crucial. And with that, you can do almost any function whether you've seen it or not. So, it's really nice, it works out really well. Also, a big thing with a V lookup is your data source has
to be sorted and they might forget that. But if they've done what I told them, and they read these instructions, it says, “The table must be sorted in ascending order.” So, if they forgot and
they read that, then they’re like, “Oh, wait, I've got to stop for a minute and make sure that my tables are in the correct order before I get started.” And also, with this pulled up, as I
began filling in my information to the right of the equal sign, it should be showing information. So, for this V lookup, I want to be looking up baby food, then where it says equal, it should
be showing the word baby food there. Or if it was a number, or if it was a range, it should show all the values in that range. And I tell them that that way they can tell step-by-step, if
they're maybe doing it correct or not because they can get immediate feedback before they even click OK. Even if my students think they don't need a dialog box and make them open it up. So,
this really helps. And I do it like a contest. I’m very competitive. I'm not an athlete, I just love a challenge. I'll have students in Zoom, and I'll have students in my class. The ones in
Zoom, as soon as I get an answer, they put it in chat just to me. And I'll give a shout out to that first person with the correct answer. For those in class, one thing that's always been
really fun is, I’ll say, “When you think you’ve got the right answer, turn off your monitor.” And they will turn off their monitors, they'll be jumping around and see if anybody else has
their monitor off. And the one thing I do is have them all turn on their monitors before it starts to become uncomfortable for the student that doesn't have it. So, I only use it for that
very first [student],… and we really do it like a game. I will tell them how to do a function to begin with, but after that, I won’t even tell him what function we're going to use if we've
already studied it. I want them to learn to figure out what function should they use.
One of the more interesting things I found with Expert is that when they have been used to doing functions but [in certain scenarios] it's just a simple formula that they need, they try to
overcomplicate it. And so, I frequently incorporate things that are very simple as well to get them out of thinking, “Oh, we're doing Ifs right now.” And it really works well. I really did
have a good time with my class, and they find it fun. I would say, for those that are not wanting to do it on the certification test, then that to me, means that they're not understanding how
to figure out what they need to use. And so, you could just make a few little setups like this sheet that I have up where we're going to be doing multiple formulas and functions, and I want
them to try to figure it out. One thing that's just worked out the best for us is when I make it a challenge.
Kim: Just making them comfortable with that function dialog box is a big step. Once they can learn how to use that and read the information there, and even when they start selecting their
cells or putting in their greater than or less than symbol and it says that it's false and they're like, “Wait a second. That's supposed to be true. Oh, I got my symbol turned around.” If
they can learn how to use that then that really helps them understand it better.
Michelle: I actually start with going over what PEMDAS is so that they'll understand that we are using the same thing that they're using in a math class. And a few of them don't remember it
or didn't learn it right. But most of them are really excited about like, “Oh, this is something I can relate to.” I start with using that and then I explain the difference between a formula
that would use the things that we use in PEMDAS and a function which would use ranges and, named ranges and all those other things. It's exciting for me and I know some of the students get a
little bit worried about those formulas. The really good thing about them is that it doesn't matter what kind of computer you have or what kind of software you have, you put that equal sign
down, and you can start typing that formula or that function. I actually had a last-minute thing where I had to walk into an Apple lab because they were doing something in my lab. All of a
sudden, my computers didn’t work and the only lab available was the Apple lab. I went into the Apple Lab knowing nothing and I had to teach a Numbers class with my Excel students. And I told
the teacher, “I don't know what to do with this.” She said, “What are you teaching?” I said, “Excel.” She said, “Well, just use Numbers.” And I said, “What's Numbers?” And so, it works. You
put the equal sign down and it still worked even in that Apple lab. I've even used it with OpenOffice and Office 365 and all that. The formulas and functions can be done with almost any type
of software.
5. What project-based assignments have you used to engage students in authentic learning?
Kim: I have a few different projects. The one that I had replied about on Facebook was using our football stats. Back a few years ago, our junior high football team was on a winning streak
and we won 50 something straight games in a row. I took that real data, and we went so far as to insert the data from a text file, to creating it as a table, and then this is the solution
here. I've got the actual data file here with instructions. I usually do this as with the students as a class. But of course, with COVID you may have to revise it so the students can do it on
their own. But we will go through and work through this. And then over here in Column J, we have all the different functions that we would use like which function would we use to find out how
many years our winning streak went through, our average score, their max score, total zero-point difference. Those are just simple formulas or functions. In the actual columns, we did a
subtraction for the point difference. We figured we’d do an If function for, “Was this a mercy rule?” We don't really have enough information to know if a mercy rule is called or not. So, we
just say it is a possible mercy roll if the score difference was greater than or equal to 35. And was it a shutout? Did the opponent not make any points whatsoever? And they really like
seeing that. And as you can see in the solution, we even did some conditional formatting to highlight, “Yes, we had those shut outs.” And then we put stars beside the ones that were possibly
a mercy rule so that they could see that. Then we broke it down to find results against one of our main opponents like how many games did we have against them? You could take it further, how
much was our point difference against them? Or something like that but I just break it down so that they can see real information and it makes it so that they can understand it better. They
can count how many years we had a winning streak, or we can use a function to do it. That makes it easier for them to understand.
I also do a project where I would have them type in their information. They’d put in their gender, their age, their birthday, their brothers, sisters, favorite colors, do they plan to go to
college, and make up some test scores. Just different things like that. And then when we go through and we open this in Excel, and they can figure out how many pets every student has, what's
our average score, who likes pizza? And what functions are we going to use for that? And I do give them hints for functions. And then on the other worksheet, we would do some text functions
like the concatenate, left, right, upper, lower, and proper. And for the first ones, I give them what the function is supposed to be, what the task might actually tell them on the test, and
what they're supposed to do. And then as we go on, I leave off the function and hopefully by then, they can start trying to figure out how to decipher what that function is.
One last project that I do in the spring when we’re over into baseball, I've actually pulled up the Arkansas Razorbacks roster and got their stats. I came up with some instructions in order
to do projects so we could do some more functions and some projects. And I've got a URL to all of these projects for you that I'll share where you can take home and revise them to how ever
you want. We even do If Error functions. We do all kinds of things in order to reinforce that information. And then the main thing I wanted to do with this one was get into charts. And so,
after we did all of our functions then we started doing our charts for different scenarios. And then we took all that information and changed it to softball. So that they wouldn't have to do
any of the functions or the charts again. All they had to do is just put in the new roster with the softball player stats. And then, everything changed so they could see once you get your
worksheets set up, all you need is to change the information. It's just so simple. Those are the three main projects that I use. They really like the function data practice, where I showed
where they use their own information, and they liked the football stats one also. And then, this [last] one has been pretty good too.
Michelle: I actually start with going over what PEMDAS is so that they'll understand that we are using the same thing that they're using in a math class. And a few of them don't remember it
or didn't learn it right. But most of them are really excited about like, “Oh, this is something I can relate to.” I start with using that and then I explain the difference between a formula
that would use the things that we use in PEMDAS and a function which would use ranges and, named ranges and all those other things. It's exciting for me and I know some of the students get a
little bit worried about those formulas. The really good thing about them is that it doesn't matter what kind of computer you have or what kind of software you have, you put that equal sign
down, and you can start typing that formula or that function. I actually had a last-minute thing where I had to walk into an Apple lab because they were doing something in my lab. All of a
sudden, my computers didn’t work and the only lab available was the Apple lab. I went into the Apple Lab knowing nothing and I had to teach a Numbers class with my Excel students. And I told
the teacher, “I don't know what to do with this.” She said, “What are you teaching?” I said, “Excel.” She said, “Well, just use Numbers.” And I said, “What's Numbers?” And so, it works. You
put the equal sign down and it still worked even in that Apple lab. I've even used it with OpenOffice and Office 365 and all that. The formulas and functions can be done with almost any type
of software.
Gwen: I would like to share one thing about some useful data for the students. In North Carolina, we have a really neat program called NC Star Jobs and it gives a lot of information about
jobs in North Carolina. The really neat thing about this is I tried to explain to [the students] how useful Excel is even once you get out of high school. Almost any company you go to will
likely use Excel. Even most databases are set up to be used with Excel through CSV export or just with Excel in general. So, this is a great database that tells them about different clusters
of occupations, and what the actual pay is in North Carolina based on taxes. It's real data, and they can dig into it and get a lot of information. I'll point out to them that at the top, we
have an option to download it as Excel or as a CSV. And to begin with I always make them do CSV files. Then we later import the CSV files into Excel and leave a couple of columns out and then
when we export it and put it into Excel, the students can see how useful it is to get it out of this website and manipulate it in Excel. And we do things like look at what average pay is for
the jobs that are high school diploma or equivalent.
6. What additional resources or fun projects would you recommend for students to complete as they prepare for the test?
Michelle: I do have a couple of fun projects. They're not difficult. They are some things that I start out with at the beginning. One of them is to create a cash register, and the other one
would be to create a board where they play Battleship. In Battleship, each person sets up their monitors back-to-back so they can't see each other. And they set up a board in Excel, and they
are allowed to have one aircraft carrier that's big and takes up so many cells and other little ships that are so big and take up so many cells. And then they have to resize their cells to
make them a little bit bigger and they have to use color coding and all that. They hit at each other's screens by calling different cell numbers and they have to respond, “You hit my
battleship.” or “You didn't hit my battleship.” And when they hit one, they have to color code the cell so they know that they found it and whoever finds all the battleships for the other
team or the other person, wins.
Another that I do like a lot is I only need columns, A, B, and C, to make a restaurant. They actually use four worksheets where very basic stuff gets them started. On the sheet one, they're
going to design an entry page, so that we know what the restaurant is. Then on page two they list 10 different items: maybe a Hamburger, soda, and fries. Then, they have to put at least 10
things into their restaurant and then they have to create a price for that item. Then, they have to leave Column C blank so that they can take orders from their friends. In Column D, they
have to make a formula without anything in Column C. Sometimes, if they really don't get it, I'll let them go back and put something in Column C so that they'll understand that they have to
go to column D and put in a formula that lets them multiply how many times the number that gets ordered. And then they go and take orders. In the first friend sheet, they'll have to copy it
and go over to the next sheet and they have to rename the sheets by the person's name that they're working with. I can send them off into groups, with the Google Meet function for that, and
let them take orders from the three different people. Next, they have to format the money as accounting format. And how many hamburgers does this person want? Maybe they want two hamburgers,
one soda, and they don't want any fries. And then they'll have to AutoSum and they will get a total. They'll have to figure a tax, I usually give them 7.75. There's your subtotal. And then we
put tax. And they we’ll say equals the subtotal times the tax rate. And then you have to explain that that has to be an absolute cell reference if you're ever going to do anything with it
like use the Fill Handle with that tax rate. So, you do want to make sure that it always goes back to that same cell. And then you get a total here. Then the most fun part is for them is to
figure out the change. How much did they pay you? And then they pay them, say $10, and then they have to figure out the change. They have to know that they have to subtract the bigger number
minus the little number to figure out how much change they actually give back.
7. Is the CertMaster program enough to be proficient on the test?
Gwen: I was excited to get this question because my first thought was that I don't know for sure because I have used CertMaster somewhat supplemental. Like I said, I did not get it until
COVID. When we first got CertMaster , I had already completed regular Excel and was approaching the expert level but did not have any way to work with the students because like I said, we
weren't even one-to-one [Chromebooks/computers]. So, this is when I got CertMaster and what I did was I assigned Excel and CertMaster to my students. And they loved it. They were like, “This
was great review. I was familiar with the tasks that they were asking me to do.” I was very happy about that. Then in the fall, they used CertMaster at home completely by themselves. So, I do
grades in CertMaster . What I do is, each week they have a section that they have to do. They are required to watch every video. They are required to read all of the literature and complete
all labs. I don't do any of this with them. What I do in class is supplemental to CertMaster now. I expect them to do all of CertMaster , not to skip over any part. We do the practice tests
that are in CertMaster . We do the Form A and Form B practice test. When we come in class, we are going over these formulas and functions, but just applying what they've learned in CertMaster
to something real to them, like a car payment. A lot of times, they're like, “Oh, okay, I remember that in CertMaster and now I understand it better.”
So, the question was: Is it enough to be proficient on the test? I did not know because if they didn't completely get it in CertMaster , hopefully I’ve covered it in class. Well, I had a
unique situation where a student found out the college that she was going to required that she be certified in Excel, Word, and PowerPoint. And this was one of the students that did not go
through CTE. She was a bright student, but she was really panicking because she was going to have to take a course [to get certified]. I got permission to let her use one of our licenses and
she completed CertMaster , not in Microsoft class and with zero help whatsoever but I just told her, “You got to use CertMaster the way it was intended.” You will need to watch every video,
read all the literature, and do all the labs. When you don't know how to do something, look at how you missed the lab. Don't just say, I didn't understand it. Go figure it out. And she passed
Word, PowerPoint, and Excel, all on her own strictly using CertMaster . So, is it enough? Yes. Do most students use it the way it is intended? No. Because they're like, “I don't need to watch
those videos, I’ll go jump to the lab.” And when they do that, and they're asking for help, the very first thing I say is, “Did you watch the videos leading up to this?” And they say, “No, I
haven't watched the videos.” Then I say, “Go watch the videos and then let me know if you still need help.” They figure it out then. So CertMaster to me is an exceptional program. The detail
that has been put into place for CertMaster to prepare the students for certification is one of a kind. I don't know of anything that even closely touches what CertMaster can do. I think the
key is making the students use it as intended. I tell all of my students before they certify is that we do not have nearly the amount of time to teach you everything that Microsoft offers. I
learn stuff every day, I'm still learning things in Microsoft. So, it's just not the kind of course where I'm going to be teaching you every single thing you need to know. I am truly trying
to teach you how to use it. Excel is set up wonderfully to help you find what you want to do.
I think that if they will take it like a challenge, like a video game. I frequently say, “Are you all ready for a challenge?” And I ask them to do something I’ve never mentioned before. They
love it! They want to be the first person to figure it out. To actually be what is considered a certified Excel user, I should be able to figure things out that I've never seen before. I
should be able to look at the data and pretend that somebody has asked me out loud, what were our total sales in that region? And you should be able to figure out how we're going to determine
that. So, when they go into it with this mindset, they feel empowered. I always tell them how great they are, because they are. They're really good, and I tell them that they're so much
better than they think. I talk to them like we're in a meeting. I pretend that we're in a meeting looking at data, and I'll call on the students, and ask, “Who had the highest sales in the
east region?” And they have to go find that. I teach them that that makes them very valuable as an employee, that you can help analyze data. I do think CertMaster will sufficiently prepare
them if they use it as intended. If they try to skip around and just do the labs, I don't know that they should be certified because that's not the employee I want. And, the wonderful thing
is that it requires them to understand their data, it's not just a memorization. You need to understand the data, and that's the kind of employee I want. That has a paper piece of paper that
says, I understand how Excel works, and I can apply it to data. And if I don't know how to do it, I understand how to figure it out.
Allan: If I just might add something about that. I know that Excel prep, particularly in some areas, may require some more practice exercises apart from just going through our skills,
challenge, and applied labs and even, the practice exams at the end of the course. I just wanted to make everyone aware of additional spreadsheet resources that we have in our teaching aids
in CertMaster. We have some applied lab sample documents, Excel spreadsheet activity files, as well as some Capstone projects that you can use apart from what we have in the course already.
8. How do I get MOS Excel certified?
Michelle: You simply go to www.certiport.com. You set up an account. You probably have to work with your school system to pay for your certification test. You may have to pay for it yourself.
Hopefully, all the school systems out there realize the value of taking certification test, and they'll help you to pay for it. But one thing, I also found is that there's another
certification that is available through CertMaster . CertMaster also has its own certification. Under those Facts and Tips, I found that CertMaster has a thing called the Skills Guarantee,
which if you get employed after you take the CertMaster certification test, (which is PowerPoint, Word, and Excel [in one exam]), and the employer, after hiring and putting this person to
work for at least three weeks, finds that the person who has gotten that CertMaster certification doesn't possess the IT skills that is applicable to that certification then CertMaster is
going to pay that company up to $1000. I’m just really stunned that anybody would offer such a thing. So, there are two ways to get several certifications out there, but I was really
impressed with that CertMaster certification. And one opportunity to certify comes with every license. So, each one of our students, if they took all three Word, PowerPoint, and Excel, could
go and take that test.
Gwen: I wanted to point out something that when we've had some countywide meetings, some of the other teachers did not know this resource was also in CertMaster . When you're looking at what
is offered in CertMaster , I found that a lot of teachers were just going straight to the chapters and they thought that was everything that was there for Excel. And when I mentioned the
practice tests, they had not noticed the practice tests down at the very bottom of the course outline. But these are great ways for the students to figure out how prepared they are going into
certification. And I tell them if they're not proficient on that then they can troubleshoot back to the labs. And then, once they've kind of mastered this part, then I'll tell them to go to
the MOS practice exams, which is Section B, and these are timed. And it's the same number of questions, for the most part, that the certification test is, they have the 50 minutes and that
helps prepare them for the time. That’s one of the things that really gets them the most, is the time. And when they get used to the time then, they can take care of that stressor before they
ever began the test. And I've seen very good results from students that do well on Form A, and Form B, and then they go into certification, and they just relax at that point.
Note: Some responses have been edited for length and/or clarity.
Teacher files shared by Kim Conant:
Teacher files shared by Michelle Lewis: | {"url":"https://support.testout.com/hc/en-us/articles/18919281345556-Office-Pro-Teachers-Lounge-March-25-2021-Helping-Students-Excel-on-the-MOS-Excel-Certification-Exam","timestamp":"2024-11-03T09:40:37Z","content_type":"text/html","content_length":"78857","record_id":"<urn:uuid:af847bb9-5ada-4665-b755-0a621fc1f6ea>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00237.warc.gz"} |
Testing the Waters: Guide on Camos from the "Hard/Easy to Aim At" Perspective
Hey folks,
There is a fantastic guide (made by a unicum player with the help of their clanmates in testing) on how in-game camos affect the "comfort" of aiming at your ship with this or that camo mounted on
However the guide is not in English and consequently there would be a substantial effort to localize it into English.
Because of this, I would like to know how many players here on the forum are interested in reading this guide.
I would really like to contribute to the massive body of knowledge that has already been accumulated here. However I need to understand if the game is worth the candles.
Thank you.
Edited by 100 Krakens
• 100 Krakens changed the title to Testing the Waters: Guide on Camos from the "Hard/Easy to Aim At" Perspective
That would be a fun read.
• 2
1 hour ago, 100 Krakens said:
Hey folks,
There is a fantastic guide (made by a unicum player with the help of their clanmates in testing) on how in-game camos affect the "comfort" of aiming at your ship with this or that camo mounted on
However the guide is not in English and consequently there would be a substantial effort to localize it into English.
Because of this, I would like to know how many players here on the forum are interested in reading this guide.
I would really like to contribute to the massive body of knowledge that has already been accumulated here. However I need to understand if the game is worth the candles.
Thank you.
for 100+ languages
Yandex Translate — synchronized translation for 102 languages, predictive typing, dictionary with transcription, pronunciation, context and usage examples, and many other features."
Edited by Wolfswetpaws
• 1
What's the original language?
• 1
Count me in as interested.
• 1
Interested, though I thought the games auto aim assist would cancel out any effect of camos.
51 minutes ago, I_cant_Swim_ said:
Interested, though I thought the games auto aim assist would cancel out any effect of camos.
That's occurred to me too, but it doesn't seem to fix things if your manual aim is off, though. At least it doesn't too it for me... isn't the aim assist only for improved dispersion?
• 1
The lock on bonus is improved dispersion. Pretty much the rest of it is knowledge of your shell characteristics, flight time to target, and your own aim. But honestly lining up a shot and doing doing
a quick glance at the mini-map to check where the circle for shell destination is in relation to the ship you are aiming at usually is a pretty solid second opinion. Most of the difference between
full pens and citadels is knowing exactly how fast they are going to better line up dropping the shells in the sweet spots.
• 5
I always thought that was a thing. After all, dazzle camo was designed to throw off the enemies "Mark One" eyeball. Why shouldn't camo do the same to us in game. I'd love to see what they've
• 2
1 hour ago, Kynami said:
But honestly lining up a shot and doing doing a quick glance at the mini-map to check where the circle for shell destination is in relation to the ship you are aiming at usually is a pretty solid
second opinion.
I just recently became aware of this absolute gem of an idea. I think it was from a StatsBloke video. One of those "how to" videos on aiming. I was pretty surprised at how well it worked. This
info should be told to every noob in the game. My accuracy is better for having learned it.
Just now, Captain Slattery said:
I just recently became aware of this absolute gem of an idea. I think it was from a StatsBloke video. One of those "how to" videos on aiming. I was pretty surprised at how well it worked.
This info should be told to every noob in the game. My accuracy is better for having learned it.
It should. I don't know how of a difference it makes depending on your graphic settings and/or mods people are using. In order for any camos to actually work like camos in WoWS, the way we see the
target should remain 'consistently obscured relative to the distance and prevailing weather conditions', for want of a better expression.
• 1
1 minute ago, Admiral_Karasu said:
It should. I don't know how of a difference it makes depending on your graphic settings and/or mods people are using. In order for any camos to actually work like camos in WoWS, the way we see
the target should remain 'consistently obscured relative to the distance and prevailing weather conditions', for want of a better expression.
It works for me and I'm running on a mini-rig now sporting a graphics crushing Ryzen 7 5700U processor with Vega 8 integrated graphics. Here's a picture of it....
• 6
1 hour ago, Captain Slattery said:
I always thought that was a thing. After all, dazzle camo was designed to throw off the enemies "Mark One" eyeball. Why shouldn't camo do the same to us in game. I'd love to see what they've
@Lord_Zath once did a Twitch stream where they tested the usefulness of a French DD dropping depth-charges as a substitute for a smokescreen when attempting to dodge gunfire while sailing away from
those who are shooting at the French DD.
It was entertaining.
And it seems there is some merit to the concept, even if the results varied considerably. 🙂
• 1
55 minutes ago, Admiral_Karasu said:
I don't know how of a difference it makes depending on your graphic settings and/or mods people are using.
When I upgraded my computer, I was able to increase the detail level of the graphics I was using.
It made a positive improvement in my perception of the game environment, and I could discern details which help me aim at targets.
Smoke trails from the stacks, waves made by the hull and other clues help me discern a target's speed and direction (as @Kynami discussed in their earlier post within this topic).
• 1
2 minutes ago, Wolfswetpaws said:
When I upgraded my computer, I was able to increase the detail level of the graphics I was using.
It made a positive improvement in my perception of the game environment, and I could discern details which help me aim at targets.
Smoke trails from the stacks, waves made by the hull and other clues help me discern a target's speed and direction (as @Kynami discussed in their earlier post within this topic).
Yes, that's roughly what I meant. All that sort of thing really would need to go if we wanted 'working camos' instead of just something to tart up our ships with.
• 2
Whats the point in asking for help in translation if you don't say what language it is? Nobody can actually commit to translating this unless they know what language the original post is in.
2 minutes ago, Unlooky said:
Whats the point in asking for help in translation if you don't say what language it is? Nobody can actually commit to translating this unless they know what language the original post is in.
The OP isn't looking for translators. He's gauging interest in a translated version to see if it would be worth the effort.
• 2
1 minute ago, Captain Slattery said:
The OP isn't looking for translators. He's gauging interest in a translated version to see if it would be worth the effort.
I see, thats my error then.
• 2
23 hours ago, 100 Krakens said:
However the guide is not in English and consequently there would be a substantial effort to localize it into English.
Can You share the guide in its original language?
Let the guys interested deal with the translation part.
• 1
I’m more of a point and shoot guy that’s why I stink probably. Never used numbers on the curser or tried to figure out speed beyond looking at smoke stacks.
Funny thing is when I have the time to line up the perfect shot on a broad side ship that’s when I get 5 overpens and torpedo belt hit. When I just point and shoot fast I seem to do better the camo
doesn’t really matter to me.
Edited by clammboy
• 2
14 minutes ago, clammboy said:
I’m more of a point and shoot guy that’s why I stink probably. Never used numbers on the curser or tried to figure out speed beyond looking at smoke stacks.
Funny thing is when I have the time to line up the perfect shot on a broad side ship that’s when I get 5 overpens and torpedo belt hit. When I just point and shoot fast I seem to do better.
While simultaneously saying "Aim for the heart, Ramon"? 😉
Edited by Wolfswetpaws
• 2
My aim has improved. I use a static crosshairs and count clicks. But rng is a thing. One single solitary time [like two days ago], in a Sharny '43; I hit 5 citadels on a japanese cruiser in one full
volley. Pretty near full health. Ruined that guy's day. He basically said so in chat. I replied, 5 cits. But this never happens to me. I did my same aiming thing, waterline and counting clicks for a
fast croozer and there is that. Many times I will get one or two, but never more than that.
I do not think camo is going to help people aim. Aiming will help people aim.
Watching good streamers will help you aim, sometimes they talk about it at the time they are actually aiming. Some streamers are not pros by the way. Come to find out.
• 4
Me, I just point and go pew-pew-pew-ka-BOOM! I may need to specify that the ka-BOOM part is in reference to my own ship.
• 1
• 2
39 minutes ago, Admiral_Karasu said:
Me, I just point and go pew-pew-pew-ka-BOOM! I may need to specify that the ka-BOOM part is in reference to my own ship.
Sinking Ship Simulator: The Royal Navy's Damage Repair Instructional Unit
• 1
Hmmm ....... interesting subject.
When the camo rework gone live, conducted my own little experiments. Nothing fancy just in port and using the "zoom out in port" mod. For what is worth my findings were, the effect of camos varied
greatly, depending on environment, as lighting, weather,"terrain" (I mean water), angle and distance.
Generally speaking, open colour camos were the worst, quite unsurpisingly .
Even no camo was better...
....tho, like I said, environment....
Then come big contrast camos ....
......then patterned ones....
......which can also be repetitive, thus leading.
For PvP ( ranked) settled for this..
It is called Polygonal Steel.
One special case is this.....
..... the Steel camo. Under certain circumstances can be truly disruptive, coz shines and reflects the light thus can be tiresome. But is TX only.
Edited by Andrewbassg
• 1
• 1 | {"url":"https://www.devstrike.net/topic/4618-testing-the-waters-guide-on-camos-from-the-hardeasy-to-aim-at-perspective/","timestamp":"2024-11-07T02:15:51Z","content_type":"text/html","content_length":"365529","record_id":"<urn:uuid:62821069-bbed-45d8-98e9-b75f7ad40948>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00383.warc.gz"} |
Abelian ordered groups
An \emph{abelian ordered group} is an ordered group $\mathbf{A}=\langle A,+,-,0,\le\rangle$ such that
$+$ is commutative: $x+y=y+x$
Let $\mathbf{A}$ and $\mathbf{B}$ be abelian ordered groups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is an orderpreserving homomorphism: $h(x + y)=h(x) + h
(y)$ and $x\le y\Longrightarrow h(x)\le h(y)$.
Example 1: $\langle\mathbb Z,+,-,0,\le\rangle$, the integers with the usual ordering.
Basic results
Every ordered group with more than one element is infinite.
Finite members | {"url":"https://mathcs.chapman.edu/~jipsen/structures/doku.php?id=abelian_ordered_groups","timestamp":"2024-11-05T20:24:53Z","content_type":"application/xhtml+xml","content_length":"16947","record_id":"<urn:uuid:14bbfa74-2801-468d-85f1-7ca126697060>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00083.warc.gz"} |
11.1 Temperature and Thermal Energy
Learning Objectives
Learning Objectives
By the end of this section, you will be able to do the following:
• Explain that temperature is a measure of internal kinetic energy
• Interconvert temperatures between Celsius, Kelvin, and Fahrenheit scales
Section Key Terms
absolute zero Celsius scale degree Celsius $( °C ) ( °C )$
degree Fahrenheit $( °F ) ( °F )$ Fahrenheit scale heat
kelvin (K) Kelvin scale temperature
thermal energy
What is temperature? It’s one of those concepts so ingrained in our everyday lives that, although we know what it means intuitively, it can be hard to define. It is tempting to say that temperature
measures heat, but this is not strictly true. Heat is the transfer of energy due to a temperature difference. Temperature is defined in terms of the instrument we use to tell us how hot or cold an
object is, based on a mechanism and scale invented by people. Temperature is literally defined as what we measure on a thermometer.
Heat is often confused with temperature. For example, we may say that the heat was unbearable, when we actually mean that the temperature was high. This is because we are sensitive to the flow of
energy by heat, rather than the temperature. Since heat, like work, transfers energy, it has the SI unit of joule (J).
Atoms and molecules are constantly in motion, bouncing off one another in random directions. Recall that kinetic energy is the energy of motion, and that it increases in proportion to velocity
squared. Without going into mathematical detail, we can say that thermal energy—the energy associated with heat—is the average kinetic energy of the particles (molecules or atoms) in a substance.
Faster moving molecules have greater kinetic energies, and so the substance has greater thermal energy, and thus a higher temperature. The total internal energy of a system is the sum of the kinetic
and potential energies of its atoms and molecules. Thermal energy is one of the subcategories of internal energy, as is chemical energy.
To measure temperature, some scale must be used as a standard of measurement. The three most commonly used temperature scales are the Fahrenheit, Celsius, and Kelvin scales. Both the Fahrenheit scale
and Celsius scale are relative temperature scales, meaning that they are made around a reference point. For example, the Celsius scale uses the freezing point of water as its reference point; all
measurements are either lower than the freezing point of water by a given number of degrees (and have a negative sign), or higher than the freezing point of water by a given number of degrees (and
have a positive sign). The boiling point of water is 100$°C °C$ for the Celsius scale, and its unit is the degree Celsius $( °C ( °C$).
On the Fahrenheit scale, the freezing point of water is at 32$°F °F$, and the boiling point is at 212$°F °F$. The unit of temperature on this scale is the degree Fahrenheit $( °F ( °F$). Note that
the difference in degrees between the freezing and boiling points is greater for the Fahrenheit scale than for the Celsius scale. Therefore, a temperature difference of one degree Celsius is greater
than a temperature difference of one degree Fahrenheit. Since 100 Celsius degrees span the same range as 180 Fahrenheit degrees, one degree on the Celsius scale is 1.8 times larger than one degree on
the Fahrenheit scale (because $180 100 = 9 5 =1.8 180 100 = 9 5 =1.8$). This relationship can be used to convert between temperatures in Fahrenheit and Celsius (see Figure 11.2).
The Kelvin scale is the temperature scale that is commonly used in science because it is an absolute temperature scale. This means that the theoretically lowest-possible temperature is assigned the
value of zero. Zero degrees on the Kelvin scale is known as absolute zero; it is theoretically the point at which there is no molecular motion to produce thermal energy. On the original Kelvin scale
first created by Lord Kelvin, all temperatures have positive values, making it useful for scientific work. The official temperature unit on this scale is the kelvin, which is abbreviated as K. The
freezing point of water is 273.15 K, and the boiling point of water is 373.15 K.
Although absolute zero is possible in theory, it cannot be reached in practice. The lowest temperature ever created and measured during a laboratory experiment was $1.0× 10 −10 1.0× 10 −10$ K, at
Helsinki University of Technology in Finland. In comparison, the coldest recorded temperature for a place on Earth’s surface was 183 K (–89 °C ), at Vostok, Antarctica, and the coldest known place
(outside the lab) in the universe is the Boomerang Nebula, with a temperature of 1 K. Luckily, most of us humans will never have to experience such extremes.
The average normal body temperature is 98.6$°F °F$ (37.0$°C °C$), but people have been known to survive with body temperatures ranging from 75$°F °F$ to 111$°F °F$ (24$°C °C$ to 44$°C °C$).
Watch Physics
Comparing Celsius and Fahrenheit Temperature Scales
This video shows how the Fahrenheit and Celsius temperature scales compare to one another.
Grasp Check
Even without the number labels on the thermometer, you could tell which side is marked Fahrenheit and which is Celsius by how the degree marks are spaced. Why?
a. The separation between two consecutive divisions on the Fahrenheit scale is greater than a similar separation on the Celsius scale, because each degree Fahrenheit is equal to $1.8$ degrees
b. The separation between two consecutive divisions on the Fahrenheit scale is smaller than the similar separation on the Celsius scale, because each degree Celsius is equal to $1.8$ degrees
c. The separation between two consecutive divisions on the Fahrenheit scale is greater than a similar separation on the Celsius scale, because each degree Fahrenheit is equal to $3.6$ degrees
d. The separation between two consecutive divisions on the Fahrenheit scale is smaller than a similar separation on the Celsius scale, because each degree Celsius is equal to $3.6$ degrees
Converting Between Celsius, Kelvin, and Fahrenheit Scales
Converting Between Celsius, Kelvin, and Fahrenheit Scales
While the Fahrenheit scale is still the most commonly used scale in the United States, the majority of the world uses Celsius, and scientists prefer Kelvin. It’s often necessary to convert between
these scales. For instance, if the TV meteorologist gave the local weather report in kelvins, there would likely be some confused viewers! Table 11.1 gives the equations for conversion between the
three temperature scales.
To Convert From… Use This Equation
Celsius to Fahrenheit $T °F = 9 5 T °C +32 T °F = 9 5 T °C +32$
Fahrenheit to Celsius $T °C = 5 9 ( T °F −32 ) T °C = 5 9 ( T °F −32 )$
Celsius to Kelvin $T K = T °C + 273.15 T K = T °C + 273.15$
Kelvin to Celsius $T °C = T K −273.15 T °C = T K −273.15$
Fahrenheit to Kelvin $T K = 5 9 ( T °F −32 )+273.15 T K = 5 9 ( T °F −32 )+273.15$
Kelvin to Fahrenheit $T °F = 9 5 ( T K −273.15 )+32 T °F = 9 5 ( T K −273.15 )+32$
Converting between Temperature Scales: Room Temperature
Room temperature is generally defined to be 25 $°C . °C .$ (a) What is room temperature in $°F ? °F ?$ (b) What is it in K?
To answer these questions, all we need to do is choose the correct conversion equations and plug in the known values.
Solution for (a)
1. Choose the right equation. To convert from $°C °C$ to $°F °F$, use the equation
11.1$T °F = 9 5 T °C +32 . T °F = 9 5 T °C +32 .$
2. Plug the known value into the equation and solve.
11.2$T °F = 9 5 25°C+32=77°F T °F = 9 5 25°C+32=77°F$
Solution for (b)
1. Choose the right equation. To convert from $°C °C$ to K, use the equation
11.3$T K = T °C + 273.15 . T K = T °C + 273.15 .$
2. Plug the known value into the equation and solve.
11.4$T K = 25 °C + 273.15 = 298K T K = 25 °C + 273.15 = 298K$
Living in the United States, you are likely to have more of a sense of what the temperature feels like if it’s described as 77$°F °F$ than as 25$°C °C$ (or 298 K, for that matter).
Converting Between Temperature Scales: The Reaumur Scale
The Reaumur scale is a temperature scale that was used widely in Europe in the 18^th and 19^th centuries. On the Reaumur temperature scale, the freezing point of water is 0$°R °R$ and the boiling
temperature is 80$°R . °R .$ If “room temperature” is 25$°C °C$ on the Celsius scale, what is it on the Reaumur scale?
To answer this question, we must compare the Reaumur scale to the Celsius scale. The difference between the freezing point and boiling point of water on the Reaumur scale is 80$°R °R$. On the Celsius
scale, it is 100$°C °C$. Therefore, 100$°C = 80 °R °C = 80 °R$. Both scales start at 0$° °$ for freezing, so we can create a simple formula to convert between temperatures on the two scales.
1. Derive a formula to convert from one scale to the other.
11.5$T °R = 0.80°R °C × T °C T °R = 0.80°R °C × T °C$
2. Plug the known value into the equation and solve.
11.6$T °R = 0.80°R °C ×25°C = 20°R T °R = 0.80°R °C ×25°C = 20°R$
As this example shows, relative temperature scales are somewhat arbitrary. If you wanted, you could create your own temperature scale!
Practice Problems
Practice Problems
What is 12.0 °C in kelvins?
a. 112.0 K
b. 273.2 K
c. 12.0 K
d. 285.2 K
What is 32.0 °C in degrees Fahrenheit?
a. 57.6 °F
b. 25.6 °F
c. 305.2 °F
d. 89.6 °F
Tips For Success
Sometimes it is not so easy to guess the temperature of the air accurately. Why is this? Factors such as humidity and wind speed affect how hot or cold we feel. Wind removes thermal energy from our
bodies at a faster rate than usual, making us feel colder than we otherwise would; on a cold day, you may have heard the TV weather person refer to the wind chill.
On humid summer days, people tend to feel hotter because sweat doesn’t evaporate from the skin as efficiently as it does on dry days, when the evaporation of sweat cools us off.
Check Your Understanding
Check Your Understanding
Exercise 1
What is thermal energy?
a. The thermal energy is the average potential energy of the particles in a system.
b. The thermal energy is the total sum of the potential energies of the particles in a system.
c. The thermal energy is the average kinetic energy of the particles due to the interaction among the particles in a system.
d. The thermal energy is the average kinetic energy of the particles in a system.
Exercise 2
What is used to measure temperature?
a. a galvanometer
b. a manometer
c. a thermometer
d. a voltmeter | {"url":"https://texasgateway.org/resource/111-temperature-and-thermal-energy?book=79076&binder_id=78141","timestamp":"2024-11-14T11:17:10Z","content_type":"text/html","content_length":"74645","record_id":"<urn:uuid:28fea60d-a80e-4a52-8d50-0c05b33e089d>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00538.warc.gz"} |
Seminar Analysis and Theoretical Physics
Anderson Models, from Schrödinger operators to singular SPDEs
The name Anderson model is used to refer either to the stochastic partial differential equation (SPDE) called the parabolic Anderson model or to the corresponding operator called the Anderson
Hamiltonian. The operator is a random Schrödinger operator and in solid state physics this operator describes the evolution of a quantum state via the Schrödinger equation. On the other hand, the
operator describes a random motion in a random environment by means of the parabolic Anderson model. Therefore the solution to both equations can be described by the spectral properties of the
I will discuss the Anderson Hamiltonian and the parabolic Anderson model with a white noise potential, which, due to its low regularity, brings us into the realm of singular SPDEs. One of the
beauties of the parabolic Anderson model is the Feynman-Kac representation of the solution, by which one is able to derive the behaviour of its solution.
I will motivate the model, give a feeling for the singularity and the construction of the Anderson Hamiltonian and describe how the relation to the parabolic Anderson is used to describe the
asymptotic behaviour of its total mass. | {"url":"https://www.ifam.uni-hannover.de/en/news-talks-events/talks/news-details/news/oberseminar-analysis-und-theoretische-physik-46","timestamp":"2024-11-05T13:38:31Z","content_type":"application/xhtml+xml","content_length":"32800","record_id":"<urn:uuid:91db00b2-92af-41b7-afcd-570559113ca6>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00527.warc.gz"} |
Trapping Rain Water in Ruby: Mastering Algorithms
Solutions for Common Algorithm Questions in Ruby
The Trapping Rain Water problem is a prominent challenge in the realm of algorithms and has been a staple in many coding interviews. It presents a scenario where given an array of integers
representing elevations, the task is to determine how much rainwater can be trapped between the elevations after a downpour. In this article, we’ll delve into the problem and discuss a comprehensive
approach to solving it using Ruby.
Problem Statement
Imagine you have an elevation map where the width of each bar is 1 unit. The height of each bar represents the elevation at that point. Now, after a heavy rain, water would accumulate between these
bars. The question is — how much water can be trapped?
Consider the elevation map represented by the following array: [0,1,0,2,1,0,1,3,2,1,2,1]. If you plot this on a graph, it will form a series of peaks and valleys. Water will accumulate in the valleys
between the peaks. In this example, the total trapped rainwater is 6 units.
Visualizing the Problem
visualizing the Trapping Rain Water problem in plaintext is an excellent way to understand it. Let’s take the example array: [0,1,0,2,1,0,1,3,2,1,2,1]. | {"url":"https://patrickkarsh.medium.com/trapping-rain-water-in-ruby-mastering-algorithms-bf0d758e911d?responsesOpen=true&sortBy=REVERSE_CHRON&source=read_next_recirc-----23773bfcd82d----1---------------------cf33eb57_f9cf_4408_bc22_6be645b899f7-------","timestamp":"2024-11-09T01:34:48Z","content_type":"text/html","content_length":"89905","record_id":"<urn:uuid:cfcf9ac3-d14e-4735-95f9-abb9668d0404>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00659.warc.gz"} |
Practice Question
• Subject 2. Measures of Dispersion
CFA Practice Question
There are 676 practice questions for this topic.
CFA Practice Question
A portfolio earned the following rates of return over a period of twelve months.
What is the portfolio's co-efficient of variation?
Explanation: Co-efficient of variation = Std. dev./Average return = 0.85/0.24 = 3.54
User Contributed Comments 15
User Comment
danlan Ignore risk free rate.
anricus How is the standard deviation calculated? I've used my calculator and Sx is .85 but this is a population so variance x is applicable which I got as .81??
danlan Sample standard deviation is 0.852445 and population standard deviation is 0.81615
MUSK Why are we using sample standard deviation here rather than using population standard deviation?
Is there a quick way to calculate Std. dev. I got the answer by doing Sum of all(Observation - Average)sq divided by n-1 (11). I got the right answer but it took like 5 mins to do it...Iam
gaur not sure 5 mins is appropriate time to spend on a question even if you can get it right. Iam more inclined towards leaving this and moving on...and maybe revisiting the question if I have
time left in the end
achu Using the BA 2-Plus ,you can enter the data mode and enter 12 x-values. The calculator will then compute the mean and sample variance.
teeday thanks for the heads up on using the BA 2!
sample: because it's only for a period of 12 months
we would have taken population if we had portfolio returns for all periods held
kellyyang can someone shows how to get the #.81 and .24.
NickPash Can anybody show me how you can do this in HP12C? I spent more than 5 mins for this doing it mannually. Tnx.
NickPash Enter all xs such as 1.25 enter sigma+ ...-1.05 enter sigma+, then press, gS = .8524 , gx-bar = .24 , .85/.24=3.54
bp019j thx NickPash
endurance Easy task really - on the BAII, type in all the returns in the data (2nd data). Change to "stat" (2nd stat) and find X-bar and stddev. Divide 0,85 with 0,24 and you'll get 3.5417
hmichta Thanks endurance. Will save me loads of time :)
farhan92 guys using the ba II remember you can store values using the STO button and recall them using the RCL button so when you get to the mean hit STO 1 and then for the SD hit STO 2. Then do
RCL 2/RCL 1
You need to log in first to add your comment. | {"url":"https://analystnotes.com/cfa_question.php?p=YWLUH7UXM","timestamp":"2024-11-10T22:47:55Z","content_type":"text/html","content_length":"22862","record_id":"<urn:uuid:c9975bf3-8c74-4855-a188-f32d3564269e>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00622.warc.gz"} |
Interpretations of "probability" — LessWrong
(Written for Arbital in 2016.)
What does it mean to say that a flipped coin has a 50% probability of landing heads?
Historically, there are two popular types of answers to this question, the "frequentist" and "subjective" (aka "Bayesian") answers, which give rise to radically different approaches to experimental
statistics. There is also a third "propensity" viewpoint which is largely discredited (assuming the coin is deterministic). Roughly, the three approaches answer the above question as follows:
• The propensity interpretation: Some probabilities are just out there in the world. It's a brute fact about coins that they come up heads half the time. When we flip a coin, it has a fundamental
propensity of 0.5 for the coin to show heads. When we say the coin has a 50% probability of being heads, we're talking directly about this propensity.
• The frequentist interpretation: When we say the coin has a 50% probability of being heads after this flip, we mean that there's a class of events similar to this coin flip, and across that class,
coins come up heads about half the time. That is, the frequency of the coin coming up heads is 50% inside the event class, which might be "all other times this particular coin has been tossed" or
"all times that a similar coin has been tossed", and so on.
• The subjective interpretation: Uncertainty is in the mind, not the environment. If I flip a coin and slap it against my wrist, it's already landed either heads or tails. The fact that I don't
know whether it landed heads or tails is a fact about me, not a fact about the coin. The claim "I think this coin is heads with probability 50%" is an expression of my own ignorance, and 50%
probability means that I'd bet at 1 : 1 odds (or better) that the coin came up heads.
For a visualization of the differences between these three viewpoints, see Correspondence visualizations for different interpretations of "probability". For examples of the difference, see
Probability interpretations: Examples. See also the Stanford Encyclopedia of Philosophy article on interpretations of probability.
The propensity view is perhaps the most intuitive view, as for many people, it just feels like the coin is intrinsically random. However, this view is difficult to reconcile with the idea that once
we've flipped the coin, it has already landed heads or tails. If the event in question is decided deterministically, the propensity view can be seen as an instance of the mind projection fallacy:
When we mentally consider the coin flip, it feels 50% likely to be heads, so we find it very easy to imagine a world in which the coin is fundamentally 50%-heads-ish. But that feeling is actually a
fact about us, not a fact about the coin; and the coin has no physical 0.5-heads-propensity hidden in there somewhere — it's just a coin.
