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ManPag.es -
slarrr.f −
subroutine SLARRR (N, D, E, INFO)
SLARRR performs tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.
Function/Subroutine Documentation
subroutine SLARRR (integerN, real, dimension( * )D, real, dimension( * )E, integerINFO)
SLARRR performs tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.
Perform tests to decide whether the symmetric tridiagonal matrix T
warrants expensive computations which guarantee high relative accuracy
in the eigenvalues.
N is INTEGER
The order of the matrix. N > 0.
D is REAL array, dimension (N)
The N diagonal elements of the tridiagonal matrix T.
E is REAL array, dimension (N)
On entry, the first (N-1) entries contain the subdiagonal
elements of the tridiagonal matrix T; E(N) is set to ZERO.
INFO is INTEGER
INFO = 0(default) : the matrix warrants computations preserving
relative accuracy.
INFO = 1 : the matrix warrants computations guaranteeing
only absolute accuracy.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
September 2012
Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA
Definition at line 95 of file slarrr.f.
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Today's Quote From Steve Schmidt
"Live your life in the way...if by chance...it was someday turned into a book"
FLORIDA WOULD BAN IT !!!!!!
9 minutes ago, Rufus69 said:
"Live your life in the way...if by chance...it was someday turned into a book"
FLORIDA WOULD BAN IT !!!!!!
7 minutes ago, RedZone said:
Lmao @Rufus69 congrats on the Boards biggest racist agreeing with u, not a good look mi amigo
49 minutes ago, FreeBird said:
Lmao @Rufus69 congrats on the Boards biggest racist agreeing with u, not a good look mi amigo
But what about the homeless veterans?!?
4 minutes ago, GoBigBlack said:
But what about the homeless veterans?!?
Screw em let’s give all the resources to Ukrainine and Venezuelan citizens
6 minutes ago, FreeBird said:
Screw em let’s give all the resources to Ukrainine and Venezuelan citizens
Or the unicorns, they could use the space.
1 hour ago, Rufus69 said:
"Live your life in the way...if by chance...it was someday turned into a book"
FLORIDA WOULD BAN IT !!!!!!
1 hour ago, RedZone said:
If instances in which minors were sexually exploited for the sexual satisfaction from pedophiles, then yes it would be banned. If not then you got nothing to worry about
1 hour ago, Rufus69 said:
"Live your life in the way...if by chance...it was someday turned into a book"
FLORIDA WOULD BAN IT !!!!!!
This does seem like a dog whistle to pedophiles. We can’t stop calling it out when we see it. I wouldn’t use criticizing laws meant to prevent child sexual exploitation as jokes if I were you. Not a
good look.
2 hours ago, FreeBird said:
Lmao @Rufus69 congrats on the Boards biggest racist agreeing with u, not a good look mi amigo
Hoss...I don't have a problem with somebody being racist. It don't affect me nary a bit. I'm too old to worry about that shit.
31 minutes ago, Rufus69 said:
Hoss...I don't have a problem with somebody being racist. It don't affect me nary a bit. I'm too old to worry about that shit.
Except if the guys first name is Don
am I right or am I right 😏
58 minutes ago, FreeBird said:
Except if the guys first name is Don
am I right or am I right 😏
Trumpy The Ass Clown has a LOT more problems than just being racist.
8 hours ago, Nolebull813 said:
This does seem like a dog whistle to pedophiles. We can’t stop calling it out when we see it. I wouldn’t use criticizing laws meant to prevent child sexual exploitation as jokes if I were you.
Not a good look.
Dude, you actually live in a communist state.
2 hours ago, Wildcat Will said:
Dude, you actually live in a communist state.
Sorry trying to prevent minors from being sexually exploited makes you so upset.
5 hours ago, Nolebull813 said:
Sorry trying to prevent minors from being sexually exploited makes you so upset.
Me? Upset? Nah. Pick up a newspaper or something. Go to the store and pick up the scuttlebutt.
By the way, The states largest employer is eventually going to leave the state.
All those jobs
All that tourism
All that revenue
18 hours ago, FreeBird said:
Except if the guys first name is Don
am I right or am I right 😏
this is the first time you've ever been right!!.....Trump = racist....BINGO Champ!!.....😉
9 hours ago, Wildcat Will said:
Me? Upset? Nah. Pick up a newspaper or something. Go to the store and pick up the scuttlebutt.
By the way, The states largest employer is eventually going to leave the state.
All those jobs
All that tourism
All that revenue
Please leave. People can actually go to Orlando. Disney is a thorn in the ass of the locals. Only foreigners and creepy adults go to Disney
2 minutes ago, Nolebull813 said:
Please leave. People can actually go to Orlando. Disney is a thorn in the ass of the locals. Only foreigners and creepy adults go to Disney
Nah, it is a child magnet. You have visited quite often so should you be grouped with the creepy adults.
You do not want Disney to leave. They mean to much to central and north Florida.
The economy would suffer greatly if Disney packed it up.
19 minutes ago, concha said:
Two big assumptions:
1. DIsney leaves
2. They shut down the Disney Florida parks
Where do they go?
You need to read up on Disney. DeSantis is not their only issue at this time. The fact that they are having issues with him exacerbates things.
Business is down. Probably partially due to the fight with DeSantis, which is dumb on DeSantis' part.
4 minutes ago, Wildcat Will said:
You need to read up on Disney. DeSantis is not their only issue at this time. The fact that they are having issues with him exacerbates things.
Business is down. Probably partially due to the fight with DeSantis, which is dumb on DeSantis' part.
No. I really don't.
You just can't answer the question.
