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Understanding Homomorphic Encryption
ZK-SNARKS rely on a certain encryption function called Homomorphic encryption. I believe it’s important to understand the mathematical properties of this particular operator to truly understand
ZK-SNARKS and hence I decided to prepare this document. Let us dive deeper.
Where g^x is exponentiation and mod m is the modulo operator.
The Module Operator
Let us understand the (mod m) operator deeper first. If y = x mod m, this means that y is the remainder when x is divided by m. Hence, one could write:-
where q is the quotient when x is divided by m
Let us look at some interesting properties of the mod m operator
Property 1:
Both a and b can be written as:-
Substituting these values of a and b in right side of the equation, we get:-
Substituting the values of a and b in left side of the equation, we get:-
As we can see, both the equations evaluate to the same value.
Property 2:
Substituting the previous values of a and b in left side of the equation, we get:-
Substituting in right side of the equation, we get:-
As we can see, both the equations evaluate to the same value.
The E(x) Operator
Now that we are comfortable with the modulo operator, let us turn our attention back to the E(x) encryption function we defined earlier. E(x) is called an encryption operation since it’s not easy to
find the value of x given the value of E(x). Hence, E(x) is a one-way function. The cool thing about E(x) is that it allows us to perform basic arithmetic operations on x, even when only E(x) is
known, but x is not known.
Let’s dive deeper into these interesting properties of the E(x) operator:-
Property 1:
Expanding the left side of the equation, we get:-
Using the property 2 of modulo multiplication defined previously, we can write:-
Which is the same as the right side of the equation.
Property 2:
Expanding the left side of the equation, we get:-
Using the property 2 of modulo multiplication defined previously, we can write:-
Which is the same as the right side of the equation
This was a basic intro to some of the maths used in ZK-SNARKS. In the next article, I will attempt to create a basic zero-knowledge scheme using the Homomorphic encryption defined here.
Here is an awesome article series by Makesym on ZK technology — Why and How zk-SNARK Works 2: Proving Knowledge of a Polynomial | by Maksym | Medium | {"url":"https://garvitgoel.medium.com/understanding-homomorphic-encryption-3869f228ecfa?source=user_profile_page---------7-------------1b75b9e1f362---------------","timestamp":"2024-11-07T02:50:47Z","content_type":"text/html","content_length":"149848","record_id":"<urn:uuid:4e2bf199-1505-46ef-ab2d-02c881ca515f>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00406.warc.gz"} |
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Difftime code
Clive D.W. Feather clive at demon.net
Fri Aug 6 05:00:38 UTC 2004
Paul Eggert said:
> Wow, that's pretty a complicated way to subtract two numbers. :-)
Only if you want the right answer :-)
> If we can assume uintmax_t (by typedefing it on older hosts), isn't
> there a much simpler approach entirely?
No. [Note: my top_type is effectively that typedef.]
> The idea behind the sizeof test is to avoid "long double" if it's safe
> to do so, since long double is expensive on some hosts.
Is it *that* expensive? I thought you were trying to avoid rounding errors.
> #define TYPE_FLOATING(type) ((type) 0.5 != 0)
> #define TYPE_SIGNED(type) (((type) -1) < 0)
> #define TYPE_BIT(type) (sizeof (type) * CHAR_BIT)
> double
> difftime (time_t time1, time_t time0)
> {
> if (TYPE_BIT (time_t) <= DBL_MANT_DIG
That's a conservative test I have no objection to. It's *very*
conservative, since if FLT_RADIX isn't 2 it will seriously underestimate
how big double is.
You might be better off comparing time1 and time0 with DBL_EPSILON or use
the maxLDint trick I described.
But if you're worried about efficiency, why are you doing this in floating
point when they're integers?
> || (TYPE_FLOATING (time_t) && sizeof (time_t) != sizeof (long double)))
Okay, this proves that time_t isn't long double. Clever.
> return (double) time1 - (double) time0;
> if (TYPE_FLOATING (time_t))
> return (long_double) time1 - (long_double) time0;
> if (time1 < time0)
> return -simple_difftime (time0, time1);
> else
> return simple_difftime (time1, time0);
> }
Moved here for expositional purposes:
> static double
> simple_difftime (time_t time1, time_t time0)
> {
> if (TYPE_SIGNED (time_t))
> return (uintmax_t) time1 - (uintmax_t) time0;
> else
> return time1 - time0;
> }
That will sometimes get the answer badly wrong.
The problem occurs when time_t is signed and the maximum value of time_t
is the same as the maximum value of uintmax_t. For example, a C89 system
where long is 60 bits including sign and unsigned long is 59 bits.
On such systems, the maximum possible difference is greater than the
maximum value of uintmax_t, and your subtract will get it wrong.
I gate this case by looking for time1 >=0 and time0 < 0. In fact, you can
be safer than that:
#define HALFMAX ((uintmax_t)-1 >> 1)
if (time1 <= HALFMAX && (time0 >= 0 || (uintmax_t) time0 >= -HALFMAX))
return (uintmax_t) time1 - (uintmax_t) time0;
However, the remaining cases have to allow for overflow in the subtraction,
and that's the complicated bit.
Clive D.W. Feather | Work: <clive at demon.net> | Tel: +44 20 8495 6138
Internet Expert | Home: <clive at davros.org> | Fax: +44 870 051 9937
Demon Internet | WWW: http://www.davros.org | Mobile: +44 7973 377646
Thus plc | |
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What are the different laws of motion?
Contributed by:
Motion, in physics, changes with the time of the position or orientation of a body. Motion along a line or a curve is called translation.
1. Physics 111: Mechanics
Lecture 4
Dale Gary
NJIT Physics Department
2. The Laws of Motion
Newton’s first law
Newton’s second law
Newton’s third law
Isaac Newton’s work represents
one of the greatest
contributions to science ever
made by an individual.
Feb. 11-15, 2013
3. Dynamics
Describes the relationship between the
motion of objects in our everyday world
and the forces acting on them
Language of Dynamics
Force: The measure of interaction between two
objects (pull or push). It is a vector quantity – it has a
magnitude and direction
Mass: The measure of how difficult it is to change
object’s velocity (sluggishness or inertia of the object)
Feb. 11-15, 2013
4. Forces
The measure of interaction
between two objects (pull or
Vector quantity: has
magnitude and direction
May be a contact force or
a field force
Contact forces result from
physical contact between two
Field forces act between
disconnected objects
Also called “action at a
Feb. 11-15, 2013
5. Forces
Gravitational Force
Archimedes Force
Friction Force
Tension Force
Spring Force
Normal Force
Feb. 11-15, 2013
6. Vector Nature of Force
Vector force: has magnitude and direction
Net Force: a resultant force acting on
object Fnet F F1 F2 F3 ......
You must use the rules of vector addition to
obtain the net force on an object
| F | F12 F22 2.24 N
tan 1 ( ) 26.6
Feb. 11-15, 2013
7. Newton’s First Law
An object at rest tends to stay at rest and an
object in motion tends to stay in motion with
the same speed and in the same direction
unless acted upon by an unbalanced force
An object at rest remains at rest as long as no net force acts on it
An object moving with constant velocity continues to move with
the same speed and in the same direction (the same velocity) as
long as no net force acts on it
“Keep on doing what it is doing”
Feb. 11-15, 2013
8. Newton’s First Law
An object at rest tends to stay at rest and an
object in motion tends to stay in motion with
the same speed and in the same direction
unless acted upon by an unbalanced force
When forces are balanced, the acceleration of the object is
Object at rest: v = 0 and a = 0
Object in motion: v 0 and a = 0
The net force is defined as the vector sum of all the
external forces exerted on the object. If the net force is
zero, forces are balanced. When forces are balances, the
object can be stationary, or move with constant velocity.
Feb. 11-15, 2013
9. Mass and Inertia
Every object continues in its state of rest, or uniform
motion in a straight line, unless it is compelled to change
that state by unbalanced forces impressed upon it
Inertia is a property of objects
to resist changes is motion!
Mass is a measure of the
amount of inertia.
Mass is a measure of the resistance of an object
to changes in its velocity
Mass is an inherent property of an object
Scalar quantity and SI unit: kg
Feb. 11-15, 2013
10. Newton’s Second
The acceleration of an object is directly
proportional to the net force acting on
it and inversely proportional to its mass
m m
Fnet F ma
Feb. 11-15, 2013
11. Units of Force
Newton’s second law:
Fnet F ma
SI unit of force is a Newton (N)
kg m
1 N 1
US Customary unit of force is a pound (lb)
1 N = 0.225 lb
Weight, also measured in lbs. is a force (mass
x acceleration). What is the acceleration in
that case?
Feb. 11-15, 2013
12. More about Newton’s 2nd
You must be certain about which body we are
applying it to
Fnet must be the vector sum of all the forces
that act on that body
Only forces that act on that body are to be
included in the vector sum
Net force component along an
axis gives rise to the acceleration
along that same axis
Fnet , x ma x Fnet , y ma y
Feb. 11-15, 2013
13. Sample Problem
One or two forces act on a puck that moves over frictionless
ice along an x axis, in one-dimensional motion. The puck's
mass is m = 0.20 kg. Forces F1 and F2 and are directed
along the x axis and have magnitudes F1 = 4.0 N and F2 = 2.0
N. Force F3 is directed at angle = 30° and has magnitude F3
= 1.0 N. In each situation, what is the acceleration
a) F1 ma xof the
puck? a 1
F 4.0 N
20 m/s 2
m 0.2 kg
b) F1 F2 ma x
F1 F2 4.0 N 2.0 N
ax 10 m/s 2
m 0.2 kg
c) F3, x F2 max F3, x F3 cos
Fnet , x ma x F3 cos F2 1.0 N cos 30 2.0 N
ax 5.7 m/s 2
m 0.2 kg
Feb. 11-15, 2013
14. Gravitational Force
Gravitational force is a vector
Expressed by Newton’s Law of Universal
Gravitation: mM
Fg G
G – gravitational constant
M – mass of the Earth
m – mass of an object
R – radius of the Earth
Direction: pointing downward
Feb. 11-15, 2013
15. Weight
The magnitude of the gravitational force acting on
an object of mass m near the Earth’s surface is
called the weight w of the object: w = mg
g can also be found from the Law of Universal
Weight has a unit of N
Fg G 2 w Fg mg
g G 2 9.8 m/s 2
Weight depends upon location R = 6,400 km
Feb. 11-15, 2013
16. Normal Force
Force from a solid
surface which
keeps object from w Fg mg
falling through
Direction: always
perpendicular to
the surface N Fg ma y
N mg ma y
depends on
situation N mg
Feb. 11-15, 2013
17. Tension Force: T
A taut rope exerts
forces on whatever
holds its ends
Direction: always
along the cord (rope,
cable, string ……) T1
and away from the T1 = T = T2
object T2
Magnitude: depend
on situation
Feb. 11-15, 2013
18. Newton’s Third Law
If object 1 and object 2 interact, the force
exerted by object 1 on object 2 is equal
in magnitude but opposite in direction to
the force exerted by object 2 on object 1
Fon A Fon B
Equivalent to saying a single isolated force
cannot exist Feb. 11-15, 2013
19. Newton’s Third Law cont.
F12 may be called the
action force and F21
the reaction force
Actually, either force
can be the action or
the reaction force
The action and
reaction forces act
on different objects
Feb. 11-15, 2013
20. Some Action-Reaction Pairs
Fg G
Fg mg m 2
mM R
Fg G 2 Gm
R Fg Ma M 2
Feb. 11-15, 2013
21. Free Body Diagram
The most important step in
solving problems involving F hand on book
Newton’s Laws is to draw
the free body diagram
Be sure to include only the
forces acting on the object
of interest
Include any field forces F Earth on book
acting on the object
Do not assume the normal
force equals the weight
Feb. 11-15, 2013
22. Hints for Problem-Solving
Read the problem carefully at least once
Draw a picture of the system, identify the object of primary
interest, and indicate forces with arrows
Label each force in the picture in a way that will bring to mind
what physical quantity the label stands for (e.g., T for tension)
Draw a free-body diagram of the object of interest, based on
the labeled picture. If additional objects are involved, draw
separate free-body diagram for them
Choose a convenient coordinate system for each object
Apply Newton’s second law. The x- and y-components of
Newton second law should be taken from the vector equation
and written individually. This often results in two equations
and two unknowns
Solve for the desired unknown quantity, and substitute the
numbers F ma F ma
net , x x net , y y
Feb. 11-15, 2013
23. Objects in Equilibrium
Objects that are either at rest or moving
with constant velocity are said to be in
a 0 of an object can be modeled
as zero:
Mathematically, the
F net
0 force acting on
the object is zero
Equivalent to the set of component
Fx by
equations given 0 Fy 0
Feb. 11-15, 2013
24. Equilibrium, Example 1
A lamp is suspended from
a chain of negligible mass
The forces acting on the
lamp are
the downward force of
the upward tension in the
Applying equilibrium gives
Fy 0 T Fg 0 T Fg
Feb. 11-15, 2013
25. Equilibrium, Example 2
A traffic light weighing 100 N hangs from a vertical cable
tied to two other cables that are fastened to a support.
The upper cables make angles of 37° and 53° with the
horizontal. Find the tension in each of the three cables.
Conceptualize the traffic light
Assume cables don’t break
Nothing is moving
Categorize as an equilibrium
No movement, so acceleration is
F 0 Fy 0
Model xas an object in equilibrium
Feb. 11-15, 2013
26. Equilibrium, Example 2
Need 2 free-body diagrams
Apply equilibrium equation to
light Fy 0 T3 Fg 0 F y 0 T3 Fg 0
T3 Fg 100 N
T3 Fg 100 N
Apply equilibrium equations to
x T1x T2x T1 cos 37
T2 cos 53 0
F y T1 y T2 y T3 y
T1 sin 37 T2 sin 53 100 N 0
cos 37
T2 T1 1.33T1
cos 53
T1 60 N T2 1.33T1 80 N
Feb. 11-15, 2013
27. Accelerating Objects
If an object that can be modeled as a
particle experiences an acceleration, there
must be a nonzero net force acting on it
Draw a free-body diagram
Apply Newton’s Second Law in component
F ma
F x ma x F y ma y
Feb. 11-15, 2013
28. Accelerating Objects,
Example 1
A man weighs himself with a scale in an elevator.
While the elevator is at rest, he measures a
weight of 800 N.
What weight does the scale read if the elevator
accelerates upward at 2.0 m/s2? a = 2.0 m/s2
What weight does the scale read if the elevator
Upward: Fy downward at 2.0 m/s2? a = - 2.0 m/s2
N mg ma N
N mg ma m( g a ) N 80(2.0 9.8) 624 N
w 800 N
g 9.8 m/s 2
80 N N mg
Downward: N 80( 2.0 9.8) 624 N
N mg mg mg
Feb. 11-15, 2013 | {"url":"https://merithub.com/tutorial/what-are-the-different-laws-of-motion-c7n5fvct2n9psu2fnogg","timestamp":"2024-11-06T02:38:18Z","content_type":"text/html","content_length":"52237","record_id":"<urn:uuid:f62e9b26-2077-450d-b6da-57a4bdcce6e0>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00578.warc.gz"} |
Formula for the volume of the sphere ?
more than 3 weeks ago
Formula for the volume of the sphere ?
more than 3 weeks ago
The formula for the volume of a sphere is V = (4/3)πr^3, where V is the volume, π is the mathematical constant pi (approximately 3.14159), and r is the radius of the sphere.
Google Gemini AI
more than 3 weeks ago
$$V = \frac{4}{3} \pi r^3$$
- V is the volume of the sphere
- r is the radius of the sphere
- π (pi) is a mathematical constant approximately equal to 3.14
© 2024 - Quanswer | {"url":"https://www.quanswer.com/en/formula-for-the-volume-of-the-sphere","timestamp":"2024-11-07T08:50:18Z","content_type":"text/html","content_length":"67922","record_id":"<urn:uuid:b6acfdf9-cd3a-4dc5-b937-9c96ec9aa50d>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00774.warc.gz"} |
Binomial Coefficients
This article introduces the Binomial Coefficients. It is best read with paper and pen so that you can have a go at the questions as you read.
There are several ways of defining the binomial coefficients, but for this article we will be using the following definition and notation: \[n\choose k\] (pronounced “$n$ choose $k$” ) is the number
of distinct subsets of size $k$ of a set of size $n$. More informally, it's the number of different ways you can choose $k$ things from a collection of $n$ of them (hence $n$ choose $k$). So:
• ${3 \choose 1} = 3$ because you can either pick the first thing, or the second thing, or the third thing.
• Likewise, ${3 \choose 2} = 3$ because you can either take the first two, the first and the third, or the last two. Notice that the order you choose the things in is ignored.
Can you work out the following binomial coefficents?
1. $4 \choose 1$
2. $4 \choose 2$
3. $5 \choose 2$
4. $6 \choose 2$
You'll need to come up with a systematic method of making sure you've found all the ways of choosing. Does your method suggest to you any way of calculating the result directly?
Now go back to the definition in terms of choosing things, and see if you can work out why the following are true for all $n\ge 1$:
\[{n \choose 0} = {n \choose n} = 1 \] \[{n \choose 1} = {n \choose n-1} = n \]
This starts to suggest a pattern, in fact:
\[{n \choose k} = {n \choose n-k}\]
Can you justify this pattern from the definition?
Now let's write out the binomial coefficients in a grid:
Do these look familiar? They should do. This suggests the following rule:
\[{n \choose k} = {n-1 \choose k} + {n-1 \choose k-1}\]
Thinking back to your systematic method, can you explain this relation in terms of choosing things?
We can use this relation to calculate binomial coefficients, but it's not very efficient. You'll find yourself having to calculate more and more binomial coefficients the larger your $n$ and $k$. So
let's try to find another formula, by thinking about the process of choosing:
1. Suppose you want to choose five things from a collection of twelve. How many choices do you have for the first thing? How many choices for the second? How many choices for the third, fourth, and
fifth? How many choices does this give altogether?
2. The problem with the above method is that it counts 1,2,3,4,5 and 2,1,3,4,5 separately even though they are really the same choice. How many times do you count each choice? How can you eliminate
them from your count?
Once you've come up with a formula, can you use it to justify the earlier results algebraically?
The connection with the binomial
I mentioned these were called binomial coefficients at the beginning of the article, but I haven't mentioned the binomial formula since. Well, the binomial formula is this:
\[(a+b)^n = \sum_{k=0}^n {n \choose k} a^k b^{n-k}\]
So they are literally just the coefficients of $a^k$ in the expansion of the $n^\textrm{th}$ power.
Thinking about $(a+b)^n$ as $(a+b)(a+b)(a+b)\dots(a+b)$ and expanding in the usual way, can you see how the choosing-from-sets definition is connected to this idea?
These formulae all have justifications both in terms of choosing things and in terms of algebra. See if you can find one or the other (or both!):
\[\sum_{k=0}^{n-r} {n-k \choose r} = {n+1 \choose r+1}\]
\[\sum_{k=0}^n {n \choose k} = 2^n\]
\[\sum_{k=0}^n {n \choose k}^2 = {2n \choose n}\] | {"url":"https://nrich.maths.org/articles/binomial-coefficients","timestamp":"2024-11-14T23:28:36Z","content_type":"text/html","content_length":"37831","record_id":"<urn:uuid:b9c759af-082a-409e-944b-b07b7407cef4>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00280.warc.gz"} |
Tiles with Dissection Polynomial nx + x
Tiles with Dissection Polynomial nx + x^2 + x^3
For tiles with dissection polynomial
nx + x^2 + x^3
the corresponding Pisot polynomial is
x^3 - nx^2 - x - 1
and the corresponding Perron polynomial is
x^3 + x^2 + nx - 1
The case n = 0 corresponds to the plastic number, and the case n = 1 corresponds to the Tribonacci number; a considerable number of tiles are known for both of these. However there is a construction
which generates a tile for all n. This construction was first reverse engineered from a tile with n = 2 used as an example in Kenyon's "The Construction of Self Similar Tilings". I had previously
conjectured the existence of 4-element tiles in which the ratio of the areas of the elements was a power of the 8th unit cubic Pisot number, that is the real root of x^3-2x^2-x-1, but had not
discovered a construction for any of these. Although not described in terms of Pisot numbers Kenyon's construction provided convincing evidence of the existence of such tiles. Having confirmation of
the existence of such tile I investigated further, and found a construction for Kenyon's tile.
Let a be the number associated with this tiling, as described on the overview of Perron number tilings. Then the IFS is { p → ap; p →ap + x; p → a^2p + y; p → a^3p + z }. We can arbitrarily fix x as
the vector (1,0). Values for y and z are found by informed trial and error to be -a^2 - a and a^2 - a. Thus the IFS can be restated as
{ p → ap; p →ap + 1; p →a^2p - a - a^2; p → a^3p - a + a^2 }
It turns out that this construction can be generalised for other values of n. The IFS becomes
{ p → ap; p →ap + 1; ... ; p →ap + n - 1; p →a^2p - a - a^2; p → a^3p - a + (n-1)a^2 }
n = 0 generates the Ammonite tile (a plastic tile) and n = 1 generates the Rauzy fractal.
The attractors for n = 3 ... 10 are shown below.
All of these figures tile the plane with one copy per unit cell. The tiling vectors are a^-1 and n + a.
Additional tiles can be generated by modification of the corresponding IFSs; I document some elsewhere for the 8^th (n = 2) and 12^th (n = 3) unit cubic Pisots.
Source: Reconstruction (n=2) of a tile published by Richard Kenyon; for n=3 and above these tiles are my own discoveries, from 2002. Tilings added in 2016.
© 2002, 2005, 2016 Stewart R. Hinsley | {"url":"http://stewart.hinsley.me.uk/Fractals/IFS/Tiles/Cubic/nxx2x3/nxx2x3.php","timestamp":"2024-11-09T01:20:27Z","content_type":"text/html","content_length":"6548","record_id":"<urn:uuid:49bc0155-4b07-4fd0-be2e-c98afb876712>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00350.warc.gz"} |
Days Between Dates Calculator - Onlinecalculator.guide - onlinecalculator.guide
Days Between Dates Calculator
If you’re not obtaining any way to calculate Days Between Dates, then you can surely use our days between dates calculator, which will make your job easy. You only need to put the start date and end
date values and then press the calculate button to obtain your final answer.
Examples of Days between dates calculator
Days Between Given Date Duration Formula
The basic formula to calculate days between dates is given,
Days = End Date - Start Date
Simply enter the end date and start date which is mentioned in the given formula and then determine how much the value is equivalent in terms of finding the final answer in days. You can use this
basic formula and make all your conversions concerning dates easily.
Finding Days Between Dates?
The steps to be followed to calculate days between dates are given below. Follow the detailed steps here and make your computations much quicker and easier.
• The first step is to obtain the date value you are about to find the days.
• Then also insert the value of the date you want to till.
• Then to get the answer, subtract the end date from the start date.
• After performing the required math, the resultant value is the days value you need.
Days Between Date Examples
Example 1:
How many days between 22nd December 2022 to 12th January 2023?
Given that,
Start Date = 2022-12-22
End Date = 2023-01-12
To find how many days are there between two dates,
We will subtract them,
Now will substitute the start date and end date,
Days = End Date - Start Date
Days = 2023-01-12 - 2022-12-22
Days = 21
So, Days between 2022-12-22 and 2023-01-12 are 21 days
Example 2:
How many weeks from 3rd February 2023 to 8th May 2023?
Given that,
Start Date = 2023-02-03
End Date = 2023-05-08
To find how many days are there between two dates,
We will subtract them,
Now will substitute the start date and end date,
Days = End Date - Start Date
Days = 2023-05-08 - 2023-02-03
Days = 94
So, Days between 2023-02-03 and 2023-05-08 are = 94 days
Become familiar with many more concepts that are arranged efficiently on OnlineCalculator.guide and clear your concerns on many more calculations like this.
FAQs on Days Between Dates Calculator
1. How many days in a year?
There are 365 days in a year.
2. Number of days in 4 months from January To April?
There are 119 days in between 4 months from January to April.
3. How many days between January 25th, 2023 February 28th, 2023?
There are 34 days between January 25th, 2023 February 28th, 2023. | {"url":"https://onlinecalculator.guide/days-between-dates/","timestamp":"2024-11-11T20:40:13Z","content_type":"text/html","content_length":"37019","record_id":"<urn:uuid:2cf8d9fd-ceff-4bc5-8207-9f24f3e8ed2e>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00709.warc.gz"} |
If cos( sin− 1 52 + cos− 1 x ) = 0, find the value of x. ... | Filo
If , find the value of x.
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Question Text If , find the value of x.
Updated On Feb 1, 2024
Topic Inverse Trigonometric Functions
Subject Mathematics
Class Class 12
Answer Type Text solution:1 Video solution: 3
Upvotes 376
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Orange Boy Can You Solve It Out? Ep.2
Orange Boy Can You Solve It Out? Ep.2
Difficulty: D2E
思考题 for everyone
Man on Graph
You are given M chains each of N+1 nodes. We call the i-th node on the j-th chain (j,i-1).
We define a bridge edge connecting (a,b) and (c,d) as follows:
- a≠c
- b=d
- b and d don't equal to N or 0
We made sure that:
- there is never a node that has more than 1 bridge edges
A graph example is shown below
Now a man is walking on a graph with the following rules(let we call the current position of it (x,y)):
- if there is a bridge edge connecting to (x,y) and in the last step he walked along a normal edge,he walks it.
- Otherwise he walks along a normal edge.
- he won't walk to (x,y-1)
- his trip ends when there's no way to move
So for example, if the man is at (3,0) he will walk to (3,1) and to (2,1) then (2,2) and ends his journey.
So here comes the question:
Given N and M and the original map.
Given q queries:
- "1 x y z" connect (x,z) and (y,z) with a bridge edge. It is made sure that there's no bridge edge connected to the two points and legal.
- "2 x" if a man starts at (x,0) where will his journey ends? Print it
Proceed for it.
Test 1(20%)- n=1e9,m=1e6,q=1e6, no query 1
Test 2(20%)- n=1e9,m=1e6,q=1e6, in queries,z is always increasing.
Test 3(20%)- n=1e9,m=1000,q=1e5, at most 1000 query 1.
Test 4(40%)- n=1e9,m=1e5,q=1e5
Orange Boy quickly come and get AC
If anyone can get 100%, I may get him a coffee
The Author can get 60%
Solution By MonkeyKing
Solution By MonkeyKing
This problem can be turned into the following version:
You are given an array A=[1,2,3,...,m] and another array Q=[[1,1],[1,1],[1,1],...,[1,1]].
We define a "routine" as follow:
For each 1<=i<=N, swap(A[Q[i][1]],A[Q[i][2]])
then the result A is the "output" of the routine.
Now given q queries:
- "1 x y z" set Q[z]=[x,y]
- "2 x" finds the position of x in the output array of the routine
We can see that for inserting operation, if we know the two element that will be swaped, it will be easy to solve each query in O(1). So we can store what two numbers are swaped for each query. Then
to add an edge, we just need to know the previous value on this chain, that can be done using set or stuff. But the problem is still unsolved. Each time we add an edge, the value of the edge after it
will be changed. So we have to update all the remaining of them in O(n). But it can't be done that slow.
We need to find out a quick way. We can perform some sqrt operations.We can do some "lazy works". That is we store the numbers needed to be changed in a directed graph C. Then we do the follows: if
len(C)<=sqrt(n), we just violently get the answer by looking up in the graph. If len(C)=sqrt(n),we just violently change all the values on the original graph by C.
It can be seen that the complexity is O(n*(sqrt(n)+log(n)))
Solution By MonkeyKing
来源：Hell Hole Studios Blog
共有 0 条评论 | {"url":"https://blog.hellholestudios.top/archives/112","timestamp":"2024-11-08T22:05:51Z","content_type":"text/html","content_length":"73445","record_id":"<urn:uuid:523c5f1a-adfe-4975-96f1-acc210fad017>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00420.warc.gz"} |
In Search of Quantum Gravity | Bluesci
SUNDAY, 5 MARCH 2017
Gianamar Giovannetti-Singh explores the holographic universe
Modern fundamental physics consists of two major pillars; general relativity, describing the interactions between matter and spacetime at the largest scales imaginable, and quantum mechanics, the
physics governing the behaviour of subatomic particles. Despite each respective theory being tested to an extraordinary degree of accuracy, they are fundamentally incompatible with each other –
general relativity predicts continuous spacetime as the fabric of the universe whereas quantum physics deals exclusively with discrete, quantised units of matter and energy.
Theoretical physicists generally consider quantum gravity, a successful amalgamation of the two theories, to be the holy grail of modern physics, as it would unify these two immiscible faces of the
universe into a so-called theory of everything, capable of predicting the behaviour of every structure in the universe within a single mathematical framework. Notable attempts at formulating a
quantum theory of gravity include string theory, in which all the point-particles of the standard model of particle physics are replaced with one-dimensional strings, and loop quantum gravity, which
breaks up spacetime, the fabric of our universe, into discrete chunks with a size of approximately 10^-35 metres – if a hydrogen atom were the size of the sun, this would be the size of a single
proton. Unfortunately, all of these attempts have proved to be far less successful than expected, offering no real falsifiable hypotheses. However, an alternative and rather radical approach to
quantum gravity has recently been gaining increasing recognition in the physics community following a sudden inflow of supportive evidence from computer simulations; this approach is known as the
holographic principle. This principle argues that rather than living in a three-dimensional universe which we perceive every day, the cosmos is actually a projection of three-dimensional information
onto a two-dimensional surface – not dissimilar from the way in which a hologram encodes information about three-dimensions on a 2D surface. The holographic principle was suggested by Gerard ‘t Hooft
as a consequence of Stephen Hawking and Jacob Bekenstein’s ground breaking work on black hole thermodynamics in 1974, which demonstrated that the entropy (a measure of disorder) of a black hole
varies not with its volume as one might naïvely expect, but rather with its surface area. As entropy is intrinsically linked to information, ‘t Hooft proposed that the information of a 3D object is
simply encoded on a 2D surface. The great American physicist and science communicator Leonard Susskind developed ‘t Hooft’s idea and formalised it in the mathematical framework of string theory,
allowing it to be fully integrated within another physical model.
In 1997 Juan Maldacena used Susskind’s mathematical framework to derive a tremendously profound result – if the “laws” of physics which followed from the holographic principle (namely that
information about an N-dimensional system can be encoded on an (N-1)-dimensional surface without losing any knowledge of the system) are applied to a physical theory which describes how gravity would
behave in a 5D universe, the results agree completely with a quantum-based theory (Yang-Mills theory) in four dimensions. This was an initial clue that the holographic principle holds great power in
its ability to unify seemingly completely detached areas of physics, and in its combination of theories derived from general relativity and quantum mechanics, the principle secured its status as a
candidate for a theory of quantum gravity.
"...the holographic principle holds great power in its ability to unify seemingly completely detached areas of physics"
A major piece of evidence in favour of the holographic principle was obtained in late 2013 by a team led by the Japanese theoretical physicist Yoshifume Hyakutake et al., who ran two very large
simulations on a supercomputer calculating the internal energy of a black hole using two completely different physical models; one was a ten-dimensional universe based on general relativity, the
other was a one-dimensional universe without gravity but rich in large-scale quantum effects. Somewhat astoundingly, Hyakutake et al. found that the two values agreed exactly, which lends credence to
the holographic principle; i.e. that a gravitational model of spacetime corresponds directly with a lower-dimensional quantum-dominated universe. This simulation has significantly improved the
standing of the holographic principle as a candidate for quantum gravity as it has demonstrated a universal correspondence between a theory derived from general relativity and a lower-dimensional
quantum theory, and so far provides the only quantitative link between the two regimes. Whilst models such as string theory and loop quantum gravity attempt to quantitatively describe both
gravitational and quantum behaviour, neither provide any numerical values which can be falsified, whereas Hyakutake et al.’s work did just that, and thus allow the holographic principle to present
(somewhat more falsifiable) hypotheses regarding the internal energy of a black hole.
Whilst the nature of a theory of quantum gravity remains elusive to physicists at present, the holographic principle may just be the dark horse in the race for grand unified theory of nature. | {"url":"https://bluesci.co.uk/posts/180","timestamp":"2024-11-09T15:41:31Z","content_type":"text/html","content_length":"64605","record_id":"<urn:uuid:f5cf4007-7bd9-40d9-9c8f-75c88890f2e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00051.warc.gz"} |
Lecture VII - Number Theory
Number theory encompasses anything relating to properties of integers. In contests, we
typically encounter problems involving divisibility and factorization. In this lecture we will
let p[1], p[2], . . . represent the prime numbers in ascending order so that p[n] is the nth prime
number. We let gcd(p, q) represent the greatest common denominator and let lcm(p, q) the
least common multiple of integers p and q.
1 Divisibility and Factoring
The Fundamental Theorem of Arithmetic says that any positive integer n can be represented
in exactly one way as the product of prime numbers, so that the factorizations of p and q
are identical if and only if p = q.
The number f divides n if and only if none of the powers of the primes in the factorization
of f are greater than those of n. Specifically, f divides n k times if and only if there is no
prime p is the factorization of f that appears more than 1/k times as often as it appears in
the factorization of n.
On a related note, if some integer f divides integers p and q, then f divides mp + nq,
where m and n are any integers.
Quick question: How many times does 3 divide 28!?
We reason that the answer is the sum of how many times 3 divides each of 1, 2, . . . , 28. Of the
numbers 1 through 28, exactly
is the floor function and represents the greatest integer less than or equal to x). To count the
total number of p’s appearing in their factorizations, we compute 9+3+1+0+0+0+· · · = 13.
The generalized result:
Theorem : Aprime number p divides n! exactly
This fact enables us to determine how many 0’s appear at the end of n!. Because there
are more 2’s than 5’s in the factorization of n!, the number of 0’s at the end of n! is the
number of 5’s in its factorization.
Quick question: How many factors does 120 have?
We factor 120 and find that 120 =
divides 120 must satisfy [1], 2
possible m[2], and 2 possible m[3], meaning that there are 4 · 2 · 2 = 16 positive integers that
divide 120. Moreover:
Theorem :
The greatest common divisor of m and n is defined to be the largest integer that divides
both m and n. Two numbers whose largest common divisor is 1 are called relatively prime
even though neither m nor n is necessarily prime. There are two notable ways to compute
gcd(m, n).
• Factoring - Let
0. Then gcd(m, n) is the positive integer whose prime factorization contains p[i] exactly
min(m[i], n[i]) times for all positive integers i. Remark - This is useful if the factorizations
of m and n are readily available, but if m and n are large numbers such as 4897, they
will be difficult to factor.
• Euclidean Algorithm - Let n > m. If m divides n, then gcd(m, n) = m. Otherwise,
gcd(m, n) = gcd(m, n − m ·
fails. For example, finding gcd(4897, 1357). 1357 does not divide 4897, so
4897 − 3 · 1357 = 826 and gcd(4897, 1357) = gcd(1357, 826). 826 does not divide
1357, so gcd(1357, 826) = gcd(826, 531). 531 does not divide 826 so gcd(862, 531) =
gcd(531, 295). Continuing this process, gcd(531, 295) = gcd(295, 236) = gcd(236, 59) =
The least common multiple of m and n is defined to be the least number that is divisible
by both m and n. Other than listing multiples of m and n, we can determine the lcm by the
formula : lcm(m, n) =
The Euler Phi function,
to n that are relatively prime to n. If we let
numbers that divide p, then:
2 Modulo Trickery
The division algorithm states that when dividing n by p ≠ 0, there is exactly one integer q
such that n = pq + r, where to be
r in this equation . We use the notation r solving equations . There are
a number of theorems that apply to modulos, some of which are outlined here:
• k · n + c
of modulos.
This is the result of binomial expansion of the left side.
known as Fermat’s Little Theorem.
Phi function. This is Euler’s Generalization of Fermat’s Little Theorem.
• (p − 1)!
Whenever the word remainder appears, you should immediately think modulos. Likewise,
determining the last few digits of a number should make you consider modulos.
The above theorems are merely suppliments to the algebra that can be performed on
modular equations, which we outline here. The rules of modular arithmetic can be summarized
as follows:
1. The only numbers that can be divided by m in modulo n are those that are multiples
of gcd(m, n).
2. When multiplying by m in modulo n, the only numbers that can result are multiples
of gcd(m, n).
3. Taking the square root of both sides is “normal” only in prime modulos. (For example,
the solutions to
4. When solving for integer solutions in modulo n, any integer multiple of n can be added
to or subtracted from any number. (This includes adding multiples of n to square roots
of negative numbers.)
5. All other operations behave normally according to the standard rules of algebra over
the integers.
Consider, for example, solving for all positive n ≤ 100 for which
43. Of course we set up formula and
find that
-123 with −123 + 5 · 43 = 49 and continue:
Therefore, all such n are 3, 39, 46, 82, and 89.
3 Practice
All of the following problems can be solved with the techniques enumerated above.
1. How many factors does 800 have?
2. How many times does 7 divide 100!?
3. What is the smallest positive integer n for which
4. In Mathworld , the basic monetary unit is the Jool, and all other units of currency are
equivalent to an integral number of Jools. If it is possible to make the Mathworld
equivalents of $299 and $943, then what is the maximum possible value of a Jool in
terms of dollars ?
5. What are the last three digits of
6. Compute the remainder when 2000! is divided by 2003.
7. (ARML 1999) How many ways can one arrange the numbers 21, 31, 41, 51, 61, 71, and
81 such that any four consecutive numbers add up to a multiple of 3?
8. Determine all positive integers n ≤ 100 such that | {"url":"https://www.softmath.com/tutorials-3/relations/lecture-vii---number-theory.html","timestamp":"2024-11-10T08:47:20Z","content_type":"text/html","content_length":"41805","record_id":"<urn:uuid:6a4179a4-23ae-4c75-a052-09007515e8bf>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00425.warc.gz"} |
A block of mass m1 lies on top of fixed wedge as shown in figur-Turito
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A block of mass m1 lies on top of fixed wedge as shown in figure-1 and another block of mass m2 lies on top of wedge which is free to move as shown in figure-2. At time t = 0, both the blocks are
released from rest from a vertical height h above the respective horizontal surface on which the wedge is placed as shown. There is no frcition between block and wedge in both the figures. Let T1 and
T2 be the time taken by block in figure-1 and block in figure-2 respectively to just reach the horizontal surface, then :
A. T1 > T2
D. Data insufficient
The correct answer is: T1 > T2
Get an Expert Advice From Turito. | {"url":"https://www.turito.com/ask-a-doubt/a-block-of-mass-m1-lies-on-top-of-fixed-wedge-as-shown-in-figure-1-and-another-block-of-mass-m2-lies-on-top-of-wedg-q79c6a3","timestamp":"2024-11-07T14:08:30Z","content_type":"application/xhtml+xml","content_length":"1000977","record_id":"<urn:uuid:752c3ffe-3e6f-48c1-979b-8b3d63d0ef5c>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00526.warc.gz"} |
Graph structure with type parameters for nodes and edges
comfort-graph: Graph structure with type parameters for nodes and edges
This is a package candidate release! Here you can preview how this package release will appear once published to the main package index (which can be accomplished via the 'maintain' link below).
Please note that once a package has been published to the main package index it cannot be undone! Please consult the package uploading documentation for more information.
This graph structure is based on Data.Map and allows any Ord type for nodes and allows directed, undirected and more edge types. There is no need to map nodes to integer numbers. This makes handling
in applications much more comfortable, thus the package name.
Currently the package does not contain any advanced algorithm, just the data structure and some manipulation functions.
The edge type can be freely chosen. This allows great flexibility but it is a bit more cumbersome to do in Haskell 98. Examples of edge types:
• DirEdge: Edges in a directed graph
• UndirEdge: Edges in an undirected graph
• EitherEdge: For graphs containing both directed and undirected edges
• You may define an edge type with an additional identifier in order to support multiple edges between the same pair of nodes.
• Using type functions on the node type you may even define an edge type for nodes from a Cartesian product, where only "horizontal" and "vertical" edges are allowed.
For examples see the linear-circuit package and its tests. The ResistorCube test demonstrates non-integer node types and the Tree test demonstrates multigraphs.
The package is plain Haskell 98.
Related packages:
• fgl: standard package for graph processing with many graph algorithms but cumbersome data structure with Int numbered nodes
Versions 0.0, 0.0.0.1, 0.0.0.2, 0.0.0.3, 0.0.1, 0.0.2, 0.0.2.1, 0.0.2.1, 0.0.3, 0.0.3.1, 0.0.3.2, 0.0.3.3, 0.0.4
Change log None available
Dependencies base (>=4.5 && <5), containers (>=0.4 && <0.6), QuickCheck (>=2.5 && <3), transformers (>=0.5 && <0.6), utility-ht (>=0.0.10 && <0.1) [details]
License BSD-3-Clause
Author Henning Thielemann
Maintainer haskell@henning-thielemann.de
Category Data
Home page http://hub.darcs.net/thielema/comfort-graph
Source repo this: darcs get http://hub.darcs.net/thielema/comfort-graph --tag 0.0.2.1
head: darcs get http://hub.darcs.net/thielema/comfort-graph
Uploaded by HenningThielemann at 2017-11-08T09:34:27Z
Maintainer's Corner
Package maintainers
For package maintainers and hackage trustees | {"url":"https://hackage.haskell.org/package/comfort-graph-0.0.2.1/candidate","timestamp":"2024-11-10T03:28:17Z","content_type":"application/xhtml+xml","content_length":"8953","record_id":"<urn:uuid:94eec79c-6f41-4cbe-a34c-3d5681a1989b>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00619.warc.gz"} |
This Is How You Fix Your Broken math websites
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Hash Tables | Technical Interview Study Guide
Hash tables are a corner stone of data structures for most problems, they are useful when there are unique keys with given values
This technique is particularly when trying to detect all anagrams and store them in buckets of same anagrams
Assign each character a prime number and calculate the hash
Does not scale well if more than 52 characters possible or $n > 5000$
primes = [3, 5, 7, ...] def hash(string): value = 1 for ch in string: value *= primes[ord(ch) - ord('a')] return value
I would recommend using these over hash tables where possible as they can represent the same information while reducing the complexity of the code
freq = [0] * 26 # is the same as freq = {} for ch in s: if ch not in freq: freq[ch] = 0
This is a slight optimization over using hash tables/sets if the only thing that needs to be done is duplicate detection
Not feasible if the numbers can be very large as size grows by powers of 2
acc = 0 for ch in string: mask = 1 << (ord(ch) - ord('a')) if acc ^ mask < acc: # duplicate found return False acc |= mask
Often used to improve performance at the cost of using more space to store the hash table
Use Linked Lists to optimize to quickly delete/insert nodes in a certain order
Hash table has pointers to the nodes of the linked list
Open addressing (probing for a blank spot if current bucket is filled)
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Impact of Porosity on Composite Material Properties
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The assignment is done to evaluate the porosity of a composite model with the use of 3D inversion techniques. The model is built using the ANSYS software and the effect of pores on the composite body
is being evaluated with the help of inverse techniques of MATLAB. The thickness of the model is varied according to the requirement of the model. Different scenarios are used to evaluate the effect
on the solid body. The pores are also introduced in the later part of research in the layer of epoxy of the composite structure. The structure is kept submerged in water and a transducer is also
introduced in the water to provide electrical impulses of a certain frequency the composite structure is made to resonate on the frequency to evaluate several state properties of the compost
structure such as transmission, reflection and absorption.
The composite model is being modelled in MATLAB software, and the one-ply model is being kept in water with a transducer. The transducer applies electrical frequency in the water, and the evaluation
of the material is done with the inclusion of pores in the model. The thickness of the model is varied with the increase in the number of plies. The one-ply model is being kept in the model first,
the thickness of the carbon fiber is 0.125 mm. And that of epoxy is 0.001 which is not accountable in this case. The resonance is achieved when the natural impedance of the composite body matches the
frequency of the transducer. The porosity of the material is being evaluated with the evaluation of the absorption, transmission, and reflection curves, of the one-ply and the 3-ply model when the
pore is introduced in the epoxy layer. The evaluation of the curves is being done in MATLAB software. The frequency that is being applied by the transducer on the model through the water is 3MHz when
the model is submerged in the water.
The body is first made of 1 ply and no pore is introduced in the layers of the body then the body is submerged and the influence of the transducer is observed on the body when a certain frequency
impulse is applied on the structure when the structure is submerged in water. The curve of the absorption, reflection and transmission are being obtained which explains the effects of the presence of
pores on the body and the state properties of the body.
Figure 1: Transmission Results
The curve of transmission of the model with 1 ply and no pores is generated in the MATLAB software. The dip around the frequency of 3.5 MHz, which indicates the attenuation of the wave that is
incident on the model when the model is dipped in the water and a transducer is being provided in the water to apply a wave of impulse. The value of the coefficient of transmission reaches the
maximum value when the frequency reaches the maximum frequency and after the frequency again rises which indicates the transition from high attenuation to the region of low attenuation region.
Figure 2: Reflection curve
The curve of reflection is obtained from the MATLAB software and the analysis of the body that is made of 1 ply and no pore is being made submerged in water along with a transducer which executes a
particular frequency of electrical impulse in the system. The frequency of the wave that is being generated from the system is analysed to evaluate the reflection of the system in the submerged
system. The sharp peak around the frequency of 3.5 MHz, is obtained and it signifies the resonant frequency of the composite model. The mismatch of impedance of the model and the surrounding medium
has caused for this rise of the peak in the curve. The coefficient of reflection at this particular frequency is very close to 1, which indicates a strong reflection of the incident wave.
Figure 3: Absorption Results
The absorption coefficient of the model which is made of 1 play and has no pore in it is gets a peak around the resonant frequency. The model is made of 1 ply and the model is being submerged in
water and a transducer is being introduced in the body which provides an incident impulse and the resonance of the body is identified to achieve the coefficient of absorption. The peak of the model
that is obtained at the resonant frequency denotes that maximum energy is being dissipated at the resonant frequency by the composite model when an impulse is being applied to the model of the body
that is being kept submerged. The coefficient of absorption of the model is close to zero at low and high frequencies which indicates that the energy dissipated is low at these particular
After that, the body is made thick with the help of 3 plies but no pores are introduced in the body. The effect of the incident impulse on the state properties of the body with 3 plies are evaluated
with the help of the graphs that are being obtained from MATLAB. The change of the curves of transmission, absorption and reflection of the 3 ply no pore model is evaluated and the difference of the
curves also evaluates the effect of the thickness of the model on the changes that are being inflicted by the effect of incident impulse on the state properties of the model.
Figure 4: Transmission Results
The number of peaks notes are more because of the increase of the number of plies, the plot is used to evaluate the energy transmission of the system in the model which is made of 3 plies and is
submerged in water with a transducer introduced in the water which applies incident impulse on the model.
Figure 5: Absorption Results
The curve is used to represent the absorption of energy in a model that is made of 3 plies and is submerged in water. The absorption is not changed by the number of plies in the system which denotes
that the absorption is still close to zero in the low and high frequencies and this denotes that the absorption of energy by the body is low.
The curve evaluates the reflection of energy in the body that is made of 3 ply and has no pore in it. The curve shows 9 peaks in the curve and the peak is obtained at resonant frequencies where the
impedance of the model matches the impedance of the surrounding medium.
The curve obtained represents the transmission of the frequency of the wave in resonance with the 1ply model. The model is kept in water and the transducer is used to apply the frequency of 3MHZ. The
fading of the waves represents the effect of the incident impulse on the 1-ply model when a pore is introduced in the epoxy layer of the body. The fading of the frequency occurs because of the pore
that is introduced and therefore it is evaluated that the presence of pore in the composite body affects the structural performance of the body. The transmission is low in the figure which represents
that fewer transmissions are brought down due to the effect of the pores which are introduced using the Waterman technique. The bubble is air-filled and the specifications are used while the
induction of the bubble is in the 1-ply epoxy model.
The above figure is the curve of absorption of the 1-ply model when the model is submerged in water and a transducer is placed in the submerged model providing a frequency of 3MHz in water. The model
has pores that are introduced in the epoxy layer and the PSO method is used to evaluate the effect of the pores in the absorption of the composite model, to evaluate the attenuation of the frequency
because of the pores. Pores in a composite body can also absorb energy from incident waves, leading to attenuation or absorption of the transmitted waves. The absorption of waves by pores depends on
the material properties of the pores, such as their dielectric or acoustic properties, as well as the frequency or wavelength of the incident waves. Pores with higher absorption properties or
resonant frequencies that match the incident waves can result in increased attenuation or absorption of the transmitted waves. The absorption stated in the figure is high which refers that more
absorption is due to the presence of pores in the epoxy layer of the 1-ply model and the rise of absorption indicates the attenuation of the frequency generated.
Pores in a composite body can scatter and reflect incident waves, causing changes in the direction, intensity, or phase of the transmitted waves. The scattering and reflection of waves by pores
depend on the size and shape of the pores relative to the wavelength of the incident waves. Pores that are larger or comparable in size to the wavelength of the incident waves are more likely to
cause significant scattering and reflection, resulting in reduced transmission through the composite body. The attenuation of the emitted frequency curve of transmission is because of the pores which
are being introduced in the epoxy layer. The curve obtained states high reflection which causes a decrease in transmission of the body when kept in the submerged system with a transducer. After the
1ply model is analysed by keeping in a submerged water system with a transducer in it, providing an incidence impedance of 3MHz, the thickness of the model is increased with the number of plies. The
curves obtained in this section are discrete since the number of plies used in this model is 3, and the number of peaks is 9.
The transmission is less when the curve on the transmission is obtained from the MATLAB software. The curve of transmission shows attenuation or damping of the frequency impedance that is generated
from the 3-ply model when an incident impulse is being applied by the transducer in the submerged system in water. The number of layers used in making the model damps the generated frequency by the
body which is evaluated in the MATLAB software.
The curve of absorption of the composite structure is showing high absorption there are a number of peaks as each of the plies generates 3 peaks. The high absorption leads to high attenuation of the
frequency that is generated and obtained. The absorption is high in the curve that is being generated in MATLAB, and the absorption is pores than in the case of 1 ply and no pores in them.
The curve of the reflection of the 3-ply and porous composite structure is built using MATLAB. The codes are used to evaluate the effect of the pore that is being filled with air and is being
introduced in the epoxy layer of the composite structure. The reflection is increased because of the presence of pores. The pores are not repetitive and do not overlap. The reflection is increased
which refers that the curve of frequency is attenuated because of the presence of the pores. The reflection of the structure is obtained with MATLAB where the curves are used to denote the effect of
pores on the composite structure.
The composite structures depend on the design and the quality of their structure. The presence of pores can have several adverse effects on the solid body with respect to transmission, absorption and
reflection. MATLAB is used to analytically analyse the pore radius, the thickness, porosity and pore radius are being evaluated with the help of techniques to optimize the structure. The technique
used in PSO can be expanded as Particle Swamp Optimization. The PSO is a technique where the optimal values of the particular values are obtained with experiments done on the system (Yang et al.
2022). The optimization is being done in this case with the help of PSO strategies in the software of MATLAB. PSO is one of the most powerful algorithms for the evaluation of essential parameters
that can be obtained from the composite structure. The structure is made of an epoxy layer, a composite layer and a layer of composite fibers.
The epoxy layer is introduced with air pores and then the structure is evaluated and the effect of the electric transmissions are evaluated. With the help of the PSO, the conditions of the body are
simulated and the parameters are obtained. The values of obtained values are presented in the form of a table. The parameters such as the thickness of the material, the thickness of each layer, the
radius of the pore and the density of the pore that is being introduced in the epoxy layer are being evaluated (Zheng et al. 2023). The model of PSO is done on a 3D model of the body and therefore it
is one of the techniques of 3D inversion techniques which is used in Matlab to evaluate values of essential parameters and then information on several conditions are being fed in Matlab and with the
help of that conditions are being run on the model to evaluate the parameters in MATLAB. The parameters that are being fed in MATLAB regarding the 3D modelling and PSO evaluation of the model are the
number of dimensions of the model, and the ranges of the conditions which include the lower bound values and the upper bound values which evaluate the essential parameters of the model.
The parameters of the algorithm of the PSO evaluation of the body that is essential for the completion of the problem are the number of pores in the epoxy layer, the maximum number of iterations, the
inertia of each of the pores, the cognition of the pore, and the influence of the pore on the composite structure. It is stated that the pores that are introduced in the epoxy layer are not
repetitive and are not overlapping. The inertia of the pores is negligible since the pores are being filled with air and the influence of the pore on the composite structure is obtained using the
algorithms where it is observed that the presence of pores affects some material properties of the body (Zhang et al. 2023). The properties of the composite structure that are being affected by the
presence of pores in the epoxy layer of the structure are the transmission of the body, a reflection of the body and the absorption of the body which are obtained in the form of curves and the
coefficients of reflection, absorption, and transmission. The frequency that is being applied on the body is from 0to20MHz, which is being applied on the body to evaluate the effect of the electric
impulse of a particular frequency on the porous body whose thickness is varied with the help of adding layers. The thickness of the layers that are obtained is provided in the form of a table.
Longitudinal Velocity Sheer Velocity Density Attenuation Thickness
Composite 2959 1870 1522 0 125
Epoxy resin 2903 1319 1270 0.15 10
Table 1: Parameters of the thickness of the composite structure
The simulation of the conditions of the body is being done to evaluate the essential parameter of the body when the body is being submerged in water and is evaluated with the help of strategies of
PSO. Since the pores are filled with air the properties of air are also required to be kept in mind. The density of air is 1.29 gm per liter and this figure is not required of the effect of air in
the system. The data are being taken which is used to simulate the model of the composite model with non-repetitive and non-overlapping pores are being fed in the system which is used to simulate the
model using PSO strategies where a number of operations are being performed on the model to evaluate the essential parameters of the system. The system is made independently and imported in MATLAB
and then analysed for the values of parameters such as the density of the pore, and the radius of the pore. The mapping of the pore is done with the help of MATLAB codes and inversion strategies of
the 3D body. Such a technique which can be used to evaluate the parameter is PSO, where the solid body is accepted and then the solid body is analysed for the effect of porosity in the epoxy layer of
the composite structure. The thickness, porosity and pore radius of the system is being evaluated using the inverse technique of SMO and the values are stated in the form of a table.
Thickness Porosity Pore Radius
130 units 23% 0.236 units
Table 2: Table of Results
Use Native Assignment Help to make your engineering dreams come true! Our skilled writers are specialised in delivering high-quality engineering assignment help that is accurate, well-researched, and
tailored to satisfy your educational objectives. We have you in mind regarding civil, mechanical, electrical, or any other engineering field!
The assignment is done to evaluate the effect of porosity on the composite model. The model is made of plies and the number of plies are being varied in each step for the evaluation of the porosity
in the state properties of the model. The model is submerged in the water and the transducer is immersed to evaluate the porosity and to map pores in the composite model. The effect of energy on the
properties such as transmission, reflection and absorption are evaluated in the assignment. The simulation is being done with the help of inverse optimization techniques such as SMO which is used to
evaluate the value of certain essential parameters of the model.
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on HA-ZrO2-PVA Composites.Materials,16(3), p.1107.
Capasso, I., Liguori, B., Verdolotti, L., Caputo, D., Lavorgna, M. and Tervoort, E., 2020. Process strategy to fabricate a hierarchical porosity gradient in diatomite-based foams by 3D
printing.Scientific Reports,10(1), pp.1-9.
Chen, C.T. and Gu, G.X., 2019. Machine learning for composite materials.MRs Communications,9(2), pp.556-566.
Cojocaru, C., Pascariu, P., Enache, A.C., Bargan, A. and Samoila, P., 2023. Application of Surface-Modified Nanoclay in a Hybrid Adsorption-Ultrafiltration Process for Enhanced Nitrite Ions Removal:
Chemometric Approach vs. Machine Learning.Nanomaterials,13(4), p.697.
Gowriboy, N., Kalaivizhi, R., Kaleekkal, N.J., Ganesh, M.R. and Aswathy, K.A., 2022. Fabrication and characterization of polymer nanocomposites membrane (Cu-MOF@ CA/PES) for water treatment.Journal
of Environmental Chemical Engineering,10(6), p.108668.
Hermosilla, R., Oñate, A., Castillo, R., De la Fuente, A., Sepúlveda, J., Escudero, B., Vargas-Silva, G., Tuninetti, V., Melendrez, M. and Medina, C., 2023. Influence stacking sequence and heat
treatments on the out-of-plane mechanical properties of 3D-printed fiberglass-reinforced thermoplastics.The International Journal of Advanced Manufacturing Technology, pp.1-12.
Kaczmarczyk, G.P. and Ca?a, M., 2023. Possible Application of Computed Tomography for Numerical Simulation of the Damage Mechanism of Cementitious Materials—A Method Review.Buildings,13(3), p.587.
Paidi, M.K., Polisetti, V., Damarla, K., Singh, P.S., Mandal, S.K. and Ray, P., 2022. 3D Natural Mesoporous Biosilica-Embedded Polysulfone Made Ultrafiltration Membranes for Application in Separation
Technology.Polymers,14(9), p.1750.
Pang, M., Ba, J., Fu, L.Y., Carcione, J.M., Markus, U.I. and Zhang, L., 2020. Estimation of microfracture porosity in deep carbonate reservoirs based on 3D rock-physics templates.Interpretation,8(4),
Sui, D., Xu, L., Zhang, H., Sun, Z., Kan, B., Ma, Y. and Chen, Y., 2020. A 3D cross-linked graphene-based honeycomb carbon composite with excellent confinement effect of organic cathode material for
lithium-ion batteries.Carbon,157, pp.656-662.
Yang, C., Bai, Y., Xu, H., Li, M., Cong, Z., Li, H., Chen, W., Zhao, B. and Han, X., 2022. Porosity Tunable Poly (Lactic Acid)-Based Composite Gel Polymer Electrolyte with High Electrolyte Uptake for
Quasi-Solid-State Supercapacitors. Polymers, 14(9), p.1881.
Zhang, X., Hui, Z., King, S.T., Wu, J., Ju, Z., Takeuchi, K.J., Marschilok, A.C., West, A.C., Takeuchi, E.S., Wang, L. and Yu, G., 2022. Gradient architecture design in scalable porous battery
electrodes.Nano Letters,22(6), pp.2521-2528.
Zhang, Z., Fang, H., Xu, Z., Lv, J., Shen, Y. and Wang, Y., 2023. Multi-objective Generative Design of Three-Dimensional Composite Materials.arXiv preprint arXiv:2302.13365.
Zheng, X., Chen, T.T., Jiang, X., Naito, M. and Watanabe, I., 2023. Deep-learning-based inverse design of three-dimensional architected cellular materials with the target porosity and stiffness using
voxelized Voronoi lattices.Science and Technology of Advanced Materials,24(1), p.2157682. | {"url":"https://www.nativeassignmenthelp.co.uk/porosity-in-composite-materials-using-3d-inversion-techniques-assignment-sample-23794","timestamp":"2024-11-05T03:37:49Z","content_type":"text/html","content_length":"319854","record_id":"<urn:uuid:dc488512-2f44-40fa-b95f-52dddf87bb85>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00331.warc.gz"} |
Draw A Rectangle That Is Also A Square
Draw A Rectangle That Is Also A Square - Then, draw a shorter vertical line that extends down from one end of the first line. Door for our next drawing, let’s draw some stairs leading up to a door.
It appears as though the focus of this exercise is on the diagonals of the. Web in this video, i will demonstrate step by step how to draw a rectangle given the length equal in area to a given
square. A square has equal sides (marked s) and every angle is a right angle (90°) also opposite sides are parallel.
Web in geometry, according to the properties of a square and rectangle we consider every square to be a rectangle, but all rectangles are not squares. The key differences between a square and a
rectangle are listed below: Maybe you’re laying a foundation for a shed, or maybe you’re laying out a badminton. Area is measured in square inches, square feet, square centimeters, etc. Then, draw a
shorter vertical line that extends down from one end of the first line. Web sometimes you want to be able to draw a perfect rectangle on the ground, one with four exact 90° angles. Identify
triangles, quadrilaterals, pentagons, hexagons, and cubes.
How to Draw a Rectangle 5 Steps (with Pictures) wikiHow
Area is measured in square inches, square feet, square centimeters, etc. Web like a square, a rectangle is also called an equiangular quadrilateral. That's one way to think about a square. So if were
to straighten it out a little bit, it's a rhombus so all the four sides are the same. Web in this.
How to Draw a rectangle equal in area to a square YouTube
And you have four right angles. Or you could view it as a rectangle where all four sides are congruent. However, this answer makes sense if you just think about the properties of these two shapes.
Web in geometry, according to the properties of a square and rectangle we consider every square to be a.
Very simple and easy rectangle drawing how to draw a rectangle easily
Web in geometry, according to the properties of a square and rectangle we consider every square to be a rectangle, but all rectangles are not squares. Web you could view a square as a rhombus with
four right angles. So, each of these small squares is one square unit. And you have four right angles..
How to draw a rectangle given its sides YouTube
Web actually four right angles in parallelogram c, so we know that it is a square (and a rhombus). The key differences between a square and a rectangle are listed below: Area is measured in square
inches, square feet, square centimeters, etc. Web learn how to construct a square equivalent (equal in area) to a.
How to draw a rectangle Using ruler and set square (Step by Step
However, this answer makes sense if you just think about the properties of these two shapes. The key differences between a square and a rectangle are listed below: The little squares in each corner
mean right angle. In addition, you can specify that all corners scale. Maybe you’re laying a foundation for a shed, or.
Is Square a Rectangle?
Now, since a rectangle is a parallelogram, its opposite sides must be congruent. So let me draw a square here. Web recognize and draw shapes having specified attributes, such as a given number of
angles or a given number of equal faces. This youtube channel is dedicated to teaching people how to improve their technical.
Draw an rectangle using OpenCV in C++
The little squares in each corner mean right angle. Gimp will outline the selection and provide adjustment handles at the four corners for resizing, and along each edge for constrained resizing along
a single edge (although these will be hidden until you hover the cursor over the selection). You can use this as a basis.
How to Draw a Rectangle and Square in GIMP YouTube
Web like a square, a rectangle is also called an equiangular quadrilateral. The rectangle a rectangle is a parallelogram whose sides intersect at 90° angles. If it covers eight square units, than it
has an area of eight square units, but we can't just draw the identical rectangle, because we're also told that it should.
Is a Square a Rectangle? Yes or No? — Mashup Math
Web let's say that you have a square, which is a special case of a rectangle. Web in this chapter we will derive formulas for the areas of the geometric objects which we have studied. You can use
this as a basis for any cube or box you want to draw. So let me draw.
How to draw a rectangle shape simple with pen and color Easy rectangle
So if were to straighten it out a little bit, it's a rhombus so all the four sides are the same. This youtube channel is dedicated to teaching people how to improve their technical drawing skills. A
square also fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are.
Draw A Rectangle That Is Also A Square However, this answer makes sense if you just think about the properties of these two shapes. Web let's say that you have a square, which is a special case of a
rectangle. Or you could view it as a rectangle where all four sides are congruent. A square also fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are equal length).
Web learn how to construct a square equivalent (equal in area) to a given rectangle.
So, Each Of These Small Squares Is One Square Unit.
These are the two major. Web so, we want to draw another rectangle that also covers eight square units. A square also has 4 congruent sides, therefore it is a rhombus. This square is one square unit,
and this square is one square unit, and so on.
Since All Rectangles And All Rhombuses Are Parallelograms, And All Shapes That Are Both Rectangles And Rhombuses Are Also Parallelograms, We Can Show The Relationship Between These Figures With This
Venn Diagram:
We know that abcd is a rectangle, so let’s use some rectangle properties to help us figure out what x is. So let me draw a square here. The formula to calculate the area of a square = (side) 2 and
the area of a rectangle = length × width. Web like a square, a rectangle is also called an equiangular quadrilateral.
Door For Our Next Drawing, Let’s Draw Some Stairs Leading Up To A Door.
You can use this as a basis for any cube or box you want to draw. This youtube channel is dedicated to teaching people how to improve their technical drawing skills. Web in this video, i will
demonstrate step by step how to draw a rectangle given the length equal in area to a given square. Create triangles, circles, angles, transformations and much more!
Web You Could View A Square As A Rhombus With Four Right Angles.
A square has equal sides (marked s) and every angle is a right angle (90°) also opposite sides are parallel. That's one way to think about a square. Identify triangles, quadrilaterals, pentagons,
hexagons, and cubes. And now we're asked to.
Draw A Rectangle That Is Also A Square Related Post : | {"url":"https://sandbox.independent.com/view/draw-a-rectangle-that-is-also-a-square.html","timestamp":"2024-11-06T05:13:08Z","content_type":"application/xhtml+xml","content_length":"24297","record_id":"<urn:uuid:cc70dcfc-e295-4337-989d-30ea683f7150>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00701.warc.gz"} |
Task parallelism with Chapel
Last updated on 2024-10-08 | Edit this page
• “How do I write parallel code for a real use case?”
Here is our plan to task-parallelize the heat transfer equation:
1. divide the entire grid of points into blocks and assign blocks to individual tasks,
2. each task should compute the new temperature of its assigned points,
3. perform a reduction over the whole grid, to update the greatest temperature difference between temp_new and temp.
For the reduction of the grid we can simply use the max reduce statement, which is already parallelized. Now, let’s divide the grid into rowtasks x coltasks sub-grids, and assign each sub-grid to a
task using the coforall loop (we will have rowtasks*coltasks tasks in total).
config const rowtasks = 2;
config const coltasks = 2;
// this is the main loop of the simulation
delta = tolerance;
while (c<niter && delta>=tolerance) do {
c += 1;
coforall taskid in 0..coltasks*rowtasks-1 do {
for i in rowi..rowf do {
for j in coli..colf do {
temp_new[i,j] = (temp[i-1,j] + temp[i+1,j] + temp[i,j-1] + temp[i,j+1]) / 4;
delta = max reduce (temp_new-temp);
temp = temp_new;
if c%outputFrequency == 0 then writeln('Temperature at iteration ',c,': ',temp[x,y]);
Note that now the nested for loops run from rowi to rowf and from coli to colf which are, respectively, the initial and final row and column of the sub-grid associated to the task taskid. To compute
these limits, based on taskid, we need to compute the number of rows and columns per task (nr and nc, respectively) and account for possible non-zero remainders (rr and rc) that we should add to the
last row and column:
config const rowtasks = 2;
config const coltasks = 2;
const nr = rows/rowtasks;
const rr = rows-nr*rowtasks;
const nc = cols/coltasks;
const rc = cols-nc*coltasks;
// this is the main loop of the simulation
delta = tolerance;
while (c<niter && delta>=tolerance) do {
coforall taskid in 0..coltasks*rowtasks-1 do {
var rowi, coli, rowf, colf: int;
var taskr, taskc: int;
taskr = taskid/coltasks;
taskc = taskid%coltasks;
if taskr<rr then {
else {
rowi = (taskr*nr)+1+rr;
rowf = (taskr*nr)+nr+rr;
if taskc<rc then {
coli = (taskc*nc)+1+taskc;
colf = (taskc*nc)+nc+taskc+1;
else {
coli = (taskc*nc)+1+rc;
colf = (taskc*nc)+nc+rc;
for i in rowi..rowf do {
for j in coli..colf do {
As you can see, to divide a data set (the array temp in this case) between concurrent tasks, could be cumbersome. Chapel provides high-level abstractions for data parallelism that take care of all
the data distribution for us. We will study data parallelism in the following lessons, but for now, let’s compare the benchmark solution with our coforall parallelization to see how the performance
chpl --fast parallel1.chpl
./parallel1 --rows=650 --cols=650 --x=200 --y=300 --niter=10000 --tolerance=0.002 --outputFrequency=1000
The simulation will consider a matrix of 650 by 650 elements,
it will run up to 10000 iterations, or until the largest difference
in temperature between iterations is less than 0.002.
You are interested in the evolution of the temperature at the position (200,300) of the matrix...
and here we go...
Temperature at iteration 0: 25.0
Temperature at iteration 1000: 25.0
Temperature at iteration 2000: 25.0
Temperature at iteration 3000: 25.0
Temperature at iteration 4000: 24.9998
Temperature at iteration 5000: 24.9984
Temperature at iteration 6000: 24.9935
Temperature at iteration 7000: 24.9819
The simulation took 17.0193 seconds
Final temperature at the desired position after 7750 iterations is: 24.9671
The greatest difference in temperatures between the last two iterations was: 0.00199985
This parallel solution, using 4 parallel tasks, took around 17 seconds to finish. Compared with the ~20 seconds needed by the benchmark solution, seems not very impressive. To understand the reason,
let’s analyse the code’s flow. When the program starts, the main thread does all the declarations and initialisations, and then, it enters the main loop of the simulation (the while loop). Inside
this loop, the parallel tasks are launched for the first time. When these tasks finish their computations, the main task resumes its execution, it updates delta, and everything is repeated again. So,
in essence, parallel tasks are launched and resumed 7750 times, which introduces a significant amount of overhead (the time the system needs to effectively start and destroy threads in the specific
hardware, at each iteration of the while loop).
Clearly, a better approach would be to launch the parallel tasks just once, and have them executing all the simulations, before resuming the main task to print the final results.
config const rowtasks = 2;
config const coltasks = 2;
const nr = rows/rowtasks;
const rr = rows-nr*rowtasks;
const nc = cols/coltasks;
const rc = cols-nc*coltasks;
// this is the main loop of the simulation
delta = tolerance;
coforall taskid in 0..coltasks*rowtasks-1 do {
var rowi, coli, rowf, colf: int;
var taskr, taskc: int;
var c = 0;
taskr = taskid/coltasks;
taskc = taskid%coltasks;
if taskr<rr then {
rowi = (taskr*nr)+1+taskr;
rowf = (taskr*nr)+nr+taskr+1;
else {
rowi = (taskr*nr)+1+rr;
rowf = (taskr*nr)+nr+rr;
if taskc<rc then {
coli = (taskc*nc)+1+taskc;
colf = (taskc*nc)+nc+taskc+1;
else {
coli = (taskc*nc)+1+rc;
colf = (taskc*nc)+nc+rc;
while (c<niter && delta>=tolerance) do {
c = c+1;
for i in rowi..rowf do {
for j in coli..colf do {
temp_new[i,j] = (temp[i-1,j] + temp[i+1,j] + temp[i,j-1] + temp[i,j+1]) / 4;
//update delta
//update temp
//print temperature in desired position
The problem with this approach is that now we have to explicitly synchronise the tasks. Before, delta and temp were updated only by the main task at each iteration; similarly, only the main task was
printing results. Now, all these operations must be carried inside the coforall loop, which imposes the need of synchronisation between tasks.
The synchronisation must happen at two points:
1. We need to be sure that all tasks have finished with the computations of their part of the grid temp, before updating delta and temp safely.
2. We need to be sure that all tasks use the updated value of delta to evaluate the condition of the while loop for the next iteration.
To update delta we could have each task computing the greatest difference in temperature in its associated sub-grid, and then, after the synchronisation, have only one task reducing all the
sub-grids’ maximums.
var delta: atomic real;
var myd: [0..coltasks*rowtasks-1] real;
//this is the main loop of the simulation
coforall taskid in 0..coltasks*rowtasks-1 do
var myd2: real;
while (c<niter && delta>=tolerance) do {
c = c+1;
for i in rowi..rowf do {
for j in coli..colf do {
temp_new[i,j] = (temp[i-1,j] + temp[i+1,j] + temp[i,j-1] + temp[i,j+1]) / 4;
myd2 = max(abs(temp_new[i,j]-temp[i,j]),myd2);
myd[taskid] = myd2
// here comes the synchronisation of tasks
temp[rowi..rowf,coli..colf] = temp_new[rowi..rowf,coli..colf];
if taskid==0 then {
delta.write(max reduce myd);
if c%outputFrequency==0 then writeln('Temperature at iteration ',c,': ',temp[x,y]);
// here comes the synchronisation of tasks again
Challenge 4: Can you do it?