The other two interpretations are both self-consistent, and give rise to pragmatically different statistical techniques, and there has been much debate as to which is preferable. The subjective
interpretation is more generally applicable, as it allows one to assign probabilities (interpreted as betting odds) to one-off events.
Frequentism vs subjectivism
As an example of the difference between frequentism and subjectivism, consider the question: "What is the probability that Hillary Clinton will win the 2016 US presidential election?", as analyzed in
the summer of 2016.
A stereotypical (straw) frequentist would say, "The 2016 presidential election only happens once. We can't observe a frequency with which Clinton wins presidential elections. So we can't do any
statistics or assign any probabilities here."
A stereotypical subjectivist would say: "Well, prediction markets tend to be pretty well-calibrated about this sort of thing, in the sense that when prediction markets assign 20% probability to an
event, it happens around 1 time in 5. And the prediction markets are currently betting on Hillary at about 3 : 1 odds. Thus, I'm comfortable saying she has about a 75% chance of winning. If someone
offered me 20 : 1 odds against Clinton — they get $1 if she loses, I get $20 if she wins — then I'd take the bet. I suppose you could refuse to take that bet on the grounds that you Just Can't Talk
About Probabilities of One-off Events, but then you'd be pointlessly passing up a really good bet."
A stereotypical (non-straw) frequentist would reply: "I'd take that bet too, of course. But my taking that bet is not based on rigorous epistemology, and we shouldn't allow that sort of thinking in
experimental science and other important venues. You can do subjective reasoning about probabilities when making bets, but we should exclude subjective reasoning in our scientific journals, and
that's what frequentist statistics is designed for. Your paper should not conclude "and therefore, having observed thus-and-such data about carbon dioxide levels, I'd personally bet at 9 : 1 odds
that anthropogenic global warming is real," because you can't build scientific consensus on opinions."
...and then it starts getting complicated. The subjectivist responds "First of all, I agree you shouldn't put posterior odds into papers, and second of all, it's not like your method is truly
objective — the choice of "similar events" is arbitrary, abusable, and has given rise to p-hacking and the replication crisis." The frequentists say "well your choice of prior is even more
subjective, and I'd like to see you do better in an environment where peer pressure pushes people to abuse statistics and exaggerate their results," and then down the rabbit hole we go.
The subjectivist interpretation of probability is common among artificial intelligence researchers (who often design computer systems that manipulate subjective probability distributions), Wall
Street traders (who need to be able to make bets even in relatively unique situations), and common intuition (where people feel like they can say there's a 30% chance of rain tomorrow without
worrying about the fact that tomorrow only happens once). Nevertheless, the frequentist interpretation is commonly taught in introductory statistics classes, and is the gold standard for most
scientific journals.
A common frequentist stance is that it is virtuous to have a large toolbox of statistical tools at your disposal. Subjectivist tools have their place in that toolbox, but they don't deserve any
particular primacy (and they aren't generally accepted when it comes time to publish in a scientific journal).
An aggressive subjectivist stance is that frequentists have invented some interesting tools, and many of them are useful, but that refusing to consider subjective probabilities is toxic. Frequentist
statistics were invented in a (failed) attempt to keep subjectivity out of science in a time before humanity really understood the laws of probability theory. Now we have theorems about how to manage
subjective probabilities correctly, and how to factor personal beliefs out from the objective evidence provided by the data, and if you ignore these theorems you'll get in trouble. The frequentist
interpretation is broken, and that's why science has p-hacking and a replication crisis even as all the wall-street traders and AI scientists use the Bayesian interpretation. This "let's compromise
and agree that everyone's viewpoint is valid" thing is all well and good, but how much worse do things need to get before we say "oops" and start acknowledging the subjective probability
interpretation across all fields of science?
The most common stance among scientists and researchers is much more agnostic, along the lines of "use whatever statistical techniques work best at the time, and use frequentist techniques when
publishing in journals because that's what everyone's been doing for decades upon decades upon decades, and that's what everyone's expecting."
See also Subjective probability and Likelihood functions, p-values, and the replication crisis.
Which interpretation is most useful?
Probably the subjective interpretation, because it subsumes the propensity and frequentist interpretations as special cases, while being more flexible than both.
When the frequentist "similar event" class is clear, the subjectivist can take those frequencies (often called base rates in this context) into account. But unlike the frequentist, she can also
combine those base rates with other evidence that she's seen, and assign probabilities to one-off events, and make money in prediction markets and/or stock markets (when she knows something that the
market doesn't).
When the laws of physics actually do "contain uncertainty", such as when they say that there are multiple different observations you might make next with differing likelihoods (as the Schrodinger
equation often will), a subjectivist can combine her propensity-style uncertainty with her personal uncertainty in order to generate her aggregate subjective probabilities. But unlike a propensity
theorist, she's not forced to think that all uncertainty is physical uncertainty: She can act like a propensity theorist with respect to Schrodinger-equation-induced uncertainty, while still
believing that her uncertainty about a coin that has already been flipped and slapped against her wrist is in her head, rather than in the coin.
This fully general stance is consistent with the belief that frequentist tools are useful for answering frequentist questions: The fact that you can personally assign probabilities to one-off events
(and, e.g., evaluate how good a certain trade is on a prediction market or a stock market) does not mean that tools labeled "Bayesian" are always better than tools labeled "frequentist". Whatever
interpretation of "probability" you use, you're encouraged to use whatever statistical tool works best for you at any given time, regardless of what "camp" the tool comes from. Don't let the fact
that you think it's possible to assign probabilities to one-off events prevent you from using useful frequentist tools!
It's just a different way of arriving at the same conclusions. The whole project is developing game-theoretic proofs for results in probability and finance.
The pitch is, rather than using a Dutch Book argument as a separate singular argument, they make those intuitions central as a mechanism of proof for all of probability (or at least the core of it,
thus far).
The claim "I think this coin is heads with probability 50%" is an expression of my own ignorance, and 50% probability means that I'd bet at 1 : 1 odds (or better) that the coin came up heads.
Just a minor quibble - using this interpretation to define one's subjective probabilities is problematic because people are not necessarily indifferent about placing a bet that has an expected value
of 0 (e.g. due to loss aversion).
Therefore, I think the following interpretation is more useful: Suppose I win [some reward] if the coin comes up heads. I'd prefer to replace the winning condition with "the ball in a roulette wheel
ends up in a red slot" for any roulette wheel in which more than 50% of the slots are red.
(I think I first came across this type of definition in this post by Andrew Critch)
Frequentist statistics were invented in a (failed) attempt to keep subjectivity out of science in a time before humanity really understood the laws of probability theory
I'm a Bayesian, but do you have a source for this claim? It was my understanding that Frequentism was mostly promoted by Ron Fisher in the 20th century, well after the work of Bayes.
Synthesised from Wikipedia:
While the first cited frequentist work (the weak law of large numbers, 1713, Jacob Bernoulli, Frequentist probability) predates Bayes' work (edited by Price in 1763, Bayes' Theorem), it's not by
much. Further, according to the article on "Frequentist Probability", "[Bernoulli] is also credited with some appreciation for subjective probability (prior to and without Bayes theorem)."
The ones that pushed frequentism in order to achieve objectivity were Fisher, Neyman and Pearson. From "Frequentist probability": "All valued objectivity, so the best interpretation of probability
available to them was frequentist". Fisher did other nasty things, such as using the fact that causality is really hard to soundly establish to argue that tobacco was not proven to cause cancer. But
nothing indicates that this was done out of not understanding the laws of probability theory.
AI scientists use the Bayesian interpretation
Sometimes yes, sometimes not. Even Bayesian AI scientists use frequentist statistics pretty often.
This post makes it sound like frequentism is useless and that is not true. The concepts of: a stochastic estimator for a quantity, and looking at whether it is biased, and its variance; were
developed by frequentists to look at real world data. AI scientists use it to analyse algorithms like gradient descent, or approximate Bayesian inference schemes, but the tools are definitely useful.
The difference is (to the naive view; I don’t necessarily endorse it) that in the case where the coin has landed, I do not know how it landed, but there’s a sense in which I could, in theory, know;
there is, in any case, something to know; there is a fact of the matter about how the coin has landed, but I do not know that fact. So the “probability” of it having landed heads, or tails—the
uncertainty—is, indeed, entirely in my mind.
But in the case where the coin has yet to be tossed, there is as yet no case of the matter about whether it’s heads or tails! I don’t know whether it’ll land heads or tails, but nor could I know;
there’s nothing to know! (Or do you say the future is predetermined?—asks the naive interlocutor—Else how else may one talk about probability being merely “in the mind”, for something which has not
happened yet?)
Whatever the answers to these questions may be, they are certainly not obvious or simple answers… and that is my objection to the OP: that it attempts to pass off a difficult and confusing conceptual
question as a simple and obvious one, thereby failing to do justice to those who find it confusing or difficult.
the coin is already heads or tails, no matter that I don’t know which it is
it's worse than that. All you know that the coin has landed. You need further observations to learn more. Maybe it will slip from your hand and fall on the ground. Maybe you will be distracted with
reading LW and forget to check. Maybe you don't remember which side to check, the wrist or the hand side. You can insist that the coin has already landed and therefore it has landed either heads or
tails, but that is not a useful supposition until you actually look. Think just a little way back: the coin is about to land, but not quite yet. Is it the same as the coin has landed? Almost, but not
quite. what about a little ways further back? The uncertainty about the outcome is even more. So, there is nothing special about the landed coin until you actually look, beyond a certain level of
probabilities. A pragmatic approach (I refuse to wade into the ideological debate between militant frequentists and militant Bayesians) would be to use all available information to make the best
prediction possible, depending on the question asked.
He never said "will land heads", though. He just said "a flipped coin has a chance of landing heads", which is not a timeful statement. EDIT: no longer confident that this is the case
Didn't the post already counter your second paragraph? The subjective interpretation can be a superset of the propensity interpretation.
If you are interested in the objective probability of the coin flip,the it only has one value because it is only one event. In a deterministic universe the objective probability is 1, in a suitably
indeterministic universe it is always 0.5.
If you think the questions "what will it be" and "what was it" are different, you are dealing with subjective probability, because the difference the passage of time makes is a difference in the
information available to you, the subject.
Failing to distinguish objective and subjective probability leads to confusion. For instance, the sleeping beauty paradox is only a paradox if you expect all observers to calculate the same
probability despite the different information available to them.
But the same is true of coin flips.
When you say "all days similar to this one", are you talking about all real days or all possible days? If it's "all possible days", then this seems like summing over the measures of all possible
worlds compatible with both your experiences and the hypothesis, and dividing by the sum of the measures of all possible worlds compatible with your experiences. (Under this interpretation,
jessicata's response doesn't make much sense; "similar to" means "observationally equivalent for observers with as much information as I have", and doesn't have a free variable.)
There are also two schools of bayesian thinking: "It is popular to divide Bayesians into two main categories, “objective” and “subjective” Bayesians. The divide is sometimes made formal, there are
conferences labelled as one but not the other, for example.
A caricature of subjective Bayes is that all probabilities are just opinion, and the best we can do with an opinion is make sure it isn’t self contradictory, and satisfying the rules of probability
is a way of ensuring that. A caricature of objective Bayes is that there exists a correct probability for every hypothesis given certain information, and that different people with the same
information should make exactly the same probability judgments."
Must the frequentist refuse to assign probabilities to one-off events? Consider the question 'will it rain tomorrow'. The frequentist can define some abstract class of events, say the class of
possible weathers. She can then assume that every day the actual weather is randomly sampled from this imaginary population. She can then look at some past weather records and calculate a random
sample from this hypothetical population. Suppose there that in this large sample 30% of days were sunny; we can then say that approximately 30% of the hypothetical weathers in this population are
sunny and hence the probability of drawing a sunny day tomorrow is approx 30%.
Obviously the answer she gets hinders on the model assumptions she specifies. She can, for instance, model the weather as some stationary, autoregressive process (then the actual weather is sampled
from an abstract population of weather time series), run her regression, calculate the estimates and arrive at a completely different answer. That is still the case for Bayesians though, since they
also have to specify their priors and models and their answers depend on how they do it. My point is only that the above line of thought would allow a frequentist to make statements about
probabilities of one-offs.
It seems to me that this kind of philosophy is often employed in social science. When political scientists estimate the effect of democracy on GDP, what they are trying to find, statistically
speaking, is the expected difference in GDP between a democratic and a non-democratic country drawn from their respective populations, all else equal. Those populations are not the *real world*
populations of democratic and non-democratic countries, but some abstract populations which real-world countries are assumed to be drawn from. I have never seen this logic explicitly spilled out, but
it seems to be implicitly assumed and is required for applying frequentist techniques to social science questions.
I don't think using likelihoods when publishing in journals is tractable.
1. Where did your priors come from? What if other scientists have different priors? Justifying the chosen prior seems difficult.
2. Where did your likelihood ratios come from? What if other scientists disagree.
P values may bave been a failed attempt at objectivity, but they're a better attempt than moving towards subjective probabilities (even though the latter is more correct).
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The idea that "probability" is some preexisting thing that needs to be "interpreted" as something always seemed a little bit backwards to me. Isn't it more straightforward to say:
1. Beliefs exist, and obey the Kolmogorov axioms (at least, "correct" beliefs do, as formalized by generalizations of logic (Cox's theorem), or by possible-world-counting). This is what we refer to
as "bayesian probabilities", and code into AIs when we want to them to represent beliefs.
2. Measures over imaginary event classes / ensembles also obey the Kolmogorov axioms. "Frequentist probabilities" fall into this category.
Personally I mostly think about #1 because I'm interested in figuring out what I should believe, not about frequencies in arbitrary ensembles. But the fact is that both of these obey the same
"probability" axioms, the Kolmogorov axioms. Denying one or the other because "probability" must be "interpreted" as exclusively either #1 or #2 is simply wrong (but that's what frequentists
effectively do when they loudly shout that you "can't" apply probability to beliefs).
Now, sometimes you do need to interpret "probability" as something -- in the specific case where someone else makes an utterance containing the word "probability" and you want to figure out what they
meant. But the answer there is probably that in many cases people don't even distinguish between #1 and #2, because they'll only commit to a specific number when there's a convenient instance of #2
that make #1 easy to calculate. For instance, saying 1/6 for a roll of a "fair" die.
People often act as though their utterances about probability refer to #1 though. For instance when they misinterpret p-values as the post-data probability of the null hypothesis and go around
believing that the effect is real...
You might be interested in some work by Glenn Shafer and Vladimir Vovk about replacing measure theory with a game-theoretic approach. They have a website here, and I wrote a lay review of their first
book on the subject here.
I have also just now discovered that a new book is due out in May, which presumably captures the last 18 years or so of research on the subject.
This isn't really a direct response to your post, except insofar as I feel broadly the same way about the Kolmogorov axioms as you do about interpreting their application to phenomena, and this is
another way of getting at the same intuitions.
There's a Q&A with one of the authors here which explains a little about the purpose of the approach, mainly talks about the new book.
I clicked this because it seemed interesting, but reading the Q&A:
In atypical game we consider, one player offers bets, another decides how to bet, and a third decides the outcome of the bet. We often call the first player Forecaster, the second Skeptic, and
the third Reality.
How is this any different from the classical Dutch Book argument, that unless you maintain beliefs as probabilities you will inevitably lose money?
The subjective interpretation: Uncertainty is in the mind, not the environment. If I flip a coin and slap it against my wrist, it’s already landed either heads or tails. The fact that I don’t
know whether it landed heads or tails is a fact about me, not a fact about the coin. The claim “I think this coin is heads with probability 50%” is an expression of my own ignorance, and 50%
probability means that I’d bet at 1 : 1 odds (or better) that the coin came up heads.
Hold on, you’re pulling a fast one here—you’ve substituted the question of “what is the probability that this coin which I have already flipped but haven’t looked at yet has already landed heads” for
the question of “what is the probability that this coin which I am about to flip will land lands”!
It is obviously easy to see what the subjective interpretation means in the case of the former question—as you say, the coin is already heads or tails, no matter that I don’t know which it is. But it
is not so easy to see how the subjective interpretation makes sense when applied to the latter question—and that is what people generally have difficulty with, when they have trouble accepting
Doesn't it mean the same thing in either case? Either way, I don't know which way the coin will land or has landed, and I have some odds at which I'll be willing to make a bet. I don't see the
(Though my willingness to bet at all will generally go down over time in the "already flipped" case, due to the increasing possibility that whoever is offering the bet somehow looked at the coin in
the intervening time.)
Actually, the assignment of probability 1 to an event that has happened is also subjective. You don't know that it had to occur with complete inevitability, ie you don't know that it had a
conditional probability of 1 relative to the preceding state of the universe. You are setting it to 1 because it is a given as far as you are concerned
The question is not “what is the probability that the coin would have landed heads”. The question is, “what is the probability that the coin has in fact landed heads”!
The subjectivist interpretation of probability is common [in] … common intuition (where people feel like they can say there’s a 30% chance of rain tomorrow without worrying about the fact that
tomorrow only happens once)
Why do you say that “there’s a 30% chance of rain tomorrow” is an example of the subjective interpretation? Isn’t it just as readily interpreted as saying “on 30% of all days similar to this one [in
meteorological conditions, etc.], it rains”?
Besides, “this coin flip that I am going to do right now” only happens once, too (any subsequent coin flips will be other, different, coin flips, and not that specific coin flip). Surely you don’t
conclude from this that when someone says “this coin has a 50% chance of coming up heads”, it means they’re taking the subjectivist view of the coin’s behavior?
There are 0 other days "similar to" this one in Earth's history, if "similar to" is strict enough (e.g. the exact pattern of temperature over time, cloud patterns, etc). You'd need a precise, more
permissive definition of "similar to" for the statement to be meaningful. | {"url":"http://www.lesswrong.com/posts/BhSL973CGivhDJ4DH/interpretations-of-probability","timestamp":"2024-11-09T10:51:07Z","content_type":"text/html","content_length":"565666","record_id":"<urn:uuid:4af8981c-986f-4687-afeb-65a2b559a07b>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00007.warc.gz"} |
Staircase - DeriveIt
We want to find $\texttt{N(n)}$, the number of ways to get to the $\texttt{n}^{\th}$ stair.
1. Recursion
To write $\texttt{N}$ recursively, we want to write $\texttt{N(n)}$ in terms of $\texttt{N(\small{something else})}$.
If you ever get stuck on this, it always helps to write the very end result recursively. The very end result above is $\texttt{N(4)}$, and we can get to step 4 from either step 3 or step 2. So $\
texttt{N(4)} = \texttt{N(3)} + \texttt{N(2)}$. The same idea applies to any stair. In general, the recursion is:
2. Base case
The recursion calls smaller and smaller $\tt n$, and we need to find a stopping point for it. We need two base cases, because $\texttt{N(n)}$ depends on the previous two function calls $\texttt{N
(n-1)}$ and $\texttt{N(n-2)}$.
We can stop when we get to the very first stair. We know the number of ways to get to the first stair is 1 because we start there, so $\texttt{N(0) = 1}$. We can also stop when we get to stair -1.
There are no ways of getting to that step, so we can set the number of ways equal to zero, $\texttt{N(-1) = 0}$.
3. Code
To code this up, you can combine the recursion and base case as usual:
This solution is theoretically correct, but it's very slow, because it makes a lot of redundant function calls. To optimize, we have to use Dynamic Programming. To use Dynamic Programming, you
compute function calls in the order that they're needed, so that you don't compute the same call twice, called "Tabulation".
To do this, you can realize that the function $\texttt{N(n)}$ depends on function calls that use smaller inputs $\texttt{N(n-1)}$ and $\texttt{N(n-2)}$. This means we can compute the function on
smaller inputs first, and larger inputs last, going from $\texttt{N(-1), N(0), N(1), \dots, N(n)}$. We only have to keep track of the previous two values we've computed, because $\tt N$ only depends
on the previous two function calls. Here's the code for doing this:
Time Complexity $O(n)$
Space Complexity $O(1)$ | {"url":"https://deriveit.org/coding/staircase-163","timestamp":"2024-11-12T03:01:41Z","content_type":"text/html","content_length":"106817","record_id":"<urn:uuid:1df5d3e1-0196-4124-8100-f4bbaa7e0717>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00368.warc.gz"} |
What Is Avogadro's Law (Avogadro's Hypothesis Or Avogadro's Principle)?
What Is Avogadro’s Law (Avogadro’s Hypothesis Or Avogadro’s Principle)?
Written by Mahak JalanLast Updated On: 19 Oct 2023Published On: 2 Oct 2018
Table of Contents (click to expand)
Avogadro’s law states that under conditions of constant pressure and temperature, there is a direct relationship between the number of moles and volume of a gas. This was Avogadro’s initial
hypothesis. This law was applicable to ideal gases, while real gases show a slight deviation from it.
The modern definition of Avogadro’s law is that for a particular mass of an ideal gas, the amount (number of moles) and volume of the gas are directly proportional, provided the temperature and
pressure conditions are constant.
Avogadro’s Law Formula
Avogadro’s law’s mathematical formula can be written as:
V ∝ n or V/n = k
Where “V” is the volume of the gas, “n” is the amount of the gas (number of moles of the gas) and “k” is a constant for a given pressure and temperature.
Avogadro’s law formula describes how equal volumes of all gases contain the same number of molecules, under the same conditions of pressure and temperature. In other words, it describes that equal
volumes of two different gases will have the same number of molecules as long as the temperature and pressure are the same.
Amadeo Avogadro was an Italian scientist of the 19^th century. He is known for making major contributions to chemistry, when it was just becoming a separate science field. His work came around the
same as that of Jacques Charles (Charles Law), Robert Boyle (Boyle’s Law), etc. In fact, Avogadro’s Law, the hypothesis set by him, was among the laws on which the Ideal Gas Law is based.
An ideal gas can be defined as one in which the collisions between the molecules of the gas are elastic – i.e. there is no loss of kinetic or of momentum, and the molecules don’t have any
intermolecular forces of attraction, i.e. they don’t have any interactions between them with the exceptions of the randomized collisions.
Before we get into understanding his work however, let us go over some basics.
A mole is a measure of the quantity of a substance. One mole of a substance is defined as that quantity which has as many units as the number of carbon atoms in 12 grams of C-12 carbon.
Another thing to remember is that a lot of these laws use STP or standard temperature and pressure. For STP, the value of temperature is 273.15 K (which is 0℃) while the value of pressure is 1atm or
Amedeo Avogadro (Image Credit: Wikimedia Commons)
Also Read: What Is Charles’s Law?
Avogadro’s Number
Avogadro’s number is the number of molecules of gas in a mole. This number is huge, the current figure being 6.022 x 10^23. The unit for Avogadro’s number is mol^-1. This means that the measure of
the entity in question is per mole. Avogadro’s number is usually symbolised by N.
It’s interesting to note that contrary to popular belief, Avogadro’s number was not discovered by Amedeo Avogadro. The concept of mole and the determination of the value of Avogadro’s number happened
after Avogadro’s death. In fact, Avogadro’s number is so called in honour of his discovery and his work.
The first person to calculate the total number of particles present in a substance was an Austrian high school teacher called Josef Loschmidt, who, after a few years, became a professor at the
University of Vienna.
Using kinetic molecular theory, Loschmidt was able to estimate the number of particles present in one cubic centimeter of gas at standard conditions of pressure and temperature. The value he
calculated back in 1865 is known as the Loschmidt constant today, and its value is 2.6867773 x 1025 m-3.
The term ‘Avogadro’s number’ was first used by Jean Baptiste Perrin – a French physicist. He reported an estimate of the Avogadro’s number in 1909 based on his work on Brownian motion. For the
uninitiated, Brownian motion is the random, haphazard movement of microscopic particles suspended in a gas/liquid.
Accurate determination of the Avogadro’s number only became possible for the first time when Robert Millikan – an American physicist – successfully measured the charge on an electron. Prior to this,
the charge on a mole of electrons was already known (it’s a constant called ‘Faraday’, which is equal to 96,485.3383 coulombs per mole of electrons).
Certain parallels have been drawn to understand how humongous this number is. One of the easiest to comprehend is if this number of unpopped kernels of popcorn were spread across the area of the
United States, after popping, the popcorn would cover the country up to a depth of 9 miles (For reference, the area of the United States is 3.797 square miles!!)
You will have noticed that I mentioned the current figure of Avogadro’s number. This is because over the years since the value was first determined, different methods have been used to calculate it.
While each method gives approximately the same answer, there are slight variations. Therefore, based on the most recent calculations, that is the accepted figure. The first person to make this
calculation, however, was Loschmidt.
Also Read: Grams To Moles: How To Convert Grams To Moles?
Moles To Grams
Moles can be converted to grams, which is another very popular measurement of quantity, and vice versa, by the following formula
Moles = grams/molar mass
To calculate the molar mass of a substance, one has to employ the use of the ever efficient periodic table. It be calculated by simply adding the mass number of the individual atoms in the substance.
For instance, if one has to calculate the molar mass of NaCl –
Mass number of Na = 22.99 g/mol
Mass number of Cl = 35.45 g/mol
Therefore molar mass of NaCl is 22.99 + 35.45 = 58.44 g/mol
Avogadro’s number has a lot of applications in chemistry and physics. Certain generalization’s have also been drawn. For instance, the volume of 1 mole of a gas at STP is 22.4L. These are very handy
in calculations. Avogadro’s number is an entity used by chemists worldwide. Although he may not have determined it, his work was the precedence for these calculations.
How well do you understand the article above!
Can you answer a few questions based on the article you just read?
About the Author
Mahak Jalan has a BSc degree in Zoology from Mumbai University in India. She loves animals, books and biology. She has a general assumption that everyone shares her enthusiasm about the human body!
An introvert by nature, she finds solace in music and writing.
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Homology software
This page is a part of CVprimer.com, a wiki devoted to computer vision. It focuses on low level computer vision, digital image analysis, and applications. It is designed as an online textbook but the
exposition is informal. It geared towards software developers, especially beginners, and CS students. The wiki contains mathematics, algorithms, code examples, source code, compiled software, and
some discussion. If you have any questions or suggestions, please contact me directly.
Homology software
From Computer Vision Primer
CHomP [1], Computational Homology Project. This is a software package developed by Computational Homology Project group at Georgia Tech, now at Rutgers. CHomP runs on Windows and consists of 38
individual programs for homology computations in n-dimensions. CHomP can compute the Betti numbers and homology of a simplicial and cubical complex as well as the maps induced in homology. All the
algorithms are also explained in detail in T. Kaczynski, K. Mischaikow, and M. Mrozek, Computational Homology, Appl. Math. Sci. Vol. 157, Springer Verlag, NY 2004. It is a very powerful, yet very
easy package to use, with no installation necessary. The programs are called from a command prompt.
PLEX [2]. This is a set of routines written for MATLAB, developed by Vin de Silva at Stanford University with the help of Gunnar Carlson. PLEX computes the Betti numbers of simplicial complexes as
well as the Betti numbers of maps. One may only enter data as a list of vertices. After defining an epsilon value (similar to the alpha-value in Persistent Homology), PLEX calculates the appropriate
edges for the vertex set.
Alpha Shapes [3]. The package consists of several programs (Pdb2alf, Delcx, Mkalf, Alvis, Volbl) by the National Center for Supercomputing Applications and the Department of Computer Science at the
University of Illinois at Urbana-Champaign. This program runs only in Linux/Unix/Sun. These programs have been used in the study of proteins, and have also been modified for commercial use in surface
reconstruction by Geomagic. Alpha Shapes can compute the Betti numbers of a simplicial complex, but only in dimension three. For the purpose of computing the Betti numbers, this software has a
weakness in that you can only look at one Betti number at a time. This means that if you want to look at how the Betti number change as alpha grows (Persistent Homology), you must enter the values in
another program by hand. This can be a very time consuming process.
CGAL [4], Computational Geometry Algorithms Library. This is a geometry software package created as a collaborative effort from several sites in Europe and Israel. CGAL is basically code to be used
in C++ under IRIX/Linux/Unix/Windows. CGAL does not directly compute homology, but can provide the tools necessary to compute triangulations and alpha complexes from which homology can easily be
calculated incrementally. However, CGAL can only be used in dimensions two and three.
GAP homology package [5]. It runs only on Linux/Unix. The two purposes behind this project were to implement efficient algorithms for the calculation of Smith Normal Forms of sparse matrices with
integer entries and simplicial homology of simplicial complexes. Three uses have been suggested for this program. 1. Calculation of homology groups for a large, complex simplicial complex. 2.
Calculation of homology for many small simplicial complexes. 3. Use as a teaching aide in homology course.
Moise [6]. A topology package for Maple.
Bottom lime, these software packages are not very user friendly.
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How Einstein tried to model the shape of the Universe
Not even Einstein immediately knew the power of the equations he gave us.
Credit: Annelisa Leinbach / Big Think; Getty Images
In 1917, just two years after Albert Einstein proposed the general theory of relativity — his revolutionary new theory of gravity — he took a bold step forward and decided to apply his theory to the
Universe as a whole. His question was simple but incredibly bold: Can we model the shape of the Universe? To answer, Einstein made use of his new, powerful theory that described gravity as the
curvature of spacetime around a mass. The more massive a body, the more warped the geometry around it is, and the slower time ticks.
Einstein’s reasoning was crystal clear. Since his theory allowed him to calculate how the Sun’s mass bends space around it, if he modeled how mass is distributed in the Universe, he could calculate
its shape. His theory was not limited to any particular location in the Universe — it could measure the Universe itself. Imagine that: a human mind computing the geometry of the Cosmos.
Einstein’s madhouse cosmology
Einstein was the first to recognize how controversial his ideas might be. In a letter to physicist and friend Paul Ehrenfest in early 1917, Einstein wrote, “I have…again perpetrated something about
gravitational theory which somewhat exposes me to the danger of being confined in a madhouse.” Einstein’s proposal inaugurated a new era in cosmology, one that started with the application of general
relativity to the Universe as a whole and allowed scientists to study the structure and evolution of the Cosmos.
But the equations of general relativity are very complex, and to find solutions one needs to impose simplifications. This happens often in physics, especially now that most of the simpler, linear
problems have been dealt with. Before computers allowed us to tackle nonlinear systems, physics was the art of effective approximations. Even when a problem in its full complexity could not be
solved, you were in business if you could keep its main features and introduce “easy” equations to solve.
But in 1917, Einstein had a huge task ahead of him. He had to simplify the Universe, fit it into a version of his equations that he could solve by hand. At that time, no one thought seriously that
the Universe was expanding — in other words, that it was changing in time. There were small-scale motions like the local displacements of stars, but these did not reveal any overall trend. There was
no compelling evidence that large-velocity motions existed in the Universe. It would take until 1929 for Edwin Hubble to confirm cosmic expansion, a topic we explored here recently.
Universal homogeneity
What Universe would Einstein theorize? The less data is available, the more a scientist is free to speculate. This is fascinating from a cultural aspect, because the choices a scientist makes with
such freedom reveal a lot about their worldview. Einstein, like most everyone else at the time, believed the Universe to be static. He thought that most matter was part of the Milky Way. Only in 1924
would it become clear that our galaxy was one among billions of others — again thanks to Hubble’s work.
Einstein was not comfortable with the notion of an infinite Universe that contained a finite amount of matter. He believed that a spatially bounded, and thus finite, Universe was a much more natural
choice from the point of view of general relativity. It was also the simplest choice and the most mathematically elegant one. It pictures the Universe as a perfect balloon.
The geometry of the Universe is uniquely determined by its total mass (and/or its energy, as a consequence of special relativity, described by Einstein’s earlier theory). Remember that we are looking
here for simplifications. Well, Einstein’s first simplification became known as the cosmological principle. It told us that the Universe on average looks the same everywhere in all directions. At
large enough volumes, the Universe is homogeneous (the same everywhere) and isotropic (the same in all directions). There is no preferred point or direction in the Universe. If we look within small
volumes, such as in the neighborhood of the Sun, we will see stars that are not really spread out in the same way in all directions. But if we take a large enough chunk of the Universe and compare it
to another large chunk, according to this principle, they will look about the same. A useful image is to think of a crowded beach on a summer afternoon. If you walk around, you will see a lot of
variation, with some empty spots here and there. But from afar the beach is homogeneous, presenting a mass and a mess of humans across its breadth.
Collapsing universal logic
Once homogeneity and isotropy are factored in, it becomes much easier to solve Einstein’s equations. Einstein’s Universe is spherical, and its geometry is determined by a single parameter — the
radius of the Universe. Because Einstein’s is a static Universe, the distribution of matter does not change in time, hence neither does the geometry.
Einstein, then, assumed a finite, spherical, and static Universe, one with a closed geometry characterized by a three-dimensional generalization of the surface of a sphere. As such it had a radius,
which was determined by the total mass of the Universe. This is as it should be, since matter bends geometry. As he proudly announced in 1922, “The complete dependence of the geometrical upon the
physical properties becomes clearly apparent by means of this equation.”
Much to Einstein’s disappointment, this solution came with a high price tag. If the Universe is finite and static, and gravity is an attractive force, matter will tend to collapse on itself unless it
has negative pressure, which is a weird property. When filled with a constant density of matter that has zero or positive pressure, this Universe simply could not exist. Something else was needed.
To keep his Universe static, Einstein added a term into the equations of general relativity, one he initially dubbed a negative pressure. It soon became known as the cosmological constant.
Mathematics allowed the concept, but it had absolutely no justification from physics, no matter how hard Einstein and others tried to find one. The cosmological constant clearly detracted from the
formal beauty and simplicity of Einstein’s original equations of 1915, which achieved so much without any need for arbitrary constants or additional assumptions. It amounted to a cosmic repulsion
chosen to precisely balance the tendency of matter to collapse on itself. In modern parlance we call this fine tuning, and in physics it is usually frowned upon.
Einstein knew that the only reason for his cosmological constant to exist was to secure a static and stable finite Universe. He wanted this kind of Universe, and he did not want to look much further.
Quietly hiding in his equations, though, was another model for the Universe, one with an expanding geometry. In 1922, the Russian physicist Alexander Friedmann would find this solution. As for
Einstein, it was only in 1931, after visiting Hubble in California, that he accepted cosmic expansion and discarded at long last his vision of a static Cosmos.
Einstein’s equations provided a much richer Universe than the one Einstein himself had originally imagined. But like the mythic phoenix, the cosmological constant refuses to go away. Nowadays it is
back in full force, as we will see in a future article.
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What does it mean when a parabola has a horizontal axis?
If a parabola has a horizontal axis, the standard form of the equation of the parabola is this: (y – k)2 = 4p(x – h), where p≠ 0. The vertex of this parabola is at (h, k). The focus is at (h + p, k).
The directrix is the line x = h – p. The axis is the line y = k.
What is a horizontal axis of symmetry?
A line through a shape so that each side is a mirror image. When the shape is folded in half along the axis of symmetry, then the two halves match up.
Will a parabola have an axis of symmetry?
All parabolas have exactly one axis of symmetry (unlike a circle, which has infinitely many axes of symmetry).
What is a horizontal parabola?
The squaring of the variables in the equation of the parabola determines where it opens: When y is squared and x is not, the axis of symmetry is horizontal and the parabola opens left or right. For
example, x = y2 is a horizontal parabola; it’s shown in the figure.
What is axis of symmetry in parabola?
The axis of symmetry is the vertical line that goes through the vertex of a parabola so the left and right sides of the parabola are symmetric. To simplify, this line splits the graph of a quadratic
equation into two mirror images.
How do you tell if a parabola is horizontal or vertical?
If the x is squared, the parabola is vertical (opens up or down). If the y is squared, it is horizontal (opens left or right). If a is positive, the parabola opens up or to the right. If it is
negative, it opens down or to the left.
Can a horizontal parabola be a function?
Sideways Stuff As you can see, these parabolas are not functions, since there is more than one y value for each x value.
What is axis of symmetry of a line?
The axis of symmetry is an imaginary straight line that divides a shape into two identical parts, thereby creating one part as the mirror image of the other part. When folded along the axis of the
symmetry, the two parts get superimposed.
How many lines of symmetry does a parabola have?
Every parabola has an axis of symmetry which is the line that divides the graph into two perfect halves.
What type of symmetry does a parabola have?
A parabola is the graph of a quadratic function. Each parabola has a line of symmetry. Also known as the axis of symmetry, this line divides the parabola into mirror images. The line of symmetry is
always a vertical line of the form x = n, where n is a real number.
Where is the axis of symmetry?
The axis of symmetry is the vertical line that goes through the vertex of a quadratic equation.
What is the definition of axis of symmetry?
axis of symmetry. noun. Mathematics. a straight line for which every point on a given curve has corresponding to it another point such that the line connecting the two points is bisected by the given | {"url":"https://nikoismusic.com/what-does-it-mean-when-a-parabola-has-a-horizontal-axis/","timestamp":"2024-11-10T15:07:19Z","content_type":"text/html","content_length":"53924","record_id":"<urn:uuid:edd2aa06-07d3-45df-af0a-7c3a7c9ed23a>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00853.warc.gz"} |
PROC REG: Computations for Ridge Regression and IPC Analysis :: SAS/STAT(R) 9.22 User's Guide
Computations for Ridge Regression and IPC Analysis
In ridge regression analysis, the crossproduct matrix for the independent variables is centered (the NOINT option is ignored if it is specified) and scaled to one on the diagonal elements. The ridge
constant k (specified with the RIDGE= option) is then added to each diagonal element of the crossproduct matrix. The ridge regression estimates are the least squares estimates obtained by using the
new crossproduct matrix.
Let X be an Y be an D be a
For IPC analysis, the smallest m eigenvalues of m is specified with the PCOMIT= option) are omitted to form the estimates.
For information about ridge regression and IPC standardized parameter estimates, parameter estimate standard errors, and variance inflation factors, refer to Rawlings (1988), Neter, Wasserman, and
Kutner (1990), and Marquardt and Snee (1975). Unlike Rawlings (1988), the REG procedure uses the mean squared errors of the submodels instead of the full model MSE to compute the standard errors of
the parameter estimates. | {"url":"http://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/statug_reg_sect046.htm","timestamp":"2024-11-12T12:57:27Z","content_type":"application/xhtml+xml","content_length":"12700","record_id":"<urn:uuid:dee88233-f8b3-485f-bc13-b71d0a22a7c4>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00891.warc.gz"} |
Autumn School on Correlated Electrons
E. Pavarini Orbital Ordering in Materials
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R. Eder Multiplets in Transition Metal Ions and Introduction to Multiband Hubbard Models
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E. Pavarini LDA+DMFT: Multi-Orbital Hubbard Models
H. Tjeng Determining Orbital Wavefunctions using Core-Level Non-Resonant Inelastic X-Ray Scattering
H.G. Evertz DMRG for Multiband Impurity Solvers
F. Assaad Quantum Monte Carlo Methods for Fermion-Boson Problems
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H. Hafermann Introduction to Diagrammatic Approaches
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M. Eckstein Correlated Electrons out of Equilibrium: Non-Equilibrium DMFT
R. Martin Electronic Structure Computation Meets Strong Correlation: Guiding Principles
R. Scalettar Insulator, Metal, or Superconductor: The Criteria
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R. Eder Introduction to the Hubbard Model
P. Prelovšek The Finite Temperature Lanczos Method and its Applications
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R. Scalettar An Introduction to the Hubbard Hamiltonian
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F. Manghi Multi-Orbital Cluster Perturbation Theory for Transition-Metal Oxides
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B. Keimer Recent Advances in Experimental Research on High-Temperature Superconductivity
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O. Gunnarsson Strongly Correlated Electrons: Estimates of Model Parameters
R. Zeller DFT-based Green Function Approach for Impurity Calculations
F. Anders The Kondo Effect
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F. Aryasetiawan The Constrained RPA Method for Calculating Hubbard U from First-Principles
E. Koch The Lanczos Method
N. Blümer Hirsch-Fye Quantum Monte Carlo Method for Dynamical Mean-Field Theory
P. Werner Continuous-Time Impurity Solvers
A. Lichtenstein Non-Local Correlations in Solids: Beyond DMFT
H. Ebert Multiple-Scattering Formalism for Correlated Systems: A KKR-DMFT Approach
K. Held Hedin Equations, GW, GW+DMFT, and all That
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Joint Radar-Communication Using PMCW and OFDM Waveforms
This example shows how to model a joint radar-communication (JRC) system using the Phase Array System Toolbox. Two modulation schemes for combining radar and communication signals are presented. The
first scheme uses the phase-modulated continuous wave (PMCW) waveform. The second scheme uses the orthogonal frequency division multiplexing (OFDM) waveform. For each modulation scheme the example
shows how to process the received signal by both the radar and the communication receivers.
JRC System Parameters
JRC is one of many different radar and communication coexistence strategies. Such strategies have emerged recently as a response to an increasingly congested frequency spectrum and a demand of radar
and communication systems for a large bandwidth. In a JRC system both the radar and the communication share the same platform and have a common transmit waveform. This example shows two approaches to
utilizing a single waveform for both functions. The first approach uses PMCW. Such a JRC system can be considered as radar-centric, since PMCW is a waveform tailored to radar applications. The second
approach uses OFDM. This approach is communication-centric since a communication waveform is used to perform the radar function.
Consider a JRC system that operates at a carrier frequency of 24 GHz and has a bandwidth of 100 MHz.
% Set the random number generator for reproducibility
fc = 24e9; % Carrier frequency (Hz)
B = 100e6; % Bandwidth (Hz)
Set the peak transmit power to 10 mW and the transmit antenna gain to 20 dB.
Pt = 0.01; % Peak power (W)
Gtx = 20; % Tx antenna gain (dB)
Let the JRC transmitter and the radar receiver be colocated at the origin. Create a phased.Platform object to represent the JRC system.
jrcpos = [0; 0; 0]; % JRC position
jrcvel = [0; 0; 0]; % JRC velocity
jrcmotion = phased.Platform('InitialPosition', jrcpos, 'Velocity', jrcvel);
Assume the radar receiver has the following parameters.
Grx = 20; % Radar Rx antenna gain (dB)
NF = 2.9; % Noise figure (dB)
Tref = 290; % Reference temperature (K)
Let the maximum range of interest and the maximum observed relative velocity be 200 m and 60 m/s respectively.
Rmax = 200; % Maximum range of interest
vrelmax = 60; % Maximum relative velocity
Consider a scenario with three moving radar targets and a single downlink user. Create a phased.Platform object to represent moving targets and a phased.RadarTarget object to represent the target's
radar cross sections (RCS).
tgtpos = [110 45 80; -10 5 -4; 0 0 0]; % Target positions
tgtvel = [-15 40 -32; 0 0 0; 0 0 0]; % Target velocities
tgtmotion = phased.Platform('InitialPosition', tgtpos, 'Velocity', tgtvel);
tgtrcs = [4.7 3.1 2.3]; % Target RCS
target = phased.RadarTarget('Model', 'Swerling1', 'MeanRCS', tgtrcs, 'OperatingFrequency', fc);
Create a position vector to represent the position of the downlink user.
userpos = [100; 20; 0]; % Downlink user position
Visualize the JRC scenario using helperPlotJRCScenario helper function.
helperPlotJRCScenario(jrcpos, tgtpos, userpos, tgtvel);
JRC System Using a PMCW Waveform
The first modulation scheme considered in this example uses a PMCW waveform. A PMCW radar repeatedly transmits a selected phase-coded sequence. The duration of the transmitted sequence is called a
modulation period. Since the duty cycle of a PMCW radar is equal to one, the next modulation period starts right after the previous.
This example uses a maximum length pseudo-random binary sequence (PRBS) also frequently referred to as an m-sequence. Maximum length sequences have low autocorrelation sidelobes that result in a good
range estimation performance. The communication data is modulated on top of the radar waveform such that each modulation period carries a single bit. This is achieved by simply multiplying the entire
modulation period by the corresponding BPSK symbol.
Using the helperMLS helper function create a PRBS.
% Generate an m-sequence. The sequence lengths has to be 2^p-1, where p is
% an integer.
p = 8;
prbs = helperMLS(p);
Nprbs = numel(prbs); % Number of chips in PRBS
Given that the chip duration of a phase-modulated waveform is an inverse of the total bandwidth, compute the modulation period.
Tchip = 1/B; % Chip duration
Tprbs = Nprbs * Tchip; % Modulation period
Generate the communication data to be transmitted. Encode the generated data bits using the BPSK modulation.
Mpmcw = 256; % Number of transmitted data bits
dataTx = randi([0, 1], [Mpmcw 1]); % Binary data
bpskTx = pskmod(dataTx, 2); % BPSK symbols
Since the communication channel is unknown, the JRC system must perform channel sounding. One way to do it is to send out a single period of the PRBS sequence unmodulated with the communication data.
Thus, a transmission of one data bit would require transmitting the PRBS sequence twice. The first time to sound the channel, the second time to transmit the data. Create the PMCW waveform by
stacking two modulation periods together, the first period is just the generated PRBS and the second period is the generated PRBS multiplied by the corresponding BPSK symbol.