Disney has been fucking up for awhile now. Four million lost Disney+ subscriptions, lower ad revs, hundreds of millions lost due to forcing woke bullshit into kids/family movies, the failed Star Wars
DIsney can't afford to close the park.
They can move some woke-fuck execs to Cali. Fine. Let them see the hit from California cost-of-living. The housing costs, the taxes... the hell holes the downtowns have become...
1 hour ago, concha said:
Two big assumptions:
1. DIsney leaves
2. They shut down the Disney Florida parks
Where do they go?
Key West
56 minutes ago, concha said:
No. I really don't.
You just can't answer the question.
Disney has been fucking up for awhile now. Four million lost Disney+ subscriptions, lower ad revs, hundreds of millions lost due to forcing woke bullshit into kids/family movies, the failed Star
Wars hotel...
DIsney can't afford to close the park.
They can move some woke-fuck execs to Cali. Fine. Let them see the hit from California cost-of-living. The housing costs, the taxes... the hell holes the downtowns have become...
Damn, didn't I just say this?
DeSantis is no match for Isner. Disney wins.
12 hours ago, concha said:
Two big assumptions:
1. DIsney leaves
2. They shut down the Disney Florida parks
Where do they go?
They need to buy enough land somewhere to hold the twenty-two Disney World Hotels that the helicopters are going to fly in. 😉
Most of the people who work at are probably not qualified enough to work at a better job than Disney so if Disney moved, then those employees would have a pretty simple time finding new employment
because whatever job they did look for wouldn’t be hard to get. I’m not dissing them. I always respect people who work and pay taxes. I would never mistreat or make fun of anyone doing any type of
job. But from a qualification standpoint, Disney is in the ballpark of teenage jobs making minimum wage. Grocery stores and fast food levels. So it wouldn’t be hard to find work. | {"url":"https://www.prepgridiron.com/topic/37462-todays-quote-from-steve-schmidt/","timestamp":"2024-11-05T02:42:37Z","content_type":"text/html","content_length":"399582","record_id":"<urn:uuid:8f65e4b1-a40a-4ec4-91aa-73a99b32796f>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00607.warc.gz"} |
sorm2r: overwrites the general real m by n matrix C with Q * C if SIDE = aqLaq and TRANS = aqNaq, or Qaq* C if SIDE = aqLaq and TRANS = aqTaq, or C * Q if SIDE = aqRaq and TRANS = aqNaq, or C * Qaq if SIDE = aqRaq and TRANS = aqTaq, - Linux Manuals (l)
sorm2r (l) - Linux Manuals
sorm2r: overwrites the general real m by n matrix C with Q * C if SIDE = aqLaq and TRANS = aqNaq, or Qaq* C if SIDE = aqLaq and TRANS = aqTaq, or C * Q if SIDE = aqRaq and TRANS = aqNaq, or C * Qaq
if SIDE = aqRaq and TRANS = aqTaq,
SORM2R - overwrites the general real m by n matrix C with Q * C if SIDE = aqLaq and TRANS = aqNaq, or Qaq* C if SIDE = aqLaq and TRANS = aqTaq, or C * Q if SIDE = aqRaq and TRANS = aqNaq, or C * Qaq
if SIDE = aqRaq and TRANS = aqTaq,
SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO )
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDA, LDC, M, N
REAL A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
SORM2R overwrites the general real m by n matrix C with where Q is a real orthogonal matrix defined as the product of k elementary reflectors
= H(1) H(2) . . . H(k)
as returned by SGEQRF. Q is of order m if SIDE = aqLaq and of order n if SIDE = aqRaq.
SIDE (input) CHARACTER*1
= aqLaq: apply Q or Qaq from the Left
= aqRaq: apply Q or Qaq from the Right
TRANS (input) CHARACTER*1
= aqNaq: apply Q (No transpose)
= aqTaq: apply Qaq (Transpose)
M (input) INTEGER
The number of rows of the matrix C. M >= 0.
N (input) INTEGER
The number of columns of the matrix C. N >= 0.
K (input) INTEGER
The number of elementary reflectors whose product defines the matrix Q. If SIDE = aqLaq, M >= K >= 0; if SIDE = aqRaq, N >= K >= 0.
A (input) REAL array, dimension (LDA,K)
The i-th column must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by SGEQRF in the first k columns of its array argument A. A is modified by the
routine but restored on exit.
LDA (input) INTEGER
The leading dimension of the array A. If SIDE = aqLaq, LDA >= max(1,M); if SIDE = aqRaq, LDA >= max(1,N).
TAU (input) REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by SGEQRF.
C (input/output) REAL array, dimension (LDC,N)
On entry, the m by n matrix C. On exit, C is overwritten by Q*C or Qaq*C or C*Qaq or C*Q.
LDC (input) INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK (workspace) REAL array, dimension
(N) if SIDE = aqLaq, (M) if SIDE = aqRaq
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value | {"url":"https://www.systutorials.com/docs/linux/man/l-sorm2r/","timestamp":"2024-11-12T06:24:22Z","content_type":"text/html","content_length":"11140","record_id":"<urn:uuid:58cd3584-2187-4bb7-be99-39f1f7689122>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00192.warc.gz"} |
Shaders, Raymarching
Here is the minimal picture i have of how a graphics card works.
There are two main stages, the vertex shader and the fragment shader. The vertex shader is given a list of triangles basically. It computes and applies the transformations necessary to rotate and
translate the camera position for the vertices and compute normal vectors and some other geometrical things.
The fragments shader is then passed info from the vertex shader. It has to output a color by setting a variable fragColor. There are variables called that are given the type annotation varyings that
are automatically interpolated in the vertex (think of smoothly rotating vectors or colors between vertices). The fragment shader is called once per pixel on the screen.