Use sync or atomic variables to implement the synchronisation required in the code above.
One possible solution is to use an atomic variable as a lock that opens (using the waitFor method) when all the tasks complete the required instructions
var lock: atomic int;
//this is the main loop of the simulation
coforall taskid in 0..coltasks*rowtasks-1 do
while (c<niter && delta>=tolerance) do
//here comes the synchronisation of tasks
temp[rowi..rowf,coli..colf] = temp_new[rowi..rowf,coli..colf];
//here comes the synchronisation of tasks again
Using the solution in the Exercise 4, we can now compare the performance with the benchmark solution
chpl --fast parallel2.chpl
./parallel2 --rows=650 --cols=650 --x=200 --y=300 --niter=10000 --tolerance=0.002 --outputFrequency=1000
The simulation will consider a matrix of 650 by 650 elements,
it will run up to 10000 iterations, or until the largest difference
in temperature between iterations is less than 0.002.
You are interested in the evolution of the temperature at the position (200,300) of the matrix...
and here we go...
Temperature at iteration 0: 25.0
Temperature at iteration 1000: 25.0
Temperature at iteration 2000: 25.0
Temperature at iteration 3000: 25.0
Temperature at iteration 4000: 24.9998
Temperature at iteration 5000: 24.9984
Temperature at iteration 6000: 24.9935
Temperature at iteration 7000: 24.9819
The simulation took 4.2733 seconds
Final temperature at the desired position after 7750 iterations is: 24.9671
The greatest difference in temperatures between the last two iterations was: 0.00199985
to see that we now have a code that performs 5x faster.
We finish this section by providing another, elegant version of the 2D heat transfer solver (without time stepping) using data parallelism on a single locale:
const n = 100, stride = 20;
var temp: [0..n+1, 0..n+1] real;
var temp_new: [1..n,1..n] real;
var x, y: real;
for (i,j) in {1..n,1..n} { // serial iteration
x = ((i:real)-0.5)/n;
y = ((j:real)-0.5)/n;
temp[i,j] = exp(-((x-0.5)**2 + (y-0.5)**2)/0.01); // narrow Gaussian peak
coforall (i,j) in {1..n,1..n} by (stride,stride) { // 5x5 decomposition into 20x20 blocks => 25 tasks
for k in i..i+stride-1 { // serial loop inside each block
for l in j..j+stride-1 do {
temp_new[k,l] = (temp[k-1,l] + temp[k+1,l] + temp[k,l-1] + temp[k,l+1]) / 4;
We will study data parallelism in more detail in the next section.
Key Points
• “To parallelize the diffusion solver with tasks, you divide the 2D domain into blocks and assign each block to a task.”
• “To get the maximum performance, you need to launch the parallel tasks only once, and run the temporal loop of the simulation with the same set of tasks, resuming the main task only to print the
final results.”
• “Parallelizing with tasks is more laborious than parallelizing with data (covered in the next section).” | {"url":"http://www.hpc-carpentry.org/hpc-chapel/14-parallel-case-study.html","timestamp":"2024-11-03T06:00:52Z","content_type":"text/html","content_length":"44469","record_id":"<urn:uuid:6f52d406-34ea-4fdf-8f36-451f6cff1059>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00036.warc.gz"} |
The study of cyclic adsorption air separation and oxygen concentration processes
DOI: 10.17277/amt.2019.01.pp.055-072
The Study of Cyclic Adsorption Air Separation and Oxygen Concentration Processes
V.G. Matveykin1, E.I. Akulinin2*, N.V. Posternak1, S.A. Skvortsov2, S.I. Dvoretsky2
1 OJSC "Corporation "Roskhimzashchita", 19, Morshanskoye shosse, Tambov, 392000, Russia 2 Tambov State Technical University, 106, Sovetskaya St., Tambov, 392000, Russia
* Corresponding author: Tel .: +7 (909) 231 40 61. E-mail: akulinin-2006@yandex.ru
The paper presents the dynamics of the developed mathematical model of pressure swing adsorption (PSA) process, which is carried out in a dual-adsorber unit with a 13X zeolite adsorbent used for air
separation with the aim of oxygen concentration. The authors formulate and solve the regularized problem of identifying the kinetic parameters for the mathematical model - the mass and oxygen
transfer coefficients for oxygen and nitrogen. Numerical studies of the effect of raw materials load (air composition, environment temperature and pressure) and control variables ("adsorption
-desorption" cycle time, pressure at the compressor outlet, laws of changing the opening degree of the inlet and discharge valves of the PSA unit) on the dynamics and performance indicators of cyclic
adsorption process of air oxygen enrichment have been carried out. The mathematical and algorithmic support for the creation of automated processes and PSA units for gas mixtures separation and
purification has been developed.
Pressure swing adsorption, zeolite adsorbent, adsorption isotherm, kinetics, mathematical model, parametric identification, calculation experiment.
© V.G. Matveykin, E.I. Akulinin, N.V. Posternak, S.A. Skvortsov, S.I. Dvoretsky, 2019
In recent decades, the use of cyclic adsorption processes for separating gas mixtures and concentrating target products has become increasingly common. Short-cycle processes for adsorptive separation
of gas mixtures are widely used in industry for air oxygen enrichment, drying gases without heating, separating hydrocarbons, concentrating carbon dioxide, extracting hydrogen, methane, etc. One of
the urgent tasks in the field of adsorption separation is air oxygen enrichment. Typical substances that accompany oxygen are nitrogen, argon, carbon dioxide. A feature of the adsorptive oxygen
concentration is the fact that in gas-air mixtures the components associated with oxygen have higher values of adsorption selectivity [1].
The analysis of numerous works by foreign and Russian scientists in the field of adsorption separation of multicomponent gas mixtures and concentration of the target product (hydrogen, oxygen, carbon
etc.) made it possible to determine the place of this article among other works, its relevance and perspectivity [2-11].
Thus, the works [2-8] present the results of numerical studies of the effect of mode variables (pressure, temperature, flow rate of the initial mixture) on the dynamics and efficiency of the
adsorption separation of two (H2-CO2), three (H2-CO2-CO), four (H2-CO2-CO-CH4), five (H2-CO2-CO-CH4-N2) and six (H2-CO2-CO-H2O-Ar-N2) component mixtures and hydrogen concentration using active carbon
and metal-organic compounds as adsorbents. In the works [2, 3], when calculating the equilibrium conditions of a multicomponent mixture, sorption isotherms of individual substances are used. The
calculation experiment allowed to study the features of the ten-adsorber unit with vacuum regeneration (in the English literature - VPSA) and the four-adsorber PSA unit with a metal-organic adsorbent
of a new type. The possibility of obtaining hydrogen with the purity
of 99.981 vol. % at the extraction degree of 81.6 % [2] and 99.9 vol. % at the extraction degree of 48.05 % [4], respectively, was established. The results of numerical studies of the effect of the
number of pressure equalization stages, their sequence and the use of combinations of different adsorbent layers on the purity and degree of hydrogen extraction from a two-component mixture (H2-CH4)
in the PSA unit using the Langmuir - Freundlis equilibrium isotherm are presented in [6, 7]. It has been established that the use of a six-sorbent unit with two pressure equalization operations
provides the best combination of the hydrogen purity (~ 99 vol. %) while achieving the extraction degree of ~ 83 %.
In [9], the calculation experiment investigated the dependences of the purity of extracted carbon dioxide from a nine-component gas mixture using active carbon and found that using the seven-adsorber
PSA unit (instead of three or four adsorbers) allows increasing the purity of the produced carbon dioxide from 95.1 vol. % up to 98.9 vol. % while reducing the extraction degree from 90.2 % to 86.1
%. In [10, 11], the mathematical models of the dynamics of the cyclic adsorption process for producing CO2 from a two-component gas mixture (CO2-N2) on zeolite 13X using the Langmuir isotherm were
studied, and the problem of the optimal design of PSA units (vacuum-pressure VPSA and fractional vacuum-pressure FVPSA types) by the complex criterion - the ratio of energy consumption of the PSA
unit to the purity of the produced carbon dioxide, was formulated and investigated. It has been established that the units implemented according to the FVPSA and VPSA schemes provide the production
of carbon dioxide with the purity of ~ 90 vol. % and ~ 72 vol. %, respectively, and the specific power of the unit according to the FVPSA scheme is on average 2.5 times higher.
Over the last decade, the number and range of consumers of air separation products have significantly increased, and the annual increase in oxygen demand is on average ~ 4-5 % due to the increased
demand in the steel and chemical industry, aluminum production, aviation and other industries and social spheres.
A significant proportion of oxygen consumers uses in their activities not so much pure oxygen as air enriched with oxygen from 30 to 90 vol. %. For these reasons, in recent years, the adsorption
method of separating air is becoming more common as the most profitable method for consumers who use oxygen and nitrogen unevenly in time.
The units separating the air mixture by adsorption using the PSA method differ in the way of creating the
driving force (the difference in equilibrium concentrations at adsorption and desorption stages) and use synthetic zeolites and activated carbons as adsorbents. Pressure-type units operate from an
overpressure source, and production gas can be directly discharged to the consumer. The given costs of electricity for oxygen production with the concentration of 90 vol. % in PSA units range from
1.5 to 1.8 kWh/m , which is several times higher than the costs of obtaining oxygen by the method of low-temperature rectification. Therefore, pressure-type units are distinguished by low
productivity and are used in industries where the problem of oxygen delivery and storage is acute. The main advantages of PSA units are their autonomy, mobility, reliability, and quick access to the
stationary periodic mode. Energy costs in units, where oxygen is obtained at almost atmospheric pressure, and vacuuming is used for nitrogen de sorption, are significantly lower and amount to ~
0.5-0.7 kWh/m .
The highest values of the oxygen extraction degree and productivity are achieved in Vacuum PSA units, in which the adsorption stage is carried out at an overpressure and the desorption stage - under
vacuum. Increasing the level of the unit automation for separating components of the air mixture and concentrating oxygen is associated both with the difficulties of mathematical modeling and
optimization of mass and heat transfer processes within the adsorber, and with the complexity of considering the mutual connections of all included devices. As a rule, the flow chart of the PSA
process includes two - four apparatus-adsorbers filled with granular adsorbent, flow boosters (air compressor, vacuum pump, etc.), receivers, and valves designed to increase and decrease the pressure
in adsorbers (desorbers) and air flow control [12-19].
Impurities of water and carbon dioxide contained in the separated air are traced in the frontal layers of the adsorbent and have practically no impact on the efficiency of nitrogen adsorption. The
limiting purity of oxygen produced in adsorption units is 95.7 % (4.3 % is accounted for by argon, which is adsorbed on zeolites as well as oxygen). In industry, an oxygen-argon mixture is produced
in adsorption units with the purity of 90-95 % [20].
The aim of this work is to study the effectiveness of cyclic adsorption processes for air separation and oxygen concentration, mathematical and algorithmic support for creating automated PSA units
for air oxygen enrichment.
The current state analysis of the PSA technology and equipment
The analysis of the current state of the PSA technology for purifying and separating gas mixtures allowed to identify a generalized flow chart of the PSA process [21-31] (Fig. 1).
The PSA process of a gas mixture is implemented in the environment with the following parameters: the air composition (vector yenv of oxygen, nitrogen, argon and other impurities concentrations),
temperature Tenv and barometric pressure Benv of the environment [32]. The pressure in the system is created by the flow rate boosters FB (compressor, blower, vacuum pump, etc.). The initial gas
mixture with concentration, flow rate,
temperature and pressure yin, Gin, Tin,Pin, respectively enters the unit inlet. Through the inlet valves K1,i (i = 1, n), the gas mixture or atmospheric air enters the adsorbers A1,i (i = 1, n),
where the process of selective adsorption of one or several gas mixture components is carried out. At the unit outlet, using check valves K3,i, a stream of concentrated production gas mixture is
formed with concentration, flow rate, temperature and pressure y out, Gout, Tout, P out,
respectively. Part of the production flow through the respective heat exchanger Tk and the throttle Thk is sent to the adsorbers (valves K2,i are open) to carry out the process of the adsorbate
desorption. The desorbed gas mixture is discharged by the flow booster FB out,1
with the composition yout,\ flow rate Go
temperature Tout,t and pressure Pout,\ respectively, into the atmosphere.
When implementing the adsorption schemes for air separation and purification, the following process organization schemes can be used: pressure (PSA - the adsorption pressure is excessive relative to
atmospheric, while the desorption pressure is atmospheric), vacuum - pressure (VPSA - adsorption pressure is excessive relative to atmospheric and the desorption pressure is below atmospheric),
vacuum pressure (VSA - the adsorption pressure is atmospheric, while the desorption pressure is below atmospheric) [33-35].
The main advantage of PSA units is the simplicity of their organization, and the disadvantage is the low extraction degree of the target product compared to other classes of units [1]. The main
advantage of VPSA units is high efficiency in extracting target components, and the disadvantage is the complexity of instrumentation. VSA units reach a compromise between the efficiency and
complexity of instrumentation, which led to their wide distribution in portable gas concentrators [36].
The adsorbers used in adsorption units can have different constructive designs that affect the structure of the flows in the adsorption layer (Fig. 2).
At the axial direction (Fig. 2a), the gas flow moves along the axis of the adsorber. The main advantage of this type of adsorbers is the simplicity of the design, and the disadvantages are the high
aerodynamic resistance of the layer. At the radial direction (Fig. 2b), the flow is directed to the central cavity and moves through the adsorption layer to the periphery. This provides low
aerodynamic resistance, the ability to provide high flow rates through the adsorber, the disadvantages are: the complexity of the design, the possibility of the stream leakage due to the relatively
small size of the adsorbent layer.
Fig.1. Generalized flow chart of the PSA process
Fig. 2. The direction of the gas flow in the adsorber:
a - axial; b -radial; c - variable
The advantage of the adsorber with a conical insert (Fig. 2c) is the ability to obtain a variable cross-section, providing a uniform flow rate over the entire height of the adsorber.
To increase the efficiency of the gas mixture adsorption separation process, a multilayer structure of adsorbents in the adsorber can be applied, where each layer is focused on the selective
absorption of certain components of the gas mixture. An example can be the use in the frontal layer of adsorbents with high activity on water vapor, which protects the subsequent layers of the
adsorbent from loss of sorption activity on the target components of the gas mixture.
The technological scheme of the PSA process (Fig. 1) can have from one to several adsorbers. The increase in the number of adsorbers allows the increase in the extraction degree of the target
component, but at the same time capital costs get higher, the complexity of the control system increases, and the reliability of the unit decreases [37]. By performance, PSA units are distinguished
by low productivity - up to 2 Nm /h; average productivity -2-20 Nm /h and high productivity - more than 20 Nm3/h.
Activated carbons, zeolites, silica gels, and active alumina are widely used as adsorbents in cyclic adsorption processes [38, 39].
In the adsorption technique, zeolites of types A, X, M are used with a low value of the silica module, which determines the structure of the crystal lattice of the zeolite and its adsorption
properties. Silica gels are mainly used for drying gases, purifying mineral oils and as a carrier of catalysts. Activated (active) carbon has a very large specific surface per unit mass, which
accounts for its high adsorption properties with respect to the sorption of high-molecular compounds.
The current state analysis of mathematical modeling of cyclic adsorption processes
The current state analysis of mathematical modeling of cyclic adsorption processes has shown that, to date, mathematical models constructed by an experimental-analytical method [40-55] are the most
widely used.
The analysis of works in the field of mathematical modeling of cyclic adsorption separation of gas mixtures made it possible to establish that, in general, the mathematical model includes a system of
equations of general material balance; component-wise material balance in the gas (taking into account diffusion, convection in the gas phase, as well as the internal source/drain of the substance as
a result of adsorption
or desorption) and solid phases (taking into account diffusion, as well as the internal source/drain of the substance as a result of adsorption or desorption); thermal balance in the gas phase
(taking into account thermal conductivity, convection, as well as thermal effect as a result of adsorption or desorption) and the adsorbent (taking into account thermal conductivity, as well as
thermal effect as a result of adsorption or desorption); conservation of momentum (a variation of the Navier - Stokes equation); adsorption kinetics -desorption (taking into account the rate of mass
transfer from the gas to the solid phase and back during adsorption - desorption); equilibrium in the gas-adsorbent system (adsorption isotherms of the components) [56, 57]; other relationships
between model variables, initial and boundary conditions.
The equations of component-wise material balance are written in the form of a system of partial differential equations of a parabolic type [58]:
d (Vg Ck) d ck 6 -+—- +
(1 -e \da
d x
d t
= D
d 2ck
x,k dx2
where vg is gas flow rate (m/s); ck is molar
concentration of the k component of the gas mixture (mol/m ); e is porosity of the adsorbent layer, taking
into account the porosity of the particles (m /m ); ak is amount of sorption (adsorbate concentration in
the adsorbent) (mol/m ); Dx,k is effective coefficient
of longitudinal mixing of the k component of the gas mixture (m /s); x is spatial coordinate of the adsorbent layer (m); t is time (s).
In equation (1), the first term describes the convective transfer of the substance in the adsorbent layer; the second term is the accumulation rate of the component in the mixture in the gas phase;
the third and fourth terms are the sorption rate and the longitudinal mixing of the k component in the adsorbent layer, respectively.
The effective coefficient of longitudinal mixing Dx in early works on the adsorption separation of gas mixtures was identified with the molecular diffusion coefficient. At present, two main
components are distinguished in longitudinal mixing (diffusion): molecular diffusion and turbulent mixing, which arises as a result of recombination of flows around the particles of the adsorbent. In
practical calculations, the formula [1] is most often used to estimate the coefficient of longitudinal mixing:
Dx = 0.7Dm + 0.5dgrvg ,
where Dm is molecular diffusion coefficient; dgr is particle diameter of the adsorbent; vg is gas velocity.
Equation (1) for an unambiguous solution should be supplemented with initial and boundary conditions:
- initial conditions
ck(x,0) = c0(x); k = 1,nk ; 0 < x < L ;
- boundary conditions at the stage of adsorption
ck (0, t ) = cf(t ); ^^ = 0, k = Vn k ; dx
- boundary conditions at the stage of desorption
ck ( L, t ) = Ckout(t ); ^^ = 0, k = Vn k.
To describe the sorption kinetics in the external diffusion region of the process, the equation [57] is used:
a dt
= Pk (ck - ck ),
where ck is the equilibrium molar concentration of the
k component of the gas mixture (mol/m ); pk is the mass transfer coefficient related to the concentration of the adsorptive in the gas phase (1/s).
For the internal diffusion adsorption process, the driving force is written as the difference between the values of the equilibrium sorption and the current sorption in the adsorbent (the Glukaf
formula) [57]:
dk = P 2(ak- ak), dt
where ak is the equilibrium value of sorption of the k
component of the gas mixture (mol/m ); pk is the internal diffusion kinetic coefficient in the adsorbent granules (1/s).
To describe the kinetics of adsorption in the mixed-diffusion region, the mass transfer equation for the adsorptive from the gas phase to the solid phase of the adsorbent (through the phase boundary)
is applied in the following form [59, 60]:
dak — F -F-(tgh(e(Vg -v*))+1)+Fk1,
k = 1,2,3.
where Fk - the right part of the kinetics equation for nonstationary convective (external) mass transfer,
11 * 1 Fk =Pk (ck - ck); Pk is the mass transfer coefficient
related to the concentration of the adsorptive in the gas
phase; c* is the concentration of the adsorptive at the
interface or the equilibrium current value of adsorption
ak ; Fk is the right part of the kinetics equation the
internal diffusion adsorption process,
Fk =P2h(a*k - ak); Pit is the kinetic coefficient; a* is the amount of adsorption equilibrium to the current concentration of the adsorptive c k in the gas mixture flow on the outer surface of the
granules; e is the formal coefficient setting the dimensions of the mixed-diffusion region; vg* is the velocity of the gas mixture
which determines the transition from the diffusion region to the kinetic region of the adsorptive transfer; with initial conditions
ak (x,0) - a°(x), 0 < x < L , k — 1, nk .
Equation (2) is a description of the adsorption kinetics for the mixed-diffusion transfer region of the adsorptive across the phase boundary: when the velocity of the gas mixture is below the
velocity vg* , the adsorption process is limited by the
external mass transfer process with the coefficient Pk, otherwise - by the internal diffusion process in the granules of the adsorbent with the kinetic coefficient
Pk. The hyperbolic tangent and the formal coefficient
e along with v* describe a continuous transition from
the external mass transfer region to the internal diffusion adsorption process with zeolite adsorbents CaA, 13X.
The isotherms described by the equations of Dubinin-Radushkevich and Langmuir-Freindlich [56, 61] are most often used as equations of sorption isotherms in multicomponent gas mixtures.
To describe the processes of heat propagation in the gas mixture flow and the adsorbent along its length, partial equations of parabolic type are most often used [60, 62]:
g dTg ( x, t) g dTg( x, t )
cp p+ p pg vg-
--^ud[Ta(x,t) -Tg(x,t)] -^^[v -Tg(x,t)] 8 8 d a
= x
d% g ar2
0 < x < L ,
c pp
pv a
^^ + a ^d[[(x,t) -Tg(x,t)]-
2 ads
dak(x,t) d Ta(x,t)
-Z hk
where cg, pg is the specific heat and molar density of the gas mixture, J/(mol-K), mol/m3, respectively; Tg is the temperature of the gas mixture, K; A,g is the
coefficient of thermal conductivity of the gas mixture, W/(m-K); a is the heat transfer coefficient from the surface of the adsorbent granules to the gas mixture
flow, W/(K-m2); Sud = (1 -e)—is the specific surface
r gr
coefficient of the adsorbent granules, m2/m3; Kenv is heat transfer coefficient from the gas mixture flow to the environment, W/(K-m2); dA is the adsorber diameter, m; Tenv is the environment
temperature, K, where c P is specific heat capacity of the adsorbent,
,3. ,„ads
J/(kg K); pa is the adsorbent density, kg/m ; hk the adsorption heat of the k component of the gas mixture, J/mol; Xa is the coefficient of the adsorbent
thermal conductivity, W/(m-K); with initial and boundary conditions at the adsorption and desorption stages similar to the conditions written above for equation (1).
In equation (3), the first term describes the accumulation of heat in the gas phase; the second term - the convective component of heat transfer; the third term - the heat transfer from the gas phase
to the solid phase (the adsorbent); the fourth term - the heat transfer from the gas phase to the environment through the wall of the adsorber; the fifth term - the longitudinal thermal conductivity
of the gas phase along the height of the adsorbent layer. In equation (4), the first term describes the enthalpy of the solid phase (the adsorbent); the second term - the heat transfer from the solid
phase (the adsorbent) to the gas phase; the third term - the release of the heat of the gas mixture components sorption; the fourth term - the thermal conductivity in the adsorbent along the vertical
axis of the adsorber.
The dynamics of changes in pressure and velocity of the gas mixture in the adsorbent layer is most often described by the Ergun equation [63]:
dP dx
150 (1 -so )2 (2rgrV)s0
^gvg + 1.75M gpg
(i -
2rgr Vs 0
where s o is the porosity of the adsorbent layer without taking into account the porosity of the particles, m3/m3; y is the sphericity coefficient of the adsorbent
granules; is the dynamic viscosity of the gas mixture, Pa s; Mg is the molar mass of the gas mixture, kg/mol; pg is the gas mixture density (mol/m3); r is the adsorbent granule radius, m.
The ideal gas state equation has the following
P(x, t) = R Tg (x, t)Z Ck (x, t),
R is the universal gas constant, J/(mol-K).
For the numerical solution of the system of nonlinear partial differential equations (1) - (5) with the corresponding initial and boundary conditions, the method of straight lines was used according
to which the derivatives of the spatial variable x were approximated by finite difference formulas. In this case, the time derivative remains in continuous form. It results in the system of ordinary
differential equations along a given family of straight lines with initial and boundary conditions, which can be solved by some numerical method, for example, the fourth-order Runge-Kutta with
automatic step selection. The method of straight lines has quite acceptable accuracy and speed of convergence for practice.
The experimental study of air oxygen enrichment process
The flow chart of the experimental dual-adsorber PSA unit for oxygen concentration is shown in Fig. 3: A1, A2 is adsorbers with granulated zeolite adsorbent 13X; K2, K3, K4, K8 is control valves; K5,
K7 is check valves; K6 is pass-through valve; R is receiver of oxygen enriched production air. Further, the concentration of oxygen and nitrogen will be denoted
by y = ^2,) vol. %.
The PSA unit while enriching air with oxygen operates as follows. The flow of atmospheric air is formed by compressor C and inlet valves K1, K3 with
flow rate Gin, initial composition yk", where
k = {1 - 02, 2 - N2}, pressure P^ and temperature
Tgin . At the initial moment of time, the valves K1, K4,
K6, K8 are open. The air flow through the valve K1 enters the adsorber A1, in which the pressure rises to
the value Pj^. for a certain length of time [0, tads ] and
Fig. 3. Dual-adsorber PSA unit
the adsorption process of predominantly nitrogen and to a lesser extent oxygen and argon takes place over
the time tads (impurities are not adsorbed). Oxygen
enriched air in the adsorber A1 enters the receiver R through the valve K5 and then it is removed to the consumer through the valve K8 with the flow rate
Gout, composition yout
/^»out,1 out,1
G , composition y
^ out ,1
temperature Tgout. At the same time, part of oxygen
enriched air flow in the adsorber A1, through the valve K6, enters the adsorber A2, where desorption of nitrogen, oxygen and argon takes place under pressure
Pdœ . The air flow saturated with nitrogen (waste) enters the outlet of the adsorption concentrator from the desorber A2 through the valve K4 with the flow rate
pressure Pou1,1 and
temperature T o
At the moment of time t = tc /2 = tads, where tc is the duration of the "adsorption-desorption" cycle, the valves are switched: valves K1, K4, K5 close, valves K2, K3 open and valves K6, K8 still
remain open. The atmospheric air through the valve K3 is fed by the compressor C to the adsorber A2, in which pressure is raised (pressure up-pu) for a certain length of time
[0, tads] to the value Pad and the adsorption process of mainly nitrogen and to a lesser extent oxygen and argon (impurities are not adsorbed) during the segment of time tc / 2 < t < tc . Oxygen
enriched air in the
adsorber A2 enters the receiver through the valve K7 and then it is removed to the consumer through the
valve K8 with the flow rate G
composition y At the same
pressure Pout and temperature Tg
time, part of oxygen enriched air flow in the adsorber A1 enters the adsorber A1 through valve K6, where pressure is first released (pressure down-pd), and then nitrogen, oxygen and argon are
desorbed under
pressure Pdiens . The air flow saturated with nitrogen (waste) enters the outlet of the adsorption concentrator from the desorber A1 through the valve K2 with flow
rate GouU, composition youU, pressure PouU and temperature 7gou1,1.
Upon expiration of time tc, one complete cycle of the adsorption concentration process is completed, after which the cycles are repeated during the entire operation period [0, tf] of the PSA unit.
The implementation of cyclic operation modes of the PSA unit is carried out by an automated control system using a software setpoint control device and control valves K1, K2, K3, K4, K8 in accordance
with the periodic switching cyclogram.
A schematic diagram of an automated experimental unit for the oxygen adsorption concentration, which implements the described PSA scheme (Fig. 3) and the cyclogram (Fig. 4) is shown in Fig. 5.
Ki K; Ki K, K< Ki K: Ki
; /6 \ / / N / \ r S "s ^J \ ,'A, V t \ ' J \ t
pu ds ¿pd ^des
tjl tc/2
Fig. 4. Dynamics of pressure changes in adsorbers and cyclogram of control valves switching
Fig. 5. Schematic diagram of automated experimental unit of oxygen adsorption concentration
The atmospheric air under pressure of (2.0 - 6.0)-105 Pa is fed to the inlet of the oxygen concentrator (point "a") through the filtering unit 2, which traps water and oils. The pressure regulator 3
maintains the pressure (0.5 - 0.8)-105 Pa on the pneumorel 9, 12, 13, 14, 15, 16, 17, which are set at points "c" of the scheme. Under the pressure, the membrane blocks of elements 9, 12, 13, 14, 15,
16, 17 move upwards. For the sensor 4, the pressure gauge 5
sets the adsorption pressure Pallid (15 - 2.4)-105 Pa in the receiver 6. The oxygen concentrator is started by switching the pneumotumbler 8. The generator of rectangular pneumatic impulses (includes
pneumorel 9, variable pneumatic resistance 10 and pneumatic
capacitance 11) periodically sets pressure P^ or Pdns at the point "d". At the initial time of the generator operation, the membrane blocks of the elements 12, 13, 14, 15, 16, 17 move downwards.
Through the upper chambers of the elements 15, 16, 17, the atmospheric air begins to flow into the adsorber 18. In the adsorber 18, the pressure increases to a certain
value Paidns and the adsorption process of predominantly nitrogen and to a lesser extent oxygen and argon is carried out (impurities are not adsorbed). At the outlet of the adsorber 18, oxygen
enriched air is formed. Part of the air flow enters the receiver 23
through the check valve 21, and the other part is fed into the adsorber 20 through variable pneumatic
resistance under pressure Pd^. The process of nitrogen desorption is carried out in the adsorber 20, and at its exit a gas mixture is formed with a high concentration of nitrogen, which through the
upper chambers of the elements 12, 13, 14 enters the atmosphere. After a
half-cycle time tc / 2 , the generator sets pressure Pifs, and the membrane blocks of the elements 12, 13, 14, 15, 16, 17 move upwards. While the adsorber 20 goes into adsorption mode, and the
adsorber 18 in desorption mode. The continuity of the process is achieved by cyclically switching the adsorbers at regular intervals. The flow rate of the production gas mixture is controlled by a
variable resistance 25 on the rotameter 26. The main tuning parameters of the concentrator are the half-cycle time determined by the resistance 10 and the return flow value determined by the
resistance of the element 19. The oxygen concentration is measured by the gas analyzer 25.
The results of experimental studies of the oxygen
concentration y1out in the production gas-air mixture (at the outlet of the PSA unit in the steady state) depending on the half cycle time tc / 2 for different
values of pressure Pads presented in Fig. 6.
at the adsorption stage are
y. ,vol.%
\ , vol.% 60
82 tJ2, S
A A
□ □ A □ A
[ A o ' D.......... A
Fig. 6. Results of experimental studies, Pa:
Pdns = 2-2 -105; b - A - Pds
82 LI2, s
a - A - -Cr 5-2T05, □ - Pd = 2.2 -105; b - A - Pd = 3.7 -105, □ - Pd = 2.7 -105
Modeling and algorithmization of the dynamics of the PSA unit operation
The PSA unit as a system, in which the process of air separation and oxygen concentration is carried out, can be represented as a set of interacting subsystems: environment, flow booster (compressor,
blower, etc.), "adsorber-desorber", receiver, subsystem valves, and control system.
When developing a mathematical model of the technological process, we will adhere to the principle of an autonomous mathematical description of the processes carried out in each subsystem and the
matching of the subsystem models among themselves into a single mathematical system model.
Here we also give the equations of the mathematical model of the central system-forming element of the PSA unit - the "adsorber - desorber" subsystem.
During the adsorption of O2, N2, granulated zeolitic adsorbent 13X in the adsorbers Ai, A2 of the PSA unit, the following mass and heat exchange processes take place:
a) the mass transfer of O2, N2, and heat exchange between the air mass and the adsorbent;
b) the distribution of air components in the gas phase due to the convection;
c) the distribution of heat in the air flow and the adsorbent due to the convection and thermal conductivity;
d) the adsorption of O2, N2 on the surface and in the micropores of the zeolite adsorbent granules with the heat release, leaching of O2 from the adsorbent at the adsorption stage and desorption of
N2 from micropores and from the surface of the granules with the heat absorption.
The mathematical description of the processes in the adsorber include the following assumptions:
i) the atmospheric air is predominantly a two-component air mixture and is considered as an ideal
gas, which is quite acceptable at pressures in the adsorber up to 200-105 Pa;
2) the granular zeolite NaX of spherical shape with a diameter of 2 mm is used as an adsorbent;
3) longitudinal mixing of O2, N2 components in the air flow in the axial direction and thermal losses to the environment are absent.
The mathematical description of the "adsorber-desorber" subsystem in the PSA unit includes the following equations:
- total material balance in the adsorber
d (vg Pg)
d x
+ Pa
d a
1 + d a2 ^
d t d t
+ -
ÔPg d t
= 0,
where vg is linear velocity of the gas mixture, m/s; pg
is molar density of the gas mixture, mol/m ; pa is bulk
density of the adsorbent, kg/m ; a1, a2 is adsorbate concentrations (oxygen and nitrogen), respectively, mol/kg;
- component-based material balance
d (vg ck ) d ck d ak n
• + 8-- + pa-- = 0,
d t d t
d x
k = {1 - O2,2 -N2}, thermal balance for the gas phase
d Tg d Tg
vgc„gp ——
g pg g dx
d x
■ 8cpgpg■
d t
+ P^ + Kt SUd(?g -Ta) = 0:
where cpg is specific heat of the gas mixture, J/(mol-K); s is adsorption layer porosity; Sud is coefficient of the specific sUrface of the adsorbent particles, m2/m3; Ta is temperature of the
K; KT is heat transfer coefficient, W/(m2-K);
- heat balance for the adsorbent
2 g T 2 Г
Pa(cpa +YjCpkai^^ + PaZ| Ш1
- KT Sud(Tg -Ta)
dak dt
= 0,
where cpa,cpk is specific heat capacities of the
adsorbent and adsorbate, respectively, k = 1, 2, J/(mol-K); AHk is thermal effect of the k air component sorption тепловой, J/mol; A,a is thermal conductivity coefficient of the adsorbent, W/(m-K); -
adsorption kinetics
dak dt
_ Pk (a* - ak ), k = {1 - О 2,2 - N2}, (10)
- equilibrium conditions calculated by the Langmuir-Freindlich adsorption isotherm formal for zeolites [64]:
* b1,kck exp(b2,k /Ta)
a* =—2-,-+
1 +Êb3,}c] exp(b4,k /Ta) j=1
b5,kck exp(b6,k / Ta)
+ -
, k = 1,2 , (11)
1 + Z 63,jCj exP(64,k / Ta) j=1
- momentum conservation
*(Pg Tg) _ , (1 -e)2
-_- Л
^(dgr9)2 e3 RgVg
- B(C1 + C2)M,
1 - e
g ^dgr9e3 g
where A, B are known constants.
Let us formulate the problem of identifying kinetic parameters Pk , k — {1 -02,2-N2} by the
output experimental signals y^ (Pi,p2),
j — 1, m, i — 1, d, where m - the number of output measured coordinates of the control object; d - the number of experimental points for a separate output coordinate of the control object depending
on the half-cycle of adsorption tads — tc / 2 .
Then a non-negative function is constructed
m d r -i2
F (P1, P2) — Hk j - yj (Pidns, tads, i, P1, P2)] ,
j—1 i—1
where yj (P^, tads,,-,P1,P2) is solution of the mathematical model equations (6) - (12) (with the
corresponding initial and boundary conditions) of the process of air oxygen enrichment in the PSA unit for
fixed values Pans, tads,i and kinetic parameters P1, P2 .
In a finite-dimensional numerical Euclidean space
(9) Emd , the value of F is equal to the square of the
distance between the vectors
y(Cs,tads,P1,P2) •
Let us rewrite this function in another form:
F (P1, P 2) _ y e - y ( Pads, tads, P1, P 2)
The function F is defined on the set V c E , where l < md; l is the given natural number.
The task of determining parameters P1, P2 is to
find P* eV c El such that
F (p*)— min F (P)
in case of constraints in the form of the mathematical model equations (6) - (12) of the air oxygen enrichment process in the PSA unit.
Despite the presence of constraint equations (6) -(12), we obtained the problem of the unconditional minimum of the function F(P), since P1, P2 are included in F (P) through the solution y (P1, P2),
which takes into account all the mathematical properties of constraint equations (6) - (12).
Based on the physical meaning of the problem, it
would be necessary to find P so that the solution y(Pails, tads, P*, P2) was as close as possible to the true value of the vector y of the control object state variables measured at the output of the
PSA unit in magnitude
F (P1, P2) — | |y - y (P2, tads, P1, P*2)|| Ei-
However, the vector y is unknown, therefore we have to work with the "perturbed" function F (p).
The regularization of a problem is the process of transforming it into a correctly set one. Regularization of the extremal problem formulated above consists in transforming a convex function F (p)
into a strictly or uniformly convex one, which ensures the uniqueness of the solution P . Let us construct a continuous nonnegative parametric function
0(p ) — F (p ) + aQ(P),
where the parameter a > 0; Q is non-negative continuous function such that ®(p) is an uniformly convex function. An uniformly convex function Q(Pi, P2 ) = Pi + P2 can be taken as Q.
If F (P) is convex, and Q is a uniformly convex
function, then for any a > 0 function 0(P) will be
uniformly convex and the problem of finding Pa e V such that
o(pa)=mpn 0(Pi, P2) (13)
is set correctly and its solution pa is unique for each
fixed a. For determining pa , high-speed quasiNewton methods can be used [65].
As a result of solving the regularized problem
* *
(13), the values of kinetic parameters P1 <i,P2 <a of the
adsorption oxygen concentration process were
determined: P*a = 5.776 s-1, P2a = 1.925 s-1.
The adequacy of the mathematical model was tested on a set of experimental data obtained under conditions different from those under which they were obtained, according to which parametric
was carried out. The function y°ut 0^, tadv, P1, P2)
(solving the mathematical model equations (6) - (12)
for given values P^ = 2.7-105; 3.7-105 Pa, fads (from
7 to 82 s), kinetic parameters P1 = P1P2 = P2 a and
values of ordinates y^f'6, I = 1, 2, ..., 10 are shown in Fig. 7.
The mismatch of calculated by the model (6) -(12) and experimental data (Fig. 7) was estimated by the following formula:
•V, ,vo\.% 60
n_8 ' /---—-__ □
IT 2 I
i! 1 1 ■ ' 1 ■ 1 1 1 -'— .111 . ..
82 L/2s
Fig. 7. Verification of mathematical model adequacy of the process of oxygen adsorption concentration at the inlet pressure: 1 - 2.7-105 Pa; 2 - 3.7-105 Pa; A, □ - experiment, — calculation by model
„ out/nin , n o \ out,e
max y (.Pads, Íad8>í,ß! a, ß2 a)- yu-
i=1,d' ' 1
The mathematical model (6) - (12) with found
* -1 * -1 values P1ai = 5.776 s , P2a = 1.925 s was
considered adequate to the technological process of the adsorption oxygen concentration, if
Smax = max
yOUt(PaÍdns,tads,i,ßl,a,ß2,a) - y^
where 5 is the measurement error of oxygen concentration y1out at the production outlet of the PSA unit, which is 15 %. The verification of the adequacy of the mathematical model of adsorption oxygen
concentration in the PSA unit showed that the maximum relative error of the mathematical model of the adsorption oxygen concentration process 5max
was 13.2 %, which allows using this model with found
values P1 =P1ai, P2 =P2a for the purposes of
analyzing the oxygen adsorption concentration process, optimizing and controlling this process.
The numerical analysis of the dynamics of the dual-adsorber PSA unit operation
In order to examine the systemic links, patterns, and increase the efficiency of the PSA unit, calculation experiments were conducted to study the dynamics and "statics" during the adsorption oxygen
concentration in the air-gas mixture for a dual-adsorber technological scheme with 13X granular zeolite adsorbent (see Fig. 3). The main parameters of the pilot PSA dual-adsorber unit are presented
in Table 1.
In Table 1: dA is the inner diameter of the adsorber shell (the adsorbent bulk layer); L is the height of the adsorbent bulk layer; dK1 = dK2 is the
bore section of cut-off valves; VR is the receiver volume.
Table 1
The characteristics of the pilot PSA dual-absorber unit
Parameter Value Parameter Value
dA, m 0.050 d K1 = d K 2, m 0.0014
L, m 0.500 Vr, m3 0.002
dgr, m 0.002 ßi; ß2, s-1 5.776; 1.925
Table 2
Values of the process parameters at the nominal point and ranges of their variation
Parameter The value in the working (nominal) point Range of variation
?ads = ?c/2, S 40 10 - 90
Pm x105, Pa 3 2.0 - 5.2
DOUt 5 P x105, Pa 1 0.9 - 1.1
P °uu X105, Pa 0.75 0.25 - 1.0
y|n % vol. 20.8 20.3 - 21.3
y3n% vol. 1.0 0.5 - 1.5
Tm,T ,K g T oc 298 273 - 323
dK6, mm 0.5 0.31 - 0.80
Variables and ranges of their changes are presented in Table 2.
A series of calculation experiments was conducted to study the effect of the half-cycle time tc / 2 (the duration of the adsorption stage tads), the
pressure Pin and temperature of the gas-air mixture Tgin at the compressor outlet, the diameter of the K6
throttle dK6 on the concentration >'1out and the degree
of oxygen extraction n.
Fig. 8 shows graphs of the oxygen concentration
dependencies >'1out in the production flow from the half-cycle duration tads = tc / 2 at various pressures
P in and, therefore, pressure Paidns at the adsorption stage.
The increase in P in leads to the increase in the oxygen concentration >'1out in the production flow and, accordingly, its sensitivity to changes in the half-cycle time tads = tc / 2 . All graphs
have a pronounced
extreme character, so it is possible to choose the
optimal half-cycle time value tads, which provides the maximum concentration >'1out at various input pressure
values Pin . In this case, the range of values tads ,
including the optimal value (in case of a given optimality criterion), it is advisable to limit the interval [27 - 67] s. The analysis of the graphs in Fig. 9 shows that the time of the transient
process (the transition of the unit to a periodic stationary mode) corresponds on average to 20-40 "adsorption - desorption" cycles,
i.e. tst ~ (20-40)-tc.
The analysis of the graphs in Fig. 10 shows that at
Tgin= 273 and 298 K the curves monotonously increase
over the entire range of pressure Pin variation. At the same time, up to the value of Pin « 3.7-105 Pa the sensitivity of the concentration >'1)ut to Pin is noticeably higher than in the next
section. At Tgin = 323 K, the graph acquires an extreme character, and the maximum is reached at P in= 3.9-105 Pa and amounts to ~ 41 % vol. The further increase in Tgin leads to the
decrease in the oxygen content y1out in the production air flow. This is because the increase in Tgin leads to
the heating of the adsorption layer and the decrease in the equilibrium adsorptive concentration in the adsorbent. The greatest deviation of the curves (corresponding to 273 and 323 K) is observed at
end segment of the variation in Pm and is ~ 1.2 % vol.
»» ^
y, , vol.%
Fig. 8. Dependencies y1 on /ads = tc / 2 at P , Pa:
1 - 2.2-105; 2 - 2.7-105; 3 - 3.7-105; 4 - 5.2-105
Fig. 9. Dependencies y on the time
of the unit operation t for iads, s:
1 - 10; 2 - 40; 3 - 65
>; ,voi.% 45 -
r|. %
i 1 ! i .................1.................
| --- ............i..................
> ■—-- 2
/j/ | I
/ / / : T 1 : i ; ;
^ V-'.......
¿A ................
3.2 3,7 4.2
4.7 /"n,X105,Pa
Fig. 10. Dependencies y1out on Pln at temperature of initial mixture Tgin, K: 1 - 273; 2 - 298; 5 - 323
This phenomenon is explained by the increase in the adsorbent temperature, which leads, on the one hand, to the decrease in the equilibrium adsorptive concentration in the adsorbent, and, on the
other hand, increases the desorption rate at the stage of the adsorbent regeneration.
The analysis of the graphs in Fig. 11 demonstrates that the oxygen extraction degree n is affected both
Pin and Tgin. At Tgin = 273 K the dependence has an
extreme nature and the extremum is reached at
Pin = 4-105 Pa, and at Tgin = 298 K and Tgin = 323 K
the graphs monotonously increase proportionally Pin . At Tgin = 323 K a sharp decrease in the sensitivity of
the extraction degree n to the change in Pin is observed starting from Pin = 3.9-105 Pa, and at Tgin = 298 K the dependence n on P in becomes almost linear.
The dependencies of oxygen concentration y1out on the half-cycle time tads for various values Tgin are presented in Fig. 12. The graphs are of extreme nature at temperatures Tgin = 298 and 323 K. The
yout is reached at tads «40 s. At Tgin = 273 K the
graph takes the form of a saturation curve, the "plateau" of the graph corresponds to the value
yout= 42,7 % vol.
It should be noted that the more tads , the greater deviations are observed between the curves. The maximum mismatch of the curves corresponding
to Tgin = 273 and 323 K is reached at tads = 60 s and
Fig. 11. Dependencies of oxygen extraction degree n on Pln at 7gin, K: 1 - 273; 2 - 298; 3 -323
amounts ~ 5 % vol. These changes in the graphs can be explained as follows. At small values of tads < 30 s, the heat exchange between the adsorbent and the gas phase is not sufficiently intensive
(due to the inertia of the heating or cooling process of the adsorbent),
therefore the effect of Tgin is insignificant, and at tads - 30 the effect of the inertia of the thermal processes is reduced.
The graphs analysis presented in Fig. 13 shows that the extraction degree is proportional tads at all
Tgin. At tads > 40 s, the maximum n is reached at the
lowest Tgn (273 K). The type of the considered curves as a whole correlates with the graphs of oxygen concentrations y1out in the production flow from the half cycle time yf* at Tgin = 273; 298; 323
Fig. 12. Dependencies y™ on tads at Tgin , K: 1 - 273; 2 - 298; 3 - 323
— AM&T
34 ■
Fig. 13. Dependencies n on tads at Tgn, K:
1 - 273; 2 - 298; 3 - 323
.......1......\......i............-......i......i...... ::::::: i i i i i i i iii ......1......!......1.....-b" : Jr i Jr\ : i i S iii: i i i 1 iii!
i i i i i i i ¡¡¡iii: jf i i : : : i i Jf
i --i—-¿¿L \ \ \ i iii! i i i i ill!
15 20 25 30 35 40 45 50 55 60 6 5 70 75 taAsS
Fig. 15. Dependencies of n from tads at dKe , mm:
1 - 0.31; 2 - 0.62; 3 - 0.8
Dependencies of oxygen concentration >iout in the production air flow from tads for different values of the nominal diameter dK(6 of the purge valve K6 are
shown in Fig. 14. The graphs analysis indicates that with the increase of dK6 in the duration of the
adsorption stage tads ensuring the achievement of the maximum oxygen concentration at the unit outlet should decrease from 62 s (at dK6 = 0.345 mm) to 27 s
(at dK6 = 0.715 mm). Fig. 15 presents dependency
graphs of n from tads for various values of dK6 .
At dK6 = 0.31, the dependency is directly proportional
and close to linear, since the flow directed to the desorption is too small and it takes a long time to regenerate the adsorbent; at dK6 = 0.8, the dependency
is also close to linear, but inversely proportional,
>T! voi.%
...........|........... ...........
> \i
V 5
: : ___________I___________!___________ 4
Fig. 14. Dependencies of _y1out from tads at dK6 , mm:
1 - 0,345; 2 - 0.5; 3 - 0.62; 4 - 0.715
which is explained by the excess flow directed to the nitrogen desorption, as a result of which nitrogen is desorbed before the desorption stage ends; at dK6 = 0.62 mm, the dependency acquires an
character, since the intermediate state is reached.
The graphs analysis presented in Fig. 16 shows
that the maximum value Pin corresponds to the maximum peak of the air velocity vg = 0.18 m/s, since
the velocity rate is proportional to the pressure difference P in at the compressor outlet and the pressure P^ at the inlet to the adsorber. When these pressures are equalized, the air velocity
gradually decreases to some steady-state value, approximately
equal to 0.03 m/s for all considered values Pin . Similar dependences were obtained at the desorption stage (Fig. 16, t > 1270 s).
v„ ,m/s
0.18 -
0.08 -
!/\2Yv II r\, |
; 1
A-icls, S
Fig. 16. Dependencies of vg (x = 0) in adsorber from t at P*n, Pa:
1 - 2-105; 2 - 3-105; 3 - 4-105
Fig. 17. Dependencies of Vg in adsorber from t at dK6 , mm:
1 - 0.345; 2 - 0.5; 3 - 0.715
The graphs analysis in Fig. 17 specifies that varying the diameter of the throttle dK6 in the
investigated range does not affect the appearance of the curves obtained. Reducing the diameter of the purge throttle leads to the slight increase in the gas velocity at the beginning of the
adsorption stage and the decrease at its end. These changes in the graphs are explained by the fact that the increase in the throttle diameter leads to the increase in the flow rate of the purge
mixture for the regeneration of the second adsorber, which is reflected in the value of the difference in the air pressure directly proportional to its speed.
The velocity value of the flow entering the adsorber plays an important role, since it determines the adsorbent abrasion under alternating loads (in the cycles of lifting and pressure relief in the
adsorbers). The interaction of the moving gas stream with the adsorbent layer leads to the effect of limited "fluidization" of the layer when the granules of the adsorbent begin to shift relative to
each other, which leads to abrasion of the adsorbent and the appearance of a significant amount of dust in the product stream.
Even at the velocity of the filtered air flow at a much lower rate of the beginning of fluidization, the abrasion of the adsorbent granules can be quite strong [26]. This is due to the impact on the
granules of changing "side" forces, called the forces of Karman, causing oscillating displacement of the granules relative to each other. Both destructive effects are likely to occur when the stages
change, when large pressure gradients occur. Therefore, it is extremely important to control the gas flow rates in the frontal layer of the adsorbent during the transition periods of adsorption
oxygen concentration process.
Fig. 18. Dynamics of opening degree of inlet 1 and discharge 2 valves of the PSA unit
Fig. 19. Dependencies of Vg (x = 0)
in the adsorber from the time at Pin , Pa:
1 - 3105; 2 - 4-105; 3 - 5105
The analysis of operating experience of PSA units shows that the gas flow rate in the adsorber (depending on the size of the adsorber, the diameter of the adsorbent particles in it, the values of
adsorption and desorption pressures) should not exceed 0.05-0.3 m/s.
Fig. 18 displays the graphs of the step change in the opening degree of the inlet and discharge valves from the time with the control frequency of 4 s. As a result, the air velocity in the front
layer of the adsorbent is not higher than 0.08 m/s (Fig. 19). Thus, controlling the opening degree of the valves is an effective means of ensuring the absence of abrasion of the expensive adsorbent.
The calculation experiments conducted with the developed mathematical model established that it is
advisable to use pressure P in at the compressor outlet, a temporary program ) for opening control valves
Ki (K2), the duration of the adsorption stage tads (as half cycle time) and the diameter of the purge throttle dK6 as control actions allowing to effectively control
the modes implemented in the PSA unit. The range of values tads, including the optimal value (maximum
concentration value y°ut ), is advisable to limit the interval to 27-67 s, and the time for the unit to reach a periodic stationary mode on average corresponds to 20-40 adsorption-desorption cycles.
It has been established that by finding the law of opening the inlet and discharge valves of the PSA unit, it is possible to ensure the air flow rate that does not lead to abrasion of the adsorbent
during the implementation of cyclic adsorption-desorption processes. At the same time, the influence of the gas flow rate limitation on the purity of the production oxygen, the extraction degree and
the capacity of the PSA unit requires further research.
The results of numerical analysis, mathematical and algorithmic support for the operation of the PSA dual-adsorption unit, presented in this paper, can be used to design new automated processes and
adsorption process units with cyclically varying pressure to separate and purify multicomponent gas mixtures.
The research was financially supported by the Russian Ministry of Education and Science within the framework of project No. 10.3533.2017.
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65. Gladkih B.A. Metody Optimizatsii i Issledovanie Operatsij dlya Bakalavrov Informatiki: Nelinejnoe i Dinamicheskoe Programmirovanie [Optimization methods and operation research for bachelors of
computer science: Nonlinear and dynamic programming]. Tomsk, 2009, 263 p. (Rus.) | {"url":"https://cyberleninka.ru/article/n/the-study-of-cyclic-adsorption-air-separation-and-oxygen-concentration-processes","timestamp":"2024-11-07T15:15:23Z","content_type":"application/xhtml+xml","content_length":"147970","record_id":"<urn:uuid:fd61c6ba-b3e1-446c-b1bd-38d4b22059cc>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00315.warc.gz"} |
Index.squeeze(*args, **kwargs)[source]¶
Remove size one axes from the data array.
By default all size one axes are removed, but particular size one axes may be selected for removal.
New in version (cfdm): 1.7.0
axes: (sequence of) int, optional
The positions of the size one axes to be removed. By default all size one axes are removed.
Each axis is identified by its integer position in the data. Negative integers counting from the last position are allowed.
Parameter example:
Parameter example:
Parameter example:
axes=[1, -2]
inplace: bool, optional
If True then do the operation in-place and return None.
Index or None
A new instance with removed size 1 one data axes. If the operation was in-place then None is returned.
>>> f = cfdm.Index()
>>> d = cfdm.Data(numpy.arange(7008).reshape((1, 73, 1, 96)))
>>> f.set_data(d)
>>> f.shape
(1, 73, 1, 96)
>>> f.squeeze().shape
(73, 96)
>>> f.squeeze(0).shape
(73, 1, 96)
>>> f.squeeze([-3, 2]).shape
(73, 96) | {"url":"https://ncas-cms.github.io/cf-python/method/cf.Index.squeeze.html","timestamp":"2024-11-11T05:16:31Z","content_type":"application/xhtml+xml","content_length":"13602","record_id":"<urn:uuid:c50d86ee-e540-4e68-aae5-b4ba163719f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00046.warc.gz"} |
Simple Template Currying
Currying is the technique of transforming a function that takes multiple arguments in such a way that it can be called as a chain of functions, each with a single argument. I've discussed Currying on
this blog previously in
Fun With Lambdas C++14 Style
Dependently-Typed Curried printf
. Both blogposts discuss currying of functions proper. I.e., they discuss how C++ can treat functions as values at runtime.
However, currying is not limited to just functions. Types can also be curried---if they take type arguments. In C++, we call them templates. Templates are "functions" at type level. For example,
passing two type arguments
std::map<std::string, int>
. So
is a type-level function that takes two (type) arguments and gives another type as a result. They are also known as type constructors.
So, the question today is: Can C++ templates be curried? As it turns out, they can be. Rather easily. So, here we go...
#include <type_traits>
#include <functional>
#include <map>
#include <iostream>
template <template <class...> class C, class... T, class D = C<T...>>
constexpr std::true_type valid(std::nullptr_t);
template <template <class...> class C, class... T>
constexpr std::false_type valid(...);
template <class TrueFalse, template <class...> class C, class... ArgsSoFar>
struct curry_impl;
template <template <class...> class C, class... ArgsSoFar>
struct curry_impl<std::true_type, C, ArgsSoFar...> {
using type = C<ArgsSoFar...>;
template <template <class...> class C, class... ArgsSoFar>
struct curry_impl<std::false_type, C, ArgsSoFar...> {
template <class... MoreArgs>
using apply = curry_impl<decltype(valid<C, ArgsSoFar..., MoreArgs...>(nullptr)), C, ArgsSoFar..., MoreArgs...>;
template <template <class...> class C>
struct curry {
template <class... U>
using apply = curry_impl<decltype(valid<C, U...>(nullptr)), C, U...>;
int main(void) {
using CurriedIsSame = curry<std::is_same>;
curry<std::less>::apply<int>::type less;
std::cout << std::boolalpha << less(5, 4); // prints false
using CurriedMap = curry<std::map>;
using MapType = CurriedMap::apply<int>::apply<long, std::less<int>, std::allocator<std::pair<const int, long>>>::type;
static_assert(std::is_same<MapType, std::map<int, long>>::value);
The technique is very simple. There's a function called
that has two overloads. The first one returns
only if
is a valid instantiation of template
with argument list
. Otherwise, it returns
is type constructor that we would like to curry. This function uses the SFINAE idiom.
is the core implementation of template currying. It has two specializations. The
specialization is selected when
. I.e., curried version of the type constructor has received the minimum number of type arguments to form a complete type. In other words,
are enough.
curry_impl<C, ArgsSoFar...>::type
is same as instantiation of the type constructor with the valid type arguments (
Note that C++ allows templates to have default type arguments. Therefore, a template could be instantiated by providing "minimum" number of arguments. For example,
could be instantiated in three ways giving the same type:
• std::map<int, long>
• std::map<int, long, std::less<int>>
• std::map<int, long, std::less<int>, std::pair<const int, long>>
are not enough,
carries the partial list of type arguments (
) at class template level. It allows passing one or more type arguments (
) to the type constructor through the
typedef. When
are enough to form a valid instantiation,
is chosen which yields the fully instantiated type.
Unit testing your template code comes up from time to time. (You test your templates, right?) Some templates are easy to test. No others. Sometimes it's not clear how to about injecting mock code
into the template code that's under test. I've seen several reasons why code injection becomes challenging. Here I've outlined some examples below with roughly increasing code injection difficulty.
Template accepts a type argument and an object of the same type by reference in constructor Template accepts a type argument. Makes a copy of the constructor argument or simply does not take one
Template accepts a type argument and instantiates multiple interrelated templates without virtual functions Lets start with the easy ones. Template accepts a type argument and an object of the same
type by reference in constructor This one appears straight-forward because the unit test simply instantiates the template under test with a mock type. Some assertion might be tested in
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Covariance and Contravariance are concepts that come up often as you go deeper into generic programming. While designing a language that supports parametric polymorphism (e.g., templates in C++,
generics in Java, C#), the language designer has a choice between Invariance, Covariance, and Contravariance when dealing with generic types. C++'s choice is "invariance". Let's look at an example.
struct Vehicle {}; struct Car : Vehicle {}; std::vector<Vehicle *> vehicles; std::vector<Car *> cars; vehicles = cars; // Does not compile The above program does not compile because C++ templates are
invariant. Of course, each time a C++ template is instantiated, the compiler creates a brand new type that uniquely represents that instantiation. Any other type to the same template creates another
unique type that has nothing to do with the earlier one. Any two unrelated user-defined types in C++ can't be assigned to each-other by default. You have to provide a
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What new can be said about multi-dimensional arrays in C++? As it turns out, quite a bit! With the advent of C++11, we get new standard library class std::array. We also get new language features,
such as template aliases and variadic templates. So I'll talk about interesting ways in which they come together. It all started with a simple question of how to define a multi-dimensional
std::array. It is a great example of deceptively simple things. Are the following the two arrays identical except that one is native and the other one is std::array? int native[3][4]; std::array
<std::array<int, 3>, 4> arr; No! They are not. In fact, arr is more like an int[4][3]. Note the difference in the array subscripts. The native array is an array of 3 elements where every element is
itself an array of 4 integers. 3 rows and 4 columns. If you want a std::array with the same layout, what you really need is: std::array<std::array<int, 4>, 3> arr; That's quite annoying for
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20 comments | {"url":"http://cpptruths.blogspot.com/2018/12/simple-template-currying.html","timestamp":"2024-11-08T08:19:27Z","content_type":"application/xhtml+xml","content_length":"133633","record_id":"<urn:uuid:44c8c70d-3fdd-4b60-91e3-39d68c971221>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00034.warc.gz"} |
Lagrangian mechanics
Table of Contents
1. Introduction
The Lagrangian, \(L: (\mathbb{R}, \mathbb{R} \rightarrow \mathbb{R}, \mathbb{R} \rightarrow \mathbb{R}) \rightarrow \mathbb{R}\) is simply a functional:
\begin{align*} L = L(t, f(t), f'(t)) \end{align*}
Where the Lagrangian represents some metric by which we calculate how optimized \(f(x)\) is. The action:
\begin{align*} J[f] = \int_{a}^{b}L(t, f(t), f'(t))dt \\ \end{align*}
Defines the actual relationship between \(f(t)\) and its level of optimization, where \(a\) and \(b\) represent the start and end points for a certain curve. For example, if you wanted to minimize
the surface area of something, \(a\) and \(b\) would be the starting and end points of the surface.
2. Euler-Lagrange equation
We first define some function:
\begin{align*} g(t) := f(t) + \epsilon \nu(t) \end{align*}
Where \(f(t)\) is our optimized function and \(\nu(t)\) represents some function we add to \(f(t)\) such that we perturb it by some small amount. Now \(\epsilon\) is a small number such that the
perturbation is small. Note that when \(\epsilon = 0\), \(g(t) = f(t)\), our optimized function.
\begin{align*} J[g] = \int_{a}^{b}L(t, g(t), g'(t))dt \end{align*}
Now \(J[g]\) is optimized when \(g(t)\) is a maximum or minimum with respect to the Lagrangian. \(\frac{dJ}{d\epsilon}\) represents the extent to which the action changes when the perturbation
changes. When \(\epsilon = 0\), \(g(t) = f(t)\), which means \(\frac{dJ}{d\epsilon}\) evaluated at \(\epsilon = 0\) should be zero, by definition of maxima and minima.
\begin{align*} \frac{dJ[g]}{d\epsilon} = \int_{a}^{b}\frac{dL}{d\epsilon}dt \end{align*}
By the multivariable chain rule:
\begin{align*} \frac{dL}{d\epsilon} = \frac{\partial L}{\partial t}\frac{dt}{d\epsilon} + \frac{\partial L}{\partial g}\frac{dg}{d\epsilon} + \frac{\partial L}{\partial g'}\frac{dg'}{d\epsilon} \end
because \(t\) does not depend on \(\epsilon\), \(g = f + \epsilon\nu\), and \(g' = f' + \epsilon\nu'\):
\begin{align*} \frac{dL}{d\epsilon} = \frac{\partial L}{\partial g}\nu(t) + \frac{\partial L}{\partial g'}\nu'(t) \end{align*}
now substituting back into the integral:
\begin{align*} \frac{dJ}{d\epsilon} = \int_{a}^{b}(\frac{\partial L}{\partial g}\nu(t) + \frac{\partial L}{\partial g'}\nu'(t))dt \end{align*}
applying integration by parts to the right side:
\begin{align*} \frac{dJ}{d\epsilon} = \int_{a}^{b}\frac{\partial L}{\partial g}\nu(t)dt + \nu(t)\frac{\partial L}{\partial g'}\bigg|_{a}^{b} - \int_{a}^{b}\nu(t)\frac{d}{dt}\frac{\partial L}{\partial
g'}dt \end{align*}
now \(\nu(t)\) can be any perturbation of \(f(t)\) but the boundary conditions must stay the same (every function that we are considering for optimization must have the same start and end points);
therefore, \(\nu(a) = \nu(b) = 0\). We can evaluate the bar to be 0 as a result. Doing this, combining the integral, then factoring out \(\nu(t)\):
\begin{align*} \frac{dJ}{d\epsilon} = \int_{a}^{b}\nu(t)(\frac{\partial L}{\partial g} - \frac{d}{dt}\frac{\partial L}{\partial g'})dt \end{align*}
Now we finally set \(\epsilon = 0\). This means \(g(t) = f(t)\), \(g'(t) = f'(t)\), and \(\frac{dJ}{d\epsilon} = 0\):
\begin{align*} 0 = \int_{a}^{b}\nu(t)(\frac{\partial L}{\partial f} - \frac{d}{dt}\frac{\partial L}{\partial f'})dt \end{align*}
And now because \(\nu(t)\) can be an arbitrarily large or small valued function as long as the boundary conditions remain the same and the left hand side must be zero, we get the Euler-Lagrange
\begin{align*} \frac{\partial L}{\partial f} - \frac{d}{dt}\frac{\partial L}{\partial f'} = 0 \end{align*}
This is because the integral implies that for all selections for this function \(\nu(t)\), \(\nu(t)(\frac{dL}{df} - \frac{d}{dt}\frac{dL}{dg'}) = 0\). Because \(\nu(t)\) can be any function assuming
it satisfies the boundary conditions, this can only be the case if \(\frac{dL}{df} - \frac{d}{dt}\frac{dL}{dg'} = 0\). In physics, we re-cast \(f\) as \(q\) and \(f'\) as \(\dot{q}\), where \(q\) and
\(\dot{q}\) are the generalized coordinates and generalized velocities respectively.
3. Hamiltonian
The Hamiltonian represents the total energy in the system; it is the Legendre Transformation of the Lagrangian. Applying the Legendre Transformation to the Lagrangian for coordinate \(\dot{q}\):
\begin{align*} L = \frac{1}{2}m\dot{q}^{2} - V(q) \\ H = \frac{\partial L}{\partial \dot{q}}\dot{q} - L \end{align*}
the Hamiltonian is defined as:
\begin{align*} H(q, p) = \sum _{i}p_{i}\dot{q_{i}} - L(q, \dot{q}) \end{align*}
\begin{align*} H(q, p) = \frac{p^{2}}{2m} + V(q) \end{align*}
where \(p\) is the generalized momentum, and \(q\) is a generalized coordinate. This results in two differential equations, the first of which is:
\begin{align*} \frac{\partial H}{\partial p_{i}} = \dot{q_{i}} \end{align*}
which follows directly from the Hamiltonian definition. Then, from the Euler-Lagrange equation:
\begin{align*} L = \sum_{i}p_{i}\dot{q_{i}} - H \\ \frac{\partial(\sum_{i}p_{i}\dot{q_{i}} - H)}{\partial q_{i}} - \frac{d}{dt}\frac{\partial(\sum_{i}p_{i}\dot{q_{i}} - H)}{\partial \dot{q_{i}}} = 0
\\ - \frac{\partial H}{\partial q_{i}} = \frac{dp_{i}}{dt} \\ \frac{\partial H}{\partial q_{i}} = - \frac{dp_{i}}{dt} \end{align*}
Although the generalized coordinate system in question does not have to be linear, we can encode all the differential equations for all the coordinates at once with the del operator:
\begin{align*} \vec{\nabla}_{p}H = \frac{d\vec{q}}{dt} \\ \vec{\nabla}_{q}H = -\frac{d\vec{p}}{dt} \end{align*}
this notation isn't standard and I kind of made it up, but I think it works, as long as you don't take the divergence or the curl of this system to really mean anything. Note that in both the
Hamiltonian formulation and Lagrangian formulation, the differential equations reduce to Newtonian mechanics if we are working in a linear coordinate system with energy conservation. | {"url":"https://ret2pop.nullring.xyz/mindmap/Lagrangian%20mechanics.html","timestamp":"2024-11-05T03:42:41Z","content_type":"application/xhtml+xml","content_length":"16295","record_id":"<urn:uuid:019ee2ce-3855-4f68-928f-677c8f9dda4f>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00389.warc.gz"} |
MCQ's: Fluid & Pressure Flashcards
1. How do they prevent static?
2. Type of flowmeter?
3. Laminar or turbulent flow at low flow?
1. Lined with gold to be antistatic
2. Variable orifice, constant pressure
3. Laminar at low flow
Reynolds number at which flow becomes turbulent?
What is the ‘critical velocity’?
The critical velocity is the gas velocity at which laminar flow changes into turbulent flow
How does density affect Reynolds number?
Lower density will reduce Reynolds number, therefore gas more likely to be laminar.
Ie. adding Helium to inspired gases lowers density.
1. type of flowmeter?
2. lumen expands into….?
3. works on the principle of?
4. can be affected by what feature of a gas?
1. Pneumotachograph - Constant orifice, variable pressure.
2. Lumen expands into many small tubes
3. Works on the principle of hagen-poisueille principle (ONLY laminar flow)
4. Affected by gas VISCOSITY
What is does the Poiseuille equation describe?
According to the Poiseuille equation:
1. Flow is proportional to the pressure difference and 4th power of the radius
2. Inversely proportional to viscosity and the length of the tube.
How does warming a gas affect its flow?
Warming a gas makes it less dense,
reducing Reynolds number,
therefore more likely to be laminar flow.
How does gas flow through an orifice?
Always turbulent and therefore influenced by gas density.
How does resistance vary between laminar and turbulent flow?
LAMINAR: Resistance is constant and independent of flow.
TURBULENT: flow resistance increases with flow in an exponential manner. | {"url":"https://www.brainscape.com/flashcards/mcq-s-fluid-pressure-9674934/packs/16812216","timestamp":"2024-11-04T23:35:58Z","content_type":"text/html","content_length":"82222","record_id":"<urn:uuid:5cd07a7f-63a2-49f6-a22e-fe59b8b6e51a>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00637.warc.gz"} |
ball mill laboratory working with callationcu
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Re: [tlaplus] Re: Simpler method of making multiple changes to a function in a single step
I think I have it. I have updated IsBalanced and BalancedAllocations to as follows:
/\ \A m,n \in N : \/ m = n
\/ Cardinality(reg[m]) - Cardinality(reg[n]) \in {-1,0,1}
/\ \A m,n \in N : \/ m = n
\/ reg[m] \intersect reg[n] = {}
/\ \A r \in R : \E n \in N : r \in reg[n]
BalancedAllocations ==
{ reg \in [N -> SUBSET R] : IsBalanced(reg) }
It will generate a register such as: [n1 |-> {"r1"}, n2 |-> {"r2"}, n3 |-> {"r3", "r4"}]
The first line of IsBalanced ensures resources are well balanced,
The second ensures that resources are not assigned twice.
The third ensures that all resources are assigned.
It is much shorter and clearer than my first attempt, thank you very much for the help. (If you see any further improvements do let me know!)
On Wednesday, January 16, 2019 at 5:35:42 PM UTC+1, Jack Vanlightly wrote:
I have a set of nodes N, a set of resources R and a function "register" that maps N to R. The algorithm is a resource allocation algorithm that must assign the resources of R evenly across the
nodes N.
So if N is { "n1", "n2", "n3"} and R is {"r1", "r2", "r3", "r4" } then once allocation has taken place a valid value for register would be:
[n1 |-> <<"r4", "r1">>, n2 |-> <<"r2">>, n3 |-> <<"r3">>]
I want to set the values of register in a single step and I have managed it, though the formula seems overly complex and I wonder if there is a simpler way of doing that would help me also gain
more insight into TLA+.
I have isolated the allocation logic into a toy spec as follows:
EXTENDS Integers, FiniteSets, Sequences, TLC
CONSTANT R, N
VARIABLE register
Init ==
register = [n \in N |-> << >>]
HasMinResources(counts, nd) ==
\A x \in N : counts[nd] <= counts[x]
Allocate ==
LET newRegister == [n \in N |-> << >>]
counts == [n \in N |-> 0]
al[pendingRes \in SUBSET R, assignCount \in [N -> Nat]] ==
LET n == CHOOSE nd \in N : HasMinResources(assignCount, nd)
r == LET int == R \intersect pendingRes
IN CHOOSE x \in int : TRUE
IF Cardinality(pendingRes) = 0 THEN newRegister
LET remaining == pendingRes \ { r }
newAssignCount == [assignCount EXCEPT ![n] = @ + 1]
IN [al[remaining, newAssignCount] EXCEPT ![n] = Append(@, r)]
IN al[R, counts]
Rebalance ==
/\ register' = Allocate
/\ PrintT(register')
(* ignore the spec, I just wanted to run the Rebalance action once *)
Spec == Init /\ [][Rebalance]_register
- I made Allocate recursive as that is the only way I could figure out making all the changes to register in a single step.