% Transmit waveform
xpmcw = [prbs * ones(1, Mpmcw); prbs * bpskTx.'];
Thus, the duration of one PMCW block is twice the modulation period.
The block duration can be longer if the same data symbol has to be repeated multiple times in order to improve the signal-to-noise ratio (SNR) by integrating multiple periods at the receiver.
Assume that the coherent processing interval is long enough to coherently process Mpmcw blocks. Compute the duration of a single PMCW frame that carries Mpmcw data bits.
Radar Signal Simulation and Processing
This section of the example shows how to simulate the radar sensing part of the JRC system with the PMCW waveform. First, generate received PMCW blocks.
% Set the sample rate equal to the bandwidth
fs = B;
% Setup the transmitter and the radiator
transmitter = phased.Transmitter('Gain', Gtx, 'PeakPower', Pt);
% Assume the JRC is using an isotropic antenna
ant = phased.IsotropicAntennaElement;
radiator = phased.Radiator('Sensor', ant, 'OperatingFrequency', fc);
% Setup the collector and the receiver
collector = phased.Collector('Sensor', ant, 'OperatingFrequency', fc);
receiver = phased.ReceiverPreamp('SampleRate', fs, 'Gain', Grx, 'NoiseFigure', NF, 'ReferenceTemperature', Tref);
% Setup a free space channel to model the JRC signal propagation from the
% transmitter to the targets and back to the radar receiver
radarChannel = phased.FreeSpace('SampleRate', fs, 'TwoWayPropagation', true, 'OperatingFrequency', fc);
% Preallocate space for the signal received by the radar receiver
ypmcwr = zeros(size(xpmcw));
% Transmit one PMCW block at a time assuming that all Mpmcw blocks are
% transmitted within a single CPI
for m = 1:Mpmcw
% Update sensor and target positions
[jrcpos, jrcvel] = jrcmotion(Tpmcw);
[tgtpos, tgtvel] = tgtmotion(Tpmcw);
% Calculate the target angles as seen from the transmit array
[tgtrng, tgtang] = rangeangle(tgtpos, jrcpos);
% Transmit signal
txsig = transmitter(xpmcw(:, m));
% Radiate signal towards the targets
radtxsig = radiator(txsig, tgtang);
% Apply free space channel propagation effects
chansig = radarChannel(radtxsig, jrcpos, tgtpos, jrcvel, tgtvel);
% Reflect signal off the targets
tgtsig = target(chansig, false);
% Receive target returns at the receive array
rxsig = collector(tgtsig, tgtang);
% Add thermal noise at the receiver
ypmcwr(:, m) = receiver(rxsig);
Now the received signals can be processed to obtain estimates of the targets ranges and velocities. The following diagram shows the signal processing steps needed to compute the range-Doppler map.
Start by removing the communication data. Multiply the modulation periods that carry the data by the corresponding BPSK symbols.
ypmcwr1 = ypmcwr(Nprbs+1:end, :) .* (bpskTx.');
The modulation period used for the communication channel sounding can be integrated with the modulation period that carried the data.
ypmcwr1 = ypmcwr1 + ypmcwr(1:Nprbs, :);
It is more computationally efficient to perform matched filtering in the frequency domain. First, compute the frequency domain representation of the integrated signal then, multiply the result by the
complex-conjugate version of the DFT of the used PRBS.
% Frequency-domain matched filtering
Ypmcwr = fft(ypmcwr1);
Sprbs = fft(prbs);
Zpmcwr = Ypmcwr .* conj(Sprbs);
Use phased.RangeDopplerResponse to compute and plot the range-Doppler map.
% Range-Doppler response object computes DFT in the slow-time domain and
% then IDFT in the fast-time domain to obtain the range-Doppler map
rdr = phased.RangeDopplerResponse('RangeMethod', 'FFT', 'SampleRate', fs, 'SweepSlope', -B/Tprbs,...
'DopplerOutput', 'Speed', 'OperatingFrequency', fc, 'PRFSource', 'Property', 'PRF', 1/Tpmcw,...
'ReferenceRangeCentered', false);
plotResponse(rdr, Zpmcwr, 'Unit', 'db');
xlim([-vrelmax vrelmax]);
ylim([0 Rmax]);
Communication Signal Simulation and Processing
This example uses a stochastic Rician channel model to simulate the propagation of the JRC signal from the transmitter to the downlink receiver. Create a communication channel object and pass the
PMCW signal through the channel.
% Line-of-sight between the JRC transmitter and the downlink user
dlos = vecnorm(jrcpos - userpos);
taulos = dlos/physconst('LightSpeed');
% Delays and gains due to different path in the communication channel
pathDelays = taulos * [1 1.6 1.1 1.45] - taulos;
averagePathGains = [0 -1 2 -1.5];
% Maximum two-way Doppler shift
fdmax = 2 * speed2dop(vrelmax, freq2wavelen(fc));
commChannel = comm.RicianChannel('PathGainsOutputPort', true, 'DirectPathDopplerShift', 0,...
'MaximumDopplerShift', fdmax/2, 'PathDelays', pathDelays, 'AveragePathGains', averagePathGains, ...
'SampleRate', fs);
% Pass the transmitted signal through the channel
ypmcwc = commChannel(xpmcw(:));
For simplicity, model the downlink receiver using additive gaussian noise and assuming that the signal-to-noise ratio (SNR) is 40 dB.
SNRu = 40; % SNR at the downlink user's receiver
ypmcwc = awgn(ypmcwc, SNRu, "measured");
ypmcwc = reshape(ypmcwc, 2*Nprbs, []);
Use the first PRBS repetition in each PMCW block to estimate the channel frequency response.
ysound = ypmcwc(1:Nprbs, :);
Hpmcw = fft(ysound)./Sprbs;
Once the channel estimates are available, the received signal can be processed to extract the user data. The diagram below show the required processing steps.
Since a radar waveform is used to carry the user data, the matched filtering is the first step in the demodulation process. Perform matched filtering in the frequency domain.
% Compute the frequency domain representation of the received signal
Ypmcwc = fft(ypmcwc(Nprbs + 1:end, :));
% Multiply by the complex-conjugate version of the DFT of the used PRBS
Zpmcwc = Ypmcwc .* conj(Sprbs);
Multiply the frequency representation of the matched filtered signal by the inverse of the estimated channel frequency response to remove the channel effects.
Demodulate the signal to first obtain the BPSK symbols and then the binary data.
zpmcwc = ifft(Zpmcwc);
[~, idx] = max(abs(zpmcwc), [], 'linear');
bpskRx = zpmcwc(idx).';
bpskRx = bpskRx./Nprbs;
dataRx = pskdemod(bpskRx, 2);
refconst = pskmod([0 1], 2);
constellationDiagram = comm.ConstellationDiagram('NumInputPorts', 1, ...
'ReferenceConstellation', refconst, 'ChannelNames', {'Received PSK Symbols'});
Compute the bit error rate.
[numErr, ratio] = biterr(dataTx, dataRx)
JRC System Using an OFDM Waveform
OFDM is a second modulation scheme considered in this example. OFDM signals are widely used for wireless communication. OFDM is a multicarrier signal composed of a set of orthogonal complex
exponentials known as subcarriers. The complex amplitude of each subcarrier can be used to carry communication data allowing for a high data rate. OFDM waveforms were also noted to exhibit Doppler
tolerance and a lack of range-Doppler coupling. These OFDM features are attractive to radar applications. This section of the example shows how to use an OFDM waveform to perform both the radar
sensing and communication.
Let the number of OFDM subcarriers be 1024. Compute the subcarrier separation given that all subcarriers must be equally spaced in frequency.
Nsc = 1024; % Number of subcarriers
df = B/Nsc; % Separation between OFDM subcarriers
Tsym = 1/df; % OFDM symbol duration
According to the OFDM system design rules, the subcarrier spacing must be at least ten times larger than the maximum Doppler shift experienced by the signal. This is required to ensure the
orthogonality of the subcarrier. Verify that this condition is met for the scenario considered in this example.
In order to avoid intersymbol interference, the OFDM symbols are prepended with cyclic prefix. Compute the cyclic prefix duration based on the maximum range of interest.
Tcp = range2time(Rmax); % Duration of the cyclic prefix (CP)
Ncp = ceil(fs*Tcp); % Length of the CP in samples
Tcp = Ncp/fs; % Adjust duration of the CP to have an integer number of samples
Tofdm = Tsym + Tcp; % OFDM symbol duration with CP
Nofdm = Nsc + Ncp; % Number of samples in one OFDM symbol
Designate some of the subcarriers to act as guard bands.
nullIdx = [1:9 (Nsc/2+1) (Nsc-8:Nsc)]'; % Guard bands and DC subcarrier
Nscd = Nsc-length(nullIdx); % Number of data subcarriers
Generate binary data to be transmitted. Since each OFDM subcarrier can carry a complex data symbol, the generated binary data first must be modulated. Use QAM-modulation to modulate the generated
binary data. Finally, OFDM-modulate the QAM-modulated symbols.
bps = 6; % Bits per QAM symbol (and OFDM data subcarrier)
K = 2^bps; % Modulation order
Mofdm = 128; % Number of transmitted OFDM symbols
dataTx = randi([0,1], [Nscd*bps Mofdm]);
qamTx = qammod(dataTx, K, 'InputType', 'bit', 'UnitAveragePower', true);
ofdmTx = ofdmmod(qamTx, Nsc, Ncp, nullIdx);
Assume that the coherent processing interval is long enough to coherently process Mofdm symbols. Compute the duration of a single OFDM frame and a total number of data bits in one frame.
Tofdm*Mofdm % Frame duration (s)
Nscd*bps*Mofdm % Total number of bits
Notice that the OFDM frame duration is close to the PMCW frame duration, however the total number of transmitted bits is larger by several orders of magnitude.
Radar Signal Simulation and Processing
Use the same system as in the PMCW case to simulate transmission of the OFDM signal by the JRC and the reception of the target returns by the radar receiver.
xofdm = reshape(ofdmTx, Nofdm, Mofdm);
% OFDM waveform is not a constant modulus waveform. The generated OFDM
% samples have power much less than one. To fully utilize the available
% transmit power, normalize the waveform such that the sample with the
% largest power had power of one.
xofdm = xofdm/max(sqrt(abs(xofdm).^2), [], 'all');
% Preallocate space for the signal received by the radar receiver
yofdmr = zeros(size(xofdm));
% Reset the platform objects representing the JRC and the targets before
% running the simulation
% Transmit one OFDM symbol at a time assuming that all Mofdm symbols are
% transmitted within a single CPI
for m = 1:Mofdm
% Update sensor and target positions
[jrcpos, jrcvel] = jrcmotion(Tofdm);
[tgtpos, tgtvel] = tgtmotion(Tofdm);
% Calculate the target angles as seen from the transmit array
[tgtrng, tgtang] = rangeangle(tgtpos, jrcpos);
% Transmit signal
txsig = transmitter(xofdm(:, m));
% Radiate signal towards the targets
radtxsig = radiator(txsig, tgtang);
% Apply free space channel propagation effects
chansig = radarChannel(radtxsig, jrcpos, tgtpos, jrcvel, tgtvel);
% Reflect signal off the targets
tgtsig = target(chansig, false);
% Receive target returns at the receive array
rxsig = collector(tgtsig, tgtang);
% Add thermal noise at the receiver
yofdmr(:, m) = receiver(rxsig);
Process the target returns to obtain a range-Doppler map. The signal processing steps are shown in the block diagram.
Remove the cyclic prefix and compute the DFT in the fast-time domain using the ofdmdemod function.
yofdmr1 = reshape(yofdmr, Nofdm*Mofdm, 1);
% Demodulate the received OFDM signal (compute DFT)
Yofdmr = ofdmdemod(yofdmr1, Nsc, Ncp, Ncp, nullIdx);
The complex amplitudes of the subcarriers that comprise the received OFDM signal are the transmitted QAM symbols. Therefore, to remove the communication data divide the DFT of the received signal by
the transmitted QAM symbols.
Use phased.RangeDopplerResponse to compute and plot the range-Doppler map.
% Range-Doppler response object computes DFT in the slow-time domain and
% then IDFT in the fast-time domain to obtain the range-Doppler map
rdr = phased.RangeDopplerResponse('RangeMethod', 'FFT', 'SampleRate', fs, 'SweepSlope', -B/Tofdm,...
'DopplerOutput', 'Speed', 'OperatingFrequency', fc, 'PRFSource', 'Property', 'PRF', 1/Tofdm, ...
'ReferenceRangeCentered', false);
plotResponse(rdr, Zofdmr, 'Unit', 'db');
xlim([-vrelmax vrelmax]);
ylim([0 Rmax]);
Communication Signal Simulation and Processing
Use the same stochastic Rician channel model as in the case of the PMCW to model the signal propagation from the JRC transmitter to the downlink user.
[yofdmc, pathGains] = commChannel(ofdmTx);
Add the receiver noise.
yofdmc = awgn(yofdmc, SNRu, "measured");
Compute the OFDM channel response to perform channel equalization.
channelInfo = info(commChannel);
pathFilters = channelInfo.ChannelFilterCoefficients;
toffset = channelInfo.ChannelFilterDelay;
Hofdm = ofdmChannelResponse(pathGains, pathFilters, Nsc, Ncp, ...
setdiff(1:Nsc, nullIdx), toffset);
Perform demodulation and equalization. The block diagram of the processing steps is shown below.
zeropadding = zeros(toffset, 1);
ofdmRx = [yofdmc(toffset+1:end,:); zeropadding];
qamRx = ofdmdemod(ofdmRx, Nsc, Ncp, Ncp/2, nullIdx);
qamEq = ofdmEqualize(qamRx, Hofdm(:), 'Algorithm', 'zf');
Demodulate the QAM symbols to obtain the received binary data.
dataRx = qamdemod(qamEq, K, 'OutputType', 'bit', 'UnitAveragePower', true);
refconst = qammod(0:K-1, K, 'UnitAveragePower', true);
constellationDiagram = comm.ConstellationDiagram('NumInputPorts', 1, ...
'ReferenceConstellation', refconst, 'ChannelNames', {'Received QAM Symbols'});
% Show QAM data symbols comprising the first 10 received OFDM symbols
Compute the bit error rate.
[numErr,ratio] = biterr(dataTx, dataRx)
This example simulates a JRC system using two different modulation schemes for generating a transmit waveform: PMCW and OFDM. The transmit waveforms are generated such that they can be used to sense
radar targets and to carry communication data. The example shows how to simulate transmission, propagation, and reception of these waveforms by the radar receiver and by the downlink user. For each
modulation scheme the example shows how to perform radar signal processing to obtain a range-Doppler map of the observed scenario. The example also shows how to process the received signal at the
downlink receiver to obtain the transmitted communication data.
1. Sturm, Christian, and Werner Wiesbeck. "Waveform design and signal processing aspects for fusion of wireless communications and radar sensing." Proceedings of the IEEE 99, no. 7 (2011):
2. de Oliveira, Lucas Giroto, Benjamin Nuss, Mohamad Basim Alabd, Axel Diewald, Mario Pauli, and Thomas Zwick. "Joint radar-communication systems: Modulation schemes and system design." IEEE
Transactions on Microwave Theory and Techniques 70, no. 3 (2021): 1521-1551.
Supporting Functions
function seq = helperMLS(p)
% Generate a pseudorandom sequence of length N=2^p-1, where p is an
% integer. The sequence is generated using shift registers. The feedback
% coefficients for the registers are obtained from the coefficients of an
% irreducible, primitive polynomial in GF(2p).
pol = gfprimdf(p, 2);
seq = zeros(2^p - 1, 1);
seq(1:p) = randi([0 1], p, 1);
for i = (p + 1):(2^p - 1)
seq(i) = mod(-pol(1:p)*seq(i-p : i-1), 2);
seq(seq == 0) = -1; | {"url":"https://au.mathworks.com/help/phased/ug/joint-radar-communication-using-pmcw-and-ofdm-waveforms.html","timestamp":"2024-11-02T15:44:34Z","content_type":"text/html","content_length":"106397","record_id":"<urn:uuid:ff573683-b989-4793-a9d6-44685f117edc>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00685.warc.gz"} |
Part II: PNGs and Moore
Last time I showed how we can render an automaton in your browser using existing tools.
This time we're going to roll a few of our own, so we can render fancier things. The SVG we generated last time was just too slow for many users and some folks complained that they couldn't see it at
all on an iPad, or that it crashed Firefox.
To rectify those concerns, we'll start off by writing a PNG generator!
... but I'll take a bit of a circuitous path to get there.
A couple of weeks back, Gabriel Gonzales posted about his foldl library. In that he used the following type to capture the essence of a left fold:
data Fold a b = forall x . Fold (x -> a -> x) x (x -> b)
I want to take a bit of a digression to note a few things about this type, and then show that it is just a presentation of something we already know pretty well in computer science!
Gabriel proceeded to supply an Applicative for his Fold type that looked something like:
instance Functor (Fold a) where
fmap f (Fold rar r rb) = Fold rar r (f.rb)
data Pair a b = Pair !a !b
instance Applicative (Fold a) where
pure b = Fold (\() _ -> ()) () (\() -> b)
{-# INLINABLE pure #-}
Fold sas s0 s2f <*> Fold rar r0 r2x = Fold
(\(Pair s r) a -> Pair (sas s a) (rar r a))
(Pair s0 r0)
(\(Pair s r) -> s2f s (r2x r))
{-# INLINABLE (<*>) #-}
But there is actually a fair bit more we can say about this type!
Being Applicative, we can lift numeric operations directly into it:
instance Num b => Num (Fold a b) where
(+) = liftA2 (+)
(-) = liftA2 (-)
(*) = liftA2 (*)
abs = fmap abs
signum = fmap signum
fromInteger = pure . fromInteger
instance Fractional b => Fractional (Fold a b) where
recip = fmap recip
(/) = liftA2 (/)
fromRational = pure . fromRational
But we can also note that it is contravariant in its first argument and covariant in its second, and therefore it must form a Profunctor.
instance Profunctor Fold where
dimap f g (Fold rar r0 rb) = Fold (\r -> rar r . f) r0 (g . rb)
All this does is let us tweak the inputs and/or outputs to our Fold.
But what perhaps isn't immediately obvious is that Fold a forms a Comonad!
instance Comonad (Fold a) where
extract (Fold _ r rb) = rb r
duplicate (Fold rar r0 rb) = Fold rar r0 $ \r -> Fold rar r rb
Notice that the duplicate :: Fold b a -> Fold b (Fold b a) sneaks in and generates a nested fold before the final tweak at the end that destroys our accumulator is applied! It works a bit like a last
second pardon from the governor, a stay of execution if you will.
(this is skippable)
It also forms a somewhat scarier sounding (strong) lax semimonoidal comonad, which just is to say that (<*>) is well behaved with regards to extract, so we can say:
instance ComonadApply (Fold a) where
(<@>) = (<*>)
This enables our Comonad to work with the codo sugar in Dominic Orchard's codo-notation package. I won't be doing that today, but you may want to download and modify one of the later examples to use
it, just to get a feel for it. It is pretty neat.
(this part of mostly skippable too)
I'll get back to the actual usecases for this Comonad shortly, but first I want to start with ways I could have come up with the definition.
It turns out there are a few comonads very closely related to Gabriel's left Fold!
If we take the definition of Gabriel's Fold and rip off the existential, we get:
data FoldX x a b = Fold (x -> a -> x) x (x -> b)
If we look through the menagerie supplied by the comonad-transformers package, we can pattern match on that with some effort and find:
FoldX x a b is isomorphic to both EnvT (x -> a -> x) (Store x) b and StoreT x (Env (x -> a -> x)) b. That it matches both of these types isn't surprising.
With Monad transformers, State, Reader and Writer all commute. In the space of Comonad transformers, Store, Env, and Traced all commute similarly.
Store is our old friend from the previous post, but Env and EnvT is something we haven't looked at before.
Env is also pretty much the easiest comonad to derive yourself.
Give it a shot!
{-# LANGUAGE DeriveFunctor, ScopedTypeVariables #-}
-- show
import Control.Comonad
-- /show
import Control.Exception
import Control.Monad
-- show
data Env e a = Env e a deriving (Eq,Ord,Show,Read,Functor)
instance Comonad (Env e) where
-- extract :: Env e a -> a
extract (Env e a) = error "unimplemented exercise"
-- duplicate :: Env e a -> Env e (Env e a)
duplicate (Env e a) = error "unimplemented exercise"
-- /show
main = do
test "extract" $ extract (Env 1 2) == 2
test "duplicate" $ duplicate (Env 1 2) == Env 1 (Env 1 2)
test :: String -> Bool -> IO ()
test s b = try (return $! b) >>= \ ec -> case ec of
Left (e :: SomeException) -> putStrLn $ s ++ " failed: " ++ show e
Right True -> putStrLn $ s ++ " is correct!"
Right False -> putStrLn $ s ++ " is not correct!"
When we bolt an extra bit of environment onto our Store from the first part, we get
data StoreAndEnv s e a = StoreAndEnv e (s -> a) s
If we fix e = (s -> b -> s), we get
data StoreAndStep s b a = StoreAndStep (s -> b -> s) (s -> a) s
then if we existentially tie off the s parameter to keep the end-user from fiddling with it we get back to
data Fold a b = forall s. Fold (s -> b -> s) (s -> a) s
which we could shuffle around into the right place.
You can view tying off s as taking a coend if you are so categorically inclined.
It was somewhat unsatisfying that we had to take a coend and make something existential in that type. Can we do without it?
It turns out we can, as noted by Elliott Hird, we just need to turn to another Comonad!
A Moore machine is one of the two classic ways to represent a deterministic finite automaton (DFA). The definition we'll use here is going to allow for deterministic infinite automata for free.
That sort of thing happens a lot in Haskell.
A Moore machine gives you a result associated with each state in the automaton rather than each edge. We'll make the Moore machine itself represent the state implicitly.
data Moore b a = Moore a (b -> Moore b a)
You can play around deriving its extract method below:
{-# LANGUAGE DeriveFunctor, ScopedTypeVariables #-}
-- show
import Control.Comonad
-- /show
import Control.Exception
import Control.Monad
-- show
data Moore b a = Moore a (b -> Moore b a) deriving Functor
instance Comonad (Moore b) where
-- extract :: Moore b a -> a
extract (Moore a as) = error "unimplemented exercise"
-- duplicate :: Moore b a -> Moore b (Moore b a)
duplicate w@(Moore _ as) = Moore w (duplicate <$> as)
-- extend :: (Moore b a -> c) -> Moore b a -> Moore b c
extend f w@(Moore _ as) = Moore (f w) (extend f <$> as)
-- /show
main = do
test "extract" $ 1 == extract (Moore 1 $ error "you don't need to look in the tail")
test :: String -> Bool -> IO ()
test s b = try (return $! b) >>= \ ec -> case ec of
Left (e :: SomeException) -> putStrLn $ s ++ " failed: " ++ show e
Right True -> putStrLn $ s ++ " is correct!"
Right False -> putStrLn $ s ++ " is not correct!"
If you have an eye for this sort of thing, you may have noted that Moore is a Cofree Comonad!
That is to say, Moore b a ~ Cofree ((->) b) a.
Moore machines are supplied in my machines package.
We can also derive an Applicative for Moore and all the machinery from the Fold package, plus our new toys above.
Here is where I'd love to be able to say that, reformulating things in this simpler way pays off and everything gets faster from using this encoding. Alas, that is not to be.
The Moore machine formulation is about 50% slower than the Fold representation in part due to the fact that it has hidden information about the environment for our machine from the optimizer. With
Fold, the explicit s can be manipulated by the inliner very easily.
Moreover applying an fmap is clearly done at the end, and so you pay no real cost for it until after the last iteration of the loop.
However, with the Moore representation, we pay for each fmap, because it winds up entangled in our core loop forever and we have to 'step over it' to get to the actual core of work we want to do. If
we apply the co-Yoneda lemma to our Moore machine, we get
data YonedaMoore a b = forall r. YonedaMoore (r -> a) (Moore a r)
Then you get rid of the overhead for each fmap, but we've brought back the existential and just made the optimizer's job harder.
What we do gain is flexibility in exchange for a bit of speed and no need for extensions.
A Moore machine can represent a mixture of strict and lazy left folds without extra boxes. The Fold type we started with can only represent one or the other easily, but otherwise must use a box
around the intermediate value type. The choice is made when you go to apply the Fold. Gabriel has chosen (rightly) to focus on strict left folds.
With Moore we can define the embedding to either be lazy
moorel :: (a -> b -> a) -> a -> Moore b a
moorel f = go where
go a = Moore a (go . f a)
or strict
moorel' :: (a -> b -> a) -> a -> Moore b a
moorel' f = go where
go !a = Moore a (go . f a)
and because we don't have an explicit 's' parameter, we don't have to put a Box around it if we want the lazy version.
Then the kinds of combinators supplied by Fold can be implemented as
total :: Num a => Moore a a
total = moorel' (+) 0
count :: Num a => Moore b a
count = moorel' (\a _ -> a + 1) 0
Fold and Moore are equivalent in expressive power, so another way to think about a Fold is as Cofree ((->) a) represented with an explicit seed in the style of Nu from my recursion-schemes package!
If we redefine our Moore machine using record syntax:
data Moore b a = Moore { this :: a, less :: b -> Moore b a }
then we can run one of our Moore machines by continually calling less with new inputs and then extracting the answer for its final result state.
more :: Foldable t => t b -> Moore b a -> a
more xs m = extract (F.foldl' less m xs)
Note that even though I'm using foldl' here, the thing that is being strictly updated is the Moore machine, not its member, which is only strict if you built the Moore machine using moorel' above.
more xs is now a Cokleisli arrow for our Comonad, just like rule 110 was for our Store Comonad in the last post.
We can construct a similar version of more for Fold using Gabriel's fold combinator.
more :: Foldable t => t b -> Fold b a -> a
more xs m = extract (fold m xs)
The Comonad for Fold a or Moore enables us to partially apply a Fold or Moore machine to some input and then resume it later.
If we extend (more xs) we get the ability to resume it with additional input, having partially driven our Fold!
If we turn to (=<=) from Control.Comonad.
(=<=) :: Comonad w => (w b -> c) -> (w a -> b) -> w a -> c
f =<= g = f . extend g
Then we can express the laws for more:
extract = more []
more as =<= more bs = more (as ++ bs)
So more provides us a monoid homomorphism between Cokleisli composition and concatenation.
Operationally, it sneaks in before you apply the last step to convert from your intermediate accumulator to the final result and lets you continue to do more work on the accumulator.
This strikes me as not intuitively obvious, because unless you look at it carefully, it isn't immediately obvious that you can resume something like a hash function because at the end, you usually
tweak the result before giving it to the user. Here because we have access to the internals of the Comonad, we can duplicate them into the result before closing it off.
This is where the explicit seed pays off, because that duplicate incurs no overhead during the actual traversal under Gabriel's representation.
This same existential construction works for foldMap- and foldr-based folds as well, though most of the "stream fusion" benefits require you to be able to stream and so foldMap-like structures,
sadly, get little benefit.
Let us consider a couple of CRC-like functions, to have something non-trivial to fold.
data Adler32 = Adler32 {-# UNPACK #-} !Word32 {-# UNPACK #-} !Word32
adler32 :: Moore Word8 Word32
adler32 = done <$> moorel' step (Adler32 1 0) where
step (Adler32 s1 s2) x = Adler32 s1' s2' where
s1' = mod (s1 + fromIntegral x) 65521
s2' = mod (s1' + s2) 65521
done (Adler32 s1 s2) = unsafeShiftL s2 16 + s1
In Adler32, the final step of hashing destroys the separation of information between s1 and s2, but we can sneak in with the comonad before we destroy it and resume!
Similarly, but less catastrophically, in CRC32 the final step is to complement the input.
crc32 :: Moore Word8 Word32
crc32 = complement <$> moorel' step 0xffffffff where
step r b = unsafeShiftR r 8 `xor` (crcs Unboxed.! fromIntegral (xor r (fromIntegral b) .&. 0xff))
crcs :: Unboxed.Vector Word32
crcs = Unboxed.generate 256 (go.go.go.go.go.go.go.go.fromIntegral) where
go c = unsafeShiftR c 1 `xor` if c .&. 1 /= 0 then 0xedb88320 else 0
We can describe similar Moore machines (or Folds) for common hashing functions, and then we don't need to make up separate functions for initializing the state, feeding them some incremental
additional data and finally cleaning up when we're done.
The Moore machine provides you with all of that, and the entire API necessary to interact with them comes down to feeding it more, extending after doing so to accept more input!
This strikes me as an incredibly clean implementation pattern for HMACs such as MD5 and SHA in Haskell. You don't need to name 3 separate pieces.
You just name the HMAC itself as the Moore machine that produces it. Then you can feed it more data, extending it as needed until you finally go to look at the last result.
So let's put our code where our mouth is and show that we can use this to do some software engineering by writing some code to produce a PNG image from scratch in Haskell.
A bit over a year ago, Keegan McAllister wrote a nice post on how to generate a minimal uncompressed PNG using python. We'll copy his development here, except we'll switch out to the nicer
table-based crc32 above.
As he noted, you need to implement two hash functions to actually get through writing an uncompressed PNG. Hrmm. We appear to have those.
We'll use Data.Binary to write out the results, mostly because PNG is an annoyingly introspective format, so we'll have to talk about the lengths of fragments we're generating as we go.
We can write the ability to put a PNG 'chunk' out, which consists of a 4 byte header followed by some data, but which first encodes the length of just the data, then emits the header, then the data,
and finally closes off the chunk with the CRC32 of both.
Let's generalize more to work over any Fold (in the lens sense this time!) that yields the input type.
moreOf :: Getting (Endo (Endo (Moore b a))) s b -> s -> Moore b a -> a
moreOf l xs m = extract (foldlOf' l less m xs)
That somewhat baroque seeming type can be read as a more liberal version of:
moreOf :: Fold s b -> s -> Moore b a -> a
that just happens to get better inference due to the lack of rank-2 types.
Now we can use it directly on the lazy bytestring fragments we get along the way
putChunk :: Lazy.ByteString -> Put -> Put
putChunk h (runPut -> b) = do
putWord32be $ fromIntegral (Lazy.length b)
putLazyByteString h
putLazyByteString b
putWord32be $ moreOf bytes h =<= moreOf bytes b $ crc32
To write out a PNG file, we need to be able to emit the IHDR chunk, 1 or more IDAT chunks of zlib compressed data, and an IEND chunk.
We can break up our zlib data into uncompressed blocks. However, zlib only allows uncompressed runs of 64k at a time, so we let's define the encoding for a nested uncompressed deflate block.
deflated :: Bool -> Lazy.ByteString -> Put
deflated final b | l <- fromIntegral (Lazy.length b) = do
putWord8 $ if final then 1 else 0
putWord16le l -- yep, now it's little endian!
putWord16le (complement l)
putLazyByteString b
Then we just rip our input up into 64k blocks, embed each of those blocks in one enormous IDAT block, then finally seal everything up with the Adler32 checksum that we so helpfully supplied as an
example above!
zlibbed :: Lazy.ByteString -> Put
zlibbed bs = do
putWord8 0x78
putWord8 0x01
go bs
putWord32be $ moreOf bytes bs adler32
go (Lazy.splitAt 0xffff -> (xs, ys)) | done <- Lazy.null ys = do
deflated done xs
M.unless done (go ys)
Now we can write out a PNG header, loop through the data, state that we're not applying any transformation for each row:
png :: Int -> [Int -> (Word8, Word8, Word8)] -> Lazy.ByteString
png w fs = runPut $ do
putLazyByteString "\x89PNG\r\n\x1a\n"
putChunk "IHDR" $ do
putWord32be $ fromIntegral w
putWord32be $ fromIntegral (List.length fs)
putWord8 8 -- 8 bit color depth
putWord8 2 -- RGB
putWord8 0
putWord8 0
putWord8 0
putChunk "IDAT" $ zlibbed (runPut rows)
putChunk "IEND" $ return ()
rows = forM_ fs $ \f -> do
putWord8 0
forM_ [0..w-1] (put . f)
Here I've chosen to tell the PNG the width, but leave height implicit in the length of the list of functions from horizontal position to pixel color. I may revisit that later, but it was the fastest
thing I could think of to write.
This lets png nicely fit into the recursion pattern from the previous post.
But we've written a lot of code, so it'd be nice to check that we generated a valid PNG.
-- show
{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE ViewPatterns #-}
{-# LANGUAGE TemplateHaskell #-}
{-# LANGUAGE QuasiQuotes #-}
{-# LANGUAGE TypeFamilies #-}
{-# OPTIONS_GHC -Wall #-}
import Control.Applicative
import Control.Comonad
import Control.Lens
import qualified Control.Monad as M
import Data.Bits
import Data.Binary
import Data.Binary.Put
import qualified Data.ByteString.Lazy as Lazy
import Data.ByteString.Lens
import Data.Monoid
import qualified Data.Vector.Unboxed as Unboxed
import Data.Foldable as F
import Data.List as List
import Yesod
-- * Moore machines
data Moore b a = Moore { this :: a, less :: b -> Moore b a }
instance Num a => Num (Moore b a) where
(+) = liftA2 (+)
(-) = liftA2 (-)
(*) = liftA2 (*)
abs = fmap abs
signum = fmap signum
fromInteger = pure . fromInteger
instance Fractional a => Fractional (Moore b a) where
recip = fmap recip
(/) = liftA2 (/)
fromRational = pure . fromRational
instance Functor (Moore b) where
fmap f = go where go (Moore a k) = Moore (f a) (go . k)
instance Comonad (Moore b) where
extract = this
duplicate w@(Moore _ as) = Moore w (duplicate . as)
instance ComonadApply (Moore b) where
(<@>) = (<*>)
instance Applicative (Moore b) where
pure a = as where as = Moore a (const as)
Moore f fs <*> Moore a as = Moore (f a) $ \b -> fs b <*> as b
instance Profunctor Moore where
dimap f g (Moore a as) = Moore (g a) (dimap f g . as . f)
moorel :: (a -> b -> a) -> a -> Moore b a
moorel f = go where go a = Moore a (go . f a)
moorel' :: (a -> b -> a) -> a -> Moore b a
moorel' f = go where go !a = Moore a (go . f a)
moreOf :: Getting (Endo (Endo (Moore b a))) s b -> s -> Moore b a -> a
moreOf l xs m = extract (foldlOf' l less m xs)
-- * Adler 32
data Adler32 = Adler32 {-# UNPACK #-} !Word32 {-# UNPACK #-} !Word32
adler32 :: Moore Word8 Word32
adler32 = done <$> moorel' step (Adler32 1 0) where
step (Adler32 s1 s2) x = Adler32 s1' s2' where
s1' = mod (s1 + fromIntegral x) 65521
s2' = mod (s1' + s2) 65521
done (Adler32 s1 s2) = unsafeShiftL s2 16 + s1
-- * CRC32
crc32 :: Moore Word8 Word32
crc32 = complement <$> moorel' step 0xffffffff where
step r b = unsafeShiftR r 8 `xor` crcs Unboxed.! fromIntegral (xor r (fromIntegral b) .&. 0xff)
crcs :: Unboxed.Vector Word32
crcs = Unboxed.generate 256 (go.go.go.go.go.go.go.go.fromIntegral) where
go c = unsafeShiftR c 1 `xor` if c .&. 1 /= 0 then 0xedb88320 else 0
-- * PNG
putChunk :: Lazy.ByteString -> Put -> Put
putChunk h (runPut -> b) = do
putWord32be $ fromIntegral (Lazy.length b)
putLazyByteString h
putLazyByteString b
putWord32be $ moreOf bytes h =<= moreOf bytes b $ crc32
deflated :: Bool -> Lazy.ByteString -> Put
deflated final b | l <- fromIntegral (Lazy.length b) = do
putWord8 $ if final then 1 else 0
putWord16le l -- yep, now it's little endian!
putWord16le (complement l)
putLazyByteString b
zlibbed :: Lazy.ByteString -> Put
zlibbed bs = do
putWord8 0x78
putWord8 0x01
go bs
putWord32be $ moreOf bytes bs adler32
go (Lazy.splitAt 0xffff -> (xs, ys)) | done <- Lazy.null ys = do
deflated done xs
M.unless done (go ys)
png :: Int -> [Int -> (Word8, Word8, Word8)] -> Lazy.ByteString
png w fs = runPut $ do
putLazyByteString "\x89PNG\r\n\x1a\n"
putChunk "IHDR" $ do
putWord32be $ fromIntegral w
putWord32be $ fromIntegral (List.length fs)
putWord8 8 -- 8 bit color depth
putWord8 2 -- RGB
putWord8 0
putWord8 0
putWord8 0
putChunk "IDAT" $ zlibbed (runPut rows)
putChunk "IEND" $ return ()
rows = forM_ fs $ \f -> do
putWord8 0
forM_ [0..w-1] (put . f)
-- * Yesod
data App = App
instance Yesod App
mkYesod "App" [parseRoutes| / ImageR GET |]
main :: IO ()
main = warpEnv App
-- /show
-- show
getImageR :: MonadHandler m => m TypedContent
getImageR = sendResponse $ toTypedContent (typePng, toContent img) where
img = png 500 $ take 300 $ pixel <$> [0..]
pixel y x = (fromIntegral x,fromInteger y,0)
-- /show
That image matches up byte for byte with the output of Keegan's sample, so we seem to have an end-to-end test that works.
A lot of this code is redundant, however.
For instance all of the Moore code could be taken from the machines package, which provides Data.Machine.Moore along with all of these instances! Then with a bit of tightening of exposition and
removing unnecessary detours we could generate the whole thing in a lot less code.
Of course, this is supposed to be a series about cellular automata. So let's draw one.
1.) I'll be switching to a 4 line minimalist version of Gabriel's foldl library, rather than using the Moore representation, since we don't need any of the instances. I've renamed his Fold to L here
to avoid conflicts with the Lens library.
2.) We don't need to use the Comonad for the fold type we spent all that time above building up. Here we're working with lazy bytestrings, so let's just append them in the one case we need!
2.) I'll also be using the Context comonad from the lens package rather than continuing to roll our own Store. That'll be useful next time when I want to abuse the separate indices.
3.) I've tweaked the memoization rule to use
`haskell loop f = iterate (tab . extend f) . tab `
instead of
`haskell loop f = iterate (extend f . tab) `
to get slightly better memoization. I also switched to representable-tries, because it'll make it easier to switch to new topologies later.
4.) Finally, to reduce the footprint of the PNGs we generate we'll let the existing zlib bindings for Haskell do the compression rather than manually deflate. This reduces the footprint of the
generated images a great deal.
Click Run!
{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE ExistentialQuantification #-}
{-# LANGUAGE Rank2Types #-}
{-# LANGUAGE TemplateHaskell #-}
{-# LANGUAGE QuasiQuotes #-}
{-# LANGUAGE TypeFamilies #-}
import Control.Applicative
import Control.Comonad
import Codec.Compression.Zlib
import Control.Lens.Internal.Context
import Control.Lens as L
import Data.Bits
import Data.Bits.Lens as L
import Data.Monoid
import Data.Binary
import Data.Binary.Put
import qualified Data.ByteString.Lazy as Lazy
import qualified Data.Vector.Unboxed as Unboxed
import Data.Foldable as F
import Data.MemoTrie
import Yesod
rule :: Num s => Word8 -> Context s s Bool -> Bool
rule w (Context f s) = testBit w $ 0 L.& partsOf L.bits .~ [f (s+1), f s, f (s-1)]
loop :: HasTrie s => (Context s s a -> a) -> Context s s a -> [Context s s a]
loop f = iterate (tab . extend f) . tab where
tab (Context k s) = Context (memo k) s
data L b a = forall x. L (x -> b -> x) x (x -> a)
more :: Lazy.ByteString -> L Word8 a -> a
more bs (L xbx x xa) = xa (Lazy.foldl' xbx x bs)
crc32 :: L Word8 Word32
crc32 = L step 0xffffffff complement where
step r b = unsafeShiftR r 8 `xor` crcs Unboxed.! fromIntegral (xor r (fromIntegral b) .&. 0xff)
crcs :: Unboxed.Vector Word32
crcs = Unboxed.generate 256 (go.go.go.go.go.go.go.go.fromIntegral) where
go c = unsafeShiftR c 1 `xor` if c .&. 1 /= 0 then 0xedb88320 else 0
putChunk :: Lazy.ByteString -> Lazy.ByteString -> Put
putChunk h b = do
putWord32be $ fromIntegral (Lazy.length b)
putLazyByteString h
putLazyByteString b
putWord32be $ more (h <> b) crc32
png :: Int -> Int -> [Int -> (Word8, Word8, Word8)] -> Lazy.ByteString
png w h fs = runPut $ do
putLazyByteString "\x89PNG\r\n\x1a\n"
putChunk "IHDR" $ runPut $ do
putWord32be (fromIntegral w)
putWord32be (fromIntegral h)
putWord8 8 -- 8 bit color depth
putWord8 2 -- RGB
putWord8 0
putWord8 0
putWord8 0
putChunk "IDAT" $
compressWith defaultCompressParams { compressLevel = bestSpeed } $
runPut $ forM_ (take h fs) $ \f -> do
putWord8 0
forM_ [0..w-1] (put . f)
putChunk "IEND" mempty
data App = App
instance Yesod App
mkYesod "App" [parseRoutes| / ImageR GET |]
main = warpEnv App
-- /show
-- show
getImageR :: MonadHandler m => m TypedContent
getImageR = sendResponse $ toTypedContent (typePng, toContent img) where
img = png 150 150 $ draw <$> loop (rule 110) (Context (==149) 149)
draw (Context p _) x = if p x then (0,0,0) else (255,255,255)
-- /show
That weighs in somewhere around 75 lines, and includes our compressed PNG generator, all the logic for running Wolfram's 2-color rules as before, and our embedded Yesod server. You can feel free to
tweak the output above.
In the real world you'd probably just use JuicyPixels.
Now that we're not shackled by the SVG rendering speed we can generalize this to other topologies and maybe try to improve on our other bottlenecks in future updates.
September 1, 2013 | {"url":"https://www.schoolofhaskell.com/user/school/to-infinity-and-beyond/pick-of-the-week/part-2","timestamp":"2024-11-02T23:38:47Z","content_type":"text/html","content_length":"56146","record_id":"<urn:uuid:25b087a0-24b0-4520-8a55-8c9d8de3cb06>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00743.warc.gz"} |
Four Operations Intermediate Vocabulary
The Four Operations Intermediate Vocabulary resource from Cazoom Maths is a comprehensive tool designed to enhance students' understanding of key mathematical operations. Aimed at school years 7 to
9, this resource facilitates a deeper grasp of maths vocabulary associated with addition, subtraction, multiplication, and division. Offering a free PDF printable version, it serves as an essential
aid for both teachers and parents in reinforcing intermediate maths concepts.
What Is Four Operations Intermediate Vocabulary?
Four Operations Intermediate Vocabulary is a structured educational tool that outlines essential terms and definitions related to the four basic maths operations. It simplifies complex concepts,
making them accessible and understandable for students. This printable PDF resource includes an illustrative image, and a succinct description, and identifies the relevant school year groups,
ensuring its effective use in educational settings.
Why Is the Topic of Four Operations Important?
Mastery of the four operations is fundamental to maths. It underpins more complex mathematical reasoning and problem-solving. A solid understanding of these operations and their associated vocabulary
prepares students for advanced maths topics and real-life applications, enhancing their numerical literacy and confidence in maths.
Why Is This Teaching Resource Helpful for Learning?
The Four Operations Intermediate Vocabulary resource supports learning by breaking down complex terms into digestible information. It aids memory retention and understanding of key maths concepts,
encouraging a more engaging and effective learning experience. This tool is particularly useful for visual learners and those who benefit from a clear, concise presentation of information. It acts as
a valuable reference for students, enriching their maths vocabulary and comprehension.
In conclusion, incorporating the Four Operations Intermediate Vocabulary into educational practices can significantly enhance students' mathematical understanding and confidence, making it a vital
component of any educational toolkit.
Also, have a look at our wide range of worksheets that are specifically curated to help your students practice their skills in the four operations vocabulary. These teaching resources and worksheets
are in PDF format and can be downloaded easily. | {"url":"https://www.cazoommaths.com/teaching-resource/four-operations-intermediate-vocabulary/","timestamp":"2024-11-10T05:49:07Z","content_type":"text/html","content_length":"440636","record_id":"<urn:uuid:bc1bce43-3593-4617-b167-ad5f08b6005c>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00649.warc.gz"} |
Chi-Square Test of Independence in R
Chi-Square Test of Independence in R
The chi-square test of independence is used to analyze the frequency table (i.e. contengency table) formed by two categorical variables. The chi-square test evaluates whether there is a significant
association between the categories of the two variables. This article describes the basics of chi-square test and provides practical examples using R software.
Data format: Contingency tables
We’ll use housetasks data sets from STHDA: http://www.sthda.com/sthda/RDoc/data/housetasks.txt.