Mostly information is passed around via pointer rather than function return (I guess that is kind of a common C paradigm and it does make sense.). What does suck about that is you’ll see variables
appear out of nowhere. They are basically global variables from your codes perspective. I assume there is a limited number of them so you get used to it.
shadertoy.com is awesome. It draws a big rectangle and gives you easy access to the fragment shader with some useful extra variables available to you.
It feels like a big map in the functional sense. You write a shader function and then the gpu runs
pixelcolor = map shader pixelinfo
This is refreshing. The api I’m used to is an imperative api where you consecutively mutate some screen object by called line or point or circle on it. Not that that’s bad, necessarily. But I do like
the newness.
This will draw a white circle.
void mainImage( out vec4 fragColor, in vec2 fragCoord )
vec2 coord = fragCoord.xy - iResolution.xy/2.0;
if(length(coord) < 100.0){
fragColor = vec4(1);
fragColor = vec4(0);
Also here is a raymarched sphere with a little lighting. Ray marching uses the distance function to push the ray smart distances. There are a ton of things you could do here. Could optimize the loop
to break when rays get close enough, or when they fly off to infinity.
float sphere( in vec3 p){
return length(p) - 0.5;
void mainImage( out vec4 fragColor, in vec2 fragCoord )
vec2 uv = fragCoord.xy / iResolution.y;
vec2 st = 2.0*uv -1.0;
fragColor = vec4(uv,0.5+0.5*sin(iGlobalTime),1.0);
vec3 cam = vec3(0.0, 0.0, -1.0);
vec3 ray = vec3(st.x, st.y, 1);
ray = normalize(ray);
float depth = 0.0;
for(int i=0; i<64; i++){
vec3 p = cam + depth * ray;
depth = depth += sphere(p);
vec3 p = cam + depth * ray;
if(abs(sphere(p)) < 0.1){
fragColor = vec4(0.3+dot(normalize(p), normalize(vec3(1.0,1.0,0.0)))); | {"url":"https://www.philipzucker.com/shaders-raymarching/","timestamp":"2024-11-08T20:57:06Z","content_type":"text/html","content_length":"9588","record_id":"<urn:uuid:b9455e54-9080-42ca-9f66-edddafbe32bb>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00524.warc.gz"} |
the operating speed of a ball mill should be critical speed
The ball mill speed is closely related to the ball loading rate and there are two different scenarios: working below the critical speed and operating at supercritical speed. So far, the majority
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Speed of rotation of mill At low speeds, the balls simply roll over one another and little grinding is obtained while at very high speeds, the balls are simply carried along the walls of the
shells and little or no grinding takes place, so for an effective grinding, the ball mill should be operated at a speed that is optimum speed equal to ...
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22 May, 2019. The ball mill consists of a metal cylinder and a ball. The working principle is that when the cylinder is rotated, the grinding body (ball) and the object to be polished (material)
installed in the cylinder are rotated by the cylinder under the action of friction and centrifugal force. At a certain height, it will automatically ...
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BALL MILL Objective: To determine the (a) Critical speed (b) Actual speed (c) Optimum speed (d) Reduction ratio (e) Constants for i. Rittinger's Law ii. Kick's Law iii. ... No. of balls N= 20
Observations: 1. Feed size D:= mm 2. Feed mass = 708 gm = ton 3. Power consumption under operating condition = * 106 KW 4. Product size D:o ...
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The critical speed n (rpm) when the balls remain attached to the wall with the aid of centrifugal force is: n = D m. ... The company clams this new ball mill will be helpful to enable extreme
highenergy ball milling at rotational speed reaches to 1100 rpm. This allows the new mill to achieve sensational centrifugal accelerations up to 95 ...
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In fact, when the rotation speed was close to the critical speed of the mill, the balls, due to centrifugal force, adhered to the mill wall and led to a decrease in impulses. In this case, the
defect was visible in the spectrogram at a mean severity level,, 1530% of the degradation rate.
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By using following relation you can find out the critical speed of ball mill. Nc = 1/2π.√g/Rr The operating speed/optimum speed of the ball mill is between 50 and 75 percent of the critical
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When a ball mill having a proper crushing load is rotated at the critical speed, the balls strike at a point on the periphery about 45° below horizontal, or S in Fig. 1. An experienced operator
is able to judge by the sound whether a mill is crushing at maximum efficiency, or is being over or underfed.
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A Slice Mill is the same diameter as the production mill but shorter in length. Request Price Quote. Click to request a ball mill quote online or call to speak with an expert at Paul O. Abbe® to
help you determine which design and size ball mill would be best for your process. See our Size Reduction Options.
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The video contain definition, concept of Critical speed of ball mill and step wise derivation of mathematical expression for determining critical speed of b...
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dimensionality variation and mill operating variables on its occurrence. It is shown that varying the operating conditions, specifically the load fraction critical speed, can reduce the risk of
overload for existing operations; while appro priate decreases in LID ratio can minimize risk design new circuits. Ball mill overload
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The power, which is given by the product of this torque curve with the mill speed, has a much rounder peak centered on N=100% before dropping sharply for higher rotation rates. For a mill
operating at a typical 80% of critical, the power predicted by the model is 3×10 5 W/m of mill length. For a typical 7mlong mill, the power required is ...
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3. Results and discussion. Fig. 1 shows as an example the pressuretime record corresponding to a milling experiment performed using the following experimental conditions: ω d = 250 rpm, k = and
BPR = 24, from which a t ig value of 110 min and 22 s was determined. As can be seen, the pressure spike at ignition is very intense and the t ig value can be determined accurately.