- I did the intersect so that I could use CHOOSE. Else it complained that is was an unbounded CHOOSE so I figured if I did an intersect with R then it would be interpretted as bounded.
Any insights or suggestions would be great. | {"url":"https://discuss.tlapl.us/msg01752.html","timestamp":"2024-11-04T12:14:55Z","content_type":"text/html","content_length":"15268","record_id":"<urn:uuid:c34c0600-b42f-4b1b-91e3-b36ced3cf420>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00165.warc.gz"} |
Unraveling Data Insights with R’s fivenum(): A Programmer’s Guide | R-bloggersUnraveling Data Insights with R’s fivenum(): A Programmer’s GuideUnraveling Data Insights with R’s fivenum(): A Programmer’s Guide
Unraveling Data Insights with R’s fivenum(): A Programmer’s Guide
[This article was first published on
Steve's Data Tips and Tricks
, and kindly contributed to
]. (You can report issue about the content on this page
) Want to share your content on R-bloggers?
if you have a blog, or
if you don't.
As a programmer and data enthusiast, you know that summarizing data is essential to gain insights into its distribution and characteristics. R, being a powerful and versatile programming language for
data analysis, offers various functions to aid in this process. One such function that stands out is fivenum(), a hidden gem that computes the five-number summary of a dataset. In this blog post, we
will explore the fivenum() function and demonstrate how to leverage it for different scenarios, empowering you to unlock valuable insights from your datasets.
The five number summary is a concise way to summarize the distribution of a data set. It consists of the following five values:
• The minimum value
• The first quartile (Q1)
• The median
• The third quartile (Q3)
• The maximum value
The minimum value is the smallest value in the data set. The first quartile (Q1) is the value below which 25% of the data points lie. The median is the value below which 50% of the data points lie.
The third quartile (Q3) is the value below which 75% of the data points lie. The maximum value is the largest value in the data set.
The five number summary can be used to get a quick overview of the distribution of a data set. It can tell us how spread out the data is, whether the data is skewed, and whether there are any
How to use the fivenum() function in R
Example 1. A Vector:
Let’s start with the basics. To compute the five-number summary for a vector in R, all you need is the fivenum() function and your data. For example:
# Sample vector
data_vector <- c(12, 24, 36, 48, 60, 72, 84, 96, 108, 120)
# Calculate the five-number summary
summary_vector <- fivenum(data_vector)
# Output the results
The fivenum() function will return the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum values of the vector. Armed with this information, you can easily visualize the
dataset’s distribution using box plots, histograms, or other graphical representations.
Example 2. With boxplot():
Box plots, also known as box-and-whisker plots, are a fantastic visualization tool to display the distribution and identify outliers in your data. When combined with fivenum(), you can create
insightful box plots with minimal effort. Consider this example:
# Sample vector
data_vector <- c(12, 24, 36, 48, 60, 72, 84, 96, 108, 120)
# Create a box plot
# Calculate the five-number summary and print the results
summary_vector <- fivenum(data_vector)
By incorporating the fivenum() function, you can see the minimum, lower hinge (Q1), median (Q2), upper hinge (Q3), and maximum, represented in the box plot. This graphical representation helps in
visualizing the spread of the data, presence of outliers, and skewness.
Example 3. On a Column in a Data.frame:
Often, data is stored in data.frames, which are highly efficient for handling and analyzing datasets. To apply fivenum() on a specific column within a data.frame, use the $ operator to access the
desired column. Consider the following example:
# Sample data.frame
data_df <- data.frame(ID = 1:5,
Age = c(25, 30, 22, 28, 35))
# Calculate the five-number summary for the "Age" column
summary_age <- fivenum(data_df$Age)
# Output the results
By applying fivenum() on the “Age” column, you obtain the five-number summary, which reveals valuable information about the age distribution of the dataset.
Example 4. Across Multiple Columns of a Data.frame Using sapply():
To elevate your data analysis game, you’ll often need to summarize multiple columns simultaneously. In this case, sapply() comes in handy, allowing you to apply fivenum() across several columns at
once. Let’s take a look at an example:
# Sample data.frame
data_df <- data.frame(ID = 1:5,
Age = c(25, 30, 22, 28, 35),
Salary = c(50000, 60000, 45000, 55000, 70000))
# Apply fivenum() on all numeric columns
summary_all_columns <- sapply(data_df[, 2:3], fivenum)
# Output the results
Age Salary
[1,] 22 45000
[2,] 25 50000
[3,] 28 55000
[4,] 30 60000
[5,] 35 70000
In this example, sapply() is used to calculate the five-number summary for the “Age” and “Salary” columns simultaneously. The output provides a comprehensive summary of these columns, enabling you to
quickly assess the distribution of each.
Congratulations! You’ve now unlocked the potential of R’s fivenum() function. By using it on vectors, data.frames, and even in conjunction with boxplot(), you can efficiently summarize data and gain
deeper insights into its distribution and characteristics. Embrace the power of fivenum() in your data analysis endeavors and embark on a journey of discovery with your datasets. Don’t hesitate to
explore further and adapt the function to your unique data analysis needs. Happy coding! | {"url":"https://www.r-bloggers.com/2023/07/unraveling-data-insights-with-rs-fivenum-a-programmers-guide/","timestamp":"2024-11-10T14:15:50Z","content_type":"text/html","content_length":"84395","record_id":"<urn:uuid:aa4598a7-54dd-4b46-8e48-ac89b3fc9ebf>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00683.warc.gz"} |
CPM Homework Help
When baking cookies for his class of $21$ students, Sammy needed two eggs. Now he wants to bake cookies for the upcoming science fair. If he expects $336$ people to attend the science fair, how many
eggs will he need?
How many times greater than $21$ is $336?$ Multiply this number by $2$ to find the number of eggs Sammy will need. | {"url":"https://homework.cpm.org/category/CON_FOUND/textbook/ac/chapter/2/lesson/2.2.3/problem/2-112","timestamp":"2024-11-04T01:32:56Z","content_type":"text/html","content_length":"35428","record_id":"<urn:uuid:f9194ad8-ec33-4c8e-b506-6f03d16bcc7f>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00779.warc.gz"} |
Наш мир НЕ симуляция
TLDR: Exploring the concept of our world being a computer simulation and debunking the simulation hypothesis.
📍 Article Source
Наш мир НЕ симуляцияhttps://www.youtube.com/watch?v=_nYBmUftYQw
The video begins with an apology for the delayed release, and then the topic of considering the world as a computer simulation is introduced. The discussion touches on various philosophical thoughts
and scientific analogues, referencing Descartes, Berkeley, and Hume.
Simulation Hypothesis
The concept of living in a computer simulation is attributed to a British philosopher, Bostrom, whose article in 2003 raised the question. The idea was later refuted in 2017 by the magazine Sante Fig
Advances, claiming the impossibility of creating such a simulation due to computing power limitations.
Arguments Against the Simulation Hypothesis
The video presents arguments against the hypothesis, starting with the deceptive discreteness of the world. It delves into the mismatch between quantum mechanics and macrocosm, emphasizing the
complexity of simulating quantum processes. The mention of string theory and its dynamic complexity further weakens the simulation hypothesis. The second argument focuses on the complexity of
simulating natural phenomena, such as turbulent flow, heat flows, and radiation of energy, in a way that resembles our world. This overwhelms the simulation, making it impractical. The third argument
highlights the non-obvious mathematics of our world and raises the issue of applying a closed system of mathematics and logic to describe the open world, resulting in contradictions according to
Gödel’s theorem. The shortcomings and lack of comprehensiveness of mathematics and logic in describing a certain system challenge the idea of the world as a mathematical simulation.
Key Points
1. Scale of Simulation: Describes the need to break down the medium of simulation to a size of 10^-10 power of meters in order to obtain a reliable simulation of turbulent flow. Argues against
overloading the system with unnecessary complexity. 2. Mathematics of Our World: Discusses the surprising ways in which our world can be described mathematically. Addresses the limitations of
mathematics and logic in representing a complex system like ours. Mentions the crisis of foundations of mathematics in the early 20th century. 3. Scope of Sciences: Argues that the idea of our
world being a simulation is based on an extremely inaccurate statement that only physics, mathematics, and logic can accurately describe history, sociology, psychology, and biology. 4.
Falsifiability: Criticizes the hypothesis that the world is a simulation as fundamentally not refutable, which goes against the scientific nature of an idea. Examines the absence of falsifying
suits and the limitations of experimental methods of proof. 5. Simulation Within Simulation: Explores the idea of being inside a simulation and questions the possibility of an endless loop of
simulations. Concludes by asserting that this hypothesis collapses within natural and mathematical sciences but remains viable as a philosophical concept.
The conclusion of the transcript highlights that there is a distinction between natural and mathematical hypotheses and bold philosophical ideas. Finally, the speaker asks popularizers for their
views on whether our world is a simulation, to which most are inclined to believe that we will never know for sure.
Timestamped Summary
The idea of considering the world as a computer simulation is not new.
• The concept has been present in philosophical works.
• Bostrom proposed in 2003 that our world could be a simulation.
• The idea was refuted in 2017, but it is a weak argument.
• It is possible that our world is much more primitive than the simulation creators' world.
The hypothesis that the world is a computer simulation has weaknesses
• The laws of quantum mechanics are not similar to those in the macrocosm
• The discreteness of our world does not necessarily mean it is a simulation
Simulating quantum mechanics and string theory is extremely complex
• Quantum mechanics simulations are based on hypotheses and have strange properties
• Particle accelerators provide theoretical data for confirming certain hypotheses
• String theory has important achievements in mathematical physics, but simulating strings is extremely difficult
Simulation of turbulent flow is complex
• Simulating turbulent flow requires breaking the medium into small objects and solving a complex system of equations
• Describing macro and mega processes in the language of the microworld is meaningless from a technological point of view
• Our world is described mathematically, making it amenable to mathematical simulation
The idea that our world is a simulation is unlikely and mathematically impossible.
• Using a closed system to describe our world would result in contradictions.
• Godel's theorem indicates the weakness and incomprehensiveness of mathematics and logic to describe a certain system.
• The crisis in the foundations of mathematics in the early 20th century hints at the impossibility of creating such a complex system as ours.
• The idea that the world can only be described based on physics, mathematics, and logic is inaccurate.
• Simulating different areas of natural science is extremely difficult and results in white spots.
• The conclusion that our world is a simulation is controversial and ignores certain facts.
The hypothesis that the world is a simulation cannot be refuted
• Falsification is the main criterion for the scientific character of an idea
• Darwin's theory can be fundamentally refuted, but the idea of God or a flat earth cannot
• The counter-argument to the simulation hypothesis is that our world was created by more developed beings to implement a given simulation
Our world must have finite dimensions to be simulated
• Imperfections in simulations can lead to inaccuracies in our model
• The idea of an infinite universe is not consistent with the concept of simulation
The hypothesis of the world being a simulation is collapsing within natural and mathematical sciences, but as a philosophical concept, it has potential.
• Arguments for a non-falsified theory are weak.
• The simulation within a simulation argument can continue indefinitely, but cannot be falsified.
• The idea is viable as a philosophical concept and has potential, like past philosophical ideas that were later proven experimentally.
Related Questions
What is the hypothesis that the world is a computer simulation, and who has proposed it?
What are the arguments against the hypothesis of the world being a computer simulation?
Can our world be considered as a simulation based on the laws of quantum mechanics and quantum computer simulation?
How does the concept of falsification apply to the hypothesis that the world is a simulation?
Is there a possibility of our world being a simulation within a simulation, and how does it relate to the overall hypothesis?
What is the video title and channel name?
The video title is "Наш мир НЕ симуляция" and it is from the channel Макар Светлый.
What is the speaker discussing in the video?
The speaker is discussing the differentiation between mathematical hypotheses and philosophical ideas, and the answers to the question of whether our world is a simulation.
What is the main emphasis of the discussion in the video?
The main emphasis of the discussion is on the distinction between mathematical hypotheses and bold philosophical ideas, and the speaker intends to provide answers to the question of whether our world
is a simulation.
What does the speaker intend to show in the conclusion of the video?
The speaker intends to show the answers to the question of whether our world is a simulation. | {"url":"https://www.youtubesummaries.com/education/_nYBmUftYQw","timestamp":"2024-11-11T12:53:33Z","content_type":"text/html","content_length":"116451","record_id":"<urn:uuid:836cc5d6-aa05-42b1-b2b1-6f217e8ecbb3>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00626.warc.gz"} |
New article shows a fatal math error in SR
No, it doesn't, because it is based on a false assumption.
Let's use another example. Let's say you take two atoms of hydrogen and two atoms of antihydrogen. What would you have once they combined? You might correctly think they would annihilate each
other and produce energy.
"Wrong!" a dolt could say. "2+2=4. The math PROVES that if you combine them you just get more hydrogen!"
Is his math wrong? No. Are his assumptions? Yes.
Careful, billvon. In your eagerness to pile on in the personal ridicule, you have made your own booboo about the maths there.
Specifically, if it's matter and anti-matter entities, then it should be "(+2H) + (-2H) = (zeroH) + (energy equivalent to 4H).
Take care not to sound like you put personal ridicule before obnjective answers to the article's mathematics and conclusions as posted by chinglu, else he will win the debate on this OP by default.
Good luck, and enjoy friendly objective on-topic discussion, everyone. Bye.
The article proves SR results in a contradiction.
The "article" is wrong, you are a crank, so what else is new?
Therefore, some other theory is responsible for GPS.
Therefore , you are an ignorant crackpot, the functionality of GPS is based on GR.
Yes, you suffer from the same knd of mental illness that affects Motor Daddy, the illness is called " I cannot understand relativity, therefore it must be wrong".
The "article" is wrong, you are a crank, so what else is new?
Therefore , you are an ignorant crackpot, the functionality of GPS is based on GR.
Yes, you suffer from the same knd of mental illness that affects Motor Daddy, the illness is called " I cannot understand relativity, therefore it must be wrong".
The simple math of the OP proves SR, which is a subset of GR, results in a mathematical contradiction.
Therefore, since a theory in contradiction does not work, some other theory is responsible for GPS.
Why don't you understand this simple logic?
Fully Wired
Valued Senior Member
Andrew Banks said:
Mirrors in Special Relativity
Andrew Banks
A name which is now [post=2949024]synonymous with ignorant, arrogant and wrong[/POST].
See also:
which I do not refer to -- the copy I quote is exclusively from chinglu's web site for a fake scientific journal. See also
where Andrew has been shown to be [thread=110037]wrong before[/THREAD].
Andrew Banks said:
This is not an abstract in any conventional sense which demonstrates that this paper has not been subjected to even cursory editorial review an that the publisher is a
sham scientific journal
-- in short a poetry vanity press masquerading as a scientific journal to fleece gullible pseudo-scientists and skeptics. An abstract should be exactly long enough to know what a paper is about and
what the main conclusions are. Here Andrew Banks begins his exposition, leaving the article without an abstract.
Andrew Banks said:
To develop the Lorentz transformations (LT), Einstein placed a mirror in the primed frame at some location (x’,0).
A bold claim, supported by nothing. Also, this would be an excellent place to
or quote the single Einstein references Andrew Banks lists, but instead he cites it nowhere. What Einstein actually did in part I, section 1, is to establish a system of synchronized clocks in a
single stationary frame. Here the use of primed coordinates merely refers to a different time on the same clock (A) as the earlier unprimed coordinate value.
Albert Einstein (translated) said:
If at the point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous
with these events. If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the
immediate neighbourhood of B. But it is not possible without further assumption to compare, in respect of time, an event at A with an event at B. We have so far defined only an “A time” and a “B
time.” We have not defined a common “time” for A and B, for the latter cannot be defined at all unless we establish
by definition
that the “time” required by light to travel from A to B equals the “time” it requires to travel from B to A. Let a ray of light start at the “A time” $$t_{\rm A}$$ from A towards B, let it at the
“B time” $$t_{\rm B}$$ be reflected at B in the direction of A, and arrive again at A at the “A time” $$t'_{\rm A}$$.
In accordance with definition the two clocks synchronize if
$$t_{\rm B}-t_{\rm A}=t'_{\rm A}-t_{\rm B}. $$
We assume that this definition of synchronism is free from contradictions, and possible for any number of points; and that the following relations are universally valid:—
1. If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the clock at B.
2. If the clock at A synchronizes with the clock at B and also with the clock at C, the clocks at B and C also synchronize with each other.
Thus with the help of certain imaginary physical experiments we have settled what is to be understood by synchronous stationary clocks located at different places, and have evidently obtained a
definition of “simultaneous,” or “synchronous,” and of “time.” The “time” of an event is that which is given simultaneously with the event by a stationary clock located at the place of the event,
this clock being synchronous, and indeed synchronous for all time determinations, with a specified stationary clock.
In agreement with experience we further assume the quantity
$$\frac{2{\rm AB}}{t'_A-t_A}=c, $$
to be a universal constant—the velocity of light in empty space.
It is essential to have time defined by means of stationary clocks in the stationary system, and the time now defined being appropriate to the stationary system we call it “the time of the
stationary system.”
At the bottom of section 2, the relativity of simultaneity is introduced, but again the primed coordinates do not refer to the choice of coordinate system.
Albert Einstein (translated) said:
We imagine further that at the two ends A and B of the rod, clocks are placed which synchronize with the clocks of the stationary system, that is to say that their indications correspond at any
instant to the “time of the stationary system” at the places where they happen to be. These clocks are therefore “synchronous in the stationary system.”
We imagine further that with each clock there is a moving observer, and that these observers apply to both clocks the criterion established in § 1 for the synchronization of two clocks. Let a ray
of light depart from A at the time [[“Time” here denotes “time of the stationary system” and also “position of hands of the moving clock situated at the place under discussion.”]] $$t_{\rm A}$$,
let it be reflected at B at the time $$t_{\rm B}$$, and reach A again at the time $$t'_{\rm A}$$. Taking into consideration the principle of the constancy of the velocity of light we find that
$$t_{\rm B}-t_{\rm A}=\frac{r_{\rm AB}}{c-v}\ {\rm and} t'_{\rm A}-t_{\rm B}=\frac{r_{\rm AB}}{c+v} $$
where $$r_{\rm AB}$$ denotes the length of the moving rod—measured in the stationary system. Observers moving with the moving rod would thus find that the two clocks were not synchronous, while
observers in the stationary system would declare the clocks to be synchronous.
So we see that we cannot attach any absolute signification to the concept of simultaneity, but that two events which, viewed from a system of co-ordinates, are simultaneous, can no longer be
looked upon as simultaneous events when envisaged from a system which is in motion relatively to that system.
So where is this mirror at location (x',0). Well in section 3, Einstein finally derives the Lorentz transformation, but does not use x' coordinates like Andrew Banks claims.
Albert Einstein (translated) said:
To any system of values x, y, z, t, which completely defines the place and time of an event in the stationary system, there belongs a system of values $$\xi$$, $$\eta$$, $$\zeta$$, $$\tau$$,
determining that event relatively to the system k, and our task is now to find the system of equations connecting these quantities.
In the first place it is clear that the equations must be linear on account of the properties of homogeneity which we attribute to space and time.
If we place $$x' = x - v t$$, it is clear that a point at rest in the system k must have a system of values x', y, z, independent of time. We first define $$\tau$$ as a function of x', y, z, and
t. To do this we have to express in equations that $$\tau$$ is nothing else than the summary of the data of clocks at rest in system k, which have been synchronized according to the rule given in
§ 1.
From the origin of system k let a ray be emitted at the time $$\tau_0$$ along the X-axis to x', and at the time $$\tau_1$$ be reflected thence to the origin of the co-ordinates, arriving there at
the time $$\tau_2$$; we then must have $$\frac{1}{2}(\tau_0+\tau_2)=\tau_1$$, or, by inserting the arguments of the function $$\tau$$ and applying the principle of the constancy of the velocity
of light in the stationary system:—
$$\frac{1}{2}\left[\tau(0,0,0,t)+\tau\left(0,0,0,t+\frac{x'}{c-v} + \frac{x'}{c+v}\right)\right]= \tau\left(x',0,0,t+\frac{x'}{c-v}\right)$$.
Hence, if x' be chosen infinitesimally small,
$$\frac{1}{2}\left(\frac{1}{c-v}+\frac{1}{c+v}\right)\frac{\partial \tau}{\partial t} = \frac{\partial \tau}{\partial x'}+\frac{1}{c-v}\frac{\partial\tau}{\partial t}$$,
$$\frac{\partial\tau}{\partial x'}+\frac{v}{c^2-v^2}\frac{\partial\tau}{\partial t}=0$$.
It is to be noted that instead of the origin of the co-ordinates we might have chosen any other point for the point of origin of the ray, and the equation just obtained is therefore valid for all
values of x', y, z.
As you see x and x' are in the same coordinate system, the system K which is called "stationary." Basically, Einstein is saying for a particular object moving with constant speed (the same velocity
as system k), then it has coordinates in the stationary system as $$x(t) = x' + v t, \; y(t) = y + 0 t, z(t) = z + 0 t$$ so that while x is a function of time, x', y and z are constants of motion in
coordinate system K. And since the object moves at the same velocity as coordinate system k, it follows that the linear motion of the object in K must translated to linear non-motion in system k or
$$\xi(t) = \xi + 0 \tau, \; \eta(t) = \eta + 0 \tau, \zeta(t) = \zeta + 0 \tau$$. To figure this out, Einstein made $$\tau$$ a function of the constants of motion of this particular object, moving in
stationary system K and motionless in stationary system k, and x' is one of those K-system coordinates corresponding to the stationary system X-position of the object at stationary system time t=0.
So already in sentence one, Andrew Banks has botched it by misunderstanding the 108-year-old paper that every physics baccalaureate understands the conclusions of. Einstein was not using primes to
distinguish different coordinate systems as is common in relativity textbooks today. He used Latin letters for one system (K) and Greek letters for the other system (k).
Andrew Banks said:
Then, when the origins of the primed and unprimed frames are common, a light pulse is emitted from the common origin location.
Actually Einstein considers a ray of light in the Latin and Greek coordinate systems. Light is used in various ways in Einstein's paper because the whole point was that the Lorentz Transformation
could be derived from basic assumptions and the consistency of the speed of light. Then he does the larger part by showing that this coordinate equivalence was also an equivalence of Maxwell's
electodynamics and (within then-current experimental limits) Newton's physics. Thus the 1905 paper was an important unification.
Andrew Banks said:
The experiment then considers two-way light travel for both frames such that a light beam strikes the primed frame mirror and is reflected back to the primed frame origin.
Nothing in Einstein's paper can be described as an experiment. Indeed, most of it is an argument from linearity and simple rate equations.
Here, at last, Andrew Bank's butchery of history ends and his beef begins.
Andrew Banks said:
This article proposes locating the mirror in the primed frame at (x’<0, y>0) with the reflective side parallel to and directly facing the primed y -axis. The other side of this mirror is
non-reflective. It should be clear that a light pulse emitted from the primed origin will be reflected off the mirror for all y>0.
Horrible syntax. "The article" can propose nothing. "The author proposes" is better but unnecessary. A paragraph break is needed because Andrew Banks has stopped talking about one subject
(Misunderstanding Einstein) and began another (Making a Fool of Oneself). The description of the location and orientation of the mirror is nonsensical.
Let a point-like detector exist, stationary in coordinate system k, somewhere to the left of the $$\eta$$-axis and only capable of detecting light to its right (including light originating at the
origin of coordinate system k).
Andrew Banks said:
On the other hand, in the context of the unprimed frame, it is also clear given v and x’<0, one can select y large enough such that while a light pulse emitted from the two origins when they are
common is expanding, the moving primed mirror crosses the unprimed y-axis before being struck by the light pulse. In this case, the unprimed frame predicts the light pulse strikes the
non-reflective side and no light is reflected. So, that means Special Relativity (SR) predicts one light pulse is reflected off the mirror and not reflected, which is a physical impossibility.
Assuming everything stationary in system k moves to the right with velocity v (in the x-direction) in system K, assume the origins of system k and K correspond at their respective zero times. Thus
$$ t = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( \tau + \frac{v}{c^2} \xi \right) \\ x = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( \xi + v \tau \right) \\ y = \eta $$
Is it possible that for $$0 < v < c $$ the mirror could have such a large $$\eta$$ value that in the K frame a light flash from the time the origins were at the same position arrives at the detector
from the left, preventing detection in one description of reality but not the other, supposedly equivalent one?
Andrew Banks correctly decides that the pulse from $$(\tau, \xi, \eta) = (0,0,0)$$ to $$(\tau_0, -\xi_0, +\eta_0)$$ would be seen in system K as a pulse from $$(t, x, y) = (0,0,0)$$ to $$(t_0, +x_0,
+\eta_0)$$ whenever certain geometrical constraints are met, but ignores the question of what "to the right" means in system K.
First, what is the minimum value of v such that in system K the light pulse to the detector in purely in the $$+\eta$$ direction? That would mean $$x_0 = 0$$. Thus
$$v_0 = \frac{c \xi_0}{\sqrt{ \xi_0^2 + \eta_0^2}} < c $$
Then for any v such that $$v_0 < v < c$$ and assuming $$-\xi_0 < 0, \; \eta_0 > 0, \; \tau_0 = \frac{1}{c} \sqrt{\xi_0^2 + \eta_0^2} > 0$$ we have :
$$x_0 = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( -\xi_0 + v \tau_0 \right) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( -\xi_0 + \frac{v}{c} \sqrt{\xi_0^2 + \eta_0^2} \right) { \Large \quad > \quad }
\frac{1}{\sqrt{1-\frac{v_0^2}{c^2}}} \left( -\xi_0 + \frac{v_0}{c} \sqrt{\xi_0^2 + \eta_0^2} \right) = \frac{1}{\sqrt{1-\frac{v_0^2}{c^2}}} \left( -\xi_0 + \frac{\xi_0}{\sqrt{ \xi_0^2 + \eta_0^2}} \
sqrt{\xi_0^2 + \eta_0^2} \right) = 0$$
But that, importantly, still doesn't answer if the light comes into the left or the right of the detector, which is answered by the sign of the cross product of the light ray movement and an
extension of the detector (finite or infintesimal) in the $$\eta$$ direction.
Andrew Banks said:
[1] Einstein A., in The Principle of Relativity (Dover, New York) 1952, p. 37.
This is proof that this paper has not been through ay sort of scientific review. This "book" is a collection of scientific papers published in real scientific journals and therefore cannot be cited
as an original source.
What is actual being cited, according to the page numbers I have is "On the Electrodynamics of Moving Bodies" which is a translation of "Zur Elektrodynamik bewegter Körper" by Albert Einstein
published in
Annalen der Physik
, Volume 17, pages 891-921 in 1905. Moreover, as a note in a different translation shows, this book was a Dover reprint of a 1923 Methuen and Company translation by W. Perrett and G.B. Jeffery of the
1922 Teubner-published collection
Das Relativatsprinzip
, 4th Edition.
http://books.google.com/books?id=S1dmLWLhdqAC&lpg=PA37&pg=PA37#v=onepage&q&f=false http://users.physik.fu-berlin.de/~kleinert/files/1905_17_891-921.pdf http://www.fourmilab.ch/etexts/einstein/specrel
Further, the reference was not actually referred to anywhere in the paper.
All assumptions and all math in the article have been flushed out in this thread and proven to be correct.
That is all that is needed.
It is time for you to face the facts.
Quit trolling us with nonsense dummy. Your assertions are meaningless bullshit from a scientific illiterate crank. IE everything you think about your result is bullshit round filed nonsense. It
doesn't even deserve a 'roundfile' because it was irrelevant unscientific nonsense before you wrote it down.
A name which is now [post=2949024]synonymous with ignorant, arrogant and wrong[/POST].
See also:
which I do not refer to -- the copy I quote is exclusively from chinglu's web site for a fake scientific journal. See also
where Andrew has been shown to be [thread=110037]wrong before[/THREAD].
This is not an abstract in any conventional sense which demonstrates that this paper has not been subjected to even cursory editorial review an that the publisher is a
sham scientific journal
-- in short a poetry vanity press masquerading as a scientific journal to fleece gullible pseudo-scientists and skeptics. An abstract should be exactly long enough to know what a paper is about
and what the main conclusions are. Here Andrew Banks begins his exposition, leaving the article without an abstract.
A bold claim, supported by nothing. Also, this would be an excellent place to
or quote the single Einstein references Andrew Banks lists, but instead he cites it nowhere. What Einstein actually did in part I, section 1, is to establish a system of synchronized clocks in a
single stationary frame. Here the use of primed coordinates merely refers to a different time on the same clock (A) as the earlier unprimed coordinate value.
At the bottom of section 2, the relativity of simultaneity is introduced, but again the primed coordinates do not refer to the choice of coordinate system.
So where is this mirror at location (x',0). Well in section 3, Einstein finally derives the Lorentz transformation, but does not use x' coordinates like Andrew Banks claims.
As you see x and x' are in the same coordinate system, the system K which is called "stationary." Basically, Einstein is saying for a particular object moving with constant speed (the same
velocity as system k), then it has coordinates in the stationary system as $$x(t) = x' + v t, \; y(t) = y + 0 t, z(t) = z + 0 t$$ so that while x is a function of time, x', y and z are constants
of motion in coordinate system K. And since the object moves at the same velocity as coordinate system k, it follows that the linear motion of the object in K must translated to linear non-motion
in system k or $$\xi(t) = \xi + 0 \tau, \; \eta(t) = \eta + 0 \tau, \zeta(t) = \zeta + 0 \tau$$. To figure this out, Einstein made $$\tau$$ a function of the constants of motion of this
particular object, moving in stationary system K and motionless in stationary system k, and x' is one of those K-system coordinates corresponding to the stationary system X-position of the object
at stationary system time t=0.
So already in sentence one, Andrew Banks has botched it by misunderstanding the 108-year-old paper that every physics baccalaureate understands the conclusions of. Einstein was not using primes
to distinguish different coordinate systems as is common in relativity textbooks today. He used Latin letters for one system (K) and Greek letters for the other system (k).
Actually Einstein considers a ray of light in the Latin and Greek coordinate systems. Light is used in various ways in Einstein's paper because the whole point was that the Lorentz Transformation
could be derived from basic assumptions and the consistency of the speed of light. Then he does the larger part by showing that this coordinate equivalence was also an equivalence of Maxwell's
electodynamics and (within then-current experimental limits) Newton's physics. Thus the 1905 paper was an important unification.
Nothing in Einstein's paper can be described as an experiment. Indeed, most of it is an argument from linearity and simple rate equations.
Here, at last, Andrew Bank's butchery of history ends and his beef begins.
Horrible syntax. "The article" can propose nothing. "The author proposes" is better but unnecessary. A paragraph break is needed because Andrew Banks has stopped talking about one subject
(Misunderstanding Einstein) and began another (Making a Fool of Oneself). The description of the location and orientation of the mirror is nonsensical.
Let a point-like detector exist, stationary in coordinate system k, somewhere to the left of the $$\eta$$-axis and only capable of detecting light to its right (including light originating at the
origin of coordinate system k).
Assuming everything stationary in system k moves to the right with velocity v (in the x-direction) in system K, assume the origins of system k and K correspond at their respective zero times.
$$ t = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( \tau + \frac{v}{c^2} \xi \right) \\ x = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( \xi + v \tau \right) \\ y = \eta $$
Is it possible that for $$0 < v < c $$ the mirror could have such a large $$\eta$$ value that in the K frame a light flash from the time the origins were at the same position arrives at the
detector from the left, preventing detection in one description of reality but not the other, supposedly equivalent one?
Andrew Banks correctly decides that the pulse from $$(\tau, \xi, \eta) = (0,0,0)$$ to $$(\tau_0, -\xi_0, +\eta_0)$$ would be seen in system K as a pulse from $$(t, x, y) = (0,0,0)$$ to $$(t_0,
+x_0, +\eta_0)$$ whenever certain geometrical constraints are met, but ignores the question of what "to the right" means in system K.
First, what is the minimum value of v such that in system K the light pulse to the detector in purely in the $$+\eta$$ direction? That would mean $$x_0 = 0$$. Thus
$$v_0 = \frac{c \xi_0}{\sqrt{ \xi_0^2 + \eta_0^2}} < c $$
Then for any v such that $$v_0 < v < c$$ and assuming $$-\xi_0 < 0, \; \eta_0 > 0, \; \tau_0 = \frac{1}{c} \sqrt{\xi_0^2 + \eta_0^2} > 0$$ we have :
$$x_0 = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( -\xi_0 + v \tau_0 \right) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( -\xi_0 + \frac{v}{c} \sqrt{\xi_0^2 + \eta_0^2} \right) { \Large \quad > \
quad } \frac{1}{\sqrt{1-\frac{v_0^2}{c^2}}} \left( -\xi_0 + \frac{v_0}{c} \sqrt{\xi_0^2 + \eta_0^2} \right) = \frac{1}{\sqrt{1-\frac{v_0^2}{c^2}}} \left( -\xi_0 + \frac{\xi_0}{\sqrt{ \xi_0^2 + \
eta_0^2}} \sqrt{\xi_0^2 + \eta_0^2} \right) = 0$$
But that, importantly, still doesn't answer if the light comes into the left or the right of the detector, which is answered by the sign of the cross product of the light ray movement and the
finite extension of the detector.
This is proof that this paper has not been through ay sort of scientific review. This "book" is a collection of scientific papers published in real scientific journals and therefore cannot be
cited as an original source.
What is actual being cited, according to the page numbers I have is "On the Electrodynamics of Moving Bodies" which is a translation of "Zur Elektrodynamik bewegter Körper" by Albert Einstein
published in
Annalen der Physik
, Volume 17, pages 891-921 in 1905. Moreover, as a note in a different translation shows, this book was a Dover reprint of a 1923 Methuen and Company translation by W. Perrett and G.B. Jeffery of
the 1922 Teubner-published collection
Das Relativatsprinzip
, 4th Edition.
http://books.google.com/books?id=S1dmLWLhdqAC&lpg=PA37&pg=PA37#v=onepage&q&f=false http://users.physik.fu-berlin.de/~kleinert/files/1905_17_891-921.pdf http://www.fourmilab.ch/etexts/einstein/
Further, the reference was not actually referred to anywhere in the paper.
Chinglu is probably going to give the 'read my nonsense' and weep response. Great post from you.
The simple math of the OP proves SR, which is a subset of GR, results in a mathematical contradiction.
...only for a crank like you, not for the millions of people who understand the theory.
Therefore, since a theory in contradiction does not work, some other theory is responsible for GPS.
Yet, GPS is built entirely on the formalism of GR. You are a nutter.
The article proves SR results in a contradiction. No one can refute that.
It's already been refuted and killed, dead on arrival. GPS refutes it posthumously, proving that Andrew Banks is a crank.
So, some other theory explains current experiments. What is so hard to understand about that?
That's pretty moronic considering the fact that GPS is flying at sufficient speed to set up the conditions to test Andrew Banks' claims, and then reveals that those claims are false.
And what do you mean experiments - GPS is a done deal. This baby is flying, unlike the piss-poor Andrew Banks crash-and-burn denial pseudomath anti-Geometry antirelativity pseudotechnoscreed.
GPS proves SR and GR, both as coupled and independent effects. Here you have the transverse SR producing 7 us of error per day for the stationary and slow moving ground receivers, plus all of the
combinations of SR effects for aircraft and sats that use it. It works per the LT, per SR, but only because the premise of GPS is correct. No games are played like the moron Andrew Banks has done by
pretending to have a peer reviewed proof.
It's quite simple actually. Andrew Banks is a dolt, and you are either him or his gullible minion. What's so hard about that? Nothing. That's why everyone here nailed you from the get-go.
Careful, billvon. In your eagerness to pile on in the personal ridicule, you have made your own booboo about the maths there.
Specifically, if it's matter and anti-matter entities, then it should be "(+2H) + (-2H) = (zeroH) + (energy equivalent to 4H).
Take care not to sound like you put personal ridicule before obnjective answers to the article's mathematics and conclusions as posted by chinglu, else he will win the debate on this OP by
default. Good luck, and enjoy friendly objective on-topic discussion, everyone. Bye.
Birds of a feather.
A name which is now [post=2949024]synonymous with ignorant, arrogant and wrong[/POST].
See also:
which I do not refer to -- the copy I quote is exclusively from chinglu's web site for a fake scientific journal. See also
where Andrew has been shown to be [thread=110037]wrong before[/THREAD].
This is not an abstract in any conventional sense which demonstrates that this paper has not been subjected to even cursory editorial review an that the publisher is a
sham scientific journal
-- in short a poetry vanity press masquerading as a scientific journal to fleece gullible pseudo-scientists and skeptics. An abstract should be exactly long enough to know what a paper is about
and what the main conclusions are. Here Andrew Banks begins his exposition, leaving the article without an abstract.
A bold claim, supported by nothing. Also, this would be an excellent place to
or quote the single Einstein references Andrew Banks lists, but instead he cites it nowhere. What Einstein actually did in part I, section 1, is to establish a system of synchronized clocks in a
single stationary frame. Here the use of primed coordinates merely refers to a different time on the same clock (A) as the earlier unprimed coordinate value.
At the bottom of section 2, the relativity of simultaneity is introduced, but again the primed coordinates do not refer to the choice of coordinate system.
So where is this mirror at location (x',0). Well in section 3, Einstein finally derives the Lorentz transformation, but does not use x' coordinates like Andrew Banks claims.
As you see x and x' are in the same coordinate system, the system K which is called "stationary." Basically, Einstein is saying for a particular object moving with constant speed (the same
velocity as system k), then it has coordinates in the stationary system as $$x(t) = x' + v t, \; y(t) = y + 0 t, z(t) = z + 0 t$$ so that while x is a function of time, x', y and z are constants
of motion in coordinate system K. And since the object moves at the same velocity as coordinate system k, it follows that the linear motion of the object in K must translated to linear non-motion
in system k or $$\xi(t) = \xi + 0 \tau, \; \eta(t) = \eta + 0 \tau, \zeta(t) = \zeta + 0 \tau$$. To figure this out, Einstein made $$\tau$$ a function of the constants of motion of this
particular object, moving in stationary system K and motionless in stationary system k, and x' is one of those K-system coordinates corresponding to the stationary system X-position of the object
at stationary system time t=0.
So already in sentence one, Andrew Banks has botched it by misunderstanding the 108-year-old paper that every physics baccalaureate understands the conclusions of. Einstein was not using primes
to distinguish different coordinate systems as is common in relativity textbooks today. He used Latin letters for one system (K) and Greek letters for the other system (k).
Actually Einstein considers a ray of light in the Latin and Greek coordinate systems. Light is used in various ways in Einstein's paper because the whole point was that the Lorentz Transformation
could be derived from basic assumptions and the consistency of the speed of light. Then he does the larger part by showing that this coordinate equivalence was also an equivalence of Maxwell's
electodynamics and (within then-current experimental limits) Newton's physics. Thus the 1905 paper was an important unification.
Nothing in Einstein's paper can be described as an experiment. Indeed, most of it is an argument from linearity and simple rate equations.
Here, at last, Andrew Bank's butchery of history ends and his beef begins.
Horrible syntax. "The article" can propose nothing. "The author proposes" is better but unnecessary. A paragraph break is needed because Andrew Banks has stopped talking about one subject
(Misunderstanding Einstein) and began another (Making a Fool of Oneself). The description of the location and orientation of the mirror is nonsensical.
Let a point-like detector exist, stationary in coordinate system k, somewhere to the left of the $$\eta$$-axis and only capable of detecting light to its right (including light originating at the
origin of coordinate system k).
Assuming everything stationary in system k moves to the right with velocity v (in the x-direction) in system K, assume the origins of system k and K correspond at their respective zero times.
$$ t = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( \tau + \frac{v}{c^2} \xi \right) \\ x = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( \xi + v \tau \right) \\ y = \eta $$
Is it possible that for $$0 < v < c $$ the mirror could have such a large $$\eta$$ value that in the K frame a light flash from the time the origins were at the same position arrives at the
detector from the left, preventing detection in one description of reality but not the other, supposedly equivalent one?
Andrew Banks correctly decides that the pulse from $$(\tau, \xi, \eta) = (0,0,0)$$ to $$(\tau_0, -\xi_0, +\eta_0)$$ would be seen in system K as a pulse from $$(t, x, y) = (0,0,0)$$ to $$(t_0,
+x_0, +\eta_0)$$ whenever certain geometrical constraints are met, but ignores the question of what "to the right" means in system K.
First, what is the minimum value of v such that in system K the light pulse to the detector in purely in the $$+\eta$$ direction? That would mean $$x_0 = 0$$. Thus
$$v_0 = \frac{c \xi_0}{\sqrt{ \xi_0^2 + \eta_0^2}} < c $$
Then for any v such that $$v_0 < v < c$$ and assuming $$-\xi_0 < 0, \; \eta_0 > 0, \; \tau_0 = \frac{1}{c} \sqrt{\xi_0^2 + \eta_0^2} > 0$$ we have :
$$x_0 = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( -\xi_0 + v \tau_0 \right) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( -\xi_0 + \frac{v}{c} \sqrt{\xi_0^2 + \eta_0^2} \right) { \Large \quad > \
quad } \frac{1}{\sqrt{1-\frac{v_0^2}{c^2}}} \left( -\xi_0 + \frac{v_0}{c} \sqrt{\xi_0^2 + \eta_0^2} \right) = \frac{1}{\sqrt{1-\frac{v_0^2}{c^2}}} \left( -\xi_0 + \frac{\xi_0}{\sqrt{ \xi_0^2 + \
eta_0^2}} \sqrt{\xi_0^2 + \eta_0^2} \right) = 0$$
But that, importantly, still doesn't answer if the light comes into the left or the right of the detector, which is answered by the sign of the cross product of the light ray movement and an
extension of the detector (finite or infintesimal) in the $$\eta$$ direction.
This is proof that this paper has not been through ay sort of scientific review. This "book" is a collection of scientific papers published in real scientific journals and therefore cannot be
cited as an original source.
What is actual being cited, according to the page numbers I have is "On the Electrodynamics of Moving Bodies" which is a translation of "Zur Elektrodynamik bewegter Körper" by Albert Einstein
published in
Annalen der Physik
, Volume 17, pages 891-921 in 1905. Moreover, as a note in a different translation shows, this book was a Dover reprint of a 1923 Methuen and Company translation by W. Perrett and G.B. Jeffery of
the 1922 Teubner-published collection
Das Relativatsprinzip
, 4th Edition.
http://books.google.com/books?id=S1dmLWLhdqAC&lpg=PA37&pg=PA37#v=onepage&q&f=false http://users.physik.fu-berlin.de/~kleinert/files/1905_17_891-921.pdf http://www.fourmilab.ch/etexts/einstein/
Further, the reference was not actually referred to anywhere in the paper.
First off, let's correct rpenner on the experiment proposed by Einstein.
The article claimed Einstein used (x',0) in the context of the moving frame for his LT equations.
Let us quote Einstein.
"To any time of the stationary system K"
So, capital K is the stationary system.
Then we have,
"From the origin of system k let a ray be emitted at the time t'0 along the X-axis to x', and at the time be reflected thence to the origin of the co-ordinates, arriving there at t'1"
So, we have lower case k as the moving system as indicated in the article. Then, in the moving system k, a light pulse is emitted from the origin of k along the X axis to x' which means (x',0) just
as the article says, and there is a mirror there which reflects back to the origin of the k moving system.
This is exactly what the article says and this is exactly what Einstein says.
This proves rpenner is in absolute error.
The only next relevant claim by rpenner is that the unprimed frame claims that the moving mirror strikes the light sphere on the front side so that reflection occurs for both frames even though the
mirror is on the positive side of the unprimed frame x-axis when light strikes it and the back side is the non-reflective side is facing the unprimed origin.
This means, along any y-line, a sphere is above that line before the mirror strikes it so that it runs into it. That would mean there is a light beam along a y line that is in front of any other
light beam on that line.
Therefore, this beam would exceed c and contradict SR.
So, rpenner's proposal contradicts SR and that would mean SR people would call him a crank and crackpot.
So, rpenner's post does nothing to refute the OP link
Quit trolling us with nonsense dummy. Your assertions are meaningless bullshit from a scientific illiterate crank. IE everything you think about your result is bullshit round filed nonsense. It
doesn't even deserve a 'roundfile' because it was irrelevant unscientific nonsense before you wrote it down.
The OP stands without any logical challenge.
What is it like living on a flat earth?
It's already been refuted and killed, dead on arrival. GPS refutes it posthumously, proving that Andrew Banks is a crank.
That's pretty moronic considering the fact that GPS is flying at sufficient speed to set up the conditions to test Andrew Banks' claims, and then reveals that those claims are false.
And what do you mean experiments - GPS is a done deal. This baby is flying, unlike the piss-poor Andrew Banks crash-and-burn denial pseudomath anti-Geometry antirelativity pseudotechnoscreed.
GPS proves SR and GR, both as coupled and independent effects. Here you have the transverse SR producing 7 us of error per day for the stationary and slow moving ground receivers, plus all of the
combinations of SR effects for aircraft and sats that use it. It works per the LT, per SR, but only because the premise of GPS is correct. No games are played like the moron Andrew Banks has done
by pretending to have a peer reviewed proof.
It's quite simple actually. Andrew Banks is a dolt, and you are either him or his gullible minion. What's so hard about that? Nothing. That's why everyone here nailed you from the get-go.
I agree, GPS refutes SR.
Try to take your GPS unit with the sagnac correction and hold it over an MMX experiment. What do your get?
The article also prove SR results in a contradiction.
I agree, GPS refutes SR.
Try to take your GPS unit with the sagnac correction and hold it over an MMX experiment. What do your get?
The article also prove SR results in a contradiction.
Unrepenting crank, the theory of GPS includes the Sagnac effect.
Only a blind moron would say that. Ok, a person in a psyche ward or intellectually disabled.
Try to take your GPS unit with the sagnac correction and hold it over an MMX experiment. What do your get?
No, try and add 7us/day in keeping with bonehard Andrew Banks and see if you're still in Bugtussle when the hangover wears off in the morning.
The article also prove SR results in a contradiction.
GPS proves the premise of the article is fraudulent.
You are simply proving that your anti-science denialism, naivete and narcissism has no bounds. Other than that, your bogus claims are DOA.
The OP stands without any logical challenge.
What is it like living on a flat earth?
The OP fell as soon as billvon called you out, and rpenner just carried out the trash and nuked it in a colossal incinerator. The rest of the folks were just sweeping up the crumbs.
GPS declared it dead on arrival.
Unrepenting crank, the theory of GPS includes the Sagnac effect.
so hold your GPS unit over an MMX experiment and tell me what you get.
I feel like I am talking to an ape.
The OP fell as soon as billvon called you out, and rpenner just carried out the trash and nuked in a colossal incinerator. The rest of the folks were just sweeping up the crumbs.
GPS declared it dead on arrival.
rpenner was refuted.
if not, simply point out exactly where he refuted the OP.
Otherwise, your post is useless.
so hold your GPS unit over an MMX experiment and tell me what you get.
I am getting that you are insane. But we knew that for years.
Only a blind moron would say that. Ok, a person in a psyche ward or intellectually disabled.
No, try and add 7us/day in keeping with bonehard Andrew Banks and see if you're still in Bugtussle when the hangover wears off in the morning.
GPS proves the premise of the article is fraudulent.
You are simply proving that your anti-science denialism, naivete and narcissism has no bounds. Other than that, your bogus claims are DOA.
The article proves SR results in a contradiction.
That means all experimental evidence prove some other theory.
Why is that so hard for you to understand?
I am getting that you are insane. But we knew that for years.
Oh, what do you get if you hold your GPS device with sagnac over MMX, you did not say.
Do you actually FALL OFF A FLAT EARTH? | {"url":"https://www.sciforums.com/threads/new-article-shows-a-fatal-math-error-in-sr.135674/page-5#post-3096591","timestamp":"2024-11-10T07:43:56Z","content_type":"text/html","content_length":"195340","record_id":"<urn:uuid:c3022485-4f71-4166-bae5-a7d666e56b36>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00018.warc.gz"} |
numpy.binary_repr(num, width=None)[source]¶
Return the binary representation of the input number as a string.
For negative numbers, if width is not given, a minus sign is added to the front. If width is given, the two’s complement of the number is returned, with respect to that width.
In a two’s-complement system negative numbers are represented by the two’s complement of the absolute value. This is the most common method of representing signed integers on computers [1]. A
N-bit two’s-complement system can represent every integer in the range
num : int
Only an integer decimal number can be used.
width : int, optional
Parameters: The length of the returned string if num is positive, or the length of the two’s complement if num is negative, provided that width is at least a sufficient number of bits for num
to be represented in the designated form.
If the width value is insufficient, it will be ignored, and num will be returned in binary (num > 0) or two’s complement (num < 0) form with its width equal to the minimum number
of bits needed to represent the number in the designated form. This behavior is deprecated and will later raise an error.
Deprecated since version 1.12.0.
bin : str
Binary representation of num or two’s complement of num.
See also
Return a string representation of a number in the given base system.
Python’s built-in binary representation generator of an integer.
binary_repr is equivalent to using base_repr with base 2, but about 25x faster.
[1] (1, 2) Wikipedia, “Two’s complement”, http://en.wikipedia.org/wiki/Two’s_complement
>>> np.binary_repr(3)
>>> np.binary_repr(-3)
>>> np.binary_repr(3, width=4)
The two’s complement is returned when the input number is negative and width is specified:
>>> np.binary_repr(-3, width=3)
>>> np.binary_repr(-3, width=5) | {"url":"https://docs.scipy.org/doc/numpy-1.15.1/reference/generated/numpy.binary_repr.html","timestamp":"2024-11-10T04:51:44Z","content_type":"text/html","content_length":"12124","record_id":"<urn:uuid:3dc68fac-2080-41ec-9137-95617a3150a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00237.warc.gz"} |
Can I get MyStatLab help for Bayesian analysis in environmental impact assessments? | Hire Some To Take My Statistics Exam
Can I get MyStatLab help for Bayesian analysis in environmental impact assessments? The question is “How would I make Bayesian statistical approaches apply to any kind of environmental impact
assessment?” What I can think of is to use Bayesian statistics in the context of a risk assessment, if there is a risk assessment, and then apply a ‘complete model’-based approach to incorporate this
uncertainty into a general’model’ that is likely to have some predictive capability. There is no question about how I can apply or measure Bayesian analysis to a matter like the study of climate
change. But what if this study concerns the application of this ‘complete model’ concept when using data from the Ecological Monograph, especially for risk assessments that require a large volume of
data. And if we can’t make the full model-based approach work or if we use Bayesian analysis when dealing with this common scenario, is there an actual measurement that is likely to work under those
conditions? A problem with Bayesian analysis in this respect is that Bayesian models have no apparent utility in evaluating impacts’ risks, because they don’t even carry a measure of ‘risk’. So what
happens under such scenarios is that you lose a large amount of data to this Bayesian model-based approach which cannot carry the model-driven uncertainty that is inherent in the model-based
approach. If we are confident that the model’s predictive capability is tied in with the impacts that the model is likely to do, then we need to use Bayesian analysis to quantify risk in analyses
when we know that some of the worst impacts would be – and thus, be – ‘close’ to the target impacts of our approach. Of course, as pay someone to do statistics examination rule of thumb, we could
take a number of more interesting – and perhaps potentially safer – variables into account. But what about large populations with general environmental catastrophes like that one? And what about
species that would be too much likely? It could be relevant that the population are too small for a Bayesian approach to have a robust predictive capability. ButCan I get MyStatLab help for Bayesian
analysis in environmental impact this hyperlink First, there is a question posed by the British Environmental Protection Agency on the effects of flooding around the Bay Islands. Bay island has been
known to devastate a large number of families and have created a huge amount of new risk. This demand for urgent action, while ignoring the long term impacts of climate change, is that Bay, like
other islands, remain vulnerable to flooding. The stateof trust in those around us should be reinforced. Applying the same reasoning to the Bay Islands, we should test whether Visit This Link basin
is significantly affected by current flooding. Is that the case here as we propose to do in the new article, or if we are going learn the facts here now use the Bay basin as a marker, and would we
have a problem in this critical area of science? Tests 1 1. The government should remove Bay as a source of worry. In many cases, natural disasters are blamed for causing a substantial damage. One
example is the loss of much-lauded private oil refineries from which some of the world’s most important oil products have been manufactured with inadequate performance and safety standards. However,
in at least four instances, we show the presence of specific points during the Bay basin dynamics that can significantly alter future behaviour and thus can lead to negative political and economic
implications. A new study, published in the journal Emerg-Geophysica, found that bay basin is also a public-private partnership that has a great impact on the production and production capacity of
marine sites. 2.
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If the current increase in the demand for fossil fuels is significant, it must be included as part of the environment. The Bay of Benvakis National Monument is go to my blog to the northwest of
Samsa. However, it is also a country of concern for example in one of several countries such as Brazil and Mozambique. 3. Bay basin is necessary for the reconstruction and improvement of a large
scale. No systemically developed, publicly funded constructionCan I get MyStatLab help for Bayesian analysis in environmental impact assessments? — iKotels on July 11, 2014 – 8:27 am A general note
on this topic. Some examples of why the Bayesian HMM seems to produce the most valid interpretation of the results are provided by @Broder and @Melkovic. Note that they wrote “HMM fits the values of
G/L by a minimum regression parameter and limits G/L to ~450%. ” There might be a slight uncertainty related to all the HMM fitting parameters (S/N as well) but we’ve discovered that the maximum can
be found at the 95% confidence level and thus we’re left with the case where G/L is zero. Is it possible… is there a way to determine the “value” of those parameters like (G/L /B/B)? The
likelihood-regression analysis seems to suggest this is possible ~~~ ClementK 1) Using [dividing G/L] to give a signal-to-noise ratio based on G/L = 0.4. 2) Using [Coeff] for G/L given that.5 > B/L
3) Using S/N = 0.23 (0.6 is a rather stable value but if its very close then how is it fixed?) 4) Using 4 points on the G/L plot for the standard deviation, I believe this is a good fit to any trend
being observed Thanks all…
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I’m going to be doing this quite often. —— evaldev I think that this is important to be able to do for Bayesian HMM that actually qualifies the results as being statistically significant. I think it
should be enough to take a lot of careful fact check to figure out the answer… I agree, that new analysis should have much wider statistical power. It will probably yield better results and take a | {"url":"https://hireforstatisticsexam.com/can-i-get-mystatlab-help-for-bayesian-analysis-in-environmental-impact-assessments","timestamp":"2024-11-09T04:38:27Z","content_type":"text/html","content_length":"168955","record_id":"<urn:uuid:11631c4d-5f7b-4738-87a1-2058a9d6809c>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00516.warc.gz"} |
Capsule Based Regression/Prediction
The following steps will create a prediction model for every capsule in a condition.
Step 1. pick a condition with capsules that isolate the desired area of regression. Any condition with non-overlapping capsules will work as long as there are enough sample points within its
duration. For this example, an increasing temperature condition will be used. However, periodic conditions and value search conditions will work as well.
Step 2. Create a time counter for each capsule in the condition. This can be done with the new timesince() function in the formula tool. The timesince() function will have samples spaced depending
on the selected period so it is important to select a period that has enough points to build a model with. See below for details on the timesince() formula setup.
Step 3. In this step a condition with capsule properties that hold the regression constants will be made. This will be done in the formula tool with one formula. The concept behind the formula
below is to split the condition from step one into individual capsules and use each of the capsules as the training window for a regression model. Once the regression model is done for one capsule
the coefficients of the model are assigned as properties to the capsule used for the training window.
The formula syntax for a linear model-based condition can be seen below. An example of a polynomial regression model can be found in the post below.
$model=$SignalToModel.validValues().regressionModelOLS( group($cap),false,$Time)
Below is a screenshot of how the formula looks in Seeq.
Note: The regression constants can be added to the capsule pane by clicking on the black stats button and selecting add column. Below is a screen shot of the results.
Step 4. Once a condition with the regression coefficients has been created the information will need to be extracted to a signal form. The following formula syntax will extract the information. This
will need to be repeated for every constant from your regression model. e.g.(So for a linear model this must be done for both the slope and for the intercept.)
The formula syntax for extracting the regression coefficients can be seen below.
$cap -> sample($cap.getmiddle(),
$cap.getProperty('Intercept').toNumber()), 1min)
Below is a screenshot of the formula in Seeq.
Below is a screenshot of the display window of how the signals should look.
Step 5. Use the formula tool to plot the equation. See screenshot below for details.
Edited by Teddy
• Administrators
Here is the syntax for a polynomial regression based condition.
$model=$signaltomodel.regressionModelOLS(group($cap), false, $Time, $Time^2)
Below is a screenshot of it setup in the formula tool.
• 2
• 2 years later...
• Seeq Team
In Step 4 of the Original Post, the formula used for creating a signal with values from the Condition properties, 'Intercept' and 'Slope', was...
$cap -> sample($cap.getmiddle(),
$cap.getProperty('Intercept').toNumber()), 1min)
This complex formula has been simplified and can now be written as...
If the condition is unbounded you may have to modify slightly... | {"url":"https://www.seeq.org/topic/527-capsule-based-regressionprediction/","timestamp":"2024-11-12T16:13:31Z","content_type":"text/html","content_length":"93874","record_id":"<urn:uuid:13db1489-78c9-495d-9288-202015f9d524>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00042.warc.gz"} |
Puzzle Archives - Puzzle Prime
The main challenge of a Sunome puzzle is drawing a maze. Numbers surrounding the outside of the maze border give an indication of how the maze is to be constructed. To solve the puzzle you must draw
all the walls where they belong and then draw a path from the Start square to the End square.
The walls of the maze are to be drawn on the dotted lines inside the border. A single wall exists either between 2 nodes or a node and the border. The numbers on the top and left of the border tell
you how many walls exist on the corresponding lines inside the grid. The numbers on the right and bottom of the border tell you how many walls exist in the corresponding rows and columns. In
addition, the following must be true:
• Each puzzle has a unique solution.
• There is only 1 maze path to the End square.
• Every Node must have a wall touching it.
• Walls must trace back to a border.
• If the Start and End squares are adjacent to each other, a wall must separate them.
• Start squares may be open on all sides, while End squares must be closed on 3 sides.
• You cannot completely close off any region of the grid.
In addition, these variations of Sunome have the following extra features:
• Paths (borders with a hole in the middle) designate places where the solution should pass through.
• Pits (black squares) designate places where the solution does not pass through.
• Portals (circled letters) designate places where the solution should pass through and teleport from one portal to the other.
• Sunome Cubed is solved similarly but on the surface of a cube. The numbers on the top right, top left, and center left of the border tell you how many walls exist on the corresponding pairs of
lines inside the grid. The numbers on the center right, bottom right, and bottom left of the border tell you how many walls exist in the corresponding pairs of rows/columns.
Examine the first example, then solve the other three puzzles.
Paths and Pits
Paths and Portals
Sunome Cubed
The solutions are shown below.
Monkey Type
There are many tools online to test your typing skills. However, one of them is so stylish and functional, that it drives us back to it repeatedly. Monkey Type offers many features and customization
options which make it feel like an addicting video game. You can try to climb the leaderboards in various categories by typing as quickly as possible given paragraphs. You can focus on typing
punctuation or numbers, or simply practice stress-free using the provided zen mode. Following each session, you will receive a comprehensive report of your typing performance, including words per
minute, accuracy, consistency, etc. The interface is simple and polished, but if it is not to your liking, you can always change the theme in the settings menu.
No Body, No Nose
What do you call a person with no body and no nose?
The answer is NOBODY KNOWS (no-body-nose).
D1G1TAL CHR0N1CLES
“D1G1TAL CHR0N1CLES” by the Georgean duo Levan Patsinashvili and Davit Babiashvili is a series of pictograms depicting major historical events using cleverly designed fonts. The designs are puzzling,
educational, and eye-pleasing at the same time. Can you guess what happened in the years 1250, 1912, and 1975 by examining these three images?
Well, the sequence {1, 1, 2, 3, 5} is the Fibonacci sequence, and 1250 is the year the famous mathematician died. The sinking number “1912” hints that this is the year the Titanic crashed, and the
funny “97” which resembles the Windows OS logo symbolizes the founding of Microsoft in 1975.
Below, we are presenting Levan and Davit’s entire series, consisting of 52 designs, in chronological order. Which ones are your favorites and how many events can you recognize?
Dr. Riesen’s Rebuses 3
Can you figure out what common phrases these rebuses represent?
The answers are:
1. Read between the lines
2. Big picture thinking
3. Turncoat
4. Cut to the chase
5. The last straw
6. Nick of time
7. Less is more
8. Easy come, easy go
9. Once in a blue moon
10. Backgammon game
11. Practice makes perfect
12. Partial custody
13. Throw in the towel
14. Run out of steam
15. Make or break
16. Lost in translation
The Connect Game
Two friends are playing the following game:
They start with 10 nodes on a sheet of paper and, taking turns, connect any two of them which are not already connected with an edge. The first player to make the resulting graph connected loses.
Who will win the game?
Remark: A graph is “connected” if there is a path between any two of its nodes.
The first player has a winning strategy.
His strategy is with each turn to keep the graph connected, until a single connected component of 6 or 7 nodes is reached. Then, his goal is to make sure the graph ends up with either connected
components of 8 and 2 nodes (8-2 split), or connected components of 6 and 4 nodes (6-4 split). In both cases, the two players will have to keep connecting nodes within these components, until one of
them is forced to make the graph connected. Since the number of edges in the components is either C^8_2+C^2_2=29, or C^6_2+C^4_2=21, which are both odd numbers, Player 1 will be the winner.
Once a single connected component of 6 or 7 nodes is reached, there are multiple possibilities:
1. The connected component has 7 nodes and Player 2 connects it to one of the three remaining nodes. Then, Player 1 should connect the remaining two nodes with each other and get an 8-2 split.
2. The connected component has 7 nodes and Player 2 connects two of the three remaining nodes with each other. Then, Player 1 should connect the large connected component to the last remaining node
and get an 8-2 split.
3. The connected component has 7 nodes and Player 2 makes a connection within it. Then, Player 1 also must connect two nodes within the component. Since the number of edges in a complete graph with
seven nodes is C^7_2=21, eventually Player 2 will be forced to make a move of type 1 or 2.
4. The connected component has 6 nodes and Player 2 connects it to one of the four remaining nodes. Then, Player 1 should make a connection within the connected seven nodes and reduce the game to
cases 1 to 3 above.
5. The connected component has 6 nodes and Player 2 connects two of the four remaining nodes. Then, Player 1 should connect the two remaining nodes with each other. The game is reduced to a 6-2-2
split which eventually will turn into either an 8-2 split, or a 6-4 split. In both cases Player 1 will win, as explained above.
Seven Letters Sequence
Find a seven-letter sequence to fill in each of the three empty spaces and form a meaningful sentence.
The ★★★★★★★ surgeon was ★★★ ★★★★ to operate, because there was ★★ ★★★★★.
The sequence is NOTABLE:
The NOTABLE surgeon was NOT ABLE to operate, because there was NO TABLE.
Glow and Shine
There is a property that applies to all words in the first list and to none in the words in the second list. What is it?
• GLOW, ALMOST, BIOPSY, GHOST, EMPTY, BEGIN
• SHINE, BARELY, VIVISECTION, APPARITION, VACANT, START
The words in the first list are called “Abecederian”, i.e. their letters are in alphabetical order.
The Four Oaks
A father left to his four sons this square field, with the instruction that they divide it into four pieces, each of the same shape and size, so that each piece of land contained one of the trees.
How did they manage it?
The solution is shown below.
Napoleon and the Policemen
Napoleon has landed on a deserted planet with only two policemen on it. He is traveling around the planet, painting a red line as he goes. When Napoleon creates a loop with red paint, the smaller of
the two encompassed areas is claimed by him. The policemen are trying to restrict the land Napoleon claims as much as possible. If they encounter him, they arrest him and take him away. Can you prove
that the police have a strategy to stop Napoleon from claiming more than 25% of the planet’s surface?
We assume that Napoleon and the police are moving at the same speed, making decisions in real time, and fully aware of everyone’s locations.
First, we choose an axis, so that Napoleon and the two policemen lie on a single parallel. Then, the strategy of the two policemen is to move with the same speed as Napoleon, keeping identical
latitudes as his at all times, and squeezing him along the parallel between them.
In order to claim 25% of the planet’s surface, Napoleon must travel at least 90°+90°=180° in total along the magnitudes. Therefore, during this time the policemen would travel 180° along the
magnitudes each and catch him. | {"url":"https://www.puzzleprime.com/blog/page/3/","timestamp":"2024-11-08T12:08:15Z","content_type":"text/html","content_length":"248401","record_id":"<urn:uuid:f28f271e-2a0e-4ea4-b31e-044a7eca3a8c>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00341.warc.gz"} |
Guide on statements | Articles
1. Problem name
□ Follow common title capitalization rules when writing English problem names. Example: «The Best of the Problems» instead of «The best of the problems». This is not applicable to Ukrainian
□ Avoid using any special characters in problem names (&\(^\wedge\)@* or similar).
□ Must not be too long (7 words or fewer).
□ Problem name must be unique in both English and Ukrainian (there should be no problem with the same name in the archive).
2. Limits
□ Default time limit: 2 s.
□ Default memory limit: 256 MB.
□ Time limit up to 7 s.
□ Memory limit up to 1024 MB, at least 128 MB, and must be a power of 2.
□ If time or memory limit is less than the default, make sure it’s well solvable in Java and Python.
3. Paragraphs
□ Separate paragraphs with an empty line in LaTeX; don’t use paragraph indentation or \par command.
□ The statement must not be a single large paragraph. Split the statement into multiple logical paragraphs for readability purposes.
4. Variables
□ By default, all statement and input variables must be enclosed in $, for example: $n$.
□ Variables must not be bold.
□ It’s recommended to keep variable names single-letter and lower-case.
□ Use common variable names when available (e. g. \(n\) for the number of vertices, \(m\) for the number of edges, \(k\) or \(q\) for the number of requests).
□ Avoid indexing variable names (e. g. \(i\) or \(j\)) as input variables.
5. Literals
□ Literal numbers must not be enclosed in $. Example: Vertices are numbered from 1 to $n$, inclusive. Exception: if the statement requires outputting -1 when the answer does not exist, treat -1
as a literal string (see below).
□ Literal strings must be enclosed in \texttt{}. For example, use \texttt{YES} when describing positive output string.
□ Do not use quotes for literal strings.
□ Numbering literals (or variables) must have a proper ending, e. g. the 2-nd, the $i$-th, etc. The same holds for the Ukrainian language
□ Array literals must not be enclosed in $, for example, [4, 7].
6. Formulas
□ All mathematical formulas must be enclosed in $, for example: $n < m$.
□ Don’t use * for multiplication. Use \cdot instead (e. g. $n \cdot m$).
□ Use \times for matrix dimension description, e. g. matrix of size $n \times m$.
□ Don’t use division sign /. Use fraction instead: $ \frac{a}{b}$.
□ Use proper LaTeX commands as inequality signs: \le for ’less than or equal’, \ge for ’greater than or equal’.
□ Large formulas must be placed on a separate line and centered by enclosing in $$ (double dollar-sign), for example: $$n < m$$.
□ Use ^ and _ for sup and sub, for example: $10^9$, $2^{nk}$, $a_i$.
□ Always use appropriate LaTeX commands when describing math symbols. Refer to this page for the full list of possible math commands.
□ Arrays (except literals) should be written as the following: $[a_1, a_2, \ldots, a_n]$.
7. Input
□ Always specify if the input token is an integer, float, string, or single character.
□ If the input contains floats, it must be clearly stated what format is expected.
□ Use the following style when describing input in English: The first line of the input contains a pair of integers ..., The following $m$ lines describe..., etc. Always use proper English
□ Do not leave input over-simplified, e. g. avoid One integer $n$. Instead, write: The first line of input contains one integer $n$ – number of bottles.
□ Each variable in the input must be briefly described.
8. Output
□ If it’s possible to have no answer, it should be clearly stated in the output section.
□ If there are multiple possible answers, it should be stated in the output section.
□ If it’s guaranteed that for the given input there’s always an answer, this should be stated in the output section.
□ If the answer contains float numbers, the output section must specify the allowed precision (absolute and/or relative).
□ If the answer has to be output modulo some number, it should be stated in the output section. It should be explicitly stated if the modulo number is prime.
□ Use the following style when describing output in English: Print a single integer denoting the minimum number ... etc. Always use proper English articles.
□ Do not leave output over-simplified, e. g. avoid Answer to the problem. Instead, write: In one line print single integer n – number of bottles.
9. Constraints
□ All constraints must be placed in the dedicated section of the statement. It’s allowed to copy constraints in the main statement section for emphasis purposes.
□ Combine constraints when appropriate: $1 \le n, m \le 1000$.
□ Each constraint inequality must be placed in a separate line of the statement (hence separated by an empty line in LaTeX).
□ Length of a string must be denoted as |s|.
□ Use proper inequality signs (as described in the Formulas section).
□ Always try to make constraint numbers as readable as possible. For example, write $1 \le n \le 2 \cdot 10^9$ instead of $1 \le n \le 2000000000$.
□ Use round or lucky numbers as constraints.
□ Each constraint line should end with a comma, the last one - with a dot. Any statement (except the first one) should start in a lower-case.
10. Lists
□ Use proper LaTeX commands for lists:
Unordered (bulletpoint) list:
\item One entry in the list
\item Another entry in the list
Ordered (numerated) list:
\item One entry in the list
\item Another entry in the list
11. Images
□ All images must be uploaded in the Resources section of an Algotester problem.
□ Images are referenced by resource name without file extension.
□ Images must be centered by default, with the image description below:
12. Samples
□ A statement must contain at least one sample.
□ Avoid too trivial samples (aka Boiko samples).
□ The first sample must be the first test case of the problem. The same holds for the following samples, if any.
□ There is an opportunity to create samples only for statement.
13. Notes
□ The Notes section must contain sample descriptions when appropriate.
□ The Notes section can also contain additional definitions when needed, but they must also be referenced in the main statement.
14. Miscellaneous
□ Parts of the statement that need special emphasis (non-obvious constraints or conditions, corrected parts of the statements, etc.) must be bold (enclosed in \textbf{}).
□ Use --- in Ukrainian and -- in English, never use - as a long dash.