# Import the data
file_path <- "http://www.sthda.com/sthda/RDoc/data/housetasks.txt"
housetasks <- read.delim(file_path, row.names = 1)
# head(housetasks)
An image of the data is displayed below:
The data is a contingency table containing 13 housetasks and their distribution in the couple:
• rows are the different tasks
• values are the frequencies of the tasks done :
• by the wife only
• alternatively
• by the husband only
• or jointly
Graphical display of contengency tables
Contingency table can be visualized using the function balloonplot() [in gplots package]. This function draws a graphical matrix where each cell contains a dot whose size reflects the relative
magnitude of the corresponding component.
To execute the R code below, you should install the package gplots: install.packages(“gplots”).
# 1. convert the data as a table
dt <- as.table(as.matrix(housetasks))
# 2. Graph
balloonplot(t(dt), main ="housetasks", xlab ="", ylab="",
label = FALSE, show.margins = FALSE)
Note that, row and column sums are printed by default in the bottom and right margins, respectively. These values can be hidden using the argument show.margins = FALSE.
It’s also possible to visualize a contingency table as a mosaic plot. This is done using the function mosaicplot() from the built-in R package garphics:
mosaicplot(dt, shade = TRUE, las=2,
main = "housetasks")
• The argument shade is used to color the graph
• The argument las = 2 produces vertical labels
Note that the surface of an element of the mosaic reflects the relative magnitude of its value.
• Blue color indicates that the observed value is higher than the expected value if the data were random
• Red color specifies that the observed value is lower than the expected value if the data were random
From this mosaic plot, it can be seen that the housetasks Laundry, Main_meal, Dinner and breakfeast (blue color) are mainly done by the wife in our example.
There is another package named vcd, which can be used to make a mosaic plot (function mosaic()) or an association plot (function assoc()).
# install.packages("vcd")
# plot just a subset of the table
assoc(head(dt, 5), shade = TRUE, las=3)
Chi-square test basics
Chi-square test examines whether rows and columns of a contingency table are statistically significantly associated.
• Null hypothesis (H0): the row and the column variables of the contingency table are independent.
• Alternative hypothesis (H1): row and column variables are dependent
For each cell of the table, we have to calculate the expected value under null hypothesis.
For a given cell, the expected value is calculated as follow:
\[ e = \frac{row.sum * col.sum}{grand.total} \]
The Chi-square statistic is calculated as follow:
\[ \chi^2 = \sum{\frac{(o - e)^2}{e}} \]
• o is the observed value
• e is the expected value
This calculated Chi-square statistic is compared to the critical value (obtained from statistical tables) with \(df = (r - 1)(c - 1)\) degrees of freedom and p = 0.05.
• r is the number of rows in the contingency table
• c is the number of column in the contingency table
If the calculated Chi-square statistic is greater than the critical value, then we must conclude that the row and the column variables are not independent of each other. This implies that they are
significantly associated.
Note that, Chi-square test should only be applied when the expected frequency of any cell is at least 5.
Compute chi-square test in R
Chi-square statistic can be easily computed using the function chisq.test() as follow:
chisq <- chisq.test(housetasks)
Pearson's Chi-squared test
data: housetasks
X-squared = 1944.5, df = 36, p-value < 2.2e-16
In our example, the row and the column variables are statistically significantly associated (p-value = 0).
The observed and the expected counts can be extracted from the result of the test as follow:
# Observed counts
Wife Alternating Husband Jointly
Laundry 156 14 2 4
Main_meal 124 20 5 4
Dinner 77 11 7 13
Breakfeast 82 36 15 7
Tidying 53 11 1 57
Dishes 32 24 4 53
Shopping 33 23 9 55
Official 12 46 23 15
Driving 10 51 75 3
Finances 13 13 21 66
Insurance 8 1 53 77
Repairs 0 3 160 2
Holidays 0 1 6 153
# Expected counts
Wife Alternating Husband Jointly
Laundry 60.55 25.63 38.45 51.37
Main_meal 52.64 22.28 33.42 44.65
Dinner 37.16 15.73 23.59 31.52
Breakfeast 48.17 20.39 30.58 40.86
Tidying 41.97 17.77 26.65 35.61
Dishes 38.88 16.46 24.69 32.98
Shopping 41.28 17.48 26.22 35.02
Official 33.03 13.98 20.97 28.02
Driving 47.82 20.24 30.37 40.57
Finances 38.88 16.46 24.69 32.98
Insurance 47.82 20.24 30.37 40.57
Repairs 56.77 24.03 36.05 48.16
Holidays 55.05 23.30 34.95 46.70
Nature of the dependence between the row and the column variables
As mentioned above the total Chi-square statistic is 1944.456196.
If you want to know the most contributing cells to the total Chi-square score, you just have to calculate the Chi-square statistic for each cell:
\[ r = \frac{o - e}{\sqrt{e}} \]
The above formula returns the so-called Pearson residuals (r) for each cell (or standardized residuals)
Cells with the highest absolute standardized residuals contribute the most to the total Chi-square score.
Pearson residuals can be easily extracted from the output of the function chisq.test():
round(chisq$residuals, 3)
Wife Alternating Husband Jointly
Laundry 12.266 -2.298 -5.878 -6.609
Main_meal 9.836 -0.484 -4.917 -6.084
Dinner 6.537 -1.192 -3.416 -3.299
Breakfeast 4.875 3.457 -2.818 -5.297
Tidying 1.702 -1.606 -4.969 3.585
Dishes -1.103 1.859 -4.163 3.486
Shopping -1.289 1.321 -3.362 3.376
Official -3.659 8.563 0.443 -2.459
Driving -5.469 6.836 8.100 -5.898
Finances -4.150 -0.852 -0.742 5.750
Insurance -5.758 -4.277 4.107 5.720
Repairs -7.534 -4.290 20.646 -6.651
Holidays -7.419 -4.620 -4.897 15.556
Let’s visualize Pearson residuals using the package corrplot:
corrplot(chisq$residuals, is.cor = FALSE)
For a given cell, the size of the circle is proportional to the amount of the cell contribution.
The sign of the standardized residuals is also very important to interpret the association between rows and columns as explained in the block below.
1. Positive residuals are in blue. Positive values in cells specify an attraction (positive association) between the corresponding row and column variables.
• In the image above, it’s evident that there are an association between the column Wife and the rows Laundry, Main_meal.
• There is a strong positive association between the column Husband and the row Repair
2. Negative residuals are in red. This implies a repulsion (negative association) between the corresponding row and column variables. For example the column Wife are negatively associated (~ “not
associated”) with the row Repairs. There is a repulsion between the column Husband and, the rows Laundry and Main_meal
The contribution (in %) of a given cell to the total Chi-square score is calculated as follow:
\[ contrib = \frac{r^2}{\chi^2} \]
• r is the residual of the cell
# Contibution in percentage (%)
contrib <- 100*chisq$residuals^2/chisq$statistic
round(contrib, 3)
Wife Alternating Husband Jointly
Laundry 7.738 0.272 1.777 2.246
Main_meal 4.976 0.012 1.243 1.903
Dinner 2.197 0.073 0.600 0.560
Breakfeast 1.222 0.615 0.408 1.443
Tidying 0.149 0.133 1.270 0.661
Dishes 0.063 0.178 0.891 0.625
Shopping 0.085 0.090 0.581 0.586
Official 0.688 3.771 0.010 0.311
Driving 1.538 2.403 3.374 1.789
Finances 0.886 0.037 0.028 1.700
Insurance 1.705 0.941 0.868 1.683
Repairs 2.919 0.947 21.921 2.275
Holidays 2.831 1.098 1.233 12.445
# Visualize the contribution
corrplot(contrib, is.cor = FALSE)
The relative contribution of each cell to the total Chi-square score give some indication of the nature of the dependency between rows and columns of the contingency table.
It can be seen that:
1. The column “Wife” is strongly associated with Laundry, Main_meal, Dinner
2. The column “Husband” is strongly associated with the row Repairs
3. The column jointly is frequently associated with the row Holidays
From the image above, it can be seen that the most contributing cells to the Chi-square are Wife/Laundry (7.74%), Wife/Main_meal (4.98%), Husband/Repairs (21.9%), Jointly/Holidays (12.44%).
These cells contribute about 47.06% to the total Chi-square score and thus account for most of the difference between expected and observed values.
This confirms the earlier visual interpretation of the data. As stated earlier, visual interpretation may be complex when the contingency table is very large. In this case, the contribution of one
cell to the total Chi-square score becomes a useful way of establishing the nature of dependency.
Access to the values returned by chisq.test() function
The result of chisq.test() function is a list containing the following components:
• statistic: the value the chi-squared test statistic.
• parameter: the degrees of freedom
• p.value: the p-value of the test
• observed: the observed count
• expected: the expected count
The format of the R code to use for getting these values is as follow:
# printing the p-value
# printing the mean
This analysis has been performed using R software (ver. 3.2.4).
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Learning Analytics Methods and Tutorials
Learning Analytics Methods and Tutorials: A Practical Guide Using R
Learning Analytics Methods and Tutorials: In today’s data-driven world, educational institutions and learning environments are increasingly leveraging analytics to improve student outcomes, optimize
teaching strategies, and make informed decisions. Learning analytics is the application of data analysis and data-driven approaches to education, where insights are extracted from student data to
enhance learning experiences and outcomes. If you’re looking to dive into learning analytics, R, a powerful programming language for statistical computing and data analysis, is an ideal tool for the
job. This article will introduce some fundamental learning analytics methods and offer a practical guide using R.
What is Learning Analytics?
Learning analytics refers to the measurement, collection, analysis, and reporting of data about learners and their contexts, for the purpose of understanding and optimizing learning and the
environments in which it occurs. It involves the analysis of various types of student data, including academic performance, learning behaviors, and even social interactions within online platforms.
The primary goals of learning analytics are:
• To understand student learning processes.
• To identify struggling students and offer timely interventions.
• To personalize learning experiences.
• To enhance educational design and teaching methods.
Why Use R for Learning Analytics?
R is an open-source language widely used for statistical analysis and visualization. It’s particularly popular in educational research and learning analytics because of its flexibility, extensive
library support, and ability to handle large datasets. Using R, educators and data analysts can build customized analytics pipelines, perform detailed statistical tests, and generate insightful
visualizations to better understand learning behaviors and trends.
Some advantages of using R for learning analytics include:
• Data wrangling and manipulation: R excels at cleaning and transforming data.
• Statistical analysis: R offers a wide range of statistical techniques, from basic descriptive statistics to advanced machine learning methods.
• Visualization: R’s packages like ggplot2 make it easy to create compelling and informative visualizations of data.
• Extensibility: R has numerous packages specifically designed for educational data analysis, such as eddata or LAK2011.
Now, let’s walk through some common learning analytics methods and explore how you can apply them using R.
Learning Analytics Methods and Tutorials A Practical Guide Using R
1. Descriptive Analytics
The first step in learning analytics is often descriptive analysis, where we summarize and describe the data to understand general trends and patterns. For example, we might want to know the average
grades of students, attendance rates, or the distribution of time spent on assignments.
Practical Example in R:
# Load required packages
# Simulate a dataset
student_data <- tibble(
student_id = 1:100,
grades = runif(100, min = 50, max = 100),
attendance_rate = runif(100, min = 0.5, max = 1)
# Summary statistics
# Visualize grades distribution
ggplot(student_data, aes(x = grades)) +
geom_histogram(binwidth = 5, fill = "blue", color = "white") +
theme_minimal() +
labs(title = "Distribution of Student Grades", x = "Grades", y = "Count")
This script generates a dataset and provides a summary of the grades and attendance rates. Additionally, it creates a histogram to visually represent the grade distribution.
2. Predictive Analytics
Predictive analytics uses statistical models and machine learning techniques to predict future outcomes. For instance, you may want to predict whether a student is likely to fail a course based on
their previous performance and engagement in class.
Practical Example in R:
# Load the caret package for machine learning
# Create a binary outcome: 1 = pass, 0 = fail
student_data$pass <- ifelse(student_data$grades >= 60, 1, 0)
# Split the data into training and testing sets
trainIndex <- createDataPartition(student_data$pass, p = 0.7, list = FALSE)
train_data <- student_data[trainIndex,]
test_data <- student_data[-trainIndex,]
# Train a logistic regression model
model <- glm(pass ~ grades + attendance_rate, data = train_data, family = binomial)
# Make predictions on the test set
predictions <- predict(model, newdata = test_data, type = "response")
test_data$predicted <- ifelse(predictions > 0.5, 1, 0)
# Evaluate model accuracy
confusionMatrix(as.factor(test_data$predicted), as.factor(test_data$pass))
This example demonstrates how to use logistic regression in R to predict whether a student will pass or fail a course based on their grades and attendance rates. The caret package is used for
splitting the data into training and testing sets and evaluating the model.
3. Social Network Analysis (SNA)
In collaborative learning environments, social interactions can play a significant role in student success. Social Network Analysis (SNA) allows us to analyze relationships and interactions among
students, such as discussion forum participation or group projects.
Practical Example in R:
# Load igraph package for network analysis
# Create a simple social network (student interactions)
edges <- data.frame(from = c(1, 2, 3, 4, 5, 6),
to = c(2, 3, 4, 1, 6, 5))
# Create a graph object
g <- graph_from_data_frame(edges, directed = FALSE)
# Plot the network
plot(g, vertex.size = 30, vertex.label.cex = 1.2,
vertex.color = "lightblue", edge.color = "gray")
In this example, we create a social network graph representing student interactions. This is a basic use case of SNA, which can be extended to analyze larger and more complex networks of student
4. Text Mining and Sentiment Analysis
With the rise of online learning platforms and digital assessments, textual data has become a valuable resource for learning analytics. Text mining and sentiment analysis can help us understand the
tone of student feedback, identify common topics in discussion forums, and even detect potential areas of improvement in the curriculum.
Practical Example in R:
# Load text mining and sentiment analysis libraries
# Sample student feedback
feedback <- c("The course was great!", "I found the assignments too difficult.", "Loved the teacher's approach.")
# Create a corpus for text mining
corpus <- Corpus(VectorSource(feedback))
corpus <- tm_map(corpus, content_transformer(tolower))
corpus <- tm_map(corpus, removePunctuation)
# Perform sentiment analysis
sentiments <- sentiment(feedback)
This code performs basic sentiment analysis on student feedback, giving us insight into how students feel about different aspects of a course. Such analysis can provide valuable qualitative data
alongside more traditional numerical measures.
Learning analytics has the potential to revolutionize education by providing data-driven insights that inform teaching practices and improve student outcomes. With tools like R, educational data
analysts can explore a wide variety of methods, from simple descriptive statistics to advanced predictive modeling and network analysis. As you dive into learning analytics, consider starting with
the basic methods described above and gradually expanding your toolkit to include more sophisticated approaches.
By mastering learning analytics with R, educators and researchers can unlock new ways to personalize learning, increase student engagement, and ultimately foster better educational experiences.
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High-resolution analysis of the physicochemical characteristics of sandstone media at the lithofacies scale
Articles | Volume 11, issue 4
© Author(s) 2020. This work is distributed under the Creative Commons Attribution 4.0 License.
High-resolution analysis of the physicochemical characteristics of sandstone media at the lithofacies scale
The prediction of physicochemical rock properties in subsurface models regularly suffers from uncertainty observed at the submeter scale. Although at this scale – which is commonly termed the
lithofacies scale – the physicochemical variability plays a critical role for various types of subsurface utilization, its dependence on syndepositional and postdepositional processes is still
subject to investigation.
The impact of syndepositional and postdepositional geological processes, including depositional dynamics, diagenetic compaction and chemical mass transfer, onto the spatial distribution of
physicochemical properties in siliciclastic media at the lithofacies scale is investigated in this study. We propose a new workflow using two cubic rock samples where eight representative
geochemical, thermophysical, elastic and hydraulic properties are measured on the cubes' faces and on samples taken from the inside. The scalar fields of the properties are then constructed by means
of spatial interpolation. The rock cubes represent the structurally most homogeneous and most heterogeneous lithofacies types observed in a Permian lacustrine-deltaic formation that deposited in an
intermontane basin. The spatiotemporal controlling factors are identified by exploratory data analysis and geostatistical modeling in combination with thin section and environmental scanning electron
microscopy analyses.
Sedimentary structures are well preserved in the spatial patterns of the negatively correlated permeability and mass fraction of Fe[2]O[3]. The Fe-rich mud fraction, which builds large amounts of the
intergranular rock matrix and of the pseudomatrix, has a degrading effect on the hydraulic properties. This relationship is underlined by a zonal anisotropy that is connected to the observed
stratification. Feldspar alteration produced secondary pore space that is filled with authigenic products, including illite, kaolinite and opaque phases. The local enrichment of clay minerals implies
a nonpervasive alteration process that is expressed by the network-like spatial patterns of the positively correlated mass fractions of Al[2]O[3] and K[2]O. Those patterns are spatially decoupled
from primary sedimentary structures. The elastic properties, namely P-wave and S-wave velocity, indicate a weak anisotropy that is not strictly perpendicularly oriented to the sedimentary structures.
The multifarious patterns observed in this study emphasize the importance of high-resolution sampling in order to properly model the variability present in a lithofacies-scale system. Following this,
the physicochemical variability observed at the lithofacies scale might nearly cover the global variability in a formation. Hence, if the local variability is not considered in full-field projects –
where the sampling density is usually low – statistical correlations and, thus, conclusions about causal relationships among physicochemical properties might be feigned inadvertently.
Received: 04 Feb 2020 – Discussion started: 30 Mar 2020 – Revised: 04 Jun 2020 – Accepted: 06 Jul 2020 – Published: 13 Aug 2020
The utilization of the subsurface in disciplines such as petroleum reservoir engineering, geothermal heat extraction, mining, carbon capture and storage or nuclear waste disposal requires highly
accurate spatial predictions of relevant physical or geochemical properties in order to assess the economic feasibility of a target region (Landa and Strebelle, 2002; Heap et al., 2017; Kushnir
et al., 2018; Rodrigo-Ilarri et al., 2017). Although most of these types of utilizations take place at full-field scales, geological variability present at the submeter scale may play an important
role during the development process. The scale we are speaking of is commonly termed the lithofacies scale (Miall, 1985). Geological heterogeneities at the lithofacies scale might constitute
undesirable features in the subsurface such as flow barriers in reservoirs (Landa and Strebelle, 2002; Ringrose et al., 1993; Medici et al., 2016, 2019), pathways in radionuclide repository sites (
Kiryukhin et al., 2008) and in contaminated sites (Tellam and Barker, 2006) or geochemical anomalies in mining areas (Wang and Zuo, 2018). Hence, the controlling factors of submeter variability
should be understood and at least roughly quantified before starting the development in the subsurface region.
In sedimentary bodies, the spatial distribution of the properties is mainly controlled by depositional and diagenetic processes (McKinley et al., 2011, 2013). The spatial characteristics of
physicochemical properties in sedimentary rock media are complex due to strongly intersecting and interacting processes during sediment transport, deposition and diagenesis (McKinley et al., 2011).
Multiple studies aimed to quantify the variability at the lithofacies scale, most of which concentrated on reservoir properties such as permeability and porosity in 2D spaces (McKinley et al., 2011;
Hornung et al., 2020). A 2D analysis is well suited to identifying nonvisible patterns related to microbedding structures at multiple scales even in very homogeneous sandstones (McKinley et al., 2004
). That perspective, however, involves simplifications of the physicochemical variability in 3D spaces since specific rock properties such as permeability are dependent on the Cartesian direction.
Also, consideration of geostatistical parameters such as variographic direction, range, sill and nugget revealed differences in 3D compared to 2D spaces (Landa and Strebelle, 2002; Hurst and Rosvoll
, 1991).
With proper knowledge of the statistical and causal relationships among physicochemical rock properties at different scales, prognostic property models can be significantly enhanced by the
integration of small-scale uncertainty into upscaling or conditional simulation algorithms (Lake and Srinivasan, 2004; Verly, 1993). Especially since multivariate geostatistics can account for
interrelationships between rock properties, those relationships can be used as trends or drifts in geostatistical predictions in order to optimize their accuracy in space and time (Hudson and
Wackernagel, 1994).
In order to quantify the spatial variability and the multidimensional relationships among physicochemical properties at the 3D lithofacies scale, the quasi-continuous scalar fields of two rock cubes
are modeled by means of spatial interpolation, which is constrained by laboratory measurements. The rock cubes have volumes of 0.0156 and 0.008m^3 and have been sampled from a Permian
lacustrine-deltaic sandstone formation that deposited in the intermontane Saar–Nahe basin during the Cisuralian series.
The lithological characteristics of the sandstones are analyzed, and both isotropic and anisotropic properties, including bulk rock geochemistry, thermophysical, hydraulic and elastic rock
properties, are measured on the cubes' faces. In addition, the intrinsic gas permeability under an infinite pressure gradient, the effective porosity, the elemental composition, the thermal
conductivity, and the thermal diffusivity together with the P-wave and S-wave velocity are measured on 108 rock cylinders taken from the inside of the cubes representative for each Cartesian
direction in order to account for anisotropic patterns.
The measurements are used to interpolate the full 3D field of each property. Prior to interpolation, the discrete measurements are statistically analyzed for correlation and formal relationships.
Interpolations are conducted using deterministic and geostatistical methods, including the inverse distance weighting (IDW) and simple kriging (SK) interpolation. The models are evaluated through
cross validation, and the observed spatial patterns are categorized. The interpolation results providing the lowest cross validation error are statistically analyzed again and compared with the
aforementioned statistical patterns. Eventually, the geological processes, which produced the observed patterns, are interpreted and discussed with the help of qualitative thin section and
environmental scanning electron microscope (ESEM) analyses.
The research outputs of this study lie between the scale of a core plug measurement and a wireline log and/or pumping test (Medici et al., 2018). Hence, we aim to contribute to estimating the
uncertainty that must be accounted for when performing up- or downscaling between those two scales of investigation (Zheng et al., 2000; Jackson et al., 2003; Corbett et al., 2012; Hamdi et al., 2014
2.1Sedimentological characterization and rock sampling
In order to cover multiple varieties of sedimentary lithofacies types, a quarry in Obersulzbach (Rhineland-Palatinate, Germany) in the Saar–Nahe basin was selected for the investigations (Fig. 1).
The quarry belongs to the lacustrine-deltaic Disibodenberg Formation that is assigned to the inner Variscan Rotliegend group and comprises four lithofacies types. This formation is deeply buried
(1995 to 2380mbelow ground surface) in the northern Upper Rhine Graben in southwestern Germany (Becker et al., 2012) and constitutes a potential target unit for hydrothermal exploitation (Aretz
et al., 2015). The maximum past overburden of the field site can be estimated to be between 1950 and 2400m, as indicated by shale-compaction analyses which were performed by Henk (1992). The outcrop
has been chosen in order to estimate the variability of physicochemical properties that could be expected in this formation as an uncertainty factor if it is targeted in a deep geothermal project.
Two rock cubes of $\mathrm{0.2}×\mathrm{0.2}×\mathrm{0.2}$m (OSB2_c) and $\mathrm{0.25}×\mathrm{0.25}×\mathrm{0.25}$m (OSB1_c) were extracted from the outcrop wall using a rock chainsaw. According
to the outcrop's coordinate system, one edge of the cuboid runs east–west (x), one north–south (y) and one in an altitudinal (z) direction. The irregular cuboids were reworked to regular cubes with a
stationary rock saw. We selected two types of lithofacies (Fig. 1e) – both sandstones – with one representing a heterogeneous, compartmentalized variety (OSB1_c) and the other one being a homogeneous
variety (OSB2_c). The cubes were both extracted from a distributary mouth bar that is building a foreset in a fluviatile-dominated lacustrine delta. OSB1_c (Fig. 2) was taken from the high energetic
basal part, whereas OSB2_c was taken from the lower energetic top. The sedimentological characteristics, including grain size, sorting, angularity, sedimentary structures and mineral content, were
determined by visual inspection, thin section and ESEM analyses. Two different types of zonal anisotropy and spatial patterns were expected to be found with the aforementioned sampling strategy. In
other studies, such as McKinley et al. (2011), measurements were directly conducted in the field. This approach, however, often provides a drawback with the accuracy and precision, especially in
permeability measurements. In order to address this issue, we performed analyses on the faces of the cubes under laboratory conditions. In the next step, the cubes were cut into rock slabs from which
cylinder samples were extracted. In total, 108 rock cylinders – 79 from OSB1_c and 29 from OSB2_c – were extracted from the rock cubes. It was ensured that at least five samples were produced that
were representative for each Cartesian direction. By applying the formula for calculating a cylinder's volume V[c] with the following:
$\begin{array}{}\text{(1)}& {V}_{\mathrm{c}}=h×\mathit{\pi }×{r}^{\mathrm{2}},\end{array}$
where h is the height of the cylinder and r the radius, the relative volume covered by the rock cylinders in the rock cubes was calculated to be 25.4% for OSB1_c and 18.2% for OSB2_c, respectively.
Eventually, target meshes are needed to interpolate the full 3D scalar fields. Therefore, both cubes were modeled in 3D using a regular grid consisting of 27000 hexahedral orthogonal cells. The
elementary cell of OSB1_c has a volume of $\mathrm{5.7}×{\mathrm{10}}^{-\mathrm{7}}$m^3, whereas OSB2_c's elementary cells have a volume of $\mathrm{3}×{\mathrm{10}}^{-\mathrm{7}}$m^3.
2.2Laboratory experiments
First, a local metric coordinate system was defined, where each edge of the cube represents an axis in the Cartesian coordinate system in order to reference each measurement to a point in space. The
sampling points were set in a raster of 9×9 points on each face for OSB1_c and 5×5 for each face of OSB2_c. All measurements were conducted in the laboratory of the Institute of Applied Geosciences
in Darmstadt, Germany. After drying the rock cubes at 60^∘C, noninvasive measurements were conducted on each face of the cube. On the cubes' faces, the P-wave and S-wave velocity and elemental mass
fractions were determined (Fig. 3).
After the extraction, the rock cylinders were ovendried at 105^∘C and measured in order to determine the intrinsic gas permeability, effective porosity, P-wave and S-wave velocity, elemental mass
fractions, thermal conductivity, and the thermal diffusivity in unsaturated conditions. Those properties can be considered key properties of the rock matrix in porous aquifers with regard to
hydrothermal systems (Agemar et al., 2014) since they constitute input variables for the governing equations for heat transfer and fluid flow in the subsurface (Carslaw and Jaeger, 1959).
The permeability was measured with the Hassler cell permeameter, which is described in Filomena et al. (2014). The Hassler cell is a gas-driven permeameter which measures the permeability of a
cylinder-shaped rock sample under steady-state gas flow. This technique allows for the estimation of the intrinsic gas permeability, which is the permeability at an infinite pressure gradient. The
permeameter was set to accept a measurement if 15 consecutive readings did not deviate by more than 5%. The measurement error, however, can exceed that value, especially in low permeable
lithologies. Effective porosity measurements were conducted using an envelope density analyzer (GeoPyc 1360). The accuracy is given by the manufacturer to be within ±0.55% (Micromeritics, 1998).
Thermal properties under unsaturated conditions, namely the thermal conductivity and thermal diffusivity, were determined with a thermal conductivity scanner (TCS) according to the work of Popov
et al. (1999). The measurement error is quantified to be ≤3% for thermal conductivity and ≤8% for thermal diffusivity (Popov et al., 1999). The elastic properties of P-wave and S-wave velocity in
the rock media were measured with the sonic wave generator UKS-D (Geotron Elektronik) by sending a sonic wave pulse from a pulse-providing test head (UPG-S) to a receiver (UPG-E). The wave velocity
is a function of the travel length and time, together with the density of the material. The initial occurrence of the P wave or S wave must be picked manually after visual inspection by the operator.
Thus no measurement error can be provided since user bias cannot be assessed quantitatively. Bulk elemental analysis using the Bruker S1 TITAN handheld portable X-ray fluorescent (pXRF) analyzer was
used to find correlations between the elemental composition and the petrophysical properties. The measurement device works on the basis of energy-dispersive X-ray fluorescence (EDXRF) and estimates
the elemental mass fractions of a sample. This device produces an ionizing X-ray beam with a diameter of 1.2cm and quantifies the elemental composition based on the energy emitted by the ionized
elements in the targeted area. The portable device can measure the fraction of elements with an ordinal number ≥12 and ≤235 if the threshold value, defined by the measurement error for the specific
element in the sample, is exceeded. For this study, the device was operated in GeoChem, Dual Mining mode, allowing for the detection of the major oxides of SiO[2], Al[2]O[3], Fe[2]O[3] and K[2]O and
a wide range of other elements. The device was calibrated with international standards. We used the previously mentioned major oxides for analyses since those can provide insight into the iron oxide
and clay mineral distribution, which can significantly impact the petrophysical properties. More details on the measurement devices can be found in the works of Hornung and Aigner (2002), Sass and
Götz (2012), Filomena et al. (2014), and Aretz et al. (2015).
2.3Data analysis and spatial modeling
The experimental semivariogram represents the cumulative dissimilarity of a discrete set of point pairs x, with n[c] as the count of point pairs within the distance classes h of identical distance
increments (Eq. 2).
$\begin{array}{}\text{(2)}& \mathit{\gamma }\left(\mathbit{h}\right)=\frac{\mathrm{1}}{\mathrm{2}{n}_{c}}\sum _{\mathit{\alpha }=\mathrm{1}}^{{n}_{\mathrm{c}}}\left(z\left({x}_{\mathit{\alpha }}+\
mathbit{h}\right)-z\left({x}_{\mathit{\alpha }}\right){\right)}^{\mathrm{2}}.\end{array}$
The continuous counterpart, represented by the variogram model, is an approximation of the experimental semivariogram that assumes z(x) to be a stationary random field (Wackernagel, 2003). A
variogram model γ[theo] is represented by a covariance function c, with the relationship ${\mathit{\gamma }}_{\mathrm{theo}}\left(\mathbit{h}\right)=c\left(\mathrm{0}\right)-c\left(\mathbit{h}\right)
$, where c is a positive definite even function. Six covariance models are mostly used to fit the experimental semivariogram, namely the spherical, gaussian, exponential, power, cardinal sine and the
linear model (Armstrong, 1998; Ringrose and Bentley, 2015). In this study, we only observe spherical relationships with a nugget effect. This model is calculated as follows:
$\begin{array}{}\text{(3)}& {c}_{\mathrm{sph}}\left(\mathbit{h}\right)=\begin{array}{ll}n+b\cdot \left(\mathrm{1}-\frac{\mathrm{3}|\mathbit{h}|}{\mathrm{2}a}+\frac{|\mathbit{h}{|}^{\mathrm{3}}}{\
mathrm{2}{a}^{\mathrm{3}}}\right)& \mathrm{for}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathrm{0}\le |\mathbit{h}|<a\\ n& \mathrm{for}\phantom{\rule{0.25em}{0ex}}\phantom{\rule
{0.25em}{0ex}}|\mathbit{h}|\ge a,\end{array}\end{array}$
with the variables nugget (n), range (a) and sill (b). Semivariograms can be used to quantify the spatial or time correlation of a random property (Ringrose and Bentley, 2015; Gu et al., 2017; Rühaak
et al., 2015). Further on, the differences in range and sill in dissimilar directional semivariograms can quantify the zonal and geometric anisotropy of a property (Ringrose and Bentley, 2015). The
resulting covariance function is an input variable for geostatistical interpolation algorithms.
2.3.2Rock property interpolation
Spatial inter- and extrapolation can be generated with deterministic and geostatistical techniques. All interpolations are based on the assumption that a point x[k] with a known value z(x[k]) has a
weight on a discrete point x[0] in space with an unknown value z(x[0]). The global known points, however, can be reduced to a local neighborhood of x[0].
For deterministic interpolation, the p value inverse distance weighting (IDW; Shepard, 1968) interpolation is used. The IDW interpolation generally calculates an unknown value z(x[0]) at point x[0]
by weighting the distance of that point to each known value point (x[k]) in space. The underlying formula for IDW is as follows:
$\begin{array}{}\text{(4)}& z\left({x}_{\mathrm{0}}\right)=\frac{{\sum }_{k=\mathrm{1}}^{n}z\left({x}_{k}\right)/{d}_{k}^{p}}{{\sum }_{k=\mathrm{1}}^{n}\mathrm{1}/{d}_{k}^{p}},\end{array}$
where d is the Euclidean distance between the point with the known value x[k] and the point with the unknown value x[0], and p is an exponent factor to bias the weights nonlinearly. The p value is
mostly used for smoothing the results by controlling the distance decay effect (Lu and Wong, 2008). IDW is a reliable and widely applied method to interpolate static rock properties in a 1D to 3D
space (Rühaak, 2006).
For geostatistical interpolation, simple kriging (SK) is used. Kriging in general is a popular technique for interpolating geological properties in space (Goovaerts, 1997; Rühaak, 2015; Malvić et al.
, 2019). Through kriging, the value z(x[0]) at an unknown point x[0] is calculated by weighting the neighboring known values and building a linear combination of those via the following formula:
$\begin{array}{}\text{(5)}& z\left({x}_{\mathrm{0}}\right)=\sum _{k=\mathrm{1}}^{n}{w}_{k}\cdot z\left({x}_{k}\right),\end{array}$
where w[k] is the weight of the known point x[k] with the value z(x[k]). SK requires knowledge of the stationary mean μ (Deutsch and Journel, 1998), which modifies Eq. (5) into the following:
$\begin{array}{}\text{(6)}& z\left({x}_{\mathrm{0}}{\right)}_{\mathrm{SK}}=\sum _{k=\mathrm{1}}^{n}{w}_{k}\cdot z\left({x}_{k}\right)+\left(\mathrm{1}-\sum _{k=\mathrm{1}}^{n}{w}_{k}\right)\cdot \
mathit{\mu }.\end{array}$
To obtain the simple kriging weights, a set of n equations has to be solved. This set of equations can be written as follows:
$\begin{array}{}\text{(7)}& \begin{array}{rl}& \left(\begin{array}{ccc}c\left({x}_{\mathrm{1}}-{x}_{\mathrm{1}}\right)& \mathrm{\cdots }& c\left({x}_{\mathrm{1}}-{x}_{n}\right)\\ \mathrm{⋮}& \mathrm
{\ddots }& \mathrm{⋮}\\ c\left({x}_{n}-{x}_{\mathrm{1}}\right)& \mathrm{\cdots }& c\left({x}_{n}-{x}_{n}\right)\end{array}\right)\left(\begin{array}{c}{w}_{\mathrm{1}}\\ \mathrm{⋮}\\ {w}_{n}\end
{array}\right)\\ & =\left(\begin{array}{c}c\left({x}_{\mathrm{1}}-{x}_{\mathrm{0}}\right)\\ \mathrm{⋮}\\ c\left({x}_{n}-{x}_{\mathrm{0}}\right)\end{array}\right),\end{array}\end{array}$
with c as the covariance function and x[n] as the point with a known value (Wackernagel, 2003). The quality of kriging interpolation is dependent on the variogram model, defined neighborhood,
sampling density and goodness of fit to the experimental values.
2.4Cross validation
Cross validation can be used to assess the quality of a model. During cross validation, p randomly selected samples are removed from the input data set of size n with $\mathrm{0}<p<n$, and the
interpolation is performed without those samples (Celisse, 2014). The measures of goodness of fit being used in this study include the root mean square error (RMSE) as follows:
$\begin{array}{}\text{(8)}& \mathrm{RMSE}=\sqrt{\frac{\mathrm{1}}{n}\sum _{k=\mathrm{1}}^{n}\left(\stackrel{\mathrm{^}}{z}\left({x}_{k}\right)-z\left({x}_{k}\right){\right)}^{\mathrm{2}}},\end{array}
and the mean absolute error (MAE) as follows:
$\begin{array}{}\text{(9)}& \mathrm{MAE}=\frac{\mathrm{1}}{n}\sum _{k=\mathrm{1}}^{n}|\stackrel{\mathrm{^}}{z}\left({x}_{k}\right)-z\left({x}_{k}\right)|,\end{array}$
with $\stackrel{\mathrm{^}}{z}\left({x}_{k}\right)$ as estimated value at point x[k]. Those parameters allow for the quantitative assessment of an interpolation's quality. They might be prone to bias
if the sampling density in the target domain is extremely scarce.
Anisotropy describes the dependence of a physical property on a direction. Rock properties such as stiffness, permeability or thermal conductivity are anisotropic in most cases. Hence, measurements
of those properties might show differing magnitudes in different directions if the medium is polar anisotropic. The intrinsic permeability, for example, provides typical ranges for the ratio between
the vertical (k[v]) and horizontal permeability (k[h]) of 10^−5 to 1 (Ringrose and Bentley, 2015). Anisotropy in geological media is generated by the preferred orientation of mineral grains or cracks
and by the intrinsic anisotropy of single crystals (Thomsen, 1986).
In the following, we will provide an exemplary description of the anisotropy of elasticity, and we will provide measures for anisotropy quantification under the simplifying assumption of transverse
isotropy. The elastic modulus tensor can be expressed as a fourth rank tensor as follows:
$\begin{array}{}\text{(10)}& \mathbf{C}=\left(\begin{array}{cccccc}{C}_{\mathrm{11}}& {C}_{\mathrm{11}}-\mathrm{2}{C}_{\mathrm{66}}& {C}_{\mathrm{13}}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ {C}_{\
mathrm{11}}-\mathrm{2}{C}_{\mathrm{66}}& {C}_{\mathrm{11}}& {C}_{\mathrm{13}}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ {C}_{\mathrm{13}}& {C}_{\mathrm{13}}& {C}_{\mathrm{33}}& \mathrm{0}& \mathrm{0}& \
mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& {C}_{\mathrm{44}}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& {C}_{\mathrm{44}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \
mathrm{0}& \mathrm{0}& \mathrm{0}& {C}_{\mathrm{66}}\end{array}\right),\end{array}$
where C[ij] represents an elasticity modulus and the indices are related to the directional P-wave and S-wave velocity, under the assumption that z is the symmetry axis. The velocities can be
calculated by the following:
$\begin{array}{}\text{(11)}& {v}_{\mathrm{p}}^{z}=\sqrt{\frac{{C}_{\mathrm{33}}}{\mathit{\rho }}}\text{(12)}& {v}_{\mathrm{s}}^{z}=\sqrt{\frac{{C}_{\mathrm{44}}}{\mathit{\rho }}},\end{array}$
where v[p] is the P-wave velocity and v[s] is the S-wave velocity parallel to the symmetry axis and ρ is the bulk density (Yang et al., 2020). The anisotropy, here exemplarily expressed for the
P-wave polar anisotropy, can be quantified with the Thomsen parameters (Thomsen, 1986). For example, ϵ can be expressed as follows:
$\begin{array}{}\text{(13)}& \mathit{ϵ}=\frac{{C}_{\mathrm{11}}-{C}_{\mathrm{33}}}{\mathrm{2}{C}_{\mathrm{33}}}.\end{array}$
If ϵ≪1, the material can be classified as weakly anisotropic.
2.4.2Correlation and regression analysis
In order to quantify the linear statistical relationship between two independent variables x and y, the Pearson linear product–moment correlation coefficient (R) can be used. R is expressed as
$\begin{array}{}\text{(14)}& R=\frac{\sum _{k=\mathrm{1}}^{n}\left({x}_{k}-\stackrel{\mathrm{‾}}{x}\right)\left({y}_{k}-\stackrel{\mathrm{‾}}{y}\right)}{\left(\sum _{k=\mathrm{1}}^{n}{x}_{k}^{\mathrm
{2}}-n\cdot {\stackrel{\mathrm{‾}}{x}}^{\mathrm{2}}\right)\left(\sum _{k=\mathrm{1}}^{n}{y}_{k}^{\mathrm{2}}-n\cdot {\stackrel{\mathrm{‾}}{y}}^{\mathrm{2}}\right)},\end{array}$
with n representing the number of compared point pairs and $\stackrel{\mathrm{‾}}{x}$ and $\stackrel{\mathrm{‾}}{y}$ standing for the arithmetic mean of x and y.
Regression aims at finding a fitting function between samples of two or more random variables. For curvilinear regression, a function of a degree >1 will be approximated for a discrete set of
values. A second-degree polynomial function f(x), for instance, would be described as follows:
$\begin{array}{}\text{(15)}& f\left(x\right)={b}_{\mathrm{0}}+{b}_{\mathrm{1}}x+{b}_{\mathrm{2}}{x}^{\mathrm{2}}.\end{array}$
Thus, we would need to find n+1 regression coefficients, where n is the degree of f(x). In general, the regression model yields the following:
$\begin{array}{}\text{(16)}& f\left(x{\right)}_{i}={b}_{\mathrm{0}}+{b}_{\mathrm{1}}{x}_{i}+{b}_{\mathrm{2}}{x}_{i}^{\mathrm{2}}+\mathrm{\cdots }+{b}_{n}{x}_{i}^{n},\end{array}$
with $i=\mathrm{1},\mathrm{2},\mathrm{\dots },n$. The regression coefficients b[m] are obtained through solving a system of linear equations as follows:
$\begin{array}{}\text{(17)}& \left(\begin{array}{c}{y}_{\mathrm{1}}\\ {y}_{\mathrm{2}}\\ \mathrm{⋮}\\ {y}_{n}\end{array}\right)=\left(\begin{array}{cccc}\mathrm{1}& {x}_{\mathrm{1}}^{\mathrm{1}}& \
mathrm{\cdots }& {x}_{\mathrm{1}}^{m}\\ \mathrm{1}& {x}_{\mathrm{2}}^{\mathrm{1}}& \mathrm{\cdots }& {x}_{\mathrm{2}}^{m}\\ \mathrm{⋮}& \mathrm{⋮}& \mathrm{\dots }& \mathrm{⋮}\\ \mathrm{1}& {x}_{n}^
{\mathrm{1}}& \mathrm{\cdots }& {x}_{n}^{m}\end{array}\right)\left(\begin{array}{c}{b}_{\mathrm{0}}\\ {b}_{\mathrm{1}}\\ \mathrm{⋮}\\ {b}_{m}\end{array}\right),\end{array}$
where x and y are the samples. The function approximations, as produced in regression analyses, are commonly evaluated by the coefficient of determination (R^2), which is calculated through the
$\begin{array}{}\text{(18)}& {R}^{\mathrm{2}}=\mathrm{1}-\frac{{s}_{\mathrm{res}}}{{s}_{\mathrm{tot}}}\in \left[\mathrm{0},\mathrm{1}\right],\end{array}$
$\begin{array}{}\text{(19)}& {s}_{\mathrm{res}}=\sum _{k=\mathrm{1}}^{n}\left({y}_{k}-f\left(x{\right)}_{k}{\right)}^{\mathrm{2}}\end{array}$
is the explained sum of squares, and
$\begin{array}{}\text{(20)}& {s}_{\mathrm{tot}}=\sum _{k=\mathrm{1}}^{n}\left({y}_{k}-\stackrel{\mathrm{‾}}{y}{\right)}^{\mathrm{2}}\end{array}$
is the total sum of squares.
2.4.3Spatial modeling and statistical analyses
The spatial dependence of the discrete values is evaluated through experimental semivariograms. The semivariograms are generated for the single rock faces, where measurements are available, and for
the plug measurements. The empirical semivariogram is fitted to a variogram model, which is then used for the geostatistical interpolation. Interpolation analyses are performed as IDW and SK
realizations (Fig. 4) that are assessed through cross validation. The power parameter for IDW is chosen to be three since this constant provides the lowest RMSE among the realizations. The search
radii for each prediction is chosen to be 0.2m, in x and y direction, and 0.15m, in z direction, in OSB1_c to account for the sedimentary structures. For OSB2_c, the search radii are chosen to be
isotropic with a length of 0.2m. To make the methods comparable, we selected the maximum number of neighboring points to be 25 to represent between 5% and 95% of the measurements.
We decided to waive sequential simulation as large amounts of the cubes' volumes are covered by rock samples. Thus, we do not expect a relevant kriging variance. With this in mind, the simulations
are assumed to capture most of the total variance from the measurements themselves. The interpolation results that provide the lowest cross validation error are used for statistical analyses in order
to derive correlations and regression functions between the scalar fields. Eventually, significant correlations are compared with the noninterpolated data sets. Both the spatial modeling and the
statistical analyses are performed with the open source software called Geological Reservoir Virtualization (GeoReVi; Linsel, 2020a). This software tool provides functionality for multidimensional
subsurface characterization using the concept of knowledge discovery in databases, which is helpful when handling huge data sets such as those produced in this study.
3.1Sedimentological characteristics
The sandstones belong to a clinothem strata deposited in a fluvial-dominated lacustrine delta. More specifically, the architectural element represents a distributary mouth bar formed by rapid
sandstone deposition in sheet-like bodies, as described in Fongngern et al. (2018). The base of those bodies is typically erosive, which is why muddy rip-up clasts commonly occur above the base.