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Recommended Mill Operating Speed RPM. Here is a table of typically recommended ball mill speed or rod mill speed as a % of critical will operate at. In summary, the larger the mill, the slower
you will want the RPM to be set at and the mill to turn.
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Problem Calculate the operating speed of a ball mill from the following data: (i) Diameter of ball mill=500 mm (ii) Diameter of ball= 50mm Operating speed of ball mill is 35 % of critical speed.
Problem Calculate the power required in horsepower to crush 150000 kg of feed, if 80 % of feed passes through 2 inch screen and 80% of product ...
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Critical speed formula of ball mill. Nc = 1/2π √g/R r The operating speed/optimum speed of the ball mill is between 50 and 75% of the critical speed. Also Read: Hammer Mill Construction and
Wroking Principal. Take these Notes is, Orginal Sources: Unit OperationsII, KA Gavhane
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Mill speed and air classifier speed were the investigated parameters for the closed cycle mill. Almost six speed level are used in the closed cycle mill are 750, 800, 830,850, 900, 950 rpm.
Blaine is the important characteristic of ball mill which is influenced by the mill speed and separator speed. Ball mill is grinding equipment which is used ...
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Critical speed is defined as the point at which the centrifugal force applied to the grinding mill charge is equal to the force of gravity. At critical speed, the grinding mill charge clings to
the mill inner surface and does not tumble. Most ball mills operate at approximately 75% critical speed, as this is determined to be the optimum speed ...
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Afton claims that the variable mill speed has been useful for preventing liner wear while processing soft ore (Pazour, 1978). Sydvaranger processes iron ore through a 21 by 33foot variable speed
ball mill. The speed range of this mill is from 62 to 82 percent of critical, with normal operating speed at 78 percent.
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In this research, in order to find a suitable range for the number of lifters in the liner of ball mills, the DEM method is utilized. Initially, a pilotscale ball mill with dimensions of m × ...
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Mill critical speed is defined as that rotational speed at which an infinitely small particle will centrifuge, assuming no slippage between the particle and the shell. The critical speed, Nc, in
revolutions per minute, is a function of the mill diameter, D, expressed as: Nc = /√D (meters) or Nc = /√D (feet)
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determine the critical speed for any given size Mill, we use the following formula: divided by the square root of the radius in feet. The smaller the Mill the faster in RPM it must run to attain
critical speed. Our " diameter Specimen Jar has a critical speed of 125 RPM, and our 90" diameter Ball Mill 28 RPM.
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According to available literature, the operating speeds of AG mills are much higher than conventional tumbling mills and are in the range of 8085% of the critical speed. SAG mills of comparable
size but containing say 10% ball charge (in addition to the rocks), normally, operate between 70 and 75% of the critical speed.
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Power draw is related directly to mill length, and, empirically to the diameter to the power Theoretically, this exponent should be (Bond, 1961). Power draw is directly related to mill speed (in,
or, fraction of critical speed) over the normal operating range.
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There is a specific operating speed for most efficient ... and our size #00 90" diameter Ball Mill 28 RPM. ... the Mills are run almost a critical speed so that the balls are drooped to give the
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At normal operating speed (70% to 80% critical speed), about 80% load is en masse. Balls in the en masse region are neither close packed nor in discrete layers. Slippage between adjacent layers
may occur. ... distance from ball center to mill center, i, layer of balls. r b. radius of ball, m. R d. radius of Davis circle, m. R m.
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Find out the critical speed of ball mill by using the following data: Diameter of the ball mill is 450mm and Diameter of the ball is 25 mm. 2. Calculate the operating speed of the ball mill from
the given data below Diameter of the ball mill is 800 mm and Diameter of the ball is 60 mm. if critical speed is 40% less than the operating speed.
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Φ c = fraction of critical speed. The Rowland and Kjos equation indicates that power draw is a function of the fraction of the critical speed for the mill. Marcy mills are recommended to operate
a peripheral speeds governed by the following relationship : Peripheral Speed= D meters/ min. D = mill diameter in meters
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Speed rate refers to the ratio of the speed of the mill to the critical speed, where the critical speed is n c = 30 / R. In practice, the speed rate of the SAG mill is generally 65% to 80%.
Therefore, in the experiment, the speed was set to vary in 50% and 90% of the critical speed ( rad/s) for the crossover test as shown in Table 2.
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Thus we have found the value of critical speed of rotation of ball mill ( In revolution per second). STEP2:Using the above relation we can now solve our given question as follows; ... We know
that the operating speed of the ball mill is 50 to 75 % of the critical speed.
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More bad news for renewables and hydrogen
One embarrassing fact for the many tireless advocates for renewables is that, despite claims that electricity from such sources is cheaper than conventional power, it seems that the more renewables
there are on a grid around the world the more expensive the electricity. There are exceptions to this, notably those grids that use a lot of hydropower which counts as a renewable but has none of the
disadvantages of intermittent wind and solar. There are also a number of US states that buck the trend unless analysts take a closer look. This article in the Manhattan Contrarian looks at the
claims of Mark Z Jacobson, a professor at Stanford University who points to various states including South Dakota, Montana, Iowa and four others as having high levels of renewables but cheaper power
prices. These critics point out in the article and the comments at the end that they cannot reconcile Professor Jacobson's figures with reality and that the cheaper prices may well be because the
states are selling the renewable power to other states looking to meet their targets.