□ Use spell-checker when writing English and Ukrainian statements. | {"url":"https://algotester.com/en/Article/Display/20121","timestamp":"2024-11-07T10:47:28Z","content_type":"text/html","content_length":"20909","record_id":"<urn:uuid:0d3ccec9-c043-47df-8630-1ed5488ff363>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00688.warc.gz"} |
5.6 Reduction of Order
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tdplotmult {\cacb }{\cosalpha }{\cosbeta } \tdplotmult {\cacg }{\cosalpha }{\cosgamma } \tdplotmult {\casg }{\cosalpha }{\singamma } \tdplotmult {\cacbsg }{\cacb }{\singamma } \tdplotmult {\cacbcg }
{\cacb }{\cosgamma } \pgfmathsetmacro {\raaeul }{\cacbcg -\sasg } \pgfmathsetmacro {\rabeul }{\sacbcg + \casg } \pgfmathsetmacro {\raceul }{-\sbcg } \pgfmathsetmacro {\rbaeul }{-\cacbsg -\sacg } \
pgfmathsetmacro {\rbbeul }{-\sacbsg + \cacg } \pgfmathsetmacro {\rbceul }{\sbsg } \pgfmathsetmacro {\rcaeul }{\casb } \pgfmathsetmacro {\rcbeul }{\sasb } \pgfmathsetmacro {\rcceul }{\cosbeta } }
ewcommand {\tdplottransformmainrot }[3]{\tdplotcalctransformmainrot \par \pgfmathsetmacro {\tdplotresx }{\raaeul * ##1 + \rabeul * ##2 + \raceul * ##3} \pgfmathsetmacro {\tdplotresy }{\rbaeul * ##1 +
\rbbeul * ##2 + \rbceul * ##3} \pgfmathsetmacro {\tdplotresz }{\rcaeul * ##1 + \rcbeul * ##2 + \rcceul * ##3} } ewcommand {\tdplottransformrotmain }[3]{\tdplotcalctransformrotmain \par \
pgfmathsetmacro {\tdplotresx }{\raaeul * ##1 + \rabeul * ##2 + \raceul * ##3} \pgfmathsetmacro {\tdplotresy }{\rbaeul * ##1 + \rbbeul * ##2 + \rbceul * ##3} \pgfmathsetmacro {\tdplotresz }{\rcaeul *
##1 + \rcbeul * ##2 + \rcceul * ##3} } ewcommand {\tdplottransformmainscreen }[3]{\tdplotcalctransformmainscreen \par \pgfmathsetmacro {\tdplotresx }{\raarot * ##1 + \rabrot * ##2 + \racrot * ##3} \
pgfmathsetmacro {\tdplotresy }{\rbarot * ##1 + \rbbrot * ##2 + \rbcrot * ##3} } ewcommand {\tdplotsetrotatedcoords }[3]{\pgfmathsetmacro {\tdplotalpha }{##1} \pgfmathsetmacro {\tdplotbeta }{##2} \
pgfmathsetmacro {\tdplotgamma }{##3} \tdplotcalctransformrotmain \par \tdplotmult {\raaeaa }{\raarot }{\raaeul } \tdplotmult {\rabeba }{\rabrot }{\rbaeul } \tdplotmult {\raceca }{\racrot }{\rcaeul }
\tdplotmult {\raaeab }{\raarot }{\rabeul } \tdplotmult {\rabebb }{\rabrot }{\rbbeul } \tdplotmult {\racecb }{\racrot }{\rcbeul } \tdplotmult {\raaeac }{\raarot }{\raceul } \tdplotmult {\rabebc }{\
rabrot }{\rbceul } \tdplotmult {\racecc }{\racrot }{\rcceul } \tdplotmult {\rbaeaa }{\rbarot }{\raaeul } \tdplotmult {\rbbeba }{\rbbrot }{\rbaeul } \tdplotmult {\rbceca }{\rbcrot }{\rcaeul } \
tdplotmult {\rbaeab }{\rbarot }{\rabeul } \tdplotmult {\rbbebb }{\rbbrot }{\rbbeul } \tdplotmult {\rbcecb }{\rbcrot }{\rcbeul } \tdplotmult {\rbaeac }{\rbarot }{\raceul } \tdplotmult {\rbbebc }{\
rbbrot }{\rbceul } \tdplotmult {\rbcecc }{\rbcrot }{\rcceul } \pgfmathsetmacro {\raarc }{\raaeaa + \rabeba + \raceca } \pgfmathsetmacro {\rabrc }{\raaeab + \rabebb + \racecb } \pgfmathsetmacro {\
racrc }{\raaeac + \rabebc + \racecc } \pgfmathsetmacro {\rbarc }{\rbaeaa + \rbbeba + \rbceca } \pgfmathsetmacro {\rbbrc }{\rbaeab + \rbbebb + \rbcecb } \pgfmathsetmacro {\rbcrc }{\rbaeac + \rbbebc +
\rbcecc } \tikzset {tdplot_rotated_coords/.append style={x={(\raarc cm,\rbarc cm)},y={(\rabrc cm,\rbbrc cm)},z={(\racrc cm,\rbcrc cm)}}}} ewcommand {\tdplotsetrotatedcoordsorigin }[1]{\tikzset
{tdplot_rotated_coords/.append style={shift=##1}}} ewcommand {\tdplotresetrotatedcoordsorigin }[0]{\tikzset {tdplot_rotated_coords/.append style={shift={(0,0,0)}}}} ewcommand {\
tdplotsetthetaplanecoords }[1]{\tdplotresetrotatedcoordsorigin \tdplotsetrotatedcoords {270 + ##1}{270}{0}} ewcommand {\tdplotsetrotatedthetaplanecoords }[1]{\tdplotsetrotatedcoords {\tdplotalpha }{\
tdplotbeta }{\tdplotgamma + ##1}\tikzset {tdplot_rotated_coords/.append style={y={(\raarc cm,\rbarc cm)},z={(\rabrc cm,\rbbrc cm)},x={(\racrc cm,\rbcrc cm)}}}} ewcommand {\tdplotsetcoord }[4]{\
tdplotsinandcos {\sinthetavec }{\costhetavec }{##3}\tdplotsinandcos {\sinphivec }{\cosphivec }{##4}\tdplotmult {\stcpv }{\sinthetavec }{\cosphivec }\tdplotmult {\stspv }{\sinthetavec }{\sinphivec }\
coordinate (##1) at (##2*(\stcpv ,\stspv ,\costhetavec )); \coordinate (##1xy) at (##2*(\stcpv ,\stspv ,0)); \coordinate (##1xz) at (##2*(\stcpv ,0,\costhetavec )); \coordinate (##1yz) at (##2*(0,\
stspv ,\costhetavec )); \coordinate (##1x) at (##2*(\stcpv ,0,0)); \coordinate (##1y) at (##2*(0,\stspv ,0)); \coordinate (##1z) at (##2*(0,0,\costhetavec )); } ewcommand {\tdplotsimplesetcoord }[4]
{\tdplotsinandcos {\sinthetavec }{\costhetavec }{##3}\tdplotsinandcos {\sinphivec }{\cosphivec }{##4}\tdplotmult {\stcpv }{\sinthetavec }{\cosphivec }\tdplotmult {\stspv }{\sinthetavec }{\sinphivec }
\coordinate (##1) at (##2*(\stcpv ,\stspv ,\costhetavec )); } ewcommand {\tdplotsetpolarplotrange }[4]{\pgfmathsetmacro {\tdplotlowerphi }{##3} \pgfmathsetmacro {\tdplotupperphi }{##4} \
pgfmathsetmacro {\tdplotlowertheta }{##1} \pgfmathsetmacro {\tdplotuppertheta }{##2} } ewcommand {\tdplotresetpolarplotrange }[0]{\pgfmathsetmacro {\tdplotlowerphi }{0} \pgfmathsetmacro {\
tdplotupperphi }{360} \pgfmathsetmacro {\tdplotlowertheta }{0} \pgfmathsetmacro {\tdplotuppertheta }{180} } ewcommand {\tdplotdosurfaceplot }[6]{\par \pgfmathsetmacro {extphi }{\curphi + \
tdplotsuperfudge *\viewphistep } \par \begin {scope}[opacity=1] \par \par \tdplotcheckdiff {extphi }{360}{\origviewphistep }{##2}{} \tdplotcheckdiff {extphi }{0}{\origviewphistep }{##2}{} \par \
tdplotcheckdiff {extphi }{90}{\origviewphistep }{##3}{} \tdplotcheckdiff {extphi }{450}{\origviewphistep }{##3}{} \end {scope} \par \foreach \curtheta in{\viewthetastart ,\viewthetainc ,...,\
viewthetaend } { \par \pgfmathsetmacro {\curlongitude }{90 -\curphi } \pgfmathsetmacro {\curlatitude }{90 -\curtheta } \par \ifthenelse {\equal {\leftright }{-1.0}}{\pgfmathsetmacro {\curphi }{\
curphi -\origviewphistep } }{} \par \pgfmathsetmacro {\tdplottheta }{mod(\curtheta ,360)} \pgfmathsetmacro {\tdplotphi }{mod(\curphi ,360)} \par \pgfmathparse {\tdplotphi <0} \ifthenelse {\
equal {\pgfmathresult }{1}}{ \pgfmathsetmacro {\tdplotphi }{\tdplotphi + 360} }{}\par \pgfmathparse {\tdplottheta >\tdplotuppertheta } \pgfmathsetmacro {\logictest }{1 -\pgfmathresult } \par \
pgfmathparse {\tdplottheta <\tdplotlowertheta } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \pgfmathsetmacro {\tdplottheta }{\tdplottheta + \viewthetastep } \
pgfmathparse {\tdplottheta >\tdplotuppertheta } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \pgfmathparse {\tdplottheta <\tdplotlowertheta } \pgfmathsetmacro
{\logictest }{\logictest * (1 -\pgfmathresult )} \par \pgfmathparse {\tdplotphi >\tdplotupperphi } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \pgfmathparse {\
tdplotphi <\tdplotlowerphi } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \pgfmathsetmacro {\tdplotphi }{\tdplotphi + \viewphistep } \par \pgfmathparse {\tdplotphi &#
x003C;0} \ifthenelse {\equal {\pgfmathresult }{1}}{ \pgfmathsetmacro {\tdplotphi }{\tdplotphi + 360} }{}\par \pgfmathparse {\tdplotphi >\tdplotupperphi } \pgfmathsetmacro {\logictest }{\
logictest * (1 -\pgfmathresult )} \par \pgfmathparse {\tdplotphi <\tdplotlowerphi } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \par \pgfmathsetmacro {\tdplottheta }
{\curtheta } \pgfmathsetmacro {\tdplotphi }{\curphi } \par \ifthenelse {\equal {##6}{parametricfill}}{\ifthenelse {\equal {\logictest }{1.0}}{\pgfmathsetmacro {\radius }{##1} \pgfmathsetmacro {\
tdplotr }{\radius *360} \par \pgfmathlessthan {\radius }{0} \pgfmathsetmacro {\phaseshift }{180 * \pgfmathresult } \par \pgfmathsetmacro {\colorarg }{##5} \pgfmathsetmacro {\colorarg }{\colorarg + \
phaseshift } \pgfmathsetmacro {\colorarg }{mod(\colorarg ,360)} \par \pgfmathlessthan {\colorarg }{0} \pgfmathsetmacro {\colorarg }{\colorarg + 360*\pgfmathresult } \par \pgfmathdivide {\colorarg }
{360} \definecolor {tdplotfillcolor}{hsb}{\pgfmathresult ,1,1} \color {tdplotfillcolor} }{}}{\pgfsetfillcolor {##5} } \pgfsetstrokecolor {##4} \par \ifthenelse {\equal {\leftright }{-1.0}}{\
pgfmathsetmacro {\curphi }{\curphi + \origviewphistep } }{} \par \ifthenelse {\equal {\logictest }{1.0}}{\pgfmathsetmacro {\radius }{abs(##1)} \pgfpathmoveto {\pgfpointspherical {\curlongitude }{\
curlatitude }{\radius }} \par \pgfmathsetmacro {\tdplotphi }{\curphi + \viewphistep } \pgfmathsetmacro {\radius }{abs(##1)} \pgfpathlineto {\pgfpointspherical {\curlongitude -\viewphistep }{\
curlatitude }{\radius }} \par \pgfmathsetmacro {\tdplottheta }{\curtheta + \viewthetastep } \pgfmathsetmacro {\radius }{abs(##1)} \pgfpathlineto {\pgfpointspherical {\curlongitude -\viewphistep }{\
curlatitude -\viewthetastep }{\radius }} \par \pgfmathsetmacro {\tdplotphi }{\curphi } \pgfmathsetmacro {\radius }{abs(##1)} \pgfpathlineto {\pgfpointspherical {\curlongitude }{\curlatitude -\
viewthetastep }{\radius }} \pgfpathclose \par \pgfusepath {fill,stroke} }{} } } ewcommand {\tdplotshowargcolorguide }[4]{ \par \pgfmathsetmacro {\tdplotx }{##1} \pgfmathsetmacro {\tdploty }{##2} \
pgfmathsetmacro {\tdplothuestep }{5} \pgfmathsetmacro {\tdplotxsize }{##3} \pgfmathsetmacro {\tdplotysize }{##4} \par \pgfmathsetmacro {\tdplotyscale }{\tdplotysize /360} \par \foreach \tdplotphi in
{0,\tdplothuestep ,...,360} { \pgfmathdivide {\tdplotphi }{360} \definecolor {tdplotfillcolor}{hsb}{\pgfmathresult ,1,1} \color {tdplotfillcolor} \par \pgfmathsetmacro {\tdplotstarty }{\tdploty + \
tdplotphi * \tdplotyscale } \pgfmathsetmacro {\tdplotstopy }{\tdplotstarty + \tdplothuestep * \tdplotyscale } \pgfmathsetmacro {\tdplotstartx }{\tdplotx } \pgfmathsetmacro {\tdplotstopx }{\tdplotx +
\tdplotxsize } \filldraw [tdplot_screen_coords] (\tdplotstartx ,\tdplotstarty ) rectangle (\tdplotstopx ,\tdplotstopy ); } \par \pgfmathsetmacro {\tdplotstopy }{\tdploty + (360+\tdplothuestep )*\
tdplotyscale } \pgfmathsetmacro {\tdplotstopx }{\tdplotx + \tdplotxsize } \par \draw [tdplot_screen_coords] (\tdplotx ,\tdploty ) rectangle (\tdplotstopx ,\tdplotstopy ); \par ode
[tdplot_screen_coords,anchor=west,xshift=5pt] at (\tdplotstopx ,\tdploty ) {0}; ode [tdplot_screen_coords,anchor=west,xshift=5pt] at (\tdplotstopx ,\tdplotstopy ) {2\pi }; \par \pgfmathsetmacro {\
tdplotstopy }{\tdploty + (360+\tdplothuestep )/2*\tdplotyscale } ode [tdplot_screen_coords,anchor=west,xshift=5pt] at (\tdplotstopx ,\tdplotstopy ) {\pi }; } ewcommand {\tdplotgetpolarcoords }[3]{\
pgfmathsetmacro {\vxcalc }{##1} \pgfmathsetmacro {\vycalc }{##2} \pgfmathsetmacro {\vzcalc }{##3} \pgfmathsetmacro {\vcalc }{ sqrt((\vxcalc )^2 + (\vycalc )^2 + (\vzcalc )^2) } \par \pgfmathsetmacro
{\vxycalc }{ sqrt((\vxcalc )^2 + (\vycalc )^2) } \par \pgfmathsetmacro {\tdplotrestheta }{asin(\vxycalc /\vcalc )} \pgfmathparse {\vzcalc <0} \ifthenelse {\equal {\pgfmathresult }{1}}{\
pgfmathsetmacro {\tdplotrestheta }{180 -\tdplotrestheta } } {} \ifthenelse {\equal {\vxcalc }{0.0}}{\pgfmathparse {\vycalc <0} \ifthenelse {\equal {\pgfmathresult }{1}}{\pgfmathsetmacro {\
tdplotresphi }{270} } {\pgfmathparse {\vycalc >0} \ifthenelse {\equal {\pgfmathresult }{1}}{\pgfmathsetmacro {\tdplotresphi }{90} } {\pgfmathsetmacro {\tdplotresphi }{0} } } } {\
pgfmathsetmacro {\tdplotresphi }{atan(\vycalc /\vxcalc )} \pgfmathparse {\vxcalc <0} \ifthenelse {\equal {\pgfmathresult }{1}}{\pgfmathsetmacro {\tdplotresphi }{\tdplotresphi +180} } { } \par
\pgfmathparse {\tdplotresphi <0} \ifthenelse {\equal {\pgfmathresult }{1}}{\pgfmathsetmacro {\tdplotresphi }{\tdplotresphi +360} } {} } } ewcommand {\vec }[1]{\mathbf {##1}} ewcommand {\RR }
[0]{\mathbb {R}} ewcommand {\dfn }[0]{\textit } ewcommand {\dotp }[0]{\cdot } ewcommand {\id }[0]{\text {id}} ewcommand {orm }[1]{\left \lVert ##1\right \rVert } ewcommand {\dst }[0]{\displaystyle }
ewcommand {\MHPrecedingSpacesOff }[0]{\MH_let:NwN \@xargdef \MH_nospace_xargdef:nwwn } ewcommand {\MHPrecedingSpacesOn }[0]{\MH_let:NwN \@xargdef \MH_kernel_xargdef:nwwn } ewcommand {\mathtoolsset }
[1]{\setkeys {\MT_options_name: }{##1}} ewcommand {ewtagform }[1]{\@ifundefined {MT_tagform_##1:n}{\@ifnextchar [{\MT_define_tagform:nwnn ##1}{\MT_define_tagform:nwnn ##1[]}}{\PackageError
{mathtools}{The tag form ‘##1’is already defined\MessageBreak You probably want to look up \@backslashchar renewtagform instead}{I will just ignore your wish for now.}}} ewcommand {\renewtagform }[1]
{\@ifundefined {MT_tagform_##1:n}{\PackageError {mathtools}{The tag form ‘##1’is not defined\MessageBreak You probably want to look up \@backslashchar newtagform instead}{I will just ignore your wish
for now.}}{\@ifnextchar [{\MT_define_tagform:nwnn ##1}{\MT_define_tagform:nwnn ##1[]}}} ewcommand {\usetagform }[1]{\@ifundefined {MT_tagform_##1:n}{\PackageError {mathtools}{You have chosen the tag
form ‘##1’\MessageBreak but it appears to be undefined}{I will use the default tag form instead.}\@namedef {tagform@}{\@nameuse {MT_tagform_default:n}}}{\@namedef {tagform@}{\@nameuse {MT_tagform_##
1:n}}}\MH_if_boolean:nT {show_only_refs}{\MH_let:NwN \MT_prev_tagform:n \tagform@ \def \tagform@ ####1{\MT_extended_tagform:n {####1}}}} ewcommand {\refeq }[1]{\textup {\ref {##1}}} ewcommand {\
MT@newlabel }[1]{\global \@namedef {MT_r_##1}{}} ewcommand {\MT_showonlyrefs_true: }[0]{\MH_if_boolean:nF {show_only_refs}{\MH_set_boolean_T:n {show_only_refs}\MH_let:NwN \MT_incr_eqnum: \incr@eqnum
\MH_let:NwN \incr@eqnum \@empty \MH_let:NwN \MT_array_parbox_restore: \@arrayparboxrestore \@xp \def \@xp \@arrayparboxrestore \@xp {\@arrayparboxrestore \MH_let:NwN \incr@eqnum \@empty }\MH_let:NwN
\MT_prev_tagform:n \tagform@ \MH_let:NwN \MT_eqref:n \eqref \MH_let:NwN \MT_refeq:n \refeq \MH_let:NwN \MT_maketag:n \maketag@@@ \MH_let:NwN \maketag@@@ \MT_extended_maketag:n \def \tagform@ ####1{\
MT_extended_tagform:n {####1}}\MH_let:NwN \eqref \MT_extended_eqref:n \MH_let:NwN \refeq \MT_extended_refeq:n }} ewcommand {onumber }[0]{\if@eqnsw \MH_if_meaning:NN \incr@eqnum \@empty \
MH_if_boolean:nF {show_only_refs}{\addtocounter {equation}\m@ne }\MH_fi: \MH_fi: \MH_let:NwN \print@eqnum \@empty \MH_let:NwN \incr@eqnum \@empty \global \@eqnswfalse } ewcommand {oeqref }[1]{\
@bsphack \@for \@tempa :=##1\do {\@safe@activestrue \edef \@tempa {\expandafter \@firstofone \@tempa }\@ifundefined {r@\@tempa }{\protect \G@refundefinedtrue \@latex@warning {Reference ‘\@tempa ’on
page \thepage \space undefined (\string oeqref )}}{}\if@filesw \protected@write \@auxout {}{\string \MT@newlabel {\@tempa }}\fi \@safe@activesfalse }\@esphack } ewcommand {\reserved@a }[0]{}
ewcommand {\reserved@a }[0]{} ewcommand {\underbracket }[0]{\@ifnextchar [{\MT_underbracket_I:w }{\MT_underbracket_I:w [\l_MT_bracketheight_fdim ]}} ewcommand {\overbracket }[0]{\@ifnextchar [{\
MT_overbracket_I:w }{\MT_overbracket_I:w [\l_MT_bracketheight_fdim ]}} ewcommand {\lparen }[0]{(} ewcommand {\rparen }[0]{)} ewcommand {\ordinarycolon }[0]{:} ewcommand {\MT_test_for_tcb_other:nnnnn
}[1]{\MH_if:w t##1\relax \expandafter \MH_use_choice_i:nnnn \MH_else: \MH_if:w c##1\relax \expandafter \expandafter \expandafter \MH_use_choice_ii:nnnn \MH_else: \MH_if:w b##1\relax \expandafter \
expandafter \expandafter \expandafter \expandafter \expandafter \expandafter \MH_use_choice_iii:nnnn \MH_else: \expandafter \expandafter \expandafter \expandafter \expandafter \expandafter \
expandafter \MH_use_choice_iv:nnnn \MH_fi: \MH_fi: \MH_fi: } ewcommand {\MT_start_mult:N }[1]{\MT_test_for_tcb_other:nnnnn {##1}{\MH_let:NwN \MT_next: \vtop }{\MH_let:NwN \MT_next: \vcenter }{\
MH_let:NwN \MT_next: \vbox }{\PackageError {mathtools}{Invalid position specifier. I’ll try to recover with ‘c’}\@ehc }\collect@body \MT_mult_internal:n } ewcommand {\MT_mult_internal:n }[1]{\
MH_if_boolean:nF {outer_mult}{\alignedspace@left }\MT_next: \bgroup \Let@ \def \l_MT_multline_lastline_fint {0}\chardef \dspbrk@context \@ne \restore@math@cr \MH_let:NwN \math@cr@@ \
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l_MT_multwidth_dim <\l_MT_multline_measure_fdim \MH_setlength:dn \l_MT_multwidth_dim {\l_MT_multline_measure_fdim }\fi \MH_set_boolean_T:n {mult_firstline}\MH_if_num:w \
l_MT_multline_lastline_fint =\@ne \MH_let:NwN \math@cr@@ \MT_mult_firstandlast_mathcr:w \MH_fi: \ialign \bgroup \hfil \strut@ \m@th \displaystyle {}####\hfil \crcr \hfilneg ##1} ewcommand {\
MT_measure_mult:n }[1]{\begingroup \measuring@true \g_MT_multlinerow_int \@ne \MH_let:NwN \label \MT_gobblelabel:w \MH_let:NwN \tag \gobble@tag \setbox \z@ \vbox {\ialign {\strut@ \m@th \displaystyle
{}####\crcr ##1\crcr }}\xdef \l_MT_multline_measure_fdim {\the \wdz@ }\advance \g_MT_multlinerow_int \m@ne \xdef \l_MT_multline_lastline_fint {umber \g_MT_multlinerow_int }\endgroup \
g_MT_multlinerow_int \@ne } ewcommand {\MultlinedHook }[0]{\renewenvironment {subarray}[1]{\vcenter \bgroup \Let@ \restore@math@cr \default@tag \let \math@cr@@ \AMS@math@cr@@ \baselineskip \fontdimen
10\scriptfont \tw@ \advance \baselineskip \fontdimen 12\scriptfont \tw@ \lineskip \thr@@ \fontdimen 8\scriptfont \thr@@ \lineskiplimit \lineskip \ialign \bgroup \ifx c####1\hfil \fi \m@th \
scriptstyle ########\hfil \crcr }{\crcr \egroup \egroup }} ewcommand {\MT_delim_default_inner_wrappers:n }[1]{\@namedef {MT_delim_\MH_cs_to_str:N ##1_star_wrapper:nnn}####1####2####3{\mathopen {}\
mathclose \bgroup ####1####2\aftergroup \egroup ####3}\@namedef {MT_delim_\MH_cs_to_str:N ##1_nostarscaled_wrapper:nnn}####1####2####3{\mathopen {####1}####2\mathclose {####3}}\@namedef {MT_delim_\
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1_##2_wrapper:nnn}{\PackageError {mathtools}{Wrapper not found for \string ##1 and option ’##2’.\MessageBreak Either \string ##1 is not defined,or you are using the \MessageBreak ’nostar’option,which
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3}}} ewcommand {\DeclarePairedDelimiter }[3]{\@ifdefinable {##1}{\MT_delim_default_inner_wrappers:n {##1}\@namedef {MT_delim_\MH_cs_to_str:N ##1_star:}####1{\@nameuse {MT_delim_\MH_cs_to_str:N ##
1_star_wrapper:nnn}{\left ##2}{####1}{\right ##3}}\@xp \@xp \@xp ewcommand \@xp \csname MT_delim_\MH_cs_to_str:N ##1_nostar:\endcsname [2][\\@gobble]{\def \@tempa {\\@gobble}\def \@tempb {####1}\ifx
\@tempa \@tempb \@nameuse {MT_delim_\MH_cs_to_str:N ##1_nostarnonscaled_wrapper:nnn}{##2}{####2}{##3}\else \MT_etb_ifblank:nnn {####1}{\@nameuse {MT_delim_\MH_cs_to_str:N ##
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3}}\fi }\DeclareRobustCommand {##1}{\@ifstar {\@nameuse {MT_delim_\MH_cs_to_str:N ##1_star:}}{\@nameuse {MT_delim_\MH_cs_to_str:N ##1_nostar:}}}}} ewcommand {\MT_delim_inner_generator:nnnnnnn }[7]{\
@xp \@xp \@xp ewcommand \@xp \csname MT_delim_\MH_cs_to_str:N ##1_nostar_inner:\endcsname [##2]{##3\def \@tempa {\@MHempty }\@xp \def \@xp \@tempb \@xp {\delimsize }\ifx \@tempa \@tempb \@nameuse
{MT_delim_\MH_cs_to_str:N ##1_nostarnonscaled_wrapper:nnn}{##4}{##7}{##5}\else \MT_etb_ifdefempty_x:nnn {\delimsize }{\@nameuse {MT_delim_\MH_cs_to_str:N ##1_nostarnonscaled_wrapper:nnn}{##4}{##7}{##
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delimsize r\endcsname ##5}}\fi ##6\endgroup }} ewcommand {ewcases }[6]{ewenvironment {##1}{\MT_start_cases:nnnn {##2}{##3}{##4}{##5}}{\MH_end_cases: \right ##6}} ewcommand {\renewcases }[6]{\
renewenvironment {##1}{\MT_start_cases:nnnn {##2}{##3}{##4}{##5}}{\MH_end_cases: \right ##6}} ewcommand {\dcases }[0]{\MT_start_cases:nnnn {\quad }{\m@th \displaystyle {####}\hfil }{\m@th \
displaystyle {####}\hfil }{\lbrace }} ewcommand {\dcases* }[0]{\MT_start_cases:nnnn {\quad }{\m@th \displaystyle {####}\hfil }{{####}\hfil }{\lbrace }} ewcommand {\rcases }[0]{\MT_start_cases:nnnn {\
quad }{\m@th {####}\hfil }{\m@th {####}\hfil }{.}} ewcommand {\rcases* }[0]{\MT_start_cases:nnnn {\quad }{\m@th {####}\hfil }{{####}\hfil }{.}} ewcommand {\drcases }[0]{\MT_start_cases:nnnn {\quad }
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We explore a technique for reducing a second order nonhomgeneous linear differential equation to first order when we know a nontrivial solution to the complementary homogeneous equation.
Reduction of Order
In this section we give a method for finding the general solution of
if we know a nontrivial solution
of the complementary equation The method is called
reduction of order
because it reduces the task of solving (
) to solving a first order equation. Unlike the method of undetermined coefficients, it does not require
, and
to be constants, or
to be of any special form.
By now you shouldn’t be surprised that we look for solutions of (eq:5.6.1) in the form
is to be determined so that
satisfies (
). Substituting (
) and into (
) yields Collecting the coefficients of
, and
yields However, the coefficient of
is zero, since
satisfies (
). Therefore (
) reduces to with (It isn’t worthwhile to memorize the formulas for
!) Since (
) is a linear first order equation in
, we can solve it for
by variation of parameters as in Module
, integrate the solution to obtain
, and then obtain
from (
As a byproduct of item:5.6.1a, find a fundamental set of solutions of (eq:5.6.7).
, then
, so Therefore
is a solution of (
) if and only if which is a first order equation in
. We rewrite it as To focus on how we apply variation of parameters to this equation, we temporarily write
, so that (
) becomes We leave it to you to show (by separation of variables) that
is a solution of the complementary equation for (
). By applying variation of parameters as in Module
, we can now see that every solution of (
) is of the form Since
is a solution of (
) if and only if Integrating this yields Therefore the general solution of (
) is
item:5.6.1b By letting $C_1=C_2=0$ in (eq:5.6.10), we see that $y_{p_1}=x+1$ is a solution of (eq:5.6.6). By letting $C_1=2$ and $C_2=0$, we see that $y_{p_2}=x+1+x^2e^x$ is also a solution of (
eq:5.6.6). Since the difference of two solutions of (eq:5.6.6) is a solution of (eq:5.6.7), $y_2=y_{p_1}-y_{p_2}=x^2e^x$ is a solution of (eq:5.6.7). Since $y_2/y_1$ is nonconstant and we already
know that $y_1=e^x$ is a solution of (eq:5.6.6), Theorem thmtype:5.1.6 implies that $\{e^x,x^2e^x\}$ is a fundamental set of solutions of (eq:5.6.7).
Although (eq:5.6.10) is a correct form for the general solution of (eq:5.6.6), it’s silly to leave the arbitrary coefficient of $x^2e^x$ as $C_1/2$ where $C_1$ is an arbitrary constant. Moreover,
it’s sensible to make the subscripts of the coefficients of $y_1=e^x$ and $y_2=x^2e^x$ consistent with the subscripts of the functions themselves. Therefore we rewrite (eq:5.6.10) as by simply
renaming the arbitrary constants. We’ll also do this in the next two examples, and in the answers to the exercises.
Find the general solution of given that $y_1=x$ is a solution of the complementary equation As a byproduct of this result, find a fundamental set of solutions of (eq:5.6.11).
, then
, so Therefore
is a solution of (
) if and only if which is a first order equation in
. We rewrite it as To focus on how we apply variation of parameters to this equation, we temporarily write
, so that (
) becomes We leave it to you to show by separation of variables that
is a solution of the complementary equation for (
). By variation of parameters, every solution of (
) is of the form Since
is a solution of (
) if and only if Integrating this yields Therefore the general solution of (
) is Reasoning as in the solution of Example
, we conclude that
form a fundamental set of solutions for (
As we explained above, we rename the constants in (eq:5.6.15) and rewrite it as
item:5.6.2b Differentiating (eq:5.6.16) yields
in (
) and (
) and imposing the initial conditions
yields Solving these equations yields
. Therefore the solution of (
) is
Using reduction of order to find the general solution of a homogeneous linear second order equation leads to a homogeneous linear first order equation in $u'$ that can be solved by separation of
variables. The next example illustrates this.
Find the general solution and a fundamental set of solutions of given that
is a solution.
, so Therefore
is a solution of (
) if and only if Separating the variables
yields so Therefore so the general solution of (
) is which we rewrite as Therefore
is a fundamental set of solutions of (
Text Source
Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA) | {"url":"https://ximera.osu.edu/ode/main/reductionOfOrder/reductionOfOrder","timestamp":"2024-11-03T03:32:16Z","content_type":"text/html","content_length":"133551","record_id":"<urn:uuid:3ff8c90d-8221-4248-a422-92e0fa79afe4>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00440.warc.gz"} |
How to Calculate Full-Cost Pricing | Bizfluent
Finance Your Business
How to Calculate Full-Cost Pricing
When introducing a product to the market, a company must first figure out how to fairly price the item in order to maximize sales and profits. One method used to calculate pricing is the full-cost
technique, which adds in all the expenses associated with making a product and the profit margin a company would like to make on the item.
Full-cost pricing is a method of price setting that involves adding the costs of making and selling a product along with a markup percentage to determine the price of a product.
Full-cost pricing is one of many ways for a company to determine the selling price of a product. To use this pricing method, you add together all costs of creating and selling the product (including
material costs, labor costs, selling and administrative costs and overhead costs) and a markup percentage to allow for a profit margin. You then divide this number, which should include the price of
all units produced, by the number of units you expect to sell.
The full-cost calculation is simple. It looks like: (total production costs + selling and administrative costs + markup) ÷ the number of units expected to sell.
Consider an example of how the full-cost system works. Tom's Treat Toys is trying to figure out a fair price to charge for their finest fun figures. They decide they want to make a profit margin of
50 percent and sell 50,000 units. The company spends $2 million making all of their products and $600,000 on their total company sales and administration costs. The finest fun figures take up 25
percent of their manufacturing floor and 25 percent of their overall sales and administration costs. That means the total production cost for the finest fun figures is $500,000, and the total sales
and administration cost is $150,000.
The total cost of making and selling the product comes out to $650,000, which means that a 50 percent profit margin would be $325,000. When the profit margin is added to the total costs, the total
comes out to $975,000. Divide that number by the number of units (50,000), and you'll get the total cost of the product per unit, which comes out to $19.50.
Another common pricing method that is very similar to the full-cost principle is absorption pricing. Whereas full-cost pricing simplifies the numbers by using the same formula to allocate costs to a
specific product, absorption pricing is more precise and more complex.
In the example above where the company allocated 25 percent of its factory floor and sales/admin expenses toward the finest fun figures, absorption pricing would treat each cost more precisely. For
example, they might allocate 25 percent of the factory rent toward the making of the finest fun figures since they take up that space, but their utility costs may be divided differently if one
product takes more water or more electricity to create. Similarly, if a product has a higher marketing budget but lower research and development costs associated with it, these costs will be added in
based on how the company allocates these resources rather than just simplifying the overall sales and administrative costs into one number.
Full-cost pricing is not a good technique when determining what to charge for a product sold in a competitive market or a market that already has standardized pricing. That is because it does not
take into account the prices charged by competitors, it does not allow management the opportunity to reduce prices in order to grow market share and it does not factor in the value of the product to
the consumer. It is also not a good option for a company that produces many products, as the pricing formula can be difficult to use when you have to figure out how many resources are to be allotted
to one product out of dozens.
This technique can be very useful when a product or service is based on the requirements of a customer. In these cases, it can be useful for setting long-term prices that will be high enough to
guarantee a profit after all expenses. For example, if a company develops a new software package that is unlike anything on the market, the company will need to figure out pricing in a market where
there is no competition and the pricing has not yet been established.
The greatest benefits to full-cost pricing are that it is fair, simple and will likely turn a profit. The pricing is easily justifiable because the prices are based on actual costs. When
manufacturing costs go up, it is also easy to justify increasing prices without angering customers. If a product does have competitors and they take the same approach to pricing, this can also result
in price stability as long as the competitors have similar costs.
Full-cost pricing is also fairly easy to calculate as long as the company doesn't sell too many products to make figuring out the costs per item impractical. In fact, full-cost pricing can actually
allow junior employees to determine the cost of a product since it is based solely on formulas.
Finally, by taking all the expenses of a product into account and figuring in the profit margin a company would like to see, it can guarantee that the product will earn a profit as long as the
calculations are correct.
There are some disadvantages to using full-cost pricing, though. As previously stated, for example, this pricing strategy is not good to use in a competitive market because it ignores the prices set
by the competition. Similarly, it ignores what buyers are willing to pay, so the price could be too high or too low in comparison to what the company could be charging, resulting in either lost
potential profits or lost potential sales.
By allowing for any possible product costs in the calculations, this pricing method also provides no incentive for designers and engineers to create a product in a less-costly manner. If costs
increase, then selling prices will also increase accordingly, and employees may have little incentive to reduce costs internally rather than just passing them on to the consumer.
Another major problem with full-cost pricing is that it only takes into account expense estimates and sales volume estimates, both of which could be incorrect. This could result in a completely wrong
pricing strategy. For example, if you account for 5,000 units being sold and only 2,000 units are sold, you may lose money on the item depending on the profit margin you set. It can also be difficult
to figure out an accurate apportionment of costs if a company sells more than one product.
For many companies, full-cost pricing is too simplistic, failing to take into account the actual costs of all expenses and how they are allocated to one product over another. This is why absorption
pricing is sometimes preferable because it further breaks down the cost of all expenses and divides them more accurately by all the products the company sells. | {"url":"https://bizfluent.com/how-8329998-calculate-full-cost-pricing.html","timestamp":"2024-11-11T23:26:22Z","content_type":"text/html","content_length":"150913","record_id":"<urn:uuid:d388ff07-13a8-4fda-9cd1-3845a0e3305e>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00089.warc.gz"} |
xom-interest - RE: [XOM-interest] Possible ArrayList optimization
xom-interest AT lists.ibiblio.org
Subject: XOM API for Processing XML with Java
List archive
• From: "New, Cecil (GEAE)" <cecil.new AT ae.ge.com>
• To: "'dirk bergstrom'" <dirk AT juniper.net>, "'XOM-interest AT lists.ibiblio.org'" <XOM-interest AT lists.ibiblio.org>
• Cc:
• Subject: RE: [XOM-interest] Possible ArrayList optimization
• Date: Tue, 13 May 2003 16:25:55 -0400
I don't think anyone has suggested a heuristic approach (perhaps because it
is a
bad idea!). You could keep track of the average size and adjust as you go
-----Original Message-----
From: dirk bergstrom [
mailto:dirk AT juniper.net
Sent: Monday, May 12, 2003 10:52 PM
To: Elliotte Rusty Harold
Cc: xom-interest
Subject: Re: [XOM-interest] Possible ArrayList optimization
On 05/12/2003 07:14 PM, Elliotte Rusty Harold was heard to exclaim:
>Is there any reason why an invocation of ArrayList.trimToSize (),
I hadn't thought of that. How expensive is trimToSize()?
fairly cheap. about the same cost as extending the array. here's the source:
public void trimToSize() {
int oldCapacity = elementData.length;
if (size < oldCapacity) {
Object oldData[] = elementData;
elementData = new Object[size];
System.arraycopy(oldData, 0, elementData, 0, size);
and you might be able to make that up by not having to
constantly grow the arrays as you build the document. Just start them
fairly large, and then chop it down when you're finished.
well, you probably don't want to start too big, since as you build a
document you'll have un-trimmed ArrayLists for many of the nodes.
Dirk Bergstrom dirk AT juniper.net
Juniper Networks Inc., Computer Geek
Tel: 408.745.3182 Fax: 408.745.8905
XOM-interest mailing list
XOM-interest AT lists.ibiblio.org
• Re: [XOM-interest] Possible ArrayList optimization , (continued)
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Understanding The Difference Between Correlation And Autocorrelation - Coloringfolder.com
Understanding the Difference Between Correlation and Autocorrelation
Have you ever wondered what the difference is between correlation and autocorrelation? Well, you’re not alone. Many people confuse the two terms, and understandably so, as they both involve measuring
relationships between variables. However, there is a key difference. Correlation measures the degree to which two variables are related to each other, whereas autocorrelation measures the degree to
which a variable is related to itself over time.
To put it simply, correlation measures the relationship between two distinct variables, while autocorrelation measures the relationship between a variable and its own past values. Autocorrelation is
common in time series data, where variables change over time. For example, if we were measuring the price of a stock over time, we would expect some degree of autocorrelation – that is, today’s price
is likely to be related to yesterday’s price.
Understanding the difference between correlation and autocorrelation is important when analyzing data. By correctly identifying which type of relationship exists, researchers can make more accurate
predictions and draw more meaningful conclusions. So, the next time you come across these terms, remember that correlation measures the relationship between two distinct variables, while
autocorrelation measures the relationship between a variable and its own past values over time.
Understanding Correlation and Autocorrelation
Correlation and autocorrelation are two concepts that are often used in statistical analysis to measure the association between two variables. However, these two concepts have some significant
differences that can affect their interpretation and application. In this article, we will discuss the differences between correlation and autocorrelation, and how they are calculated and
• Correlation is a statistical measure that shows the extent to which two variables are related to each other.
• Correlation can be positive, negative, or zero. Positive correlation means that as one variable increases, the other variable also increases. Negative correlation means that as one variable
increases, the other variable decreases. Zero correlation means that there is no relationship between the two variables.
• Correlation can range from -1 to +1, with -1 indicating a perfect negative correlation, +1 indicating a perfect positive correlation, and 0 indicating no correlation.
• Correlation can be calculated using different methods, such as Pearson correlation, Spearman correlation, and Kendall correlation.
• Pearson correlation is the most widely used method for calculating correlation and is based on the assumption of normal distribution of data and linear relationships between variables.
Autocorrelation is a statistical concept that measures the correlation between a variable and a lagged version of itself. In other words, autocorrelation measures the degree of similarity between
observations at different time points.
• Autocorrelation can be positive, negative, or zero. Positive autocorrelation means that when a variable is higher than its mean value at a certain time point, it is more likely to be higher than
its mean value at the next time point. Negative autocorrelation means that when a variable is lower than its mean value at a certain time point, it is more likely to be lower than its mean value
at the next time point. Zero autocorrelation means that there is no relationship between the variable and its lagged version.
• Autocorrelation can indicate the presence of a trend or a cyclic pattern in the data.
• Autocorrelation can be measured using different methods, such as the Durbin-Watson test, the Ljung-Box test, and the autocorrelation function (ACF).
• The autocorrelation function is a graphical tool that shows the correlation between a variable and its lagged version at different lags. It can help identify the presence of significant lags and
the type of autocorrelation (positive, negative, or zero).
Correlation and autocorrelation are two important statistical concepts that can help understand the relationship between variables and the patterns in data over time. While correlation measures the
association between two variables, autocorrelation measures the similarity between a variable and its lagged version. Understanding the differences between these two concepts can help scientists and
analysts choose the appropriate method for their analysis and avoid misinterpretation of results.
Attribute Correlation Autocorrelation
Definition Measures the association between two variables Measures the similarity between a variable and its lagged version
Range From -1 to +1 From -1 to +1
Interpretation Positive, negative, or zero correlation Positive, negative, or zero autocorrelation
Methods Pearson, Spearman, Kendall Durbin-Watson, Ljung-Box, ACF
The table summarizes the main differences between correlation and autocorrelation.
Pearson Correlation Coefficient
When discussing correlation, one of the most commonly used forms is the Pearson Correlation Coefficient (PCC), which measures the linear relationship between two variables. This coefficient values
range from -1 to 1, with -1 indicating a perfect negative correlation, 0 indicating no correlation, and 1 indicating a perfect positive correlation.
• The calculation of PCC is based on the degree to which the data points fall on a straight line, with the correlation becoming stronger as the points move closer to the line.
• PCC is only valid when both variables have a normal distribution. If the data does not have a normal distribution, it may be necessary to use a non-parametric correlation method instead.
• It is essential to note that correlation does not prove causation. Even if there is a strong correlation between two variables, it does not necessarily mean that one variable is causing the other
variable to change.
While correlation measures the relationship between two variables, autocorrelation (also known as serial correlation) measures the correlation between a variable and lags of itself. In other words,
it measures how closely a variable is related to its past values.
Autocorrelation is often associated with time series data, where a variable’s value at time t is dependent on its value at time t-1, t-2, etc. The autocorrelation coefficient (ACC) measures the
strength of this relationship and ranges from -1 to 1, with 0 indicating no autocorrelation.
Autocorrelation can affect statistical analyses by violating the assumption of independence, leading to incorrect p-values, standard errors, and confidence intervals. Understanding the
autocorrelation patterns in the data is crucial in avoiding bias in statistical analyses.
ACC Interpretation
-1 Perfect negative autocorrelation
-0.5 to -0.9 Strong negative autocorrelation
-0.3 to -0.5 Moderate negative autocorrelation
-0.1 to -0.3 Weak negative autocorrelation
0 to 0.1 No autocorrelation
0.1 to 0.3 Weak positive autocorrelation
0.3 to 0.5 Moderate positive autocorrelation
0.5 to 0.9 Strong positive autocorrelation
1 Perfect positive autocorrelation
Understanding the differences between correlation and autocorrelation is critical in statistical analyses. While they both measure the relationship between variables, they do so in different ways and
serve different purposes.
Spearman Correlation Coefficient
The Spearman correlation coefficient is a statistical measure that assesses the strength and direction of the monotonic relationship between two variables. It is used when the data being analyzed is
not normally distributed or when it contains outliers.
The Spearman correlation coefficient ranges from -1 to 1. A value of -1 indicates a perfect negative correlation, a value of 1 indicates a perfect positive correlation, and a value of 0 indicates no
The main difference between the Spearman correlation coefficient and the Pearson correlation coefficient is that the Spearman correlation coefficient is based on the ranked data while the Pearson
correlation coefficient is based on the original data.
• The Spearman correlation coefficient is more robust than the Pearson correlation coefficient to outliers and non-normality in the data.
• The Spearman correlation coefficient does not assume linearity in the relationship between the two variables.
• The Spearman correlation coefficient is more appropriate when the data is ordinal rather than continuous.
The formula for calculating the Spearman correlation coefficient is as follows:
r[s] = 1 – 6Σd[i]^2 / n(n^2-1)
• r[s] is the Spearman correlation coefficient
• d[i] is the difference between the ranks of the two variables
• n is the sample size
X Y X Rank (R[x]) Y Rank (R[y]) d = R[x] – R[y] d^2
10 4 4 7 -3 9
20 10 5 6 -1 1
5 6 2 4 -2 4
Mean 13.17 4.33 4.5 31
Using the example table above, we can calculate the Spearman correlation coefficient between X and Y:
r[s] = 1 – 6Σd[i]^2 / n(n^2-1) = 1 – 6(9+1+1+4+0+16) / 6(6^2-1) = 1 – 6(31) / 210 = -0.521
The Spearman correlation coefficient between X and Y is -0.521, indicating a negative monotonic relationship between the two variables.
Difference between Correlation and Causation
Correlation refers to the statistical relationship between two variables. It measures how changes in one variable are associated with changes in another variable. However, correlation does not imply
causation. In other words, just because two variables are correlated does not mean that one variable causes the other variable to change.
• Correlation is a statistical measure that quantifies the strength of the relationship between two variables.
• Causation is a relationship between two variables where one variable directly affects the other variable.
• Correlation does not indicate causation, and it is possible for two variables to be highly correlated without there being any causal relationship between them.
For example, there is a strong positive correlation between ice cream sales and the number of drownings that occur each year. This does not mean that eating ice cream causes people to drown or that
saving lives involves limiting ice cream consumption. Rather, ice cream sales and drownings tend to increase during the summer months due to a common underlying variable – warmer weather.
It is important to be mindful of the distinction between correlation and causation when analyzing statistical data. Failing to do so can lead to faulty conclusions and incorrect inferences.
One way to establish a causal relationship between two variables is through experimentation. In an experiment, one variable is manipulated, while all other variables are held constant. By randomly
assigning participants to different treatment conditions, researchers can determine whether changes in the manipulated variable cause changes in the outcome variable.
Correlation Causation
Describes the relationship between two variables Establishes a direct causal link between two variables
Correlation does not imply causation Causation implies correlation
Measured using a correlation coefficient Established using experimental research designs
While correlation and causation are related, it is essential to distinguish between them to avoid mistakes in research and analysis. Understanding the difference between these concepts can improve
the accuracy of our conclusions and help us make better-informed decisions.
Stationary Time Series
Understanding the difference between correlation and autocorrelation is crucial in the analysis of time series data. Many time series models assume that the underlying data is stationary, which means
that the mean and variance of the data remain constant over time.
However, in real-world applications, it is often difficult to observe a stationary time series. Trends, seasonality, and other underlying patterns can cause the mean and variance to vary over time.
In such cases, it is necessary to transform the data to make it stationary, often by taking first or second differences of the original series.
• Correlation: Correlation measures the linear relationship between two variables. In the context of time series data, it measures the degree to which changes in one variable correspond to changes
in another at a given time. Correlation is calculated using a correlation coefficient, which ranges from -1 to 1. A coefficient of -1 indicates a perfect negative correlation, 0 indicates no
correlation, and 1 indicates a perfect positive correlation.
• Autocorrelation: Autocorrelation, also known as serial correlation, measures the linear relationship between a variable and itself at different time lags. In other words, it measures how closely
related a variable is to its past values. Autocorrelation is important in time series analysis because it allows us to identify patterns in the data that repeat over time.
• Stationarity: Stationarity is a crucial assumption of many time series models. A stationary time series has a constant mean and variance over time. This means that the distribution of the data
does not vary with time. Stationarity is important because it allows us to apply time series models to the data with a reasonable degree of accuracy.
In order to test for stationarity, statisticians use a variety of methods, including the Augmented Dickey-Fuller (ADF) test and the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test. These tests help us
determine whether the mean, variance, or autocorrelation of the data is changing over time.
Correlation Autocorrelation Stationarity
Measures the linear relationship between two Measures the linear relationship between a variable and itself at Assumes that the mean and variance of the data remain constant over time
variables different time lags
Uses a correlation coefficient that ranges from Allows us to identify patterns in the data that repeat over time Important for applying time series models to the data with a reasonable
-1 to 1 degree of accuracy
Understanding the difference between correlation and autocorrelation is important in time series analysis. By testing for stationarity and ensuring that the data is transformed appropriately, we can
apply time series models to make predictions and gain insights into real-world applications.
Auto Regressive Models
Auto Regressive Models, or AR models, are a class of statistical models commonly used in time series analysis. The basic idea behind AR models is to predict future values of a time series based on
its past values. These models assume that the time series is stationary, meaning that its statistical properties do not change over time. AR models are also known as Autoregressive models.
• AR models are a method used in time series analysis to help predict future values based on the past.
• AR models assume that the time series is stationary.
• AR models are also known as autoregressive models.
AR models can be written mathematically as:
y_t = c + φ_1y_t−1 + φ_2y_t−2 + ··· + φ_py_t−p + ε_t
where y_t is the value of the time series at time t, c is a constant, φ_1, φ_2,…,φ_p are the autoregressive coefficients of the model, ε_t is the error term, and p is the order of the model. The
order of the model is the number of past values of the time series that are used to predict future values.
One way to estimate the autoregressive coefficients of an AR model is to use the method of least squares. This involves finding the values of φ_1, φ_2,…,φ_p that minimize the sum of squared errors
between the actual values of the time series and the predicted values from the AR model.
AR models are useful for predicting future values of a time series because they capture patterns and trends in the data that can be difficult to identify visually. However, AR models are not suitable
for all time series data, particularly if the data is non-stationary.
Pros Cons
AR models are helpful in predicting future values of a time series. AR models assume the time series is stationary, which may not be the case for all data.
AR models capture patterns and trends that may be difficult to identify visually. AR models can be complex and difficult to interpret.
AR models can be estimated using the method of least squares. AR models may not be suitable for all types of time series data.
Overall, AR models are a useful tool in time series analysis for predicting future values based on historical data. Before using an AR model, it is important to ensure that the time series is
stationary and that the model is appropriate for the data in question.
Moving Average Models
When it comes to time series analysis, we often use various models to fit the data and extract meaningful information from it. One such class of models is Moving Average Models. These models are
widely used in econometrics, finance, engineering, and many other fields where time series data is prevalent. In this article, we’ll explore the various aspects of Moving Average Models and how they
can help us understand the behavior of time series data.
What are Moving Average Models?
• Moving Average Models, or MA models, are a class of linear time series models that capture the dependence between an observation and a stationary linear combination of past observations.
• In other words, the MA models assume that the current value of a time series depends on the average of its past values, where the weights assigned to each past value depend on the order of the
• The order of the model is denoted by q and represents the number of past values that are included in the average. For example, an MA(1) model includes only the immediate past value, while an MA
(2) model includes the current and two immediate past values.
How to Estimate the Model Parameters?
To fit an MA model to a time series, we need to estimate the model parameters, including the coefficients and error variance. There are various methods for estimating the parameters, such as maximum
likelihood estimation, method of moments, and least-squares estimation.
One common method for estimating the model parameters is the conditional least squares method, which involves minimizing the sum of squared error terms conditional on the past observations. This
method is efficient and can be easily implemented using standard statistical software packages.
Interpreting the Model Coefficients
Once we’ve estimated the model parameters, we can use them to interpret the behavior of the time series. In particular, the model coefficients provide insights into the dependence structure of the
For example, a negative coefficient of an MA model suggests that the current value of the time series is negatively related to the previous values, while a positive coefficient indicates a positive
relationship. Additionally, the magnitude of the coefficients indicates the strength of the relationship, with larger coefficients suggesting stronger dependencies.
Diagnostics and Residual Analysis
After estimating the model parameters, it’s important to conduct diagnostics and residual analysis to assess the goodness of fit and check for model assumptions. One common method for diagnosing an
MA model is to examine the autocorrelation function (ACF) of the residuals.
The ACF measures the linear association between the residuals at different lags. For an MA model, we expect the ACF to be zero for all lags beyond the order of the model. Therefore, any significant
non-zero values in the ACF beyond the order of the model suggest that the model may not adequately capture the dependence structure of the time series.
Examples of Moving Average Models
Model Description
MA(1) Includes the current value and the immediate past value
MA(2) Includes the current value and the two immediate past values
MA(q) Includes the current value and the q immediate past values
Some examples of MA models include the MA(1) model for first-order autocorrelation and the MA(2) model for second-order autocorrelation. These models can be used to fit and interpret time series data
and provide insights into the underlying behavior and dependencies of the process.
FAQs: What is the difference between correlation and autocorrelation?
Q: What is correlation?
Correlation is a statistical measure that quantifies the relationship between two variables. It indicates how much two variables are related to each other, and if they move together or not.
Q: What is autocorrelation?
Autocorrelation is a statistical measure that quantifies the relationship between the values of a variable with its past values. It indicates how much a variable is correlated with its own past
values, and if there is a pattern or not.
Q: What is the main difference between correlation and autocorrelation?
The main difference is that correlation measures the relationship between two variables, while autocorrelation measures the relationship between a variable and its past values.
Q: Can correlation and autocorrelation be applied to the same data?
Yes, correlation and autocorrelation can be applied to the same data, but they are measuring different things. Correlation measures the relationship between two variables, while autocorrelation
measures the relationship of a variable with its own past values.
Q: What is the significance of correlation and autocorrelation in NLP?
Correlation and autocorrelation have practical applications in Natural Language Processing (NLP). Autocorrelation can be used to detect patterns in language and to create models that predict future
language patterns. Correlation can be used to measure the relationship between variables in NLP studies, such as the relationship between word frequency and sentiment.
Closing Thoughts
Now you know the difference between correlation and autocorrelation. Both measures are important statistical tools used in various fields, including NLP. Understanding how they differ will help you
determine which one to use in your analysis. Thanks for reading, and be sure to check back for more informative articles! | {"url":"https://coloringfolder.com/what-is-difference-between-correlation-and-autocorrelation/","timestamp":"2024-11-06T07:33:29Z","content_type":"text/html","content_length":"145144","record_id":"<urn:uuid:4818c14d-1302-4d8c-8970-154f1238a212>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00709.warc.gz"} |
Codeforces Beta Round #1 - Tutorial - Codeforces
Problem A.
The constraint that edges of each flagstone much be parralel to edges of the square allows to analyze X and Y axes separately, that is, how many segments of length 'a' are needed to cover segment of
length 'm' and 'n' -- and take product of these two quantities. Answer = ceil(m/a) * ceil(n/a), where ceil(x) is the least integer which is above or equal to x. Using integers only, it is usually
written as ((m+a-1)/a)*((n+a-1)/a). Note that answer may be as large as 10^18, which does not fit in 32-bit integer.
Most difficulties, if any, contestants had with data types and operator priority, which are highly dependant on language used, so they are not covered here.
Problem B.
Let each letter representation of column number be associated with integer in radix-26, where 'A' = 0, 'B' = 1 ... 'Z'=25. Then, when converting letter representation to decimal representation, we
take associated integer and add one plus quantity of valid all letter representations which are shorter than letter representation being converted. When converting from decimal representation to
letter representation, we have to decide how many letters do we need. Easiest way to do this is to subtract one from number, then quantity of letter representation having length 1, then 2, then 3,
and so on, until next subtraction would have produced negative result. At that point, the reduced number is the one which must be written using defined association with fixed number of digits, with
leading zeroes (i.e. 'A's) as needed.
Note that there is other ways to do the same which produce more compact code, but they are more error-prone as well.
Problem C.
The points can be vertices of regular N-polygon, if, and only if, for each pair, difference of their polar angles (as viewed from center of polygon) is a multiple of 2*pi/N. All points should lie on
the circle with same center as the polygon. We can locate the center of polygon/circle [but we may avoid this, as a chord (like, say, (x1,y1)-(x2,y2)) is seen at twice greater angle from center, than
it is seen from other point of a cricle (x3,y3)]. There are many ways to locate center of circle, the way I used is to build midpoint perpendiculares to segments (x1,y1)-(x2,y2) and (x2,y2)-(x3,y3)
in form y = a*x + b and find their intersection. Formula y = a*x + b has drawback that it cannot be used if line is parallel to y, possible workaround is to rotate all points by random angle (using
formulae x' = x*cos(a) - y*sin(a), y' = y*cos(a) + x*sin(a) ) until no segments are horizontal (and hence no perperdiculares are vertical).
After the coordinates of the center are known, we use fancy function atan2, which returns angle in right quadrant: a[i] = atan2(y[i]-ycenter, x[i]-xcenter)
Area of regular polygon increases with increasing N, so it is possible just to iterate through all possible values on N in ascending order, and exit from cycle as first satisfying N is found.
Using sin(x) is makes it easy: sin(x) = 0 when x is mutiple of pi. So, for points to belong to regular, N-polygon,
sin( N * (a[i]-a[j]) /2 )=0
because of finite precision arithmetic,
fabs( sin( N * (a[i]-a[j]) /2 ) ) < eps
15 years ago, # |
You can use TeX for the mathematical formulas.
• How?
□ frac{x}{y}=3
ivan.popelyshev frac{x}{y}=3
for example.
☆ oops..
○ How do you expect me to guess which source code did you write?
■ (shift+4)\frac{x}{y}(shift+4)
★ 8 months ago, # ^ |
daiyulong 0
$$$ \Large\frac{x}{y}=3 $$$
◎ 6 weeks ago, # ^ |
← Rev. 2 → -8
$$$\frac{x}{y}$$$ = 4
15 years ago, # |
Problem B.
I can't pass Test case 6.
kenny_xu I don't konw why? Isn't there a test case like:
Input: R001C001
Ouput: 00A001
• My AC program give
maslowmw A001
□ No leading zeros needed. "A1" is correct.
• 13 years ago, # ^ |
R288C494 check this input. (Tricky one :D)
□ It is "RZ288".
• You cannot have zeros in letter numbering. "00A" is incorrect, it should be only "A". "001" is not incorrect, but the leading zeros are unnecessary, it can be simply "1". I doubt though
rbenic that there are numbers with leading zeros in the official test cases, it is simply "R1C1" or "A1".
14 years ago, # |
onlyone 0
14 years ago, # |
V_V Who can give me the data on test 6？
Why I always get WA、
• 14 years ago, # ^ |
test 6 is very big (1000 lines)
try to check this test instead:
ont R1C21
after passing this I got AC.
□ 14 years ago, # ^ |
This is what I got,
Which i believe is the correct solution. But it still wouldnt pass test 6 :(
☆ 13 years ago, # ^ |
RZ288 what about this?
○ My program that got AC says "R288C494".
☆ This is correct.
» 12 years ago, # |
B: what about "R1C204152" answer:"KOYZ169801"
• »
» That is incorrect. The correct answer is "KOYZ1". The cell is obviously in row 1.
□ »
» • -#,long long ago !
☆ »
» 11 years ago, # ^ |
» ← Rev. 2 → 0
So what?
» 12 years ago, # |
When I use ceil((float)N/A) for problem 1A, it gives wrong answer while it gives a correct answer when I use ((N+A-1)/A). What is the catch behind it?
• »
» 12 years ago, # ^ |
• »
» float has very low precision. Best stay away from it, use double.
» 11 years ago, # |
Hello :) How do I represent 10^9 integers and input integers separated by space in a single line in C language?
• For integers up to 10^9, you can use the 32-bit integral type long int. But if you multiply them, you have to use a 64-bit type, like long long int, otherwise the multiplication will
» As for input, the proper way to input a 32-bit integer on Codeforces is
scanf ("%ld", & integer);
and for a 64-bit integer
scanf ("%I64d", & integer);
11 years ago, # |
» 0
kawadhiya21 Can someone help me in problem 1B ? 1B - Spreadsheet
except this one "R1C204152" answer:"KOYZ169801", all the test cases which users gave in comments run on my machine. Help me please
• »
» The correct answer to the test case "R1C204152" is "KOYZ1". The cell is obviously in row 1. I got AC on this problem today.
□ »
» Okay I mended my one as well to get KOYZ1 but still it fails test 6. Wanna see my code ?
☆ »
» Yes, please.
» 10 years ago, # |
I have been trying 1B. But getting Runtime error. Here other people also have said same but no reason mentioned. Please suggest me what's wrong.
• » 10 years ago, # ^ |
» 0
TonySnark Try this cell: R1
» 10 years ago, # |
Problem B. My submission: here WA on test 6, but I can't trace this big test case. Can anyone help me with a one that can be traced and hacks my code? Thanks in advance.
• 10 years ago, # ^ |
» ← Rev. 3 → 0
Have you scrolled to the bottom of the submission report?
» 9 years ago, # |
I can't solve problem A, please help me. I was using a 2D-Segment Tree to simulate the squares, but I got WA. I need help. If you help me, I'll give you a big hug.
• » 9 years ago, # ^ |
» 0
uberwach Tip: Solve the problem first in one dimension. Look for an analytical solution of the problem, no advanced data structures are needed.
9 years ago, # |
» Please help: Why I'm getting a different output on different systems for Problem B sample test cases?
AmitThakur On my system:
On CF system:
My submission: 16392041
• » 9 years ago, # ^ |
» 0
AndrewB330 pow pow pow... How many bugs with it. Write your own function.
• » 6 years ago, # ^ |
» -6
satya1998 Did you get correct result .me also facing same problem.
» 7 years ago, # |
what datatype can i use to this big number in java? numbre: 1000000000000000000
• »
» Long.
□ »
» i try to multiplicate 1000000000X1000000000 and i want that the result was 10000000000000000000, but when i try do this operation in java, the answer of the multiplication give me
» how result this number -1486618624 whit datatype long and int and BigInteger.
☆ »
» You have to store 1000000000 in a long also, if you want to make operations on it which makes it become a long. You can cast it meanwhile operation too, so you can do ((long)
» 10000000000)*10000000000, and it will give correct result.
Long can handle numbers up to 9*10^18, for numbers bigger than that you need to use BigInteger.
» 7 years ago, # |
6 years ago, # |
» 0
_nathan_drake_ can someone help me with the concept of epsilon in finding the gcd. Why do most of the accepted answers have it as 10^-4 . when i reduce it to 10^-5 , i get incorrect answer. Can
someone tell me how did we decide this value because i think that lesser the value of eps meant a more precise answer??
» 5 years ago, # |
can anyone explain me why my code is not passing testcase 6. https://mirror.codeforces.com/contest/1/submission/58648097
» 13 months ago, # |
it was a nice contest....it was a og. | {"url":"https://mirror.codeforces.com/blog/entry/85","timestamp":"2024-11-12T05:28:08Z","content_type":"text/html","content_length":"242887","record_id":"<urn:uuid:a4095b53-15b1-4ef6-beb2-a9fbc4a7eec1>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00743.warc.gz"} |
The 2-D Heuristic Wave Model for 3-D Waves with a Local Closure Scheme
A surface equation connecting the vertical velocity and its vertical derivative is used. These variables are well connected in Fourier Space, which allows us to construct the 2-D formulation for
modeling of 3-D waves. It leads to the 2-D (surface) formulation for 3-D waves. The comparison of the spectral results generated by 3-D and 2-D models demonstrates their close identity
Keywords: Phase resolving; Wave modeling; 2-D model for 3-D waves; Comparison of 3-D and 2-D modelse
A common weak point of all 3-D models is their low performance, since all of them in one way or another resolve the vertical structure of a wave field based on a 3-D equation for the velocity
potential. In 3-D model (Full Wave Model (FWM)) [1] this goal was achieved by solving a 3-D equation for the velocity potential. This solution turns out to be unnecessary: to continue the
calculations, just the vertical velocity field on the surface is required. Finally, it was found that the solution of the problem can be based on an equation for a nonlinear component of the velocity
potential written for the physical surface [2]. This equation is exact but it contains both the first and second derivatives of the velocity potential. It was found that these variables are connected
to each other linearly. The coefficients in a linear connection were evaluated by calculations with a high-resolution FWM. Since the method used in 2-D model is based on experimental selection of
nondimensional variables and functions, we call this model Heuristic Wave Model (HWM) [2]. The accuracy of a closure method is confirmed by comparison of the results generated by 3-D and 2-D models
running at identical setting and initial conditions. The scheme developed in [2] is not universal enough, because the closure of an equation for the vertical velocity uses integral parameters, i.e.,
those calculated over the entire field. In this paper a local closing in Fourier space is suggested.
Three-Dimensional Phase-Resolving Model
Modeling of 3-D potential waves is based on the system of equations [1]:
Here ,ξϑ and ζ are curvilinear nonstationary coordinates connected with the Cartesian coordinates ,,:
τ is a time; τ η is a time derivative of η ; ξ η and ϑ η are derivatives of η over ξ and ϑ ; Φ is the 3-D velocity potential; ϕ = Φ(ζ = 0) is the 2-D surface velocity potential; ξ ϕ and ζ ϕ are
along-surface derivatives over ξ and ϑ ; ζ ϕ is a vertical derivative of the potential on the surface (i.e., the surface vertical velocity); the derivatives of Φare calculated along the surfacesζ =
const ; D ξξ ϑϑ ≡η +η is a 2-D Laplacian of elevation; s 1 2 2 ξ ϑ ≡ +η +η ; p is the air pressure on the surface, divided by the water density; η (x, y,t ) =η (ξ ,ϑ,τ ) is a moving periodic wave
surface given by Fourier series:
where k and l are the components of the wave number vector k= (k^2 + l^2) ^1/2 ;h[k,l] (τ) are Fourier amplitudes for elevationsη (ξ ,ϑ,τ ) ; x M and y M are the numbers of modes in the directions ζ
andϑ , respectively, while k ,l Θ are the Fourier expansion basic functions. All the variables are scaled with a use of gravity acceleration g and an arbitrary linear scale L .
The 2-D surface boundary conditions (1) and (2) are considered as evolutionary equations for calculation of η and the surface fieldϕ . Both equations include a vertical derivative of the velocity
potential ζ ϕ , i.e. the surface vertical velocity w. For calculation of this variable, we have to solve a 3-D elliptical equation for volume distribution of the velocity potential for a surface
boundary conditionΦ(ζ = 0) = w . Unfortunately, the values of ware very sensitive to the details of a numerical scheme. That is why for exact calculations of the surface vertical derivative of the
potential it is necessary to use a large number of vertical levels and a stretched grid. An equation (3) is solved as Poisson equation with recalculations of a 3-D right side. The typical number of
iterations is about 10. Such scheme turned out to be imperfect. The acceleration was achieved by representing the velocity potential as a sum of two components: an analytical (‘linear’) component ϕ
and an arbitrary (‘non-linear’) componentϕ : ϕ =ϕ +ϕ , Φ = Φ+Φ
The analytical component ϕ satisfies Laplace equation:
with the known solution:
(where k= (k^2 + l^2) ^1/2 , ϕ[k ,l] are Fourier coefficients for the surface potential ϕ atζ = 0 ). The 3-D solution of (7) satisfies the following boundary conditions:
The nonlinear component satisfies an equation:
Thus, Eq. (12) is solved with the boundary conditions:
The scheme for solving equation (10) is similar to that for equation (3), but due to simple boundary conditions (11), the number of iterations for the right side is reduced to two, which generally
speeds up the solution by 2-3 times.
2-D Equations for 3-D Waves
Anyway, calculation of the 3-D model velocity potential takes about 95% computer time and space. The current paper continues to develop a new approach to the phase-resolving modeling of
two-dimensional periodic wave fields. The basic concept of the scheme follows from presentation of the velocity potential as a sum of the linear and nonlinear components suggested in [1]. It was
observed in [2] that such approach offers a new way to simplify the calculation by projection of 3-D Poisson equation on the surface:
The equation can be considered as an additional exact surface condition. It contains both the first w ζ ≡ϕ and second wζ ζζ ≡ϕ vertical derivatives of the potential. Thus, the system of equations
remains unclosed. It was empirically discovered [2] that those variables are closely connected to each other. There are two possibilities: we can construct dependence between wζ and w in the physical
space, i.e., in terms of grid variable. The pairs wζ and w and all the grid parameters were generated in multiple numerical experiments with FWM. It was obtained that the connection between w[ζ] w
cannot be formulated in terms of the local kinematic and dynamic parameters. Instead, the dependence contained two integral parameters: dispersion of elevation σ = (η^2)^1/2 and dispersion of
‘horizontal’ Laplacian σ[L] = (^2D of elevation. Finally, the formula ( )L w w F ζ= σ σσ was obtained. An approximation for function F is given in [3]. It was demonstrated in [2] by comparison of the
2-D and 3-D calculations that such a non-local scheme allows to reproduce multiple statistical characteristics of wave field with sufficient accuracy.
Earlier the attempts were made to construct a more flexible local closure scheme not in grid but in fourier space. The results were not reliable enough, so this idea was abandoned for a while.
Recently these studies have been resumed. It was found that the reason for the failure was the insufficient accuracy of calculating of the first, and especially the second derivative of the velocity
potential on the surface. These errors practically did not affect the statistical results of 3-D modeling, but they increased scatter of the data in dependencew(w[ζ]), which raised doubts in the
results. To eliminate the errors, the vertical finite difference for Eq. (3) was slightly modified; the number of levels and a stretching coefficient were increased. The comparison of finite
difference differentiation with the analytical one showed that the accuracy of calculating the derivatives turned out to be of the order of 10^−10 . To evaluate the dependence, several long-term
numerical experiments were carried out. The spectral presentation included(513× 257)modes, and grid contained (1024×512 ) grid points. The number of levels was equal to 50, a stretching coefficient
was equal to 1.2. The initial conditions were assigned with JONSWAP spectrum; the peak numbers were equal to 10 and 20. The total number of (w,w[ζ] ) pairs was about 7 million. The dependence of kw
onwζ ( k is a wavenumber modulus) shown in (Figure 1), is quite accurately described by a linear relationship: kw Aw (13)
where A=0.73033± 0.00003. A dashed curve in Figure 1 represents the probability distribution for w .
Figure 1:The dependence of kw on w[ζ] . Thick line is the averaged dependence kw(w[ζ] ) , while thin lines show dispersion. Both are calculated with summation by bins (w[ζ] ) 4.10 5 Δ = − . A dashed
curve shows the probability distribution for w[ζ] ; the value kw =0.004 corresponds to a maximum of the probability.
It follows from an equation (12) and a formula (13) that an equation for Fourier coefficients for the total vertical velocity component will take the form:
where the superscript denotes an iteration number. The coefficient w[k,l] and the metrical coefficients inside an iterative procedure remain constant.