Also, the beds, which deposited after the intraclast-rich basal beds, typically show trough or ripple-cross stratification with set heights of 5–15cm. The vertical orientation of rip-up clasts can
be observed in matrix-rich debrites or turbidites deposited under high-energy turbulent hyperpycnal to homopycnal flow conditions (Li et al., 2017). Those are unconformably overlying lacustrine,
laminated mud strata from the prodelta environment. Accordingly, Bouma A–E layers (Bouma, 1962; Middleton, 1993) with a prograding trend can be identified in the outcrop. With ongoing sedimentation,
the depositional energy in a Bouma sequence typically decreases, which leads to massive sandstones. OSB1_c was taken from a basal bed of the Bouma A interval characterized by a high number of rip-up
intraclasts, normal grading and subhorizontal pseudolayering, which may occur in a Bouma A interval if the rip-up clasts experienced buoyancy during transport. OSB2_c was taken from the topmost bed,
which corresponds to a Bouma E interval that is characterized by a massive structure.
The average grain size in both cubes ranges from fine to very coarse sand (200–1400µm). While the grain size distribution in OSB2_c does not show a significant variability – mainly characterized by
medium to coarse sand – a normal grading is observable in OSB1_c. Here, the grain size gradually decreases from very coarse sand at the base to medium sand at the top. Likewise, sorting increases
from poor to moderate. In OSB2_c the sorting is moderate throughout the entire sample volume. The components provide a low to medium sphericity, while the grain shapes vary between subangular and
subrounded. Locally, pelitic rip-up clasts occur with diameters of up to 4cm. The rip-up clasts show a very low textural maturity and are subvertically oriented with respect to bedding.
The original rigid detrital components consist of 50%–60% quartz, 20%–30% strongly altered feldspar and 10%–25% lithic fragments. Mica grains are often bent between more rigid grains. The rock
matrix accounts for approximately 10%–20% and is built up by detrital grains, coated by iron oxides, ductile, autochthonous pelite grains and fine-grained quartz. According to the geochemical
analyses, the rocks can be classified as lithic arenites to arkoses or wackes (Fig. 5), respectively, if the matrix content exceeds 15% based on the classification of Herron (1988).
Thin section analysis (Fig. 6a) reveals that most of the pore space is secondary due to grain dissolution. The secondary pores are undeformed, indicating that grain dissolution took place during
structural inversion – probably during telogenesis, according to the concept of Worden and Burley (2003). Most of the primary intergranular volume was destroyed during mechanical compaction. ESEM
analysis (Fig. 6b) confirms the presence of quartz accompanied by coprecipitated calcite, opaque phases – mainly iron oxides – and authigenic clay minerals including kaolinite and illite in the
cement fraction. Thus, chemical compaction had taken place due to iron oxide, quartz and clay mineral precipitation during diagenesis. Here, the earliest cement phase is represented by the opaque
phases comprising a high number of iron oxides. Thereafter, kaolinite is formed, mainly in the secondary pore space, and overgrown by illite. Often, the early cement is overgrown syntaxially by
quartz. The source of SiO[2] might be internal and related to feldspar dissolution.
3.2Exploratory data analysis
In order to provide full comparability, the following section will provide an overview of the measurements derived from the rock cylinder analyses. For each property, 79 rock samples from OSB1_c and
29 from OSB2_c were investigated. An overview of the rock properties' ranges is provided in the box and whisker charts shown in Fig. 7.
The local variability of OSB1_c is significantly higher than that of OSB2_c. The intrinsic permeability of OSB1_c provides a coefficient of variation of 0.3 and a Dykstra–Parsons coefficient of 0.4,
while measurements from OSB2_c show values of 0.2 for the coefficient of variation and 0.18 for the Dykstra–Parsons coefficient, respectively. According to the classification provided by Corbett and
Jensen (1992), the intrinsic permeability of both rock cubes can be classified as being very homogeneous. Also, the intrinsic permeability does not show a significant anisotropy.
The range of values in OSB1_c for each property is greater than the range of those in OSB2_c. OSB1_c provides lower values in P-wave and S-wave velocity, thermal conductivity and mass fraction of Fe
[2]O[3] compared to OSB2_c. Intrinsic permeability and porosity, in turn, are greater. The mass fraction of silicon oxide and thermal diffusivity provides similar statistical parameters in both
cubes; however, the ranges are marginally larger in OSB1_c. The measurements of the elastic rock properties revealed a weak anisotropy of the P-wave attenuation, especially in rock cube OSB2_c. The
Thomsen parameter ϵ is 0.047 for OSB1_c and 0.096 for OSB2_c. It should be noted that OSB1_c provides visible bedding structures in contrast to OSB2_c; hence, the observed degree of anisotropy is not
connectable to the bedding features in this case.
Statistically significant linear correlations (Fig. 8), in the sense of passing a two-tailed significance test at the 0.05 level, were found between porosity and permeability, permeability and Fe[2]O
[3], v[p] and v[s], v[p] and SiO[2], v[p] and Al[2]O[3], v[p] and K[2]O, Fe[2]O[3] and SiO[2], and K[2]O and Al[2]O[3]. The strongest positive linear correlation can be observed between v[p] and v[s]
(R=0.88), K[2]O and Al[2]O[3] (R=0.70), and porosity and permeability (R=0.31). The strongest negative correlation can be observed between permeability and Fe[2]O[3] ($R=-\mathrm{0.56}$). Properties
not being mentioned do not provide significant statistical correlations to others.
3.3Submeter-scale spatial correlation
The spatial dependence of the discrete measurements is estimated using experimental semivariograms. Therefore, the geochemical representatives SiO[2] (Fig. 9a) and Fe[2]O[3] (Fig. 9b) that were
measured on each of the rock faces of OSB1_c are given an exemplary analysis. The experimental semivariograms greatly vary from rock face to rock face in OSB1_c. The nugget effect for each
experimental variogram is very low. The range of each semivariogram varies between 0.05 and 0.3m. In the experimental semivariograms of SiO[2], two types of patterns can be identified. The XY base,
XZ back and YZ front of the rock faces show ranges of approximately 0.08m and a sill between 8%^2 and 10%^2, until the semivariance exponentially increases when exceeding a lag distance of 0.2m.
The semivariance on the other rock faces runs similarly, with ranges of 0.2m and a sill of 4.7%^2. The semivariogram for Fe[2]O[3] shows some similarities. Here, the XY base, YZ front and XZ front
of the rock faces show very low ranges between 0.05 and 0.15m and a sill between 0.1%^2 and 0.15%^2 again, with an exponential increase when exceeding a lag distance from 0.2 to 0.25m. In
contrast, the semivariance of YZ back has the highest sill, with 0.21%^2 and a range of 0.15m; however, semivariance drops after exceeding a lag distance of 0.2m. XZ back provides the highest
degree of similarity with a range of 0.3m and a sill of 0.09%^2, using a spherical approximation. Both geochemical properties show a zonal anisotropy where the sill shows different magnitudes along
different directions (Wackernagel, 2003; Allard et al., 2016).
3.4Spatial pattern analysis
The spatial distributions of the rock properties are interpolated with Shepard's inverse distance weighting (IDW) and simple kriging (SK). Both realizations of a single scalar field provide
comparable patterns, which is due to the high sampling density. The interpolation errors are also located in similar ranges; however, IDW seems to be more sensitive to outliers, resulting in much
higher interpolation errors with regard to properties like P-wave velocity or mass fraction of SiO[2] (Table 1). IDW tends to underestimate the maximum and minimum values in the scalar fields. Thus,
petrophysical and geochemical contrasts are more distinctly reproduced in the geostatistical approach. Also, the IDW realization shows the bull's eye effect, which is a typical artifact of IDW
interpolations (Shepard, 1968). Accordingly, the simple kriging realizations are used for further analyses.
The rock properties exhibit a multitude of spatial patterns. Here, discrete, layered and homogeneous patterns, both connected and disconnected to primary sedimentary structures, could be observed in
the interpolations.
3.4.1Patterns connected to sedimentary structures
A bedding-connected pattern is exhibited in the intrinsic permeability and Fe[2]O[3] interpolation results of OSB1_c. The mass fraction of Fe[2]O[3] varies between 1.25% and 5% in OSB1_c. In the
histogram displayed in Fig. 11, outliers were removed according to Tukey's outlier-detection method (Tukey, 1977). The local histogram of OSB1_c's intrinsic permeability shows a bimodal distribution
ranging from 0.7 to 3.9mD. The application of Tukey's method revealed no statistical outliers in this scalar field.
The bedding structures in OSB1_c are well reflected by the spatial pattern of the interpolated intrinsic permeability gradually increasing from low values, between 0.7 and 2mD, in the lower beds to
higher values, between 2 and 4mD, in the upper beds (Fig. 10).
The spatial distribution of the mass fraction of Fe[2]O[3] in OSB1_c provides a reciprocal trend compared to the permeability. Here, the lowermost bed shows a significantly higher content compared to
the upper beds. Both scalar fields show zonal anisotropy. The Fe[2]O[3] content is an indicator of the detrital matrix, pseudomatrix and cement content that, in turn, would explain the reciprocal
relationship with the permeability measurements. In siliciclastic systems, iron can be contained in clay minerals (up to 30wt%; Brigatti et al., 2006), mafic components or in iron-rich oxides,
hydroxides or carbonates. Local excesses in the Fe[2]O[3] content exist in the spatial distribution. Those can be explained by clay-rich intraclasts observed on the rock faces. When comparing the
pattern to Fig. 2 at both XZ-oriented cube faces, rip-up clasts can be observed where high Fe[2]O[3] mass fractions occur. Those areas provide the maximum values of the Fe[2]O[3] distribution.
3.4.2Patterns decoupled from sedimentary structures
Other scalar fields are decoupled from depositional bounding surfaces. For instance, the geochemical mass fractions of K[2]O (Fig. 12) and Al[2]O[3] (Fig. 13) provide a significant positive
correlation unconnected to visible structural boundaries. Typically, those geochemical properties are indicative of the presence of orthoclase feldspar (KAlSi[3]O[8]) and/or illite (KAl[3]Si[3]O[10]
(OH)[2]) in siliciclastic environments. The mass ratio of both components is roughly 1:3 to 1:4, which is in accordance to the illite fraction that was observed in the thin section and ESEM
analyses. Only minor amounts of orthoclase feldspar could be found in the thin sections. Thus, we assume that the correlation of K[2]O and Al[2]O[3] can be traced back to the illite phases.
Higher fractions of Al[2]O[3] are supposedly due to higher kaolinite (Al[2]Si[2]O[5](OH)[4]) fractions in the clay mineral assemblages. The patterns are diffuse, showing autocorrelated areas of
slightly enriched and depleted mass fractions. Enriched areas seem to be connected, building network-like patterns, while depleted areas are more isolated.
The overall aim of this study was to quantify the 3D interdependencies of thermophysical, hydraulic, elastic and geochemical scalar fields in sandstone media at the lithofacies scale and to identify
the controlling factors for the property distributions. With a high-resolution study at the lithofacies scale, statistical and spatial interrelationships between characteristic physicochemical fields
could be discovered and traced back to depositional and diagenetic processes.
4.1Petrophysical and geochemical characteristics
Recent multiscale modeling approaches without the use of local constraints show that the prediction of permeability and porosity in siliciclastic systems is still challenging (Nordahl et al., 2014).
Geological sampling almost never includes the entire domain that is investigated. With sampling densities of 25.4% and 18.2%, we reached a very high degree of coverage. Studies such as Hurst and
Rosvoll (1991) showed that a very high sampling density is necessary to cover the entire variance of permeability at the lithofacies scale. The interpolations performed in this study reproduce the
global histogram properly and outliers are also accounted for. This, in fact, implies that the sampling density was selected adequately in order to capture the total variability present in the
physical and geochemical scalar fields. This condition is typically only fulfilled in sequential simulations (Robertson et al., 2006) rather than in conventional interpolations.
Although statistically significant correlations may imply a natural relationship between physicochemical properties, this relationship could also be based on random processes requiring causality to
be verified. Weak correlations were found between the effective porosity and the intrinsic permeability, which are usually positively correlated (Pape et al., 1999). This relationship can be traced
back to the Kozeny–Carman equation that connects the permeability with the effective pore throat radius ${r}_{\mathrm{eff}}^{\mathrm{2}}$ and a formation factor F as follows:
$\begin{array}{}\text{(21)}& k={r}_{\mathrm{eff}}^{\mathrm{2}}/\left(\mathrm{8}\cdot F\right).\end{array}$
The formation factor is defined as the ratio of tortuosity and porosity showing that porosity and permeability provide a positive formal relationship empirically. A high number of secondary pores,
produced by feldspar dissolution, did not significantly contribute to the permeability in the investigated sandstones since those pores are often hydraulically isolated. Consequently, secondary
porosity did not necessarily lead to increasing radii of the effective pore throats rather than increasing tortuosity. Also, recrystallized quartz cement – blocking a large number of the pore throats
– must be taken into account. Both effects, in turn, resulted in a degraded permeability. In addition to the geometrical aspects previously mentioned, the alteration products in the form of clay
minerals occupy the pore space, which lead to larger adhesive effects that hinder the ability to transport fluids as well. This observation is in good agreement with observations made by Molenaar
et al. (2015) in Rotliegend rocks from the Donnersberg Formation. In addition, these observations are well reflected by the very low values of the intrinsic permeability in both rock cubes. Another
reason for the very low intrinsic permeability is the high amount of primary clay and the low maturity of deltaic sheet-like distributary mouth bar deposits (Tye and Hickey, 2001).
The linear correlation analysis revealed a significant negative relationship between hydraulic and geochemical properties that fits to a polynomial regression (Fig. 14). It should be considered that
the geochemical measurements cover a very different measurement area – represented by a spot with a 1.2cm diameter and around 0.5cm penetration depth compared to the hydraulic measurements
performed on an entire rock cylinder with a 40mm height and diameter. Additionally, instead of using highly precise stationary X-ray fluorescence devices for measurements, a portable, faster device
was used to efficiently derive spatial trends in the objects of investigation. This technique weakens the implications for absolute values; however, the trends observed in the measurements from the
portable device are in good agreement with trends observed by stationary devices. Also, the observed geochemical characteristics are in accordance with geochemical properties of quartz-rich sandstone
varieties that were investigated in Bhatia (1983) or Baiyegunhi et al. (2017).
Geochemical analyses, in contrast to petrographic ones, limit the interpretations of geological processes as mineral phases can only be assumed and not determined for certain. A high mass fraction of
Fe[2]O[3] may imply that the rock is rich in iron-bearing minerals like clay minerals, hematite, magnetite, goethite, lepidocrite or ferrihydrite (Costabel et al., 2018), however, a precise
classification of the mineral phase is not possible. Iron oxides are more common in secondary precipitates that usually form during eo- and mesodiagenesis (Pettijohn et al., 1987). The degrading
impact of iron-oxide-rich coatings on permeability and porosity in unconsolidated sand and gravel has been shown in studies like Costabel et al. (2018). The number of detrital iron-rich phases, such
as hematite, which are present in the rock matrix, is typically less (Walker et al., 1981; Turner et al., 1995; Ixer et al., 1979) when compared to the secondary amount. In our case, however, thin
section and ESEM analyses revealed that a high degree of the intergranular matrix is still preserved, especially at the base of OSB2_c where high amounts of mud and mud intraclasts are incorporated
from basal erosion. The small grain size of the matrix offers a great surface area for iron-oxide-rich precipitates, which might have further enforced degradation of porosity and permeability. The
primary matrix typically plugs the pore throats of porous, matrix-rich media. This reduces the ability to conduct fluids compared to matrix-free ones. However, due to progressive compaction, we
cannot quantify for certain how large the size of the primary matrix is compared to the pseudomatrix produced by the plastic compaction of ductile, clay-rich grains and by feldspar dissolution.
A significant correlation between K[2]O and Al[2]O[3] could be detected. The spatial distribution resembles a network-like structure that might be either a product of diffusive mass transport during
meso- or telodiagenesis or it might reflect the distribution of feldspar grains and its residues in the sandstone. During feldspar alteration, SiO[2] is dissolved and K remains in the alteration
products, which could be an implication for the mesoscale network-like structure into which pore fluids could have had migrated. This relationship is underlined by a negative, yet nonsignificant,
correlation of K[2]O with SiO[2].
Significant nonintuitive relationships between the physical and geochemical scalar fields at the lithofacies scale have been revealed with a deductive approach of spatial field modeling and
statistical data analysis. All in all, the following conclusions can be drawn from this study:
1. As specific properties such as the mass fraction of Fe[2]O[3] preserve sedimentological textures well in their spatial distribution, other properties seem to be completely decoupled from
depositional bounding surfaces. These scalar fields probably reflect processes that might have taken place during the diagenetic overprint of the rocks as a result of burial and exhumation. These
processes produce diffuse patterns, as discussed with regard to the correlation of K[2]O and Al[2]O[3].
2. This study demonstrates that the observation of bedding structures does not necessarily indicate a stronger polar anisotropy compared to macroscopically unstructured lithologies. Here, the
microscopic characteristics, like the amount of secondary porosity, might play a more important role in the attenuation of physical waves than the bounding surfaces.
3. It could be shown that hydraulic properties are dependent on the intergranular matrix and cement amount, which are in turn controlled by depositional processes and eogenetic precipitates. Those
findings are not new (see Wilson and Pittman, 1977 or Nordahl et al., 2014); however, they have not been evaluated in lithofacies-scale 3D environments yet. We assume that primary matrix and
ductile grain content has the most detrimental effect on rock permeability. Ductile grains were mechanically deformed during compaction, leading to plugged pore throats. Feldspar dissolution has
a highly productive effect on porosity but not on permeability.
4. We demonstrate that the strength of statistical correlation can be preserved in spatial interpolations as long as the sampling density is sufficient. If the sampling density is too low, a
statistical correlation might be inadvertently feigned.
5. As shown in this study, the local geological variability should not be underestimated as an uncertainty factor in spatial predictions and upscaling procedures. In fact, the local geological
variability of physicochemical properties might nearly cover the variability being present in an entire formation. Therefore, a high-resolution analysis of physicochemical rock properties can
assist in assessing the uncertainty of field-scale property models which is induced by the local geological variability at the lithofacies scale.
The investigated rock samples are available at the Institute of Applied Geosciences, TU Darmstadt, and can be requested from linsel@geo.tu-darmstadt.de. Also, the samples are registered in the System
for Earth Sample Registration (SESAR; http://www.geosamples.org, last access: 11 August 2020; registration numbers are provided in the data set of Linsel, 2020b).
AL conceptualized and prepared the paper. AL and SW conducted the laboratory and field measurements. JH contributed to the conceptualization of the study. MH was the overall supervisor of the study.
The authors declare that they have no conflict of interest.
The authors are grateful for the permission to work in the sandstone quarry of Konrad Müller GmbH in Obersulzbach, Germany. Also, we would like to thank Reimund Rosmann and Institut IWAR (Technische
Universität Darmstadt, Germany) for the preparation of the rock cubes. We are extremely thankful to Mattia Pizzati and Giacomo Medici for their time and effort in putting together constructive
reviews. Adrian Linsel has received financial support from the Friedrich-Ebert-Stiftung, Germany, which is gratefully acknowledged.
This research has been supported by the Friedrich-Ebert-Stiftung, Germany.
This paper was edited by Kei Ogata and reviewed by Giacomo Medici and Mattia Pizzati.
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At Mathswizz we believe every child has the potential to become a wizz at maths to improve their mental maths. All that is required is a strong foundation of the basics. This is why we have created
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challenge them to improve on that time at every try. Please Note Problem Solving worksheets don't have answers as there might be more than one correct answer. | {"url":"https://mathswizz.co.uk/","timestamp":"2024-11-14T10:39:24Z","content_type":"text/html","content_length":"18930","record_id":"<urn:uuid:b8578861-9f8d-4c76-890b-c1c7383508f0>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00700.warc.gz"} |
Confidence interval - Biochemia Medica
Every science research starts with a detailed analysis of the field of interest, defining the clear aim and hypothesis, by thorough planning of the research design and the way of data collecting and
data analysis, as well as the reporting of the results. For a successful research the choice of the representative sample is crucial (1). A good sample assures reliability of our results and
conclusions that arise from them. By analysing some sample characteristics we actually want to have an overview of the situation in the population. In the most cases it is impossible to examine the
population as a whole. Therefore, based on what we find out about our sample using the descriptive analysis we make conclusions for the population as a whole (2). This is the basic concept of, so
called, inferential statistics. What is hereby understood is the fact that everything we conclude about the sample is reliably applicable to the whole population.
Let us presume, for example, that we want to know what the average cholesterol concentration in the population is. To answer this question we select a sample for which we believe that represents the
population and in this sample we calculate the mean of cholesterol concentration. By random selection we have decided for the sample (N = 121) and we determine the mean of cholesterol concentration:
5.7 ± 1.4 mmol/L. In this case we use the arithmetic mean as a point estimate of cholesterol concentration in the population. The thing we want to know is the answer to the question: can we consider
this calculated mean as a good estimate for cholesterol concentration in the population. Which statistical indicator points to that? What does the reliability of this estimate depend on? The
confidence interval (CI) gives the answer to all these questions.
What is confidence interval?
In statistics, for any statistical measure, a confidence interval presents a range of possible values within which, with some certainty, we can find the statistical measure of the population.
Let us go back to the example of the average cholesterol concentration in the population. By random selection we have chosen one sample of 121 individuals and on this sample we have calculated the
mean cholesterol concentration. Subsequently, we repeat the procedure and we take a new sample with a new arithmetic mean. We repeat sampling for 100 times and each time we calculate a certain
arithmetical mean of cholesterol concentration. To each arithmetical mean we give an appropriate confidence interval. Out of a total of hundred related 95%-confidence intervals, 95% of them will
contain the actual arithmetical mean of the population (μ). This is also the most accurate definition of a confidence interval.
As such, confidence interval is a realistic estimate of (in)precision and sample size of certain research (3). Therefore, we can consider confidence interval also as a measure of a sample and
research quality. Many journals therefore require providing the key results with respective confidence intervals (4,5).
Confidence interval is defined by its margins of error. Depending on the confidence level that we choose, the interval margins of error also change. The most used confidence intervals in the
biomedical literature are the 90%, 95%, 99% and not so often 99.9% one. The narrower the margins of an interval are, the higher is the estimate accuracy. The 95% confidence interval is traditionally
the most used interval in the literature and this relates to the generally accepted level of statistical significance P < 0.05. There is a rule for same sized samples: the smaller the confidence
level is, the higher is the estimate accuracy. Let us now see how the range of confidence interval and its margins of error change depending on the confidence level in our example of cholesterol
concentration estimate in the population (Figure 1).
Figure 1. a) 90% confidence interval; b) 95% confidence interval; c) 99% confidence interval of mean cholesterol concentration (N=121)
We can claim with 90% confidence that the cholesterol concentration in the population lies within the interval margins of 5.49–5.91 mmol/L. In other words: if we select randomly hundred times a
sample of 121 individuals and calculate the mean cholesterol concentration and the confidence interval of that estimate in the sample, then in ten out of these hundred samples the confidence interval
will not include the actual mean of the population. The thing we do not know is – which 10 samples are we talking about? Exactly that is what makes our estimate (un)reliable.
If we decide for the 95% confidence, the interval margins are 5.46–5.96 mmol/L and in five randomly selected samples the mean of population will not be included. The largest range belongs to the 99%
confidence (5.38–6.03 mmol/L). The larger the confidence interval is, the higher is the possibility that this interval includes also the mean of cholesterol concentration in the population.
Only the studies with a large sample will give a very narrow confidence interval, which points to high estimate accuracy with a high confidence level.
How to calculate a confidence interval?
A confidence interval can be attributed to almost every statistical measure: to a correlation coefficient (6), to odds ratio (OR) and e.g. to the measures of diagnostic accuracy such as sensitivity,
specificity and some others (7). Although there are some other ways of calculating it, the confidence interval is generally and most frequently calculated using standard error. Standard error is a
standard deviation of sample means, calculated out of hundred random population samples (8).
First we have to determine the confidence level for estimating the mean of a parameter in a population. In other words, we ask ourselves to what extent we accept a wrong estimate. Most often we
decide for the 95% confidence what means that we will allow that only in 5% cases the actual mean of population does not fall into our interval.
Margins of error for a confidence interval are calculated using Z value, standard sample deviation (SD) and sample size (N) according to the formula:
The lower margin of a confidence interval is calculated by deducting the previous formula result from the mean. The upper margin is calculated by adding the formula result to the mean. The definition
of a confidence interval is therefore:
The Z value depends on the chosen confidence level. It should be kept in mind that a confidence interval is accurate only for the samples that follow a normal distribution, whereas it is
approximately accurate for large samples that are not distributed normally. For small samples (N < 30) the t value should be used instead of the Z value in the formula for confidence interval, with
N-1 degrees of freedom (9). The t value comes from the Student’s t-distribution and depends on the sample size. For small samples the t value is higher than the Z value what logically means that the
confidence interval for smaller samples with the same confidence level is larger. Z values for matching 90%, 95%, 99% and 99.9% confidence levels are listed in the Table 1. Many statistical textbooks
contain tables with t values for matching confidence level and different degrees of freedom (1).
Table 1. Z values for the most frequent confidence levels
In our sample example, with the mean cholesterol concentration in the population, the confidence interval would be calculated by using the Z value because of the large size of the sample (N=121) that
is normally distributed.
In our example we would report the mean cholesterol concentration with appropriate 95% confidence interval as 5.7 mmol/L (95% CI = 5.46–5.96).
Due to the fact that today there are a lot of statistical softwares that calculate and provide confidence intervals for the majority of statistical indicators, we shall rarely calculate a confidence
interval manually. However, it is important to know the input based on which the confidence interval is calculated, so we could better understand its meaning and interpretation.
Do the P value and the confidence interval have the same meaning?
The P value and the confidence interval are two complementary statistical indicators. They describe the same thing, but in two different ways and are complementary to each other. The P value
describes probability that the observed phenomenon (deviation) occurred by chance, whereas the confidence interval provides margins of error within which it is possible to expect the value of that
Confidence interval can be calculated for difference or ratio between any two statistical indicators, so we could examine if this difference or ratio is of any statistical significance. Let us go
back to our example of cholesterol concentration in the population to see how the confidence interval can be used for estimating statistical significance of the difference between two means.
The difference in the cholesterol concentration between men and women in our sample is 0.22 mmol/L. Is this difference statistically significant? Do women in our sample have indeed a lower
cholesterol concentration then men, or did the observed difference only occur by chance? The answer to this question gives us the Student t-test. The P value calculated by the t-test is 0.426 proving
that the difference of the cholesterol concentration between men and women is not statistically significant. The same thing, only in a different way, we can see in 95% confidence interval of
difference of mean cholesterol concentration between men and women, which is (-0.322) to (+0.757) (Table 2).
Table 2. Difference in the cholesterol concentration between men and women (N = 121)
Which conclusion can we draw from that confidence interval? Let us remember what the definition of confidence interval was: it defines margins of error within which we can expect the actual value
with 95% confidence. Our confidence interval contains also a zero (0) meaning that it is quite possible that the actual value of the difference will equal zero, namely, that there is no difference
between the cholesterol concentration between men and women.
How do we define the confidence interval when it is related to a ratio as, for example, in OR? Let us imagine that we have assessed the degree of carotid artery stenosis for all our individuals from
the sample (N = 121) using echo-colour Doppler sonography analysis. We divide them into two groups: the ones with the absence of carotid artery stenosis and the others with stenosis (> 50% lumen) of
at least one carotid artery. We are interested whether those two groups are different in average cholesterol concentration and is the cholesterol concentration a risk factor for development of
carotid artery stenosis. The answer to this question gives us the OR (Table 3).
Table 3. OR and 95% confidence interval of cholesterol concentration in individuals with/without carotid artery stenosis
OR is higher than 1, but the confidence interval for OR includes the number 1 as well. What does it mean? It means that the odds of cholesterol concentration being and not being a risk factor for
carotid artery stenosis are even. That is, for any cholesterol concentration the odds of that person having or not having carotid artery stenosis are the same. This also confirms the percentage of
correctly classified individuals considering the cholesterol concentration (50.41%). Only half of the individuals are correctly categorised in an adequate group – so the selection does not depend on
cholesterol concentration but on a pure chance.
Confidence interval can be attributed to almost every statistical measure. In the last twenty years, every day there are more journals that require reporting of the confidence intervals for each of
their key results. Reporting of this confidence interval provides additional information about the sample and the results. It is, moreover, above all useful and irreplaceable supplement to a
classical hypothesis testing and to the generally accepted P value. It should become a standard of all scientific journals to report key results with respective confidence intervals because it
enables better understanding of the data to an interested reader. | {"url":"https://biochemia-medica.com/en/journal/18/2/10.11613/BM.2008.015","timestamp":"2024-11-14T12:16:41Z","content_type":"text/html","content_length":"84113","record_id":"<urn:uuid:d6d19a09-1e87-4c44-8d3a-5a68756d2910>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00221.warc.gz"} |
Average Monthly Payment
• • The average monthly student loan payment is $393.
• • 33% of student loan borrowers pay between $200 and $299 per month.
• • 11% of student loan borrowers pay between $300 and $399 per month.
• • The median monthly student loan payment is $222.
• • 22% of student loan borrowers pay over $500 per month.
• • 29% of borrowers pay between $100 and $199 per month.
• • The average monthly payment for student loans is $393.
• • Women pay more than men on average for student loan payments.
• • 60% of borrowers pay less than $300 per month on student loans.
• • 19% of borrowers pay between $400 and $499 per month.
• • The average monthly student loan payment for borrowers aged 20 to 30 is $351.
• • 42% of borrowers are in an income-driven repayment plan.
• • The average monthly student loan payment for borrowers aged 31 to 40 is $406.
• • 38% of borrowers are in standard repayment plans.
• • 7.8% of student loan borrowers pay over $1000 per month.
Move over avocado toast, student loan payments are the new millennial budget buster! With the average monthly payment for student loans clocking in at a hefty $393, it seems like we might need to
start swapping lattes for financial austerity. From gender disparities to age breakdowns, and the surprising number of borrowers in income-driven repayment plans, this blog post dives deep into the
student loan payment landscape. So grab your calculators and prepare to crunch the numbers with a side of wit and wisdom!
$100 $199
• 29% of borrowers pay between $100 and $199 per month.
It seems that for nearly a third of student loan borrowers, their monthly payments fall somewhere between the cost of a fancy dinner and a weekend shopping spree. While this amount may not break the
bank for some, it highlights the ongoing financial burden that many face in pursuing higher education. The numbers may seem manageable, but the toll they take on individuals' wallets and overall
economic well-being should not be overlooked.
$186 falls into the category "Less than $200"
• The average monthly payment for student loan borrowers with high school diplomas is $186.
The average monthly payment for student loan borrowers with just a high school diploma is about $186. Perhaps this is the universe's way of reminding us that while knowledge may be priceless, the
cost of education can definitely add up. It serves as a reminder that investing in one's education can open doors to opportunities, but it comes with a price tag that many young individuals are
grappling with. So, as we navigate the intricacies of student loan debt, let's remember that with a little bit of wit and a whole lot of determination, we can chip away at those numbers like the
savvy scholars we are.
$200 $299
• 33% of student loan borrowers pay between $200 and $299 per month.
• The median monthly student loan payment is $222.
• 60% of borrowers pay less than $300 per month on student loans.
• The average monthly payment for private student loans is $228.
• 30% of student loan borrowers pay between $200 and $299 per month.
• 28% of student loan borrowers pay between $200 and $299 per month.
• The average monthly student loan payment for borrowers with associate degrees is $231.
• 41% of student loan borrowers pay less than $300 per month.
• The average monthly payment for student loan borrowers with some college education but no degree is $268.
• 36% of student loan borrowers pay between $200 and $299 per month.
In a swirling sea of statistics and dollar signs, one thing is clear: student loan payments are as common as complaints about the cafeteria food. While some may be cruising comfortably with $200
monthly installments, others find themselves diving deeper into debt at $300 and beyond. It's a financial rite of passage for many, a monthly dance with the devil dressed in interest rates and
repayment schedules. The numbers may vary, but the struggle is universal as graduates navigate the choppy waters of student loan repayment. Welcome to the club, where the membership fee is more than
just a diploma.
$300 $399
• The average monthly student loan payment is $393.
• 11% of student loan borrowers pay between $300 and $399 per month.
• The average monthly payment for student loans is $393.
• The average monthly student loan payment for borrowers aged 20 to 30 is $351.
• 42% of borrowers are in an income-driven repayment plan.
• 38% of borrowers are in standard repayment plans.
• The average monthly payment for federal student loans is $393.
• 25% of student loan borrowers pay between $300 and $399 per month.
• The average monthly student loan payment for borrowers aged 61 and older is $396.
• The average monthly payment for student loan borrowers with graduate degrees is $393.
• The average monthly payment for borrowers with bachelor's degrees is $351.
• 33% of student loan borrowers pay between $300 and $399 per month.
In a world where student loan payments have become as ubiquitous as avocado toast, the numbers reveal a complex financial puzzle for borrowers of all ages and education levels. The statistics paint a
picture of a generation juggling monthly payments akin to a high-stakes game of financial Tetris, with some managing to fit comfortably in the $300 to $399 bracket while others struggle to keep up.
As income-driven and standard repayment plans vie for attention, the average monthly payment of $393 looms large over borrowers, with age and education level adding intriguing twists to the
narrative. It seems that in the game of student loans, everyone's playing, but not everyone's winning.
$400 $499
• Women pay more than men on average for student loan payments.
• 19% of borrowers pay between $400 and $499 per month.
• The average monthly student loan payment for borrowers aged 31 to 40 is $406.
• The average monthly student loan payment for borrowers aged 41 to 50 is $471.
• 17% of student loan borrowers pay between $400 and $499 per month.
In the symphony of student loan payments, it seems women are hitting a higher note than men, with statistics revealing they carry a heavier financial burden in the form of larger monthly payments. As
borrowers in their 30s play the tune of $406 on average, their counterparts in their 40s seem to be improvising with a $471 melody. Perhaps it's time for a remix in the student loan orchestra to
ensure everyone is harmonizing on an equal playing field.
$500 and above
• 22% of student loan borrowers pay over $500 per month.
• 7.8% of student loan borrowers pay over $1000 per month.
• 13% of student loan borrowers pay between $500 and $599 per month.
• The average monthly student loan payment for borrowers aged 51 to 60 is $506.
• 8% of student loan borrowers pay over $1000 per month.
• 21% of student loan borrowers pay over $500 per month.
• The average monthly payment for student loan borrowers in the healthcare field is $724.
• 7% of student loan borrowers pay over $1000 per month.
Well, it seems like the student loan payment landscape is as varied as a university cafeteria menu. From the affordable ramen noodle budget of $500 to the gourmet lobster tail expense of over $1000
per month, student loan borrowers are navigating a range of financial flavors. Whether you're aged 51 to 60 and dishing out an average of $506 a month or working in the healthcare field where the
bill climbs to $724, one thing is clear - student loan debt is no fast food drive-thru order; it's a full-course financial feast that many are struggling to digest.
$821 falls into the category of $500 and above
• The average monthly payment for student loan borrowers in the legal field is $821.
In the legal field, the average monthly student loan payment of $821 is a stark reminder that the cost of knowledge often comes with a hefty price tag. While these lawyers may be skilled at
negotiating in the courtroom, navigating their own financial obligations requires a different set of skills. It seems that in the world of law, even the scales of student debt tip heavily in favor of
the lenders.
Less than $200
• 39% of student loan borrowers pay less than $200 per month.
• 32% of student loan borrowers pay less than $200 per month.
In a startling revelation that highlights the precarious financial situation many Americans find themselves in, it appears that a significant portion of student loan borrowers are only able to afford
payments that are less than the price of a monthly gym membership. With 39% and 32% of borrowers struggling to make ends meet by shelling out less than $200 each month towards their education debt,
it's clear that the burden of student loans is weighing heavily on the shoulders of a large segment of our population. It's a stark reminder of the urgent need for more accessible education financing
options and comprehensive debt relief measures. | {"url":"https://gitnux.org/average-monthly-payment-for-student-loans/","timestamp":"2024-11-07T16:00:13Z","content_type":"text/html","content_length":"101046","record_id":"<urn:uuid:424e37b6-37fd-4713-8ae8-1de9662c5b02>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00855.warc.gz"} |
Jonathan J. Heckman
You have reached the website of Jonathan Heckman
I am an Associate Professor in the Department of Physics and Astronomy at the University of Pennsylvania.
I mainly work on aspects of high energy theoretical physics.
More details:
Dramatic progress in string theory over the past few decades has produced a suite of powerful tools for addressing conceptual questions connected with quantum field theory and quantum gravity. As an
added bonus, string theory also provides a framework for motivated stringy extensions of particle physics. Combining these threads is of central importance in determining the mathematical
consistency of string theory, and its potential role as a theory of the real world.
The broad questions my work aims to address are:
What is quantum field theory, and what are its limitations as a tool in describing Nature?
What are the building blocks of matter and spacetime?
String theory provides a promising way to address these issues since it is the only known viable approach to unifying quantum theory with gravity.
Currently, my efforts are concentrated in three directions:
1) The study of quantum field theory using the extra dimensions of string theory.
2) The study of formal and phenomenological aspects of string compactification, and in particular F-theory.
3) Conceptual questions connected with the embedding of field theoretic UV cutoffs in string theory. | {"url":"https://www.jjheckman.com/","timestamp":"2024-11-07T20:03:28Z","content_type":"text/html","content_length":"27023","record_id":"<urn:uuid:e8a01c3b-3f6c-49b9-831a-c12657a1565e>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00582.warc.gz"} |
Supply Chain View
Storage capacity calculator
This is a simple storage capacity calculator, similar to something I put together a few years ago for warehouse design projects. It takes a set of product dimensions, a list of possible storage
modules (pallets, stillages, bins, shelves, etc.) of different sizes, and calculates the number of products that will fit in each module. Providing a desired quantity to store allows the calculator
to suggest the best module.
I have not tested this to destruction, so I would appreciate any comments. Please see the notes on use below.
Credit should go to my colleague John Bradon who came up with the original algorithm.
Click here to open the Storage Capacity Calculator (in a new window).
(The user interface for this widget was developed using the Ext framework.)
Notes on use
The quick-start guide: enter the dimensions of a product, optionally a product weight, and the quantity you are aiming to store. Click on the Calculate button and see the module capacities calculated
and the best module highlighted.
Data security
Any data that you enter in the calculator stays on your PC – no data is transmitted over the internet, no data comes to Supply Chain View.
Metric/US measure?
You will note that there are no units of measure when specifying dimensions. You can use any units of measure as long as you are consistent. The Lin Bin examples are in mm.
Weights and loading
If you omit product weights, then the calculator ignores loadings. If you specify a weight you should specify the loading limits of each module. The calculator will respect the loading limits when
calculating the capacities.
Product dimension assumptions
The calculator assumes the product is a basic, regular cuboid, such as a cardboard carton. When I have more time I will test and upload an improved version of the calculator which deals with
cylindrical products and products that stack (such as buckets).
Default module list
I have provided a few standard modules as a demonstration, but you can use the Delete Module and New Module buttons to remove these and add new modules. Double click on the names and dimensions of
the modules to edit them. Remember to use the internal dimensions of the modules.
What quantity to store?
This deserves a quick explanatory post of its own. In brief, for small parts or slow-moving inventory this will be the maximum stock level you aim to hold. For faster movers the planned standard
order quantity is a good number to start with (and once you have the capacity of a pallet that can feed back into the planned order quantity calculation…)
How do I do this for all my products?
Aha! This is just a little web widget, so the answer is, you can’t. It is possible to write this kind of logic in Excel for example, or Access, but you will need some help. I may at some point
upgrade the calculator to allow people to upload a list of products and dimensions in a comma-delimited format (easy to export from Excel) and generate output in the same way – watch this space.
How it works
The capacity calculation takes into account all the possible orientations of a product (there are 6 for a cuboid carton) and finds the best fit. It also takes any free space and works out whether we
can fit any more products in that space if we rotate them by 90 degrees. In this way it is a true calculation of capacity, something that a simple cubic volume calculation will overestimate
(drastically if the product is large compared with the module).
Comment from Stewart Arbuckle
Time 20 November 2007 at 8:29 am
We deal with the warehousing sector and if your software was easy to work with I would trial it on my website. Let me know if this is of interest? I did try using it aboe, but could not fully get the
most of it.
Comment from Vaibhav
Time 10 April 2008 at 6:26 am
better then others | {"url":"http://www.supplychainview.com/blog/storage-capacity-calculator/","timestamp":"2024-11-08T17:40:41Z","content_type":"application/xhtml+xml","content_length":"21596","record_id":"<urn:uuid:60d5422a-4a6d-49fe-9c53-a713c2fdb1d1>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00820.warc.gz"} |
Consider the line y=8x-2. What is the equation of the line that is parallel to this line and passes through the point (-5,6)? | Socratic
Consider the line #y=8x-2#. What is the equation of the line that is parallel to this line and passes through the point (-5,6)?
1 Answer
All lines with the same slope are parallel to each other.
All lines with a slope $8$ have equation
$y = 8 x + C$
where $C$ is some constant.
Let's find a constant $C$ such that point $\left(- 5 , 6\right)$ lies on a line:
$6 = 8 \left(- 5\right) + C$
from which follows:
$C = 46$
Therefore, line $y = 8 x + 46$ is parallel to a given line $y = 8 x - 2$ and passes through point $\left(- 5 , 6\right)$.
Impact of this question
12094 views around the world | {"url":"https://socratic.org/questions/consider-the-line-y-8x-2-what-is-the-equation-of-the-line-that-is-parallel-to-th#193352","timestamp":"2024-11-03T09:58:48Z","content_type":"text/html","content_length":"34058","record_id":"<urn:uuid:5b691f30-05ad-4b10-8b89-67c54416cd4b>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00373.warc.gz"} |
Standing Waves
1. Draw a sine wave. On this graph, indicate what the Amplitude
and Period...
Standing Waves 1. Draw a sine wave. On this graph, indicate what the Amplitude and Period...
Standing Waves
1. Draw a sine wave. On this graph, indicate what the Amplitude and Period are. If someone just gave you this graph, how could you find the Frequency?
2. Imagine that two water waves moving in opposite directions run into each other. What will the resulting wave look like?
3. When 2 waves interfere, can the resulting wave have a larger amplitude than either of the two original waves? When?
4. What is the definition of a node?
5. For musical instruments, the sound waves produced are of a certain frequency. This frequency corresponds to the pitch of the notes that your ears hear. How can one instrument make so many
different sounds? | {"url":"https://justaaa.com/physics/383531-standing-waves-1-draw-a-sine-wave-on-this-graph","timestamp":"2024-11-11T10:35:02Z","content_type":"text/html","content_length":"37234","record_id":"<urn:uuid:a1decc6d-bf8f-45d2-aced-8f16ebe620be>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00323.warc.gz"} |
6.6.5. Generalised derived instances for newtypes
6.6.5. Generalised derived instances for newtypes¶
Since: 6.8.1. British spelling since 8.6.1.
Enable GHC’s cunning generalised deriving mechanism for newtypes
When you define an abstract type using newtype, you may want the new type to inherit some instances from its representation. In Haskell 98, you can inherit instances of Eq, Ord, Enum and Bounded by
deriving them, but for any other classes you have to write an explicit instance declaration. For example, if you define
newtype Dollars = Dollars Int
and you want to use arithmetic on Dollars, you have to explicitly define an instance of Num:
instance Num Dollars where
Dollars a + Dollars b = Dollars (a+b)
All the instance does is apply and remove the newtype constructor. It is particularly galling that, since the constructor doesn’t appear at run-time, this instance declaration defines a dictionary
which is wholly equivalent to the Int dictionary, only slower!
DerivingVia (see Deriving via) is a generalization of this idea.
6.6.5.1. Generalising the deriving clause¶
GHC now permits such instances to be derived instead, using the extension GeneralizedNewtypeDeriving, so one can write
newtype Dollars = Dollars { getDollars :: Int } deriving (Eq,Show,Num)
and the implementation uses the same Num dictionary for Dollars as for Int. In other words, GHC will generate something that resembles the following code
instance Num Int => Num Dollars
and then attempt to simplify the Num Int context as much as possible. GHC knows that there is a Num Int instance in scope, so it is able to discharge the Num Int constraint, leaving the code that GHC
actually generates
One can think of this instance being implemented with the same code as the Num Int instance, but with Dollars and getDollars added wherever necessary in order to make it typecheck. (In practice, GHC
uses a somewhat different approach to code generation. See the A more precise specification section below for more details.)
We can also derive instances of constructor classes in a similar way. For example, suppose we have implemented state and failure monad transformers, such that
instance Monad m => Monad (State s m)
instance Monad m => Monad (Failure m)
In Haskell 98, we can define a parsing monad by
type Parser tok m a = State [tok] (Failure m) a
which is automatically a monad thanks to the instance declarations above. With the extension, we can make the parser type abstract, without needing to write an instance of class Monad, via
newtype Parser tok m a = Parser (State [tok] (Failure m) a)
deriving Monad
In this case the derived instance declaration is of the form
instance Monad (State [tok] (Failure m)) => Monad (Parser tok m)
Notice that, since Monad is a constructor class, the instance is a partial application of the newtype, not the entire left hand side. We can imagine that the type declaration is “eta-converted” to
generate the context of the instance declaration.