As for H2 this academic paper A review of the challenges with using the natural gas system for hydrogen points to the huge problems of trying to convert natural gas systems to H2. In effect, when you
get beyond a mixture of 10-20 per cent by volume (far less by energy contribution) then the enginers have to totally redesign and rebuild the pipeline and replace all the gas fittings. It has some
interesting comments on storage, transportation and as safety. None of it is really new but it is a handy summary of the huge problems involved in this one aspect of using H2 in energy systems.
notsonice + 1,255
3 hours ago, markslawson said:
One embarrassing fact for the many tireless advocates for renewables is that, despite claims that electricity from such sources is cheaper than conventional power, it seems that the more
renewables there are on a grid around the world the more expensive the electricity. There are exceptions to this, notably those grids that use a lot of hydropower which counts as a renewable but
has none of the disadvantages of intermittent wind and solar. There are also a number of US states that buck the trend unless analysts take a closer look. This article in the Manhattan
Contrarian looks at the claims of Mark Z Jacobson, a professor at Stanford University who points to various states including South Dakota, Montana, Iowa and four others as having high levels of
renewables but cheaper power prices. These critics point out in the article and the comments at the end that they cannot reconcile Professor Jacobson's figures with reality and that the cheaper
prices may well be because the states are selling the renewable power to other states looking to meet their targets.
As for H2 this academic paper A review of the challenges with using the natural gas system for hydrogen points to the huge problems of trying to convert natural gas systems to H2. In effect, when
you get beyond a mixture of 10-20 per cent by volume (far less by energy contribution) then the enginers have to totally redesign and rebuild the pipeline and replace all the gas fittings. It has
some interesting comments on storage, transportation and as safety. None of it is really new but it is a handy summary of the huge problems involved in this one aspect of using H2 in energy
when you get beyond a mixture of 10-20 per cent by volume (far less by energy contribution) then the enginers have to totally redesign and rebuild the pipeline and replace all the gas fittings????
no one is proposing putting hydrogen into existing nat gas pipelines above 20 percent .......
The main uses of Green Hydrogen will be used in industrial processes IE to make ammonia to then make fertilizer, in steelmaking and chemical processes ( to replace the production of grey hydrogen)
and a 20 percent replacement of methane
According to available data, the global hydrogen production today is approximately 75 million metric tons of pure hydrogen per year, with an additional 45 million metric tons produced as part of a
gas mixture, totaling around 120 million metric tons annually
and finally production of green hydrogen from renewables at the sites of existing gas fired power plants and storing enough hydrogen (like a days worth) to run the plants during peak demands....
it seems that the more renewables there are on a grid around the world the more expensive the electricity????????? thanks for trying to BS us again......reality today is Solar and Wind power does not
cause electricity to become more expensive
states including South Dakota, Montana, Iowa and four others as having high levels of renewables but cheaper power prices......??????? Mark my bet is you have never lived in or been in any of these
states , because if you did you would know they have the cheapest power prices in the US
1 hour ago, notsonice said:
when you get beyond a mixture of 10-20 per cent by volume (far less by energy contribution) then the enginers have to totally redesign and rebuild the pipeline and replace all the gas
no one is proposing putting hydrogen into existing nat gas pipelines above 20 percent .......
The main uses of Green Hydrogen will be used in industrial processes IE to make ammonia to then make fertilizer, in steelmaking and chemical processes ( to replace the production of grey
hydrogen) and a 20 percent replacement of methane
According to available data, the global hydrogen production today is approximately 75 million metric tons of pure hydrogen per year, with an additional 45 million metric tons produced as part of
a gas mixture, totaling around 120 million metric tons annually
Actually, they are proposing to put H2 in high quantities into town gas systems, or at least saying we have to do it to get to net zero - read the paper. You probably haven't heard about it because
no one has been dumb enough to do it. The paper's author was pointing to all the impossibilities, which is where we should leave that part of the discussion. I see you tried repeating the nonsense
about renewables being cheaper. Read the material linked. You'll see there are big questions about the few places where renewable advocates can point to a high use of renewables and cheaper power
prices, or at least prices that are not outrageously high. Renewables are certainly cheaper on a straight cost comparison but not if you put them into a grid. Then they become very, very expensive -
not so much the power they generate but all the extra stuff that has to be done to run the grid around them.
notsonice + 1,255
10 hours ago, markslawson said:
Actually, they are proposing to put H2 in high quantities into town gas systems, or at least saying we have to do it to get to net zero - read the paper. You probably haven't heard about it
because no one has been dumb enough to do it. The paper's author was pointing to all the impossibilities, which is where we should leave that part of the discussion. I see you tried repeating the
nonsense about renewables being cheaper. Read the material linked. You'll see there are big questions about the few places where renewable advocates can point to a high use of renewables and
cheaper power prices, or at least prices that are not outrageously high. Renewables are certainly cheaper on a straight cost comparison but not if you put them into a grid. Then they become very,
very expensive - not so much the power they generate but all the extra stuff that has to be done to run the grid around them.
they are proposing to put H2 in high quantities into town gas systems????
who is they???? Mark, no is proposing this....Your posts are that of a Drama Queen in a quest to stop something that is not even being proposed. You post no proposals (try posting real links to real
proposals , not requotes from other Drama Queens that also do not link to anything real)
You constantly post made up scenarios with nothing to back yourself up
such as
Renewables are certainly cheaper on a straight cost comparison but not if you put them into a grid. Then they become very, very expensive - not so much the power they generate but all the extra stuff
that has to be done to run the grid around them.
My power is now over 20 percent renewables and my power bill is the same it was over 10 years ago ( when inflation is factored in it is 30 percent less)
I now have a night time rate that also is lower than any rate I had available for the past 10 years.
Maybe if you would post some real numbers and facts, someone might take you seriously. As you keep posting the same garbage without any facts or real links to real information you lose all
4 hours ago, notsonice said:
Renewables are certainly cheaper on a straight cost comparison but not if you put them into a grid. Then they become very, very expensive - not so much the power they generate but all the extra
stuff that has to be done to run the grid around them.