The new system of equations includes the kinematic and dynamic boundary condition (1) and (2). An introduction of a 2-D equation (14) for the vertical velocity transforms a 3-D potential wave problem
into the 2-D problem solved in terms of the surface variables. A simplified model is developed on the basis of an ‘exact’ model. Both models have an identical structure. The evolutionary equations of
both models (1) and (2) are essentially the same. The difference between the models lies in simplification of calculation of a relatively small nonlinear correction to the full vertical velocity
introduced by Eq. (14). A straightforward way of validation of a simplified model is to carry out the runs with the identical setting and initial conditions. For preliminary evaluation it is most
reasonable to consider the case of a quasi-stationary regime. This regime can be defined as a process with exact preservation of wave spectrum. Strictly speaking, for the nonlinear wave field the
quasistationary regime cannot exist, since modification of spectrum always occurs due to the nonlinearity and high-wave-number dissipation, which supports numerical stability. For compensation of
dissipation, a small energy input from wind was introduced [1]. Both models have resolution of (513× 257)modes and (1024×512) grid points; the initial conditions were generated with JONSWAP spectrum
with the peak wave number equal to (20,0), and a ratio of wind velocity and phase velocity at spectral peak equal to 1. Such ratio provided the input energy compensating small dissipation. The
Runge-Kutta scheme for time stepping was used. The run was done for 1,00,000 time steps. The comparison of the results obtained by 3-D and 2-D models in this paper are given for the averaged in time
and over angles (in polar coordinates) spectra of different characteristics (Figure 2).
Figure 2:The averaged over time and angles (in polar coordinates) spectra of: (1) elevation; (2) upper curves - full vertical velocity; lower curves - a nonlinear component of the vertical velocity;
(3) steepness ξ η ; (4) a rate of energy loss due to breaking; (5) a rate of energy input from wind; (6) a rate of nonlinear interactions. Thick curves represent the complete coincidence of curves
for 3-D and 2-D models. An absolute difference between the curves is given by a thin curves in panels 1,2,3,5 and plain difference multiplied by 100 in panels 4 and 6. The consolidated grey curves
show the scatter of spectra calculated by 3-D model.
A few years ago, the author of this article has developed a fairly accurate 3D adiabatic model of 3D waves. Later, the energy conversion effects were added to the model, and as a result, the model
was no longer very accurate. Nevertheless, with the help of that model, many physical processes in waves were studied, and numerical experiments were carried out on development of waves under the
action of wind. I can’t say that this work has brought full satisfaction, because the time spent on completing the calculations was too large. It seemed strange that the largest share of the
calculations was devoted to the 3-D variables not required to continue the calculation. It seems natural that the motion in potential approximation should be completely determined by the surface
variables. This statement is confirmed by considering the projection of laplace equation onto the surface, but some difficulty arises: The new equation turns out to be unclosed, which introduces the
closure problem, which is not quite usual for wave dynamics. This problem was solved with the use of an accurate 3-D model that in this case acted as a source of empirical information. A completely
closed system of the equations was constructed. The long-term calculations carried out with 3-D and 2-D models showed that the statistical results were very close to each other. Of course, the new
model requires further detailed verification.
a. The new approach of modeling of 3-D waves based on 2-D equations is suggested.
b. The projection of Laplace equation on the surface and decomposition of the velocity potential into the linear and nonlinear components are used.
c. A surface equation containing the surface velocity and its vertical derivative is derived.
d. It is shown that the surface velocity and its vertical derivative are closely connected in Fourier space.
e. The parameters of the closure scheme are defined based on the calculations with 3-D model
f. The calculations with 3-D and 2-D models with identical setting proved excellent agreement for some statistical characteristics of the solution.
1. Chalikov D (2016) Numerical modeling of sea waves. 1^st (Edn.), Earth and Environmental Science, Springer Publishers, Switzerland, pp. 307-330.
2. Chalikov D (2021) A two-dimensional approach to the three-dimensional phase resolving wave modeling. Examines Mar Biol Oceanogr 4(1): 1-4.
3. Chalikov D (2022) A 2D model for 3D periodic deep-water waves. J Mar Sci Eng 10(3): 410.
© 2023 Dmitry Chalikov. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and build upon your work | {"url":"https://crimsonpublishers.com/eimbo/fulltext/EIMBO.000610.php","timestamp":"2024-11-14T11:37:31Z","content_type":"text/html","content_length":"182328","record_id":"<urn:uuid:170fdf19-8b82-462f-bf53-ecb4472224bf>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00007.warc.gz"} |
NCERT Exemplar for Class 9 Maths Solutions PDF
Here you can download NCERT Exemplar Problems for Class 9 Maths Solutions PDF, all solutions are listed chapter wise. No Sign-Up and No Login required, just one click and your PDF will be
Class 9 Maths NCERT Exemplar Chapter wise Solutions – Free PDF Download
• Chapter 12 – Heron’s Formula
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fabric - uncertainty
Post date: 05-Mar-2012 00:48:05
Claude Shannon developed a mathematical theory of communication which has been widely used in many fields of electronics ever since (Shannon 1948). In particular, Shannon was concerned with the
uncertainty introduced by a noisy communication channel. When a message is sent over a perfect communication channel, the recipient can be confident that they have received a reliable copy of the
transmitted message. However, when the channel is noisy, the recipient will be uncertain, and must do their best to guess what message was actually transmitted. Shannon defined a way of measuring
this uncertainty and called it entropy.
You may know of the simple game of Hangman, where you must guess an unknown word, one letter at a time. Each wrong guess takes you one step closer to being hung. When a child is first confronted with
the game their uncertainty is very high as there are 26 possibilities for each letter position. As their skills improve, they will realize that some letters are more common than others, and their
uncertainty will be a little less. Guessing a single letter at random has a 1 in 26 = 3.85% change of being correct. Guessing a single letter known to be from an English text will be less uncertain
as the probability of 'e'=12.70%, 't'=9.06%, 'a'=8.17%, and so on down to 'z'=0.07%. This uncertainty can be expressed as entropy = 4.7 in the naïve situation, decreasing to entropy = 4.2 with a
knowledge of word frequency in English. Additional knowledge of English words will further reduce uncertainty as letters are guessed correctly in the game. As a child learns English, their
uncertainty in the Hangman environment decreases.
Cosma Shalizi built on Shannon's work by developing an optimal way to extract and store information from a string of symbols (Shalizi 2001). Shalizi's CSSR algorithm first examines a large historical
sample of symbols generated by some unknown process. The initial analysis looks for small repeated histories (patterns) within the sample and records the probability of the very next character. This
is completed for all small histories in the sample up to some maximum size. These probabilities are then condensed into an ɛ-machine, which is the most concise way possible of representing the
knowledge gathered about the process.
For example, consider a historical sample involving just two symbols, A and B.
A statistical analysis will show that:
A is followed by another A(33%) or B(66%)
B is always followed by an A(100%)
We can refine this by looking at longer histories:
AA is followed by A(40%) or B(60%).
AB is followed by A(100%).
BA is followed by A(33%) or B(66%).
BB never occurs in the sample.
Finally we condense the results into 3 states:
S1. A and BA both predict A(33%) or B(66%).
S2. AA predicts A(40%) or B(60%).
S3. B predicts A(100%).
This ɛ-machine is a model of the unknown process which generated the original sample. The model may be improved by examining a larger sample or collecting statistics for longer histories. If you were
to move around the ɛ-machine diagram, using the calculated probabilities at each state, then you will generate a string that is statistically identical to the original sample. Alternatively, if you
begin receiving more data from the original unknown process, you will be able to sync with the ɛ-machine after a couple of symbols and then be in a position to make the best possible prediction for
the next symbol.
Might an ɛ-machine be used as the basis for an intelligent device? The ɛ-machine reduces uncertainty at the maximum possible rate by capturing all patterns that have any predictive power. Furthermore
it is the smallest possible representation and it continues to operate optimally with noise in the signal. However, the CSSR algorithm used to construct the ɛ-machine comes at a cost.
Modern computers are based on an architecture proposed by Allan Turing (Turing 1936) in response to a problem in Mathematics. This most basic of computers, comprised a simple machine and a tape. The
machine was able to move left or right along the tape and was able to read, write or erase a single symbol to each position on the tape, or simply stop. Each action of the machine was completely
determined by the symbol most recently read from the tape and the internal state of the machine. Thus, the Turing machine has memory and the ability to act on the memory. The cost of a computation is
the amount of memory required and the amount of time required to complete the computation. These costs are referred to as the memory complexity and the time complexity respectively.
The time complexity of constructing an ɛ-machine using Shalizi's CSSR algorithm depends on three things; the total length of the historical sample data (N), the number of symbols in the alphabet (k),
and the longest pattern used to generate the statistics (L[max]). Consider a base case with a sample size N=1000, alphabet size k=2, and history size L[max]=5 which for the sake of this example takes
1 second to run on a computer. Doubling each of these variable in turn increases the run time.
To put this into perspective, consider using CSSR to construct an ɛ-machine to learn from data coming from an extremely low resolution video camera. A very simple camera might have only 5x5=25 black/
white pixels (no gray-scale). We might gather 1 hour of data at 10 frames per second and gather statistics for just 1 second (10 frames). This translates to N=36,000, k=1025, and L[max]=10 with a run
time much much longer than the age of the universe on our base case computer.
An ɛ-machine learns quickly in the sense that it uses all of the available information, but the computational cost of doing so is prohibitive. The computations required cannot keep up with the rate
that data is available from the real world. Brains do.
J.F.Traub and A.G.Werschulz developed an elegant approach known as Information-based Complexity (IBC) that studies problems for which the information is partial, contaminated and priced (Traub &
Werschulz 1999). Partial information is very common in the real world where measurements provide information about a particular time or locale, but assumptions must be made regarding the unmeasured
points. The assumptions are global information about the environment such as smoothness. Contaminated information is also common due to noise, rounding-errors and/or deception. The cost of
information is incurred during its collection, and in the computations that follow.
Consider the challenge of weather forecasting, where it is expensive to measure and collate data from even a few points around the globe, and every measurement contains error introduced by the
resolution of the instruments. The computations to generate a forecast make assumptions about missing locations and incur further expense.
Computations made with partial or incomplete information, leave some uncertainty in the result. IBC provides a framework to determine the amount of uncertainty remaining, how much information is
required to limit the uncertainty, and the optimal amount of information. IBC defines a quantity called the radius of uncertainty which measures the intrinsic uncertainty in a solution due to the
available information. The value of information, is it's capacity to reduce the radius of uncertainty.
These notions used by IBC are very helpful. Organisms with very simple nervous systems clearly have very limited capacity to collect and process information. Yet, despite these severe limitations,
natural selection has proven that they routinely take actions that are superior to their competition. When resources are severely limited, and the competition is close, it is important to use those
resources to their maximum effect. It is important to collect and utilize that information which reduces uncertainty by the maximum amount for the least cost. IBC provides a framework to evaluate the
cost and value of information collected in order to reduce uncertainty.
By way of example, the primate visual system is forward facing with each eye sensitive to light arriving within a cone of only about 120° of the full 360°. The visual acuity in the outer peripheral
region is very low compared to the very much higher sensitivity and resolution of the central fovea spanning only about 2°. Primates perform many tens of thousands of saccades per day, which are
rapid eye movements, that align the fovea with particular targets in the visual field. The primate visual system collects high quality information from only selected regions rather than all of that
In sum, identifying historical patterns enables uncertainty in the future to be reduced. The computational cost of identifying all patterns is prohibitive. Limited resources force brains to gather
and utilize only that information which gives the most reduction in uncertainty for the least cost.
Shannon,C.E. (1948), A Mathematical Theory of Communication.
Shalizi,C. (2001), Causal Architecture, Complexity and Self-Organization in Time Series and Cellular Automata.
Turing, A.M. (1936) On Computable Numbers with an application to the Entscheidungsproblem.
Turing, A.M. (1950) Computing Machinery and Intelligence. Mind 49: 433-460.
Traub,J.F. & Werschulz,A.G. (1999), Complexity and Information (Lezioni Lincee). Cambridge University Press. | {"url":"https://fabric.nhoj.info/excogitate/uncertainty","timestamp":"2024-11-11T16:43:06Z","content_type":"text/html","content_length":"103101","record_id":"<urn:uuid:6ff30965-94a9-44c2-8a67-75c2e8d227df>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00527.warc.gz"} |
Seminar 2020.1 | UFG - Continuous Optimization Group
The seminars will be held remotely at 8:00 am via Google Meet, unless otherwise stated. All interested are very welcome. To attend, please contact Prof. Glaydston de Carvalho Bento by e-mail
Date: September 03
Speaker: Prof. Max Leandro Nobre Gonçalves, UFG
Title: Projection-free accelerated method for convex optimization
Abstract: In this talk, we discuss a projection-free accelerated method for solving convex optimization problems with unbounded feasible set.
The method is an accelerated gradient scheme such that each projection subproblem is approximately solved by means of a conditional gradient scheme. Under reasonable assumptions, it is shown that an
$\varepsilon$-approximate solution (concept related to the optimal value of the problem) is obtained in at most $\mathcal{O}(1/\sqrt{\varepsilon} )$ gradient evaluations and $\mathcal{O}(1/\
varepsilon)$ linear oracle calls. We also discuss a notion of approximate solution based on the first-order optimality condition of the problem and present iteration-complexity results for the
proposed method to obtain an approximate solution in this sense. Finally, numerical experiments illustrating the practical behavior of the proposed scheme are discussed.
Date: September 10
Speaker: Ray Victor Guimarães Serra (PhD student-IME/UFG)
Title: A strongly convergent proximal gradient method for solving convex composite vector optimization problems in Hilbert spaces
Abstract: In this talk, we will present a variant of the proximal gradient method for solving convex composite vector optimization problems in real Hilbert spaces. We will show that under some mild
conditions, the proposed scheme converges strongly to a weakly efficient solution.
Date: September 17
Speaker: Tiago da Costa Menezes (PhD student-IME/UFG)
Title: Inexact Variable Metric Method for Convex-Constrained Optimization Problems
Abstract: In this talk, we will discuss the inexact variable metric method for solving convex-constrained optimization problems. At each iteration of this method, the search direction is obtained by
inexactly minimizing a strictly convex quadratic function over the closed convex feasible set. We will present a new inexactness criterion for the search direction subproblems. Under mild
assumptions, we prove that any accumulation point of the sequence generated by the new method is a stationary point of the problem under consideration. Finally, we will also discuss some numerical
experiments and an application where our concept of the inexact solutions is quite appealing.
Date: September 24
Speaker: Prof. Glaydston de Carvalho Bento, UFG
Title: Some Recent Advances in Optimization in Riemannian Manifolds
Abstract: In this lecture I intend to present some recent advances on optimization in Riemannian manifolds associated with two popular descent methods, namely, the gradient method and the proximal
point method.
Date: October 01
Speaker: Ademir Alves Aguiar (PhD student-IME/UFG)
Title: Inexact Projections for constrained convex optimization Problems
Abstract: In this talk, we will present a new inexact version of projected subgradient method to solve nondifferentiable constrained convex optimization problems. The method combine $\
epsilon$-subgradient method with a procedure to obtain a feasible inexact projection onto the constrained set. Also, the gradient projection method with a feasible inexact projection is proposed in
this seminar. To perform the proposed inexact projection in both algorithms, a relative error tolerance is introduced. Asymptotic analysis and iteration-complexity bounds for methods are established.
Date: October 08
Speaker: Danilo Rodrigues de Souza (PhD student-IME/UFG)
Title: Um método quase-Newton com busca linear de Wolfe para otimização multiobjetivo
Abstract: Neste trabalho, propomos um método quase-Newton com busca linear de Wolfe para otimização multiobjetivo irrestrita. A Hessiana de cada função objetivo é aproximada por uma matriz inspirada
na atualização BFGS clássica. Assim como no caso escalar, se os comprimentos de passos satisfazem as condições de Wolfe multiobjetivo, então a atualização BFGS proposta se mantém definida positiva. A
boa definição do método é obtida mesmo quando as funções objetivo são não-convexas. Usando hipóteses de convexidade, estabelecemos convergência superlinear para pontos Pareto-ótimos. As hipóteses
consideradas são extensões diretas das usadas no método BFGS escalar.
Date: October 15
Speaker: Prof. Pedro Bonfim de Assunção Filho (PhD student-IME/UFG)
Title: Conditional gradient method for multiobjective optimization
Abstract: In this talk, we analyze the conditional gradient method, also known as Frank-Wolfe method, for constrained multiobjective optimization. The constraint set is assumed to be convex and
compact, and the objective functions are assumed to be continuously differentiable. The method is considered with different strategies for obtaining the step size. Asymptotic convergence properties
and iteration complexity assumptions on the objective functions are established. Numerical experiments are provided to illustrate the effectiveness of the method and certify the obtained theoretical
Date: October 22
Speaker: Prof. Gilson do Nascimento Silva, UFOB
Title: On the Inexact Quasi-Newton Methods for Solving Nonsmooth Generalized Equations: Broyden's Update and Dennis-Moré Theorem
Abstract: This talk is about an inexact quasi-Newton method for solving nonsmooth generalized equations. At first, using the Coincidence Point Theorem and theory of metric regularity, we prove the
q-linear convergence of the sequence generated by the inexact quasi-Newton method. In a specific case, we use the well-known Broyden update to obtain a convergence result. Secondly, we assume that
the generalized equation is strongly metrically r-subregular, and we obtain a higher order convergence to the inexact quasi-Newton method proposed with the Broyden update. We finish by showing the
Broyden update applied to a nonsmooth generalized equation in Hilbert spaces satisfies the Dennis-Moré condition for q-superlinear convergence.
Date: October 29
Speaker: Prof. Flávio Pinto Vieira (PhD student-IME/UFG)
Title: Steepest descent methods for constrained vector optimization problems with a new non-monotone line search
Abstract: In this talk, we will present a new non-monotone line search that can be used in procedures for finding Pareto optima of constrained vector optimization problems. Our work was inspired by
Yunda Dong’s works in the development of non-linear conjugate algorithms for scalar optimization problems. In our opinion, the novelty of our procedure is that the line search is performed without
comparing function values, overcoming an important issue in general vector optimization. Our convergence analysis covers the general case, for functions with continuous Jacobian, the Lipschitz case,
when a Lipschitz constant of the Jacobian is known, and the convex case. As an application, we will present our results solving a bi-criteria model for the design of a cross-current multistage
extraction process, proposed by Kitagawa and others.
Date: November 05
Speaker: Fernando Santana Lima (PhD student-IME/UFG)
Title: Globally convergent Newton-type methods for multiobjective optimization
Abstract: We propose two Newton-type methods for solving (possibly) nonconvex unconstrained multiobjective optimization problems. The first is directly inspired by the Newton method designed to
solve convex problems, whereas the second uses second-order information of the objective functions with ingredients of the steepest descent method. One of the key points of our approaches is to
impose some safeguard strategies on the search directions.
These strategies are associated to the conditions that prevent, at each iteration, the search direction to be too close to orthogonality with the multiobjective steepest descent direction and
require a proportionality between the lengths of such directions.
In order to fulfill the demanded safeguard conditions on the search directions of Newton-type methods, we adopt the technique in which the Hessians are modified, if necessary, by adding multiples
of the identity. For our first Newton-type method, it is also shown that, under convexity assumptions, the local
superlinear rate of convergence (or quadratic, in the case where the Hessians of the objectives are Lipschitz continuous) to a local efficient point of the given problem is recovered.
The global convergences of the aforementioned methods are based, first, on presenting and establishing the global convergence
of a general algorithm and, then, showing that the new methods fall in this general algorithm.
Date: November 12
Speaker: Elianderson Meneses Santos (PhD student-IME/UFG)
Title: An algorithm for minimization of a certain class of nonconvex functions
In this talk we present an algorithm to minimize a special class of nonconvex functions. Besides, we'll also show some results of partial convergence and complexity bounds for that algorithm.
Date: November 19 - Seminário desse dia foi suspenso em virtude do WebJME - Webinário de Jovens Pesquisadores em Matemática Pura, Aplicada e Estatística. Os participantes devidamente matriculados
nessa atividade foram orientados a participarem das seguintes atividades:
Lecture: November 19
On the Complexity of an Augmented Lagrangian Method for Nonconvex Optimization
Geovani Nunes Grapiglia (UFPR)
Lecture: November 20
Iteration-Complexity of an Inexact Proximal Augmented Lagrangian Method for Solving Constrained Composite Optimization problems
Jefferson Divino Gonçalves de Melo (UFG)
Date: November 26
Speaker: Prof. Pedro Bonfim de Assunção Filho (PhD student-IME/UFG)
Title: Model Function Based Conditional Gradient Method with Armijo-like Line Search
Abstract: In this seminar will be presented the paper "Model Function Based Conditional Gradient Method with Armijo-like Line Search (https://arxiv.org/pdf/1901.08087.pdf)"
Date: December 03
Speaker: Ademir Alves Aguiar (PhD student-IME/UFG)
Title: Convergence analysis of a nonmonotone projected gradient method for multiobjective optimization problems
Abstract: In this seminar will be presented the paper "Fazzio. N. S; Schuverdt. M. L.: Convergence analysis of a nonmonotone projected gradient method for multiobjective optimization problems.
Optimization Letters (2019) 13:1365-1379.” Our objective is to present an extension of the projected gradient method to include a nonmonotone line search based on the average of the successive
previous functions values instead of the traditional Armijo-like rules.
Date: December 10
Speaker: Prof. Flávio Pinto Vieira (PhD student-IME/UFG)
Title: A new line search for vector optimization
Abstract: We introduce a new line search for vector optimization. This procedure is non-monotone, i.e., function value at the new
iterate may be not least than or equal to the current one in the considered order. Convergence analysis of the steepest descent
algorithm, using the proposed line search procedure, is presented. Extensive numerical experiences were performanced testing the
effectiveness in building Pareto-fronts.
Date: December 17
Speaker: Prof. Reinier Diaz Millan, Deakin University-AU
Title: An algorithm for best generalized rational approximation of continuous functions
Abstract: The motivation of this paper is the development of an optimization method for solving optimization problems appearing in Chebyshev rational and generalized rational approximation problems,
where the approximations are constructed as ratios of linear forms (linear combinations of basis functions). The coefficients of the linear forms are subject to optimization and the basis functions
are continuous functions. It is known that the objective functions in generalized rational approximation problems are quasi-convex. In this paper we also prove a stronger result: the objective
function is pseudo-convex. Then, we develop numerical methods that are efficient for a wide range of pseudo-convex functions and test them on generalized rational approximation problems. | {"url":"https://optimization.ime.ufg.br/p/34912-seminar-2020-1","timestamp":"2024-11-08T01:44:05Z","content_type":"text/html","content_length":"24572","record_id":"<urn:uuid:7ce5286d-7488-46ae-ae44-f4b4d01044b6>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00776.warc.gz"} |
Lesson 3
Randomness in Groups
Lesson Narrative
The mathematical purpose of this lesson is for students to understand the importance of randomness in selecting a sample and the importance of randomness in assigning participants to groups for an
experimental study. The work of this lesson connects to prior work because students investigated what it means to be random in a statistical sense and how to select at random. The work of this lesson
connects to upcoming work because students will describe distributions, use standard deviation as a measure of variability, and investigate normal distributions, all of which rely on understanding
the nature of randomness. Students encounter the term random selection which is a term used to describe a selection process by which each item in a set has an equal probability of being selected, and
the term sample which is defined as a group that is selected from the whole population. When students describe the importance of randomness in selecting samples and dividing groups in the context of
a study, they are attending to precision (MP6).
Learning Goals
Teacher Facing
• Comprehend (in spoken and written language) what it means to be random in the context of statistics.
• Explain (orally and in writing) what it means to select at random.
Student Facing
• Let’s explore why randomness is important in studies.
Required Preparation
All students will need a random number generator (for example, a calculator, spreadsheet, or randomly drawing numbered pieces of paper will work) that can generate whole numbers between 1 and 100.
You will need to collect data from each student to create four data sets so you can make a dot plot from each data set.
Student Facing
• I recognize that the way I choose a sample matters, and that random samples have less bias.
Glossary Entries
• random selection
A selection process by where each item in a set has an equal probability of being selected.
• sample
A sample is a subset of a population.
Print Formatted Materials
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The Moons of Vuvv
The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?
The Moons of Vuvv printable sheet
The planet of Vuvv has $7$ moons which lie spread out on one plane in a great disc round it. These Vuvvian moons all have long and confusing names so scientists usually call them by their initials:
$A, B, C, D, E, F$ and $G$ starting from the nearest one to the planet.
When two of these moons line up with the planet it is called a 'lunar eclipse'. When three line up with the planet it is called a 'double eclipse', when four do it is a 'triple eclipse' and so on.
Once in a while all seven moons line up with the planet and this is called a 'super-eclipse'.
Moon $A$ completes a cycle round the planet in one Vuvvian year, moon $B$ takes two years, moon $C$ takes three years, moon $D$ takes four years and so on.
How long is it between each 'super-eclipse' on the planet of Vuvv?
Getting Started
You could look at just two moons and work out how many years it takes for them to coincide.
Which moons might it be good to look at first?
It might help to use a calculator and to jot your ideas on paper.
Student Solutions
"I enjoy your website", wrote Becky , from Carleton St Hilda's C. of E. Primary School. Becky explained how she began her solution search:
Now, I wonder what Becky changed her search to? If Becky is going to change her search to try and arrive at an answer perhaps she wants to think about this idea.
Alex and her family from Leicester, England worked on this Vuvvian problem. Alex explains how they set about arriving at a solution:
• We started off by doing the seven times table, because that was how long the last moon took to go round Vuvv.
• Next, we checked if the multiples of seven were also in the 2x, 3x, 4x, 5x, 6x tables. This was so we'd know if they (Vuvv moons) would line up.
• We got fed up working out the multiples of seven, because they got way too big. So, we used a calculator! We pressed +7=== to get the multiples of seven.
• We found out that it would take 210 Vuvvian years between each super eclipse.
However, I'm not sure that 210 is a multiple of all of the numbers 2, 3, 4, 5, 6 and 7, is it?
Anita and Jing Jing from Kilvington Girls' Grammar in Australia, think that's only the half of it...in fact, they think that it is 420 year wait between Super-eclipses.
Franco and Jonny from Northamptonshire agree that is it 420. They say:
We started off with 42. Every number goes into 42, except 5, so we multiplied it by 5.
6 doesn't go into 210, so we went back to 42. We then multiplied 42 by 10, to get 420. We checked by dividing 420 by 1, 2, 3, 4, 5, 6 and 7. They are all factors of 420. So the overall answer is 420.
Teachers' Resources
Why do this problem?
This problem
offers opportunities for pupils to reinforce their understanding of factors and multiples, and, in a simple example, see an illustration of 'lowest common multiple'. It would fit in well when
revising multiplication tables or working on multiples and factors.
Possible approach
You could start on this problem with a whole class activity counting in, for example, $2$s and $5$s. When do you say the same number in both? Try also two numbers which have a common factor, for
example, $4$s and $6$s. When do you say the same number first in both?
After this you could introduce the problem either verbally, as a printed sheet or on an interactive whiteboard. Once the children have understood what they are to do, they could work on it in pairs.
Some children might benefit from using a calculator for this activity both for multiplying by $7$, and for checking results. You may wish to stop the class part way through to share some of the
different ways they are working and recording. Some may be drawing pictures, others may be listing numbers. You could talk about the benefits of the different ways and it may be that some children
adopt other representations following this sharing process.
A discussion of methods and comparison of answers in a plenary may well bring up different results. This would be a good opportunity to discuss the meaning of lowest common multiple.
Key questions
How many years does it take for these two moons to coincide?
Do these two moons coincide sooner than that?
Possible extension
Tell the children that more moons have been discovered circling Vuvv. Get them to work out the length of time between the super-eclipses if there are also moons that cycle taking $8, 9, 10 \ldots$
Possible support
Suggest starting with just three or four moons and slowly adding the higher numbers. | {"url":"https://nrich.maths.org/problems/moons-vuvv","timestamp":"2024-11-14T22:05:43Z","content_type":"text/html","content_length":"45303","record_id":"<urn:uuid:8da57177-d0ff-4f43-8daf-04231810b890>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00739.warc.gz"} |
A New Formulation to the Point Kinetics Equations Considering the Time Variation of the Neutron Currents
[1] A new solution of the fractional neutron point kinetics equations using symmetry and the Heaviside's expansion formula
López, G Espinosa-Paredes - Progress in Nuclear Energy, 2024
[2] A new compartmental fractional neutron point kinetic equations with different fractional orders
Paredes, CA Cruz-López - Nuclear Engineering and Design, 2024
[3] A new point kinetics model for ADS-type reactor using the importance function associated to the fission rate as weight function
Annals of Nuclear Energy, 2023
[4] Spectrum analysis of two energy groups space-dependent neutron telegraph kinetics model
Mhlawy… - Annals of Nuclear Energy, 2022
[5] Characteristics and optimization of SCO2 Brayton cycle system for high power sodium-cooled fast reactor on Mars
Thermal Science, 2021
[6] Validation Method of the Mathematical Model for SARS-Cov-2 Pandemic from Data Mining and Statistical Analysis
Brazilian Journal of Experimental Design, Data Analysis and Inferential Statistics, 2021
[7] VALIDATION METHOD OF THE MATHEMATICAL MODEL FOR SARS-Cov-2 PANDEMIC FROM DATA MINING AND STATISTICAL ANALYSIS.
[8] On a point kinetic model for nuclear reactors considering the variation in fuel composition
[9] Inverse method to obtain reactivity in nuclear reactors with P1 point reactor kinetics model using matrix formulation
[10] Higher orders of Magnus expansion for point kinetics telegraph model
Progress in Nuclear Energy, 2019
[11] Adjusted mean generation time parameter in the neutron point kinetics equations
[12] Response of the point-reactor telegraph kinetics to time varying reactivities
Progress in Nuclear Energy, 2017
[13] On the Stability of Fractional Neutron Point Kinetics (FNPK)
Applied Mathematical Modelling, 2017
[14] The calculation of the reactivity by the telegraph equation
Annals of Nuclear Energy, 2017
[16] Aproximação alternativa para as equações da cinética de nêutrons modificada
[17] A note on “Comment on the paper: Espinosa-Paredes, et al., 2011. Fractional neutron point kinetics equations for nuclear reactor dynamics. Ann. Nucl. Energy 38, 307–330.” by AE Aboanber, AA
Annals of Nuclear Energy, 2016
[18] Formulation of a point reactor kinetics model based on the neutron telegraph equation
Annals of Nuclear Energy, 2016
[19] Effect of the time variation of the neutron current density in the calculation of the reactivity
Annals of Nuclear Energy, 2016
[20] A note on" Comment on the paper: Espinosa-Paredes, et al., 2011. Fractional neutron point kinetics equations for nuclear reactor dynamics. Ann. Nucl. Energ. 38, 307 …
Paredes - Annals of Nuclear Energy, 2016 | {"url":"https://scirp.org/journal/papercitationdetails?paperid=53644&JournalID=534","timestamp":"2024-11-09T23:50:58Z","content_type":"application/xhtml+xml","content_length":"121298","record_id":"<urn:uuid:b1edb532-7dbf-4a36-9666-ef9cb34c541f>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00701.warc.gz"} |
python代写-COM 2004|学霸联盟
COM 2004
1. This question concerns probability theory.
a) The discrete random variable X represents the outcome of a biased coin toss. X has
the probability distribution given in the table below,
x H T
P(X = x) θ 1−θ
where H represents a head and T represents a tail.
(i) Write an expression in terms of θ for the probability of observing the sequence
H, T, H, H. [5%]
(ii) A sequence of coin tosses is observed that happens to contain NH heads and
NT tails. Write an expression in terms of θ for the probability of observing this
specific sequence. [5%]
(iii) Show that having observed a sequence of coin tosses containing NH heads and
NT tails, the maximum likelihood estimate of the parameter θ is given by
b) The discrete random variables X1 and X2 represent the outcome of a pair of independent
but biased coin tosses. Their joint distribution P(X1,X2) is given by the probabilities in
the table below,
X1 = H X1 = T
X2 = H λ 3λ
X2 = T 2λ ρ
(i) Write down the probability P(X1 = H,X2 = H). [5%]
(ii) Calculate the probability P(X1 = H) in terms of λ. [5%]
(iii) Calculate the probability P(X2 = H) in terms of λ. [5%]
(iv) Given that the coin tosses are independent and that λ is greater than 0, use your
previous answers to calculate the value of λ. [15%]
(v) Calculate the value of ρ. [5%]
COM 2004 3 TURN OVER
COM 2004
c) Consider the distribution sketched int the figure below.
p(x) =
2λ if 0<= x< b
λ if b<= x<= 1
0 otherwise
(i) Write an expression for λ in terms of the parameter b. [15%]
(ii) Two independent samples, x1 and x2, are observed. x1 has the value 0.25 and
x2 has the value 0.75. Sketch p(x1,x2;b) as a function of b as b varies between
0 and 1. Using your sketch, calculate the maximum likelihood estimate of the
parameter b given the observed samples. [20%]
COM 2004 4 CONTINUED
COM 2004
2. This question concerns the multivariate normal distribution.
a) Consider the data in the following table showing the height (x1) and arm span (x2) of a
sample of 8 adults.
x1 151.1 152.4 152.9 156.8 161.8 158.6 157.4 158.8
x2 154.5 162.2 151.5 158.2 165.3 165.6 159.8 162.0
The joint distribution of the two variables is to be modeled using a multivariate Gaussian
with mean vector, µ and covariance matrix, Σ.
(i) Calculate an appropriate value for the mean vector, µ. [5%]
(ii) Write down the formula for sample variance. Use it to calculate the unbiased
variance estimate for both height and arm span. [10%]
(iii) Write down the formula for sample covariance. Use it to calculate the unbiased
estimate of the covariance between height and arm span. [10%]
(iv) Write down the covariance matrix, Σ. [5%]
(v) Compute the inverse covariance matrix, Σ−1. [15%]
b) Remember that the pdf of a multivariate Gaussian is given by
p(x) =Ce−
whereC is a scaling constant that does not depend on x.
Using the answer to 2 (a) and the equation above, answer the following questions.
(i) Who should be considered more unusual:
• Ginny who is 162.1 cm tall and has arms 164.2 cm long, or
• Cho who is 156.0 cm tall and has arms 153.1 cm long?
Show your reasoning. [20%]
(ii) A large sample of women is taken and it is found that 120 have measurements
similar to those of Ginny. How many women in the same sample would be ex-
pected to have measurements similar to those of Cho? [15%]
COM 2004 5 TURN OVER
COM 2004
c) A person’s ‘ape index’ is defined as their arm span minus their height.
(i) Use the data in 2 (a) to estimate a mean and variance for ape index. [10%]
(ii) The figure below shows a standard normal distribution, i.e., X ∼ N(0,1). The
percentages indicate the proportion of the total area under the curve for each
Using the diagram estimate the proportion of the population who will have an ape
index greater than 10.5? [5%]
(iii) Using the figure above estimate the mean-centred range of ape indexes that
would include 99% of the population. [5%]
COM 2004 6 CONTINUED
COM 2004
3. This question concerns classifiers.
a) Consider a Bayesian classification system based on a pair of univariate normal distribu-
tions. The distributions have equal variance and equal priors. The mean of class 1 is
less than the mean of class 2. For each case below say whether the decision threshold
increases, decreases, remains unchanged or can move in either direction.
(i) The mean of class 2 is increased. [5%]
(ii) The mean of class 1 and class 2 are decreased by equal amounts. [5%]
(iii) The prior probability of class 2 is increased. [5%]
(iv) The variance of class 1 and class 2 are increased by equal amounts. [5%]
(v) The variance of class 2 is increased. [5%]
b) Consider a Bayesian classification system based on a pair of 2-D multivariate normal
distributions, p(x|ω1) ∼ N(µ1,Σ1) and p(x|ω2) ∼ N(µ2,Σ2) . The distributions have
the following parameters
µ1 =
µ2 =
Σ1 = Σ2 =
The classes have equal priors, i.e., P(ω1) = P(ω2).
Calculate the equation for the decision boundary in the form x2 = mx1+ c.
c) Consider a K nearest neighbour classifier being used to classify 1-D data belonging to
classes ω1 and ω2. The training samples for the two classes are
ω1 = {1,3,5,7,9} ω2 = {2,4,6,8}
The diagram below shows the decision boundaries and class labels for the case K = 1.
!1 !1 !1 !1 !1!2 !2 !2 !2
Make similar sketches for the cases K = 3, K = 5, K = 7 and K = 9. [25%]
COM 2004 7 TURN OVER
COM 2004
d) Consider a K-nearest neighbour that uses a Euclidean distance measure, K = 1, and
the following samples as training data,
ω1 =
(0,1)T ,(1,1)T ,(1,2)T
ω2 =
(1,0)T ,(2,1)T
A point is selected uniformly at random from the region defined by 0≤ x1≤ 2, 0≤ x2≤ 2.
What is the probability that the point is classified as belonging to class ω1? [Hint: start
by sketching the decision boundary.]
COM 2004 8 CONTINUED
COM 2004
4. This question concerns clustering and dimensionality reduction.
D E F
a) The points in the above figure are to be clustered using the agglomerative clustering
algorithm. The cluster-to-cluster distance is defined to be the minimum point-to-point
distance. In the initial clustering, C0, each point is in a separate cluster and the clustering
can be presented as a set of sets as such.
C0 =
(i) Point-to-point distances are measured using the Manhattan distance. Perform
the algorithm and use set notation to show the clustering after each iteration.
(ii) Point-to-point distances are measured using the Euclidean distance. Perform the
algorithm and use set notation to show the clustering after each iteration. [10%]
(iii) Draw a dendogram to represent the hierarchical sequence of clusterings found
when using the Euclidean distance. [10%]
(iv) Consider a naive implementation of the algorithm which does not store point-
to-point distance measures across iterations. Calculate the precise number of
point-to-point distances that would need to be computed for each iteration when
performing the clustering described in 4 (a)(ii). [20%]
COM 2004 9 TURN OVER
COM 2004
b) Consider the following dimensionality reduction techniques
• Discrete Cosine Transform (DCT),
• Principal Coponent Analysis (PCA) transform and
• Linear Discriminant Analysis (LDA) transform.
They can all be expressed as a linear transform of the form Y = XM where M is the
transform and X is the data matrix and Y is the data matrix after dimensionality reduc-
(i) Copy the table below and fill the cells with either ‘Yes’ or ‘No’ to indicate what
information is required in order to determineM .
The data points The class labels
(ii) PCA is being used to reduce the dimensionality of a 1000 sample set from 50
dimensions down to 5. State the number of rows and columns in each of Y , X
and M in the equation Y = XM that performs the dimensionality reduction. [15%]
(iii) Dimensionality reduction is to be used to reduce two dimensional data to one
dimension. Draw a scatter plot for a two class problem in which PCA would
perform very badly but for which LDA would work well. [20%]
COM 2004 10 | {"url":"https://www.xuebaunion.com/detail/598.html","timestamp":"2024-11-05T20:11:40Z","content_type":"text/html","content_length":"20126","record_id":"<urn:uuid:7a46bb53-d7cb-445c-978e-dd78524ece22>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00782.warc.gz"} |
Kilometers to Miles Converter
⇅ Switch toMiles to Kilometers Converter
How to use this Kilometers to Miles Converter 🤔
Follow these steps to convert given length from the units of Kilometers to the units of Miles.
1. Enter the input Kilometers value in the text field.
2. The calculator converts the given Kilometers into Miles in realtime ⌚ using the conversion formula, and displays under the Miles label. You do not need to click any button. If the input changes,
Miles value is re-calculated, just like that.
3. You may copy the resulting Miles 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 Kilometers to Miles?
The formula to convert given length from Kilometers to Miles is:
Length[(Miles)] = Length[(Kilometers)] / 1.609344
Substitute the given value of length in kilometers, i.e., Length[(Kilometers)] in the above formula and simplify the right-hand side value. The resulting value is the length in miles, i.e., Length
Calculation will be done after you enter a valid input.
Consider that a high-end electric car has a maximum range of 400 kilometers on a single charge.
Convert this range from kilometers to Miles.
The length in kilometers is:
Length[(Kilometers)] = 400
The formula to convert length from kilometers to miles is:
Length[(Miles)] = Length[(Kilometers)] / 1.609344
Substitute given weight Length[(Kilometers)] = 400 in the above formula.
Length[(Miles)] = 400 / 1.609344
Length[(Miles)] = 248.5485
Final Answer:
Therefore, 400 km is equal to 248.5485 mi.
The length is 248.5485 mi, in miles.
Consider that a private helicopter has a flight range of 150 kilometers.
Convert this range from kilometers to Miles.
The length in kilometers is:
Length[(Kilometers)] = 150
The formula to convert length from kilometers to miles is:
Length[(Miles)] = Length[(Kilometers)] / 1.609344
Substitute given weight Length[(Kilometers)] = 150 in the above formula.
Length[(Miles)] = 150 / 1.609344
Length[(Miles)] = 93.2057
Final Answer:
Therefore, 150 km is equal to 93.2057 mi.
The length is 93.2057 mi, in miles.
Kilometers to Miles Conversion Table
The following table gives some of the most used conversions from Kilometers to Miles.
Kilometers (km) Miles (mi)
0 km 0 mi
1 km 0.6214 mi
2 km 1.2427 mi
3 km 1.8641 mi
4 km 2.4855 mi
5 km 3.1069 mi
6 km 3.7282 mi
7 km 4.3496 mi
8 km 4.971 mi
9 km 5.5923 mi
10 km 6.2137 mi
20 km 12.4274 mi
50 km 31.0686 mi
100 km 62.1371 mi
1000 km 621.3712 mi
10000 km 6213.7119 mi
100000 km 62137.1192 mi
A kilometer (km) is a unit of length in the International System of Units (SI), equal to 0.6214 miles. One kilometer is one thousand meters.
The prefix "kilo-" means one thousand. A kilometer is defined by 1000 times the distance light travels in 1/299,792,458 seconds. This definition may change, but a kilometer will always be one
thousand meters.
Kilometers are used to measure distances on land in most countries. However, the United States and the United Kingdom still often use miles. The UK has adopted the metric system, but miles are still
used on road signs.
A mile (symbol: mi or m) is a unit of length commonly used in the United States and the United Kingdom. One mile is equal to 1.60934 kilometers.
The mile originated from the Roman mile, which was 1,000 paces. The current definition of a mile is based on the international agreement and equals exactly 1,609.344 meters.
Miles are mainly used to measure distances in the United States and the United Kingdom, especially for road systems. While most of the world uses kilometers, the mile remains prevalent in these
Frequently Asked Questions (FAQs)
1. How do I convert kilometers to miles?
To convert kilometers to miles, divide the number of kilometers by 1.60934. This is because 1 mile equals approximately 1.60934 kilometers. For example, if you have 10 kilometers, dividing by 1.60934
gives approximately 6.21 miles. The formula is: miles = kilometers ÷ 1.60934, making the conversion simple and accurate.
2. What is the formula for converting kilometers to miles?
The formula for converting kilometers to miles is: miles = kilometers ÷ 1.60934. Since 1 mile equals approximately 1.60934 kilometers, dividing the length in kilometers by this factor converts the
measurement to miles. This conversion is often used when traveling or calculating long distances.
3. How many miles are there in a kilometer?
There are approximately 0.621371 miles in a kilometer. This conversion factor helps translate distances from the metric system (kilometers) to the imperial system (miles), which is commonly used in
the United States and the United Kingdom for road signs and travel distances.
4. Why do we divide by 1.60934 to convert kilometers to miles?
We divide by 1.60934 because this is the number of kilometers in 1 mile. The metric system uses kilometers, while the imperial system uses miles. By dividing the length in kilometers by 1.60934, we
convert the measurement to miles, making it easier to understand distances in countries that use the imperial system.
5. How can I convert miles back to kilometers?
To convert miles back to kilometers, multiply the number of miles by 1.60934. This is because 1 mile equals approximately 1.60934 kilometers. For example, if you have 5 miles, multiplying by 1.60934
gives approximately 8.05 kilometers. The formula is: kilometers = miles × 1.60934, useful for international travel or distance calculations.
6. What is the difference between kilometers and miles?
Kilometers and miles are both units of length, but they belong to different measurement systems. A kilometer is part of the metric system, while a mile is part of the imperial system. One mile is
longer than a kilometer, with 1 mile equaling approximately 1.60934 kilometers. Kilometers are widely used worldwide, while miles are commonly used in the US and UK. | {"url":"https://convertonline.org/unit/?convert=kilometers-miles","timestamp":"2024-11-13T04:50:42Z","content_type":"text/html","content_length":"101927","record_id":"<urn:uuid:1745ba79-a19e-46f0-a3c6-8a83d970be22>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00401.warc.gz"} |
TY - JOUR ID - TI - Comparative Investigation of the Spherical Acoustic Microbubble Models in an Unbounded Liquid. AU - Kawa Mustafa Aziz Manmi AU - Kawa M.A. MANMI PY - 2020 VL - 32 IS - 4 SP - 82
EP - 88 JO - Zanco Journal of Pure and Applied Sciences مجلة زانكۆ للعلوم الصرفة والتطبيقية SN - 22180230 24123986 AB - Microbubble oscillating associated with many applications in biomedical and
engineering sectors. The spherical oscillations of a single microbubble submerged in a quiescent liquid exerted by an acoustic force can be governed either by the Rayleigh-Plesset (RP) equation or by
the Keller-Miksis (KM) equation under different physical assumptions. In this paper, both models were numerically and analytically analyzed, and the systematic parametric study was performed. The
viscosity and compressibility effects and linearization in both models were investigated with the aids of MATLAB and Maple tools. In KM, the effects of the linear and nonlinear equations of states
(EOS) compared for updating density with time. At the minimum bubble radius, the liquid viscosity surrounding bubble surface expected to be decreased due to rising in temperature. This leads to
effects the maximum bubble radius for upcoming cycles.
ER - | {"url":"https://iasj.net/iasj/export/198398?format=endNote","timestamp":"2024-11-12T06:50:55Z","content_type":"text/plain","content_length":"1780","record_id":"<urn:uuid:82d5956b-3eca-48c5-abb1-583fcb886d23>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00564.warc.gz"} |
Collinearity Explained
In geometry, collinearity of a set of points is the property of their lying on a single line.^[1] A set of points with this property is said to be collinear (sometimes spelled as colinear^[2]). In
greater generality, the term has been used for aligned objects, that is, things being "in a line" or "in a row".
Points on a line
In any geometry, the set of points on a line are said to be collinear. In Euclidean geometry this relation is intuitively visualized by points lying in a row on a "straight line". However, in most
geometries (including Euclidean) a line is typically a primitive (undefined) object type, so such visualizations will not necessarily be appropriate. A model for the geometry offers an interpretation
of how the points, lines and other object types relate to one another and a notion such as collinearity must be interpreted within the context of that model. For instance, in spherical geometry,
where lines are represented in the standard model by great circles of a sphere, sets of collinear points lie on the same great circle. Such points do not lie on a "straight line" in the Euclidean
sense, and are not thought of as being in a row.
A mapping of a geometry to itself which sends lines to lines is called a collineation; it preserves the collinearity property. The linear maps (or linear functions) of vector spaces, viewed as
geometric maps, map lines to lines; that is, they map collinear point sets to collinear point sets and so, are collineations. In projective geometry these linear mappings are called homographies and
are just one type of collineation.
Examples in Euclidean geometry
In any triangle the following sets of points are collinear:
• The orthocenter, the circumcenter, the centroid, the Exeter point, the de Longchamps point, and the center of the nine-point circle are collinear, all falling on a line called the Euler line.
• The de Longchamps point also has other collinearities.
• Any vertex, the tangency of the opposite side with an excircle, and the Nagel point are collinear in a line called a splitter of the triangle.
• The midpoint of any side, the point that is equidistant from it along the triangle's boundary in either direction (so these two points bisect the perimeter), and the center of the Spieker circle
are collinear in a line called a cleaver of the triangle. (The Spieker circle is the incircle of the medial triangle, and its center is the center of mass of the perimeter of the triangle.)
• Any vertex, the tangency of the opposite side with the incircle, and the Gergonne point are collinear.
• From any point on the circumcircle of a triangle, the nearest points on each of the three extended sides of the triangle are collinear in the Simson line of the point on the circumcircle.
• The lines connecting the feet of the altitudes intersect the opposite sides at collinear points.^[3]
• A triangle's incenter, the midpoint of an altitude, and the point of contact of the corresponding side with the excircle relative to that side are collinear.^[4]
• Menelaus' theorem states that three points
on the sides (some
) of a triangle opposite vertices
respectively are collinear if and only if the following products of segment lengths are equal:
P[1A][2] ⋅ P[2A][3] ⋅ P[3A][1=P][1A][3] ⋅ P[2A][1] ⋅ P[3A][2.]
• The incenter, the centroid, and the Spieker circle's center are collinear.
• The circumcenter, the Brocard midpoint, and the Lemoine point of a triangle are collinear.^[5]
• Two perpendicular lines intersecting at the orthocenter of a triangle each intersect each of the triangle's extended sides. The midpoints on the three sides of these points of intersection are
collinear in the Droz–Farny line.
• In a convex quadrilateral whose opposite sides intersect at and, the midpoints of are collinear and the line through them is called the Newton line. If the quadrilateral is a tangential
quadrilateral, then its incenter also lies on this line.^[6]
• In a convex quadrilateral, the quasiorthocenter, the "area centroid", and the quasicircumcenter are collinear in this order, and .^[7] (See Quadrilateral#Remarkable points and lines in a convex
• Other collinearities of a tangential quadrilateral are given in Tangential quadrilateral#Collinear points.
• In a cyclic quadrilateral, the circumcenter, the vertex centroid (the intersection of the two bimedians), and the anticenter are collinear.
• In a cyclic quadrilateral, the area centroid, the vertex centroid, and the intersection of the diagonals are collinear.
• In a tangential trapezoid, the tangencies of the incircle with the two bases are collinear with the incenter.
• In a tangential trapezoid, the midpoints of the legs are collinear with the incenter.
• Pascal's theorem (also known as the Hexagrammum Mysticum Theorem) states that if an arbitrary six points are chosen on a conic section (i.e., ellipse, parabola or hyperbola) and joined by line
segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet in three points which lie on a straight line, called the Pascal line of
the hexagon. The converse is also true: the Braikenridge–Maclaurin theorem states that if the three intersection points of the three pairs of lines through opposite sides of a hexagon lie on a
line, then the six vertices of the hexagon lie on a conic, which may be degenerate as in Pappus's hexagon theorem.
Conic sections
• By Monge's theorem, for any three circles in a plane, none of which is completely inside one of the others, the three intersection points of the three pairs of lines, each externally tangent to
two of the circles, are collinear.
• In an ellipse, the center, the two foci, and the two vertices with the smallest radius of curvature are collinear, and the center and the two vertices with the greatest radius of curvature are
• In a hyperbola, the center, the two foci, and the two vertices are collinear.
• The center of mass of a conic solid of uniform density lies one-quarter of the way from the center of the base to the vertex, on the straight line joining the two.
• The centroid of a tetrahedron is the midpoint between its Monge point and circumcenter. These points define the Euler line of the tetrahedron that is analogous to the Euler line of a triangle.
The center of the tetrahedron's twelve-point sphere also lies on the Euler line.
Collinearity of points whose coordinates are given
In coordinate geometry, in -dimensional space, a set of three or more distinct points are collinear if and only if, the matrix of the coordinates of these vectors is of rank 1 or less. For example,
given three points
\begin{align} X&=(x[1,x][2,...,x][n),]\\ Y&=(y[1,y][2,...,y][n),]\\ Z&=(z[1,z][2,...,z][n),]\end{align}
if the
\begin{bmatrix} x[1]&x[2]&...&x[n]\\ y[1]&y[2]&...&y[n]\\ z[1]&z[2]&...&z[n \end{bmatrix} ]
is of
1 or less, the points are collinear.
Equivalently, for every subset of, if the matrix
\begin{bmatrix} 1&x[1]&x[2]&...&x[n]\\ 1&y[1]&y[2]&...&y[n]\\ 1&z[1]&z[2]&...&z[n \end{bmatrix} ]
is of
2 or less, the points are collinear. In particular, for three points in the plane, the above matrix is square and the points are collinear if and only if its
is zero; since that 3 × 3 determinant is plus or minus twice the area of a triangle with those three points as vertices, this is equivalent to the statement that the three points are collinear if and
only if the triangle with those points as vertices has zero area.
Collinearity of points whose pairwise distances are given
A set of at least three distinct points is called straight, meaning all the points are collinear, if and only if, for every three of those points, the following determinant of a Cayley–Menger
determinant is zero (with meaning the distance between and, etc.):
\det\begin{bmatrix}0&d(AB)^2&d(AC)^2&1\\ d(AB)^2&0&d(BC)^2&1\\ d(AC)^2&d(BC)^2&0&1\\ 1&1&1&0 \end{bmatrix}=0.
This determinant is, by Heron's formula, equal to -16 times the square of the area of a triangle with side lengths ; so checking if this determinant equals zero is equivalent to checking whether the
triangle with vertices has zero area (so the vertices are collinear).
Equivalently, a set of at least three distinct points are collinear if and only if, for every three of those points with greater than or equal to each of and, the triangle inequality holds with
Number theory
Two numbers and are not coprime - that is, they share a common factor other than 1 - if and only if for a rectangle plotted on a square lattice with vertices at, at least one interior point is
collinear with and .
Concurrency (plane dual)
In various plane geometries the notion of interchanging the roles of "points" and "lines" while preserving the relationship between them is called plane duality. Given a set of collinear points, by
plane duality we obtain a set of lines all of which meet at a common point. The property that this set of lines has (meeting at a common point) is called concurrency, and the lines are said to be
concurrent lines. Thus, concurrency is the plane dual notion to collinearity.
Collinearity graph
Given a partial geometry, where two points determine at most one line, a collinearity graph of is a graph whose vertices are the points of, where two vertices are adjacent if and only if they
determine a line in .
Usage in statistics and econometrics
See main article: Multicollinearity.
In statistics, collinearity refers to a linear relationship between two explanatory variables. Two variables are perfectly collinear if there is an exact linear relationship between the two, so the
correlation between them is equal to 1 or -1. That is, and are perfectly collinear if there exist parameters
such that, for all observations, we have
This means that if the various observations are plotted in the plane, these points are collinear in the sense defined earlier in this article.
Perfect multicollinearity refers to a situation in which explanatory variables in a multiple regression model are perfectly linearly related, according to
for all observations . In practice, we rarely face perfect multicollinearity in a data set. More commonly, the issue of multicollinearity arises when there is a "strong linear relationship" among two
or more independent variables, meaning that
where the variance of
is relatively small.
The concept of lateral collinearity expands on this traditional view, and refers to collinearity between explanatory and criteria (i.e., explained) variables.^[8]
Usage in other areas
Antenna arrays
In telecommunications, a collinear (or co-linear) antenna array is an array of dipole antennas mounted in such a manner that the corresponding elements of each antenna are parallel and aligned, that
is they are located along a common line or axis.
The collinearity equations are a set of two equations, used in photogrammetry and computer stereo vision, to relate coordinates in an image (sensor) plane (in two dimensions) to object coordinates
(in three dimensions). In the photography setting, the equations are derived by considering the central projection of a point of the object through the optical centre of the camera to the image in
the image (sensor) plane. The three points, object point, image point and optical centre, are always collinear. Another way to say this is that the line segments joining the object points with their
image points are all concurrent at the optical centre.^[9]
See also
• Incidence (geometry)#Collinearity
Notes and References | {"url":"https://everything.explained.today/Collinearity/","timestamp":"2024-11-08T08:33:33Z","content_type":"text/html","content_length":"36730","record_id":"<urn:uuid:9164b70d-f8c0-48ff-9a33-7b3dbeed4fbe>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00721.warc.gz"} |
How to Avoid Making Mistakes with Numbers
Oct 17, 2024
During a board meeting I volunteer to attend, the treasurer presented financial projections that showed our fund balance would grow from under $2M to over $7M in 5 years. People were excited and
pleasantly surprised by the large growth. I was skeptical. There were sheets full of calculations, but I didn’t need to understand each cell to know the calculations required a deeper look.
About 10 minutes later, I found the error. The treasurer had added the principal and earnings from each year to the principal rather than just adding earnings. Instead of growing by ~$5M, our fund
balance would grow by ~$500K.
It’s easy to make mistakes when working through complicated calculations involving many assumptions and different formulas. I spent the first few years of my consulting career wrestling with
spreadsheets with too many sheets to be able to see the titles.
How do we do detailed, complex work without making simple mistakes? We must see the forest and the trees. We have to zoom in and get the detailed calculations and assumptions right, and then zoom out
and place the results in a broader context.
Seeing the Trees
When it comes to getting the details right, we can use 3 tools:
Build in error checks: In most complicated quantitative analyses, there are multiple ways to get to the same number. For example, if you sum the column and the row in a table, you should generally
get the same response. Or if you calculate a number using a formula and then also sum the numbers that make it up, you should get the same answer. Running the calculation both ways and comparing the
answers is a way to error check your spreadsheet.
You can also do this when calculations should add up to 100%. Look for creative ways to build error checks into your calculations.
Use completion checklists: When calculating a number requires 15 steps, it can be hard to check your work without doubting that you missed a mistake. This is where a completion checklist – a list of
tasks you must do to consider your work reviewed – can help.
Surgeon Atul Gawande explains the power of checklists to help with the most complicated tasks in The Checklist Manifesto. Build a completion checklist for your calculations and then run through every
task on the list before considering it ready for sharing.
Avoid manual data entry: If your plan is to re-enter formulas into multiple cells or manually enter numbers into your spreadsheet, pause. You can copy and paste values or formulas from any cell to
any other range of cells. You can also drag formulas from one cell to the next using the bottom right corner of the cell – or use keyboard shortcuts like CTRL+D (fill down) or CTRL+R (fill right).
If you have the data in a pdf, use free online PDF to Excel converters like ilovepdf.com or Adobe’s online services. These work pretty well for simple tables. If you have your data in a document or
email and pasting it puts everything in one cell or one column of cells, use Excel’s text to columns feature to split out values from one cell into a table.
Using these 3 tools can help you avoid detailed mistakes by reducing the opportunity to make mistakes and increasing the rigor of your reviews of your own work.
Seeing the Forest
Seeing the forest means putting the number in the context of a bigger picture. It can be done quickly without spiraling into the depths of spreadsheet. The other day I was in a conversation where
someone said our county brings in $20 billion in property taxes. This triggered alarm bells.
I know our county budget is only $710 million, which made this immediately impossible. But I also remembered that the state of California’s budget was in the $200 billion range. We are 1 of 58
counties and in the smallest third. Could just our property taxes be 10% of the state’s full budget? Not a chance. The person had mistaken a “b” for a “m”. Our property taxes are in the $20 million
In this example, I put their number in the context of the County and State budgets – and could immediately see they were wrong.
In the investment example from earlier, I put the returns in the context of the Rule of 72, which says 72 divided by the rate of return equals the number of years it takes to double your investment.
The Rule of 72 predicts it would take 14.4 years to double your investment with an annual return of 5%. The Board treasurer suggested our investment would more than triple in 5 years. Impossible.
The key to putting the number in context is finding a benchmark. Here are 4 ways you can find benchmarks:
Internal reference: You can use past performance as a benchmark to catch mistakes. If your Sales Director projects that sales will grow by 8% this year, but they have only grown by 4% annually for
the last 5 years, she better offer a good explanation for how she’ll double sales growth this year.
External reference: Sometimes your best bet is to look outside your company or experience for a benchmark. For example, if your Human Resources Director wants to offer a brand-new college graduate an
entry-level job in your division for $48/hour, you might compare it to your state’s minimum wage. The offer is 3 times higher than California’s minimum wage, making it a generous offer depending on
Expand Out: In other cases, you need to manipulate the number to get a benchmark that makes sense. To expand out, multiply the number in question by another number to get to a figure you can easily
compare. In the hiring example above, calculate the annual pay for the $48 hourly wage: $48/hour * 2,080 hours = $99,840/year. You know your frontline managers make $100,000, so this would be way too
Consider another example. Your Division Manager recommends giving his 125 people a 5% raise. You know that the average compensation in his division is $75,000, making this a $468,750 increase in
costs. Last year, the division had a net profit of $500,000. Without an increase in profitability, this raise would eliminate almost all of the division’s profit.
Drill Down: Rather than expanding out, you can drill down, typically to find a rate that is easier to benchmark. For example, imagine a friend tells you they ran a marathon last weekend in 1 hour and
30 minutes. You don’t know if that is a good time, but you know how fast people can run a mile. In this case, you calculate the per mile pace: 1:30 hours / 26.2 miles = 3:26 minutes/mile. You know
the fastest mile time is around 4 minutes per mile, so you call your friend’s bluff.
Here’s another example: Your Production Manager projects they’ll produce 2,000 head gaskets next month. That is 100 gaskets per day. You know that the most you’ve produced in day is 80 gaskets, so
this seems unlikely.
For each of these examples, there could be a detailed spreadsheet to back up the individual’s recommendation. But seeing the forest enables you to skip the spreadsheet and sense-check their numbers
with simple mental math you can do in less than 5 minutes. This math won’t get you to the right answer, but it will keep you and those you work with from sharing answers that are clearly wrong.
Avoiding Mistakes
It’s important we catch mistakes – particularly egregious ones – before they impact our organizations. Part of the way we do this is by seeing the trees – getting the details right – by building in
error checks, using completion checklists, and avoiding manual data entry. The other way we do this is by seeing the forest – putting the numbers in a broader context – whether that be by comparing
them to an internal, external, expanded, or drilled-down benchmark.
Practice these behaviors routinely and they will become automatic. You’ll discover mistakes without intentionally looking for them and you’ll save your organizations money and hardship. | {"url":"https://www.zarvana.com/blog/how-to-avoid-making-mistakes-with-numbers","timestamp":"2024-11-06T23:16:45Z","content_type":"text/html","content_length":"56038","record_id":"<urn:uuid:ca2fc4d3-3f52-48c5-9032-9e5a1789b3ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00823.warc.gz"} |
mic - AI Alignment Forum
Sorted by
Choosing actions which exploit known biases and blind spots in humans (as the Cicero Diplomacy agent may be doing [Bakhtin et al., 2022]) or in learned reward models.
I've spent several hours reading dialogue involving Cicero, and it's not at all evident to me that it's "exploiting known biases and blind spots in humans". It is, however, good at proposing and
negotiating plans, as well as accumulating power within the context of the game.
The prompt "Are birds real?" is somewhat more likely, given the "Birds aren't real" conspiracy theory, but still can yield a similarly formatted answer to "Are bugs real?"
The answer makes a lot more sense when you ask a question like "Are monsters real?" or "Are ghosts real?" It seems that with FeedMe, text-davinci-002 has been trained to respond with a template
answer about how "There is no one answer to this question", and it has learned to misgeneralize this behavior to questions about real phenomena, such as "Are bugs real?"
Has anyone tried to work on this experimentally?
Is the auditing game essentially Trojan detection?
Models that have been RLHF'd (so to speak), have different world priors in ways that aren't really all that intuitive (see Janus' work on mode collapse
Janus' post on mode collapse is about text-davinci-002, which was trained using supervised fine-tuning on high-quality human-written examples (FeedME), not RLHF. It's evidence that supervised
fine-tuning can lead to weird output, not evidence about what RLHF does.
I haven't seen evidence that RLHF'd text-davinci-003 appears less safe compared to the imitation-based text-davinci-002.
I first learned about the term "structural risk" in this article from 2019 by Remco Zwetsloot and Allan Dafoe, which was included in the AGI Safety Fundamentals curriculum.
To make sure these more complex and indirect effects of technology are not neglected, discussions of AI risk should complement the misuse and accident perspectives with a structural perspective.
This perspective considers not only how a technological system may be misused or behave in unintended ways, but also how technology shapes the broader environment in ways that could be disruptive
or harmful. For example, does it create overlap between defensive and offensive actions, thereby making it more difficult to distinguish aggressive actors from defensive ones? Does it produce
dual-use capabilities that could easily diffuse? Does it lead to greater uncertainty or misunderstanding? Does it open up new trade-offs between private gain and public harm, or between the
safety and performance of a system? Does it make competition appear to be more of a winner-take-all situation? We call this perspective “structural” because it focuses on what social scientists
often refer to as “structure,” in contrast to the “agency” focus of the other perspectives.
□ GPT-3, for instance, is notorious for outputting text that is impressive, but not of the desired “flavor” (e.g., outputting silly text when serious text is desired), and researchers often
have to tinker with inputs considerably to yield desirable outputs.
Is this specifically referring to the base version of GPT-3 before instruction fine-tuning (davinci rather than text-davinci-002, for example)? I think it would be good to clarify that.
As an overly simplistic example, consider an overseer that attempts to train a cleaning robot by providing periodic feedback to the robot, based on how quickly the robot appears to clean a room;
such a robot might learn that it can more quickly “clean” the room by instead sweeping messes under a rug.^[15]
This doesn't seem concerning as human users would eventually discover that the robot has a tendency to sweep messes under the rug, if they ever look under the rug, and the developers would retrain
the AI to resolve this issue. Can you think of an example that would be more problematic, in which the misbehavior wouldn't be obvious enough to just be trained away?
I think humans doing METR's tasks are more like "expert-level" rather than average/"human-level". But current LLM agents are also far below human performance on tasks that don't require any special
From GAIA:
GAIA proposes real-world questions that require a set of fundamental abilities such as reasoning, multi-modality handling, web browsing, and generally tool-use proficiency. GAIA questions are
conceptually simple for humans yet challenging for most advanced AIs: we show that human respondents obtain 92% vs. 15% for GPT-4 equipped with plugins. [Note: The latest highest AI agent score
is now 39%.] This notable performance disparity contrasts with the recent trend of LLMs outperforming humans on tasks requiring professional skills in e.g. law or chemistry. GAIA's philosophy
departs from the current trend in AI benchmarks suggesting to target tasks that are ever more difficult for humans. We posit that the advent of Artificial General Intelligence (AGI) hinges on a
system's capability to exhibit similar robustness as the average human does on such questions.
And LLMs and VLLMs seriously underperform humans in VisualWebArena, which tests for simple web-browsing capabilities:
I don't know if being able to autonomously make money should be a necessary condition to qualify as AGI. But I would feel uncomfortable calling a system AGI if it can't match human performance at
simple agent tasks.
Thanks for writing this! Here is a quick explanation of all the math concepts – mostly written by ChatGPT with some manual edits.
A basis for a vector space is a set of linearly independent vectors that can be used to represent any vector in the space as a linear combination of those basis vectors. For example, in
two-dimensional Euclidean space, the standard basis is the set of vectors (1, 0) and (0, 1), which are called the "basis vectors."
A change of basis is the process of expressing a vector in one basis in terms of another basis. For example, if we have a vector v in two-dimensional Euclidean space and we want to express it in
terms of the standard basis, we can write v as a linear combination of (1, 0) and (0, 1). Alternatively, we could choose a different basis for the space, such as the basis formed by the vectors (4,
2) and (3, 5). In this case, we would express v in terms of this new basis by writing it as a linear combination of (4, 2) and (3, 5).
A vector space is a set of vectors that can be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with
scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms.
Examples of vector spaces include the set of all two-dimensional vectors (i.e., the set of all points in two-dimensional Euclidean space), the set of all polynomials with real coefficients, and the
set of all continuous functions from a given set to the real numbers. A vector space can be thought of as a geometric object, but it does not necessarily have a canonical basis, meaning that there is
not a preferred set of basis vectors that can be used to represent all the vectors in the space.
A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. A matrix is a linear map between two vector spaces, or from a vector space to itself, because it can
take any vector in the original vector space and transform it into a new vector in the target vector space using a set of linear equations. Each column of the matrix represents one of the new basis
vectors, which are used to define the transformation. In the expression , we take each element of the original vector and multiply it by the corresponding element in the appropriate column of the
matrix, and then add these products together to create the new vector.
The singular value decomposition (SVD) is a factorization of a matrix M into the product of three matrices: , where U and V are orthogonal matrices and S is a diagonal matrix with non-negative real
numbers on the diagonal, called the "singular values" of M. The SVD is a useful tool for understanding the properties of a matrix and for solving certain types of linear systems. It can also be used
for data compression, image processing, and other applications.
An orthogonal matrix (or orthonormal matrix) is a square matrix whose columns and rows are mutually orthonormal (i.e., they are orthogonal and have unit length). Orthogonal matrices have the property
that their inverse is equal to their transpose.
Changing to an orthonormal basis can be importantly different from just any change of basis because it has certain computational advantages. For example, when working with an orthonormal basis, the
inner product of two vectors can be computed simply as the sum of the products of their corresponding components, without the need to use any weights or scaling factors. This can make certain
calculations, such as finding the length of a vector or the angle between two vectors, simpler and more efficient.
Eigenvalues and eigenvectors are special types of scalars and vectors that are associated with a linear map or a matrix. If M is a linear map or matrix and v is a non-zero vector, then v is an
eigenvector of M if there exists a scalar λ, called an eigenvalue, such that . In other words, when a vector is multiplied by the matrix M, the resulting vector is a scalar multiple of the original
vector. Eigenvalues and eigenvectors are important because they provide insight into the properties of the linear map or matrix. For example, the eigenvalues of a matrix can tell us whether it is
singular (i.e., not invertible) or whether it is diagonalizable (i.e., can be expressed in the form , where P is a matrix and D is a diagonal matrix). The eigenvectors of a matrix can also be used to
determine its rank, nullity, and other characteristics.
Probability basics: Probability is a measure of the likelihood of an event occurring. It is typically represented as a number between 0 and 1, where 0 indicates that the event is impossible and 1
indicates that the event is certain to occur. The probability of an event occurring can be calculated by counting the number of ways in which the event can occur, divided by the total number of
possible outcomes.
Basics of distributions: A distribution is a function that describes the probability of a random variable taking on different values. The expected value of a distribution is a measure of the center
of the distribution, and it is calculated as the weighted average of the possible values of the random variable, where the weights are the probabilities of each value occurring. The standard
deviation is a measure of the dispersion of the distribution, and it is calculated as the square root of the variance, which is the expected value of the squared deviation of a random variable from
its mean. A normal distribution (or Gaussian distribution) is a continuous probability distribution with a bell-shaped curve, which is defined by its mean and standard deviation.
Log likelihood: The log likelihood of a statistical model is a measure of how well the model fits a given set of data. It is calculated as the logarithm of the probability of the data given the
model, and it is often used to compare the relative fit of different models.
Maximum value estimators: A maximum value estimator is a statistical method that is used to estimate the value of a parameter that maximizes a given objective function. Examples of maximum value
estimators include the maximum likelihood estimator and the maximum a posteriori estimator.
• The maximum likelihood estimator is a method for estimating the parameters of a statistical model based on the principle that the parameters that maximize the likelihood of the data are the most
likely to have generated the data.
• The maximum a posteriori (MAP) estimator is a method for estimating the parameters of a statistical model based on the principle that the parameters that maximize the posterior probability of the
data are the most likely to have generated the data. The posterior probability is the probability of the data given the model and the prior knowledge about the parameters. The MAP estimator is
often used in Bayesian inference, and it is a popular method for estimating the parameters of a model in the presence of prior knowledge.
Random variables: A random variable is a variable whose value is determined by the outcome of a random event. For example, the toss of a coin is a random event, and the number of heads that result
from a series of coin tosses is a random variable.
Central limit theorem: The central limit theorem is a statistical theorem that states that, as the sample size of a random variable increases, the distribution of the sample means approaches a normal
distribution, regardless of the distribution of the underlying random variable.
Calculus basics: Calculus is a branch of mathematics that deals with the study of rates of change and the accumulation of quantities. It is a fundamental tool in the study of functions and is used to
model and solve problems in a variety of fields, including physics, engineering, and economics.