We can even derive instances of multi-parameter classes, provided the newtype is the last class parameter. In this case, a “partial application” of the class appears in the deriving clause. For
example, given the class
class StateMonad s m | m -> s where ...
instance Monad m => StateMonad s (State s m) where ...
then we can derive an instance of StateMonad for Parser by
newtype Parser tok m a = Parser (State [tok] (Failure m) a)
deriving (Monad, StateMonad [tok])
The derived instance is obtained by completing the application of the class to the new type:
instance StateMonad [tok] (State [tok] (Failure m)) =>
StateMonad [tok] (Parser tok m)
As a result of this extension, all derived instances in newtype declarations are treated uniformly (and implemented just by reusing the dictionary for the representation type), except Show and Read,
which really behave differently for the newtype and its representation.
It is sometimes necessary to enable additional language extensions when deriving instances via GeneralizedNewtypeDeriving. For instance, consider a simple class and instance using UnboxedTuples
{-# LANGUAGE UnboxedTuples #-}
module Lib where
class AClass a where
aMethod :: a -> (# Int, a #)
instance AClass Int where
aMethod x = (# x, x #)
The following will fail with an “Illegal unboxed tuple” error, since the derived instance produced by the compiler makes use of unboxed tuple syntax,
{-# LANGUAGE GeneralizedNewtypeDeriving #-}
import Lib
newtype Int' = Int' Int
deriving (AClass)
However, enabling the UnboxedTuples extension allows the module to compile. Similar errors may occur with a variety of extensions, including:
6.6.5.2. A more precise specification¶
A derived instance is derived only for declarations of these forms (after expansion of any type synonyms)
newtype T v1..vn = MkT (t vk+1..vn) deriving (C t1..tj)
newtype instance T s1..sk vk+1..vn = MkT (t vk+1..vn) deriving (C t1..tj)
• v1..vn are type variables, and t, s1..sk, t1..tj are types.
• The (C t1..tj) is a partial applications of the class C, where the arity of C is exactly j+1. That is, C lacks exactly one type argument.
• k is chosen so that C t1..tj (T v1...vk) is well-kinded. (Or, in the case of a data instance, so that C t1..tj (T s1..sk) is well kinded.)
• The type t is an arbitrary type.
• The type variables vk+1...vn do not occur in the types t, s1..sk, or t1..tj.
• C is not Read, Show, Typeable, or Data. These classes should not “look through” the type or its constructor. You can still derive these classes for a newtype, but it happens in the usual way, not
via this new mechanism. Confer with Default deriving strategy.
• It is safe to coerce each of the methods of C. That is, the missing last argument to C is not used at a nominal role in any of the C‘s methods. (See Roles.)
• C is allowed to have associated type families, provided they meet the requirements laid out in the section on GND and associated types.
Then the derived instance declaration is of the form
instance C t1..tj t => C t1..tj (T v1...vk)
Note that if C does not contain any class methods, the instance context is wholly unnecessary, and as such GHC will instead generate:
instance C t1..tj (T v1..vk)
As an example which does not work, consider
newtype NonMonad m s = NonMonad (State s m s) deriving Monad
Here we cannot derive the instance
instance Monad (State s m) => Monad (NonMonad m)
because the type variable s occurs in State s m, and so cannot be “eta-converted” away. It is a good thing that this deriving clause is rejected, because NonMonad m is not, in fact, a monad — for the
same reason. Try defining >>= with the correct type: you won’t be able to.
Notice also that the order of class parameters becomes important, since we can only derive instances for the last one. If the StateMonad class above were instead defined as
class StateMonad m s | m -> s where ...
then we would not have been able to derive an instance for the Parser type above. We hypothesise that multi-parameter classes usually have one “main” parameter for which deriving new instances is
most interesting.
Lastly, all of this applies only for classes other than Read, Show, Typeable, and Data, for which the stock derivation applies (section 4.3.3. of the Haskell Report). (For the standard classes Eq,
Ord, Ix, and Bounded it is immaterial whether the stock method is used or the one described here.)
6.6.5.3. Associated type families¶
GeneralizedNewtypeDeriving also works for some type classes with associated type families. Here is an example:
class HasRing a where
type Ring a
newtype L1Norm a = L1Norm a
deriving HasRing
The derived HasRing instance would look like
instance HasRing (L1Norm a) where
type Ring (L1Norm a) = Ring a
To be precise, if the class being derived is of the form
class C c_1 c_2 ... c_m where
type T1 t1_1 t1_2 ... t1_n
type Tk tk_1 tk_2 ... tk_p
and the newtype is of the form
newtype N n_1 n_2 ... n_q = MkN <rep-type>
then you can derive a C c_1 c_2 ... c_(m-1) instance for N n_1 n_2 ... n_q, provided that:
• The type parameter c_m occurs once in each of the type variables of T1 through Tk. Imagine a class where this condition didn’t hold. For example:
class Bad a b where
type B a
instance Bad Int a where
type B Int = Char
newtype Foo a = Foo a
deriving (Bad Int)
For the derived Bad Int instance, GHC would need to generate something like this:
instance Bad Int (Foo a) where
type B Int = B ???
Now we’re stuck, since we have no way to refer to a on the right-hand side of the B family instance, so this instance doesn’t really make sense in a GeneralizedNewtypeDeriving setting.
• C does not have any associated data families (only type families). To see why data families are forbidden, imagine the following scenario:
class Ex a where
data D a
instance Ex Int where
data D Int = DInt Bool
newtype Age = MkAge Int deriving Ex
For the derived Ex instance, GHC would need to generate something like this:
instance Ex Age where
data D Age = ???
But it is not clear what GHC would fill in for ???, as each data family instance must generate fresh data constructors.
If both of these conditions are met, GHC will generate this instance:
instance C c_1 c_2 ... c_(m-1) <rep-type> =>
C c_1 c_2 ... c_(m-1) (N n_1 n_2 ... n_q) where
type T1 t1_1 t1_2 ... (N n_1 n_2 ... n_q) ... t1_n
= T1 t1_1 t1_2 ... <rep-type> ... t1_n
type Tk tk_1 tk_2 ... (N n_1 n_2 ... n_q) ... tk_p
= Tk tk_1 tk_2 ... <rep-type> ... tk_p
Again, if C contains no class methods, the instance context will be redundant, so GHC will instead generate instance C c_1 c_2 ... c_(m-1) (N n_1 n_2 ... n_q).
Beware that in some cases, you may need to enable the UndecidableInstances extension in order to use this feature. Here’s a pathological case that illustrates why this might happen:
class C a where
type T a
newtype Loop = MkLoop Loop
deriving C
This will generate the derived instance:
instance C Loop where
type T Loop = T Loop
Here, it is evident that attempting to use the type T Loop will throw the typechecker into an infinite loop, as its definition recurses endlessly. In other cases, you might need to enable
UndecidableInstances even if the generated code won’t put the typechecker into a loop. For example:
instance C Int where
type C Int = Int
newtype MyInt = MyInt Int
deriving C
This will generate the derived instance:
instance C MyInt where
type T MyInt = T Int
Although typechecking T MyInt will terminate, GHC’s termination checker isn’t sophisticated enough to determine this, so you’ll need to enable UndecidableInstances in order to use this derived
instance. If you do go down this route, make sure you can convince yourself that all of the type family instances you’re deriving will eventually terminate if used!
Note that DerivingVia (see Deriving via) uses essentially the same specification to derive instances of associated type families as well (except that it uses the via type instead of the underlying
rep-type of a newtype). | {"url":"https://downloads.haskell.org/ghc/9.0.2/docs/html/users_guide/exts/newtype_deriving.html","timestamp":"2024-11-04T05:25:44Z","content_type":"text/html","content_length":"52872","record_id":"<urn:uuid:ced8a22d-e234-413f-8d10-5971defadd01>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00526.warc.gz"} |
suppose you want to buy a new car and trying to choose between two models model a costs 15 000 and its gas mileage is 25 miles per gallon and its insurance is 200 per year model b costs 22 000 and its gas mileage is 50 miles per gallon and its ins - Cheap Nursing Writers
Suppose you want to buy a new car and trying to choose between two models:
• Model A: costs $15,000 and its gas mileage is 25 miles per gallon and its insurance is $200 per year.
• Model B: costs $22,000 and its gas mileage is 50 miles per gallon and its insurance is $400 per year.
If you drive approximately 40,000 miles per year and the gas costs $3 per gallon:
• Find a formula for the total cost of owning Model A where the number of years is the independent variable.
• Find a formula for the total cost of owning Model B where the number of years is the independent variable.
• Find the total cost for each model for the first five years.
• If you plan to keep the car for four years, which model is more economical? How about if you plan to keep it for six years?
• Find the number of years in which the total cost to keep the two cars will be the same.
• Identify the number of months where neither car holds a cost of ownership advantage.
• What effect would the cost of gas doubling have on cost of ownership? Graph or show hand calculations.
• If you can sell neither car for 40% of its value at any time, how does the analysis change? Graph or show hand calculations.
1. Paper must be written in 3rd person.
2. Your paper should be 4-5 pages in length (counting the title page and references page) and cite and integrate at least one credible outside source. The CSU-Global Library (Links to an external
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3. Include a title page, introduction, body, conclusion, and a reference page.
4. The introduction should describe or summarize the topic or problem. It might discuss the importance of the topic or how it affects you or society as a whole, or it might discuss or describe the
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Printable Math Worksheets For 6Th Grade
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A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms
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(ed. D.D. Kouvatsos), River Publishers, (2009), 215-243.
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and Next Generation Internet, Lecture Notes in Computer Science," 5233 (2011), 357-392.doi: 10.1007/978-3-642-02742-0.
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LNCS issue in Honour of Guenter Haring," University of Vienna, 2011, to appear.
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[66] "Tsallis Statistics, Statistical Mechanics for Non-extensive Systems and Long-Range Interactions,", Notebook, (2007). | {"url":"https://www.aimsciences.org/article/doi/10.3934/naco.2011.1.781","timestamp":"2024-11-03T06:00:23Z","content_type":"text/html","content_length":"143632","record_id":"<urn:uuid:dac48745-e7f3-4828-b246-869338dd31c2>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00811.warc.gz"} |
Lesson 16
Interpreting Inequalities
Problem 1
Priya looks at the inequality \(12-x>5\) and says “I subtract a number from 12 and want a result that is bigger than 5. That means that the solutions should be values of \(x\) that are smaller than
Do you agree with Priya? Explain your reasoning and include solutions to the inequality in your explanation.
Problem 2
When a store had sold \(\frac25\) of the shirts that were on display, they brought out another 30 from the stockroom. The store likes to keep at least 150 shirts on display. The manager wrote the
inequality \(\frac35x+30 \geq 150\) to describe the situation.
1. Explain what \(\frac35\) means in the inequality.
2. Solve the inequality.
3. Explain what the solution means in the situation.
Problem 3
You know \(x\) is a number less than 4. Select all the inequalities that must be true.
Problem 4
Here is an unbalanced hanger.
1. If you knew each circle weighed 6 grams, what would that tell you about the weight of each triangle? Explain your reasoning.
2. If you knew each triangle weighed 3 grams, what would that tell you about the weight of each circle? Explain your reasoning.
Problem 5
Match each sentence with the inequality that could represent the situation.
Problem 6
At a skateboard shop:
1. The price tag on a shirt says $12.58. Sales tax is 7.5% of the price. How much will you pay for the shirt?
2. The store buys a helmet for $19.00 and sells it for $31.50. What percentage was the markup?
3. The shop pays workers $14.25 per hour plus 5.5% commission. If someone works 18 hours and sells $250 worth of merchandise, what is the total amount of their paycheck for this pay period? Explain
or show your reasoning. | {"url":"https://im.kendallhunt.com/MS/teachers/2/6/16/practice.html","timestamp":"2024-11-13T17:41:07Z","content_type":"text/html","content_length":"74430","record_id":"<urn:uuid:54f775b8-97ad-4426-8765-95cc3cf34e8c>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00501.warc.gz"} |
General Term: Definition, Examples
The general term (sometimes called the nth term) is a formula that defines a sequence.
For example, for the sequence defined by a[n] = 1/n, the first four terms are found by plugging in 1, 2, 3, 4 for “n”:
1/1, 1/2, 1/3 and 1/4.
Different sequences have different formulas.
The general term is one way to define a sequence. The other way is the recursive definition of a sequence, which defines terms by way of other terms. For example, A[n] = A[n-1] + 4.
General Term for Arithmetic Sequences
The general term for an arithmetic sequence is a[n] = a[1] + (n – 1) d, where d is the common difference.
Example question: What is the general term of the sequence 2, 5, 8,…?
1. Find d by subtracting the second from first term: d = 5 – 2 = 3.
2. Plug d into the general formula: a[n] = a[1] + (n – 1) 3
3. Plug in the first term for a[1]: a[n] = 2 + (n – 1) 3
The general term is a[n] = 2 + (n – 1) 3
General Term for Geometric Sequences
For a geometric sequence, the formula is a[n] = a[1] r^n – 1, where r is the common ratio.
Example question: What is the general term of the geometric sequence 8, 4, 2,…?
1. Find r the ratio of any two consecutive terms. I’ll use the second and third terms in this example: r = 2/4 = ½.
2. Plug r into the general formula: a[n] = a[1] ½^n – 1
3. Plug in the first term for a[1]: a[n] = 8 (½)^n – 1
The general term is a[n] = 8 (½)^n – 1.
Using Algebra
If you don’t know what kind of sequence you have, you may have to use a little logic and your knowledge of algebra to get your terms into a workable form. For example, let’s say you’re asked to find
the general term for the sequence
This becomes much easier to work with if you change the denominators into exponents (giving an exponential sequence):
Which gives a[n] = ½^n
Steig, J. 1999. Sequences & Series – General terms. Retrieved January 20, 2021 from: http://www.mesacc.edu/~marfv02121/readings/seq_def/
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Each polio vaccination consist of four doses, and each measles vaccination consist of two doses. Last year, Dr. Potter gave a total of 60 vaccinations that consisted of a total 184 doses.
To solve this equation we need to set up a system of equations because we need to find out the value of 2 variables given 2 equations
Equation 1: Polio vaccination has 4 doses, measles vaccination has 2 doses, and he gave out 184 doses total
Equation 2: He gave 2 types of vaccinations: polio and measles, and he gave a total of 60 vaccinations.
To convert the word form of these equations, we first have assign polio and measles a variable to make them easier to keep track of
Lets just say that the number of polio shots = x and number of measles shots = y
Equation 1: 4 doses of x added with 2 doses of y equaled 184 doses total
4x+2y = 184
Equation 2: x vaccines added with y vaccines equaled 60 total vaccines
x + y = 60
We know have our system of equations
Equation 1: 4x+2y = 184
Equation 2: x + y = 60
To solve a system of equations, we can either do elimination or substitution, depending on which is easier.
In this case, substitution is pretty easy because for equation 2 we just need to have only 1 variable on a side then we can substitute it into equation 1
Equation 2: x + y = 60
Lets subtract x from both sides of the equation to "move" the x to the other side
y = - x + 60
And using y in terms of x, we can plug it into equation 1
Equation 1: 4x+2y = 184
4x+2 (-x + 60) = 184
Now distribute the 2 to (-x+60)
4x-2x+120 = 184
Simplify by combining like terms
2x+120 = 184
Subtract 120 from both sides of the equation so all variables are on one side and the constants on the other
2x = 64
Divide 2 from both sides of the equation to isolate x, when we want to find
x = 32
Now we know the number of polio vaccines he gave, plug this into an equation to find y - the number of measles vaccine
Equation 2: x + y = 60
32 + y = 60
Subtract 32 from both sides to isolate y
y = 28
We know now that x, the number of polio vaccinations = 32 and y, the number of measles vaccinations = 28
Dr. Potter gave 32 polio vaccinations and 28 measles vaccinations last year. | {"url":"https://cpep.org/mathematics/1906708-each-polio-vaccination-consist-of-four-doses-and-each-measles-vaccinat.html","timestamp":"2024-11-02T15:01:18Z","content_type":"text/html","content_length":"25868","record_id":"<urn:uuid:fe0a317f-3ffe-49cd-9f27-a199754298ff>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00807.warc.gz"} |
Fundamentals - Cone Reference
Let's start with the fundamental capabilities one might find in almost any program. To keep it simple, we focus first on how to read and write programs that work with numbers, such as this one which
calculates the value of pi:
import stdio::*
fn main():
imm pi = calcpi(10)
print <- "\u03C0=", calcpi(10), "\n" // <-- Change 10 to 2 for less accurate pi
// Calculate pi using arc-sine fractional sequence.
// 'nterms' is the number of fractional terms used to estimate pi
// https://en.wikipedia.org/wiki/Approximations_of_%CF%80#Arcsine
fn calcpi(mut nterms = 10) f64:
if nterms <= 0:
return 0d
// Initialize working values
mut result = 0.5d
mut seed = 1d
mut top = 1d
mut bot = 1d
mut twos = 2d
mut term = 0d
while --nterms:
// Calc a new fraction and add to result
top *= seed
bot *= seed + 1d
twos *= 2d * 2d
term = top / (bot * (seed + 2d) * twos)
result += term
seed += 2d
result * 6d
Over the next few pages, the story will unfold in this way: We begin by introducing the primitive values: numbers and variables. Using these values and various operators, we can then form expressions
that calculate new values. Multiple expressions and other statements can then be assembled into a block, which evaluates its statements in order. One or more blocks can define the logic of a
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RWM101: Foundations of Real World Math
Now that you are comfortable reducing fractions into their lowest terms, we want to build some familiarity with rewriting fractions. Specifically, we want to rewrite two fractions so that they have
the same denominator.
Consider the fractions $3/12$ and $4/12$. These two expressions have a common denominator (12). Now practice your fraction-reducing skills from the previous section to note that:
$\frac{3}{12} = \frac{1}{4}$ and $\frac{4}{12} = \frac{1}{3}$
The fractions $1/3$ and $1/4$ can each be re-expressed so that they have a common denominator of 12. Something similar is also true for the fractions $20/60$ and $24/60$; they have a common
denominator of $60$, and each of these fractions reduces to $1/3$ and $2/5$.
The process of finding a single common denominator for two fractions is, in some sense, the reverse of what we have just discussed: start with two different-denominator fractions and then rewrite
them so they have the same denominator.
The video you will watch next discusses how to find "the least common denominator". For our two previous examples, the least common denominator of $1/4$ and $1/3$ is 12, but the least common
denominator of $1/3$ and $2/5$ is 15 (not 60, the larger denominator we used above). Indeed, instead of writing:
$\frac{1}{3} = \frac{20}{60}$ and $\frac{2}{5} = \frac{24}{60}$,
we could have written:
$\frac{1}{3} = \frac{5}{15}$ and $\frac{2}{5} = \frac{6}{15}$.
To find the least common denominator, you will use the least common multiple of the denominators, a nice callback to section 2.2. | {"url":"https://learn.saylor.org/course/view.php?id=38§ion=21","timestamp":"2024-11-06T18:24:24Z","content_type":"text/html","content_length":"826376","record_id":"<urn:uuid:7d1aa681-4b8a-4f6e-af4d-b7e8c608c109>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00237.warc.gz"} |
What number is MMXXX? - The Roman numeral MMXXX as normal numbers
What number is MMXXX?
Your question is: What numbers are the Roman numerals MMXXX? Learn how to convert the Roman numerals MMXXX into the correct translation of normal numbers.
The Roman numerals MMXXX are identical to the number 2030.
MMXXX = 2030
How do you convert MMXXX into normal numbers?
In order to convert MMXXX into numbers, the number of position values (ones, tens, hundreds, thousands) is subdivided as follows:
Place value Number Roman numbers
Conversion 2000 + 30 MM + XXX
Thousands 2000 MM
Dozens 30 XXX
How do you write MMXXX in numbers?
To correctly write MMXXX as normal numbers, combine the converted Roman numbers. The highest numbers must always be in front of the lowest numbers to get the correct translation, as in the table
2000+30 = (MMXXX) = 2030
The next Roman numerals = MMXXXI
Convert another Roman numeral to normal numbers. | {"url":"https://www.whatnumberis.net/mmxxx/","timestamp":"2024-11-12T11:13:47Z","content_type":"text/html","content_length":"7784","record_id":"<urn:uuid:847d8002-77c5-40f4-aa7c-6a2cf699e6f0>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00497.warc.gz"} |
What is 0.05 as a Fraction? [Solved] | Brighterly Questions
What is 0.05 as a Fraction?
Answer: 0.05 as a Fraction is 1/20.
Understanding Decimals and Fractions
To convert a decimal into a fraction, you need to place the decimal over its place value and simplify. In this case, 0.05 equals 5/100, and when simplified, we get 1/20.
FAQ on Decimals and Fractions
What is 0.1 as a Fraction?
0.1 as a Fraction is 1/10.
What is 0.2 as a Fraction?
0.2 as a Fraction is 1/5.
What is 0.25 as a Fraction?
0.25 as a Fraction is 1/4. | {"url":"https://brighterly.com/questions/what-is-0-05-as-a-fraction/","timestamp":"2024-11-02T17:34:07Z","content_type":"text/html","content_length":"69067","record_id":"<urn:uuid:845cb6c9-d28c-4cee-a3f6-0af87d6c0bef>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00221.warc.gz"} |
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Find LCM of 2 numbers without using GCD
08-01-2018, 08:15 PM
(This post was last modified: 08-01-2018 08:22 PM by Dieter.)
Post: #21
Dieter Posts: 2,397
Senior Member Joined: Dec 2013
RE: Find LCM of 2 numbers without using GCD
Albert Chan wrote:
Quote:Sorry if it look stupid ... This is my first HP-12C program (only HP calc I own)
Register Y X --> X Mod(Y, X)
Keep pressing R/S, eventually it display 0, previous Mod was Gcd (Rotate key to get it back)
Great – I love stack-only programs. ;-)
For the record: here's a complete program that directly returns the GCD. The initial lines have been optimized.
01 ENTER
02 ENTER
03 CLX
04 +
05 R↓
06 ÷
07 LstX
08 X<>Y
09 INTG
10 x
11 -
12 X=0?
13 GTO 15
14 GTO 01
15 +
16 GTO 00
08-01-2018, 09:04 PM
(This post was last modified: 08-01-2018 11:28 PM by Albert Chan.)
Post: #22
Albert Chan Posts: 2,785
Senior Member Joined: Jul 2018
RE: Find LCM of 2 numbers without using GCD
uh, did it miss a subtract on line 3 ? looks like Y pushed away
when X and Y are both negative, the mod will stay negative all the way to the end
I still like the mod, and would like program to pull double duty.
How about it only does the loop only if X was negative ?
so, this will get the gcd
54321 CHS Enter 12345 CHS R/S --> -3
that will be awesome ...
Edit: my mistake, it was CLX not LstX. That was a nice trick ...
08-02-2018, 01:25 AM
(This post was last modified: 08-02-2018 06:58 PM by Albert Chan.)
Post: #23
Albert Chan Posts: 2,785
Senior Member Joined: Jul 2018
RE: Find LCM of 2 numbers without using GCD
Here is my revised Mod/Gcd program for HP-12C
If Y, X both negative, it does Gcd(Y, X)
If Y, X both positive, it does Mod(Y, X)
If X = 0, return Y
54321 CHS Enter 12345 CHS
R/S => -3 ; Gcd
54321 Enter 12345
R/S => 4941 ; Mod(54321, 12345)
R/S => 2463 ; Mod(12345, 4941)
R/S => 15 ; Mod(4941, 2463)
R/S => 3 ; Mod(2463, 15)
R/S => 0 ; Mod(15, 3)
R/S => 3 ; Gcd
01 X = 0
02 GOTO 17
03 ENTER
04 ENTER
05 CLX
06 +
07 R↓
08 /
09 Lstx
10 X<>Y
11 INTG
12 *
13 -
14 X <= Y ; Mod ?
15 GOTO 00
16 GOTO 01
17 +
18 GOTO 00
08-02-2018, 01:03 PM
Post: #24
Dieter Posts: 2,397
Senior Member Joined: Dec 2013
RE: Find LCM of 2 numbers without using GCD
Albert Chan wrote:
Quote:Here is my revised Mod/Gcd program for HP-12C
Fine, so this can be used for calculating the GCD as well as a simple MOD.
However, I'd suggest you use the common function names in your listings:
• "clear" is ambiguous. The 12C has CLEAR Σ, CLEAR PROGRAM, CLEAR FIN and CLEAR REG commands. You probably mean the CLX function here.
• "swap" may be clear to RPL-users, but not every 12C user may realize that this refers to the X<>Y key.
• "rotate" sounds like a rather obscure term for the "roll down" function, i.e. the R↓ key. If you can't insert an arrow, "Rv" is a common solution.
• Finally, the 12C has an "INT" key, but that's not what is meant here. The integer function is called INTG.
Sticking to standards usually is not a bad idea. ;-)
08-02-2018, 01:26 PM
Post: #25
Albert Chan Posts: 2,785
Senior Member Joined: Jul 2018
RE: Find LCM of 2 numbers without using GCD
Hi, Dieter.
Your suggestions added. Code fixed.
BTW, I was trying to use your code as template, but this forum Quote not working.
Where can I learn how to add non-ascii characters in post ?
08-02-2018, 03:52 PM
Post: #26
Thomas Okken Posts: 1,897
Senior Member Joined: Feb 2014
RE: Find LCM of 2 numbers without using GCD
(08-02-2018 01:26 PM)Albert Chan Wrote: BTW, I was trying to use your code as template, but this forum Quote not working.
What do you mean? When I press the Quote button below a post, a Post a New Reply window opens, with the original post surrounded by quote and /quote tags. Does it behave differently in your browser?
(08-02-2018 01:26 PM)Albert Chan Wrote: Where can I learn how to add non-ascii characters in post ?
My preferred way is to type the characters in Free42 (i.e. enter them into the alpha register using the ALPHA menu) and then use Copy and Paste to copy them to the Post text box. Free42 and the Forum
are both support Unicode, so that should work smoothly, but of course it is limited to the HP-42S character set. For really unusual characters, I google them by name and then use the first hit from
, and copy and paste from there. For example, google "unicode black right pointing triangle" gets you the hit
; scroll down to the Java Data section, and you can copy the character from string.toUpperCase() or string.toLowerCase().
08-02-2018, 04:35 PM
Post: #27
Dieter Posts: 2,397
Senior Member Joined: Dec 2013
RE: Find LCM of 2 numbers without using GCD
(08-02-2018 03:52 PM)Thomas Okken Wrote:
(08-02-2018 01:26 PM)Albert Chan Wrote: BTW, I was trying to use your code as template, but this forum Quote not working.
What do you mean? When I press the Quote button below a post, a Post a New Reply window opens, with the original post surrounded by quote and /quote tags. Does it behave differently in your
There was a problem with the "Quote" button and other forum functions. Obviously this has been fixed now (around noon UTC today it still was not working). Take a look at the "Forum Issues and
Administration" forum for more details.
(08-02-2018 03:52 PM)Thomas Okken Wrote:
(08-02-2018 01:26 PM)Albert Chan Wrote: Where can I learn how to add non-ascii characters in post ?
My preferred way is to type the characters in Free42 (i.e. enter them into the alpha register using the ALPHA menu) and then use Copy and Paste to copy them to the Post text box.
My method is a bit simpler: I have a textfile here that includes all the special characters that I often use: arrows, mathematical symbols, greek characters and some more. To prepare such a file I
simply copied these characters from the Windows "charmap" utility.
here are some characters that you may directly copy into a textfile for later use.
08-02-2018, 05:48 PM
Post: #28
Albert Chan Posts: 2,785
Senior Member Joined: Jul 2018
RE: Find LCM of 2 numbers without using GCD
Thanks Thomas Okken,
Quote is working now :-)
Thanks Dieter,
charmap chars helps, as I am typing from an ipad.
just discovered ipad keyboard have hidden gems, if I press the key a bit longer
this pops out when a is pressed: æ ã å ā à á â ä
08-02-2018, 05:54 PM
Post: #29
Thomas Okken Posts: 1,897
Senior Member Joined: Feb 2014
RE: Find LCM of 2 numbers without using GCD
(08-02-2018 05:48 PM)Albert Chan Wrote: [...] typing from an ipad.
just discovered ipad keyboard have hidden gems, if I press the key a bit longer
this pops out when a is pressed: æ ã å ā à á â ä
You can also add additional keyboards, under Settings -> General -> Keyboard -> Keyboards. I keep a Greek keyboard in there, so I can type Greek letters in mathematical equations without having to
copy and paste. (If you have multiple keyboards selected, you can switch between them by tapping or long-pressing the "globe" key left of the space bar.)
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Call us today to connect with a top Bridgend College Math tutor | {"url":"https://www.varsitytutors.com/gb/college_math-tutors-bridgend","timestamp":"2024-11-09T03:44:07Z","content_type":"text/html","content_length":"608154","record_id":"<urn:uuid:f4a3d6da-04d4-42d0-8741-c16a50b53890>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00162.warc.gz"} |
How much energy is
How much energy is in a gallon of oil?
How much energy is in a gallon of oil?
A BTU is the amount of energy required to raise one pound of water one degree F. It is about the heat of a birthday candle flame. Heating Oil has 138,690 BTUs per gallon.
How many BTUs are in a gallon of home heating oil?
140,000 BTU/gallon for heating oil]
What is the heating value of fuel oil?
42-47 MJ/kg
Heat Values of Various Fuels
Heat value
Dimethyl ether – DME (CH3OCH3) 29 MJ/kg
Petrol/gasoline 44-46 MJ/kg
Diesel fuel 42-46 MJ/kg
Crude oil 42-47 MJ/kg
How many BTU’s per gallon does no 2 fuel oil contain?
140,000 Btu/gal
2 fuel oil is a medium distillate that is used in diesel engines and also as heating oil. No. 2 fuel oil usually has an energy content of 140,000 Btu/gal (7% less energy per gallon than No. 6 oil).
How much energy is used in heating oil?
In 1 litre of Kerosene, there are 10.35kWh of heat – so each kWh of oil works out at 3.2 pence per kWh. This is marginally cheaper than buying a unit of gas from the mains grid which works out at
about 3.8p / kWh.
How many watts are in a gallon of oil?
Diesel Oil to Watt Hour Conversion Table
Diesel Oil [US] [gal diesel] Watt Hour [Wh]
How do you calculate high calorific value?
Goutel suggested the following formula from calculating the higher calorific value when the percentage proximate analysis of fuel is known. The formula is, cal. value = 343.3 x fixed carbon % + α x %
volatile matter kJ/kg. Goutel formula is unreliable for fuels having high percentage in oxygen.
How much energy is in a litre of heating oil?
Step One – Energy value of kerosene A quick Google search will tell you that one litre of kerosene contains circa 10.35 kWh of energy.
What is kerosene calorific value?
Therefore, the calorific value of wood is 17 KJ/g. in similar manner, when 1 gm of kerosene oil is burnt completely it produces 48 kilojoules heat. So, the calorific value of kerosene oil is 48 KJ/g.
the calorific value of different types of fuels is given in following table.
How many kWh are in a litre of heating oil?
Calculation for working out factor to convert litres of heating oil into kWh / litre Heating oil provides 138,500 British thermal units (BTU) per US gallon 1 BTU = 0.000293 kWh 1 US gallon =
3.78541178 litres
How many Btus are in a gallon of heating oil?
Heating oil provides 138,500 British thermal units (BTU) per US gallon 1 BTU = 0.000293 kWh 1 US gallon = 3.78541178 litres (138,500 * 0.000293)kWh in 3.78541178 litres (138,500 * 0.000293)/
3.78541178 kWh/litre
What is the heat or combustion value of fuel oil?
The heat or combustion value of a fuel oil can be expressed as the quantity of heat (Btu per gallon) released during the combustion process where the oxygen from the air reacts with the hydrogen and
carbon in the fuel. Combustion or heating values for some common fuels oil grades:
How much gas is in a short ton of heating oil?
1 gallon of finished motor gasoline (containing about 10% fuel ethanol by volume) = 120,286 Btu 1 gallon of heating oil (with sulfur content at 15 to 500 parts per million) = 138,500 Btu 1 short ton
(2,000 pounds) of coal (consumed by the electric power sector) = 18,856,000 Btu | {"url":"https://www.yemialadeworld.com/how-much-energy-is-in-a-gallon-of-oil/","timestamp":"2024-11-07T07:22:23Z","content_type":"text/html","content_length":"71714","record_id":"<urn:uuid:0ff2d8d0-e0a3-4be5-b973-e0864bd001ee>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00180.warc.gz"} |
Discrete groups of slow subgroup growth
It is known that the subgroup growth of finitely generated linear groups is either polynomial or at least {Mathematical expression}. In this paper we prove the existence of a finitely generated group
whose subgroup growth is of type {Mathematical expression}. This is the slowest non-polynomial subgroup growth obtained so far for finitely generated groups. The subgroup growth type n^ log n is also
realized. The proofs involve analysis of the subgroup structure of finite alternating groups and finite simple groups in general. For example, we show there is an absolute constant c such that, if T
is any finite simple group, then T has at most n^ c log n subgroups of index n.
Dive into the research topics of 'Discrete groups of slow subgroup growth'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/discrete-groups-of-slow-subgroup-growth","timestamp":"2024-11-12T23:24:22Z","content_type":"text/html","content_length":"45629","record_id":"<urn:uuid:bb3a603b-39d4-42ba-abd1-752400676cdc>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00299.warc.gz"} |
Mining parasite data using genetic programming
Genetic programming is a technique that can be used to tackle the hugely demanding data-processing problems encountered in the natural sciences. Application of genetic programming to a problem using
parasites as biological tags demonstrates its potential for developing explanatory models using data that are both complex and noisy. In many areas of biology, the ability to collect data outstrips
the ability to analyse it. Techniques are needed to mine large datasets and extract biologically meaningful relationships. Genetic programming (GP) is a stochastic optimization approach that helps to
discover comprehensible rules for data mining. It is one of a group of supervised, evolutionary programming techniques that uses darwinian concepts to generate and optimize predictive mathematical
models. This is done by mimicking ‘natural selection’ using ‘populations’ of mathematical models. Initially, a population of n models (short computer programmes) is generated, each model representing
a different, random combination of variables, constants and mathematical functions. The fitness of each model is determined (in terms of how well it solves the problem). The ‘best’ models are then
selected for ‘breeding’ to produce the next generation of ‘fitter’ models, and so on until a model is evolved that solves the problem with the required degree of accuracy or until a specified
stopping criterion is reached. During breeding, different parts of the models are recombined, and the mathematical functions and variables can be changed: the equivalent of crossover and mutation.
Because GP is a randomized algorithm, it is not deterministic, and each new run with a dataset evolves an independent model. Therefore, several alternative solutions to a problem can be evolved. For
complex problems for which there is no single answer, each run can result in a different best model, and a validation process must then be devised to select the most appropriate one.
Ôl bys
Gweld gwybodaeth am bynciau ymchwil 'Mining parasite data using genetic programming'. Gyda’i gilydd, maen nhw’n ffurfio ôl bys unigryw. | {"url":"https://research.aber.ac.uk/cy/publications/mining-parasite-data-using-genetic-programming","timestamp":"2024-11-02T18:00:25Z","content_type":"text/html","content_length":"54817","record_id":"<urn:uuid:e86e41be-ab58-4b3d-a43f-49d875326e2c>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00663.warc.gz"} |
Zero Nominal Interest Rates
A zero nominal interest rate happens when the interest rates match the inflation rates. So for an instance, if the interest rates are 4% then the inflation rates are also 4%. Characteristically
nominal interest rates are positive; therefore people get some incentives to lend money. Central banks tend to lower nominal interest rates during a recession, so as to spur investment in land,
machinery, factories, etc. they can start to approach the level of inflation if they cut interest rates too quickly. When interest rates are cut inflation will often rise, because these cuts have a
simulative effect on the economy. | {"url":"https://www.moneysocket.com/interest-rates/zero-nominal-interest-rates","timestamp":"2024-11-07T06:30:34Z","content_type":"text/html","content_length":"25011","record_id":"<urn:uuid:735d4d0a-0cac-4859-8e2d-d0cb9d54168a>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00683.warc.gz"} |
Find the electric field inside and outside of a spherical shell superposition
• Thread starter Davidllerenav
• Start date
In summary: I don't know what locus means, but I think that all points on the shell that have the same distance to the point would form a circle of radius ##|\vec r-\vec r'|##.Correct, the locus is a
circle with radius ##|\vec r-\vec r'|## centered at ##\vec r'##. This will be important when setting up the integral to calculate the electric field.
Homework Statement
Given a spherical shell of radius R and charge density ##\sigma_0##, find the electric field at point A r<R and point B r>R using the superposition principle. With this results, find the
analogous problem with a solid sphere with charge density ##\rho_0##
Relevant Equations
##\vec E=\frac{1}{4\pi\epsilon_0}\int \frac{\sigma}{r^2}\hat{r}##
Hi! I need help with this problem. I tried to solve it by saying that it would be the same as the field of a the spherical shell alone plus the field of a point charge -q at A or B. For the field of
the spherical shell I got ##E_1=\frac{q}{a\pi\epsilon_0 R^2}=\frac{\sigma}{\epsilon_0}## and for the point charge ##E_2=\frac{-q}{4\pi\epsilon_0 r^2}##, I said that -q was the same as q, and so I
could write it as ##E_2=\frac{-\sigma R^2}{\epsilon_0 r^2}##. After thaht I add them and I got ##E=\frac{\sigma}{\epsilon_0}[1-\frac{R^2}{r^2}]##. As I understand, I was meant to get ##E=0##, since
at A r<R. what am I doing wrong?
Davidllerenav said:
find the electric field at point A r<R and point B r>R using the superposition principle
If that is the litteral rendering of the problem statement, you can't pretend the field of the shell is known, as in
Davidllerenav said:
saying that it would be the same as the field of a the spherical shell alone plus the field of a point charge -q at A or B
Morerover, you can't add a charge -q, because it isn't there !
I read the exercise as: add contributions from smartly chosen parts of the shell to find E for r<R and for r>R
Then in part b, the solid sphere is a sum of shells
BvU said:
If that is the litteral rendering of the problem statement, you can't pretend the field of the shell is known, as in
Morerover, you can't add a charge -q, because it isn't there !
I read the exercise as: add contributions from smartly chosen parts of the shell to find E for r<R and for r>R
Then in part b, the solid sphere is a sum of shells
I see, I tried to do it by adding the -q charge because I tought I could solve it as a ploblem of a sphere with a cavity.
How should I apply the superposition principle? I was thinking of first using the charges ##dq## on the same line as the point, since those two would produce a field straight to A. all the other
little charges ##dq=\sigma da## would cance in pairs, such that only the horizontal component survives. Am I correct?
I would do it formally by starting with the general expression$$\vec E=\frac{1}{4\pi\epsilon_0}\int \frac{(\vec r-\vec r')}{|\vec r-\vec r'|^3}\sigma~ dA'$$Without loss of generality, you can pick
the observation point at ##\vec r = z ~\hat k## but the source point must be general, ##\vec r'=R(\sin \theta '\cos\phi'~\hat i+\sin \theta '\sin\phi'~\hat j+\cos \theta '~\hat k).## Then you have to
do three integrals, one for each component of the electric field. You should verify that the ##x## and ##y## components vanish as expected.
Last edited:
Davidllerenav said:
No. Only the non-axial components cancel.
[edit] sorry, forgot to post reply this afternoon.
Kuruman's advice is good. What is the locus of points with the same ##|\vec r-\vec r'|## ?
kuruman said:
I would do it formally by starting with the general expression$$\vec E=\frac{1}{4\pi\epsilon_0}\int \frac{(\vec r-\vec r')}{|\vec r-\vec r'|^3}\sigma~ dA'$$Without loss of generality, you can
pick the observation point at ##\vec r = z ~\hat k## but the source point must be general, ##\vec r'=R(\sin \theta '\cos\phi'~\hat i+\sin \theta '\sin\phi'~\hat j+\cos \theta '~\hat k).## Then
you have to do three integrals, one for each component of the electric field. You should verify that the ##x## and ##y## components vanish as expected.
Ok, I'll try it. I have one question, is it necessary to do it with vectors? If I know that the field should be in the radial direction, can I work with magnitudes o not?
BvU said:
No. Only the non-axial components cancel.
[edit] sorry, forgot to post reply this afternoon.
Kuruman's advice is good. What is the locus of points with the same ##|\vec r-\vec r'|## ?
All points with the same vector ##|\vec r-\vec r'|## are those on the surface of the shell, right?
Davidllerenav said:
All points with the same vector ##|\vec r-\vec r'|## are those on the surface of the shell, right?
All the source points are on the surface of the shell. How would you describe the locus of all points on the shell that have the same ##|\vec r-\vec r'|##?
kuruman said:
All the source points are on the surface of the shell. How would you describe the locus of all points on the shell that have the same ##|\vec r-\vec r'|##?
I don't know what locus means, but I think that all points on the shell that have the same distance to the point would form a circle of radius ##|\vec r-\vec r'|##.
A "locus" is the set of all points that share a common property. Example: A circle is the locus of all points that are equidistant from a single point in a two-dimensional plane. Saying they form a
circle of radius ##|\vec r-\vec r'|## is not specific enough. How is this circle oriented? Remember that all you have is a shell and a point P outside (or inside) the shell. Given only these two
items, how will you draw this circle on the shell?
kuruman said:
A "locus" is the set of all points that share a common property. Example: A circle is the locus of all points that are equidistant from a single point in a two-dimensional plane. Saying they form
a circle of radius ##|\vec r-\vec r'|## is not specific enough. How is this circle oriented? Remember that all you have is a shell and a point P outside (or inside) the shell. Given only these
two items, how will you draw this circle on the shell?
By oriented, do you mean clockwise or counterclockwise? I don't know.
When in doubt, make a sketch to clear up your thinking !
Davidllerenav said:
By oriented, do you mean clockwise or counterclockwise? I don't know.
Along what direction (positive or negative, it doesn't matter) is the normal to the plane of the circle? The suggestion by
to make a sketch is certainly helpful.
However, you can also forget about the circle of radius ##|\vec r-\vec r'|## and just do the three integrals as I suggested in #4. When you are done with the integral over ##\phi '## (do that first),
you will have verified that ##E_x=E_y=0## while the integral for ##E_z## will be the same as the one you get when you consider said circle.
FAQ: Find the electric field inside and outside of a spherical shell superposition
1. What is the formula for calculating the electric field inside a spherical shell?
The electric field inside a spherical shell can be calculated using the formula E = kQ/r^2, where k is the Coulomb's constant, Q is the total charge of the shell, and r is the distance from the
center of the shell to the point where the electric field is being measured.
2. How is the electric field inside a spherical shell affected by the presence of other charges outside the shell?
The electric field inside a spherical shell is not affected by the presence of other charges outside the shell. This is because the shell acts as a Faraday cage and the charges outside the shell do
not have any influence on the electric field inside.
3. What is the formula for calculating the electric field outside a spherical shell?
The electric field outside a spherical shell can be calculated using the formula E = kQ/r^2, where k is the Coulomb's constant, Q is the total charge of the shell, and r is the distance from the
center of the shell to the point where the electric field is being measured. This formula is the same as the one for calculating the electric field inside the shell.
4. How does the electric field outside a spherical shell compare to the electric field inside?
The electric field outside a spherical shell is stronger than the electric field inside. This is because the electric field inside the shell is canceled out by the charges on the inner surface of the
shell, while the electric field outside the shell is not affected by these charges.
5. Can the electric field inside a spherical shell be zero?
Yes, the electric field inside a spherical shell can be zero if the total charge of the shell is zero. This means that the charges on the inner and outer surface of the shell cancel each other out,
resulting in a net electric field of zero inside the shell. | {"url":"https://www.physicsforums.com/threads/find-the-electric-field-inside-and-outside-of-a-spherical-shell-superposition.978364/","timestamp":"2024-11-10T11:42:13Z","content_type":"text/html","content_length":"143371","record_id":"<urn:uuid:64ec3e92-25b9-4c3a-aa14-b0e83d00edf3>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00371.warc.gz"} |
A Hierarchy of Context-Free Languages Learnable from Positive Data and Membership Queries
Makoto Kanazawa & Ryo Yoshinaka
Link to the paper: here
Abstract: We consider a generalization of the “dual” approach to distributional learning of context-free grammars, where each nonterminal A is associated with a string set XA characterized by a
finite set C of contexts. Rather than letting XA be the set of all strings accepted by all contexts in C as in previous works, we allow more flexible uses of the contexts in C, using some of them
positively (contexts that accept the strings in XA) and others negatively (contexts that do not accept any strings in XA). The resulting more general algorithm works in essentially the same way as
before, but on a larger class of context-free languages.
Any question or comment can be made using the comment section of the page below.
10 Comments
1. This is a very impressive generalisation of distributional learning! Thanks you for this work.
Can one imagine a condition like the sounded pre-fixed in the primal approach? If yes, what may it looks like? If no, why ?