My power is now over 20 percent renewables and my power bill is the same it was over 10 years ago ( when inflation is factored in it is 30 percent less)
I now have a night time rate that also is lower than any rate I had available for the past 10 years.
This is completely meaningless. Your power bill is the result of many factors. None the less as the discussion in the paper linked in the original post makes clear there is a distinct correlation
increased use of renewables and higher power prices (Germany and the UK as opposed to France and Poland). Quoting individual examples without the details does not help, particularly if the renewables
you have are hydro. I'm sure that once you calm down - your post has indications of hysteria - you'll see what I mean. As for the H2 thing there were posts on an earlier version of this forum but, as
I said, no-one was mad enough to actually do it and the paper makes it plain that using H2 in town gas is all but impossible. Anyway, I urge you to look at these issues calmly and leave you with it.
If its any consolation, Robert Plant may have a spot open in his university of bad ideas for you to lecture..
9 hours ago, markslawson said:
This is completely meaningless. Your power bill is the result of many factors. None the less as the discussion in the paper linked in the original post makes clear there is a distinct correlation
increased use of renewables and higher power prices (Germany and the UK as opposed to France and Poland). Quoting individual examples without the details does not help, particularly if the
renewables you have are hydro. I'm sure that once you calm down - your post has indications of hysteria - you'll see what I mean. As for the H2 thing there were posts on an earlier version of
this forum but, as I said, no-one was mad enough to actually do it and the paper makes it plain that using H2 in town gas is all but impossible. Anyway, I urge you to look at these issues calmly
and leave you with it. If its any consolation, Robert Plant may have a spot open in his university of bad ideas for you to lecture..
Again you spout a lot of what you believe not what is reailty, again no links, no facts!
You are really making yourself look ignorant by continually doing so, you also have zero come back to the numerous links others post refuting your "beliefs".
Higher powergen costs in the European countries you cite were a direct cause of NG gas price hikes due to the war in Ukraine, any fool can see this apart from you it seems.
Here let me help you with the UK's over the last 12 years, notice a sudden spike???
As you can see cost is rapidly falling back to levels seen before the Ukraine war.
Oh and if you think we have lots of blackouts then think again, the only ones we ever get are due to storm damage.
You really need to try much much harder and try posting some real facts!
You do your own agenda no favours!
Edited by Rob Plant
Mark you talk of pipelines and the lack of them to transport H2
Have you even bothered to research what is going on before spouting your BS?
Let me help you again!
Which countries are building hydrogen pipelines fastest? | World Economic Forum (weforum.org)
Try reading this and learn something for a change
20 hours ago, Rob Plant said:
Again you spout a lot of what you believe not what is reailty, again no links, no facts!
You are really making yourself look ignorant by continually doing so, you also have zero come back to the numerous links others post refuting your "beliefs".
Higher powergen costs in the European countries you cite were a direct cause of NG gas price hikes due to the war in Ukraine, any fool can see this apart from you it seems.
Here let me help you with the UK's over the last 12 years, notice a sudden spike???
The trend I'm talking about was evident long before the war in Ukraine. I don't know where you got the idea that I was talking about the NG prices due to the war. Renewables mean higher prices, and
I'm talking about higher prices relative to other countries (ie France, Poland, Serbia, different states in the US and so on). Your graph is not sourced - also dunno if it's wholesale or retail - but
in any case its of no use in making your case, as even a moment's thought would have shown. You really shouldn't abuse someone like that unless you're fairly sure of your own case but, as I've shown,
you don't understand the point I was making in the first place. Leavfe it with you.
20 hours ago, Rob Plant said:
Mark you talk of pipelines and the lack of them to transport H2
Have you even bothered to research what is going on before spouting your BS?
Let me help you again!
Which countries are building hydrogen pipelines fastest? | World Economic Forum (weforum.org)
Try reading this and learn something for a change
Rob - I'm being patient here. Okay, pipelines. Now go and look at the detail of the stuff you cite. It's obviously for industrial uses, not power. I admit that the pipelines go for some distance but
it really makes no difference. H2 has long been used in industrial applications, mostly consumed at the same place that it's created. Now its transported a ways then used. Now as you're being
constantly abusive I'll leave it with you but remember what I said... as far as power applications are concerned H2 is a waste of time.
TailingsPond + 874
1 hour ago, markslawson said:
as far as power applications are concerned H2 is a waste of time.
I mostly agree with that. However, adding hydrogen to natural gas distribution systems at even a low percentage can act as a powerful bridging technology. With such a vast pipeline network it takes
a lot of hydrogen to replace even a small percentage of nat gas (if it were evenly mixed which I admit is not true). The hydrogen, in effect, acts as a chemical battery which can be turned back to
electricity at a typical nat gas power plant.
It's not perfect but it may have a place.
• 2
California Launches America’s First Hydrogen-Powered Passenger Train
California Launches America’s First Hydrogen-Powered Passenger Train | OilPrice.com
On 9/7/2024 at 6:25 AM, markslawson said:
The trend I'm talking about was evident long before the war in Ukraine. I don't know where you got the idea that I was talking about the NG prices due to the war. Renewables mean higher prices,
and I'm talking about higher prices relative to other countries (ie France, Poland, Serbia, different states in the US and so on). Your graph is not sourced - also dunno if it's wholesale or
retail - but in any case its of no use in making your case, as even a moment's thought would have shown. You really shouldn't abuse someone like that unless you're fairly sure of your own case
but, as I've shown, you don't understand the point I was making in the first place. Leavfe it with you.
How was I abusing you????
I stated you keep making yourself look ignorant by not backing up your beliefs with links or data, and once again you post your own beliefs with zero links or data to back it up.