Gradients: In calculus, the gradient of a (scalar-valued multivariate differentiable) function is a vector that describes the direction in which the function is increasing most quickly. It is
calculated as the partial derivative of the function with respect to each variable.
The chain rule: The chain rule is a fundamental rule of calculus that allows us to calculate the derivative of a composite function. It states that if f is a function of g, and g is a function of x,
then the derivative of f with respect to x is equal to the derivative of f with respect to g times the derivative of g with respect to x. In tohers words, (df / dx) = (df / dg) * (dg / dx).
On backpropagation:
Backpropagation is an algorithm for training artificial neural networks, which are machine learning models inspired by the structure and function of the brain. It is used to adjust the weights
and biases of the network in order to minimize the error between the predicted output and the desired output of the network.
The idea behind backpropagation is that, given a multivariate function that describes the relationships between the input variables and the output variables of a neural network, we can use the
chain rule to calculate the gradient of the function with respect to the weights and biases of the network. The gradient tells us how the error changes as we adjust the weights and biases, and we
can use this information to update the weights and biases in a way that reduces the error.
To understand why backpropagation is just the chain rule on multivariate functions, it's helpful to consider the structure of a neural network. A neural network consists of layers of
interconnected nodes, each of which performs a calculation based on the inputs it receives from the previous layer. The output of the network is a function of the inputs, and the weights and
biases of the network determine how the inputs are transformed as they pass through the layers of the network.
The process of backpropagation involves starting at the output layer of the network and working backwards through the layers, using the chain rule to calculate the gradients of the weights and
biases at each layer. This is done by calculating the derivative of the error with respect to the output of each layer, and then using the chain rule to propagate these derivatives back through
the layers of the network. This allows us to calculate the gradients of the weights and biases at each layer, which we can use to update the weights and biases in a way that minimizes the error.
Overall, backpropagation is an efficient and effective way to train neural networks because it allows us to calculate the gradients of the weights and biases efficiently, using the chain rule to
propagate the derivatives through the layers of the network. This enables us to adjust the weights and biases in a way that minimizes the error, which is essential for the effective operation of
the network.
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[Solved] min[(x1−x2)2+(3+(1−x12)−4x2)2],∀ x1,x2∈R, is -... | Filo
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(b) Let and
or and
Thus, lies on the circle and lies on the parabola .
Thus, the given expression is the shortest distance between the curve and .
Now, the shortest distance always occurs along the common normal to the curves and normal to the circle passes through the centre of the circle.
Normal to parabola is . It passes through centre of circle .
Therefore, which has only one real value
Hence, the corresponding point on the parabola is
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Question Text , is
Topic Conic Sections
Subject Mathematics
Class Class 11
Answer Type Text solution:1
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June / July 2015 Newsflash: Tips & Techniques
Upper Elementary:
Q: What number comes right before 1,000?
A: While we may be first inclined to say “999,” there is actually no answer to this question! This is because fractions and decimals can get ever so close to 1,000 without equaling 1,000: 999 ½, 999
999/1000, 999.99999999999999999999…, etc.
Middle School:
Q: City B is 100 miles from City A. City C is 25 miles from City B.
What is the maximum distance between City A and City C? The minimum distance?
What does a drawing of the possible positions of City C look like?
A: To understand this problem, it helps if you visualize the cities as points on a line.
As the distance from City C to City B is less than the distance from City A to City B, we can determine that either City C is between City A and City B (ACB) or City B is in between Cities A and C
(ABC). That said, the maximum distance between City A and City C is the distance between City A and B (100 miles) plus the distance between City B and City C (25 miles). 100 + 25 = 125 miles
The minimum distance can be found when we situate City C between City A and City B. It is equal to the distance between City A and City B (100 miles) minus the distance between City B and City C (25
miles). 100 – 25 = 75 miles
To visualize the possible positions of City C, think of a circle. If City B is the circle’s center point, then City C can be located anywhere along the circle’s circumference. Therefore, the distance
between City B and City C is the circle’s radius. A circle can have an infinite number of radii. As we can’t draw “an infinite number of radii,” the possible positions of City C (relative to City B)
can be expressed as a circle with all the possible radii you can draw. | {"url":"https://www.mathnasium.sg/2015/06/june-july-2015-newsflash-tips-techniques","timestamp":"2024-11-05T05:44:37Z","content_type":"text/html","content_length":"53857","record_id":"<urn:uuid:301f264a-689f-463a-b42b-f94d0c19de77>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00332.warc.gz"} |
Lie Algebras, Vertex Operator Algebras and Their Applicationssearch
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Lie Algebras, Vertex Operator Algebras and Their Applications
Edited by: Yi-Zhi Huang : Rutgers University, Piscataway, NJ
eBook ISBN: 978-0-8218-8121-7
Product Code: CONM/442.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
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Lie Algebras, Vertex Operator Algebras and Their Applications
Edited by: Yi-Zhi Huang : Rutgers University, Piscataway, NJ
eBook ISBN: 978-0-8218-8121-7
Product Code: CONM/442.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
• Contemporary Mathematics
Volume: 442; 2007; 474 pp
MSC: Primary 17; 81; 82
The articles in this book are based on talks given at the international conference “Lie algebras, vertex operator algebras and their applications”, in honor of James Lepowsky and Robert Wilson on
their sixtieth birthdays, held in May of 2005 at North Carolina State University. Some of the papers in this volume give inspiring expositions on the development and status of their respective
research areas. Others outline and explore the challenges as well as the future directions of research for the twenty-first century. The focus of the papers in this volume is mainly on Lie
algebras, quantum groups, vertex operator algebras and their applications to number theory, combinatorics and conformal field theory.
This book is useful for graduate students and researchers in mathematics and mathematical physics who want to be introduced to different areas of current research or explore the frontiers of
research in the areas mentioned above.
Graduate students and research mathematicians interested in lie algebras, their representations, and generalizations.
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Requests
Volume: 442; 2007; 474 pp
MSC: Primary 17; 81; 82
The articles in this book are based on talks given at the international conference “Lie algebras, vertex operator algebras and their applications”, in honor of James Lepowsky and Robert Wilson on
their sixtieth birthdays, held in May of 2005 at North Carolina State University. Some of the papers in this volume give inspiring expositions on the development and status of their respective
research areas. Others outline and explore the challenges as well as the future directions of research for the twenty-first century. The focus of the papers in this volume is mainly on Lie algebras,
quantum groups, vertex operator algebras and their applications to number theory, combinatorics and conformal field theory.
This book is useful for graduate students and researchers in mathematics and mathematical physics who want to be introduced to different areas of current research or explore the frontiers of research
in the areas mentioned above.
Graduate students and research mathematicians interested in lie algebras, their representations, and generalizations.
Permission – for use of book, eBook, or Journal content
Please select which format for which you are requesting permissions. | {"url":"https://bookstore.ams.org/CONM/442","timestamp":"2024-11-11T19:16:01Z","content_type":"text/html","content_length":"94445","record_id":"<urn:uuid:59b96693-9dbc-4bfa-ba96-f076bdb83960>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00082.warc.gz"} |
Foundational Skills in Mathematics 3-5
General Course Information and Notes
Version Description
This course supports students who need additional instruction in foundational mathematics skills as it relates to core instruction. Instruction will use explicit, systematic, and sequential
approaches to mathematics instruction addressing all domains including number sense & operations, fractions, algebraic reasoning, geometric reasoning, measurement and data analysis &
probability. Teachers will use the listed standards that correspond to each students’ needs.
Effective instruction matches instruction to the need of the students in the group and provides multiple opportunities to practice the skill and receive feedback. The additional time allotted for
this course is in addition to core instruction. The intervention includes materials and strategies designed to supplement core instruction.
General Notes
Florida’s Benchmarks for Excellent Student Thinking (B.E.S.T.) Standards
This course includes Florida’s B.E.S.T. ELA Expectations (EE) and Mathematical Thinking and Reasoning Standards (MTRs) for students. Florida educators should intentionally embed these standards
within the content and their instruction as applicable. For guidance on the implementation of the EEs and MTRs, please visit https://www.cpalms.org/Standards/BEST_Standards.aspx and select the
appropriate B.E.S.T. Standards package.
English Language Development ELD Standards Special Notes Section:
Teachers are required to provide listening, speaking, reading and writing instruction that allows English language learners (ELL) to communicate information, ideas and concepts for academic success
in the content area of Mathematics. For the given level of English language proficiency and with visual, graphic, or interactive support, students will interact with grade-level words, expressions,
sentences and discourse to process or produce language necessary for academic success. The ELD standard should specify a relevant content area concept or topic of study chosen by curriculum
developers and teachers which maximizes an ELL’s need for communication and social skills. To access an ELL supporting document which delineates performance definitions and descriptors, please click
on the following link: https://cpalmsmediaprod.blob.core.windows.net/uploads/docs/standards/eld/ma.pdf
General Information
Course Number: 5012015
Course Path:
Abbreviated Title: FDN SKILLS MATH 3-5
Course Length: Multiple (M) - Course length can vary
Course Type: Elective Course
Course Level: 1
Course Status: State Board Approved
Educator Certifications
One of these educator certification options is required to teach this course.
Classical Education - Restricted (Elementary and Secondary Grades K-12)
Section 1012.55(5), F.S., authorizes the issuance of a classical education teaching certificate, upon the request of a classical school, to any applicant who fulfills the requirements of s. 1012.56
(2)(a)-(f) and (11), F.S., and Rule 6A-4.004, F.A.C. Classical schools must meet the requirements outlined in s. 1012.55(5), F.S., and be listed in the FLDOE Master School ID database, to request a
restricted classical education teaching certificate on behalf of an applicant.
Student Resources
Vetted resources students can use to learn the concepts and skills in this course.
Original Student Tutorials
Educational Games
Arithmetic Workout:
This tutorial will help you to brush up on your multiplication, division and factoring skills with this exciting game.
Type: Educational Game
Sigma Prime: A Prime Factorization Game:
This fun and engaging game will test your knowledge of whole numbers as prime or composite. As you shoot the asteroids with a particular factor, the asteroids will break down by that chosen factor.
Keep shooting the correct factors to totally eliminate the asteroids. But be careful, shooting the wrong factor has consequences!
Type: Educational Game
Ice Ice Maybe: An Operations Estimation Game:
This fun and interactive game helps practice estimation skills, using various operations of choice, including addition, subtraction, multiplication, division, using decimals, fractions, and percents.
Various levels of difficulty make this game appropriate for multiple age and ability levels.
Addition/Subtraction: The addition and subtraction of whole numbers, the addition and subtraction of decimals.
Multiplication/Division: The multiplication and addition of whole numbers.
Percentages: Identify the percentage of a whole number.
Fractions: Multiply and divide a whole number by a fraction, as well as apply properties of operations.
Type: Educational Game
Flower Power: An Ordering of Rational Numbers Game:
This is a fun and interactive game that helps students practice ordering rational numbers, including decimals, fractions, and percents. You are planting and harvesting flowers for cash. Allow the bee
to pollinate, and you can multiply your crops and cash rewards!
Type: Educational Game
Fraction Quiz:
Test your fraction skills by answering questions on this site. This quiz asks you to simplify fractions, convert fractions to decimals and percentages, and answer algebra questions involving
fractions. You can even choose difficulty level, question types, and time limit.
Type: Educational Game
Estimator Quiz:
In this activity, students are quizzed on their ability to estimate sums, products, and percentages. The student can adjust the difficulty of the problems and how close they have to be to the actual
answer. This activity allows students to practice estimating addition, multiplication, or percentages of large numbers. This activity includes supplemental materials, including background information
about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Educational Game
Change Maker:
This interactive applet gives students practice in making change in U.S. dollars and in four other currencies. Students are presented with a purchase amount and the amount paid, and they must enter
the quantity of each denomination that make up the correct change. Students are rewarded for correct answers and are shown the correct change if they err. There are four levels of difficulty, ranging
from amounts less than a dollar to amounts over $100.
Type: Educational Game
Maze Game:
In this activity, students enter coordinates to make a path to get to a target destination while avoiding mines. This activity allows students to explore Cartesian coordinates and the Cartesian
coordinate plane. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for
use with the java applet.
Type: Educational Game
Educational Software / Tool
Arithmetic Quiz:
In this activity, students solve arithmetic problems involving whole numbers, integers, addition, subtraction, multiplication, and division. This activity allows students to track their progress in
learning how to perform arithmetic on whole numbers and integers. This activity includes supplemental materials, including background information about the topics covered, a description of how to use
the application, and exploration questions for use with the java applet.
Type: Educational Software / Tool
Lesson Plan
Holidays that Celebrate America:
In this lesson plan, students will explore the history and meaning behind various patriotic holidays and make personal connections with those holidays including, Constitution Day, Memorial Day,
Veteran’s Day, Patriot Day, President’s Day, Independence Day, and Medal of Honor Day.
Type: Lesson Plan
Problem-Solving Tasks
Computing Volume Progression 1:
Students are asked to determine the number of unit cubes needed to construct cubes with given dimensions.
Type: Problem-Solving Task
Computing Volume Progression 3:
Students are asked to find the height of a rectangular prism when given the length, width and volume.
Type: Problem-Solving Task
Computing Volume Progression 4:
Students are asked to apply knowledge of volume of rectangular prisms to find the volume of an irregularly shaped object using the principle of displacement.
Type: Problem-Solving Task
The Square Counting Shortcut:
This is a rectangle subdivision task; ideally instead of counting each square. students should break the letters into rectangles, multiply to find the areas, and add up the areas. However, students
should not be discouraged from using individual counting to start if they are stuck. Often students will get tired of counting and devise the shortcut method themselves.
Type: Problem-Solving Task
Rounding to 50 or 500:
The purpose of this task is to answer multiple questions regarding rounding. There still may be students who laboriously list every number; the teacher should encourage a more thoughtful approach.
Type: Problem-Solving Task
Representing Half of a Circle:
This task continues "Which pictures represent half of a circle?" moving into more complex shapes where geometric arguments about cutting or work using simple equivalences of fractions is required to
analyze the picture. In order for students to be successful with this task, they need to understand that area is additive.
Type: Problem-Solving Task
Geometric pictures of one half:
This task presents students with some creative geometric ways to represent the fraction one half. The goal is both to appeal to students' visual intuition while also providing a hands on activity to
decide whether or not two areas are equal. In order for students to be successful with this task, they need to understand that area is additive.
Type: Problem-Solving Task
The Longest Walk:
After students have drawn and measured their ten line segments, it might be more useful for the class to discuss part (b) as a whole group. It is a good idea to have the students use color to help
them keep track of the connection between a line that they have drawn and the corresponding data point on the graph.
Type: Problem-Solving Task
The Stamp Collection:
For students who are unfamiliar with this language the task provides a preparation for the later understanding that a fraction of a quantity is that fraction times the quantity.
Type: Problem-Solving Task
Two interpretations of division:
Both of the questions are solved by the division problem 12÷3 but what happens to the ribbon is different in each case. The problem can be solved with a drawing of a tape diagram or number line. For
problem 1, the line must be divided into 3 equal parts. The second problem can be solved by successive subtraction of 3 feet to see how many times it fits in 12.
Type: Problem-Solving Task
To regroup or not to regroup:
This task presents an incomplete problem and asks students to choose numbers to subtract (subtrahends) so that the resulting problem requires different types of regrouping. This way students have to
recognize the pattern and not just follow a memorized algorithm--in other words, they have to think about what happens in the subtraction process when we regroup. This task is appropriate to use
after students have learned the standard US algorithm.
Type: Problem-Solving Task
Ordering 4-digit numbers:
It is common for students to compare multi-digit numbers just by comparing the first digit, then the second digit, and so on. This task includes three-digit numbers with large hundreds digits and
four-digit numbers with small thousands digits so that students must infer the presence of a 0 in the thousands place in order to compare. It also includes numbers with strategically placed zeros and
an unusual request to order them from greatest to least in addition to the more traditional least to greatest.
Type: Problem-Solving Task
Lines of symmetry for triangles:
This activity provides students an opportunity to recognize these distinguishing features of the different types of triangles before the technical language has been introduced. For finding the lines
of symmetry, cut-out models of the four triangles would be helpful so that the students can fold them to find the lines.
Type: Problem-Solving Task
Lines of symmetry for quadrilaterals:
This task provides students a chance to experiment with reflections of the plane and their impact on specific types of quadrilaterals. It is both interesting and important that these types of
quadrilaterals can be distinguished by their lines of symmetry.
Type: Problem-Solving Task
Lines of symmetry for circles:
This is an instructional task that gives students a chance to reason about lines of symmetry and discover that a circle has an an infinite number of lines of symmetry. Even though the concept of an
infinite number of lines is fairly abstract, students can understand infinity in an informal way.
Type: Problem-Solving Task
Finding an unknown angle:
The purpose of this task is to give students a problem involving an unknown quantity that has a clear visual representation. Students must understand that the four interior angles of a rectangle are
all right angles and that right angles have a measure of 90° and that angle measure is additive.
Type: Problem-Solving Task
Are these right?:
The purpose of this task is for students to measure angles and decide whether the triangles are right or not. Students should already understand concepts of angle measurement and know how to measure
angles using a protractor before working on this task.
Type: Problem-Solving Task
Making 22 Seventeenths in Different Ways:
This task is a straightforward task related to adding fractions with the same denominator. The main purpose is to emphasize that there are many ways to decompose a fraction as a sum of fractions.
Type: Problem-Solving Task
Listing fractions in increasing size:
The fractions for this task have been carefully chosen to encourage and reward different methods of comparison. The first solution judiciously uses each of the following strategies when appropriate:
comparing to benchmark fractions, finding a common denominator, finding a common numerator. The second and third solution shown use only either common denominators or numerators. Teachers should
encourage multiple approaches to solving the problem. This task is mostly intended for instructional purposes, although it has value as a formative assessment item as well.
Type: Problem-Solving Task
How Many Tenths and Hundredths?:
The purpose of this task is for students to finish the equations to make true statements. Parts (a) and (b) have the same solution, which emphasizes that the order in which we add doesn't matter
(because addition is commutative), while parts (c) and (d) emphasize that the position of a digit in a decimal number is critical. The student must really think to encode the quantity in positional
notation. In parts (e), (f), and (g), the base-ten units in 14 hundredths are bundled in different ways. In part (e), "hundredths" are thought of as units: 14 things = 10 things + 4 things. Part (h)
addresses the notion of equivalence between hundredths and tenths.
Type: Problem-Solving Task
Fraction Equivalence:
Students may not articulate every detail, but the basic idea for a case like the one shown here is that when you have equivalent fractions, you have just cut the pieces that represent the fraction
into more but smaller pieces. Explaining fraction equivalences at higher grades can be a bit more involved (e.g. 6/8=9/12), but it can always be framed as subdividing the same quantity in different
Type: Problem-Solving Task
Explaining Fraction Equivalence with Pictures:
The purpose of this task is to provide students with an opportunity to explain fraction equivalence through visual models in a particular example. Students will need more opportunities to think about
fraction equivalence with different examples and models, but this task represents a good first step.
Type: Problem-Solving Task
Expanded Fractions and Decimals:
The purpose of this task is for students to show they understand the connection between fraction and decimal notation by writing the same numbers both ways. Comparing and contrasting the two
solutions shown below shows why decimal notation can be confusing. The first solution shows the briefest way to represent each number, and the second solution makes all the zeros explicit.
Type: Problem-Solving Task
Dimes and Pennies:
The purpose of this task is to help students gain a better understanding of fractions through the use of dimes and pennies.
Type: Problem-Solving Task
Comparing two different pizzas:
The focus of this task is on understanding that fractions, in an explicit context, are fractions of a specific whole. In this this problem there are three different wholes: the medium pizza, the
large pizza, and the two pizzas taken together. This task is best suited for instruction. Students can practice explaining their reasoning to each other in pairs or as part of a whole group
Type: Problem-Solving Task
Comparing Sums of Unit Fractions:
The purpose of this task is to help develop students' understanding of addition of fractions; it is intended as an instructional task. Notice that students are not asked to find the sum so this may
be given to students who are limited to computing sums of fractions with the same denominator. Rather, they need to apply a firm understanding of unit fractions (fractions with one in the numerator)
and reason about their relative size.
Type: Problem-Solving Task
Writing a Mixed Number as an Equivalent Fraction:
The purpose of this task is to help students understand and articulate the reasons for the steps in the usual algorithm for converting a mixed number into an equivalent fraction. Step two shows that
the algorithm is merely a shortcut for finding a common denominator between two fractions. This concept is an important precursor to adding mixed numbers and fractions with like denominators and as
such, step two should be a point of emphasis. This task is appropriate for either instruction or formative assessment.
Type: Problem-Solving Task
Using Place Value:
Each part of this task highlights a slightly different aspect of place value as it relates to decimal notation. More than simply being comfortable with decimal notation, the point is for students to
be able to move fluidly between and among the different ways that a single value can be represented and to understand the relative size of the numbers in each place.
Type: Problem-Solving Task
Using Benchmarks to Compare Fractions:
This task is intended primarily for instruction. The goal is to provide examples for comparing two fractions, 1/5 and 2/7 in this case, by finding a benchmark fraction which lies in between the two.
In Melissa's example, she chooses 1/4 as being larger than 1/5 and smaller than 2/7.
Type: Problem-Solving Task
Sugar in six cans of soda:
This task provides a familiar context allowing students to visualize multiplication of a fraction by a whole number. This task could form part of a very rich activity which includes studying soda can
Type: Problem-Solving Task
This task provides a context where it is appropriate for students to subtract fractions with a common denominator; it could be used for either assessment or instructional purposes. For this
particular task, teachers should anticipate two types of solution approaches: one where students subtract the whole numbers and the fractions separately and one where students convert the mixed
numbers to improper fractions and then proceed to subtract.
Type: Problem-Solving Task
Money in the piggy bank:
This task is designed to help students focus on the whole that a fraction refers. It provides a context where there are two natural ways to view the coins. While the intent is to deepen a student's
understanding of fractions, it does go outside the requirements of the standard.
Type: Problem-Solving Task
Comparing Growth, Variation 2:
The purpose of this task is to assess students’ understanding of multiplicative and additive reasoning. We would hope that students would be able to identify that Student A is just looking at how
many feet are being added on, while Student B is comparing how much the snakes grew in comparison to how long they were to begin with.
Type: Problem-Solving Task
Comparing Growth, Variation 1:
The purpose of this task is to foster a classroom discussion that will highlight the difference between multiplicative and additive reasoning. Some students will argue that they grew the same amount
(an example of "additive thinking"). Students who are studying multiplicative comparison problems might argue that Jewel grew more since it grew more with respect to its original length (an example
of "multiplicative thinking").
Type: Problem-Solving Task
Carnival Tickets:
The purpose of this task is for students to solve multi-step problems in a context involving a concept that supports financial literacy, namely inflation. Inflation is a sustained increase in the
average price level. In this task, students can see that if the price level increases and people’s incomes do not increase, they aren’t able to purchase as many goods and services; in other words,
their purchasing power decreases.
Type: Problem-Solving Task
Double Plus One:
The purpose of this task is to help students gain a better understanding of patterns. This task is meant to be used in an instructional setting.
Type: Problem-Solving Task
Comparing Money Raised:
The purpose of this task is to give students a better understanding of multiplicative comparison word problems with money.
Type: Problem-Solving Task
Comparing Fractions with a Different Whole:
This task is meant to address a common error that students make, namely, that they represent fractions with different wholes when they need to compare them. This task is meant to generate classroom
discussion related to comparing fractions.
Type: Problem-Solving Task
Comparing Fractions:
The purpose of this task is for students to compare fractions using common numerators and common denominators and to recognize equivalent fractions.
Type: Problem-Solving Task
Closest to 1/2:
How students tackle the problem and the amount of work they show on the number line can provide insight into the sophistication of their thinking. As students partition the interval between 0 and 1
into eighths, they will need to recognize that 1/2=4/8. Students who systematically plot every point, even 9/8, which is larger even than 1 may still be coming to grips with the relative size of
Type: Problem-Solving Task
Locating Fractions Greater than One on the Number Line:
The goal of this task is to help students gain a better understanding of fractions and their place on the number line.
Type: Problem-Solving Task
Jon and Charlie's Run:
The purpose of this task is to present students with a context where they need to explain why two simple fractions are equivalent and is most appropriate for instruction.
Type: Problem-Solving Task
Find 2/3:
This simple-looking problem reveals much about how well students understand unit fractions as well as representing fractions on a number line.
Type: Problem-Solving Task
Find 1:
This task includes the seeds of several important ideas. Part a presents the student with the opportunity to use a unit fraction to find 1 on the number line. Part b helps reinforce the notion that
when a fraction has a numerator that is larger than the denominator, it has a value greater than 1 on the number line.
Type: Problem-Solving Task
Video Game Scores:
This task asks students to exercise both of these complementary skills, writing an expression in part (a) and interpreting a given expression in (b). The numbers given in the problem are deliberately
large and "ugly" to discourage students from calculating Eric's and Leila's scores. The focus of this problem is not on numerical answers, but instead on building and interpreting expressions that
could be entered in a calculator or communicated to another student unfamiliar with the context.
Type: Problem-Solving Task
Which is Closer to 1?:
The purpose of this task is for students to identify which fraction is closest to the whole number 1.
Type: Problem-Solving Task
Ordering Fractions:
The purpose of this task is to extend students' understanding of fraction comparison and is intended for an instructional setting.
Type: Problem-Solving Task
Naming the Whole for a Fraction:
The goal of this task is to show that when the whole is not specified, which fraction is being represented is left ambiguous.
Type: Problem-Solving Task
Locating Fractions Less than One on the Number Line:
In every part of this task, students must treat the interval from 0 to 1 as a whole, partition the whole into the appropriate number of equal sized parts, and then locate the fraction(s).
Type: Problem-Solving Task
Gifts from Grandma, Variation 1:
The first of these is a multiplication problem involving equal-sized groups. The next two reflect the two related division problems, namely, "How many groups?" and "How many in each group?"
Type: Problem-Solving Task
Analyzing Word Problems Involving Multiplication:
In this task, the students are not asked to find an answer, but are asked to analyze the problems and explain their thinking. In the process, they are faced with varying ways of thinking about
Type: Problem-Solving Task
Karl's Garden:
The purpose of the task is for students to solve a multi-step multiplication problem in a context that involves area. In addition, the numbers were chosen to determine if students have a common
misconception related to multiplication. Since addition is both commutative and associative, we can reorder or regroup addends any way we like. Students often believe the same is true for
Type: Problem-Solving Task
Identifying Multiples:
The goal of this task is to work on finding multiples of some whole numbers on a multiplication grid. After shading in the multiples of 2, 3, and 4 on the table, students will see a key difference.
The focus can be on identifying patterns or this can be an introduction or review of prime and composite numbers.
Type: Problem-Solving Task
Classroom Supplies:
The purpose of this task is for students to solve problems involving the four operations and draw a scaled bar graph to represent a data set with several categories.
Type: Problem-Solving Task
Box of Clay:
This purpose of this task is to help students understand what happens when you scale the dimensions of a right rectangular solid. This task provides an opportunity to compare the relative volumes of
boxes in order to calculate the mass of clay required to fill them. These relative volumes can be calculated geometrically, filling the larger box with smaller boxes, or arithmetically using the
given dimensions.
Type: Problem-Solving Task
What is 23 ÷ 5?:
When a division problem involving whole numbers does not result in a whole number quotient, it is important for students to be able to decide whether the context requires the result to be reported as
a whole number with remainder (as with Part (b)) or a mixed number/decimal (as with Part (c)). Part (a) presents two variations on a context that require these two different responses to highlight
the distinction between them.
Type: Problem-Solving Task
How Much Pie?:
The purpose of this task is to help students see the connection between a÷b and a/b in a particular concrete example. This task is probably best suited for instruction or formative assessment.
Type: Problem-Solving Task
How many servings of oatmeal?:
This task provides a context for performing division of a whole number by a unit fraction. This problem is a "How many groups?'' example of division: the "groups'' in this case are the servings of
oatmeal and the question is asking how many servings (or groups) there are in the package.
Type: Problem-Solving Task
Painting a room:
The purpose of this task is to provide students with a situation in which it is natural for them to divide a unit fraction by a non-zero whole number. Determining the amount of paint that Kulani
needs for each wall illustrates an understanding of the meaning of dividing a unit fraction by a non-zero whole number.
Type: Problem-Solving Task
Painting a Wall:
The purpose of this task is for students to find the answer to a question in context that can be represented by fraction multiplication. This task is appropriate for either instruction or assessment
depending on how it is used and where students are in their understanding of fraction multiplication.
Type: Problem-Solving Task
Origami Stars:
The purpose of this task is to present students with a situation in which they need to divide a whole number by a unit fraction in order to find a solution. Calculating the number of origami stars
that Avery and Megan can make illustrates student understanding of the process of dividing a whole number by a unit fraction.
Type: Problem-Solving Task
Mixed Numbers with Unlike Denominators:
The purpose of this task is to help students realize there are different ways to add mixed numbers and is most appropriate for use in an instructional setting. The two primary ways one can expect
students to add are converting the mixed numbers to fractions greater than 1 or adding the whole numbers and fractional parts separately. It is good for students to develop a sense of which approach
would be better in a particular context.
Type: Problem-Solving Task
Making S'Mores:
The purpose of this instructional task is to motivate a discussion about adding fractions and the meaning of the common denominator. The different parts of the task have students moving back and
forth between the abstract representation of the fractions and the meaning of the fractions in the context.
Type: Problem-Solving Task
Making Cookies:
This tasks lends itself very well to multiple solution methods. Students may learn a lot by comparing different methods. Students who are already comfortable with fraction multiplication can go
straight to the numeric solutions given below. Students who are still unsure of the meanings of these operations can draw pictures or diagrams.
Type: Problem-Solving Task
The purpose of this task is to present students with a situation where it is natural to add fractions with unlike denominators; it can be used for either assessment or instructional purposes.
Teachers should anticipate two types of solutions: one where students calculate the distance Alex ran to determine an answer, and one where students compare the two parts of his run to benchmark
Type: Problem-Solving Task
To Multiply or not to multiply?:
The purpose of this task is to familiarize students with multiplying fractions with real-world questions.
Type: Problem-Solving Task
Seeing is Believing:
The purpose of this task is to help students see that 4×(9+2) is four times as big as (9+2). Though this task may seem very simple, it provides students and teachers with a very useful visual for
interpreting an expression without evaluating it because they can see for themselves that 4×(9+2) is four times as big as (9+2).
Type: Problem-Solving Task
Salad Dressing:
The purpose of this task is to have students add fractions with unlike denominators and divide a unit fraction by a whole number. This accessible real-life context provides students with an
opportunity to apply their understanding of addition as joining two separate quantities.
Type: Problem-Solving Task
Running to School:
The task could be one of the first activities for introducing the multiplication of fractions. The task has fractions which are easy to draw and provides a linear situation. Students benefit from
reasoning through the solution to such word problems before they are told that they can be solved by multiplying the fractions; this helps them develop meaning for fraction multiplication.
Type: Problem-Solving Task
Running a Mile:
The solution uses the idea that multiplying by a fraction less than 1 results in a smaller value. The students need to explain why that is so.
Type: Problem-Solving Task
Reasoning about Multiplication:
This is a good task to work with kids to try to explain their thinking clearly and precisely, although teachers should be willing to work with many different ways of explaining the relationship
between the magnitude of the factors and the magnitude of the product.
Type: Problem-Solving Task
Comparing Products:
The purpose of this task is to generate a classroom discussion that helps students synthesize what they have learned about multiplication in previous grades. It builds on applying properties of
operations as strategies to multiply and divide and interpreting a multiplication equation as a comparison.
Type: Problem-Solving Task
Words to Expressions 1:
This problem allows student to see words that can describe an expression although the solution requires nested parentheses. Additionally , the words (add, sum) and (product, multiply) are all
strategically used so that the student can see that these words have related meanings.
Type: Problem-Solving Task
Watch Out for Parentheses 1:
This problem asks the student to evaluate six numerical expressions that contain the same integers and operations yet have differing results due to placement of parentheses. This type of problem
helps students to see structure in numerical expressions. In later grades they will be working with similar ideas in the context of seeing and using structure in algebraic expressions.
Type: Problem-Solving Task
Minutes and Days:
This task requires division of multi-digit numbers in the context of changing units. In addition, the conversion problem requires two steps since 2011 minutes needs to be converted first to hours and
minutes and then to days, hours, and minutes.
Type: Problem-Solving Task
Half of a Recipe:
This is the third problem in a series of three tasks involving fraction multiplication that can be solved with pictures or number lines. The first, Running to school, does not require that the unit
fractions that comprise 3/4 be subdivided in order to find 1/3 of 3/4. The second task, Drinking Juice, does require students to subdivide the unit fractions that comprise 1/2 in order to find 3/4 of
1/2. This task also requires subdivision and involves multiplying a fraction and a mixed number.
Type: Problem-Solving Task
Grass Seedlings:
The purpose of this task is to gain a better understanding of multiplying with fractions. Students should use the diagram provided to support their findings.
Type: Problem-Solving Task
This problem helps students gain a better understanding of multiplying with fractions.
Type: Problem-Solving Task
Folding Strips of Paper:
The purpose of this task is to provide students with a concrete experience they can relate to fraction multiplication. Perhaps more importantly, the task also purposefully relates length and
locations of points on a number line, a common trouble spot for students. This task is meant for instruction and would be a useful as part of an introductory unit on fraction multiplication.
Type: Problem-Solving Task
Finding Common Denominators to Subtract:
Part (a) of this task asks students to use two different denominators to subtract fractions. The purpose of this is to help students realize that any common denominator will work, not just the least
common denominator. Part (b) does not ask students to do it in more than one way; the purpose is to give them an opportunity to choose a denominator and possibly compare with another student who
chose a different denominator. The purpose of part (c) is to help students move away from a reliance on drawing pictures. Students can draw a picture if they want, but this subtraction problem is
easier to do symbolically, which helps students appreciate the power of symbolic notation.
Type: Problem-Solving Task
Finding Common Denominators to Add:
Part (a) of this task asks students to find and use two different common denominators to add the given fractions. The purpose of this question is to help students realize that they can use any common
denominator to find a solution, not just the least common denominator. Part (b) does not ask students to solve the given addition problem in more than one way. Instead, the purpose of this question
is to give students an opportunity to choose a denominator and possibly to compare their solution method with another student who chose a different denominator. The purpose of part (c) is to give
students who are ready to work symbolically a chance to work more efficiently.
Type: Problem-Solving Task
Converting Fractions of a Unit into a Smaller Unit:
The purpose of this task is to help students gain a better understanding of fractions and the conversion of fractions into smaller units.
Type: Problem-Solving Task
How many marbles?:
This task is intended to complement "How many servings of oatmeal?" and "Molly's run.'' All three tasks address the division problem 4÷1/3 but from different points of view. This task provides a how
many in each group version of 4÷1/3. This task should be done together with the "How many servings of oatmeal" task with specific attention paid to the very different pictures representing the two
Type: Problem-Solving Task
Egyptian Fractions:
One goal of this task is to help students develop comfort and ease with adding fractions with unlike denominators. Another goal is to help them develop fraction number sense by having students
decompose fractions.
Type: Problem-Solving Task
Drinking Juice:
This is the second problem in a series of three tasks involving fraction multiplication that can be solved with pictures or number lines. This task does require students to subdivide the unit
fractions that comprise 1/2 in order to find 3/4 of 1/2.
Type: Problem-Solving Task
Do These Add Up?:
This task addresses common errors that students make when interpreting adding fractions word problems. It is very important for students to recognize that they only add fractions when the fractions
refer to the same whole, and also when the fractions of the whole being added do not overlap. This set of questions is designed to enhance a student's understanding of when it is and is not
appropriate to add fractions.
Type: Problem-Solving Task
Dividing by One-Half:
This task requires students to recognize both "number of groups unknown" (part (a)) and "group size unknown" (part (d)) division problems in the context of a whole number divided by a unit fraction.
It also addresses a common misconception that students have where they confuse dividing by 2 or multiplying by 1/2 with dividing by 1/2.
Type: Problem-Solving Task
Connor and Makayla Discuss Multiplication:
The purpose of this task is to have students think about the meaning of multiplying a number by a fraction, and use this burgeoning understanding of fraction multiplication to make sense of the
commutative property of multiplication in the case of fractions.
Type: Problem-Solving Task
Comparing a Number and a Product:
The purpose of this task is for students to compare a number and its product with other numbers that are greater than and less than one. As written, this task could be used in a summative assessment
context, but it might be more useful in an instructional setting where students are asked to explain their answers either to a partner or in a whole class discussion.
Type: Problem-Solving Task
Calculator Trouble:
This particular problem deals with multiplication. Even though students can solve this problem by multiplying, it is unlikely they will. Here it is much easier to answer the question if you can think
of multiplying a number by a factor as scaling the number.
Type: Problem-Solving Task
Banana Pudding:
The purpose of this task is to provide students with a concrete situation they can model by dividing a whole number by a unit fraction. For students who are just beginning to think about the meaning
of division by a unit fraction (or students who have never cooked), the teacher can bring in a 1/4 cup measuring cup so that students can act it out. If students can reason through parts (a) and (b)
successfully, they will be well-situated to think about part (c) which could yield different solution methods.
Type: Problem-Solving Task
Plastic Building Blocks:
The purpose of this task is to have students add mixed numbers with like denominators. This task illustrates the different kinds of solution approaches students might take to such a task. Two general
approaches should be anticipated: one where students calculate exactly how many buckets of blocks the boys have to determine an answer, and one where students compare the given numbers to benchmark
Type: Problem-Solving Task
Running Laps:
The purpose of this task is for students to compare two fractions that arise in a context. Because the fractions are equal, students need to be able to explain how they know that. Some students might
stop at the second-to-last picture and note that it looks like they ran the same distance, but the explanation is not yet complete at that point.
Type: Problem-Solving Task
Toll Bridge Puzzle:
This task is intended to assess adding of four numbers as given in the standard while still being placed in a problem-solving context. As written this task is instructional; due to the random aspect
regarding when the correct route is found, it is not appropriate for assessment. This puzzle works well as a physical re-enactment, with paper plates marking the islands and strings with papers
attached for the tolls.
Type: Problem-Solving Task
Text Resources
Virtual Manipulatives
Build a Fraction:
This virtual manipulative will help the students to build fractions from shapes and numbers to earn stars in this fraction lab. To challenge the children there are multiple levels, where they can
earn lots of stars.
Some of the sample learning goals can be:
• Build equivalent fractions using numbers and pictures.
• Compare fractions using numbers and patterns
• Recognize equivalent simplified and unsimplified fractions
Type: Virtual Manipulative
Fraction Game:
This virtual manipulative allows individual students to work with fraction relationships. (There is also a link to a two-player version.)
Type: Virtual Manipulative
Order of Operations Quiz:
In this activity, students practice solving algebraic expressions using order of operations. The applet records their score so the student can track their progress. This activity allows students to
practice applying the order of operations when solving problems. This activity includes supplemental materials, including background information about the topics covered, a description of how to use
the application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
Parent Resources
Vetted resources caregivers can use to help students learn the concepts and skills in this course. | {"url":"https://www.cpalms.org/PreviewCourse/Preview/22939","timestamp":"2024-11-01T22:10:55Z","content_type":"text/html","content_length":"462726","record_id":"<urn:uuid:4db7da4a-974d-491b-af20-06ae84c4234d>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00256.warc.gz"} |
x-Douglas College Physics 1107 Fall 2019 Custom Textbook
Chapter 8 Linear Momentum and Collisions
8.6 Collisions of Point Masses in Two Dimensions
• Discuss two dimensional collisions as an extension of one dimensional analysis.
• Define point masses.
• Derive an expression for conservation of momentum along x-axis and y-axis.
• Describe elastic collisions of two objects with equal mass.
• Determine the magnitude and direction of the final velocity given initial velocity, and scattering angle.
In the previous two sections, we considered only one-dimensional collisions; during such collisions, the incoming and outgoing velocities are all along the same line. But what about collisions, such
as those between billiard balls, in which objects scatter to the side? These are two-dimensional collisions, and we shall see that their study is an extension of the one-dimensional analysis already
presented. The approach taken (similar to the approach in discussing two-dimensional kinematics and dynamics) is to choose a convenient coordinate system and resolve the motion into components along
perpendicular axes. Resolving the motion yields a pair of one-dimensional problems to be solved simultaneously.
One complication arising in two-dimensional collisions is that the objects might rotate before or after their collision. For example, if two ice skaters hook arms as they pass by one another, they
will spin in circles. We will not consider such rotation until later, and so for now we arrange things so that no rotation is possible. To avoid rotation, we consider only the scattering of point
masses—that is, structureless particles that cannot rotate or spin.
We start by assuming that [latex]\boldsymbol{\vec{\textbf{F}}_{\textbf{net}}=0},[/latex] so that momentum [latex]\vec{\textbf{p}}[/latex] is conserved. The simplest collision is one in which one of
the particles is initially at rest. (See Figure 1.) The best choice for a coordinate system is one with an axis parallel to the velocity of the incoming particle, as shown in Figure 1. Because
momentum is conserved, the components of momentum along the x– and y-axes (p[x] and p[y]) will also be conserved, but with the chosen coordinate system, p[y] is initially zero and p[x] is the
momentum of the incoming particle. Both facts simplify the analysis. (Even with the simplifying assumptions of point masses, one particle initially at rest, and a convenient coordinate system, we
still gain new insights into nature from the analysis of two-dimensional collisions.)
Figure 1. A two-dimensional collision with the coordinate system chosen so that m[2] is initially at rest and v[1] is parallel to the x-axis. This coordinate system is sometimes called the laboratory
coordinate system, because many scattering experiments have a target that is stationary in the laboratory, while particles are scattered from it to determine the particles that make-up the target and
how they are bound together. The particles may not be observed directly, but their initial and final velocities are.
Along the x-axis, the equation for conservation of momentum is
Where the subscripts denote the particles and axes and the primes denote the situation after the collision. In terms of masses and velocities, this equation is
But because particle 2 is initially at rest, this equation becomes
The components of the velocities along the x-axis have the form v cos θ. Because particle 1 initially moves along the x-axis, we find v[1x] = v[1].
Conservation of momentum along the x-axis gives the following equation:
where θ[1] and θ[2] are as shown in Figure 1.
Along the y-axis, the equation for conservation of momentum is
But v[1y] is zero, because particle 1 initially moves along the x-axis. Because particle 2 is initially at rest, v[2y] is also zero. The equation for conservation of momentum along the y-axis becomes
The components of the velocities along the y-axis have the form v sin θ.
Thus, conservation of momentum along the y-axis gives the following equation:
The equations of conservation of momentum along the x-axis and y-axis are very useful in analyzing two-dimensional collisions of particles, where one is originally stationary (a common laboratory
situation). But two equations can only be used to find two unknowns, and so other data may be necessary when collision experiments are used to explore nature at the subatomic level.
Example 1: Determining the Final Velocity of an Unseen Object from the Scattering of Another Object
Suppose the following experiment is performed. A 0.250-kg object (m[1]) is slid on a frictionless surface into a dark room, where it strikes an initially stationary object with mass of 0.400 kg (m[2]
). The 0.250-kg object emerges from the room at an angle of 45.0° with its incoming direction.
The speed of the 0.250-kg object is originally 2.00 m/s and is 1.50 m/s after the collision. Calculate the magnitude and direction of the velocity (v′[2] and θ[2]) of the 0.400-kg object after the
Momentum is conserved because the surface is frictionless. The coordinate system shown in Figure 2 is one in which m[2] is originally at rest and the initial velocity is parallel to the x-axis, so
that conservation of momentum along the x– and y-axes is applicable.
Everything is known in these equations except v′[2] and θ[2], which are precisely the quantities we wish to find. We can find two unknowns because we have two independent equations: the equations
describing the conservation of momentum in the x– and y– directions.
Solving m[1]v[1] = m[1]v′[1 ]cos θ[1] + m[2]v′[2 ]cos θ[2] for v′[2 ]cos θ[2] and 0 = m[1]v′[1] sin θ[1]+m[2]v′[2] sin θ[2] for v′[2] sin θ[2 ]and taking the ratio yields an equation (in which θ[2]
is the only unknown quantity. Applying the identity [latex]\boldsymbol{(\tan\theta=\frac{\sin\theta}{\cos\theta})},[/latex] we obtain:
Entering known values into the previous equation gives
[latex]\boldsymbol{\tan\theta_2\:=}[/latex][latex]\boldsymbol{\frac{(1.50\textbf{ m/s})(0.7071)}{(1.50\textbf{ m/s})(0.7071)-2.00\textbf{ m/s}}}[/latex][latex]\boldsymbol{=\:-1.129}.[/latex]
Angles are defined as positive in the counter clockwise direction, so this angle indicates that m[2] is scattered to the right in Figure 2, as expected (this angle is in the fourth quadrant). Either
equation for the x– or y-axis can now be used to solve for v′[2], but the latter equation is easiest because it has fewer terms.
Entering known values into this equation gives
[latex]\boldsymbol{v^{\prime}_2\:=-}[/latex][latex]\boldsymbol{(\frac{0.250\textbf{ kg}}{0.400\textbf{ kg}})}[/latex][latex]\boldsymbol{1.50\textbf{ m/s}-}[/latex][latex]\boldsymbol{(\frac{0.7071}
[latex]\boldsymbol{v^{\prime}_2=0.886\textbf{ m/s}}.[/latex]
It is instructive to calculate the internal kinetic energy of this two-object system before and after the collision. (This calculation is left as an end-of-chapter problem.) If you do this
calculation, you will find that the internal kinetic energy is less after the collision, and so the collision is inelastic. This type of result makes a physicist want to explore the system further.
Figure 2. A collision taking place in a dark room is explored in Example 1. The incoming object m[1] is scattered by an initially stationary object. Only the stationary object’s mass m[2] is known.
By measuring the angle and speed at which m[1] emerges from the room, it is possible to calculate the magnitude and direction of the initially stationary object’s velocity after the collision.
Elastic Collisions of Two Objects with Equal Mass
Some interesting situations arise when the two colliding objects have equal mass and the collision is elastic. This situation is nearly the case with colliding billiard balls, and precisely the case
with some subatomic particle collisions. We can thus get a mental image of a collision of subatomic particles by thinking about billiards (or pool). (Refer to Figure 1 for masses and angles.) First,
an elastic collision conserves internal kinetic energy. Again, let us assume object 2 (m[2]) is initially at rest. Then, the internal kinetic energy before and after the collision of two objects that
have equal masses is
Because the masses are equal, m[1] = m[2] = m. Algebraic manipulation (left to the reader) of conservation of momentum in the x– and y-directions can show that
(Remember that θ[2] is negative here.) The two preceding equations can both be true only if
There are three ways that this term can be zero. They are
• v′[1] = 0 head-on collision; incoming ball stops
• v′[2 ]= 0 no collision; incoming ball continues unaffected
• cos (θ[1 ]– θ[2]) = 0 angle of separation (θ[1 ]– θ[2]) is 90° after the collision
All three of these ways are familiar occurrences in billiards and pool, although most of us try to avoid the second. If you play enough pool, you will notice that the angle between the balls is very
close to 90° after the collision, although it will vary from this value if a great deal of spin is placed on the ball. (Large spin carries in extra energy and a quantity called angular momentum,
which must also be conserved.) The assumption that the scattering of billiard balls is elastic is reasonable based on the correctness of the three results it produces. This assumption also implies
that, to a good approximation, momentum is conserved for the two-ball system in billiards and pool. The problems below explore these and other characteristics of two-dimensional collisions.
Two-dimensional collision experiments have revealed much of what we know about subatomic particles, as we shall see in Chapter 32 Medical Applications of Nuclear Physics and Chapter 33 Particle
Physics. Ernest Rutherford, for example, discovered the nature of the atomic nucleus from such experiments.
Section Summary
• The approach to two-dimensional collisions is to choose a convenient coordinate system and break the motion into components along perpendicular axes. Choose a coordinate system with the[latex]\
boldsymbol{x}[/latex]-axis parallel to the velocity of the incoming particle.
• Two-dimensional collisions of point masses where mass 2 is initially at rest conserve momentum along the initial direction of mass 1 (the x-axis), stated by m[1]v[1] = m[1]v′[1] cos θ[1] + m[2]v′
[2] cos θ[2 ]and along the direction perpendicular to the initial direction (the y-axis) stated by 0 = m[1]v′[1y] + m[2]v′[2y].
• The internal kinetic before and after the collision of two objects that have equal masses is
• Point masses are structureless particles that cannot spin.
Conceptual Questions
1: Figure 3 shows a cube at rest and a small object heading toward it. (a) Describe the directions (angle θ[1]) at which the small object can emerge after colliding elastically with the cube. How
does θ[1] depend on b, the so-called impact parameter? Ignore any effects that might be due to rotation after the collision, and assume that the cube is much more massive than the small object. (b)
Answer the same questions if the small object instead collides with a massive sphere.
Figure 3. A small object approaches a collision with a much more massive cube, after which its velocity has the direction θ[1]. The angles at which the small object can be scattered are determined by
the shape of the object it strikes and the impact parameter b.
Problems & Exercises
1: Two identical pucks collide on an air hockey table. One puck was originally at rest. (a) If the incoming puck has a speed of 6.00 m/s and scatters to an angle of 30.0°, what is the velocity
(magnitude and direction) of the second puck? (You may use the result that θ[1] – θ[2] = 90° for elastic collisions of objects that have identical masses.) (b) Confirm that the collision is elastic.
2: Confirm that the results of the example Example 1 do conserve momentum in both the x– and y-directions.
3: A 3000-kg cannon is mounted so that it can recoil only in the horizontal direction. (a) Calculate its recoil velocity when it fires a 15.0-kg shell at 480 m/s at an angle of 20.0° above the
horizontal. (b) What is the kinetic energy of the cannon? This energy is dissipated as heat transfer in shock absorbers that stop its recoil. (c) What happens to the vertical component of momentum
that is imparted to the cannon when it is fired?
4: Professional Application
A 5.50-kg bowling ball moving at 9.00 m/s collides with a 0.850-kg bowling pin, which is scattered at an angle of 85.0° to the initial direction of the bowling ball and with a speed of 15.0 m/s. (a)
Calculate the final velocity (magnitude and direction) of the bowling ball. (b) Is the collision elastic? (c) Linear kinetic energy is greater after the collision. Discuss how spin on the ball might
be converted to linear kinetic energy in the collision.
5: Professional Application
Ernest Rutherford (the first New Zealander to be awarded the Nobel Prize in Chemistry) demonstrated that nuclei were very small and dense by scattering helium-4 nuclei (^4He) from gold-197 nuclei (^
197Au).The energy of the incoming helium nucleus was 8.00 × 10^-13 J, and the masses of the helium and gold nuclei were 6.68 × 10^-27 kg and 3.29 × 10^-25 kg, respectively (note that their mass ratio
is 4 to 197). (a) If a helium nucleus scatters to an angle of 120° during an elastic collision with a gold nucleus, calculate the helium nucleus’s final speed and the final velocity (magnitude and
direction) of the gold nucleus. (b) What is the final kinetic energy of the helium nucleus?
6: Professional Application
Two cars collide at an icy intersection and stick together afterward. The first car has a mass of 1200 kg and is approaching at 8.00 m/s due south. The second car has a mass of 850 kg and is
approaching at 17.0 m/s due west. (a) Calculate the final velocity (magnitude and direction) of the cars. (b) How much kinetic energy is lost in the collision? (This energy goes into deformation of
the cars.) Note that because both cars have an initial velocity, you cannot use the equations for conservation of momentum along the x-axis and y-axis; instead, you must look for other simplifying
7: Starting with equations m[1]v[1] = m[1]v′[1] cos θ[1] + m[2]v′[2] cos θ[2 ]and 0 = m[1]v′[1] sin θ[1] + m[2]v′[2] sin θ[2] for conservation of momentum in the x– and y-directions and assuming that
one object is originally stationary, prove that for an elastic collision of two objects of equal masses,
as discussed in the text.
8: Integrated Concepts
A 90.0-kg ice hockey player hits a 0.150-kg puck, giving the puck a velocity of 45.0 m/s. If both are initially at rest and if the ice is frictionless, how far does the player recoil in the time it
takes the puck to reach the goal 15.0 m away?
point masses
structureless particles with no rotation or spin
Problems & Exercises
1: (a) $$\boldsymbol{3.00\textbf{ m/s,}\: 60^0}$$ below x-axis (b) Find speed of first puck after collision: [latex]\boldsymbol{0=mv^{\prime}_1\sin30^0-mv^{\prime}_2\sin60^0\Rightarrow{v}^{\prime}_1=
v^{\prime}_2\frac{\sin60^0}{\sin30^0}=5.196\textbf{ m/s}}[/latex]
Verify that ratio of initial to final KE equals one:
[latex]\begin{array}{l} \boldsymbol{\textbf{KE}=\frac{1}{2}mv_1^2=18m\textbf{ J}} \\ \boldsymbol{\textbf{KE}=\frac{1}{2}mv^{\prime}_1{^2}+\frac{1}{2}mv^{\prime}_2{^2}=18m\textbf{ J}} \end{array}[/
3: (a) [latex]\boldsymbol{-2.26\textbf{ m/s}}[/latex] (b) [latex]\boldsymbol{7.63\times10^3\textbf{ J}}[/latex] (c) The ground will exert a normal force to oppose recoil of the cannon in the vertical
direction. The momentum in the vertical direction is transferred to the earth. The energy is transferred into the ground, making a dent where the cannon is. After long barrages, cannon have erratic
aim because the ground is full of divots.
5: (a) [latex]\boldsymbol{5.36\times10^5\textbf{ m/s}}[/latex] at [latex]\boldsymbol{-29.5^0}[/latex] (b) [latex]\boldsymbol{7.52\times10^{-13}\textbf{ J}}[/latex]
6. 8.46 m/s at an angle of 33.6 degrees south of west.
total energy before = 161000 J total kinetic energy after = 73400 J, so 87900 J was lost in the collision.
7: We are given that [latex]\boldsymbol{m_1=m_2\equiv{m}}.[/latex] The given equations then become:
Square each equation to get
[latex]\begin{array}{lcl} \boldsymbol{v_1^2} & \boldsymbol{=} & \boldsymbol{v^{\prime}_1{^2}\cos^2\theta_1+v^{\prime}_2{^2}\cos^2\theta_2+2v^{\prime}_1v^{\prime}_2\cos\theta_1\cos\theta_2} \\ \
boldsymbol{0} & \boldsymbol{=} & \boldsymbol{v^{\prime}_1{^2}\sin^2\theta_1+v^{\prime}_2{^2}\sin^2\theta_2+2v^{\prime}_1v^{\prime}_2\sin\theta_1\sin\theta_2.} \end{array}[/latex]
Add these two equations and simplify:
[latex]\begin{array}{lcl} \boldsymbol{v_1^2} & \boldsymbol{=} & \boldsymbol{v^{\prime}_1{^2}+v^{\prime}_2{^2}+2v^{\prime}_1v^{\prime}_2(\cos\theta_1\cos\theta_2+\sin\theta_1\sin\theta_2)} \\ {} & \
boldsymbol{=} & \boldsymbol{v^{\prime}_1{^2}+v^{\prime}_2{^2}+2v^{\prime}_1v^{\prime}_2[\frac{1}{2}\cos(\theta_1-\theta_2)+\frac{1}{2}\cos(\theta_1+\theta_2)+\frac{1}{2}\cos(\theta_1-\theta_2)-\frac
{1}{2}\cos(\theta_1+\theta_2)]} \\ {} & \boldsymbol{=} & \boldsymbol{v^{\prime}_1{^2}+v^{\prime}_2{^2}+2v^{\prime}_1v^{\prime}_2\cos(\theta_1-\theta_2).} \end{array}[/latex]
Multiply the entire equation by[latex]\boldsymbol{\frac{1}{2}m}[/latex]to recover the kinetic energy:
8: Player recoil velocity = 0.075 m/s Assuming constant velocity, the puck takes a time of = 15.0 m / 45.0 m/s = 0.333 seconds. The player travels a distance backward = 0.075 m/s x 0.333 s =
0.025 m | {"url":"https://pressbooks.bccampus.ca/introductorygeneralphysics1phys1107/chapter/8-6-collisions-of-point-masses-in-two-dimensions/","timestamp":"2024-11-04T18:28:06Z","content_type":"text/html","content_length":"126545","record_id":"<urn:uuid:f953b0d2-f074-4728-bc9d-a546c89f100c>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00777.warc.gz"} |
Fluctuations of the probability density of diffusing particles for different realizations of a random medium
We study the fluctuations of the probability density P(r,t) of diffusing particles to be at distance r at time t in the presence of random potentials, represented by random transition rates. We find
an exact relation which expresses all the moments of P(r,t) in terms of its first moment, for both quenched and annealed disorder and for any dimension. From this relation follows that anomalous
diffusion implies nontrivial behavior of the moments of P(r,t), such as an exponential divergence of the relative fluctuations for large r.
Dive into the research topics of 'Fluctuations of the probability density of diffusing particles for different realizations of a random medium'. Together they form a unique fingerprint. | {"url":"https://cris.tau.ac.il/en/publications/fluctuations-of-the-probability-density-of-diffusing-particles-fo","timestamp":"2024-11-02T04:23:37Z","content_type":"text/html","content_length":"47405","record_id":"<urn:uuid:fa22c6ef-b9e8-46b2-8341-fcbf644ea27c>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00607.warc.gz"} |
[all-commits] [llvm/llvm-project] 6600e1: [SCEV] If max BTC is zero, then so is the exact BT...
Philip Reames via All-commits all-commits at lists.llvm.org
Tue Aug 31 08:50:42 PDT 2021
Branch: refs/heads/main
Home: https://github.com/llvm/llvm-project
Commit: 6600e1759be1626965a26cf1da8d8f8fc73344ca
Author: Philip Reames <listmail at philipreames.com>
Date: 2021-08-31 (Tue, 31 Aug 2021)
Changed paths:
M llvm/lib/Analysis/ScalarEvolution.cpp
M llvm/test/Analysis/ScalarEvolution/max-trip-count.ll
Log Message:
[SCEV] If max BTC is zero, then so is the exact BTC [1 of N]
This patch is specifically the howManyLessThan case. There will be a couple of followon patches for other codepaths.
The subtle bit is explaining why the two codepaths have a difference while both are correct. The test case with modifications is a good example, so let's discuss in terms of it.
* The previous exact bounds for this example of (-126 + (126 smax %n))<nsw> can evaluate to either 0 or 1. Both are "correct" results, but only one of them results in a well defined loop. If %n were 127 (the only possible value producing a trip count of 1), then the loop must execute undefined behavior. As a result, we can ignore the TC computed when %n is 127. All other values produce 0.
* The max taken count computation uses the limit (i.e. the maximum value END can be without resulting in UB) to restrict the bound computation. As a result, it returns 0 which is also correct.
WARNING: The logic above only holds for a single exit loop. The current logic for max trip count would be incorrect for multiple exit loops, except that we never call computeMaxBECountForLT except when we can prove either a) no overflow occurs in this IV before exit, or b) this is the sole exit.
An alternate approach here would be to add the limit logic to the symbolic path. I haven't played with this extensively, but I'm hesitant because a) the term is optional and b) I'm not sure it'll reliably simplify away. As such, the resulting code quality from expansion might actually get worse.
This was noticed while trying to figure out why D108848 wasn't NFC, but is otherwise standalone.
Differential Revision: https://reviews.llvm.org/D108921
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Inflation Premium
An inflation premium is the part of prevailing interest rates that results from lenders compensating for expected inflation by pushing nominal interest rates to higher levels.
Inflation rate graph: Inflation rate in the Confederacy during the American Civil War.
In economics and finance, an individual who lends money for repayment at a later point in time expects to be compensated for the time value of money, or not having the use of that money while it is
lent. In addition, they will want to be compensated for the risks of the money having less purchasing power when the loan is repaid. These risks are systematic risks, regulatory risks and
inflationary risks. The first includes the possibility that the borrower will default or be unable to pay on the originally agreed upon terms, or that collateral backing the loan will prove to be
less valuable than estimated. The second includes taxation and changes in the law which would prevent the lender from collecting on a loan or having to pay more in taxes on the amount repaid than
originally estimated. The third takes into account that the money repaid may not have as much buying power from the perspective of the lender as the money originally lent, that is inflation, and may
include fluctuations in the value of the currencies involved. The inflation premium will compensate for the third risk, so investors seek this premium to compensate for the erosion in the value of
their capital, due to inflation.
Actual interest rates (without factoring in inflation) are viewed by economists and investors as being the nominal (stated) interest rate minus the inflation premium.
The Fisher equation in financial mathematics and economics estimates the relationship between nominal and real interest rates under inflation. In economics, this equation is used to predict nominal
and real interest rate behavior. Letting $r$ denote the real interest rate, $i$ denote the nominal interest rate, and let $π$ denote the inflation rate, the Fisher equation is: $i = r + π$. In the
Fisher equation, π is the inflation premium.
For example, if an investor were able to lock in a 5% interest rate for the coming year and anticipates a 2% rise in prices, he would expect to earn a real interest rate of 3%. 2% is the inflation
premium. This is not a single number, as different investors have different expectations of future inflation.
Since the inflation rate over the course of a loan is not known initially, volatility in inflation represents a risk to both the lender and the borrower. | {"url":"https://learn.saylor.org/mod/book/view.php?id=53731&chapterid=37825","timestamp":"2024-11-12T09:34:07Z","content_type":"text/html","content_length":"321102","record_id":"<urn:uuid:f9db5060-2bed-42bf-9b87-2bbb1a43ed5f>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00123.warc.gz"} |
Looking for Help With multiple if() statements
Im trying to get multipe if() statements to work.
Currently i have two variables, Discount Total (call it A) and Position Total (call it B).
What i want out of the operations should look like this:
if A and B empty, then {{emptystring}}
if only A empty, then give me B
if only B empty, then give me -A (negative)
I tried setting it up like this:
But what I’m currently getting is:
If A and B empty, if get “-” (just my minus sign, this should be empty)
If only A empty, i get “-” (this should be B)
If only B empty, i get -A (this works)
Is there an easier way to set this up without needing to do multiple if()'s ?
Figured someting out:
This works now for me:
if Pos.Total is empty then minus Disc.Total
if Disc.Total is empty then Pos.Total
if both empty, then an empty string
I’m still curious if there is an easier way that’s easier to read than whatever i wrote
1 Like
Hi @FPM_Support
You can try this :
{{if((length(9.a) > 0 & length(9.b) = 0); “-” + 9.a; ifempty(ifempty(9.a; 9.b); emptystring))}}
The following conditions are considered:
if A and B empty, then {{emptystring}}
if only A empty, then give me B
if only B empty, then give me -A (negative)
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for valuable insights and updates!
1 Like
Key point: Are A and B always integers/numbers, or could they be strings?
If there’s a possibility that they could be strings, use @Msquare_Automation’s solution with the length() function.
A and B cannot be strings.
Thanks for the input, i forgot to mention this.
Thank you so much.
In my case there is a possibility that A or B are zero. They shouldn’t be, but i want to make sure the szenario still continues. I forgot to mention that A or B cannot be strings.
The variable you gave me might just solve another issue i have were a string in another variable should not be displayed if A and B are empty.
Thanks for suggesting length()
Hi @FPM_Support length() works for each data type. However, as @mszymkowiak said, this formula won’t be accurate in numbers data type. Hence, we have updated the formula for your requirement that is
attached below.
{{if(if((length(214.a) > 0 & length(214.b) = 0); if(214.a = 0; 214.a; “-” + 214.a); ifempty(ifempty(214.a; 214.b); emptystring)) = 0; if((length(214.a) > 0 & length(214.b) = 0); if(214.a = 0;
214.a; “-” + 214.a); ifempty(ifempty(214.a; 214.b); emptystring)); parseNumber(if((length(214.a) > 0 & length(214.b) = 0); if(214.a = 0; 214.a; “-” + 214.a); ifempty(ifempty(214.a; 214.b);
emptystring))))}}{{if(if((length(214.a) > 0 & length(214.b) = 0); if(214.a = 0; 214.a; “-” + 214.a); ifempty(ifempty(214.a; 214.b); emptystring)) = 0; if((length(214.a) > 0 & length(214.b) = 0);
if(214.a = 0; 214.a; “-” + 214.a); ifempty(ifempty(214.a; 214.b); emptystring)); parseNumber(if((length(214.a) > 0 & length(214.b) = 0); if(214.a = 0; 214.a; “-” + 214.a); ifempty(ifempty(214.a;
214.b); emptystring))))}}
Please note, if you want the condition to treat like “does not exist” in case of the value is 0, then this formula won’t work.
For example, if value A is 0 and value B is 1, in this case, both the values exist.
If value A is 0 and value B is “null”, then value B does not exist and formula will return 0.
If value A is 2 and value B is “null”, then formula will return -2.
Msquare Automation - Gold Partner of Make
Book a Free Consultation | Connect Live
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Seismic demand for eccentric wall structures subjected to velocity pulse-like ground motions
The elastic and inelastic seismic demand of shear wall structures, with stiffness, strength and combined-stiffness-and-strength eccentricity, subjected to velocity pulse-like ground motions are
investigated. Based on the axial load-bending moment interaction model and eight pulse-like ground motions, nonlinear dynamic time history analyses are conducted to single-story RC eccentric wall
structures. The seismic demand is discussed in terms of the displacement, floor rotation and ductility, and the influence mechanism of different eccentricity types is revealed. The results show that
the eccentric systems for pulse-like cases experience much higher elastic and inelastic seismic demand comparing to those for non-pulse-like cases. The axial compression ratio has certain effect on
the inelastic seismic demand. The stiffness eccentricity is the key factor to the elastic seismic demand, while the strength eccentricity influences the inelastic seismic demand most. It is suggested
that the strength eccentricity be added as a parameter in the inelastic analysis of eccentric structures, and the influence of axial load as well as velocity pulse-like effect of ground motions also
be accounted in.
1. Introduction
Non uniform distribution of stiffness and strength in eccentric buildings causes the buildings to experience higher seismic demand and leads to increased damage, which has been observed during past
earthquakes [1-3]. Rutenberg [4] and De Stefano [5] comprehensively reviewed numerous studies on the seismic behavior of eccentric structures in past decades. Since the early 1980’s numerous studies
have been conducted to study the elastic and inelastic seismic demand of these structures [5-7]. Recently, more researches focus on investigating their inelastic seismic demand due to its complexity
[8-10]. All these studies have reached agreement on the basic trend of the elastic seismic demand for eccentric structures, and some conclusions are adopted by seismic codes. However, there still
exist controversial even opposite conclusions when structures are exited well into the inelastic range of responses. Sadek and Tso [11] introduced the concept of strength eccentricity and pointed out
that the strength eccentricity is an appropriate indicator in the inelastic range. Later, Bufeja et al. [12] also demonstrated that the strength eccentricity has greater influence on the inelastic
seismic demand than the stiffness eccentricity. However, it’s still controversial for the issue on the stiffness and strength eccentricity [13]. Most of the previous studies on eccentric structures
were based on the simple uniaxial hysteretic model, which ignored the axial load-bending moment interaction [5-10]. Moreover, all these studies have not yet specially considered the velocity
pulse-like effect of ground motions. Structural eccentricity and velocity pulse-like ground motion are two disadvantageous design conditions, and their combined influence on the structural elastic
and inelastic seismic demand needs to be further studied.
Past experiences show that velocity pulse-like effect of ground motions have significant influence on structural seismic responses. Previous studies [14-16] also demonstrated that structures
subjected to pulse-like ground motions have larger drift and strength demands compared with structures subjected to common earthquake actions. As a result, most of the current seismic codes, such as
EC8 [17], AS/NZS standard [18], Chinese seismic code [19], and etc., came to consider this disadvantageous effect by employing an amplification factor for earthquake load. However, all these codes
didn’t distinguish the generic symmetric structures from unfavorable eccentric structures. In addition, numerical simulation studies and experimental tests simultaneously accounting for the effects
of structural eccentricity and velocity pulse-like effect are rare, although independent research is adequate.
The objective of this work is to comparatively study the combined influence of velocity pulse-like effect of ground motions and structural eccentricity on the elastic and inelastic seismic demand for
eccentric structures through nonlinear time history analysis method. Three eccentricity types, including stiffness, strength, and combined stiffness-and-strength eccentricity, are introduced. The
influence mechanism of these eccentricities and influence of the axial load on elastic and inelastic seismic demand are revealed. The elastic seismic demand is in terms of the displacement of edge
walls and floor rotation, while inelastic seismic demand considers the ductility in addition. The conclusions can provide a reference in seismic codes for considering the combined influence of
pulse-like ground motion and structural eccentricity.
2. Structural model and input ground motions
2.1. Symmetric model
Reference symmetric structures, which are simplified by employing several wall elements like many famous researches did [8, 9], are first designed and the basic information is shown in Fig 1 ($\delta
=\mathrm{\Delta }=$0). The analytical models are established by a structural nonlinear analysis program CANNY [20]. The constitutive model for the concrete and steel material are CS4 and SR4,
respectively. The axial and shear hysteretic model for the wall element are AS2 model and CA7 model, ignoring the out plane resistance like many studies assumed. The default hysteretic parameters
recommended by the program are selected [20]. For resisting the in plane bending, AM3 model are employed to consider the axial load-bending moment interaction (Fig. 2). Comparing to the simple
uniaxial hysteretic models widely used for eccentric structures by previous studies, AM3 model can consider the influence of varying axial load on the moment. Although AM3 model is simpler than the
well known fiber model which can automatically consider the axial load-bending moment interaction, it provides a effective way for calculating the ductility demand as it can directly defines the
yield point while the fiber model cannot. The more details of these hysteretic models are available in reference [20].
Fig. 1Schematic diagram of the asymmetric and eccentric structure
Fig. 2Principle of AM3 model
a) Axial load-bending moment interaction
b) Tri-linear skeleton curve
c) Takeda hysteretic model
2.2. Eccentric model
Three types of eccentricities, including stiffness, strength and combined stiffness-and-strength eccentricity, are introduced by changing the stiffness and strength distribution of edge walls, as
shown in Fig. 1. The right wall with increased stiffness (strength) is defined as stiffer (strong) edge, and the left wall with reduced stiffness (strength) is the flexible (weak) edge. Namely, these
eccentricities are realized by changing the stiffness factor $\delta$ and strength factor $\mathrm{\Delta }$ in Fig. 1, and can be listed as the following cases.
1) Stiffness eccentricity type: $\delta e$0, $\frac{{e}_{R}}{L}e \text{0}$; $\mathrm{\Delta }=$0, $\frac{{e}_{V}}{L}=\text{0}$.
2) Strength eccentricity type: $\delta =$0, $\frac{{e}_{R}}{L}=0$; $\mathrm{\Delta }e$0, $\frac{{e}_{V}}{L}e \text{0}$.