1. Sound pre-fixed points work for both approaches. For the original primal approach, an SPP must consist of the closures of finite sets of strings. To generalize, you could think of changing
either “closure” or “sets of strings”. There is some discussion in the concluding section of our paper. I can probably give more details during the Q/A period.
2. I have another question: as far as I understand, you do not have a result to compare FCP(k,l) and FCP(k,l,m), or about the potential hierarchy within FCP(k,l,m). What’s your intuition?
1. My original intuition was that of course more context-free languages are included because FC_L(k,l,m) contains many more sets than FC_L(k,l). But it’s very difficult to give examples of CFLs
that differentiate the two. Perhaps you don’t get any more CFLs.
3. Errata to the article: https://makotokanazawa.ws.hosei.ac.jp/publications/ICGI2021_errata.pdf
4. Nice work ! On the FCP(k,l,m) question, I have the intuition that the union of m components can be simulated using lots of rules using just m = 1 components.
i.e. if we have a nonterminal B = A_1 \cup \dots A_m then w can just expand out all the rules using B with rules using A_i etc,
But I think that may be incorrect?
1. I’m not sure how the suggestion goes. If G has an SPP whose B-component is a union of m sets of the form C_i^{\triangleleft} \cap D_i^{\overline{\triangleleft}}, are we supposed to replace B
with m new nonterminals A_1, \dots, A_m so that a new SPP will have the set C_i^{\triangleleft} \cap D_i^{\overline{\triangleleft}} as its A_i-component? What will the productions be that
have A_i as their left-hand side?
1. So for every production B \to \alpha we have m productions of the form A_i \to \alpha.
1. Doesn’t that make all the A_i equivalent?
1. Oh I am wrong here; the reduction doesn’t work.
I don’t see how to construct an example though. | {"url":"https://icgi2020.lis-lab.fr/a-hierarchy-of-context-free-languages-learnable-from-positive-data-and-membership-queries/","timestamp":"2024-11-11T20:37:14Z","content_type":"text/html","content_length":"91864","record_id":"<urn:uuid:b89b4250-60ba-4fb1-b60f-0120c7d62e2e>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00515.warc.gz"} |
Derivative-Calculator.org – Derivative of sinh(x) - Proof and Explanation
The hyperbolic sine function is defined as:
To find the derivative, we use the derivative of exponential functions. The derivative of ${e}^{x}$ is ${e}^{x}$, and the derivative of ${e}^{-x}$ is $-{e}^{-x}$.
Now, differentiating $\mathrm{sinh}\left(x\right)$:
This simplifies to:
Thus, the derivative of $\mathrm{sinh}\left(x\right)$ is:
The hyperbolic sine function, $\mathrm{sinh}\left(x\right)$, is similar to the sine function but based on exponential functions. It is defined as:
This expression represents the difference between the exponential growth ${e}^{x}$ and the exponential decay ${e}^{-x}$, divided by two.
To find the derivative, we differentiate each part of the function. The derivative of ${e}^{x}$ with respect to $x$ is ${e}^{x}$, and the derivative of ${e}^{-x}$ with respect to $x$ is $-{e}^{-x}$.
This is because of the chain rule, where the derivative of $-x$ is $-1$.
Substituting these derivatives into the formula for $\mathrm{sinh}\left(x\right)$:
This simplifies to:
This expression is exactly the definition of $\mathrm{cosh}\left(x\right)$, the hyperbolic cosine function: | {"url":"https://derivative-calculator.org/proofs/sinh/","timestamp":"2024-11-04T12:25:07Z","content_type":"text/html","content_length":"17122","record_id":"<urn:uuid:61136aea-3093-44ef-9465-257400be1375>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00520.warc.gz"} |
General theorems about higher dimensional black holes
Higher dimensional black holes seem to behave qualitatively different from 4-dimensional ones, in the sense that there are many qualitatively new solutions with properties that simply have no
counterpart in 4 dimensions. It is not easy to get an intuitive understanding of why this is so. But very broadly speaking, the reason seems to be that black holes in higher dimensions have more
possibilities to rotate in different directions, and this somehow gives rise to more complicated configurations in which rotational forces balance those of gravitational attraction. A very crude way
of explaining why this phenomenon can occur is as follows. In $(n-1)$-dimensional Euclidean space, there are $N=\lceil (n-2)/2 \rceil$ independent planes of rotation (or equivalently, the Cartan
subgroup of $SO(n-1)$ is $U(1)^N$). Thus, we have the possibility of having up to $N$ commuting rotational symmetries, with associated angular momenta $J_i$. The centrifugal barrier for the rotation
in a given plane is always of the order of
\sim \frac{J_i^2}{M^2 r^2} \ ,
where the radial dependence does not depend on the dimension because rotations take place in a 2-plane. By contrast, the gravitational potential exerted by a mass $M$ is
\sim \frac{GM}{r^{n-3}} \ ,
which does depend on the dimension. Thus, we see that the competition between the gravitational attraction and the centrifugal force depends on the dimension. For example, in $n=5$ dimensions both
have the same radial dependence, and this can be seen crudely as the reason why black rings (horizon topology $ S^1 \times S^2$) can exist in $n=5$ dimensions, but not in $n=4$. It also accounts for
the fact that in higher dimensions, one can have black objects with arbitrarily high spin (in 4 dimensions, $J$ is bounded by $\sim M$), leading e.g. to very flat “pancake-like” horizons, and in the
limit as some $J_i \to \infty$ to new solutions describing extended black objects such as “black branes”, which are also of considerable interest.
A more precise way of arguing that a black ring cannot exist in 4 dimensions is provided by Hawking’s famous topology theorem. The upshot is that the topology simply has to be $S^2$ in fairly general
theories of gravity, whereas the topology of a ring would be $T^2 = S^1 \times S^1$. In Einstein-Maxwell theory, one furthermore has the famous and much more precise black hole uniqueness theorems,
which establish that the solution must in fact be given by the known charged Kerr-Newman family of black holes. Black hole uniqueness theorems are known for certain types of higher dimensional black
hole solutions, but they are much less powerful than the 4-dimensional cousin. For this reason, and also from a more general perspective, it is of great interest to know general restrictions of the
type of the topology theorem also in higher dimensions. Another such general result in 4 dimensions, which is in fact an important stepping stone of the uniqueness theorems, is the so-called
“rigidity theorem”, which states that any stationary black holes must either be static, or rotating with a rotational symmetry. Static black holes are usually much easier to classify. For example in
vacuum general relativity, the only such solution is shown to be the famous Schwarzschild metric
g = -(1-2M/r) dt^2 + (1-2M/r)^{-1} dr^2 + r^2 d\sigma^2_{S^2}
and also in many other gravity theories, static solutions have been classified. If the spacetime is not static, we have learned from the rigidity theorem that the spacetime symmetry group must
include rotations in a single plane, i.e. must at least be $\mathbb{R} \times U(1)$ (this is in fact precisely the symmetry group of the Kerr-metric).
To what extent do the topology and rigidity theorems hold in dimensions $n>4$? Concerning the topology theorem, it is easy to see that if we replace the 2-sphere in the Schwarzschild metric by
another compact $n-2$ dimensional Einstein space $(B,\gamma)$ [i.e. $Ric_\gamma = c \gamma$], then we get another solution $g$ to Einstein’s equation, with $c$ related to $M$. This solution would not
be asymptotically flat, but Hawking’s argument is quasi-local and does not care about asymptotic conditions. If the constant $c$ is negative, then the mass of the solution would be negative, too, so
we should take $c>0$. Thus, it is clear that Hawking’s argument in higher dimensions cannot yield more restrictions on $B$ than that $B$ can be equipped with an Einstein-metric with positive constant
$c$. By going through the argument of Hawking in higher dimensions, Galloway and Schoen were in fact able to show that the restrictions on $B$ afforded by the use of the spacetime Einstein equations
(and other reasonable rather general conditions) restrict the topology of $B$ to be of “positive Yamabe type”, meaning, in essence, that $B$ should be able to carry a metric of everywhere positive
scalar curvature. In $n=4$, the dimension of $B$ is 2. If $\gamma$ is a metric on $B$ with positive scalar curvature, then the Gauss-Bonnet theorem tells us that
2 \pi \ [2 – 2 \ {\rm genus}(B)] = \int_B Scal_\gamma > 0 \ ,
whence the genus is positive and $B$ must be a 2-sphere. This is Hawking’s topology theorem. In higher dimensions, the positive Yamabe condition is increasingly less restrictive and e.g. satisfied
by any product of spheres $B \cong S^{q_1} \times … \times S^{q_i}$. There is now convincing evidence that regular, asymptotically flat black holes with such horizons indeed exist. In $n=5$
dimensions, the condition of positive Yamabe type is shown to imply that $B$ must be a quotient of $S^3$ by a discrete subgroup $\Gamma \subset SO(4)$, by $S^1 \times S^2$, or connected sums
A version of the rigidity theorem can also be shown in higher dimensions. It essentially states that, if the black hole is stationary, then it must either be static (in which case classifications
are known), or it must have at least one rotational symmetry, i.e. the symmetry group must be at least $\mathbb{R} \times U(1)$. The proof shows in interesting connection to ergodic theory on the
horizon manifold. It is also known that one cannot improve this result, in the sense that there exist stationary black holes in higher dimensions which just have that symmetry. Although the rigidity
theorem is in this sense less powerful than in 4 dimensions, it is nevertheless useful for many arguments about higher dimensional black holes. It can be combined also with the topology theorem to
give a somewhat more refined restriction on the possible black hole topologies, and this has been worked out e.g. in the case $n=5$:
Theorem:The topology of the horizon $B$ can be one of the following in 5 dimensions:
1. If the $U(1)$ symmetry has a fixed point on $B$, then the topology must be $ B \cong \# l\cdot (S^1 \times S^2) \# L(p_1,q_1) … \# L(p_k,q_k). $
2. If the $U(1)$ symmetry does not have a fixed point, then $B \cong S^3/\Gamma$, where $\Gamma$ can be certain finite subgroups of $SO(4)$, or $B \cong S^1 \times S^2$. This class of manifolds
includes the Lens-spaces $L(p,q)$, but also Prism manifolds, the Poincare homology sphere, and various other quotients. All manifolds in this class are Seifert fibred spaces over $S^2$ with
positive orbifold Euler characteristic.
Some references are: | {"url":"https://home.uni-leipzig.de/tet/?page_id=786","timestamp":"2024-11-12T00:38:02Z","content_type":"text/html","content_length":"38819","record_id":"<urn:uuid:80bdea28-e551-4796-89f4-886eddde100c>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00848.warc.gz"} |
(Hyper)Complex Seminar 2021
condensed matter Computer science general relativity and quantum cosmology mathematical physics quantum algebra data analysis, statistics and probability quantum physics
Institute of Mathematics, Polish Academy of Sciences
Audience: Researchers in the topic
Seminar series times: No fixed schedule
Organizer: Krzysztof Pomorski*
*contact for this listing
Conference topics are :
1.(Hyper)complex analysis
2.(Hyper)complex differential equations in relation to bifurcation, multifrequency, binary and ternary systems with physical applications
3.Symmetry foundations and its breakage in condensed matter systems
4. Theory of cyclostationary processes and functional data analysis
5. Clifford Algebras and their applications
6. Mathematical structure transformations in Machine Learning
7. Data analysis
8. Evolution systems
Conference is open and free.
In order Join Zoom Meetings click the link:
When asked, give the following data:
Meeting ID: 816 4174 1236
Passcode: 442493
November 11, 2021
Opening ceremony (9:00-9:50, POLISH TIME)
Session A1 (A tribute to the works of Julian Ławrynowicz), Chairmen: Dariusz Partyka and Mariusz Zubert
Session A2 (Monogenic functions), Chairman: Massimo Vaccaro
Session A3 (Hypercomplex structures), Chairman: Sergiy Plaksa
Session A4 (Complex analysis of several variables), Chairman: Lino F. Reséndis Ocampo
November 12, 2021 Session B1 (Condensed matter physics), Chairman: Łukasz Stępień
Session B2 (Ontology of quantum mechanics), Chairman: Adam Paszkiewicz
Session B3 (Theory of defects in topology of order parameter), Chairman: Marek Danielewski
Round Table Discussion I (20:00-21:00) Chairman: Ilan Roth
November 13, 2021
Session C1 (Embodied AI and evolutionary systems), Chairman: Mariusz Zubert
Session C2 (Mathematical modeling in Physics and Computer Science), Chairman: Fabio Bonsignorio
Session C3 (Data analysis and machine learning), Chairman: Maciej Jaworski
Session A6 (The inner radii problems), Chairman: Serhii Gryshchuk
Session A7 (Poster session), Chairman: Armen Grigoryan
Round Table Discussion II (20:30-21:00) Chairman: Sergiy Plaksa
November 14, 2021
Session D1 (Superconducting and semiconductor electronics), Chairman: Khrystyna Gnatenko
Session D2 (Physics of superconducting materials), Chairman: Mikhail Belogolovskii
Session D3 (Experiments with superconducting and seminconductor electronics), Chairman: Catherine Pépin
Session A8 (Complex and real analysis of one variable), Chairman: Radosław Kycia
Closing ceremony (20:00-20:15)
Upcoming talks
Past talks | {"url":"https://researchseminars.org/seminar/HCS2021","timestamp":"2024-11-07T16:03:19Z","content_type":"text/html","content_length":"18499","record_id":"<urn:uuid:67cc5b4e-aa84-4fc9-a501-a933d6aa0d38>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00567.warc.gz"} |
Degrees, Certificates, and Transfer Programs - Foothill College
Degrees, Certificates, & Transfer Programs
Archive Reminder
The listings on this page are archived Degree and Certificate Programs information through 2020-2021.
Mathematics Website | Science, Technology, Engineering & Math Division
Associate Degree for Transfer-Mathematics 2020-2021
Mathematics and related subjects play important dual roles in our culture. Although mathematics is a study in its own right, it is also an indispensable tool for expressing and understanding ideas in
the sciences, engineering and an increasing number of other fields. Students completing this degree will be able to construct appropriate models of natural phenomena, develop those models with
appropriate mathematical techniques and interpret results of those models.
The Associate in Science in Mathematics for Transfer degree will prepare students for transfer to California State Universities (CSUs). Students who complete the Associate in Science in Mathematics
for Transfer degree will be ensured preferential transfer status to many CSUs as mathematics majors and/or majors in related disciplines. The Associate in Science in Mathematics for Transfer degree
requirements will fulfill the lower division major requirements at many CSUs. Students are advised, however, to meet with a counselor to assess the course requirements for a specific CSU.
Program Learning Outcomes:
-Students will be able to clearly communicate mathematical ideas through graphs, tables of data, equations and verbal descriptions.
-Students will be able to construct appropriate mathematical models of natural phenomena, develop those models with appropriate mathematical techniques and interpret results of those models.
Units Required
Degree Requirements:
Associate in Science in Mathematics for Transfer requires completion of a minimum of 90 units to include:
-CSU General Education Breadth Requirements or the *Intersegmental General Education Transfer Curriculum (IGETC) (49-58 units) (full certification is required)
-Core and support courses (29.5-31 units, of which 20-21 units may satisfy the GE requirement)
-Transferable electives necessary to meet the 90-unit minimum requirement
NOTE: All courses pertaining to the major must be completed with a grade of "C" (or "P") or better. In addition, the student must obtain a minimum GPA of 2.0.
*IMPORTANT NOTE: Although it is possible to fulfill the requirements for the Associate Degree for Transfer by completing the IGETC for UC pattern, admission to CSU requires completion of an Oral
Communication course (IGETC Area 1C; CSU GE Area A-1); therefore, students who plan to transfer to CSU should complete this course as part of their GE or elective units.
Core Courses (20 units)
MATH 1A Calculus (5 units)
or MATH 1AH Honors Calculus I (5 units)
MATH 1B Calculus (5 units)
or MATH 1BH Honors Calculus II (5 units)
MATH 1C Calculus (5 units)
MATH 1D Calculus (5 units)
Support Courses (9.5-11 units)
Select ONE course each from List A and List B:
List A:
MATH 2A Differential Equations (5 units)
MATH 2B Linear Algebra (5 units)
List B:
C S 1A Object-Oriented Programming Methodologies in Java (4.5 units)
or C S 1AH Honors Object-Oriented Programming Methodologies in Java (4.5 units)
C S 1B Intermediate Software Design in Java (4.5 units)
C S 1C Advanced Data Structures & Algorithms in Java (4.5 units)
C S 2A Object-Oriented Programming Methodologies in C++ (4.5 units)
C S 3A Object-Oriented Programming Methodologies in Python (4.5 units)
MATH 2A* Differential Equations (5 units)
MATH 2B* Linear Algebra (5 units)
MATH 10 Elementary Statistics (5 units)
or MATH 17 Integrated Statistics II (5 units)
MATH 22 Discrete Mathematics (5 units)
or C S 18 Discrete Mathematics (5 units)
PHYS 4A General Physics (Calculus) (6 units)
* MATH 2A or 2B may be used to satisfy List B requirement if they were not used to meet the requirement for List A.
Control Information:
2020-2021 | Status: Approved | Modified: 2020-04-27 13:50:04 | Dept Code: MATH
See a Counselor!
Counseling Center
Student Services Building 8300, Room 8302 | {"url":"https://www.foothill.edu/programs-archive/programs.html?title_id=Associate%20Degree%20for%20Transfer-Mathematics","timestamp":"2024-11-04T05:56:19Z","content_type":"text/html","content_length":"32340","record_id":"<urn:uuid:0e0c3178-219f-44b4-b379-c6232c26a500>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00868.warc.gz"} |
The budget deficit forecasting sausage machine - The Kouk - Stephen Koukoulas | Speaker | Economic AdvisorThe budget deficit forecasting sausage machine
The budget deficit forecasting sausage machine
By Stephen Koukoulas|2022-01-07T01:20:56+00:0030 April 2014|
In light of the humbug of the ‘budget never returning to surplus unless we cut the tripe out of spending’, I though it interesting to revisit the sensitivity of budget forecasting to small changes to
the economic parameters.
The Commission of Audit finding that Australia will be dogged by perpetual deficits is based on a range of economic projections which assume the economy maintains an output gap over the next decade
(real GDP growth never above 3%), nominal GDP growth averaging 4% for the next three years and then only rising to 5.5% thereafter, the unemployment rate remaining at 6% for the next decade and a
falling particpation rate.
These forecasts may be right, they may not.
My simple budget forecasting spreadsheet shows that if we change slightly some of those projections and in two of the next three years, real GDP growth hits 3.5% as the output gap closes, if nominal
GDP is 0.75% higher in those two years, and the unemployment rate ticks down to 5.5% within a year and then drops to 5% by 2016-17, there are surpluses within three years and that surpluses remain
and get larger out to 2023-24.
Putting in very weak numbers and the budget deficit blows out to above 5% of GDP within five years.
And these different results make no allowance for policy changes – they assume the current policies remain in place.
This is what I call the budget deficit sausage machine.
Plug in the economic parameters you need and you will get a particular budget outcome. Indeed, any outcome you want. This is what happened with the MYEFO when the growth outlook was revised sharply
lower and therefore underpins the Commission of Audit reports.
It is something that I wrote on for Business Spectator last year. The article can be found here,
https://www.businessspectator.com.au/article/2013/8/30/politics/sausage-machine-approach-budget-estimates but it is reproduced in full below.
Just add tomato sauce and get whatever numbers you want, just like the Audit Commission and its salami of the outlook.
The sausage machine approach to budget estimates
The focus on policy costings in the election campaign has reached the point where everyone is a winner and everyone else is a loser when it comes to working out how much each policy will cost
(Abbott’s secret plan forced Rudd into error, August 30).
One would have thought that the revenue miscalculation over the mining tax or the error made by Treasury on the expected revenue from the goods and services tax over a decade ago would have
underpinned a more sober analysis of how much revenue a policy will cost or how much it will collect.
I will use a stylised example to show you how policy costings work and how the process can be abused.
Let’s assume that you want to introduce a goods and services tax of 10 per cent from the start of next year. No more, no less.
How do you work out how much revenue will be collected?
The critical, absolutely critical, starting point is having an estimate of how much consumers will spend on goods and services from next year. I hope that is obvious.
The first thing to do would be to go to the Forecasting Section in Treasury and say, “What is your current forecast for consumer spending over the next four years?”
The Forecasting Section will say, in this illustration, that in year one, consumer spending will be $100 billion; year two, it will be $105 billion; year three will be $110.25 billion and year four,
$115.76 billion. This is based on a forecast of a 5 per cent annual rise in consumer spending.
This seems fair.
The revenue for the government from the new GST will therefore be $10 billion, plus $10.5 billion, plus $11.03 billion, plus $11.58 billion.
Total revenue is $43.11 billion. Simple.
Just say that a few months later you go to the same Forecasting Section with EXACTLY the same policy proposal – that is, a 10 per cent GST.
The Forecasting Section might come back and say that because of the terms of trade decline, the unexpected strength of the Australian dollar and the softness in wages growth, in year one consumer
spending is forecast to be $95 billion; in year two $99.75 billion; in year three $104.74 billion and in year four $109.97 billion. It is still the same 5 per cent annual growth in the estimates, but
the starting point base is lower, for obvious reasons.
As a result, the total revenue from the GST will be $40.95 billion in this example – some $2.16 billion less than originally assumed.
Yowzer! Here is the first possibility of a black hole.
It is that simple.
Then, assume your weak-hearted finance spokesperson decides to exclude health spending from the GST. The Forecasting Section may note that health spending, as a proportion of total consumer spending,
is increasing. This will mean that the size of the pot from which the GST is being collected is shrinking relative to overall consumption, and thus the GST will not be a tax that keeps pace with
consumer spending. The calculation will be complex but of course is doable, and is based on assumptions about aggregate spending and, within that, spending on health services.
Let’s look at something as complex as paid parental leave – the Coaltion’s plan, that is.
We know that everyone who will get it will get up to six months’ pay with a limit of $75,000, and that is basically it. It will take effect from 1 July 2015.
But what is the estimate of the number of babies born after that? Two-hundred thousand a year? Two-hundred-and-twenty-thousand a year? How many parents will take up the offer? Ninety per cent?
Ninety-two per cent? Eighty-five per cent? What will happen to wages growth over those few years? Up an average of three per cent a year? Four per cent a year? Will some parents go back to work after
three months? Four months? How many? Five per cent? Ten per cent?
Now look at the 1.5 per cent levy on company tax to pay for part of the PPL scheme. How much will the company profit base be in 2015-16? Thirty billion dollars? Thirty-five billion? How much revenue
will be collected under each scenario? Will profits be stronger because of an economic boom, or weaker because of a downturn? Take your pick.
I hope you can see the issues with costings in these couple of examples.
This is the very point, the only point, that the Secretaries of Treasury Martin Parkinson and Finance David Tune said in their press release yesterday.
In their most polite, bureaucratic way, Parkinson and Tune said: “Different costing assumptions, such as the start date of a policy, take-up assumptions, indexation and the coverage that applies,
will inevitably generate different financial outcomes.”
And so it was with the costings they prepared earlier this year, that Treasurer Chris Bowen and Finance Minister Penny Wong released, showed a difference between Treasury estimates and the latest
Parliamentary Budget Office forecasts. There is nothing else to go on because the Coalition has refused to release details of its costings.
The press release from Parkinson and Tune highlights the point that the Coalition is hiding its costings assumptions, at the very least. What have they assumed?
We also know the PBO has not costed all of the Coalition’s policies. Who did?
It is easy to cost most policy changes. This is why a number of credible and not so credible economics consultancy firms try to make their mark with a simple sausage machine approach to budget
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About the Author: Stephen Koukoulas
Stephen Koukoulas is one of Australia’s leading economic visionaries and keynote speakers, past Chief Economist of Citibank and Senior Economic Advisor to the former Prime Minister of Australia. | {"url":"https://thekouk.com/the-budget-deficit-forecasting-sausage-machine/","timestamp":"2024-11-13T09:54:46Z","content_type":"text/html","content_length":"175670","record_id":"<urn:uuid:18f350cb-9741-4066-a3b4-01e485b25621>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00546.warc.gz"} |
A Bayesian approach to modelling spectrometer data chromaticity corrected using beam factors - I. Mathematical formalism
Monthly Notices of the Royal Astronomical Society
Accurately accounting for spectral structure in spectrometer data induced by instrumental chromaticity on scales relevant for detection of the 21-cm signal is among the most significant challenges in
global 21-cm signal analysis. In the publicly available Experiment to Detect the Global Epoch of Reionization Signature low-band data set, this complicating structure is suppressed using
beam-factor-based chromaticity correction (BFCC), which works by dividing the data by a sky-map-weighted model of the spectral structure of the instrument beam. Several analyses of these data have
employed models that start with the assumption that this correction is complete. However, while BFCC mitigates the impact of instrumental chromaticity on the data, given realistic assumptions
regarding the spectral structure of the foregrounds, the correction is only partial. This complicates the interpretation of fits to the data with intrinsic sky models (models that assume no
instrumental contribution to the spectral structure of the data). In this paper, we derive a BFCC data model from an analytical treatment of BFCC and demonstrate using simulated observations that, in
contrast to using an intrinsic sky model for the data, the BFCC data model enables unbiased recovery of a simulated global 21-cm signal from beam-factor chromaticity-corrected data in the limit that
the data are corrected with an error-free beam-factor model. | {"url":"https://repositorio.ucsc.cl/entities/publication/99fa54c1-0b9f-433b-874f-a35cd432627e","timestamp":"2024-11-08T02:28:14Z","content_type":"text/html","content_length":"656098","record_id":"<urn:uuid:e9f9dcc9-6a4e-4347-b43c-425e110650b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00381.warc.gz"} |
PHYSICS PAPER 2 Marking Scheme - 2017 MURANG'A MOCK EXAMINATION - EasyElimu: Learning Simplified
SECTION A (25 Marks)
Answer all the questions in this section in the spaces provided
1. The following diagram alongside shows two mirrors at an angle of 30o to each other. A ray of light is incident on one mirror at 300 as shown. Sketch the path of the ray until it leaves the
mirrors indicating the angles at each point of contact with the mirrors. (3mks)
2. A ray of right passes from air into a certain liquid at an angle of 500 to the normal. The ray is refracted such that the angle of refraction is 350 as it enters the liquid. Calculate the
refractive index of the liquid. (3mks)
3. State the necessary conditions for total internal reflection to take place. (2mks)
□ Light must travel from a denser medium to a less dense one.
□ The angle of incidence must be greater than the critical angle.
4. A wire of resistance 27Ω is cut into three equal lengths. If the three wires are connected in parallel, determine the effective resistance. (2mks)
ANS: ^1/[R] = ^1/[9] + ^1/[9] + ^1/[9] = ^3/[9] = ^1/[3]
R = 3Ω
5. Explain briefly how a P-type semiconductor is made. (1mk)
□ An intrinsic semiconductor is doped with a trivalent (a valency 3 element.)
6. A current of 5A is passed through a conductor whose resistance is 10Ω. How much energy is converted to heat in one hour (3mks)
□ E = I^2Rt
= 52 x 10 x (60 x 60)
= 25 x 10 x 3600
= 9 x 105J
7. An electromagnet is made by winding insulated copper wire on an iron core. State two changes that could be made to increase the strength of the electromagnet. (2mks)
□ Increase in the number of turns.
□ Increase in the current in the coil.
8. Define the term ‘line of force’ as applied to magnetic fields. (1mk)
□ It is a line drawn such that the tangent to it at any point gives the direction of the field.
9. A vibrator is sending out 8 ripples per second across a ripple tank. The ripples are observed to be 4cm apart. Calculate the velocity of the ripples. (2mks)
□ f = 8Hz; λ = 0.04m.
V = fλ
= 8 x 0.04
= 0.32 m/s
10. Give a reason why x-rays but Not radio waves are used to detect fractured bones. (1mk)
□ X – Rays have more penetration power than radio waves
□ X – Rays have high frequency/ energy.
11. State two properties of cathode rays (2mks)
□ Travel in straight line.
□ charged
□ Deflected by magnetic and electric field.
12. The figure below shows a positively charged metal plate with an Earthing connection. Using an arrow, show the direction of charges through the earth connection and explain the final charge of the
plate. (2mks)
13. What is photoelectric effect? (1mk)
□ It is the emission of electrons from the surface of a metal when illuminated with electromagnetic radiation of sufficient frequency.
SECTION B (55Mks)
1. The diagram below represents a metre bridge used to determine the resistance of an electrical component x
From the diagram,
1. Explain why wide brass strips are used as terminals. (1mk)
○ To minimize resistance due to terminals.
2. Explain why a cell of low e.m.f. is preferable. (1mk)
○ To minimize current and the resulting heating effect that would alter resistance.
3. If null deflection was obtained when L1 was 60.0cm. Calculate the resistance of component marked X. (2mks)
○ ^x/[20] = ^60/[40]
x=^60/[40 ]x 20
= 30Ω
4. State three ways of ensuring that error are minimized during the experiment. (3mks)
○ Use of short connecting wires
○ Use of a source with low emf.
○ The value pf known resistance R should be comparable to x.
2. A uniform resistance wire of length 2.0m conducts a current of 0.25A when connected in series with a cell of e.m.f. 1.6V. How much current would be conducted if the wire is now cut into two
equal lengths which are then arranged in parallel? (4 marks)
1. The distance of separation between the plates of a certain capacitor is reduced. State how this affects the capacitance of the capacitor. (1mk)
2. You are provided with the following apparatus used for studying charging of a capacitor; an uncharged capacitor, voltmeter, milliameter, 6v battery, connecting wires, a switch and a load
resistor R. (1mk)
1. Draw a circuit diagram that can be used to charge the capacitor. (2mks
2. Use the circuit diagram drawn above to explain how the capacitor gets charged. (2mks)
○ Negative charges flow from the negative terminal of the battery to the plate of the capacitor.
○ Negative charges flow from the other plate of the capacitor to the positive terminal of the cell.
○ Hence equal positive and negative charges gather on the plates opposing further flow of electrons when fully charged OR p.d across the plates is equal to that of the battery.
3. State the purpose of resistors R. (1mk)
○ To slow down the charging process so that current and voltage are observed
3. The zinc plate shown below is connected to a negatively charged electroscope and is exposed to ultra violet radiation.
1. Explain what happens to the leaf of the charged electroscope. (3mks)
○ The leaf falls.
○ When U.V falls on the zinc plate electrons are ejected/ photoelectric effect takes place.
○ The negative charges in the zinc plate and cap of the electroscope are repelled hence leaf falls
2. If the same experiment is repeated using a positively charged electroscope, explain the observation (3mk)
○ There is no effect on the leaf of the electroscope.
○ The electrons liberated by the U.V light are attracted back by the positive charges on the zinc plate/cap of electroscope hence no effect on leaf divergence.
1. State one difference between transformers and an induction coil. (1mk)
☆ A transformer uses alternative current while an induction coil uses interrupted direct current.
2. State two way through which energy is lost in a transformer. (2mks)
☆ Flux leakage.
☆ Eddy currents.
☆ Hysteris loss.
☆ Resistance of coil.
3. A transformer has 1000 turns in its secondary coil and 10 turns in its primary coil. An alternating current of 2.5A flows in the primary circuit when it is connected to a 12V a.c supply.
1. State the type of transformer (1mk)
2. Calculate the power input to the transformer. (1mk
○ P = I[p] x V[p
]= 2.5 x 12
= 30 W
3. Calculate the e.m.f across the secondary coil. (3mks)
Vs =^Ns/[Np] x V[p
]= 1000/10 x 12
= 1200V
4. Determine the maximum current that could flow in a circuit connected to the secondary coil if the transformer if its 80% efficient. (Use the e.m.f in secondary as calculated in (iii)
above). (3mks)
Ps = ^80/[100] x 30 = 24 W
24 = I[s] x 1200
Is = 0.02A
5. In transmitting power, why is it necessary to step up voltage before transmission? Explain. (2mks)
○ Minimizing energy losses.
○ Stepping up lowers the current hence minimizing energy losses.
1. Define radioactivity. (1mk)
☆ Disintegration of unstable nucleus.
2. Identify the radiations of tracks in the figures below. (2mks)
3. Identify radiations X and Y using the figure below. (2mks)
☆ x – alpha (α) particles.
y – Gamma (γ) rays.
z – Beta (β) particle.
4. The following reaction is part of radioactive series.
Identify radiation h and find figures of b and c.
☆ h – Beta (β) particles.
b = 82
c = 206
1. State Ohm’s law. (2mks)
☆ The current flowing through a metal conductor is proportional to the potential difference between its ends, provided the temperature and other physical conditions of the conductor remain
2. State two factors that affect heating by electric current (2mks)
☆ Magnitude of current.
☆ Resistance of conductor.
☆ Length of time the current passes through the conductor.
3. Determine the power of a motor which has a p.d of 240v applied across it when a current of 0.30A passes through it. (2mks)
☆ Power = VI
= 240 x 0.30
= 72 W
4. Explain why a fuse is always connected to the line wire in an electrical appliance (1mk)
☆ In case of an electrical fault, the fuse cuts of the circuit (no current flows) | {"url":"https://www.easyelimu.com/kenya-secondary-schools-pastpapers/mocks/2017/muranga-mocks/item/894-physics-paper-2-marking-scheme","timestamp":"2024-11-11T19:17:57Z","content_type":"text/html","content_length":"162245","record_id":"<urn:uuid:5866cdbf-8733-4b2a-a61d-8f020d4d42ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00821.warc.gz"} |
Parveen wanted to make a temporary shelter for her car, by making a box-like structure with tarpaulin that covers all the four sides and the top of the car (with the front face as a flap which can be rolled up). Assuming that the stitching margins are very small, and therefore negligible, how much tarpaulin would be required to make the shelter of height 2.5 m, with base dimensions 4 m × 3 m?
You must login to ask question.
NCERT Solutions for Class 9 Maths Chapter 13
Important NCERT Questions
Surface Areas and Volumes
NCERT Books for Session 2022-2023
CBSE Board, UP Board and Other state Boards
EXERCISE 13.1
Page No:213
Questions No:8 | {"url":"https://discussion.tiwariacademy.com/question/parveen-wanted-to-make-a-temporary-shelter-for-her-car-by-making-a-box-like-structure-with-tarpaulin-that-covers-all-the-four-sides-and-the-top-of-the-car-with-the-front-face-as-a-flap-which-can-be/","timestamp":"2024-11-15T00:11:10Z","content_type":"text/html","content_length":"162674","record_id":"<urn:uuid:f955d76b-a7bc-450b-8274-412a5c86c81a>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00454.warc.gz"} |
Excel Absolute Reference Shortcut For Mac - ExcelAdept
Key Takeaway:
• Excel absolute references are a powerful tool that allows you to maintain fixed references in your formulas, which is essential when dealing with large or complex spreadsheets.
• There are two main ways to create absolute references in Excel for Mac: using the F4 key or using the Function key. Both methods can save time and increase productivity by reducing the need for
manual editing.
• Examples of using absolute references in formulas and charts demonstrate how this functionality can be used to solve common spreadsheet challenges. However, it is important to remember to use
absolute references judiciously, as they can also make debugging and troubleshooting more challenging.
Have you ever been struggling with your Mac keyboard to enter an absolute reference in Excel? Don’t worry, we have the solution for you! This article provides a fast and easy shortcut to easily input
an absolute reference. You will be working faster in no time!
Excel Absolute Reference Shortcut Overview
When working with Excel on a Mac, utilizing absolute references can be a time-saver. Instead of typing cell references manually, use Excel’s built-in shortcut. By pressing the F4 key after selecting
a cell reference, you can switch between absolute, relative, and mixed cell references quickly. This shortcut helps ensure correct formulas and efficient work processes.
Using absolute references keeps formulas constant even when copying or dragging them to different locations. This ensures accurate calculations and saves time. To use the shortcut, simply select a
cell reference and press the F4 key to toggle between reference types. This is particularly useful when creating complex spreadsheets with multiple formulas and reference types.
It’s important to note that the F4 shortcut is only available on Mac keyboards with full-size function keys. Additionally, it may not be compatible with certain Excel versions or configurations.
Don’t miss out on the time-saving benefits of the Excel absolute reference shortcut. Incorporate this tool into your work process today and streamline your Excel experience.
Absolute Reference Basics
Mastering absolute reference basics in Excel for Mac? No sweat! Let us show you how to make sense of cell references and types of references. Get the hang of these concepts and you’ll have no problem
navigating the absolute reference shortcut in Excel.
Understand Cell References
Excel Cell References are essential for advanced worksheet functionality, and absolute references are vital for producing accurate calculations. By adjusting the reference type, you can ensure that
your formulas remain consistent as you copy them across cells. This is important to understand to optimize your Excel usage.
In addition to relative and mixed references, Absolute Cell References precisely identify the cell’s location in a worksheet by using dollar signs ($) before the column and row identifiers. Absolute
references remain consistent across different cell locations while copying formulas. Understanding these three types of cell references will help improve your efficiency when working with complex
To create an Absolute Reference in Excel, select a cell in which you have already entered a formula. Then, click on the formula bar and add dollar signs ($) before the row and column values you want
to make absolute. Another way of doing this is by using shortcuts – F4 button on Windows devices and Command + T on Mac devices – to convert existing cell references into an Absolute reference.
By mastering cell references in Excel worksheets, you can save time by setting up formulas that will automatically update throughout worksite changes. Remember that understanding Absolute Cell
References is crucial for advanced features like conditional formatting or organizing data more effectively in PivotTables or charts.
Cell references can be absolute, relative, or confused – just like my ex on a bad day.
Types of Cell References
A crucial aspect of Excel is Cell References, which can be categorized into distinct types based on the behavior and usage of the references.
A table with appropriate columns to explain the various types of cell references:
Type of Cell Reference Description
Relative Reference Refers to a cell’s relative location to the formula cell
Absolute Reference Refers to a specific fixed cell, indicated by ‘$’ sign
Mixed Reference Combines both Relative and Absolute References in a single cell reference
It is important to understand that absolute references are crucial when you want to lock a particular value or formula in a cell whilst copying it across other cells. Excel provides handy keyboard
shortcuts for Mac users enabling quicker easy access.
Pro Tip: Utilize absolute references shortcut keys while working on data-intensive projects.
Mac users rejoice, finally a shortcut that references quicker than your ex at a party.
Excel Absolute Reference Shortcut for Mac
Want to effortlessly make absolute references in Excel on a Mac?
Check out these 3 useful sub-sections:
1. F4 key
2. Function key
3. Basic tutorial
Find the solution to having a faster, more efficient workflow by creating absolute references in Excel with the F4 key, function key, and a basic tutorial.
Creating Absolute References
Creating an Excel Reference that never moves- Tips and Tricks you should know.
To create absolute references, users must learn how to lock a cell or row in place so that the values can be copied over while preserving their relative positioning. Here’s how:
1. Open the desired worksheet.
2. Select the cell where you want to enter the formula.
3. Enter an equal sign (=) to start your formula.
4. Enter your first reference: either click on a cell, type its name, or use arrow keys to navigate to it.
5. Press F4 on PC or Command + T on Mac once you’ve selected your reference – this will lock it down as an absolute reference.
6. Repeat the process for other variable cells and apply the formula.
Some important things to remember when creating absolute references are that dollar signs can also be used instead of function key shortcuts. To make a range of cells referred by a single dollar sign
from both top and bottom, put the dollar sign before every row number and column letter while indicating variables within range with two dollar signs on opposite sides.
Absolute references in Excel reduce errors and save time since they preserve original formulas across many calculations with periodic updates required only for specific domains dependent upon
There’s no need ever again to worry about moving cells inadvertently – these steps ensure accuracy thanks to Microsoft’s thoughtful design of their product! Say goodbye to endlessly scrolling and
clicking with F4, Excel’s absolute reference shortcut for Mac.
Using F4 to Create Absolute References
When working with Excel on a Mac, there is a shortcut that can be used to create absolute references. This shortcut allows you to easily lock in cells or ranges when copying formulas, preventing
unexpected changes to your data. Here’s how to use this feature:
1. Start by selecting the cell or range that you want to make an absolute reference.
2. Press the Fn + T command to open the formula bar.
3. Use the cursor keys or mouse pointer to navigate to the point where you want to insert the reference.
4. Press F4 once for relative value, twice for first type of absolute value ( $A$10 ) and three times if you want only row fixed ($A10).
5. Close the formula bar and proceed with your calculations.
It is important to note that while using these shortcuts, if ranges or cells have missing values, then the F4 keyboard shortcut may not work.
Using this shortcut can save time and prevent errors in data analysis for financial accounts or any other complex data-based tasks requiring precise formula work.
One interesting fact is that Excel was initially developed by Microsoft for Macintosh systems before it was later introduced for Windows in 1987.
Pressing Fn+F4 is like getting a tattoo of an absolute reference – it’s permanent and you better make sure it’s in the right spot.
Using Function Key to Create Absolute References
To lock reference values, you can utilize function keys as shortcuts. By doing so, the formula stays put when copying it to other cells.
1. Begin by highlighting the cell containing the formula you wish to use for absolute referencing.
2. Next, press ‘F4’ on your keyboard.
3. You will see dollar signs appear around the selected cell or range of cells in your formula, signifying an absolute reference is now in place.
4. You may also notice additional dollar signs surrounding other references within the same formula.
5. This indicates that all references within the same formula are locked as well, further reinforcing their absolute nature.
6. If you do not want a particular reference to be absolute, repeat step 2 until the correct combination of relative and absolute references is achieved.
Additionally, by using function keys to create absolute referencing on Mac Excel, you eliminate any potential errors that could arise from subsequent manipulation and copying of formulas.
Don’t miss out on conveniently created formulas with lesser chance for error. Start utilizing function keys for locking down your reference values in Excel today!
Get ready to Mac-et Excel with these absolute reference shortcuts – no command+C, command+V necessary.
Examples of Excel Absolute Reference Shortcut for Mac
To be a pro at Excel’s absolute reference shortcut for Mac, you must grasp its examples. To make your formula and chart creating easier, this section “Examples of Excel Absolute Reference Shortcut
for Mac” has two parts:
1. “Example 1: Absolute Reference in Formulas”
2. “Example 2: Absolute Reference in Charts”.
Example 1: Absolute Reference in Formulas
The practical implementation of Absolute Reference in Excel formulas is a valuable skill to possess for anyone who uses Excel frequently. With this knowledge, you can save time and achieve higher
accuracy in your work. In Excel, absolute references apply to fixed cells that do not change when you modify a formula. This shortcut helps avoid errors caused by accidental cell movements.
One way to make an absolute reference is by pressing "Fn+F4". Alternatively, use dollar signs ($) before each column and row to fix them into place within a formula. The dollar sign binds the column
or row so that it remains unchanged when you copy the formula across other cells.
It’s essential to understand that fixing only one reference doesn’t make it an absolute one. Instead, we need both the column and the row reference locked using the ‘$’ sign.
Once you get comfortable with experiencing this shortcut, it can save you hours of effort while enhancing your productivity. However, if this concept seems challenging at first, don’t be discouraged.
Keep practicing until you feel confident enough to use it efficiently.
I remember when I was starting with Excel and struggled with Absolute referencing. However, after continuously working with it, I realized how beneficial it was in my workflow and how much more
accurate my work had become because of it.
Why rely on wishful thinking when you can have the absolute reference shortcut in your pocket? Chart your course with ease on Mac-Excel!
Example 2: Absolute Reference in Charts
To use Excel absolute reference shortcut in charts, first select the data to be included in the chart. Then, click on ‘Insert’ in the main menu and choose the chart type. In the ‘Select Data’ option,
choose ‘Legend Entries’ and click on ‘Add’. Enter a name for the series and then select the range of cells containing the data for that series using absolute referencing.
Chart Type Select Data Legend Entries Add Series Name Select Data with Absolute Reference
Line Chart Select Data… Legend Entries (Series) Enter Series Name: =Sheet1!$B$2:$B$6
Pie Chart Edit Data… Name (Series) Add… =Sheet1!$C$2:$C$6
Using this method ensures that even if additional rows or columns are added to the source data, they will be automatically updated in the chart. Remember to use dollar signs to make an absolute
Pro Tip: Use keyboard shortcuts to speed up your workflow when working with charts in Excel for Mac. For example, Command+1 opens the formatting dialog box for selected objects in a chart.
Tips to Remember When Using Excel Absolute Reference Shortcut for Mac
When using Excel on a Mac, it is essential to know the tips for using the absolute reference shortcut effectively. This article will guide you through the essential tips to remember when manipulating
Excel absolute references on a Mac platform.
– To create an absolute reference in Excel on a Mac, use the command key with the ‘T’ key while typing ‘$’ before the cell address you want to lock.
– If you need to create an absolute range reference, use the command key with the ‘T’ key while typing ‘$’ before the column and row numbers you want to lock.
– To copy a formula with an absolute reference, make sure to copy the entire formula with the absolute reference markers before you paste.
– If you need to use relative references again, you can undo absolute referencing quickly by pressing the F4 key after selecting the cell or cell reference.
– The absolute reference shortcut for a Mac is an excellent tool when working with large tables where constant changes are necessary.