Anyway you now are specifically referring to power generation using H2 not being viable and not taken on by large multi-nationals, so I'll just add these links for you to peruse at your leisure.
Energy Transition Actions (siemens-energy.com)
#2 Transforming conventional power
Amid all the massive investments that need to be made, we cannot and should not overlook the infrastructure that already exists. This can and should be used as a bridge to carry out the transition –
even if it is based on conventional technologies. By shifting from coal to gas, we can reduce emissions faster. And by using gas turbines which can be co-fired with hydrogen, we are preparing the way
for even lower CO[2] intensity.
ACWA POWER | NEOM Green Hydrogen Project
Hydrogen Power Plants (siemens-energy.com)
Whilst I understand the laws of physics preclude H2 as a major powergen fuel source it doesnt mean it doesnt have a purpose or indeed a massive part to play in the transition away from FF!!!
Anyway keep it up it's entertaining!
Do your books have zero stuff in them to back up what you say too? | {"url":"https://community.oilprice.com/topic/40579-more-bad-news-for-renewables-and-hydrogen/?tab=comments","timestamp":"2024-11-14T10:08:25Z","content_type":"text/html","content_length":"360627","record_id":"<urn:uuid:ca552dda-5546-415e-9069-6c4aac615fb5>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00499.warc.gz"} |
Gauth: A Student’s Guide to Solving Polynomials with Descartes’ Rule
Polynomial is an important topic in algebra. Among these topics is Descartes’ Rule of Signs. This theorem provides a way to estimate the number of positive and negative real roots that a polynomial
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Understand Descartes’ Rule
Descartes’ Rule of Signs is an algebraic theorem that helps students find the number of positive and negative real roots of a polynomial by signifying the coefficients of the polynomial.
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Gauth is not a homework solver but a learning tool that helps students solve any algebraic problem on their own. When it comes to understanding and applying Descartes’ Rule of Signs, Gauth is a good
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polynomials using Descartes’ Rule. | {"url":"https://detbeerweek.com/gauth-a-students-guide-to-solving-polynomials-with-descartes-rule/","timestamp":"2024-11-11T18:01:59Z","content_type":"text/html","content_length":"20108","record_id":"<urn:uuid:226709ce-31dc-4f5f-be1a-fcb0c26c7e35>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00453.warc.gz"} |
Conditions | Tidal Cycles
This page will present you all the functions that can be used to add conditions to your patterns. Each function will be presented following the same model:
• Type signature: how the function is declared on the Haskell side.
• Description: verbal description of the function.
• Examples: a small list of examples that you can copy/paste in your editor.
Conditions on cycle numbers
Type: every :: Pattern Int -> (Pattern a -> Pattern a) -> Pattern a -> Pattern a
every is function that allows you to apply another function conditionally. It takes three inputs: how often the function should be applied (e.g. 3 to apply it every 3 cycles), the function to be
applied, and the pattern you are applying it to. For example: to reverse a pattern every three cycles (and for the other two play it normally)
d1 $ every 3 rev $ n "0 1 [~ 2] 3" # sound "arpy"
Note that if the function you're applying itself requires additional parameters (such as fast 2 to make a pattern twice as fast), then you'll need to wrap it in parenthesis, like so:
d1 $ every 3 (fast 2) $ n "0 1 [~ 2] 3" # sound "arpy"
Otherwise, the every function will think it is being passed too many parameters.
Type: every' :: Int -> Int -> (Pattern a -> Pattern a) -> Pattern a -> Pattern a
every' is a generalisation of every, taking one additional argument. The additional argument allows you to offset the function you are applying.
For example, every' 3 0 (fast 2) will speed up the cycle on cycles 0,3,6,… whereas every' 3 1 (fast 2) will transform the pattern on cycles 1,4,7,…
With this in mind, setting the second argument of every' to 0 gives the equivalent every function. For example, every 3 is equivalent to every' 3 0.
The every functions can be used to silence a full cycle or part of a cycle by using silent or mask "~". Mask provides additional flexibility to turn on/off individual steps.
d1 $ every 3 silent $ n "2 9 11 2" # s "hh27"
d1 $ every 3 (mask "~") $ n "2 9 10 2" # s "hh27"
d1 $ every 3 (mask "0 0 0 0") $ n "2 9 11 2" # s "hh27"
Type: foldEvery :: [Int] -> (Pattern a -> Pattern a) -> Pattern a -> Pattern a
foldEvery is similar to chaining multiple every functions together. It transforms a pattern with a function, once per any of the given number of cycles. If a particular cycle is the start of more
than one of the given cycle periods, then it it applied more than once.
d1 $ foldEvery [5,3] (|+ n 1) $ s "moog" # legato 1
The first moog samples are tuned to C2, C3 and C4. Note how on cycles multiple of 3 or 5 the pitch is an octave higher, and on multiples of 15 the pitch is two octaves higher, as the transformation
is applied twice.
Type: when :: (Int -> Bool) -> (Pattern a -> Pattern a) -> Pattern a -> Pattern a
Only when the given test function returns True the given pattern transformation is applied. The test function will be called with the current cycle as a number.
d1 $ when ((elem '4').show) (striate 4) $ sound "hh hc"
The above will only apply striate 4 to the pattern if the current cycle number contains the number 4. So the fourth cycle will be striated and the fourteenth and so on. Expect lots of striates after
cycle number 399.