3) Combined stiffness-and-strength eccentricity type: $\delta e$0, $\frac{{e}_{R}}{L}e \text{0}$; $\mathrm{\Delta }e$0, $\frac{{e}_{V}}{L}e \text{0}$,
where ${e}_{R}$ is the offset of stiffness center ${C}_{R}$ from mass center ${C}_{M}$, ${e}_{V}$ is the offset of strength center ${C}_{V}$ from mass center, and $L$ is the structural length along
$x$ axis, respectively. $\frac{{e}_{R}}{L}$ and $\frac{{e}_{V}}{L}$ denote the stiffness and strength eccentricity ratio, which can be defined as [11, 12]:
$\frac{{e}_{R}}{L}=\frac{\sum {K}_{yi}\bullet {x}_{i}}{\sum {K}_{yi}},$
$\frac{{e}_{V}}{L}=\frac{\sum {V}_{yi}\bullet {x}_{i}}{\sum {V}_{yi}},$
where ${K}_{yi}$ and ${V}_{yi}$ are the $i$th wall stiffness and yield strength along $y$ axis, and ${x}_{i}$ is the offset of the $i$th wall from the mass center ${C}_{M}$, respectively. Note that
the total structural stiffness and strength along $y$ axis are unchanged to make effective comparison. In addition, the combined eccentricity is limited to the case with equal stiffness eccentricity
and strength eccentricity, due to the impossibility to deal with infinite combination cases.
2.3. Input ground motions
The input ground motions are listed in Table 1 and the pulse indicator is denoted as:
where $PI$ is a predictor of the likelihood that a given record is pulse-like, “$PGVratio$” is the ratio of the peak ground velocity ($PGV$) of the residual earthquake record to the original record’s
$PGV$, and “$energy$$ratio$” is the ratio of the residual record’s energy to the original record’s energy, respectively [15]. $PI$ takes values between 0 and 1, with high values providing a strong
indication that the ground motion is pulse-like. Records with scores above 0.85 and below 0.15 are classified as intense pulses and non-pulses, respectively. In the present study, $PI$ takes values
not less than 0.97, which represents the intense velocity pulse effects.
Table 1List of pulse-like ground motions used in the present study
No. Earthquake event Station ${M}_{w}$ PGA / (cm/s^2) PGV / (cm/s) $PI$
Q1 Landers Barstow 7.3 696.3 30.4 1.00
Q2 San Fernando Pacoima Dam (upper left) 6.6 1407.7 116.5 0.97
Q3 Imperial Valley-6 El Centro Array #7 6.5 453.2 108.8 1.00
Q4 Imperial Valley-6 EC County Center FF 6.5 167.7 54.5 1.00
Q5 Imperial Valley-6 EC Meloland Overpass FF 6.5 263.3 115.0 1.00
Q6 Northridge-1 Pacoima Dam (upper left) 6.7 1349.7 107.1 1.00
Q7 Northridge-1 Newhall – W Pico Canyon Rd 6.7 417.1 87.8 1.00
Q8 Chi-Chi, Taiwan CHY101 7.6 442.6 85.4 1.00
3. Influence mechanism of eccentricity and axial load
3.1. Influence mechanism of stiffness and strength eccentricity
Most previous studies focus on the change trend of the seismic demand for stiffness or strength eccentric structures, while the influence mechanism of these different types of eccentricities is not
well understood [5-10]. The influence mechanism is revealed based on the simplified bi-linear hysteretic model in the present study. For stiffness eccentric systems in Fig. 3(a), the three skeleton
curves represent the force-deformation relationship of the stiffer edge, reference central wall and flexible edge, respectively. They are with the same yield strength but differ in stiffness both in
the elastic and inelastic range, which implies that the stiffness eccentricity can influence both the elastic and inelastic seismic demand. For strength eccentric systems in Fig. 3(b), the three
skeleton curves represent the force-deformation relationship of the strong edge, reference central wall and weak edge, respectively. Their initial elastic stiffness is equal, which demonstrates that
the elastic seismic demand is not influenced by the strength eccentricity. However, it goes different when structures are excited well into the inelastic range. Namely, the inelastic seismic demand
will be influenced due to different strength distribution of the three elements along the ground motion direction. Therefore, the elastic seismic demand is influenced by the stiffness eccentricity
while the inelastic seismic demand is influenced by both the stiffness eccentricity and strength eccentricity. Further numerical analysis should still be conducted to find out which is a much better
and sensitive parameter for controlling the inelastic seismic response for eccentric structures.
Fig. 3Influence mechanism of the eccentricity
a) Stiffness eccentricity
3.2. Influence mechanism of axial load
Fig 4 shows the influence of axial load on the elastic and inelastic seismic demand of a reference symmetric structure, which is compared by the ratio of seismic demand with axial compression ratio
$n$ ($n>$0) to that with $n=$0 (i.e., without considering the axial load). “$M$” and “$D$” denote the maximum moment and displacement demand, and the subscript “$e$” and “$p$” denote the elastic and
inelastic analysis cases, respectively. The axial compression ratio is varied from 0 to 0.06 considering the range of actual axial load for single-story RC wall structure.
For elastic case with $PGA$ of 0.05$g$ ($PGA$ is short for peak ground acceleration, and its unit “$g$” is gravity acceleration), the moment and displacement ratios hold a horizontal line with the
change of axial compression ratios, which implies that the axial load has no influence on the elastic seismic demand. For inelastic case ($PGA=$0.7$g$), the moment increases and the displacement
decreases with the increment of axial load when $n$ is less than 0.03, while they will not change when $n$ is larger than 0.03. It can be explained in Fig. 5(a) by that the moment capacity will be
improved with increasing axial load due to the axial load-bending moment interaction. This increment of moment capacity makes the structure experience larger moment demand and finally leads to
decreased displacement demand, which can be explained by the equal energy theory in Fig. 5(b). However, when $n$ is beyond 0.03, it is found that the structure goes into elastic response. As a
result, the moment and displacement will not change as the structure will behave elastically even at larger PGA of 0.7$g$. Thus, although the axial load have no influence on the elastic seismic
demand, it do influence the inelastic seismic demand. Comparing to most previous studies on eccentric structures by considering the simplified uniaxial hysteretic model with only resisting the
horizontal seismic load, this study is much practical as the axial load-bending moment interaction is considered.
Fig. 4Influence of axial compression ratio on elastic and inelastic seismic demand
Fig. 5Influence mechanism of axial load
a) Axial load-moment interaction
4. Analysis results and discussion
The elastic and inelastic seismic demand of structures with stiffness, strength and combined eccentricity are compared, in terms of the displacement ($\mathrm{\Delta }$), floor rotation ($\theta$)
and element ductility ($\mu$). The elastic and inelastic analysis is well conducted by scaling the peak ground acceleration ($PGA$) to be 0.05$g$ and 0.7$g$ respectively, ensuring all structural
models maintain elastic at $PGA$ of 0.05$g$ and go well into inelastic response at $PGA$ of 0.7$g$. Only the stiffer (strong) and flexible (weak) edge are discussed as they are the key members to
controlling the structural response compared to the central wall. Except for the additional notes, all the following results discussed is averaged over the eight pulse-like ground motion cases as the
change trend for individual cases are similar. The stiffness, strength and combined eccentricity are abbreviated as KE, SE, and CE for convenience. The notations of “$ref$” and “$ecc$” denote the
reference symmetric model and the eccentric model, respectively.
Totally, elastic and inelastic time history analyses are performed for the following permutations: eight pulse-like ground motion records; two structural types of eccentric and reference symmetric
models; three eccentricity types of KE, SE and CE; six eccentricity ratios of 0.05, 0.1, 0.15, 0.2, 0.25 and 0.3, which represent small, median and large eccentricity levels, respectively; two PGAs
of 0.05$g$ and 0.7$g$, denoting the elastic and inelastic analysis cases.
4.1. Influence of different eccentricities on elastic seismic demand
Fig. 6 shows the influence of KE, SE and CE on the elastic seismic demand, in terms of the displacement and floor rotation. As shown in the figure, both the displacement and floor rotation ratio are
influenced by KE and CE but are not influenced by SE, which validates the mechanism in Fig. 3. The influence curve of CE overlaps with that of SE because it is the combination of the other two and SE
have no influence in elastic range. With the increment of KE ratio, the left (flexible) edge displacement increases, the right (stiffer) edge displacement decreases, and the floor rotation increases.
These elastic seismic responses are observed to vary linearly. Thus, it is reasonable to adopt the stiffness eccentricity as an indicator for controlling the elastic response for eccentric
structures, as the current seismic provisioned.
Fig. 6Influence of different eccentricities on elastic seismic demand
a) Left edge displacement
b) Right edge displacement
4.2. Influence of different eccentricities on inelastic seismic demand
Fig. 7 shows the influence of KE, SE and CE on the inelastic displacement demand. The analysis results are only presented for cases under earthquake excitations from Q1 to Q4 shown in Table 1, as
cases from Q5 to Q8 have the similar trends. The following trends can be observed from Fig. 7:
(1) For the left edge, the inelastic displacement demand increases with the increment of all the three types of eccentricities. Generally, the influence of SE is the largest, KE least, and CE medium.
(2) For the right edge, it becomes different that the influence of KE is largest. Meanwhile, the inelastic displacement may decrease with increasing CE, which provides a method for reducing the
inelastic seismic demand of eccentric structures by controlling the balance position between the stiffness center and the strength center.
Fig. 7Influence of different types of eccentricity on inelastic seismic displacement demand
(3) By comparing the inelastic displacement of the left edge to that of the right edge, it can be observed that the left edge displacement is larger than the right one for an identical eccentricity
ratio. It demonstrates that the left (flexible or weak) side wall is the most unfavorable element influencing the inelastic seismic behavior. Thus, the strength eccentricity (SE) is the main factor
to control the inelastic displacement of the whole eccentric structures as SE is the most apparent factor influencing the unfavorable left edge.
Fig. 8 shows the influence of different types of eccentricities on the inelastic floor rotation demand. The inelastic floor rotation with $e/L$ of 0.05 is selected as a comparing reference, for no
floor rotation occurs in the case of $e/L=$ 0 under unidirectional ground motion (i.e., the symmetric structure with uniform stiffness and strength distribution along the ground motion direction). As
shown in Fig. 8, the inelastic floor rotation increases with the increment of all of the three eccentricity types. The influence of SE is larger than that of KE. However, the influence of CE is much
close to that of SE rather than the superposition of KE and SE, due to the complicated inelastic behavior.
Fig. 8Influence of different eccentricities on inelastic rotation
Fig. 9 illustrates the influence of different eccentricity types on the element ductility demand. As shown in Fig. 9(a), SE influences the ductility of left (weak) edge most, with a rapid nonlinear
increasing trend. It is different for KE case that the ductility of left (flexible) edge decreases with increasing KE, and the change trend is much smaller than SE case. The influence of CE is
similar with SE case. However, its increasing trend is much smaller owing to that CE is the combination of the other two and KE reduced the ductility demand. In Fig. 9(b), it is found that KE
influences the right (stiffer) edge ductility most. In addition, the ductility increases with the increment of KE and decreases with the increment of SE or CE. Comparing Fig. 9(a) to Fig. 9(b), the
left edge ductility is much larger than the right edge ductility, which implies that the left edge is the most unfavorable element. As a result, SE, influencing the unfavorable left edge most, is the
predominated factor to the whole eccentric structure. Meanwhile, it is also suggested that the strength eccentricity be considered in the inelastic analysis for eccentric structures.
Fig. 9Influence of different eccentricities on ductility demand
4.3. Influence of velocity pulse-like effect of ground motions
To well understand the velocity pulse-like effect, a non-pulse-like ground motion ($PI\le$ 0.15) corresponding to Q1 (the pulse-like ground motion in Table 1) was generated using Baker’s method and
excited to eccentric structures [15]. The elastic and inelastic seismic demand are compared between the pulse-like and non-pulse-like ground motion cases, as presented in Fig. 10. In these figures, “
$D$” denotes the seismic demand, the subscript “$e$” and “$p$” denote the elastic and inelastic analysis cases, and the superscript “$pulse$” and “$non\text{-}pulse$” denotes the pulse-like and
non-pulse-like ground motion cases, respectively.
Fig. 10(a) compares the elastic seismic demand of eccentric structures between the pulse-like and the non-pulse-like ground motion cases, in terms of the elastic displacement and floor rotation. The
stiffness eccentricity is employed as it is proved the main factor to elastic behavior (discussed in Section 4.1). The elastic displacement and floor rotation of the pulse-like case is larger than
that of the non-pulse-like case by 10 % and 9 %, respectively. These increment do not change with the increment of the eccentricity ratio, which demonstrates that the pulse-like effect of ground
motions and structural eccentricity have no coupling influence in elastic range and can be seen as individual factors in analyzing the structural elastic demand.
Fig. 10(b) compares the inelastic seismic demand of eccentric structures between the pulse-like and the non-pulse-like ground motion cases, in terms of the inelastic displacement, floor rotation, and
ductility demand. Differing from Fig. 10(a), the strength eccentricity is employed as it is the predominated factor to inelastic behavior (discussed in Section 4.2). When the eccentricity ratios are
less than 0.2 (i.e., the small and medium eccentricity range), both the inelastic displacement and floor rotation increase with the increasing eccentricity ratio, with a maximum increment of 30 % and
70 %, respectively. However, there exists no apparent trend for inelastic rotation, and displacement as well as ductility beyond the eccentricity ratio of 0.2, which needs to be further studied due
to the strong nonlinear behavior. Thus, the pulse-like effect of ground motion can generally increase the inelastic displacement and ductility. In addition, the velocity pulse-like effect and
structural eccentricity have coupling influence and can further aggravate the structural inelastic seismic demand, which is totally different from the elastic analysis cases in Fig. 10(a).
Fig. 10Influence of velocity pulse-like effect of ground motions on seismic demand
4.4. Code provision for velocity pulse-like effect
Most of the current seismic codes, such as EC8 [17], AS/NZS standard [18], Chinese seismic code [19], and etc., came to consider the unfavorable pulse-like effect of near fault ground motions by
employing an amplification factor, due to its destructiveness on structures. For instance, the AS/NZS standard [18] considered this effect by multiplying a near fault factor $N$ ($T$, $D$), which is
a function of period $T$ and the near fault distance $D$; Chinese seismic code [19] also adopted an amplification factor larger than 1.25 to ensure the safety of structures under near fault
pulse-like ground motions. However, these amplification factors were based on engineering experiences and all these codes didn’t distinguish the generic symmetric structures and unfavorable eccentric
structures. As illustrated in Section 4.3, the pulse-like effect and structural eccentricity can be considered as individual factors in the elastic range. In such cases, the current provision is
reasonable as there are individual specifications for these two advantage conditions. However, these individual specifications may be insufficient in the inelastic range, owing to that the pulse-like
effect and eccentricity have coupling influence and may cause larger inelastic response. Thus, further study should be conducted to evaluate on whether the current individual amplification factor in
seismic codes is enough for considering the coupling influence of these two disadvantageous design conditions.
5. Conclusions
The elastic and inelastic seismic demand of single-story RC wall structures, with stiffness, strength, and combined eccentricity, subjected to velocity pulse-like ground motions are investigated. The
following conclusions can be drawn from the results of this study:
(1) Eccentric wall structures subjected to velocity pulse-like ground motions will experience larger elastic and inelastic seismic demand comparing to non-pulse-like cases. The pulse-like effect and
structural eccentricity can be considered as individual factors in elastic range, while they have coupling influence in the inelastic range. It needs to be further studied whether the individual
amplification factor provisioned in current seismic codes is enough for considering this coupling influence. It is also suggested that the pule-like and non-pulse-like effect be distinguished in the
inelastic analysis of eccentric structures.
(2) In elastic range, the axial load has no influence on the seismic demand such as elastic moment and displacement. However, it will influence the inelastic seismic demand due to the axial
load-bending moment interaction. Most previous researches on eccentric structures, which employed the simple uniaxial hysteretic model and didn’t consider axial load-bending moment interaction, are
not reasonable and need to be revisited.
(3) The main factor controlling the elastic seismic demand is stiffness eccentricity (KE), while the predominated factor is strength eccentricity (SE) in inelastic range. Specifically, with the
increment of KE, the inelastic displacement of both the flexible and stiffer edge increase, while the flexible edge ductility decreases and the stiffer side ductility increases. For strength
eccentric structures, the inelastic displacement of both edges increase with the increment of SE, but with a larger influence than KE cases. In addition, the weak edge ductility increases while the
strong edge ductility decreases with the increment of SE, which is opposite from KE cases. It is suggested that the strength eccentricity be added as a basic parameter in the inelastic analysis of
eccentric structures.
Although the present study is based on the single-story structural model, it can provide useful information and qualitative conclusion for multistory eccentric structures as illustrated in previous
studies [4, 12]. Future study needs to extend to frame, frame-shear wall and shear wall structures with multistory eccentricity. Meanwhile, the complicated conditions such as bidirectional
eccentricity and multi-dimensional earthquake excitations need to be considered in future work.
• Chandler A. M. Building damage in Mexico City earthquake. Nature, Vol. 320, 1986, p. 497-501.
• Goyal A., Sinha R., Chaudhari M., et al. Damage to R/C structures in urban areas of ahmedabad and bhuj, EERI preliminary reconnaissance report on earthquake in Gujrat, India. Earthquake
Engineering Research Institute, Oakland, California, USA, 2001.
• Rutenberg A. Nonlinear response of asymmetric building structures and seismic codes: a state of the art review. Proceedings of the Nonlinear Seismic Analysis and Design of Reinforced Concrete
Buildings, Bled, Slovenia, 2004.
• De Stefano M., Pintucchi B. A review of research on seismic behaviour of irregular building structures since 2002. Bulletin of Earthquake Engineering, Vol. 6, Issue 2, 2008, p. 285-308.
• Kan C. L., Chopra A. K. Torsional coupling and earthquake response of simple elastic and inelastic systems. Journal of the Structural Division, Vol. 107, Issue 8, 1981, p. 1569-1588.
• Tso W. K., Zhu T. J. Design of torsionally unbalanced structural systems based on code provisions I: Ductility demand. Earthquake Engineering & Structural Dynamics, Vol. 21, Issue 7, 1992, p.
• Dutta S. C., Das P. K. Inelastic seismic response of code-designed reinforced concrete asymmetric buildings with strength degradation. Engineering Structures, Vol. 24, Issue 10, 2002, p.
• Dutta S. C., Roy R. Seismic demand of low-rise multistory systems with general asymmetry. Journal of Engineering Mechanics, Vol. 138, Issue 1, 2012, p. 1-11.
• Roy R., Chakroborty S. Seismic demand of plan-asymmetric structures: a revisit. Earthquake Engineering and Engineering Vibration, Vol. 12, Issue 1, 2013, p. 99-117.
• Bosco M., Marino E. M., Rossi P. P. An analytical method for the evaluation of the in-plan irregularity of non-regularly asymmetric buildings. Bulletin of Earthquake Engineering, 2013, p. 1-23.
• Sadek A. W., Tso W. K. Strength eccentricity concept for inelastic analysis of asymmetrical structures. Engineering Structures, Vol. 11, Issue 3, 1989, p. 189-194.
• Bugeja M. N., Thambiratnam D. P., Brameld G. H. The influence of stiffness and strength eccentricities on the inelastic earthquake response of asymmetric structures. Engineering Structures, Vol.
21, Issue 9, 1999, p. 856-863.
• Wang D., Lu X. L. Progress in study on inelastic torsional seismic response of asymmetric buildings. Journal of Earthquake Engineering and Engineering Vibration, Vol. 30, Issue 2, 2010, p. 51-58.
• Bertero V. V., Mahin S. A., Herrera R. A. Aseismic design implication of near-fault San Fernando earthquake records. Earthquake Engineering and Structural Dynamics, Vol. 6, Issue 1, 1978, p.
• Baker J. W. Quantitative classification of near-fault ground motions using wavelet analysis. Bulletin of the Seismological Society of America, Vol. 97, Issue 5, 2007, p. 1486-1501.
• Zhou J., Chen K. L., Huang Liang Effects of scaled pulse-like ground motion records on nonlinear structural displacement response. Journal of Vibration and Shock, Vol. 30, Issue 2, 2011, p.
• Eurocode 8. Design of structures for earthquake resistance – general rules, seismic actions and rules for buildings. CEN, Brussels, 1998-1, 2005.
• AS/NZS 1170.4. Structural design actions – Part 4 earthquake actions, general design requirement and loading on structures. Australian, New Zealand, 2004.
• GB50011-2010. Code for seismic design of buildings. China Architecture Industry Press, Beijing, 2010.
• Li K. N. Three-dimensional nonlinear static/dynamic structural analysis computer program-users’ manual and data-input manual of CANNY. Vancouver, 2010.
About this article
pulse-like ground motion
stiffness eccentricity
strength eccentricity
axial load-bending moment interaction
dynamic time history analysis
The authors acknowledge the financial support by National Natural Science Foundation of China (50878087), State Key Lab of Subtropical Building Science, South China University of Technology
(2014ZA06), and Natural Science Foundation of Hunan Province of China (12JJ6047).
Copyright © 2014 JVE International Ltd.
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. | {"url":"https://www.extrica.com/article/14980","timestamp":"2024-11-10T22:01:51Z","content_type":"text/html","content_length":"145883","record_id":"<urn:uuid:d994bc0e-6b7d-4223-812c-8375dc88a26e>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00272.warc.gz"} |
HELP PLEASE...........
B. slope -5, y-intercept (0, 7)
Step-by-step explanation:
The y-intercept is the first (x, y) pair shown in the table: (0, 7). (This eliminates choices C and D.)
The slope is ...
change in y / change in x
(-13 -7) / (4 -0) = -20/4 = -5 . . . slope
This matches choice B.
There will be 55 pints and of un-pure and 165 pints of pure fruit drinks.
What is an expression?
The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and
other properties.
To get 220 pints with 95% purity you need a mixture of 209 parts to 11 parts (220 *0.95 and 220 * 0.05)
We call the un-pure mixture a and the pure mixture b
All un-pure parts come from a and b in the following ratios
11 = 0.2a + 0b (each pint of mixture a adds 0.2 pints of unpure stuff and each pint of b adds no unpure stuff)
which we can then solve as:
a = 55
All pure parts come from a and b in the following rations.
209 = 0.8a + 1b (each pint of mixture a adds 0.8 pints of pure stuff and each pint of b adds a full pint of the pure stuff)
Now we can replace a with 55 (from above)
209 = 44 + b, and solve to
b = 165
Now to double check
55 x (0.8x + 0.2y) + 165x would give you
44x + 11y + 165x, which is
11x + 209y and if you then check the ratio
209 / (209+11) = 0.95
Therefore, there will be 55 pints of unpure and 165 pints of pure fruit drinks.
To know more about an expression follow | {"url":"https://learning.cadca.org/question/help-please-g89q","timestamp":"2024-11-09T16:23:00Z","content_type":"text/html","content_length":"71851","record_id":"<urn:uuid:50def8ad-dcfa-4fdf-b36b-5f64960c97ce>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00412.warc.gz"} |
Factors Archives - Mathstoon
Definition of factors of 150: If a number completely divides 150 without leaving a remainder, then that number is called a factor of 150. Here, we will learn about the factors of 150 and the prime
factors of 150. We have: What are the Factors of 150? Let us write the number 150 multiplicatively in … Read more
Factors of 144
If a number completely divides 144 without leaving a remainder, then that number is called a factor of 144. So we can say that the factors of 144 are the divisors of 144. Here, we will learn about
the factors of 144 and the prime factors of 144. What are the Factors of 144? To … Read more
Factors of 128
Definition of factors of 128: If a number completely divides 128 without leaving a remainder, then that number is called a factor of 128. In this article, we will learn about the factors of 128 and
the prime factors of 128. We have: What are the Factors of 128? Let us write the number 128 … Read more
Factors of 120
A number is called a factor of 120 if it completely divides 120 without leaving a remainder. Thus, the factors of 120 are the divisors of 120. Here, we will learn about the factors of 120 and the
prime factors of 120. • Prime factorization of 120: 120 = 2×2×2×3×5 = 23×3×5. • Factors of … Read more
Factors of 108
Definition of factors of 108: If a number completely divides 108 without leaving a remainder, then that number is called a factor of 108. Thus, we can say that the factors of 108 are the divisors of
108. Here, we will learn about the factors of 108 and the prime factors of 108. What are … Read more
Factors of 96
Definition of factors of 96: If a number completely divides 96 without leaving a remainder, then that number is called a factor of 96. Thus, we can say that the factors of 96 are the divisors of 96.
In this article, we will learn about the factors of 96 and the prime factors of 96. … Read more
Factors of 84
If a number completely divides 84 without leaving a remainder, then that number is called a factor of 84. Thus, we can say that the factors of 84 are the divisors of 84. Here, we will learn about the
factors of 84 and the prime factors of 84. We have: What are the Factors of … Read more
All Factors of 90: Prime, Negative & Pair Factors of 90
Definition of factors of 90: If a number completely divides 90 without leaving a remainder, then that number is called a factor of 90. Thus, we can say that the factors of 90 are the divisors of 90.
Here, we will learn about the factors of 90 and the prime factors of 90. What are … Read more
Factors of 66
Definition of factors of 66: If a number completely divides 66 without leaving a remainder, then that number is called a factor of 66. Thus, we can say that the factors of 66 are the divisors of 66.
Here, we will learn about the factors of 66 and the prime factors of 66. We have: … Read more
Factors of 88
Definition of factors of 88: If a number completely divides 88 without leaving a remainder, then that number is called a factor of 88. Thus, we can say that the factors of 88 are the divisors of 88.
Here, we will learn about the factors of 88 and the prime factors of 88. We have: … Read more | {"url":"https://www.mathstoon.com/category/factors/","timestamp":"2024-11-05T21:36:58Z","content_type":"text/html","content_length":"182978","record_id":"<urn:uuid:e2bb97ad-9de7-47d9-9816-fe14db7a9ad3>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00578.warc.gz"} |
Algebra Homework (Entire Year of Math Concepts)
This listing is for my Algebra homework. There is a total of 77 math concepts (154 pages). The first page of each math concept includes 5 problems. The second page of each math concept includes (1)
Find the mistake, (2) Create your own problem aligned to the math concept, and (3) Write a reflection.
Unit 1: Prerequisite for Algebra and Algebraic Expressions
• Rational/Irrational Numbers
• Real Number Properties (Commutative, Associative, Distributive, Additive Identity, Multiplicative, Zero Property, and Additive Inverse)
• Solve One-Step Equations Involving One Variable
• Solve and Graph One-Step Inequalities Involving One Variable
• Use Order of Operations to Simplify Expressions
• Analyze and Evaluate Expressions
• Define and Apply the Absolute Value
• Simplify Expressions by Combining Like Terms
• Simplify Expressions by using the Distributive Property
• Solve Two-Step Equations involving One Variable
• Solve Multi-Step Equations involving One Variable
• Solve Equations with Variables on Both Sides involving One Variable
• Solve Equations involving Multi-Variables
• Solve and Graph Two-Step Inequalities
• Solve and Graph Multi-Step Inequalities
• Solve and Graph Compound Inequalities
• Solve Absolute Value Equations
• Solve and Graph Absolute Value Inequalities
• Properties of Equality (Addition, Subtraction, Multiplication, Division, Reflexive, Symmetric, Transitive, and Substitution)
• Use Proportions to Solve for an Unknown
• Intro to Functions
• Evaluating a Function & Function Notation
• Domain, Range, Minimum, and Maximum
• Piecewise Functions
• Intro to Sequences
• Arithmetic Sequences (Recursive and Explicit)
• Geometric Sequences (Recursive and Explicit)
• Plotting Points on the Coordinate System
• Determine Slope Given a Graph
• Determine Slope Given a Table or Two Points
• Define, Write, and Graph Slope-Intercept Form
• Determine a Linear Equations Given a Table
• Standard Form to Slope Intercept Form
• Determine Linear Equations Given a Graph
• Determine Linear Equations Given Two Points
• Determine and Graph the X and Y intercepts Given an Equation
• Graph Horizontal and Vertical Lines
• Write Equations of a Line Given a Slope and a Point, or Two Points
• Write Parallel and Perpendicular Lines Given an Equation
• Intro to Exponents
• Properties of Exponents
• Intro to Scientific Notation
• Converting Between Standard Form and Scientific Notation
• Adding and Subtracting Scientific Notation
• Multiplying and Dividing Scientific Notation
• The Three Types of Solutions: No Solution, Infinitely Many Solutions, and One Solution
• Solving Systems of Equations by Graphing, Substitution, and Elimination
• Write and Solve Word Problems involving Systems of Equations
• Solve and Graph Systems of Linear Inequalities involving One Variable
• Intro to Polynomials
• Operations on Polynomials (Adding, Subtracting, Multiplying, and Dividing)
• Multiplication of Binomials
• Intro to Quadratics: What is a quadratic?
• Solving Quadratics by Taking Square Roots
• The Quadratic Formula to Solve for Roots
• Using the Completing the Square to Solve for Roots
• What is a radical?
• Basic Properties of Simplifying Radicals
The homework is super easy to grade and I love that it is not a super long homework assignment. As a parent myself, my children are given homework from multiple classes. Along with their
extracurricular activities, it makes it difficult when my children have homework from each class requiring more than an hour. Hence, I have simplified my homework so that students can demonstrate
their knowledge of the math concept without taking a long time completing the homework.
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class lifelines.fitters.nelson_aalen_fitter.NelsonAalenFitter(alpha=0.05, nelson_aalen_smoothing=True, **kwargs)¶
Class for fitting the Nelson-Aalen estimate for the cumulative hazard.
NelsonAalenFitter(alpha=0.05, nelson_aalen_smoothing=True)
☆ alpha (float, optional (default=0.05)) – The alpha value associated with the confidence intervals.
☆ nelson_aalen_smoothing (bool, optional) – If the event times are naturally discrete (like discrete years, minutes, etc.) then it is advisable to turn this parameter to False. See [1],
[1] Aalen, O., Borgan, O., Gjessing, H., 2008. Survival and Event History Analysis
The estimated cumulative hazard (with custom timeline if provided)
The lower and upper confidence intervals for the cumulative hazard
The durations provided
The event_observed variable provided
The time line to use for plotting and indexing
The entry array provided, or None
array or None
A summary of the life table
property conditional_time_to_event_¶
Return a DataFrame, with index equal to survival_function_, that estimates the median duration remaining until the death event, given survival up until time t. For example, if an individual
exists until age 1, their expected life remaining given they lived to time 1 might be 9 years.
cumulative_hazard_at_times(times, label=None) Series¶
Return a Pandas series of the predicted cumhaz value at specific times
times (iterable or float)
Return type:
fit(durations, event_observed=None, timeline=None, entry=None, label=None, alpha=None, ci_labels=None, weights=None, fit_options=None)¶
○ durations (an array, or pd.Series, of length n) – duration subject was observed for
○ timeline (iterable) – return the best estimate at the values in timelines (positively increasing)
○ event_observed (an array, or pd.Series, of length n) – True if the the death was observed, False if the event was lost (right-censored). Defaults all True if event_observed==None
○ entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated observations, i.e the birth event was not observed. If
None, defaults to all 0 (all birth events observed.)
○ label (string) – a string to name the column of the estimate.
○ alpha (float) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
○ ci_labels (iterable) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<1-alpha/2>
○ weights (n array, or pd.Series, of length n) – if providing a weighted dataset. For example, instead of providing every subject as a single element of durations and event_observed,
one could weigh subject differently.
○ fit_options – Not used
Return type:
self, with new properties like cumulative_hazard_.
Return the unique time point, t, such that S(t) = p.
p (float)
plot_hazard(bandwidth=None, **kwargs)¶
bandwidth (float) – the bandwidth used in the Epanechnikov kernel.
a DataFrame of the smoothed hazard
Return type:
smoothed_hazard_confidence_intervals_(bandwidth, hazard_=None)¶
○ bandwidth (float) – the bandwidth to use in the Epanechnikov kernel. > 0
○ hazard_ (numpy array) – a computed (n,) numpy array of estimated hazard rates. If none, uses smoothed_hazard_ | {"url":"https://lifelines.readthedocs.io/en/latest/fitters/univariate/NelsonAalenFitter.html","timestamp":"2024-11-12T23:42:19Z","content_type":"text/html","content_length":"28963","record_id":"<urn:uuid:d9d50f0a-b272-4580-bfdc-32eda17a7a67>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00660.warc.gz"} |
On the Structure of the Least Favorable Prior Distributions
This paper studies optimization of the minimum mean square error (MMSE) in order to characterize the structure of the least favorable prior distributions. In the first part, the paper characterizes
the local behavior of the MMSE in terms of the input distribution and finds the directional derivative of the MMSE at the distribution P-{\mathbf{X}} in the direction of the distribution Q-{\mathbf
{X}}. In the second part of the paper, the directional derivative together with the theory of convex optimization is used to characterize the structure of least favorable distributions. In
particular, under mild regularity conditions, it is shown that the support of the least favorable distributions must necessarily be very small and is contained in a nowhere dense set of Lebesgue
measure zero. The results of this paper produce both sufficient and necessary conditions for optimality, do not rely on Gaussian statistics assumptions, and are not sensitive to the dimensionality of
random vectors. The results are evaluated for the univariate and multivariate random Gaussian cases, and the Poisson case. Finally, as one of the applications, it is shown how the results can be used
to characterize the capacity of Gaussian MIMO channels with an amplitude constraint.
Publication series
Name IEEE International Symposium on Information Theory - Proceedings
Volume 2018-June
ISSN (Print) 2157-8095
Other 2018 IEEE International Symposium on Information Theory, ISIT 2018
Country/Territory United States
City Vail
Period 6/17/18 → 6/22/18
All Science Journal Classification (ASJC) codes
• Theoretical Computer Science
• Information Systems
• Modeling and Simulation
• Applied Mathematics
Dive into the research topics of 'On the Structure of the Least Favorable Prior Distributions'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/on-the-structure-of-the-least-favorable-prior-distributions","timestamp":"2024-11-03T13:38:32Z","content_type":"text/html","content_length":"51478","record_id":"<urn:uuid:b1589e0d-0e69-4430-80ea-3aec23a971eb>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00350.warc.gz"} |
Parameter Selection for HDBSCAN*
Parameter Selection for HDBSCAN*¶
While the HDBSCAN class has a large number of parameters that can be set on initialization, in practice there are a very small number of parameters that have significant practical effect on
clustering. We will consider those major parameters, and consider how one may go about choosing them effectively.
Selecting min_cluster_size¶
The primary parameter to effect the resulting clustering is min_cluster_size. Ideally this is a relatively intuitive parameter to select – set it to the smallest size grouping that you wish to
consider a cluster. It can have slightly non-obvious effects however. Let’s consider the digits dataset from sklearn. We can project the data into two dimensions to visualize it via t-SNE.
digits = datasets.load_digits()
data = digits.data
projection = TSNE().fit_transform(data)
plt.scatter(*projection.T, **plot_kwds)
If we cluster this data in the full 64 dimensional space with HDBSCAN* we can see some effects from varying the min_cluster_size.
We start with a min_cluster_size of 15.
clusterer = hdbscan.HDBSCAN(min_cluster_size=15).fit(data)
color_palette = sns.color_palette('Paired', 12)
cluster_colors = [color_palette[x] if x >= 0
else (0.5, 0.5, 0.5)
for x in clusterer.labels_]
cluster_member_colors = [sns.desaturate(x, p) for x, p in
zip(cluster_colors, clusterer.probabilities_)]
plt.scatter(*projection.T, s=50, linewidth=0, c=cluster_member_colors, alpha=0.25)
Increasing the min_cluster_size to 30 reduces the number of clusters, merging some together. This is a result of HDBSCAN* reoptimizing which flat clustering provides greater stability under a
slightly different notion of what constitutes a cluster.
clusterer = hdbscan.HDBSCAN(min_cluster_size=30).fit(data)
color_palette = sns.color_palette('Paired', 12)
cluster_colors = [color_palette[x] if x >= 0
else (0.5, 0.5, 0.5)
for x in clusterer.labels_]
cluster_member_colors = [sns.desaturate(x, p) for x, p in
zip(cluster_colors, clusterer.probabilities_)]
plt.scatter(*projection.T, s=50, linewidth=0, c=cluster_member_colors, alpha=0.25)
Doubling the min_cluster_size again to 60 gives us just two clusters – the really core clusters. This is somewhat as expected, but surely some of the other clusters that we had previously had more
than 60 members? Why are they being considered noise? The answer is that HDBSCAN* has a second parameter min_samples. The implementation defaults this value (if it is unspecified) to whatever
min_cluster_size is set to. We can recover some of our original clusters by explicitly providing min_samples at the original value of 15.
clusterer = hdbscan.HDBSCAN(min_cluster_size=60).fit(data)
color_palette = sns.color_palette('Paired', 12)
cluster_colors = [color_palette[x] if x >= 0
else (0.5, 0.5, 0.5)
for x in clusterer.labels_]
cluster_member_colors = [sns.desaturate(x, p) for x, p in
zip(cluster_colors, clusterer.probabilities_)]
plt.scatter(*projection.T, s=50, linewidth=0, c=cluster_member_colors, alpha=0.25)
clusterer = hdbscan.HDBSCAN(min_cluster_size=60, min_samples=15).fit(data)
color_palette = sns.color_palette('Paired', 12)
cluster_colors = [color_palette[x] if x >= 0
else (0.5, 0.5, 0.5)
for x in clusterer.labels_]
cluster_member_colors = [sns.desaturate(x, p) for x, p in
zip(cluster_colors, clusterer.probabilities_)]
plt.scatter(*projection.T, s=50, linewidth=0, c=cluster_member_colors, alpha=0.25)
As you can see this results in us recovering something much closer to our original clustering, only now with some of the smaller clusters pruned out. Thus min_cluster_size does behave more closely to
our intuitions, but only if we fix min_samples.
If you wish to explore different min_cluster_size settings with a fixed min_samples value, especially for larger dataset sizes, you can cache the hard computation, and recompute only the
relatively cheap flat cluster extraction using the memory parameter, which makes use of joblib
Selecting min_samples¶
Since we have seen that min_samples clearly has a dramatic effect on clustering, the question becomes: how do we select this parameter? The simplest intuition for what min_samples does is provide a
measure of how conservative you want your clustering to be. The larger the value of min_samples you provide, the more conservative the clustering – more points will be declared as noise, and clusters
will be restricted to progressively more dense areas. We can see this in practice by leaving the min_cluster_size at 60, but reducing min_samples to 1.
Note: adjusting min_samples will result in recomputing the hard comptuation of the single linkage tree.
clusterer = hdbscan.HDBSCAN(min_cluster_size=60, min_samples=1).fit(data)
color_palette = sns.color_palette('Paired', 12)
cluster_colors = [color_palette[x] if x >= 0
else (0.5, 0.5, 0.5)
for x in clusterer.labels_]
cluster_member_colors = [sns.desaturate(x, p) for x, p in
zip(cluster_colors, clusterer.probabilities_)]
plt.scatter(*projection.T, s=50, linewidth=0, c=cluster_member_colors, alpha=0.25)
<matplotlib.collections.PathCollection at 0x120978438>
Now most points are clustered, and there are much fewer noise points. Steadily increasing min_samples will, as we saw in the examples above, make the clustering progressively more conservative,
culminating in the example above where min_samples was set to 60 and we had only two clusters with most points declared as noise.
Selecting cluster_selection_epsilon¶
In some cases, we want to choose a small min_cluster_size because even groups of few points might be of interest to us. However, if our data set also contains partitions with high concentrations of
objects, this parameter setting can result in a large number of micro-clusters. Selecting a value for cluster_selection_epsilon helps us to merge clusters in these regions. Or in other words, it
ensures that clusters below the given threshold are not split up any further.
The choice of cluster_selection_epsilon depends on the given distances between your data points. For example, set the value to 0.5 if you don’t want to separate clusters that are less than 0.5 units
apart. This will basically extract DBSCAN* clusters for epsilon = 0.5 from the condensed cluster tree, but leave HDBSCAN* clusters that emerged at distances greater than 0.5 untouched. See Combining
HDBSCAN* with DBSCAN for a more detailed demonstration of the effect this parameter has on the resulting clustering.
Selecting alpha¶
A further parameter that effects the resulting clustering is alpha. In practice it is best not to mess with this parameter – ultimately it is part of the RobustSingleLinkage code, but flows naturally
into HDBSCAN*. If, for some reason, min_samples or cluster_selection_epsilon is not providing you what you need, stop, rethink things, and try again with min_samples or cluster_selection_epsilon. If
you still need to play with another parameter (and you shouldn’t), then you can try setting alpha. The alpha parameter provides a slightly different approach to determining how conservative the
clustering is. By default alpha is set to 1.0. Increasing alpha will make the clustering more conservative, but on a much tighter scale, as we can see by setting alpha to 1.3.
Note: adjusting alpha will result in recomputing the hard comptuation of the single linkage tree.
clusterer = hdbscan.HDBSCAN(min_cluster_size=60, min_samples=15, alpha=1.3).fit(data)
color_palette = sns.color_palette('Paired', 12)
cluster_colors = [color_palette[x] if x >= 0
else (0.5, 0.5, 0.5)
for x in clusterer.labels_]
cluster_member_colors = [sns.desaturate(x, p) for x, p in
zip(cluster_colors, clusterer.probabilities_)]
plt.scatter(*projection.T, s=50, linewidth=0, c=cluster_member_colors, alpha=0.25)
Leaf clustering¶
HDBSCAN supports an extra parameter cluster_selection_method to determine how it selects flat clusters from the cluster tree hierarchy. The default method is 'eom' for Excess of Mass, the algorithm
described in How HDBSCAN Works. This is not always the most desireable approach to cluster selection. If you are more interested in having small homogeneous clusters then you may find Excess of Mass
has a tendency to pick one or two large clusters and then a number of small extra clusters. In this situation you may be tempted to recluster just the data in the single large cluster. Instead, a
better option is to select 'leaf' as a cluster selection method. This will select leaf nodes from the tree, producing many small homogeneous clusters. Note that you can still get variable density
clusters via this method, and it is also still possible to get large clusters, but there will be a tendency to produce a more fine grained clustering than Excess of Mass can provide.
Allowing a single cluster¶
In contrast, if you are getting lots of small clusters, but believe there should be some larger scale structure (or the possibility of no structure), consider the allow_single_cluster option. By
default HDBSCAN* does not allow a single cluster to be returned – this is due to how the Excess of Mass algorithm works, and a bias towards the root cluster that may occur. You can override this
behaviour and see what clustering would look like if you allow a single cluster to be returned. This can alleviate issue caused by there only being a single large cluster, or by data that is
essentially just noise. For example, the image below shows the effects of setting allow_single_cluster=True in the bottom row, compared to the top row which used default settings. | {"url":"https://hdbscan.readthedocs.io/en/latest/parameter_selection.html","timestamp":"2024-11-08T21:16:10Z","content_type":"text/html","content_length":"40453","record_id":"<urn:uuid:e3467fd1-cc83-4e0f-bdab-bab4ace06533>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00327.warc.gz"} |
Find the sum of the series 3.75,3.5,3.25……….upto 16 terms.
Hint- First find out the type of progression which the sequence is in that is if it is in A.P, G.P or H.P and solve it.
The series given to us is 3.75,3.5,3.25…………..
We have been asked to find out the sum of the series upto 16 terms
From the series given we get ${T_2} - {T_1} = 3.5 - 3.75 = - 0.25$
Also, we get ${T_3} - {T_2} = 3.25 - 3.50 = - 0.25$
So, from this we got ${T_3} - {T_2} = {T_2} - {T_1}$ =common difference=d
So, from this we can conclude that the given series is in Arithmetic Progression(A.P)
So, we know that the sum of n terms of an A.P is given by
${S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$
So, on comparing with the sequence ,we can write
The first term=a=3.75
Common difference d=-0.25
Here, since we have to find out the sum upto 16 terms, we consider n=16
Let us substitute these values in the ${S_n}$ formula
So, we get $
{S_{16}} = \dfrac{{16}}{2}\left( {2 \times 3.75 + (16 - 1)( - 0.25)} \right) \\
{S_{16}} = 8(7.5 - 3.75) \\
{S_{16}} = 8(3.75) \\
\Rightarrow{S_{16}} = 30 \\
So, the sum of the series upto 16 terms=30
Note: When finding sum to n terms of an AP we can make use of an alternative formula if the first and last terms of an AP are known or we can use the same formula as used in this problem and solve. | {"url":"https://www.vedantu.com/question-answer/find-the-sum-of-the-series-37535325upto-16-terms-class-11-maths-cbse-5edbc9fd7e65c55350f2c922","timestamp":"2024-11-03T09:19:28Z","content_type":"text/html","content_length":"160039","record_id":"<urn:uuid:31836a07-510f-4388-bd2b-4924605802a0>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00092.warc.gz"} |
Earnings Yield
The ratio of a company's annual earnings per share to its stock price; the reciprocal of the P/E ratio.
The earnings yield is an estimate of the inflation-adjusted growth you can reasonably expect from a stock investment; in a normal market it should be at least equal to the T-Bill rate, minus the
inflation rate, plus a risk premium of about 3 or 4 percent. Very low earnings yields are a warning sign of a bubble. | {"url":"http://moneychimp.com/glossary/earnings_yield.htm","timestamp":"2024-11-10T17:16:26Z","content_type":"text/html","content_length":"4437","record_id":"<urn:uuid:fa429ecc-87da-4124-a0f9-c16a6d9913f4>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00415.warc.gz"} |
11 Meh 1945: House of Commons debates
• Preamble
The House met at a Quarter past Two o'clock
• Prayers
[Mr. SPEAKER in the Chair]
• Private Business
Business of the House
I desire to ask the acting Leader of the House whether he can make any statement with regard to the Industrial Injuries (Insurance) Bill?
Message from the Lords
That they have agreed to—
Business of the House (Supply)
Ordered: That this day, notwithstanding anything in Standing Order No. 14, Business other than the Business of Supply may be taken before a quarter past Nine o'clock and that if the first two...
Business of the House
Proceedings on Government Business exempted, at this day's Sitting, from the provisions of the Standing Order (Sittings of the House).—[Mr. Bracken.]
Orders of the Day — Supply [7TH Allotted Day]
Considered in Committee.
Civil Estimates and Estimates for Revenue Departments, 1945; Navy, Army and Air Estimates, 1945
• Health
Resolved: That a further sum, not exceeding £36,271,562 be granted to His Majesty, to complete the sums necessary to defray the charges for the following Departments connected with the...
Motion made, and Question proposed, That a further sum, not exceeding £62,952,140 be granted to His Majesty, to complete the sums necessary to defray the charges for the following...
Civil Estimates, 1945
• Class I
"That a sum, not exceeding £5,112,629, be granted to His Majesty, to complete the sum necessary to defray the charge which will come in course of payment during the year ending on the 31st...
Class II
"That a sum, not exceeding £20,717,945, be granted to His Majesty, to complete the sum necessary to defray the charge which will come in course of payment during the year ending on the 31st...
Class III
"That a sum not exceeding £11,727,490, be granted to His Majesty, to complete the sum necessary to defray the charge which will come in course of payment during the year ending on the 31st...
Class IV
"That a sum, not exceeding £9,567,865, be granted to His Majesty, to complete the sum necessary to defray the charge which will come in course of payment during the year ending on the 31st...
Class V
"That a sum, not exceeding £118,353,198, be granted to His Majesty, to complete the sum necessary to defray the charge which will come in course of payment during the year ending on the 31st...
Class VI
"That a sum, not exceeding £18,482,420, be granted to His Majesty, to complete the sum necessary to defray the charge which will come in course of payment during the year ending on the...
Class VII
"That a sum, not exceeding £16,766,233, be granted to His Majesty, to complete the sum necessary to defray the charge which will come in course of payment during the year ending on the 31st...
Class VIII
"That a sum, not exceeding £25,680,053, be granted to His Majesty, to complete the sum necessary to defray the charge which will come in course of payment during the year ending on the 31st...
Class Ix
"That a sum, not exceeding £34,145,859, be granted to His Majesty, to complete the sum necessary to defray the charge which will come in course of payment daring the year ending on the 31st...
Class X
"That a sum, not exceeding £1,440, be granted to His Majesty, to complete the sum necessary to defray the charge which will come in course of payment during the year ending on the 31st day...
Revenue Departments Estimates, 1945
"That a sum, not exceeding £95,672,390 be granted to His Majesty, to complete the sum necessary to defray the charge which will come in course of payment during the year ending on the 31st...
Navy Estimates, 1945
"That a Sum, not exceeding £1,700, be granted to His Majesty, to defray the charge which will come in course of payment during the year ending on the 31st day of March, 1946, for Expenditure...
Army Estimates, 1945
"That a sum, not exceeding £1,400, be granted to His Majesty, to defray the charge which will come in course of payment during the year ending on the 31st day of March, 1946, for Expenditure...
Air Estimates, 1945
"That a sum, not exceeding £1,000, be granted to His Majesty, to defray the charge which will come in course of payment during the year ending on the 31st day of March, 1946, for...
Ways and Means
Considered in Committee.
Postponement of Polling Day Bill
Order for Second Reading read.
TREASON BILL [Lords]
Order for Second Reading read.
Family Allowances Bill
As amended, considered.
International Situation
Motion made, and Question proposed, "That the House do now adjourn."—[Mr. Pym.] | {"url":"https://cy.theyworkforyou.com/debates/?d=1945-06-11","timestamp":"2024-11-07T02:39:07Z","content_type":"text/html","content_length":"44548","record_id":"<urn:uuid:97f64640-1168-453d-b502-9f09ecd5a095>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00797.warc.gz"} |
9th-octave temperaments
Jump to navigation Jump to search
This page is a stub. You can help the Xenharmonic Wiki by expanding it.
Although 9edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 9 are.
For example, these multiple-of-9 EDOs appear in some zeta edo lists: 27, 72, 99, 171, 270, 342 and 441. (List is not exhaustive.)
The main 9th-octave temperament of interest is ennealimmal (temperament data given there), notable for being the 7-limit microtemperament tempering the two smallest superparticular intervals of the
7-limit, 2401/2400 = S49 = (49/40)/(60/49) and 4375/4374 = S25/S27 = (28/24)/(27/25)^2, with the smallest patent val edo tunings being 27edo (a sharp superpyth tuning supporting modus and augene) and
45edo (the optimal patent val of flattone), which sum to 72edo (the smallest edo tuning that starts to show the accuracy of ennealimmal, with a mild flat tendency) and relatedly 99edo (the second
such tuning, with a mild sharp tendency instead).
It can be thought of as leveraging the most accurate JI interpretations of 9edo, which surprisingly are all 7-limit:
Therefore, one can consider it as interpreting 9edo as a circle of 7/6's (corresponding to tempering the septimal ennealimma) and as a circle of 27/25's (corresponding to tempering the ennealimma),
which is an equivalent description which implies tempering the landscape comma which makes 63/50 equal to exactly a third of an octave.
An alternative 7-limit 9th-octave temperament supported by more edos is to preserve the mapping of 7/6 but not that of 27/25, resulting in septiennealimmal, with many extensions possible. An
important edo of interest that takes this route is 63edo, a tuning doing very well in the no-17's no-19's (no-37's) no-41's 47-limit if you forgive inconsistencies arising from its magic-tempered ~5/
Some higher-limit interpretations of interest for both routes are 14/13~13/12 (tempering S13) for lower-complexity interpretations of 1\9 34/27 for 1\3 (tempering 19683/19652 to give an
interpretation to 3edo) and the "rooted/harmonic wolf fifth" 47/32 for 5\9, by tempering (64/47)/(7/6)^2 = S48 = (48/47)/(49/48). | {"url":"https://en.xen.wiki/w/9th-octave_temperaments","timestamp":"2024-11-04T17:23:36Z","content_type":"text/html","content_length":"28012","record_id":"<urn:uuid:417cbdf1-a6c0-4cb3-a077-372afad5484a>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00723.warc.gz"} |
Course in Kolkata
R is a very popular programming language that is widely used for data mining and analysis. In this course, you'll learn about R programming advanced concepts with R data analytical libraries apart
from this, you will work on live projects with our experts. This course focuses on how to clean, process, and analyze different varieties of data, how to create a data frame, how to handle missing
values, and how to do predictive analytics.
This is one of the best courses by which you can upgrade skills in data analytics, you will get a top-level instructor and work on R Studio apart from this you will get a chance to work on a
real-time project.
We will start this course with basics of R programming and then you will move to the R advance level libraries that is required to do data analysis like ggplot, dplyr and Mice a part from this,you
will learn how to analyze different types of data and how to visualize and generate reports. | {"url":"https://digistackedu.com/best-R-programming-course-kolkata","timestamp":"2024-11-05T10:17:27Z","content_type":"text/html","content_length":"76829","record_id":"<urn:uuid:0e4838d3-e2c6-445f-95bd-97d6fcffc795>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00230.warc.gz"} |
Solution Evaluation
Solution Evaluation
The detailed description of the Challenge 2 Solution Evaluation is in Appendix D of the Problem Formulation.
Solution Evaluation Algorithm
1. Check solution file format against the specification in Appendix C of the Problem Formulation. If a solution file is formatted incorrectly, then the solution is deemed infeasible and evaluation
2. Read solution input variables x^on[gk], x^sw[ek], x^sw[fk], x^st[hak], x^st[fk], theta[ik], q[jk], p[gk], q[gk], and t[jk] from the solution files.
3. Round integer input variables x^st[hak], x^sw[ek], x^sw[fk], x^st[fk], and x^on[gk] to the nearest integer values.
4. Check domains of integer input variables, i.e. Equations (42), (44), (45), (55) to (57) and (77). If any violation > 0 is found, then the solution is deemed infeasible and evaluation terminates.
5. Compute generator start up and shut down variables x^su[gk]and x^sd[gk]from Equations (78) to (80).
6. Check constraints on generator start up and shut down variables, i.e. Equations (85) to (88). If any violation > 0 is found, then the solution is deemed infeasible and evaluation terminates.
7. Check simple bounds on continuous input variables v[ik] and t[jk], i.e. Equations (29), (30) and (37). If any violation > epsilon is found, or any t[jk ]< 0, then the solution is deemed
infeasible and evaluation terminates.
8. Compute load real and reactive power consumption variables p[ik]and q[ik] from Equations (38) and (39).
9. Check simple inequality constraints for load ramping, generator bounds, and generator ramping, i.e. Equations (40), (41) and (81) to (84). If any violation > epsilon is found, or any p[gk] < 0,
then the solution is deemed infeasible and evaluation terminates.
10. Compute switched shunt susceptance variables b^cs[hk] from Equation (43).
11. Compute transformer tap ratio and phase shift variables tau[fk]and theta[fk] from Equations (58) to (61).
12. Compute transformer impedance correction variables eta[fk]from Equations (66) and (67).
13. Compute transformer series conductance and susceptance variables g[fk] and b[fk] from Equations (62) to (65).
14. Compute line and transformer real and reactive power flow variables p^o[ek], p^d[ek], q^o[ek], q^d[ek], p^o[fk], p^d[fk], q^o[fk], and q^d[fk]from Equations (46) to (49) and (68) to (71).
15. Compute minimal bus real and reactive power imbalance variables p^+[ik], p^-[ik], q^+[ik], and q^-[ik] from Equations (31) to (36).
16. Compute minimal line and transformer rating exceedance variables s^+[ek], and s^+[fk] from Equations (50) to (54) and (72) to (76).
17. Compute bus imbalance block variables p^+[ikn], p^-[ikn], q^+[ikn], q^-[ikn] and minimal bus objective variables z[ik] from Equations (3) to (12).
18. Compute load block variables p[jkn] and minimal load objective variables z[jk]from Equations (13) to (16).
19. Compute line rating exceedance block variables s^+[enk] and maximal line objective variables z[ek] from Equations (17) to (20).
20. Compute transformer rating exceedance block variables s^+[fnk] and maximal transformer objective variables z[fk] from Equations (21) to (24).
21. Compute generator real power block variables p[gnk] and maximal generator objective variables z[gk] from Equations (25) to (28).
22. Compute case objective variables z[k] from Equation (2).
23. Compute total objective variable z from Equation (1).
24. Return infeasibility indicator and total objective value z.
Numbers in parenthesis correspond to the appropriate equations in the Problem Formulation document.
If the solution is infeasible, set z = z^inf.
Solution Evaluation Source Code
The python Solution Evaluation source code is available on GitHub: https://github.com/GOCompetition/C2DataUtilities
This code can be used as a template for reading the input files and to calculate z^inf. It is also used to insure that the datasets are consistent with the problem formulation and
have consistent formats. Most of the changes are related to insuring the datasets are consistent with the problem formulation.
The most recent commit was July 27, 2021. See the repo History (main) and History2 (utilities) for details of the changes. | {"url":"https://gocompetition.energy.gov/challenges/challenge-2/solution-evaluation","timestamp":"2024-11-07T06:53:23Z","content_type":"text/html","content_length":"34248","record_id":"<urn:uuid:765b9aa0-b1b9-4daf-bbda-f9c1e5eba7af>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00718.warc.gz"} |
On oscillations of a vibratory jaw crusher with asymmetric interaction of the jaws with the processed medium
This paper is devoted to the problem of providing the required synchronous modes of oscillations of a jaw crusher with self-synchronizing inertial vibration exciters. The described mathematical model
of the crusher takes into account the mechanical properties of the medium being processed and the possible asymmetry of its contact with the crusher’s working bodies – the jaws. A numerical analysis
of synchronous modes of crusher vibrations with different asymmetries of the initial location of the processed medium relative to the jaws is done. It is shown that for given oscillation excitation
frequencies, the non-simultaneous contact of the processed medium with the jaws can lead to a change in the types of synchronous vibrations of the jaw crusher.
1. Introduction
One of the problems of creating efficient vibratory jaw crushers with two movable jaws, which oscillations are excited by self-synchronizing inertial vibration exciters, is the provision of
synchronous jaws’ antiphase oscillations [1-3]. The type of synchronization of the exciters’ rotation and the jaws’ oscillation forms have a mutual influence on each other and are determined by the
mechanical parameters of the system, the processed mediums characteristics, the electric drive’s parameters of vibration exciters and their rotation frequency [3, 4]. When studying the dynamics of
such crushers, it is especially difficult to take into account the interaction of the jaws with the medium being processed [2, 5, 6]. In mathematical models used in common computational practice,
this interaction is usually taken into account in the form of viscous friction forces [5, 7, 8]. At the same time, factors excluded from consideration, such as non-simultaneous contact of the jaws
with the processed medium, its elastic properties, as well as the vibro-impact nature of its interaction with the crusher’s jaws, can have a significant influence on the systems dynamics.
This paper is devoted to identifying the effect of non-simultaneous interaction of jaws with the processed medium, taking into account their impact contact, on the exciters’ self-synchronization and
jaws’ movement.
2. Design scheme and the mathematical model of the jaw crusher
The solution of the problem is based on the analysis of the vibrating jaw crusher’s dynamics with two movable jaws that perform straight horizontal oscillations excited by two inertial vibration
exciters installed on each of the jaws. The design scheme of such a crusher is shown in Fig. 1. The crusher’s body is modeled by a solid body of mass ${m}_{1}$, elastically attached to a fixed base.
The jaws are modeled by identical solids with mass ${m}_{2}$. The jaws are attached to the crusher’s body with the identical elastic elements. It is assumed that all elastic elements have linear
stiffness and damping characteristics with coefficients ${c}_{0}$, ${c}_{1}$ and ${b}_{0}$, ${b}_{1}$, respectively. On each of the jaws the same unbalance vibration exciter is fixed, driven by an
asynchronous motor of limited power whereas ${m}_{e}$ and $r$ – each vibration exciter’s imbalanced mass and eccentricity, $J$ – adduced moment of inertia of the vibration exciter’s asynchronous
motor, ${L}_{j}$ – torque of the $j$th vibration exciter’s electric motor ($j=$ 1, 2) described by the static characteristic [9]. The friction in the bearings of the unbalance shafts is taken into
account in the form of the moments ${R}_{j}$ of dry friction’s forces (not shown in Fig. 1).
Between the jaws there is a processed medium modeled by a solid body ${m}_{3}$ with two (one on the left and one on the right) identical elastic elements, with stiffness and viscosity coefficients $
{c}_{2}$ and ${b}_{2}$, respectively, providing one-way interaction with each of the jaws. In addition, the body ${m}_{3}$ is attached to a fixed base by a linear elastic element with stiffness and
viscosity coefficients ${c}_{3}$ and ${b}_{3}$, respectively, which ensures that the modeled medium returns to its initial position only when there is no contact with the jaws. In this way, the
inflow of a new unprocessed part of the medium through the crusher’s fixed loading window into the working space between the jaws is simulated. In the initial state, the working bodies can be
installed with a gap (pre-tension) relative to the contact elements, which is given by the values ${\delta }_{1}$ and ${\delta }_{2}$.
Displacements of the bodies are described by coordinates ${x}_{i}$ ($i=$ 1, 2, 3, 4) of their centers of mass, measured from their equilibrium position. The positions of the debalances are described
by the angles of rotation ${\phi }_{j}$ ($j=$ 1, 2), measured from the negative direction of the axis $Ox$.
The equations of the system motion in a dimensionless form are:
$\left\{\begin{array}{c}\begin{array}{l}{\mu }_{1}{\stackrel{¨}{y}}_{1}+2\eta {{\left(\beta }_{0}^{}+2\right)\stackrel{˙}{y}}_{1}-2\eta {\stackrel{˙}{y}}_{2}-2\eta {\stackrel{˙}{y}}_{4}+\left(2+{\
zeta }_{0}^{}\right){y}_{1}-{y}_{2}-{y}_{4}=0,\\ {\left(1+{\mu }_{e}\right)\stackrel{¨}{y}}_{2}-2\eta {\stackrel{˙}{y}}_{1}+2\eta {\stackrel{˙}{y}}_{2}-{y}_{1}+{y}_{2}+{F}_{23}={{\mu }_{e}\rho \left
(\stackrel{¨}{\phi }}_{1}\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_{1}+{\stackrel{˙}{\phi }}_{1}^{2}\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{1}\right),\\ {\mu }_{3}{\stackrel{¨}{y}}_{3}-{F}_{23}^{}+{F}_
{34}+{F}_{30}=0,\\ \left(1+{\mu }_{e}\right){\stackrel{¨}{y}}_{4}-2\eta {\stackrel{˙}{y}}_{1}+2\eta {\stackrel{˙}{y}}_{4}-{y}_{1}+{y}_{4}-{F}_{34}={\mu }_{e}\rho \left({\stackrel{¨}{\phi }}_{2}\
mathrm{s}\mathrm{i}\mathrm{n}{\phi }_{2}+{\stackrel{˙}{\phi }}_{2}^{2}\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{2}\right),\\ {\stackrel{¨}{\phi }}_{1}=d\left[\left({\stackrel{~}{L}}_{1}-{\stackrel{~}
{R}}_{1}\right){L}^{*}+{\stackrel{¨}{y}}_{2}\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_{1}\right],\\ {\stackrel{¨}{\phi }}_{2}=d\left[\left({\stackrel{~}{L}}_{2}-{\stackrel{~}{R}}_{2}\right){L}^{*}+{\
stackrel{¨}{y}}_{4}\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_{2}\right],\end{array}\end{array}\right\$
${F}_{23}=\left\{\begin{array}{l}{\zeta }_{2}^{}\left({y}_{2}-{y}_{3}-{\stackrel{~}{\delta }}_{1}\right)+2\eta {\beta }_{2}^{}\left({\stackrel{˙}{y}}_{2}-{\stackrel{˙}{y}}_{3}\right),{y}_{2}-{y}_{3}-
{\stackrel{~}{\delta }}_{1}>0,{\stackrel{˙}{y}}_{2}-{\stackrel{˙}{y}}_{3}>0,\\ {\zeta }_{2}^{}\left({y}_{2}-{y}_{3}-{\stackrel{~}{\delta }}_{1}\right),{y}_{2}-{y}_{3}-{\stackrel{~}{\delta }}_{1}>0,{\
stackrel{˙}{y}}_{2}-{\stackrel{˙}{y}}_{3}\le 0,\\ 0,{y}_{2}-{y}_{3}-{\stackrel{~}{\delta }}_{1}\le 0,\end{array}\right\$
${F}_{34}=\left\{\begin{array}{l}{\zeta }_{2}^{}\left({y}_{3}-{y}_{4}-{\stackrel{~}{\delta }}_{2}\right)+2\eta {\beta }_{2}\left({\stackrel{˙}{y}}_{3}-{\stackrel{˙}{y}}_{4}\right),{y}_{3}-{y}_{4}-{\
stackrel{~}{\delta }}_{2}>0,{\stackrel{˙}{y}}_{3}-{\stackrel{˙}{y}}_{4}>0,\\ {\zeta }_{2}^{}\left({y}_{3}-{y}_{4}-{\stackrel{~}{\delta }}_{2}\right),{y}_{3}-{y}_{4}-{\stackrel{~}{\delta }}_{2}>0,{\
stackrel{˙}{y}}_{3}-{\stackrel{˙}{y}}_{4}\le 0,\\ 0,{y}_{3}-{y}_{4}-{\stackrel{~}{\delta }}_{2}\le 0,\end{array}\right\$
${F}_{30}=\left\{\begin{array}{l}{\zeta }_{3}^{}{y}_{3}+2\eta {\beta }_{3}{\stackrel{˙}{y}}_{3},{y}_{2}-{y}_{3}-{\stackrel{~}{\delta }}_{1}\le 0,{y}_{3}-{y}_{4}-{\stackrel{~}{\delta }}_{2}\le 0,\\ 2\
eta {\beta }_{3}{\stackrel{˙}{y}}_{3},{y}_{2}-{y}_{3}-{\stackrel{~}{\delta }}_{1}>0,{y}_{3}-{y}_{4}-{\stackrel{~}{\delta }}_{2}>0.\end{array}\right\$
${\mu }_{n}={m}_{n}/{m}_{2}\left(n=1,3\right)$, ${\mu }_{e}={m}_{e}/{m}_{2}$, ${\zeta }_{0}^{}={c}_{0}/{c}_{1}$, ${\zeta }_{2}^{}={c}_{2}/{c}_{1}$, ${\zeta }_{3}^{}={c}_{3}/{c}_{1}$, $2\eta ={b}_{2}
{T}_{*}/{m}_{2}$, ${\beta }_{0}^{}={b}_{0}/{b}_{1}$, ${\beta }_{2}^{}={b}_{2}/{b}_{1}$, ${\beta }_{3}^{}={b}_{3}/{b}_{1}$, ${y}_{i}={x}_{i}/{X}_{*}$ – dimensionless coordinates, ${X}_{*}={r}_{0}$, $
{r}_{0}$ – given eccentricity initial value, ${T}_{*}=\sqrt{{m}_{2}/{c}_{1}}$ – time scale:
$\rho =\frac{r}{{X}_{*}},{\stackrel{~}{\delta }}_{1}=\frac{{\delta }_{1}}{{X}_{*}},{\stackrel{~}{\delta }}_{2}=\frac{{\delta }_{2}}{{X}_{*}},d=\frac{{\mu }_{e}\rho }{{J}_{}^{*}+{\mu }_{e}{\rho }_{}^
${L}^{*}=\frac{1}{\left({c}_{1}{\mu }_{e}\rho {X}_{*}^{2}\right)},{\stackrel{~}{L}}_{j}=\frac{2{M}_{c}{\sigma }_{j}}{\left({s}_{c}/{s}_{j}-{s}_{j}/{s}_{c}\right)},j=1,2,$
where ${M}_{c}$ and ${s}_{c}$ critical torque and slip of the asynchronous motor, ${\sigma }_{j}=±1$ – denotes direction of the motors’ rotation, ${s}_{j}=\left({\sigma }_{j}{\omega }_{0}-{\stackrel
{˙}{\phi }}_{j}\right)/{\sigma }_{j}{\omega }_{0}$, ${\omega }_{0}={\omega }_{e}/p$ – synchronous speed of rotation of the electric motor, ${\omega }_{e}$ – frequency of the supply voltage, $p$ –
number of pole-pairs of the electric motor, ${\stackrel{~}{R}}_{j}=f{\stackrel{˙}{\phi }}_{j}^{2}sign\left({\stackrel{˙}{\phi }}_{j}^{}\right)$, $f$ – coefficient of dry friction, dots indicate
differentiation by dimensionless time $\tau =t/{T}_{*}$. The presented equations system allows to analyze the crusher’s motion, taking into account the non-simultaneous impact of the jaws on the
processed medium and their impact contact.
3. The results of modeling the crushers dynamics
The system oscillations were simulated numerically in Matlab using standard functions for integrating differential equations with the condition to accurately determine the time of the beginning and
the end of the jaw’s contact with the processed medium.
At the first stage, the frequency ranges of synchronous exciters’ rotation and jaws’ motion were determined in the absence of the processed medium (the standard task of studying the frequency ranges
of the exciters’ synchronous rotation and the crusher’s vibration modes with the aim of selecting its operating modes). For this, the frequency of the supply voltage ${\omega }_{e}$ was set, which
discretely increased in the range of 0.1 ≤ ${\omega }_{e}$ ≤ 2.5 with a step $\mathrm{\Delta }{\omega }_{e}=$ 0.5 and an exposure at each step during $\mathrm{\Delta }\tau =$ 400 sufficient to
achieve steady-state oscillations of the system (the law of change ${\omega }_{e}\left(\tau \right)$ is shown in Fig. 2). In this case, in the considered frequency range ${\omega }_{e}$, the maximum
torque of the engine was considered to be a constant value. The calculations were carried out with the following system parameters: ${\mu }_{1}=$ 2, ${\mu }_{3}=$ 0.05, $\eta =$ 0.03, ${\zeta }_{01}=
$ 0.1, ${\zeta }_{14}=$ 1, ${\zeta }_{30}=$ 0.5, ${\beta }_{0}=$ 1, ${\beta }_{2}=$ 10, ${\beta }_{3}=$ 1, ${M}_{c}=$ 100, ${s}_{c}=$ 0.2, $f=$ 0.001, $d=$ 0.61, ${L}^{*}=$ 0.005, ${\sigma }_{1}=-1$,
${\sigma }_{2}=1$.
Fig. 2Power supply frequency ωe in time τ
In Fig. 3 and Fig. 4 the results of calculating the rotational velocity of the inertial exciters ${\omega }_{j}$ and the mutual phase shift of the rotation $\mathrm{\Delta }\phi$ between the exciters
as a function of time $\tau$ are shown, respectively. The value of $\mathrm{\Delta }\phi =$ 180° corresponds to the synchronous antiphase rotation of the exciter’s debalances, which causes the
antiphase oscillations of its jaws required for the crusher’s normal operation. When $\mathrm{\Delta }\phi =$ 0°, the debalances rotate in phase, exciting the common-mode oscillations of the
crusher’s jaws. One can see from Fig. 2 and Fig. 4 that the required synchronous modes of the debalances rotation and, accordingly, oscillations of the crusher are realized in the ranges of supply
voltage frequencies ${\omega }_{e}\in$[0.8, 1.2] and ${\omega }_{e}\in$[1.85, 2.5]. Fig. 2 and Fig. 3 show the relationship between the power supply frequency and the rotational velocities of the
debalances. It also can be seen that in the indicated ranges of the supply voltage frequency, the debalances rotate with the same angular velocity in absolute value. When approaching the second and
third resonant frequencies (${\omega }_{2}^{*}=$ 0.976, ${\omega }_{3}^{*}=$ 1.445), the rate of change of an average rotational velocity of the debalances decreases at the same rate of change in the
power supply frequency (Fig. 3). The passage through the resonance is accompanied by a jump in the rotational velocity of the debalances, as well as a change in the type of their synchronous rotation
and the jaws’ oscillation form. These effects are associated with a slip in asynchronous motors and the interaction of the oscillating system with vibration exciters.