– However, be careful when using the absolute reference shortcut for Mac on multiple ranges and cells in a complex spreadsheet, as it may cause errors or breakage, resulting in incorrect
When using the absolute reference shortcut for Mac, it is important to keep in mind the limitation of its effectiveness in complex spreadsheets. When working with large tables, the absolute reference
shortcut is an essential tool to save time and produce accurate results, allowing for easy maintenance of large and complex spreadsheets.
Regarding the history of the absolute reference shortcut for Mac, it has been available for use since around 2001, when Microsoft launched the first Office suite for Macs. Since then, the Excel
absolute reference shortcut for Mac has become an integral part of Apple’s productivity software ecosystem. Overall, the absolute reference shortcut has been an essential tool for saving time when
working with large tables, making it a valuable asset for Mac users.
Five Facts About Excel Absolute Reference Shortcut for Mac:
• ✅ The shortcut for absolute reference in Excel for Mac is ⌘ + $. (Source: Microsoft)
• ✅ Absolute references in Excel for Mac allow for fixed cell references that do not change when copied or moved. (Source: Excel Campus)
• ✅ The relative reference shortcut in Excel for Mac is ⌘ + R. (Source: Microsoft)
• ✅ The mixed reference shortcut in Excel for Mac is ⌘ + T. (Source: Excel Campus)
• ✅ Using absolute references can make it easier to create complex formulas in Excel for Mac. (Source: Spreadsheeto)
FAQs about Excel Absolute Reference Shortcut For Mac
What is the Excel absolute reference shortcut for Mac?
The Excel absolute reference shortcut for Mac is Command + Shift + $.
How do I use the Excel absolute reference shortcut for Mac?
To use the Excel absolute reference shortcut for Mac, simply select the cell that contains the formula you want to lock, then press Command + Shift + $.
What does the Excel absolute reference shortcut for Mac do?
The Excel absolute reference shortcut for Mac locks the cell reference in a formula, preventing it from changing when you copy the formula to other cells.
Can I use the Excel absolute reference shortcut for Mac with multiple cells?
Yes, you can use the Excel absolute reference shortcut for Mac with multiple cells by selecting the cells that contain the formulas you want to lock, then pressing Command + Shift + $.
Is there a keyboard shortcut to toggle between absolute and relative cell references in Excel for Mac?
Yes, the keyboard shortcut to toggle between absolute and relative cell references in Excel for Mac is Command + T.
Can I customize the Excel absolute reference shortcut for Mac?
Yes, you can customize the Excel absolute reference shortcut for Mac by going to System Preferences > Keyboard > Shortcuts > App Shortcuts, then adding a new shortcut for Excel with the desired
keyboard combination. | {"url":"https://exceladept.com/excel-absolute-reference-shortcut-for-mac/","timestamp":"2024-11-03T23:26:29Z","content_type":"text/html","content_length":"73715","record_id":"<urn:uuid:95e2e665-54be-4c39-ba8b-6d6a7eac6221>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00838.warc.gz"} |
What is equal to \
Hint: To solve the question, we have to apply trigonometric identities and the values of trigonometric functions to arrive at the value of \[\tan \left( \dfrac{\pi }{12} \right)\].
Complete step-by-step answer:
We know that the formula for \[\tan 2\alpha \] is given by \[\dfrac{2\tan \alpha }{1-{{\tan }^{2}}\alpha }\]
By substituting \[\alpha =\dfrac{\pi }{12}\] in the above formula we get
\[\tan \left( \dfrac{2\pi }{12} \right)=\dfrac{2\tan \left( \dfrac{\pi }{12} \right)}{1-{{\left( \tan \left( \dfrac{\pi }{12} \right) \right)}^{2}}}\]
\[\tan \left( \dfrac{\pi }{6} \right)=\dfrac{2\tan \left( \dfrac{\pi }{12} \right)}{1-{{\left( \tan \left( \dfrac{\pi }{12} \right) \right)}^{2}}}\] …….. (1)
We know that the value of \[\tan \left( \dfrac{\pi }{6} \right)\] is equal to \[\dfrac{1}{\sqrt{3}}\]
By substituting the above mentioned value in equation (1) we get,
\[\dfrac{1}{\sqrt{3}}=\dfrac{2\tan \left( \dfrac{\pi }{12} \right)}{1-{{\left( \tan \left( \dfrac{\pi }{12} \right) \right)}^{2}}}\]
Cross multiply the above expression to obtain a quadratic expression of \[\tan \left( \dfrac{\pi }{12} \right)\].
\[1-{{\left( \tan \left( \dfrac{\pi }{12} \right) \right)}^{2}}=\sqrt{3}\left( 2\tan \left( \dfrac{\pi }{12} \right) \right)\]
\[{{\left( \tan \left( \dfrac{\pi }{12} \right) \right)}^{2}}+2\sqrt{3}\left( \tan \left( \dfrac{\pi }{12} \right) \right)-1=0\] …….. (2)
We know that the solutions of the general quadratic expression \[a{{x}^{2}}+bx+c=0\] are given by \[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
On comparing the above expression with equation (2) we get,
The values of a = 1, b = \[2\sqrt{3}\], c = -1
Thus, the possible values of \[\tan \left( \dfrac{\pi }{12} \right)\] are equal to \[\dfrac{-2\sqrt{3}\pm \sqrt{{{\left( 2\sqrt{3} \right)}^{2}}-4(1)(-1)}}{2(1)}\]
We know that \[{{\left( ab \right)}^{m}}={{a}^{m}}\times {{b}^{m}}\]
\[=\dfrac{-2\sqrt{3}\pm \sqrt{\left( {{2}^{2}}\times 3 \right)+4}}{2}\]
\[=\dfrac{-2\sqrt{3}\pm \sqrt{12+4}}{2}\]
\[=\dfrac{-2\sqrt{3}\pm \sqrt{16}}{2}\]
\[=\dfrac{-2\sqrt{3}\pm 4}{2}\]
Since we know that the value of \[\sqrt{16}=\sqrt{4\times 4}=\sqrt{{{4}^{2}}}=4\].
\[\Rightarrow \tan \left( \dfrac{\pi }{12} \right)=\dfrac{2\left( -\sqrt{3}\pm 2 \right)}{2}\]
\[\tan \left( \dfrac{\pi }{12} \right)=-\sqrt{3}\pm 2\]
We know that \[\tan \alpha \] is positive in the interval \[0<\alpha <\dfrac{\pi }{2}\] . Thus, we get
\[\tan \left( \dfrac{\pi }{12} \right)=2-\sqrt{3}\]
\[\therefore \] The value of \[\tan \left( \dfrac{\pi }{12} \right)\] is equal to \[2-\sqrt{3}\]
Note: The possibility of mistake can be the calculation since the procedure of solving requires square root terms. The other possibility of mistake is not being able to choose the correct answer out
of the obtained two values. The alternative way of solving can be, to calculate the value of \[\cos \left( \dfrac{\pi }{12} \right),\sin \left( \dfrac{\pi }{12} \right)\] since we know the value of \
[\cos \left( \dfrac{\pi }{6} \right)\] is equal to \[\dfrac{\sqrt{3}}{2}\] . By substituting the values in the formula \[\cos 2\alpha =2{{\cos }^{2}}\alpha -1=1-2{{\sin }^{2}}\alpha \] , we can
calculate the value of \[\cos \left( \dfrac{\pi }{12} \right),\sin \left( \dfrac{\pi }{12} \right)\]. Using the formula \[\tan \alpha =\dfrac{\sin \alpha }{\cos \alpha }\] we can calculate the value
of \[\tan \left( \dfrac{\pi }{12} \right)\]. This method eases the procedure of solving. | {"url":"https://www.vedantu.com/question-answer/equal-to-tan-left-dfracpi-12-right-a-2sqrt-class-11-maths-cbse-5ee872bbe689f505aa8f98a3","timestamp":"2024-11-14T10:21:48Z","content_type":"text/html","content_length":"170316","record_id":"<urn:uuid:8af34da6-29cc-4778-8fef-815ab5cd770a>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00548.warc.gz"} |
NCERT Solutions Class 5 Maths Chapter 1 - The Fish Tale | Free PDF
Which Boat Gets How Much?
│Type of boat │Catch of fish in one trip (in Kg)│Speed of the boat (how far it goes in one hour) │
│Long boat │20 │4Km per hour │
│Long tail Boat│600 │12 Km per hour │
│Motor boat │800 │20 Km per hour │
│Machine boat │6000 │22 Km per hour │
a) About how many fish will each type of boat bring in seven trips?
b) About how far can a motor boat go in six hours?
c) If a long tail boat has to travel 60 km, how long will it take? | {"url":"https://www.orchidsinternationalschool.com/ncert-solutions-class-5-maths/chapter-1-the-fish-tale","timestamp":"2024-11-13T09:44:39Z","content_type":"text/html","content_length":"255407","record_id":"<urn:uuid:7582afa6-a2be-4484-890a-0691f5a50ded>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00229.warc.gz"} |
I-2 The Sets Theory
Chapter I-2 The Sets Theory
For the archives Wayback Machine from before May 2024, seek with the ancient URL:
Recently (20^th century) the Sets Theory attempted to give a fair basis to the whole logic and mathematics while keeping as free as possible of any reference to peculiar objects, whatever they are.
Let us first remind what Sets Theory is.
The Sets Theory is based on three axioms supposed to be obvious universal truths, from which all mathematical truths (calculations, theorems) can be drawn through rigorously exact logical reasoning
1st axiom: Any collection of objects for which there is a criterion making it possible to say if these objects are elements or not of the collection is a set.
2nd axiom: A set cannot be an element of itself.
3rd axiom: The collection of all the sets is not a set.
(Added July 2015, reviewed and moved November 2017: this is the Sets theory I learned in secondary school, and which is discussed in science reviews. I am well aware that wikipedia gives a different
description, quite confuse and unconnected to the other parts of mathematics. But I am more confident in the secondary school than in anonymous editors, leaving them the care of explaining themselves
publicly on the difference)
(In the Sets Theory as well as in all the following texts, «object» refers to any element on which we shall make logical reasoning, whatever it belongs to concrete domains, abstract domains, ideas,
feelings, spirit, or even imaginary)
From these three axioms we can find all the familiar mathematical objects. For example we define the empty set which does not contain any element, and is said of cardinal zero. A set which only
contains the empty set is said of cardinal one; a set which contains the empty set more the previous is said of cardinal two, a set which contains the empty set more the two previous is said of
cardinal three, and so on we define all the integers. Their relations give the arithmetic operations, addition, subtraction, multiplication, division, then with various rigorous constructions we find
the rational numbers, then real numbers... The Aristotelian logic, its AND and its OR, results from the familiar operations of intersection and union of sets. At last the study of the structures of
these sets leads to algebra, equations and vector spaces. This notion of an «abstract space» into a set, matching exactly with our everyday life notion of space, will be of a great importance in part
three on metaphysics.
I state here that the axioms of the Sets Theory, far from being absolute truths, represent already a given choice, which only selects the results that they are supposed to demonstrate.
The third axiom already arises questioning. No school child ever wondered: why the collection of all the sets would not be a set? The problem is that if we regard it as such, then the second axiom is
violated. If we do not regard it as such, then the first is violated. The choice which was however made by mathematicians has good reasons, but there is a feeling as if this third axiom is there only
to hide a paradox, a situation where logic is impotent to determine reality. Why simply not to recognize that logic has limits and that it sometimes leads to paradoxes, statements which we cannot
show if they are true or false? Instead of stating that this set is not one, it is enough to say that its existence is a paradox. And, after having a glance through the window, we would obtain a
rigorous demonstration of a fact: Paradoxes never prevented wheat to grow.
The second axiom also appeared essential to members of the Bourbaki group who formalized the Sets Theory. However some mathematicians remove it, thus leading toward another theory, known as of the
Supersets Theory (note 36). At a rough guess, some paradoxes may appear there in quantity, because of circular references, thus bringing back this discussion to that of the third axiom: in the
classical Sets Theory, we refuse to consider self-contained sets, to avoid appearance of paradoxes or unsettlable statements. We shall have occasion to speak again of paradoxes throughout this book,
and especially in the third part on metaphysics, where we found them a very good practical use.
But the main problem I want to point at now is with the first axiom, to which, as far as I know, no mathematician made any criticism. And yet I see there two completely arbitrary and very interesting
a priori:
1) The idea that objects are unavoidably separated and distinct the ones from the others.
2) The idea of absolute criterion for belonging to the set, completely true or completely false, without any nuance.
It is not the matter here to say that the first axiom would be «false», but to make the reader aware that it results from a choice among various possibilities, in favour of special objects which
would have the particular properties described above. Only this choice leads to the Sets Theory. Other choices were considered by mathematicians, but they remained poorly studied, as unworkable...
for Aristotelian logic!
What I say here is that considering objects with different properties would lead to other theories, and thus to other logics. And I dispute that Aristotelian logic (statements with univocal meaning,
without any ambiguity completely true or completely false) «is demonstrated» by the Sets Theory. This logic simply arises from the two choices above; it is implied from the beginning by these two
choices. In peculiar the second choice forcefully leads to the idea of completely true or completely false statement, which is the very basis of Aristotelian logic. If we want to demonstrate the
theorems of Aristotelian logic and to build mathematics starting from these two choices, the Sets Theory is perfect; but its basic axioms cannot be demonstrated from its results. With my opinion it
would be necessary to reformulate the first axiom of the Sets Theory, in order to show that Aristotelian logic directly arises from the two above choices.
Then is Aristotelian logic «true»? Is it «false»? Or does its veracity depends on the direction of the wind or on our goodwill, as claim those who use this kind of arguments biased by
self-centredness to dispute ethics and escape its obligations? What I state here is that there is no meaning to say that Aristotelian logic would be absolutely true or absolutely false, but that
Aristotelian logic is valid for peculiar objects which satisfy the two above criteria (choice 1 and 2), and only for them. Independently of course of our personal interests, this significant point
being discussed in detail in chapter I-8.
Now, what happens in the next cases, with other types of objects? It is what we are goint to see in the next chapter I-3.
Legal and copyright notice.
Modified in 2024
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crowd-sourcing systems.
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Equivalent Microwave Circuit Technique for Waveguide Iris Polarizers Development
Equivalent Microwave Circuit Technique for Waveguide Iris Polarizers Development
polarizer, waveguide polarizer, iris polarizer, circular polarization, scattering matrix, transmission matrix, differential phase shift, voltage standing wave ratio, axial ratio, crosspolar
The increase of information volumes, which are transmitted in modern satellite telecommunication systems, requires the development of new signal processing technologies, microwave devices, antenna
systems and methods of their analysis. In particular, polarization-adaptive antennas are widely used for this purpose. Such antennas provide the possibility to transmit and receive radio signals with
polarization of any type. Polarization processing devices of antenna systems must provide low levels of voltage standing wave ratio of waves with horizontal and vertical linear polarizations and high
crosspolar discrimination simultaneously. Therefore, there is the need to improve designs and methods of analysis of modern polarization processing devices.
Polarizers based on a square waveguides with irises are widely used due to the simplicity of their design and manufacturing by milling technology. The article considers a new mathematical model of
waveguide polarizers with reactive irises. For the example of model application we have simulated and optimized a polarizer based on a square waveguide with four irises. A mathematical model of this
waveguide polarizer was developed based on the description of microwave devices and their elements by wave scattering and transmission matrices. The general scattering matrix of the polarizer has
been obtained analytically. The main electromagnetic characteristics of the polarizer were determined based on the elements of this matrix. As a result, we have analyzed main characteristics of the
model, including differential phase shift, voltage standing wave ratio for vertical and horizontal polarizations, axial ratio and crosspolar discrimination. The optimization of the characteristics of
a polarizer has been performed using the developed mathematical model and a software based on the finite integration technique. The optimal characteristics and geometrical sizes of the structure are
in good agreement, which proves the correctness of the developed mathematical model of square waveguide iris polarizers.
Virone G., Tascone R., Peverini O. A., Addamo G. and Orta R. (2008) Combined phase shift waveguide polarizer. IEEE Microwave and Wireless Components Letters, Vol. 18, No. 8., pp. 509-511. DOI:
Ruiz-Cruz J. A., Fahmi M. M., Fouladi S. A., Mansour R. R. (2011) Waveguide antenna feeders with integrated reconfigurable dual circular polarization. IEEE Transactions on Microwave Theory and
Techniques, Vol.59, No. 12., pp. 3365-3374. DOI: 10.1109/TMTT.2011.2170581.
Hwang S.-M., Kim J.-M. and Lee K.-H. (2012) Study on design parameters of waveguide polarizer for satellite communication, IEEE Asia-Pacific Conference on Antennas and Propagation, pp. 153-154. DOI:
Chung M.-H., Je D.-H. and Han S.-T. (2014) Development of a 85-115 GHz 90-deg phase shifter using corrugated square waveguide, European Microwave Conference, pp. 1146-1149. DOI: 10.1109/
Dubrovka F. F., Piltyay S. I. (2017) Novel high performance coherent dual-wideband orthomode transducer for coaxial horn feeds. 2017 XI IEEE International Conference on Antenna Theory and Techniques
(ICATT), pp. 277-280. DOI: 10.1109/ICATT.2017.7972642.
Pollak A. W. and Jones M. E. (2018) A compact quad-ridge orthogonal mode transducer with wide operational bandwidth, IEEE Antennas and Wireless Propagation Letters, Vol. 17, No. 3., pp. 422–425. DOI:
Agnihotri I. and Sharma S. K. (2019) Design of a compact 3D metal printed Ka-band waveguide polarizer, IEEE Antennas and Wireless Propagation Letters, Vol. 18, No. 12, pp. 2726-2730. DOI: 10.1109/
Tribak A., Mediavilla A., Cano J. L., Boussouis M. and Cepero K. Ultra-broadband low axial ratio corrugated quad-ridge polarizer, European Microwave Conference, pp. 73-76. DOI: 10.23919/
Mediavilla A., Cano J.L. and Cepero K. Quasi-octave bandwidth phase matched K/Ka antenna feed subsystem for dual RHCP/LHCP polarization, 42nd European Microwave Conference, 29-31 Oct. 2012,
Amsterdam,Netherlands, pp. 1099-1102. DOI: 10.23919/EUMC.2012.6459338.
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How to Cite
Bulashenko , A. V. and Piltyay, S. I. (2020) “Equivalent Microwave Circuit Technique for Waveguide Iris Polarizers Development”, Visnyk NTUU KPI Seriia - Radiotekhnika Radioaparatobuduvannia, (83),
pp. 17-28. doi: 10.20535/RADAP.2020.83.17-28.
Electrodynamics. Microwave devices. Antennas
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Limit Cycles | JustToThePointLimit Cycles
If you’re going through hell, keep on going, Winston Churchill.
Sometimes people don’t want to hear the truth because they don’t want their illusions destroyed, Friedrich Nietzsche.
Differential equations
An algebraic equation is a mathematical statement that declares or asserts the equality of two algebraic expressions. These expressions are constructed using:
1. Dependent and independent variables. Variables represent unknown quantities. The independent variable is chosen freely, while the dependent variable changes in response to the independent
2. Constants. Fixed numerical values that do not change.
3. Algebraic operations. Operations such as addition, subtraction, multiplication, division, exponentiation, and root extraction.
Definition. A differential equation is an equation that involves one or more dependent variables, their derivatives with respect to one or more independent variables, and the independent variables
themselves, e.g., $\frac{dy}{dx} = 3x +5y, y’ + y = 4xcos(2x), \frac{dy}{dx} = x^2y+y, etc.$
It involves (e.g., $\frac{dy}{dx} = 3x +5y$):
• Dependent variables: Variables that depend on one or more other variables (y).
• Independent variables: Variables upon which the dependent variables depend (x).
• Derivatives: Rates at which the dependent variables change with respect to the independent variables, $\frac{dy}{dx}$
The Existence and Uniqueness Theorem provides crucial insight into the behavior of solutions to first-order differential equations ODEs. It states that if:
• The function f(x, y) (the right-hand side of the ODE) in y’ = f(x, y) is continuous in a neighborhood around a point (x[0], y[0]) and
• Its partial derivative with respect to y, $\frac{∂f}{∂y}$, is also continuous near (x[0], y[0]).
Then the differential equation y' = f(x, y) has a unique solution to the initial value problem through the point (x[0], y[0]) .
A first-order linear differential equation (ODE) has the general form: a(x)y' + b(x)y = c(x) where y′ is the derivative of y with respect to x, and a(x), b(x), and c(x) are functions of x. If c(x) =
0, the equation is called homogeneous, i.e., a(x)y’ + b(x)y = 0.
The equation can also be written in the standard linear form as: y’ + p(x)y = q(x) where $p(x)=\frac{b(x)}{a(x)}\text{ and }q(x) = \frac{c(x)}{a(x)}$
A second-order linear homogeneous differential equation ODE with constant coefficients is a differential equation of the form: y'' + Ay' + By = 0 where:
• y is the dependent variable (a function of the independent variable t),
• y′ and y′′ are the first and second derivatives of y with respect to t,
• t is the independent variable,
• A and B are constants.
This equation is homogeneous, meaning that there are no external forcing terms (like a function of t) on the right-hand side.
Limit Cycles in Non-Linear Autonomous System
In the study of differential equations and dynamical systems, non-linear autonomous systems play a crucial role due to their complex and rich behavior. One of the fascinating phenomena exhibited by
such systems is the occurrence of limit cycles, which are closed trajectories representing periodic solutions.
Consider a general non-linear autonomous system of the form:
$\begin{cases} x’ = f(x, y) \\ y’ = g(x, y) \end{cases}$ where:
• x’ and y’ denote the time derivatives $\frac{dx}{dt}$ and $\frac{dy}{dt}$, respectively.
• f(x, y) and g(x, y) are non-linear functions that govern the time evolution of the variables x and y. This system describes how the variables x and y change with respect to time.
• f(x, y) and g(x, y) are non-linear functions that govern the time evolution of the variables x and y.
This system describes how the variables x and y change with respect to time, based on their current values.
Visualizing the System: The Velocity Field
To better understand the behavior of this system, we can construct the velocity field $\vec{F}$ which provides a geometric interpretation of the system. The velocity field is defined as: $\vec{F} = f
(x, y)\hat{\mathbf{i}}+g(x, y)\hat{\mathbf{j}}$ where
• $f(x, y)\hat{\mathbf{i}}$ represents the velocity component in the x-direction.
• $g(x, y)\hat{\mathbf{j}}$ represents the velocity component in the y-direction.
• $\hat{\mathbf{i}}$ and $\hat{\mathbf{j}}$ are unit vectors in the x and y directions, respectively.
This vector field describes how the values of x and y change over time at every point in the plane. Each point (x, y) has an associated vector $\vec{F}(x, y)$ indicating the direction and speed at
which the system evolves from that point.
Trajectories in the Phase Plane
Solutions to the system are pairs of functions x(t) and y(t), but geometrically, they are trajectories or paths traced out by the evolving system in the xy-plane. These trajectories follow the
direction of the vector field $\vec{F}$, meaning that at any point along the trajectory, the tangent to the path is given by $\vec{F}(x, y)$.
Refer to Figure i for a visual representation and aid in understanding it.
Critical Points: Where the Velocity Field Vanishes
A critical point of the system occurs where the velocity field is zero, i.e., where the system's rate of change is zero. Mathematically, this happens when both $f(x_0, y_0) = 0, g(x_0, y_0) = 0$,
meaning that the velocity at the point (x[0], y [0]) is zero. At such points, the system is at equilibrium, meaning there is no motion, and the solution remains constant over time.
In terms of the field, these points are where: $\vec{F}(x_0, y_0) = f(x_0, y_0)\hat{\mathbf{i}}+g(x_0, y_0)\hat{\mathbf{j}} = \vec{0}$
Critical points are important because they often represent stable or unstable equilibrium states where the system tends to “settle” or from which it may “escape” in the long term, depending on the
nature of the equilibrium.
Stability of Critical Points
The behavior near critical points can be analyzed using linearization and the eigenvalues of the Jacobian matrix:
J = $(\begin{smallmatrix}\frac{∂f}{∂x} & \frac{∂f}{∂y}\\ \frac{∂g}{∂x} & \frac{∂g}{∂y}\end{smallmatrix})\bigg|_{(x_0, y_0)}$
The eigenvalues of J determine the local behavior near the critical point:
• If both eigenvalues have negative real parts, the critical point is a stable node or focus. Solutions near the critical point converge to it as t → ∞.
• If both eigenvalues have positive real parts, it’s an unstable node or focus. Solutions near the critical point diverge from it as t → ∞.
• If eigenvalues are real and of opposite signs, the critical point is a saddle point. Solutions approach the critical point along one eigenvector and move away along the other.
• If eigenvalues are purely imaginary conjugates, the critical point is a center. Solutions orbit around the critical point without converging or diverging
Closed Trajectories and Limit Cycles
A closed trajectory is a path in the xy-plane that loops back to its starting point and then repeats itself indefinitely. In other words, if a trajectory is closed, the system exhibits periodic
behavior, that is, after a certain period T, the system returns to its initial state, and this cycle repeats over and over.
Geometrically, a closed trajectory represents a periodic solution (x(t), y(t)) such that: x(t + T) = x(t), y(t + T) = y(t) for all t.
Importantly, a closed trajectory does not cross itself —the system cannot have two different velocity directions at the same point due to the uniqueness of solutions in differential equations (Refer
to Figure v for a visual representation and aid in understanding it). This property ensures that the system’s motion is uniquely determined by the initial conditions and evolves in a smooth,
continuous manner.
Example: A Simple Harmonic Oscillator
The simple harmonic oscillator is a fundamental example in differential equations and physics, representing systems that exhibit periodic motion, such as springs and pendulums under ideal conditions.
Consider the system of differential equations: $\begin{cases} x’ = y \\ y’ = -x \end{cases}$ where x = x(t) and y = y(t) are functions of time t (two variables that oscillate over time), and x’ and
y’ denote derivatives of x and y, respectively with respect to time. This system describes how the variables x and y evolve over time, with each variable depending on the other.
This system describes a simple harmonic oscillator.
The matrix representation of the system is: $\vec{x’} = Ax$ where A = $(\begin{smallmatrix}0 & 1\\ -1 & 0\end{smallmatrix})$ is the coefficient matrix, and $\vec{x} = (\begin{smallmatrix}x\\ y\end
{smallmatrix})$ is the state vector. This compact form allows us to use linear algebra techniques to solve the system.
To find the eigenvalues, we solve the characteristic equation: det(A-λI) = $det(\begin{smallmatrix}-λ & 1\\ -1 & -λ\end{smallmatrix}) = 0 ↭ (−λ)(−λ)−(1)(−1) = 0 ↭ λ^2 + 1 = 0 ↭ λ^2 = -1$. The
eigenvalues are λ[1] = i and λ[2] = -i. These complex eigenvalues indicate that the system exhibits rotational behavior (oscillatory) in the xy-plane (phase plane).
To find the eigenvectors, we substitute each eigenvalue back into $(A−λI)\vec{v} = \vec{0}$
For λ[1] = i, $(\begin{smallmatrix}-i & 1\\ -1 & -i\end{smallmatrix})(\begin{smallmatrix}a_1\\ a_2\end{smallmatrix}) = (\begin{smallmatrix}0\\ 0\end{smallmatrix})$
From the first equation: -ia[1] +a[2] = 0 ⇒ a[2] = ia[1]. From the second row: -a[1] -ia[2] = 0 ⇒[Substitute a[2] = ia[1]] -a[1] -i(ia[1]) = -a[1] + a[1] = 0. We can choose a[1] = 1 (since
eigenvectors are determined up to a scalar multiple). Therefore, the eigenvector is: $\vec{v_1}=(\begin{smallmatrix}1\\ i\end{smallmatrix})$
For λ[2] = -i, $(\begin{smallmatrix}i & 1\\ -1 & i\end{smallmatrix})(\begin{smallmatrix}a_1\\ a_2\end{smallmatrix}) = (\begin{smallmatrix}0\\ 0\end{smallmatrix})$
From the first equation: ia[1] +a[2] = 0 ⇒ a[2] = -ia[1]. Choosing a[1] = 1, we have: $\vec{v_2}=(\begin{smallmatrix}1\\ -i\end{smallmatrix})$
The general complex solution to the system in complex form is a linear combination of the eigenvectors multiplied by the exponential of their eigenvalues: $\vec{x}(t) = c_1e^{it}\vec{v_1} + +c_2e^
{-it}\vec{v_1}$ where c[1] and c[2] are complex constants determined by initial conditions.
Substitute the eigenvectors: $\vec{x}(t) = c_1e^{it}(\begin{smallmatrix}1\\ i\end{smallmatrix})+c_2e^{-it}(\begin{smallmatrix}1\\ -i\end{smallmatrix})$
To find real-valued solutions, we express the complex exponentials using Euler’s formula: $e^{it} = cos(t) + isin(t), e^{-it} = cos(t) - isin(t)$
Substitute back into the general solution: $\vec{x}(t) = c_1(cos(t)+isin(t))(\begin{smallmatrix}1\\ i\end{smallmatrix})+c_2(cos(t)-isin(t))(\begin{smallmatrix}1\\ -i\end{smallmatrix}) = (\begin
{smallmatrix}c_1(cos(t)+isin(t))+c_2(cos(t)-isin(t))\\ c_1(icos(t)-sin(t))+c_2(-icos(t)-sin(t))\end{smallmatrix})$
Let’s use the real and imaginary parts of the complex solutions to construct real solutions.
First Real Solution. Take the real part of $e^{it}\vec{v_1}$: $\mathbb{R}[e^{it}\vec{v_1}] = (\begin{smallmatrix}cos(t)\\ -sin(t)\end{smallmatrix})$
Second Real Solution. Take the imaginary part of $e^{it}\vec{v_1}$: $\mathbb{ℑ}[e^{it}\vec{v_1}] = (\begin{smallmatrix}sin(t)\\ cos(t)\end{smallmatrix})$
Therefore, the general real solution is: $c_1(\begin{smallmatrix}cos(t)\\ -sin(t)\end{smallmatrix})+c_2(\begin{smallmatrix}sin(t)\\ cos(t)\end{smallmatrix})$ where c[1] and c[2] are real constants
determined by initial conditions.
The solutions x(t) and y(t) represent sinusoidal functions with the same frequency but possibly different amplitudes and phases. The trajectories in the phase plane are closed curves, specifically
circles or ellipses, depending on the constants c[1] and c[2].
For the simple harmonic oscillator, this is a family of concentric circles centered at the origin, representing periodic motion [1]. Each trajectory is a closed curve, and the motion goes around
clockwise indefinitely, which is typical of simple harmonic oscillators. (Refer to Figure ii for a visual representation and aid in understanding it).
In linear systems like the simple harmonic oscillator, all trajectories are closed curves (circles) but are not isolated —there is a family of closed trajectories filling the phase plane.
Linear systems do not have limit cycles in the strict sense because limit cycles are a feature of non-linear systems where closed trajectories are isolated.
Circular Trajectories
For the simple harmonic oscillator, the trajectories are circles centered at the origin [1]. To see this, consider the expressions x(t) = c[1]cos(t) + c[2]sin(t), y(t) = -c[1]sin(t) + c[2]cos(t).
$x^2(t)+y^2(t) =[\text{Expanding both terms}] c_1^2·cos^2(t) + 2c_1c_2sin(t)cos(t) + c_2^2·sin^2(t) + c_1^2·sin^2(t) -2c_1c_2sin(t)cos(t) +c_2^2cos^2(t) =[\text{Simplify}] c_1^2(cos^2(t)+sin^2(t)) +
c_2^2(sin^2(t) + cos^2(t)) + (2c_1c_2sin(t)cos(t)-2c_1c_2sin(t)cos(t)) = (c_1^2+c_2^2)(cos^2(t)+sin^2(t)) = c_1^2+c_2^2 ↭ x^2(t)+y^2(t) = c_1^2+c_2^2 = R^2$ where R = $\sqrt{c_1^2+c_2^2}$ is the
radius of the circle. Therefore, the trajectories of the system are circles of radius R centered at the origin.
Limit Cycles
A limit cycle is a closed trajectory in the phase plane that is isolated, meaning that nearby trajectories are not closed and either spiral towards or away from the limit cycle.
Limit cycles are significant because they represent sustained oscillations in the system, which can be:
• A stable limit cycle is one where all neighboring trajectories approach the cycle as t → ∞. The system eventually returns to the limit cycle even if disturbed. In other words, a limit cycle
attracts nearby trajectories, causing the system to settle into a repeating, periodic behavior. Nearby trajectories spiral inward toward the limit cycle, getting closer and closer, but never
quite touching it. (Refer to Figure iii for a visual representation and aid in understanding it).
• An unstable limit cycle repels nearby trajectories. In this case, any small disturbance will cause the system to move away from the limit cycle, neighboring trajectories diverge from the limit
cycle as t → ∞. The system does not return to the periodic motion described by the limit cycle after a disturbance.
• A semi-stable is one where trajectories on one side approach the limit cycle, while those on the other side move away. This means the limit cycle is stable from one direction and unstable from
the other.
Limit cycles cannot occur in linear systems; they are unique to non-linear systems due to their complex interactions. In linear systems, any closed trajectories (such as circles or ellipses) are not
isolated —they form a continuum of closed orbits filling the phase plane, and thus, they do not satisfy the definition of a limit cycle.
One real-world example of a limit cycle is the natural process of breathing. Breathing is a periodic motion that can be modeled as a limit cycle. If the system (your breathing) is disturbed, say, by
a temporary obstruction, physical exercise, or a moment of anxiety, it will gradually return to its original, stable pattern of breathing. This resilience to disturbances makes it an example of a
stable limit cycle.
The Existence Problem of Limit Cycles
Understanding whether a system has a limit cycle is a crucial challenge in the study of non-linear dynamical systems. Unfortunately, there is no universal method to directly determine the existence
of limit cycles in every situation.
There are various approaches used by scientists to predict and identify limit cycles:
1. Intuition and Physical Insight: Systems modeled after physical phenomena can be analyzed using intuition about these phenomena. For example, in ecological models like predator-prey dynamics,
oscillatory behavior is expected, suggesting the presence of limit cycles.
2. Computer Simulations. Since there are no universal analytical methods for finding limit cycles, scientists frequently rely on numerical simulations. Numerical simulations allow visualization of
trajectories in the phase plane. By simulating the system with different initial conditions, potential closed trajectories (limit cycles) can be identified. Software tools such as MATLAB,
Mathematica, or Python libraries (like NumPy and SciPy) are commonly used.
3. Analytical Methods: Although general methods are scarce, there are some analytical tools, like the Poincaré-Bendixson Theorem and Bendixson’s Criterion, that provide useful insights into the
existence or non-existence of limit cycles in specific systems.
While finding limit cycles can be difficult, in some cases, it is possible to rule out their existence by applying specific criteria.
Bendixson’s Criterion
Bendixson’s Criterion is a method to exclude the possibility of closed trajectories (and hence, limit cycles) within a region of the plane.
Bendixson’s criterion. Let D be a simply connected region of the xy-plane. Consider a continuously differentiable vector field: $\vec{F} = f(x, y)\hat{\mathbf{i}} + g(x, y)\hat{\mathbf{j}}$ governing
the system:
$\begin{cases} x’ = f(x, y) \\ y’ = g(x, y) \end{cases}$
The divergence of the vector field is given by: $div \vec{F} = f_x + g_y = \frac{∂f}{∂x} + \frac{∂g}{∂y}$, where f(x, y) and g(x, y) are the components of the vector field governing the time
evolution of x and y, respectively. If the divergence $div(\vec{F})$ is continuous throughout the region D and does not change sign, (i.e., it is always positive or always negative) and is not
identically zero, then there are no closed trajectories (and therefore no limit cycles) lying entirely within D.
If the divergence of the vector field is always positive or always negative in the region D, it implies that the flow is either consistently diverging (spreading out) or converging (coming together)
throughout the region. This consistent behavior prevents the trajectories from closing onto themselves to form limit cycles within D.
Solved exercises
$\begin{cases} x’ = x^3 + y^3 \\ y’ = 3x + y^3 + 2y \end{cases}$
The vector field for this system is $\vec{F} = (x^3 + y^3)\hat{\mathbf{i}} + (3x + y^3 + 2y)\hat{\mathbf{j}}$
Let’s compute the divergence of the vector field: $div \vec{F} = f_x + g_y = 3x^2 + 3y^2 +2$. Notice that $div \vec{F} > 0$ in ℝ^2 (it is always positive since x^2≥0, y^2≥0 ∀x, y and there is an
additional positive term 2)⇒[By Bendixson’s criterion] Since the divergence is always positive and constant throughout the plane, there can be no closed trajectories in the xy-plane ⇒There are no
limit cycles anywhere in the system.
It’s important to note that Bendixson’s Criterion can only be used to exclude the possibility of limit cycles within a region. If the divergence changes sign or is zero somewhere in D, the criterion
does not provide any information about the existence of limit cycles; other methods must be used.
Proof (indirect proof)
Assume, for the sake of contradiction, that there exists a closed trajectory C within the region D. Let R be the region enclosed by the curve C (Refer to Figure iv for a visual representation and aid
in understanding it).
Consider the vector field $\vec{F} = f(x, y)\hat{\mathbf{i}} + g(x, y)\hat{\mathbf{j}}$ governing the system:
$\begin{cases} x’ = f(x, y) \\ y’ = g(x, y) \end{cases}$
Since C is a trajectory of the system, the vector field (velocity field) $\vec{F}$ is always tangent to the curve (the curve is a trajectory, it is supposed to be going in the direction given by the
vector field) at every point. Therefore, the normal vector and the vector field are perpendicular, i.e., their dot product is zero: $\vec{F}·\hat{\mathbf{n}} = 0$.
The flux integral of the vector field $\vec{F}$ across C is given by:
$\oint_{C} \vec{F}·\hat{\mathbf{n}}ds$ where:
• $\hat{\mathbf{n}}$ is the outward-pointing unit normal vector to C.
• ds is the differential arc length along C.
As we have previously stated, since $\vec{F}$ is tangent to C, $\vec{F}·\hat{\mathbf{n}} = 0$. Therefore, the flux across C is zero: $\oint_{C} \vec{F}·\hat{\mathbf{n}}ds = 0$
By Green’s Theorem, the flux integral over a closed curve C can be converted to a double integral over the region R enclosed by C: $\oint_{C} \vec{F}·\hat{\mathbf{n}}dS = \int\int_{R} div(\vec{F})
dA$, where:
• $div(\vec{F}) = \frac{∂f}{∂x}+\frac{∂g}{∂y}$ is the divergence of the vector field.
• dA is the differential area element.
Since the flux across C is zero, we have: $\int\int_{R} div(\vec{F})dA = 0$
However, we have already assumed that:
1. The divergence $div(\vec{F})$ is continuous in R.
2. $div(\vec{F})$ is not zero and does not change sign throughout R, meaning it is either strictly positive or strictly negative.
Under these conditions, the divergence is either $div(\vec{F}) > 0$ or $div(\vec{F}) < 0$ for all points in R. Since R has a positive area, and $div(\vec{F}) $ is non-zero of a constant sign the
integral of the divergence over the region R must be either $\int\int_{R} div(\vec{F})dA$ must be strictly positive ($div(\vec{F}) > 0$) or negative ($div(\vec{F}) < 0$) ⊥ Therefore, the assumption
that a closed trajectory C exists is false. Hence, under the given conditions, there are no closed trajectories (limit cycles) within the region D.
Critical point criterion
Critical point criterion (Poincaré-Bendixson). If a closed trajectory C exists within a region D in the xy-plane, there must be at least one critical point (equilibrium point) inside the region
enclosed by C.
A critical point is a point (x[0], y[0]) where f(x[0], y[0]) = 0, g(x[0], y[0]) = 0.
Contrapositive Logic. The contrapositive of a statement “If A, then B” (A⇒B) is the logically equivalent statement “If not B, then not A” (¬B⇒¬A). Applying this to the critical point criterion:
• Original statement: If a closed trajectory exists, then there is a critical point inside it.
• Contrapositive: If there are no critical points in a region, then there are no closed trajectories (no limit cycles) in that region.
This means that if you can identify a region D in the plane that contains no critical points, then you can conclude that no closed trajectories (limit cycles) exist within D.
Does this system has limit cycles?
Consider the following non-linear autonomous system:
$\begin{cases} x’ = x^2 + y^2 + 1 \\ y’ = x^2 -y^2 \end{cases}$
To investigate whether limit cycles exist, we can first apply Bendixson’s Criterion. It states that that if the divergence of the vector field is continuous and does not change sign (i.e., it is
either strictly positive or strictly negative) in a simply connected region D, then there are no closed trajectories (and hence no limit cycles) lying entirely within D.
Given the vector field $\vec{F} = f(x, y)\hat{\mathbf{i}}+g(x, y)\hat{\mathbf{j}}$ where $f(x, y) = x^2 + y^2 + 1, g(x, y) = x^2 -y^2$. The divergence of the vector field $\vec{F}$ is given by: $div
(\vec{F}) = \frac{∂f}{∂x} + \frac{∂g}{∂y} = 2x-2y$.
Analyzing the Divergence. Set the divergence equal to zero: 2x -2y = 0 ⇒ x -y = 0 ⇒ x = y. This tells us that along the line y = x, the divergence is zero.
Signs of the Divergence
• To the right of this line y = x (i.e., where x > y), the divergence is positive ($div(\vec{F}) = 2x−2y > 0$).
• To the left of this line (i.e., where x < y), the divergence is negative ($div(\vec{F}) =2x−2y < 0$).
Applying Bendixson’s Criterion
In regions where the divergence is strictly positive or strictly negative and continuous, Bendixson’s Criterion tells us that no closed trajectories can exist entirely within those regions.
• To the right of y = x: No closed trajectories exist.
• To the left of y = x: No closed trajectories exist. However, along the line y = x, the divergence is zero. Therefore, Bendixson’s Criterion does not rule out the possibility of closed
trajectories that cross this line or lie along it.
Next, we use the critical point criterion to check whether the system has any critical points. The Critical Point Criterion states:
• If a closed trajectory exists, there must be at least one critical point (equilibrium point) inside the region enclosed by the trajectory.
• Contrapositive: If there are no critical points inside a closed region, then there are no closed trajectories entirely within that region.
Critical points occur where both derivatives x′ and y′are zero simultaneously.
However, this is impossible because: $x^2+y^2+1 > 0$ (there are not real solutions to x^2 + y^2 = -1) ⇒ There are no critical points in the real plane system. ⇒Since there are no critical points in
the system, by the contrapositive of the critical point criterion, we can conclude that the system does not have any limit cycles.
Since $x’ > 0$ for all $(x, y)$, all trajectories will move rightward indefinitely. This alone is sufficient to rule out limit cycles.
• Consider the spring Mass damper system with a mass of one (m = 1), and stiffness of 1 (k = 1), and a non-linear damping c (c = c(x) > 0).
To determine whether limit cycles can exist in the given spring-mass-damper system with non-linear damping, we will apply Bendixson’s Criterion. This criterion helps us identify whether closed
trajectories, and hence limit cycles, are possible in a particular region of the phase plane by examining the divergence of the vector field.
The governing equation for this system is: $x’’ + c(x)x’ + x = 0.$
To analyze this system in the phase plane, we convert it to state space form by introducing a new variable y’ = x. Then, we have:
$\begin{cases} x’ = y \\ y’ = -x -c(x)y \end{cases}$
This gives us the vector field $\vec{F} = f(x, y)\hat{\mathbf{i}}+g(x, y)\hat{\mathbf{j}}$ where where f(x, y) = y, g(x, y) = -x -c(x)y.
To investigate whether limit cycles exist, we can apply Bendixson’s Criterion. It states that if the divergence of the vector field $\vec{F}$ is continuous and does not change sign (i.e., it is
either strictly positive or strictly negative) in a simply connected region D, then there are no closed trajectories (and thus no limit cycles) can exist entirely within D.
The divergence of the vector field $\vec{F}$ is given by: $div(\vec{F}) = \frac{∂f}{∂x} + \frac{∂g}{∂y} = 0 -c(x)$
Applying Bendixson’s Criterion:
Since c(x) > 0 by definition, we have: $div(\vec{F}) = -c(x) < 0$ ∀x, y.
The divergence is strictly negative for all x and y (assuming c(x) is continuous). Thus:
• The divergence does not change sign in any region of the xy-plane.
• The divergence is also continuous in the phase plane since c(x) is continuous.
• By Bendixson’s Criterion, since the divergence is strictly negative and continuous across the entire plane, no closed trajectories (and therefore no limit cycles) can exist in this system.
This content is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License and is based on MIT OpenCourseWare [18.01 Single Variable Calculus, Fall 2007].
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8. MIT OpenCourseWare [18.03 Differential Equations, Spring 2006], YouTube by MIT OpenCourseWare. | {"url":"https://justtothepoint.com/calculus/limitcycles/","timestamp":"2024-11-14T05:29:44Z","content_type":"text/html","content_length":"46058","record_id":"<urn:uuid:4f31bc84-e3e0-4c8f-9ea0-7209755928d7>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00274.warc.gz"} |