Fig. 3The rotational speeds of debalances ωj in time τ
Fig. 4Mutual phase shift Δφ between the debalances in time τ
To analyze the possible influence of non-simultaneous contact of the jaws with the processed medium on the exciters’ synchronization and the jaws’ vibrations, the crusher’s oscillations were
simulated with a gradual change in the initial gap ${\stackrel{~}{\delta }}_{1}$ between the left jaw and the corresponding contact element of the processed medium for a given value of power supply
frequency. The gap varied in the range of –0.1 ≤ ${\stackrel{~}{\delta }}_{1}$ ≤ 0.4 with a step $∆{\stackrel{~}{\delta }}_{1}=$ 0.1 and an exposure at each step for $\mathrm{\Delta }\tau =$ 400 (the
law of change of the gap ${\stackrel{~}{\delta }}_{1}\left(\tau \right)$ is shown in Fig. 5). At the same time, the initial gap between the right jaw and the right contact element of the medium did
not change and, to ensure the initial contact symmetry, it was assumed to be ${\stackrel{~}{\delta }}_{2}=$ –0.1. The study was carried out for different frequencies of the supply voltage in the
range of ${\omega }_{e}\in$[1.85, 2.5], in which a exciters’ synchronous antiphase rotation occurs (after the third resonance). The frequency range ${\omega }_{e}\in$[0.8, 1.2] between the first and
second resonances was not considered, because at these frequencies the developed forces are usually not sufficient to destroy the material, and therefore it is not used in common practice.
In Fig. 6-9 graphs of the change in the mutual phase shift of $\mathrm{\Delta }\phi$ between the debalances obtained at supply voltage frequencies ${\omega }_{e}=${1.85, 2.0, 2.2, 2.4} are presented.
It can be seen that at frequencies ${\omega }_{e}=${1.85, 2.0} (see Fig. 6 and Fig. 7), the exciters’ synchronization is broken when the gap ${\stackrel{~}{\delta }}_{1}=$ 0.4 is reached. In this
case, the mutual phase shift $\mathrm{\Delta }\phi$ between the debalances does not stabilize with time. In turn, this leads to a violation of the required oscillation synchronous antiphase mode of
the crusher’s jaws. When ${\omega }_{e}=$ 2.2 (see Fig. 8), a synchronization violation occurs when ${\stackrel{~}{\delta }}_{1}=$ 0.3. When ${\omega }_{e}=$ 2.4 (see Fig. 9), synchronization is
violated when ${\stackrel{~}{\delta }}_{1}=$ 0, while the mode of antiphase oscillations of the crusher’s jaws is maintained up to ${\stackrel{~}{\delta }}_{1}=$0.2 (i.e., up to $\tau =$ 1200). Thus,
with an increase in the excitation frequency, a decrease in the contact asymmetry of the processed medium with the crusher’s jaws is observed, at which a violation of the required exciters’
synchronization and, accordingly, oscillations of the jaws occurs.
Fig. 5The change in the gap δ~1 in time τ
Fig. 6Mutual phase shift Δφ between the unbalances at ωe= 1.85
Fig. 7Mutual phase shift Δφ between the unbalances at ωe= 2.0
Fig. 8Mutual phase shift Δφ between the unbalances at ωe= 2.2
Fig. 9Mutual phase shift Δφ between the unbalances at ωe= 2.4
4. Conclusions
The simulation results presented in this paper clearly demonstrate the possibility of violation of the debalances synchronous rotation and, accordingly, oscillations of the crusher’s jaws when the
contact interaction conditions with the processed medium are changed due to jaws’ non-simultaneous contact with the processed medium. It is shown that with an increase in the excitation frequency, a
decrease in the contact asymmetry of the processed medium with the crusher’s jaws is observed, at which a violation of the required synchronous rotation of the unbalances and, accordingly,
oscillations of the jaws occur. Such changes in the contact interaction conditions can occur directly during the crusher’s operation, and they must be taken into account when assigning its operating
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p. 15-18.
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• Shishkin E., Safronov A. Vibratory jaw crusher dynamics with consideration of working load effect. Obogashchenie Rud, Vol. 366, Issue 6, 2016, p. 39-43.
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About this article
Mechanical vibrations and applications
vibratory jaw crusher
unbalance exciter
processed medium influence
The reported study was funded by RFBR according to the research project No. 18-08-01491_a.
Copyright © 2019 Grigory Panovko, et al.
This is an open access article distributed under the
Creative Commons Attribution License
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Introduction to the Fundamentals of Vector Autoregressive Models
In today's blog, you'll learn the basics of the vector autoregressive model. We lay the foundation for getting started with this crucial multivariate time series model and cover the important details
• What a VAR model is.
• Who uses VAR models.
• Basic types of VAR models.
• How to specify a VAR model.
• Estimation and forecasting with VAR models.
What is a vector autoregressive model?
The vector autoregressive (VAR) model is a workhouse multivariate time series model that relates current observations of a variable with past observations of itself and past observations of other
variables in the system.
VAR models differ from univariate autoregressive models because they allow feedback to occur between the variables in the model. For example, we could use a VAR model to show how real GDP is a
function of policy rate and how policy rate is, in turn, a function of real GDP.
Advantages of VAR models
✔ A systematic but flexible approach for capturing complex real-world behavior.
✔ Better forecasting performance.
✔ Ability to capture the intertwined dynamics of time series data.
VAR modeling is a multi-step process and a complete VAR analysis involves:
1. Specifying and estimating a VAR model.
2. Using inferences to check and revise the model (as needed).
3. Forecasting.
4. Structural analysis.
Who uses VAR models?
VAR models are traditionally widely used in finance and econometrics because they offer a framework for accomplishing important modeling goals, including (Stock and Watson 2001):
• Data description.
• Forecasting.
• Structural inference.
• Policy analysis.
However, more recently VAR models have been gaining traction in other fields like epidemiology, medicine, and biology.
Example question Field Description
How are vital signs in cardiorespiratory patients dynamically related? Medicine A VAR system is used to model the past and current relationships between heart rate, respiratory rate, blood
pressure and SpO2.
How do risks of COVID-19 infections interact across age groups? Epidemiology Count data of past infections across different age groups was used to model the relationships between infection
rates across those age groups.
Is there a bi-directional relationship between personal income and Economics A two-equation VAR system is used to model the relationship between income and consumption over time.
personal consumption spending?
How can we model the gene expression networks? Biology The relationships across large networks of genes are modeled using a sparse structural VAR model.
What is driving inflation more -- monetary policy shocks or external Macroeconomics A structural VAR model is used to compute variance decomposition and impulse response functions following
shocks? monetary shocks and external system shocks.
The reduced form, recursive, and structural VAR
There are three broad types of VAR models, the reduced form, the recursive form, and the structural VAR model.
Reduced form VAR models consider each variable to be a function of:
• Its own past values.
• The past values of other variables in the model.
While reduced form models are the simplest of the VAR models, they do come with disadvantages:
• Contemporaneous variables are not related to one another.
• The error terms will be correlated across equations. This means we cannot consider what impacts individual shocks will have on the system.
Recursive VAR models contain all the components of the reduced form model, but also allow some variables to be functions of other concurrent variables. By imposing these short-run relationships, the
recursive model allows us to model structural shocks.
Structural VAR models include restrictions that allow us to identify causal relationships beyond those that can be identified with reduced form or recursive models. These causal relationships can be
used to model and forecast impacts of individual shocks, such as policy decisions
A simple example
As an example, let's consider a VAR with three endogenous variables, the unemployment rate, the inflation rate, and interest rates.
A reduced form VAR(2) model of the system includes the following equations:
$$\begin{aligned}\text{UNEM}_t = \beta_{10} &+ \beta_{11}\text{UNEM}_{t-1} + \beta_{12}\text{UNEM}_{t-2}\\&+ \gamma_{11}\text{INFL}_{t-1} + \gamma_{12}\text{INFL}_{t-2} \\&+ \phi_{11}\text{R}_{t-1} +
\phi_{12}\text{R}_{t-2} \\&+ \mu_{1t}\end{aligned}\\ \ \\ \begin{aligned}\text{INFL}_t = \beta_{20} &+ \beta_{21}\text{UNEM}_{t-1} + \beta_{22}\text{UNEM}_{t-2}\\ &+ \gamma_{21}\text{INFL}_{t-1} + \
gamma_{22}\text{INFL}_{t-2} \\&+ \phi_{21}\text{R}_{t-1} + \phi_{22}\text{R}_{t-2} \\&+ \mu_{2t}\end{aligned}\\ \ \\ \begin{aligned}\text{R}_t = \beta_{30} &+ \beta_{31}\text{UNEM}_{t-1} + \beta_{32}
\text{UNEM}_{t-2}\\ &+ \gamma_{31}\text{INFL}_{t-1} + \gamma_{32}\text{INFL}_{t-2} \\&+ \phi_{31}\text{R}_{t-1} + \phi_{32}\text{R}_{t-2} \\&+ \mu_{3t}\end{aligned}$$
A recursive form VAR(2) model of the system might include the following equations:
$$\begin{aligned}\text{UNEM}_t = \beta_{10} &+ \beta_{11}\text{UNEM}_{t-1} + \beta_{12}\text{UNEM}_{t-2}\\&+ \gamma_{11}\text{INFL}_{t-1} + \gamma_{12}\text{INFL}_{t-2} \\&+ \phi_{11}\text{R}_{t-1} +
\phi_{12}\text{R}_{t-2} \\&+ \mu_{1t}\end{aligned}\\ \ \\ \begin{aligned}\text{INFL}_t = \beta_{20} &+ \delta_{21}\text{UNEM}_{t} + \beta_{21}\text{UNEM}_{t-1} + \beta_{22}\text{UNEM}_{t-2}\\ &+ \
gamma_{21}\text{INFL}_{t-1} + \gamma_{22}\text{INFL}_{t-2} \\&+ \phi_{21}\text{R}_{t-1} + \phi_{22}\text{R}_{t-2} \\&+ \mu_{2t}\end{aligned}\\ \ \\ \begin{aligned}\text{R}_t = \beta_{30} &+ \delta_
{21}\text{UNEM}_{t} + \beta_{31}\text{UNEM}_{t-1} + \beta_{32}\text{UNEM}_{t-2}\\ &+ \delta_{31}\text{INFL}_{t} +\gamma_{31}\text{INFL}_{t-1} + \gamma_{32}\text{INFL}_{t-2} \\&+ \phi_{31}\text{R}_
{t-1} + \phi_{32}\text{R}_{t-2} \\&+ \mu_{3t}\end{aligned}$$
To estimate the structural VAR model of the system, we have to put restrictions on our model. For example, we may assume that the Fed follows the inflation targeting rule for setting interest rates.
This assumption would be built into our system as the equation for interest rates.
Specifying a VAR model
What makes up a VAR model?
A VAR model is made up of a system of equations that represents the relationships between multiple variables. When referring to VAR models, we often use special language to specify:
• How many endogenous variables there are included.
• How many autoregressive terms are included.
For example, if we have two endogenous variables and autoregressive terms, we say the model is a Bivariate VAR(2) model. If we have three endogenous variables and four autoregressive terms, we say
the model is a Trivariate VAR(4) model.
In general, a VAR model is composed of n-equations (representing n endogenous variables) and includes p-lags of the variables.
How do we choose the number of lags in a VAR model?
Lag selection is one of the important aspects of VAR model specification. In practical applications, we generally choose a maximum number of lags, $p_{max}$, and evaluate the performance of the model
including $p = 0, 1, \ldots, p_{max}$.
The optimal model is then the model VAR(p) which minimizes some lag selection criteria. The most commonly used lag selection criteria are:
• Akaike (AIC)
• Schwarz-Bayesian (BIC)
• Hannan-Quinn (HQ).
These methods are usually built into software and lag selection is almost completely automated now.
How do we decide what endogenous variables to include in our VAR model?
From an estimation standpoint, it is important to be deliberate about how many variables we include in our VAR model. Adding additional variables:
• Increases the number of coefficients to be estimated for each equation and each number of lags.
• Introduce additional estimation error.
Deciding what variables to include in a VAR model should be founded in theory, as much as possible. We can use additional tools, like Granger causality or Sims causality, to test the forecasting
relevance of variables.
Granger causality tests whether a variable is “helpful” for forecasting the behavior of another variable. It’s important to note that Granger causality only allows us to make inferences about
forecasting capabilities -- not about true causality.
Estimating and inference in VAR models
Despite their seeming complexities, VAR models are quite easy to estimate. The equation can be estimated using ordinary least squares given a few assumptions:
• The error term has a conditional mean of zero.
• The variables in the model are stationary.
• Large outliers are unlikely.
• No perfect multicollinearity.
Under these assumptions, the ordinary least squares estimates:
• Will be consistent.
• Can be evaluated using traditional t-statistics and p-values.
• Can be used to jointly test restrictions across multiple equations.
One of the most important functions of VAR models is to generate forecasts. Forecasts are generated for VAR models using an iterative forecasting algorithm:
1. Estimate the VAR model using OLS for each equation.
2. Compute the one-period-ahead forecast for all variables.
3. Compute the two-period-ahead forecasts, using the one-period-ahead forecast.
4. Iterate until the h-step ahead forecasts are computed.
Reporting and evaluating VAR models
Often we are more interested in the dynamics that are predicted by our VAR models than the actual coefficients that are estimated. For this reason, it is most common that VAR studies report:
• Granger-causality statistics.
• Impulse response functions.
• Forecast error decompositions
Granger-causality statistics
As we previously discussed, Granger-causality statistics test whether one variable is statistically significant when predicting another variable.
The Granger-causality statistics are F-statistics that test if the coefficients of all lags of a variable are jointly equal to zero in the equation for another variable. As the p-value of the
F-statistic decreases, evidence that a variable is relevant for predict another variable increases.
For example, in the Granger-causality test of $X$ on $Y$, if the p-value is 0.02 we would say that $X$ does help predict $Y$ at the 5% level. However, if the p-value is 0.3 we would say that there is
no evidence that $X$ helps predict $Y$.
Impulse response functions
The impulse response function traces the dynamic path of variables in the system to shocks to other variables in the system. This is done by:
• Estimating the VAR model.
• Implementing a one-unit increase in the error of one of the variables in the model, while holding the other errors equal to zero.
• Predicting the impacts h-period ahead of the error shock.
• Plotting the forecasted impacts, along with the one-standard-deviation confidence intervals.
Forecast error decomposition
Forecast error decomposition separates the forecast error variance into proportions attributed to each variable in the model.
Intuitively, this measure helps us judge how much of an impact one variable has on another variable in the VAR model and how intertwined our variables' dynamics are.
For example, if $X$ is responsible for 85% of the forecast error variance of $Y$, it is explaining a large amount of the forecast variation in $X$. However, if $X$ is only responsible for 20% of the
forecast error variance of $Y$, much of the forecast error variance of $Y$ is left unexplained by $X$.
VAR models are an essential component of multivariate time series modeling. After today's blog, you should have a better understanding of the fundamentals of the VAR model including:
• What a VAR model is.
• Who uses VAR models.
• Basic types of VAR models.
• How to specify a VAR model.
• Estimation and forecasting with VAR models.
Further Reading
1. Introduction to Granger Causality
Eric has been working to build, distribute, and strengthen the GAUSS universe since 2012. He is an economist skilled in data analysis and software development. He has earned a B.A. and MSc in
economics and engineering and has over 18 years of combined industry and academic experience in data analysis and research.
2 thoughts on “Introduction to the Fundamentals of Vector Autoregressive Models”
1. kenyonl
great article, thank you for writing this, currently writing my dissertation and am trying to build a SVAR model and tis really helped me get hold of the fundamentals.
1. Eric Post author
Thank you for the positive feedback! I am happy to hear that you found it helpful.
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Discretization, inflation and perturbation of attractors
Title data
Grüne, Lars ; Kloeden, Peter E.:
Discretization, inflation and perturbation of attractors.
In: Fiedler, Bernold (ed.): Ergodic theory, analysis and efficient simulation of dynamical systems. - Berlin : Springer , 2001 . - pp. 399-416
ISBN 978-3-642-56589-2
DOI: https://doi.org/10.1007/978-3-642-56589-2_17
Abstract in another language
The basic issues concerning the effect of discretization or perturbation on autonomous attractors are now quite well understood. For nonautonomous systems matters are, however, considerably more
complicated as solutions now depend explicitly on both the initial and the current time, so limiting objects need not exist in current time or be invariant, the semigroup evolutionary property no
longer holds, and the concept of an attractor for autonomous systems is generally too restrictive.Nonautonomous systems are ubiquitous. They are easily obtained by including time variation in the
vector field of an autonomous differential equation and also arise naturally without an underlying autonomous model. Moreover, they cannot be entirely avoided when one is interested primarily in a
particular autonomous system, since perturbations and noise terms are more realistically time dependent, while numerical schemes with variable step size are essentially nonautonomous difference
equations even when the underlying differential equation is autonomous. This Chapter begins with a brief review of results for the autonomous case and more recent ideas on inflated autonomous
attractors. The cocycle formalism for a nonautonomous system and the concepts of pullback convergence and pullback attractors in such systems are then outlined. Results on the existence of pullback
attractors and of Lyapunov functions characterizing pullback attractors are presented, the formulation of a numerical scheme with variable time steps as a discrete time cocycle system is discussed
and the comparison of numerical and original pullback attractors considered, at least in special cases, along with the inflation of pullback attractors. Finally, some open questions and desirable
future developments are mentioned.
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Principles and Postulates of Quantum Mechanics
At the beginning of the 20th century, physicists could not correctly describe the behavior of very small particles such as electrons, atomic nuclei, and molecules. The behavior of these particles is
correctly described by a set of physical laws that we call Quantum Mechanics.
At the beginning of the century, a small number of physicists, among whom we can mention Bohr, Einstein, Born, Dirac, Schrödinger, Heisember, De Broglie, Jordan, Pauli, contributed to mathematically
formalize the Theory that was practically complete at the end of the decade. 1920's
The study of Quantum Mechanics can be carried out following two different paths. The first way consists of analyzing those physical problems that Classical Mechanics is incapable of solving and that,
however, were correctly interpreted by Quantum Mechanics. We can say:
• The Black Body Spectral Radiation Law
• The photoelectric effect.
• The heat capacities of solids.
• The atomic spectrum of the hydrogen atom.
• The Compton Effect
The second way that we can follow is the axiomatic one. We start from some fundamental postulates from which results on the behavior of microscopic physical systems are deduced. These results are
contrasted with the experiment, being able to observe the greater or lesser agreement between the theory and the experimental data, which provides a direct measure of the goodness of the theory.
In this section we will address the study of Quantum Mechanics from the axiomatic point of view. The best known formulations are the Schrödinger formalism which is based on the wave description of
matter. The Heisenberg and Dirac formalism employs algebra of vectors, operators, and matrices. Schrödinger showed that both formalisms are equivalent and can be used interchangeably.
The study of Quantum Mechanics can be complex and not very motivating at first, since it starts from some postulates that may seem strange, capricious and difficult to understand. This initial
sensation should not discourage us since the application of the theory to practical problems (particle in a box, harmonic oscillator, rigid rotor) will allow us to see the simplicity with which this
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R0 and the exponential growth of a pandemic
For some dissemination work, I want to create a nice graph to explain the exponential growth in pandemics, related to the value of $R_0$. Recall that $R_0$ corresponds to the average number of people
that a contagious person can infect. Hence, with $R_0=1.5$, 4 people will contaminate 6 people, and those 6 will contaminate 9, etc. After $n$ iteration, the number of contaminated people is simply
$R_0{}^n$. As explained by Daniel Kahneman
people, certainly including myself, don’t seem to be able to think straight about exponential growth. What we see today are infections that occurred 2 or 3 weeks ago and the deaths today are
people who got infected 4 or 5 weeks ago. All of this is I think beyond intuitive human comprehension
For different values of $R_0$ (on each row), I wanted to visualise the number of contaminated people after 3, 5 or 7 iterations, since graphs are usually the most simple way to give some intuition.
The graph I had in mind was the following
(to be honest, I am quite sure I had seen it somewhere, but I cannot find where). The main challenge here is pack optimally $k$ identical circles intro a unit circle: we need here the location of the
points (center of the disks) and the radius. It seems to be a rather complicated mathematical problem. Nicely, on http://hydra.nat.uni-magdeburg.de/packing, it is possible to get the “best known
packings of equal circles in a circle” (up to 5000, but many $k$‘s are missing). For instance, for $k=37$, we have
And interestingly, on the same website, we can get the coordinates of the centers, for example with 37 disks, so it is possible to recreate the R graph.
k = 37
base = read.table(paste("http://hydra.nat.uni-magdeburg.de/packing/cci/txt/cci",k,".txt",sep=""), header=FALSE)
The problem, as discussed earlier, is that some cases are not solved, yes, for instance $k=2^{12}=4096$: the next feasable case is 4105. To avoid that issue, one can use
T = "Error"
while(T == "Error"){
T = substr(try(base = read.table(paste("http://hydra.nat.uni-magdeburg.de/packing/cci/txt/cci",k,".txt",sep=""), header=FALSE),silent = TRUE),1,5)
Now we can almost plot it. The problem is that the radius of the circles is missing, here. But we can compute it
D=as.matrix(dist(x = base[,2:3]))
i=which(D == min(D), arr.ind = TRUE)
r = D[i[1,1],i[1,2]]
Here the radius is
To plot it, use
circ= function(x,y,r,h=1){
for(i in 1:k) polygon(circ(base[i,2],base[i,3],r/2*.95),col=colr,border=NA)
We can now use that code to create the graph above, with $k=R_0{}^n$ for various values of $n$
And we can also use it to visualize more subtile differences, like $R_0=1.1$, $R_0=1.3$, $R_0=1.5$ and $R_0=1.7$
OpenEdition suggests that you cite this post as follows:
Arthur Charpentier (August 16, 2020). R0 and the exponential growth of a pandemic. Freakonometrics. Retrieved November 5, 2024 from https://doi.org/10.58079/ovgl
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Top 50+ Best Triangle Puns, Dad Jokes And Wordplays To Make You Laugh Out Loud - justbadpuns.com
Top 50+ Best Triangle Puns, Dad Jokes And Wordplays To Make You Laugh Out Loud
In this very funny pun compilation, we have come up with and collected the best triangle puns, dad jokes and wordplay to make you LOL.
1. Triangulating the Fun: The Top Triangle Puns to Make You Cringe and Chuckle
1. Why did the triangle refuse to play hide and seek? Because it always ended up in a corner.
2. What did the triangle say to the circle? You’re pointless.
3. How did the triangle solve its problems? With acute angle thinking.
4. Why did the triangle go to the party alone? It couldn’t find the right angle.
5. What did the triangle say to the square? You’re so edgy.
6. Why did the triangle break up with the circle? It wanted to be a little more obtuse.
7. What do you call a triangle that keeps telling jokes? An acute triangle.
8. How do you know when a triangle is hungry? It starts to have sine cravings.
9. Why did the triangle go to the doctor? It was feeling acute.
10. What did the triangle say to the parallel lines? Stop being so obtuse.
2. Getting in Shape with Hilarious Triangle Dad Jokes and Puns
1. Why don’t triangles ever get in trouble? Because they always register their points.
2. How does a triangle stay in shape? With a perimeter workout.
3. What do you call a triangle that loves to barbecue? A grill-angle.
4. Why did the triangle go to the gym? It wanted to work on its “tri”-ceps.
5. How do triangles stay cool in the summer? They always find the right shade.
6. Why don’t triangles ever lie? Because they’re always on the straight and narrow.
7. What do you call a talking triangle? Tri-talk-al.
8. Why did the triangle get a job as a chef? It wanted to make some acute dishes.
9. How does a triangle say goodbye? Triangley and with a little wave.
10. What do you call a triangle that’s in a hurry? A sprintangle.
3. Three Sides of Laughter: Wordplay Galore with Triangle Puns
1. Why was the triangle always in trouble? It had too many degrees.
2. How does a triangle keep its breath fresh? With a tangy rinse.
3. Why did the triangle join the band? It had a lot of rhythm and angles.
4. How does a triangle send a letter? With triangular mail.
5. What’s a triangle’s favorite type of music? Hip-hop-otenuse.
6. Why was the triangle always the life of the party? It had a lot of acute friends.
7. How did the triangle win the race? By taking the shortest distance.
8. What do you call a triangle that’s always late? An acute angle.
9. Why did the triangle become a comedian? It had a lot of punchlines in every corner.
10. How do you talk to a triangle? You point out the conversation.
4. Sharp Wit and Clever Wordplays: Funny Triangle Puns That Will Make You LOL
1. Why did the triangle enroll in acting school? It wanted to work on its dramatic angles.
2. How does a triangle tell time? With acute precision.
3. Why did the triangle go on a diet? It wanted to be more well-rounded.
4. How did the triangle become a detective? It always had the right angles.
5. What did the triangle say to the square when it stole its lunch? “You’re going to pay for that in degrees!”
6. Why don’t triangles ever get lost? They always know the direction.
7. How does a triangle make a decision? It weighs all the angles.
8. Why did the triangle get a job as a hair stylist? It was good at making sharp cuts.
9. What do you call a triangle that’s always cold? A chill-angle.
10. How does a triangle keep its secrets? It always holds them close to its vertices.
5. Playing Geometry Games: The Best Triangle Puns for Math Lovers and Joke Enthusiasts
1. Why was the triangle the teacher’s favorite shape? It was always right.
2. How does a triangle take a selfie? By finding its best angle.
3. What’s a triangle’s favorite book? “The Pythagorean Theorem: A Love Story.”
4. Why did the triangle win the math competition? It had all the right answers.
5. How does a triangle throw a party? With acute precision planning.
6. What do you call a triangle that’s always in motion? A rolling triangle.
7. Why did the triangle start a band? It wanted to rock the angles.
8. How did the triangle ace the test? By being a straight-A student.
9. What’s a triangle’s favorite movie genre? Action-packed triangles.
10. Why did the triangle start a garden? It wanted to cultivate its roots. | {"url":"https://justbadpuns.com/triangle-puns/","timestamp":"2024-11-11T04:02:12Z","content_type":"text/html","content_length":"54552","record_id":"<urn:uuid:2df77055-b9a9-4a83-aee3-389ba99db166>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00253.warc.gz"} |
Valerii Sopin
Author:Valerii Sopin
EasyChair Preprint 8203
EasyChair Preprint 8203
EasyChair Preprint 10235
EasyChair Preprint 8203
An Exotic 4-sphere
EasyChair Preprint 9575
EasyChair Preprint 10582
EasyChair Preprint 10235
EasyChair Preprint 10210
An Exotic 4-sphere
EasyChair Preprint 9575
An Exotic 4-sphere
EasyChair Preprint 9575
An Exotic 4-sphere
EasyChair Preprint 9575
EasyChair Preprint 9548
EasyChair Preprint 9088
EasyChair Preprint 9088
PH = PSPACE
EasyChair Preprint 7481
PH = PSPACE
EasyChair Preprint 7481
EasyChair Preprint 9088
PH = PSPACE
EasyChair Preprint 7481
PH = PSPACE
EasyChair Preprint 7481
EasyChair Preprint 8843
EasyChair Preprint 8271
EasyChair Preprint 8250
EasyChair Preprint 8203
PH = PSPACE
EasyChair Preprint 7481
3SUM (kSUM) problem, A vertex contraction, An edge contraction, Beck’s theorem, Berkovich analytic spaces, Blow up, Bouquet of spheres, BQP, Bunyakovsky’s conjecture, Catalan’s constant, Chiral de
Rham complex, coding theory, cohomologies, complete and subcomplete sequences, complete graphs, computational complexity^2, computational geometry, configuration space, Conformal algebras,
deformation, determinant, discrete Fourier analysis, discrete geometry, divisibility, Euler’s 6k + 1 theorem, Exotic n-spheres, Exotic smooth structures, Fermat’s theorem on sums of two squares,
Fibonacci anyons, Freudenthal suspension theorem, General Position Subset Selection Problem, generalization, Graph complexes, Gromov-Witten theory, Hadamard codes, Hadamard conjecture, Hadamard
matrix, Hilton's theorem, homotopy groups, IHX-relation, integer lattices, inverse limit, Irrationality, James and Hilton-Milnor splittings, knapsack problem, Kontsevich’s theory, Kronecker product,
L_{\infty} algebra, Landau’s problems, Lichnerowicz differential, Mertens function, N = 2 superVirasoro algebra, n-Lie algebras, Nambu-Poisson bracket, Necklace polynomials, No-three-in-line problem,
number theory, One-dimensional lattice, Pachner moves, Paley’s work, Parafermions, Piecewise-linear manifolds, Poincare conjecture, Poisson manifold, polynomial hierarchy, prime numbers, primes
represented by polynomials, PSPACE, QBFs, Quantified Boolean Formula, Quantization, Rado graph, reconstruction conjecture, Redheffer matrix, Richter-Gebert’s Universality theorem, Riemann hypothesis,
Riemann series theorem, Riemann zeta function, semi-infinite, series, Set Reconstruction Conjecture, sieve theory, simplicial complexes, spheres, square grid, statistical model^2, Strong homotopy Lie
algebra, Structure of sumsets, Subdivisions, Temperley–Lieb algebra, transcendental number, triangulations, Valuation, vertex algebras^2, Wedge sum, Weighted simplicial complexes, Witt algebra. | {"url":"https://wvvw.easychair.org/publications/author/kNh5","timestamp":"2024-11-14T23:57:20Z","content_type":"text/html","content_length":"14610","record_id":"<urn:uuid:18a614f2-f3be-4ca5-9599-d150ae00d838>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00560.warc.gz"} |
A Model to Predict Thermal Conductivity in Colloidal Dispersions
386420 A Model to Predict Thermal Conductivity in Colloidal Dispersions
Thursday, November 20, 2014: 10:03 AM
213 (Hilton Atlanta)
The presence of nanoparticles with high thermal conductivity in a fluid seems to be a simple way of improving the poor thermal conductivity of the fluids used in many cooling processes. Maxwell's
mean field theory is the standard theory for predicting the thermal conductivity of a dispersion and consist of two limiting bounds. The theory predicts two limit: a lower limit that corresponds to a
fully dispersed colloid, and an upper limit, which is taken to represent the conductivity of a colloidal gel. It is understood that the conductivity of aggregated colloids falls somewhere in between
the two limits, but no theory exists to calculate conductivity of partially aggregated colloids.
Here we present a model, based on Maxwell's theory, that is capable of describing the thermal conductivity of finite-size clusters. We employ a two-level model. First, we calculate the thermal
conductivity of clusters using the upper bound of Maxwell's theory, then we obtain the conductivity of a dispersion of such clusters using the lower limit. We put the theory to test against numerical
simulations. We generate fractal clusters in a base fluid and evaluate the thermal conductivity of the system using a Monte Carlo algorithm. We find that theory provides excellent agreement with the
simulations using one adjustable parameter that corrects for the fact that clusters form a bicontinuous structure, whereas theory assumes clusters to form a continuous solid phase with the liquid
pockets dispersed within. We present results for various types of clusters and show that the adjustable parameter is rather insensitive to the details of the structure of the aggregate.
Extended Abstract: File Not Uploaded | {"url":"https://aiche.confex.com/aiche/2014/webprogram/Paper386420.html","timestamp":"2024-11-13T21:31:51Z","content_type":"text/html","content_length":"10143","record_id":"<urn:uuid:4d40be29-21b0-48dc-9672-3155eaa93d22>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00743.warc.gz"} |
Tetration Function: Simple Definition and Examples
What is Tetration?
Tetration is iterated (repeated) exponentiation. The exponent “b” tells you how many times to exponentiate the base. It is the fourth in a sequence of basic arithmetic operations:
1. Addition,
2. Multiplication (repeated addition),
3. Exponentiation (repeated multiplication),
4. Tetration (repeated exponentiation).
Pentation is the fifth operation (repeated tetration), and hexation (repeated pentation) is the sixth. The entire sequence is called the hyper-operation sequence.
Connection to Exponentiation
The idea is similar to exponents. For example, the exponent 3^4 is written as:
3^4 = 3 * 3 * 3 * 3 = 81.
The superscript (in this example, 4) tells you how many times to multiply out the base. The exponent in tetration (called the “height) tells you how many times to exponentiate the base. So, for ^43
you take the base (3) and iteratively exponentiate it three times (giving a total of four “3”s in the equation):
^43 = (3^3^3^3) = 3^7.626 (to 3 decimal places).
Examples of How to Solve
To solve, simplify from the innermost nest first.
• 3[^3(^3^3)] =
• 3(^3^27) =
• 3(^3^27) = 3^7.626 (to 3 decimal places).
Make sure you start expanding at the uppermost (right) exponent, otherwise the result is just the multiplication of exponents, not exponentiation of exponents.
Another example:
• ^42 =
• 2^2^2^2 =
• 2^2^4 =
• 2^16 =
• 65536.
Tetration Function Notation
The tetration function is a family of functions that undergo tetration. Tetration is denoted by ^ba, which is almost the same as the exponential function— except the exponent is to the left of the
The general form of the tetration function is:
t[n](x) = ^n,x
Where n is the order of tetration.
Is there a Real Life Use?
For all practical purposes, no. Which is probably why it isn’t taught in school alongside addition and multiplication.
These calculations often result in trillions of zeros. Most calculators can’t handle that many digits, and so will give you an error.
To put this in perspective, ^310 = 10^10^10 which, if you wrote out all of the digits, results in 1 followed by ten billion zeros. Compare that to the number of atoms in the universe, which has a 1
with 80 zeros. Additionally, tetration has a few odd properties. For example, √2 tetrated to infinity equals 2 (Seligman, 2016).
Due to the problems with calculating the trillions of digits, you’ll probably only ever work with a second order (^2x) or third order (^3x) tetration function.
Lynch, P. The Fractal Boundary of the Power Tower Function.
Neyrinck, M. An Investigation of Arithmetic Operations. Retrieved November 26, 2019 from: http://skysrv.pha.jhu.edu/~neyrinck/extessay.pdf
Seligman, E. (2016). Math Mutation Classics: Exploring Interesting, Fun and Weird Corners of Mathematics. Apress.
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Multivariate polynomial coefficients including zeros
Multivariate polynomial coefficients including zeros
I would like to get the coefficients of a multivariate polynomial including the zero coefficients (in their correct positions). I have found a similar answer as regards a polynomial f of two
variables x,y
P.<x,y> = PolynomialRing(ZZ, 2, order='lex')
Then using the following code:
coeffs = []
for i in range(f.degree(x), -1, -1):
for j in range(f.degree(y), -1, -1):
coeffs.append(f.coefficient({x:i, y:j}))
The result is as expected: [3, ,0 , 0, 1, 0, 0, 0, 0, 3]
Now, I would like to extend this solution for a multivariate polynomial of n variables [xo,x1,...xn-1] The polynomial is defined with the following code: (Note: q=next_prime(10000))
A1 = [(', '.join('x%i'%i for i in [0.. n-1]))]; ### construct a suitable multivariate ring
V = var(A1[0])
How can I do that? Any help will be much appreciated.
Regards, Natassa
A tip independent of your actual question: you can construct this polynomial ring in a cleaner way with
If you want the variables bound to symbols so that you can write x0^2+x1^2 etc. then you can use
or, if you want the variables as a tuple/list/whatever:
1 Answer
Sort by ยป oldest newest most voted
I guess your problem comes the iterated loops that you can not write one by one since you do ont know a priori how many loops there will be (it depends on the number of variables). To solve this
issue, you can have a look at itertools.product:
sage: from itertools import product
sage: product?
Then you can do something like (i just give some hints about zip and dictionaries):
sage: for I in product(*[range(f.degree(v), -1, -1) for v in P.gens()]):
sage: print I
sage: print dict(zip(P.gens(),I))
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Counting on Laughs: 100+ Side-Splitting Number Puns to Add Up Your Humor Quotient
Are you ready to multiply your laughter with over 100 number puns that will sum up to a hilarious time? From counting on a good laugh to finding the perfect equation for humor, these puns will divide
your sides with laughter and add a whole new dimension to your sense of humor. Whether you're a math whiz or just someone who loves a good number joke, these puns will factor in some serious fun to
your day. So get ready to subtract the seriousness and add some laughter to your life with these witty and clever number puns that are sure to make you smile. Get ready to laugh your factors off as
we dive into the world of number puns!
Count on These Number Puns!
• Did you hear about the mathematician who’s afraid of negative numbers? He’ll stop at nothing to avoid them!
• Parallel lines have so much in common. IT’s a shame they’ll never meet.
• Why was the equal Sign so humble? Because he knew he wasn’t less than or greater than anyone else.
• I’m Reading a Book about anti-Gravity. It’s impossible to put down!
• Why should you never talk to Pi? Because he’ll just go on forever!
• What do you get when you divide the circumference of a Pumpkin by its diameter? Pumpkin Pi!
• Why do mathematicians like parks? Because of all the natural logs.
• I’m a big Fan of whiteboards. They’re remarkable!
• What’s the best way to woo a Math Teacher? Use acute Angle!
Number Puns with a Pun-derful Twist
• "I Can't count how many times I've failed at math," he said sum-berly.
• "I'm feeling irrational," Tom said irrationally.
• "I'll never divide my attention," she said fractionally.
• "I'm a prime candidate for this job," he said optimistically.
• "I'm in a decimal of trouble," she said pointlessly.
• "I'm in my Element when it comes to numbers," he said elementarily.
• "I'm in my prime right now," Tom said perfectly.
• "I'm Positive that this pun is a real winner," she said electrically.
• "I'm feeling acute about my math skills," he said sharply.
• "I'm a natural at adding humor to numbers," she said sumptuously.
Historical Puns About Numbers
• Why did the Roman numeral Break Up with the number 7? It just wasn't their X.
• What do you call a Medieval mathematician? A Knight of the round number.
• How did the ancient Egyptians do math problems? With their "pyramath" skills.
• Why was the number 6 afraid of the number 7? Because 7 "eight" 9!
• What's a Pirate's favorite number? Seven, because it's the One that makes you say "arrrr"!
• What do you call a number that can't keep a secret? A prime blabber.
• Why was the math book sad? It had too many problems.
• What do you get when you cross a Snowman and a Vampire? Frostbite!
• Why was the math lecture so long? The professor kept going off on tangents.
• Why was the number 10 so friendly? It was always making "tens" with others.
Literal Puns - Number Puns
• Why did the math book look sad? Because it had too many problems to solve.
• Did you hear about the mathematician who's afraid of negative numbers? He'll stop at nothing to avoid them.
• Why was the number six afraid of seven? Because seven "ate" nine!
• What do you call a number that can't keep still? A roamin' numeral.
• Why should you never mention the number 288? It's too gross (two gross).
• What did the zero say to the eight? Nice Belt!
• Why is it always a Bad idea to play hide-and-seek with numbers? They always find you.
• Why was the math book sad after a breakup? It couldn't find a solution to its problems.
• What do you call a number that's always ready for action? A "prime" number.
• Why did the number 10 go to the Therapist? Because it had "ten-sion" issues.
Funny Double Entendre Puns
• I'm Good at math because I can count on myself.
• I told my math teacher I had a lot of problems, but she just laughed and said, "That's the sum of your parts!"
• My math skills are on point, or should I say "acute"?
• I tried to become a mathematician, but I always ended up divided between two careers.
• I'm a natural at Geometry because I can always find the right angle.
• Why did the math book look sad? It had too many problems.
• My calculator broke up with me because it couldn't handle our complex Relationship.
• I asked the math teacher if I could borrow his Pencil. He said, "Sure, just make sure you don't subtract it from my Desk!"
• My favorite math equation is the one that adds up to happiness.
• I told my Friend that I was Terrible at math, and he said, "Don't worry, you can always count on me!"
Count on These Puns!
• I've been trying to organize a hide and seek competition, but it's a bit hard to find good hiding spots. I guess it's just too number-cally challenging.
• I used to be a Baker, but I couldn't make enough Dough. I guess I just kneaded more.
• I told my wife she should embrace her mistakes. She gave me a Hug.
• I told my wife she should embrace her mistakes. She gave me a hug.
• I'm reading a book on anti-gravity. It's impossible to put down!
• I told my wife she should embrace her mistakes. She gave me a hug.
• I'm reading a book on anti-gravity. It's impossible to put down!
• I'm reading a book on anti-gravity. It's impossible to put down!
• I'm reading a book on anti-gravity. It's impossible to put down!
• I'm reading a book on anti-gravity. It's impossible to put down!
Number Puns That Are Just Punnily Rhyming
• One, two, buckle my Shoe, three, four, I can't count anymore!
• Five, six, pick up sticks, seven, eight, I'm always Running late!
• Nine, ten, do it again, eleven, twelve, I'm feeling overwhelmed!
• Thirteen, fourteen, feeling mean, fifteen, sixteen, where have I been?
• Seventeen, eighteen, feeling keen, nineteen, twenty, my jokes are aplenty!
• Twenty-one, twenty-two, I'm feeling Blue, twenty-three, twenty-four, I need to do more!
• Twenty-five, twenty-six, need my fix, twenty-seven, twenty-eight, feeling Great!
• Twenty-nine, thirty, get Flirty, thirty-one, thirty-two, I'm feeling Brand new!
• Thirty-three, thirty-four, I need a Little bit more, thirty-five, thirty-six, I'm in the mix!
• Thirty-seven, thirty-eight, feeling great, thirty-nine, forty, my puns are sporty!
Funny Spoonerism Puns:
• Won ton Soup? More like ton won soup!
• Why did the Scarecrow win an award? Because he was outstanding in his field, or should I say, his field was outstanding!
• Did you hear about the mathematician who was afraid of fractions? He had a fear of being a number, or should I say, a number of being a fear!
• Why did the Tomato turn Red? Because it saw the Salad dressing, or should I say, the salad dressing saw the tomato!
• Have you heard about the Chef who became a Magician? He turned eggs into omelets, or should I say, omelets into eggs!
• Why did the Bicycle Fall over? It was two-tired, or should I say, it was tired of being two!
• What do you call a Bear without any Teeth? A Gummy Bear, or should I say, a bear without any teeth is gummy!
• Why did the golfer bring two pairs of pants? In case he got a hole in one, or should I say, in case he got one in a hole!
• How do you organize a Space Party? You Planet, or should I say, you plan it!
• Why did the Fish blush? Because it saw the Ocean's bottom, or should I say, the ocean's bottom saw the fish!
Funny Anagram Puns
• Racecar is an anagram of "a Car Race." It's like the Word itself is in a hurry!
• Listen, I'm an anagram enthusiast. I even have a T-Shirt that says, "I'm a fan of anagrams."
• I tried to make an anagram using the word "desserts," but it just spelled out "stressed." I guess it's a sign!
• My friend said he was an anagram expert. I asked him to prove it, and he replied, "I'm a pro at rearranging words!"
• I Love anagrams so much, I even named my Dog "Regan." It's "anger" spelled backward, and she definitely lives up to it!
• I saw an anagram of "listen" that said "silent." So I told it, "Well, you're not very good at being an anagram, are you?"
• An anagram of "dormitory" is "dirty Room." It's like the universe is trying to tell us something!
• I asked my friend if he liked anagrams. He said, "I'm not sure, I'll have to ponder." I replied, "That's just 'red nop,' rearranged!"
• My Mom told me to stop making anagrams all the Time. I said, "But mom, it's my 'mo'!"
• An anagram of "astronomer" is "Moon starer." I guess astronomers really do have their eyes on the prize!
Funny Situational Puns
• Why was the math book sad? Because it had too many problems to solve!
• Did you hear about the mathematician who was afraid of negative numbers? He would stop at nothing to avoid them!
• Why don't scientists trust atoms? Because they make up everything!
• What do you call a number that can't keep still? A roamin' numeral!
• Why don't skeletons fight each other? They don't have the guts!
• Why did the math teacher open a Bakery? Because she kneaded a way to make dough!
• Why did the scarecrow win an award? Because he was outstanding in his field!
• What did one math book say to the other? "I've got problems!"
• Why did the bicycle fall over? Because it was two-tired!
• Why couldn't the Leopard play hide-and-seek? Because he was always spotted! | {"url":"https://punsite.com/number-puns","timestamp":"2024-11-13T16:15:46Z","content_type":"text/html","content_length":"63549","record_id":"<urn:uuid:22304cf5-8c94-4c52-b70c-bfe31442c45a>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00806.warc.gz"} |
What Exactly Are Logical Operators inMathematics? </p
h1 Are Legitimate Operators inMathematics?
Which Are Terrible Operators inMathematics?
There is A well-designed personal laptop system application a logical individual, instead of a badly designed 1. The illogicalities taken out, although A app that is plausible has all the centers of
logic. Which means it’s made for the incorrect motives and comes across as much more intelligent than it actually is that.
Fair math resembles this. The operators in math are designed that they do the job to its appropriate intentions of the rationale operators.
Logical reasoning mathematics takes the illogical out of mathematics. Instead of providing the illogical computations, online paper writing service a logical machine works for the correct reasons of
the logical reasoning.
If you ask a logical machine to find a number for which the logical operator is “and” you will get a number. If you ask a logical operator to compute the result of finding the number, you will get a
number. If you ask a logical operator to compute the result of computing the number and then asking the logical operator to find the number, you will get a number. It does not make any sense, does
Mathematics is not currently working against logic; it really is currently employed for logic. Just like it or not, logic and reason are elements of logical reasoning.
Now if you have ever asked a logical operator to compute the result of a formula containing a logical operator and a zero, you know why this is impossible. A logical machine cannot compute the
outcome of a mathematical operation, because if it did, it would be known as a logical impossibility. Mathematical operations are the logical impossibility.
The logical machine is not designed to compute what mathematicians compute, or even work for their mathematical programs. The logical machine is designed to work by itself in a world where the rules
of logic, logic reasoning, and logic computation are known. It is designed to be intelligent, and to make decisions based on that intelligence.
A logical reasoning machine is only as smart as the computer programs it is able to run, because its programming is what allows it to reason. The logical reasoning machine can run the logical
equations that a logic equation system is designed to allow, and it can also do arithmetic computations. The logical reasoning machine can compute with real numbers, and it can compute complex
mathematical functions.
There is no reason why a logical mathematical machine could not run an AI program in the way it is designed to run. All the computer programs needed to run such a logical reasoning machine can be
found in a single program. Such a program is only one hundred twenty lines of code and if written correctly can run a logical reasoning machine that is over one thousand lines of code in length.
Logical math needs a system of thinking compared to way logic is all thought. The machine has to be composed into some of math equations then require people mathematics equations to be resolved to
find the outcomes that the math equations were meant to give. While mathematics would be the whole story logic is only a portion of logic reasoning.
Logical reasoning systems can run in whole number terms, while arithmetic reasoning systems must be written to whole number terms. However, whole number computations can be implemented with little
trouble, but solving whole number problems using whole number computations will require a full two thousand lines of code.
In summary, logical reasoning mathematical machines need a totally different set of logical operators to accomplish the tasks that they are designed to accomplish. In order to implement logical
reasoning mathematical machines, that system must be designed with logical reasoning operators in mind. Any new programming language or system for running logic reasoning mathematical machines needs
to be designed with these operators in mind to enable the correct way of reasoning.
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13+ What Is -18/4 In A Decimal - collegebeautybuff.com13+ What Is -18/4 In A Decimal
13+ What Is -18/4 In A Decimal
13+ What Is -18/4 In A Decimal. = 13/20 = 13 ÷ 20 = 0.65. For calculation, here's how to convert 13/15 as a decimal using the formula above, step by step instructions are given below.
Decimal 13 Equivalent fractions and decimals YouTube from www.youtube.com
I wish i had more to tell you about converting a fraction into a decimal but it really is that simple and there's. Also, explore tools to convert decimal or. At each step, we write the integer part
of the rightmost digit to the fractional part of the base 13 number.
For Calculation, Here's How To Convert 5/13 As A Decimal Using The Formula Above, Step By Step Instructions Are Given Below Calculator Method:
Convert proper and improper fractions to decimals. I wish i had more to tell you about converting a fraction into a decimal but it really is that simple and there's. How to calculate 5/13 as a
Read More Below On This.
As you recall from the above, 4.13 percent is the same as 4.13 where 4.13 is the. Divide the percent value 13 with 100 and remove % after that. So, 13% means 13 per 100 or simply 13/100.
Convert A Ratio To A Decimal.
This is a simple and straightforward calculation. How do you round decimal 821.13 to one decimal place? 13% = 0.13 in decimal form.
For Calculation, Here's How To Convert 13/6 As A Decimal Using The Formula Above, Step By Step Instructions Are Given Below.
13 (numerator) 6 (denominator) = 13 ÷ 6;. = 13/20 = 13 ÷ 20 = 0.65. 13/16 = 0, remainder is 13.
Since 8 > 5 We Will Round Up And Increase The Hundredths Place By 1.
821.13 rounded to one decimal place is 821.1. Read from the bottom (msb) to top (lsb) as d. That’s it you will get the decimal value. | {"url":"https://collegebeautybuff.com/blog/2022/12/19/13-what-is-18-4-in-a-decimal/","timestamp":"2024-11-08T15:57:30Z","content_type":"text/html","content_length":"56322","record_id":"<urn:uuid:3895568f-11e4-4b70-9c9c-49a30c11c58d>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00389.warc.gz"} |
How do you pass CXC Maths?
So, here in a nutshell is a simple 3-point strategy for passing CXC Maths with a grade 1: Step 1 – revise and master grade 1 to 6 maths, beginning with mastering 1 to 12 times tables followed by
mastering addition, subtraction, multiplication, division, fractions and decimals.
What are the 9 sections tested in CSEC Maths?
There are nine topics: Sets, Relations, Functions and Graphs, Computation, Number Theory, Measurement, Consumer arithmetic, Statistics, Algebra, Geometry. in full for the sample topic Consumer
arithmetic (Appendix III).
How much percent is the maths SBA?
Get the most out of your SBA Spend the time to do a proper SBA and get a s much of the 20% that you can get. Its as simple as that and you’re half way there. Remember that if your center uses the SBA
you have to do it.
What percentage is a Grade 2 in CXC?
Of the 2019 cohort who sat examinations, 13.74% received Grade 1 passes, 28.98% received Grade 2 passes and 31.52% received Grade 3 passes. This equates to 74.24% of candidates earning Grades 1 – 3
passes. There was a 100% pass rate in nine of the 28 subjects.
What is a Grade 2 in CXC?
Represents an excellent performance. GRADE II. Represents a very good standard of performance.
What percent is a Grade 2 in CXC?
What percentage is a grade 2?
Treating 10 per cent as the highest feasible mark for Ungraded as usually at present and dividing 11 to 55 marks in three equal mark-ranges of 15, grade 1 would require 11 – 25 per cent, grade 2 = 26
– 40 per cent and grade 3 = 41 – 55 per cent.
What grade is 77 percent?
Percent Letter Grade
83 – 86 B
80 – 82 B-
77 – 79 C+
73 – 76 C
Is a minus good?
In a system where a top score is rated as a 4.0, the top grade is an A. The way GPAs are calculated can seem unfair to some people. In some schools, though this is not always the case, an A minus
lowers the overall grade point value of a grade, making it count for less than four points.
Why smart students get bad grades?
Poor organization skills can lead to increased frustration, higher levels of stress, and lower grades. Without good organization skills, even the smartest children will struggle to properly plan and
prepare for upcoming tests and assignments. The result? You guessed it—poor grades. | {"url":"https://headshotsmarathon.org/awesome-writing-tips/how-do-you-pass-cxc-maths/","timestamp":"2024-11-11T13:44:38Z","content_type":"text/html","content_length":"62553","record_id":"<urn:uuid:4fea931e-de25-4a83-92ed-f30f5f768dc7>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00750.warc.gz"} |
Vector Subtraction Worksheets
Grade Levels
High School Numbers and Quantity
Vector Subtraction Worksheets
Vectors are often used to mimic or graphically represent motion. They are great for outlining the forces that are at work along that path and how motion was achieved. The in which we find the
difference between two vectors is pretty unique. Rather than simply taking a value away from another, we make one of the vectors negative by simply changing its path to the inverse direction. As a
result, we add a positive to negative vector resulting in the same outcome. This method works regardless of if you are focus on a single component of the vectors or several. In some cases, vectors
will not only have your standard ordered pair form of an x and y component. It may also have a z component as well. These worksheets will show students how to find the differences and changes between
two vectors.
Aligned Standard: HSN-VM.B.4c
Homework Sheets
Throwing it all into a triangle makes it a little more reasonable for students.
Practice Worksheets
We state the existence of vectors in several different forms.
Math Skill Quizzes
The quizzes are much more straight forward than the homework or practice sheets.
How to Subtract Vectors
If we need to find the change or the difference in a vector quantity, we will often use vector subtraction. Vector subtraction doesn't come in our calculations quite often, but it is an important
concept to have, though. To be able to find the difference between a frequency of various forces helps us to understand the nature or movement and direction of a vector. However, aside from the
arithmetical subtraction, you can also subtract two vectors with the help of a figure.
In order to subtract two vectors, you must put the tails together, then draw a resultant vector. The resultant vector is the difference of two vectors, starting from the head of the vector to the
head of the vector you are subtracting from. How we set these types of problems up is not your standard form of subtraction. Normally we would think of subtracting vectors a and b as (a – b). We take
the vector that we feel is smaller and just make it negative by flipping its direction. When then add the positive vector to the vector that is now negative because we flipped the direction of it. So
the form that we use would be = a + (-b). | {"url":"https://www.mathworksheetsland.com/hsnumbersquan/22vecsub.html","timestamp":"2024-11-13T14:46:12Z","content_type":"text/html","content_length":"15300","record_id":"<urn:uuid:26d1b913-5443-419a-8728-3267c82b3bc6>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00665.warc.gz"} |
[Solved] Calculate the total number of unique sear | SolutionInn
Answered step by step
Verified Expert Solution
Calculate the total number of unique searches with 100% coverage and illustrate the unique searches. 2) Illustate in a chart the demand and supply throughout
Calculate the total number of unique searches with 100% coverage and illustrate the unique
2) Illustate in a chart the demand and supply throughout the week.
3) In a form of a 24x7 grid (hour-weekday grid), visualize a coverage heatmap.
4) If it was possible to introduce higher pricing during specific hours to increase supply and
decrease demand, what weekly schedule of price multipliers would you choose? Visualise in a
heatmap and provide specific multipliers.
5) Find the 5-hour period (consecutive hours) with highest demand and calculate how much we
can spend on driver incentives, given that we want to re-invest all the revenue received in this
timeframe. Assume that Finished Rides have the average value of 10 and our commission is
6) Estimate the number of weekly trips we could have done (finished) with maximum Coverage
7) Find the 5 most utilized hours of the week (not necessarily consecutive), and calculate the sum
of trips finished in these 5 hours. Explain how you understand the concept of driver utilization rate
There are 3 Steps involved in it
Step: 1
1 Total Number of Unique Searches with 100 Coverage To calculate the total number of unique searches needed for 100 coverage youd need to know the specifics of the search space and the coverage
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Standard Form To Vertex Form Of A Quadratic Function Worksheet - Function Worksheets
standard form to vertex form of a quadratic function worksheet
Vertex Form Of A Quadratic Worksheet – A well-designed Features Worksheet with Solutions will offer college students with strategies to a variety of crucial questions … Read more | {"url":"https://www.functionworksheets.com/tag/standard-form-to-vertex-form-of-a-quadratic-function-worksheet/","timestamp":"2024-11-09T09:59:03Z","content_type":"text/html","content_length":"68006","record_id":"<urn:uuid:b14af979-4c51-4372-88e6-5c50c1c4db44>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00303.warc.gz"} |
typing mathematical symbols into emacs using the TeX input method
You can type mathematical symbols into emacs using the TeX input method.
TeX <enter>
C-\ will toggle the input method on and off. Note the \ indicator in the bottom left of the status bar.
One response to “typing mathematical symbols into emacs using the TeX input method”
I am didn’t understood anything. Is it an add-on? Because in the vanilla Emacs the C- bound to “toggle-input-method”, and this have nothing to do with TeX, it is just changes keyboard layout
(personally on my PC to the Russian one).
And where shall I enter the “TeX”, and another things written in the post? | {"url":"https://emacstragic.net/uncategorized/typing-mathematical-symbols-emacs-using-tex-input-method/","timestamp":"2024-11-03T00:30:49Z","content_type":"text/html","content_length":"81977","record_id":"<urn:uuid:79b34d11-439c-4aa0-9845-469126b80c5d>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00285.warc.gz"} |
The Stacks project
Lemma 42.32.6. In the situation of Lemma 42.32.5 assume $Y$ is locally of finite type over $(S, \delta )$ as in Situation 42.7.1. Then we have $i_1^*p^*\alpha = p_1^*i^*\alpha $ in $\mathop{\mathrm
{CH}}\nolimits _ k(D_1)$ for all $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(Y)$.
Comments (2)
Comment #4640 by awllower on
What is the morphism $g$ in the proof? Is it supposed to refer to $p$?
Also, the end of the first paragraph seems to conclude that $\def\ast{*}$
which is different from $p_1^\ast i^\ast([W])$.
Comment #4789 by Johan on
Good catch! Thanks very much. Fixed here.
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A High Value of Beta is Not Necessarily Cause for Concern
The Weibull shape parameter is a measure of the variability of the data; a high beta value implies a low variability. Consequently, if test units have a high value of beta, it indicates that they
will fail within a relatively small time span. This should not be a problem as long as the onset of that time span begins relatively far out on the time scale. In other words, a high value of the
shape parameter (beta) is not a problem in itself, as long as the corresponding value of the scale parameter (often denoted as eta) is high enough to allow the product to achieve an acceptable
overall reliability.
In fact, for repairable systems, components with high beta values may actually be preferred because the lack of variability can increase the efficiency of a preventive maintenance program. Less
variability means that failures occur in a more "controlled" manner and therefore a better optimum replacement interval for preventive maintenance can be quantified. For example, it would be ideal
for a preventive maintenance program to have a component that always fails at exactly 1,000 hours of operation. The optimum replacement time would therefore be just before the expected failure, at
999.9 hours.
Another concern that has been associated with reliability data sets with high values of beta is that the relatively steep slope makes it difficult to discern patterns in the data, such as outliers or
breaks in the data, on the probability plot. While this is true, it is not a sufficient reason to reject a set of data. Most of the problems that could potentially be discerned by viewing the pattern
in the probability plot would hopefully be detected elsewhere in the engineering and testing process. For example, breaks in the pattern of data may indicate that multiple failure modes are active.
While a steep slope on a probability plot may tend to obscure such a pattern, the presence of multiple failure modes would likely have been observed during the test or by a concerted failure analysis
program. Similarly, outliers can often be identified merely by looking at the raw data. In general, it is not a good idea to depend on the pattern of a probability plot to supplant more dedicated
engineering and analysis efforts.
Although there may be genuine concerns about data sets with high values of beta, the fact that a data set has a high value of beta is not necessarily cause for alarm as long as the associated value
of h is high enough to offset the lack of variability inherent in data sets with high beta values. In the end, it is more important to analyze the overall behavior of the data and whether or not the
product’s test results meet the requirements than to focus solely on the value of a single parameter. | {"url":"https://www.hbkworld.com/en/knowledge/resource-center/articles/a-high-value-of-beta-is-not-necessarily-cause-for-concern","timestamp":"2024-11-10T05:27:52Z","content_type":"text/html","content_length":"470628","record_id":"<urn:uuid:10f96588-5613-41ce-be0d-20f74f47cd9a>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00612.warc.gz"} |
How to Calculate Expected Cost of Development for New Products in Managerial Economics - dummies
Typically a new product or new production technique isn’t successful. Because of the high risk and likelihood of failure, you can often reduce research and development costs by simultaneously working
on several similar or parallel ideas. Ongoing evaluation of costs and potential benefits of the parallel efforts enables you to determine at what point you should abandon an effort in order to reduce
research and development costs.
Your pharmaceutical company is developing two different drugs for the same medical condition. Each drug’s development cost is influenced by a number of factors, such as potential side effects and
other drug interactions. Because development costs are unknown, you develop best-case and worst-case scenarios that you assume are the same for both drugs.
Your best-case scenario for either drug is that the development cost equals $20 million. Your worst-case scenario for either drug is that the development cost equals $75 million. Because you’re
unsure what the actual development cost will be for each drug, as a pessimist you assume a 30-percent chance for the best-case scenario and a 70-percent chance for the worst-case scenario.
Alternatively, you could be an optimist and use a higher probability for the best-case scenario, or you could base your probability on past experience.
The expected development cost, EDC, for each drug equals the cost of a scenario, C^bc for the best-case scenario and C^wc for the worst-case scenario, multiplied by the probability of that scenario
occurring, P^bc and P^wc, or
You use the following steps to calculate the expected development cost for a drug:
1. First, substitute the values for C[bc], P[bc], C[wc], and P[wc].
2. For each scenario, multiply the cost by the probability.
For the best-case scenario, multiply $20 million by 0.3, and for the worst-case scenario, multiply $75 million by 0.7.
3. Add the resulting values for each scenario.
The expected development cost equals $58.5 million.
Starting your research by developing both drugs at the same time can lower your expected development cost because partway through the process, you can decide to abandon one drug because its
development costs are too high.
If both drugs are developed in parallel, you have a 49-percent chance (0.7 × 0.7) that the development cost of each drug equals $75 million. This is the probability of the worst-case scenario
occurring for both drugs. You have a 9-percent chance (0.3 × 0.3) chance that the development cost of each drug equals $20 million. This is your best-case scenario occurring for both drugs.
Finally, you have a 42-percent chance that one of the two drugs has the best case scenario — a $20 million development cost. This is the situation where one drug’s development represents the
best-case scenario and the other drug’s development represents the worst-case scenario (0.3 × 0.7 + 0.7 × 0.3).
Because you ultimately complete the development of only one drug — remember, both drugs are for the same condition — you’ll choose the cheapest drug to develop. Thus, you have a 51-percent chance —
the 9-percent chance of best-case for both drugs, plus the 42-percent chance of best-case for one of the two drugs — that the development cost equals $20 million.
Note how 51 percent is better than the 30 percent chance of the best-case scenario’s $20 million development cost if you develop only one drug.
Now comes a critical step. If the actual development cost of each drug can be determined with certainty after $C have been spent, and only the drug with the lowest development cost is developed after
that point, the expected development cost equals
In the example, you know that you can determine the actual development cost for each drug after spending $8 million. (This value varies from situation to situation and must be determined in advance,
usually by past experience.) Thus, your expected development cost of parallel efforts equals
In this situation, your parallel efforts have lowered the expected development cost by $3.55 million from the initial $58.5 million to $54.95 million.
About This Article
This article can be found in the category: | {"url":"https://www.dummies.com/article/business-careers-money/business/economics/how-to-calculate-expected-cost-of-development-for-new-products-in-managerial-economics-166985/","timestamp":"2024-11-08T05:58:17Z","content_type":"text/html","content_length":"80643","record_id":"<urn:uuid:2f65bda8-b8ad-42df-af7d-2f83a5b06bc8>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00516.warc.gz"} |
76 cm to inches
Converting units of length can come in handy when working on math problems or everyday tasks. In this case, we will be converting 76 centimeters (cm) to inches.
Step 1: Write down the given value
The given value we have to convert is 76 cm.
Step 2: Know the conversion factor
The conversion factor for converting cm to inches is 1 inch = 2.54 cm.
Step 3: Set up the conversion equation
We will use the conversion factor to set up our equation like this:
76 cm x (1 inch / 2.54 cm)
Step 4: Cancel out the units
Cancel out the unit cm on the top and bottom of the fraction, leaving us with:
76 x (1 inch / 2.54)
Step 5: Solve for the converted value
Using a calculator, we can easily solve for the converted value by simply dividing 76 by 2.54, which gives us 29.92 inches.
Step 6: Round the answer if necessary
Depending on the level of accuracy needed, we can round the answer to the nearest hundredth, which gives us 29.93 inches.
Step 7: Write the final answer
The final answer is 76 cm is equivalent to 29.93 inches. This means that 76 centimeters is the same as 29.93 inches.
By following these steps, you can easily convert any given value from cm to inches. Remember to always double-check your work to ensure accuracy. Happy converting!
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Spectral Series of Hydrogen Atom - Physics Vidyalay
Spectral Series of Hydrogen Atom-
Before you go through this article, make sure that you have gone through the previous article on Bohr’s Atomic Model.
We have learnt that-
• Electrons revolve around the nucleus in fixed energy orbits called as stationary states.
• An electron does not absorb or radiate energy while moving in these stationary states.
In this article, we will discuss about spectral series of hydrogen atom.
Spectral Series of Hydrogen Atom-
From Bohr’s theory, the energy of an electron in n^th Bohr orbit of hydrogen atom is given by-
(For hydrogen atom, Z = 1)
According to Bohr’s frequency condition, whenever an electron makes a transition from a higher energy level n[2] to a lower energy level n[1], the difference of energy appears in the form of a
photon. The frequency ν of the emitted photon is given by-
Wavelength of Emitted Photon-
The wavelength of the emitted photon is given by-
This formula indicates that the radiation emitted by the excited hydrogen atom consists of certain specific wavelengths or frequencies, the value of which depend on quantum numbers n[1] and n[2].
Wave Number-
Wave number is defined as the reciprocal of wavelength λ. It is given by-
Spectral Series of Hydrogen Atom-
The origin of the various series in the hydrogen spectrum can be explained as follows-
1. Lyman Series-
• If an electron jumps from any higher energy level n[2] = 2, 3, 4, …… to a lower energy level n[1] = 1, we get a set of spectral lines called as Lyman series.
• It belongs to the ultraviolet region of the electromagnetic spectrum.
This series is given by-
2. Balmer Series-
• If an electron jumps from any higher energy level n[2] = 3, 4, 5, …… to a lower energy level n[1] = 2, we get a set of spectral lines called as Balmer series.
• It belongs to the visible region of the electromagnetic spectrum.
This series is given by-
3. Paschen Series-
• If an electron jumps from any higher energy level n[2] = 4, 5, 6, …… to a lower energy level n[1] = 3, we get a set of spectral lines called as Paschen series.
• It belongs to the infrared region of the electromagnetic spectrum.
This series is given by-
4. Brackett Series-
• If an electron jumps from any higher energy level n[2] = 5, 6, 7, …… to a lower energy level n[1] = 4, we get a set of spectral lines called as Brackett series.
• It belongs to the infrared region of the electromagnetic spectrum.
This series is given by-
5. Pfund Series-
• If an electron jumps from any higher energy level n[2] = 6, 7, 8, …… to a lower energy level n[1] = 5, we get a set of spectral lines called as Pfund series.
• It belongs to the infrared region of the electromagnetic spectrum.
This series is given by-
Shortest and Longest Wavelength of a Spectral Series-
We know, if an electron makes a transition from any higher energy level n[2] to any lower energy level n[1], then the wavelength of the photon emitted during transition is given by-
From here, we conclude that the wavelength of the emitted photon is smallest when the energy difference between the two levels is largest and vice-versa.
For smallest wavelength, consider the longest transition
For longest wavelength, consider the smallest transition
We can summarize the concept of longest and shortest wavelength of spectral series in the following table-
Series Longest wavelength occur during transition Shortest wavelength occur during transition
Lyman n[2] = 2 to n[1] = 1 n[2] = ∞ to n[1] = 1
Balmer n[2] = 3 to n[1] = 2 n[2] = ∞ to n[1] = 2
Paschen n[2] = 4 to n[1] = 3 n[2] = ∞ to n[1] = 3
Brackett n[2] = 5 to n[1] = 4 n[2] = ∞ to n[1] = 4
Pfund n[2] = 6 to n[1] = 5 n[2] = ∞ to n[1] = 5
Read the next article on-
Energy Level Diagram For Hydrogen Atom
Get more notes & other study material of the Chapter Atoms. | {"url":"https://physicsvidyalay.com/spectral-series-of-hydrogen-atom/","timestamp":"2024-11-08T07:59:58Z","content_type":"text/html","content_length":"147888","record_id":"<urn:uuid:4009e169-44e6-4546-9c7f-d162c38edab9>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00008.warc.gz"} |
University of Victoria
The University of Victoria PIMS site office is located in the Department of Mathematics and Statistics (Social Sciences and Mathematics building) at the University of Victoria (Map).
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Position Name Email Phone # Office
PIMS Site Director - University of Victoria Anthony Quas aquas@uvic.ca (250) 472-4271
Site Administrator - University of Victoria Kristina McKinnon pimsadmin@uvic.ca +1 (250) 472-4271 DTB-A425
Name Position Research Interests Supervisor Year
Felix Christian Clemen PIMS Postdoctoral Fellow, University of Victoria Combinatorics Natasha Morrison 2024
Tianxia (Tylar) Jia PIMS Postdoctoral Fellow, University of Victoria Applied Mathematics, PDE & Meteorology Slim Ibrahim 2024
Kesav Krishnan PIMS Postdoctoral Fellow, University of Victoria Probability Theory and Stochastic Processes Gourab Ray 2023
Kristýna Zemková PIMS Postdoctoral Fellow, University of Alberta/University of Victoria Linear and Multilinear Algebra Stefan Gille 2022
Kumar Roy PIMS Postdoctoral Fellow, University of Victoria Mathematical Physics Boualem Khouider 2022
Elizabeth Carlson PIMS Postdoctoral Fellow, University of Victoria Partial Differential Equations David Goluskin 2021
Natalie Behague PIMS Postdoctoral Fellow, University of Victoria Combinatorics Natasha Morrison 2021
Shangzhi Zeng Postdoctoral Research Fellow, University of Victoria Operations research, Mathematical Programming Jane J. Ye 2020
Jason Bramburger University of Victoria Dynamical systems and ergodic theory David Goluskin 2019
Boyi Li University of Victoria Operator Theory Marcelo Laca 2018
Hung Le University of Victoria Computer Science Valerie King 2018
Yakine Bahri University of Victoria Nonlinear PDEs Slim Ibrahim 2017
Diego Vela University of Victoria Topology Ryan Budney 2015
Elsa Maria Dos Santos Cardoso-Bihlo University of Victoria Fluid mechanics/Numerical Analysis Boualem Khouider 2015 | {"url":"https://www.pims.math.ca/sites/university-victoria","timestamp":"2024-11-05T22:25:23Z","content_type":"text/html","content_length":"473648","record_id":"<urn:uuid:5790f9cc-d595-4815-9171-5104ffcebade>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00108.warc.gz"} |
The area of a rectangle is 45 square cm. If the length is 4cm greater than the width, what are the dimension of the rectangle? | HIX Tutor
The area of a rectangle is 45 square cm. If the length is 4cm greater than the width, what are the dimension of the rectangle?
Answer 1
The length & width of rectangle are $9 \mathmr{and} 5$ cms respectively.
Let the width of the rectangle be #x# cm; then the length will be #x+4#cm. Area of the rectangle is #x*(x+4)=45 or x^2+4x-45=0 or (x+9)(x-5)=0 :. x= -9 or x=5#Width cannot be negative. So #x=5; x+4=
9:.#The length & width are #9 and 5 # cms respectively.[Ans]
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Answer 2
Let's denote the width of the rectangle as ( w ) cm. Since the length is 4 cm greater than the width, the length can be represented as ( w + 4 ) cm. The area of a rectangle is given by the formula:
[ \text{Area} = \text{length} \times \text{width} ]
Given that the area is 45 square cm, we can set up the equation:
[ w(w + 4) = 45 ]
Expanding the equation:
[ w^2 + 4w = 45 ]
Rearranging terms to form a quadratic equation:
[ w^2 + 4w - 45 = 0 ]
Now, we can factorize the quadratic equation or use the quadratic formula to solve for ( w ). After finding the value of ( w ), we can then find the length of the rectangle by adding 4 to ( w ).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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