Type: whenT :: (Time -> Bool) -> (Pattern a -> Pattern a) -> Pattern a -> Pattern a
Only when the given test function returns True the given pattern transformation is applied. It differs from when, being passed a continuous Time value instead of the cycle number. Basically, a
Rational version of when.
d1 $ whenT ((< 0.5).(flip Data.Fixed.mod' 2)) (# speed 2) $ sound "hh(4,8) hc(3,8)"
The above will apply # speed 2 only when the remainder of the current Time divided by 2 is less than 0.5.
Type: whenmod :: Int -> Int -> (Pattern a -> Pattern a) -> Pattern a -> Pattern a
whenmod has a similar form and behavior to every, but requires an additional number. It applies the function to the pattern, when the remainder of the current loop number divided by the first
parameter, is greater or equal than the second parameter. For example the following makes every other block of four loops twice as fast:
d1 $ whenmod 8 4 (fast 2) (sound "bd sn kurt")
Type: ifp :: (Int -> Bool) -> (Pattern a -> Pattern a) -> (Pattern a -> Pattern a) -> Pattern a -> Pattern a
ifp decides whether to apply one or another function depending on the result of a test function, which is passed the current cycle as a number. For example:
d1 $ ifp ((== 0).(flip mod 2))
(striate 4)
(# coarse "24 48") $
sound "hh hc"
This will apply striate 4 for every even cycle, and # coarse "24 48" for every odd one.
The test function does not rely on anything Tidal-specific, it uses plain Haskell functionality for operating on numbers. That is, it calculates the modulo of 2 of the current cycle which is either 0
(for even cycles) or 1. It then compares this value against 0 and returns the result, which is either True or False. This is what the first part of ifp's type signature signifies (Int -> Bool), a
function that takes a whole number and returns either True or False.
Conditions on ControlPatterns
Type: fix :: (ControlPattern -> ControlPattern) -> ControlPattern -> ControlPattern -> ControlPattern
The fix function applies another function to matching events in a pattern of controls. fix is contrast where the false-branching function is set to the identity id.
For example:
d1 $ slow 2 $ fix (# crush 3) (n "[1,4]") $ n "0 1 2 3 4 5 6" # sound "arpy"
The above only adds the crush control when the n control is set to either 1 or 4.
You can be quite specific, for example
fix (hurry 2) (s "drum" # n "1")
to apply the function hurry 2 to sample 1 of the drum sample set, and leave the rest as they are.
unfix is fix but only applies when the testing pattern is not a match.
Type: contrast :: (ControlPattern -> ControlPattern) -> (ControlPattern -> ControlPattern) -> ControlPattern -> ControlPattern -> ControlPattern
contrast is like a if-else-statement over patterns. For contrast t f p you can think of t al the true-branch, f as the false branch, and p as the test.
For contrast, you can use any control pattern as a test of equality: n "<0 1>", speed "0.5", or things like that. This lets you choose specific properties of the pattern you're transforming for
testing, like in the following example:
d1 $ contrast (|+ n 12) (|- n 12) (n "c") $ n (run 4) # s "superpiano"
where every note that isn't middle-c will be shifted down an octave but middle-c will be shifted up to c5.
Since the test given to contrast is also a pattern, you can do things like have it alternate between options:
d1 $ contrast (|+ n 12) (|- n 12) (s "<superpiano superchip>") $ s "superpiano superchip" # n 0
If you listen to this you'll hear that which instrument is shifted up and which instrument is shifted down alternates between cycles.
Type: contrastBy :: (a -> Value -> Bool) -> (ControlPattern -> Pattern b) -> (ControlPattern -> Pattern b) -> Pattern (Map.Map String a) -> Pattern (Map.Map String Value) -> Pattern b
contrastBy is a more general version of contrast where you can specify an abritrary boolean function that will be used to compare the control patterns. For example, contrast itself is defined as
contrastBy (==), to test for equality.
Compare the following:
d1 $ contrast (|+ n 12) (|- n 12) (n "2") $ n "0 1 2 [3 4]" # s "superpiano"
d2 $ contrastBy (>=) (|+ n 12) (|- n 12) (n "2") $ n "0 1 2 [3 4]" # s "superpiano"
In the latter example, we test for "greater than or equals to" instead of simple equality.
Choosing patterns and functions
Type: choose :: [a] -> Pattern a
The choose function emits a stream of randomly choosen values from the given list, as a continuous pattern:
d1 $ sound "drum ~ drum drum" # n (choose [0,2,3])
As with all continuous patterns, you have to be careful to give them structure; in this case choose gives you an infinitely detailed stream of random choices.
Type: chooseBy :: Pattern Double -> [a] -> Pattern a
The chooseBy function is like choose but instead of selecting elements of the list randomly, it uses the given pattern to select elements.
chooseBy "0 0.25 0.5" ["a","b","c","d"]
will result in the pattern "a b c" .
Type: wchoose :: [(a, Double)] -> Pattern a
wchoose is similar to choose, but allows you to 'weight' the choices, so some are more likely to be chosen than others. The following is similar to the previous example, but the 2 is twice as likely
to be chosen than the 0 or 3.
d1 $ sound "drum ~ drum drum" # n (wchoose [(0,0.25),(2,0.5),(3,0.25)])
Prior to version 1.0.0 of Tidal, the weights had to add up to 1, but this is no longer the case.
Type: wchooseBy :: Pattern Double -> [(a,Double)] -> Pattern a
The wchooseBy function is like wchoose but instead of selecting elements of the list randomly, it uses the given pattern to select elements.
select :: Pattern Double -> [Pattern a] -> Pattern a
Chooses between a list of patterns, using a pattern of floats (from 0 to 1).
selectF :: Pattern Double -> [Pattern a -> Pattern a] -> Pattern a -> Pattern a
Chooses between a list of functions, using a pattern of floats (from 0 to 1)
pickF :: Pattern Int -> [Pattern a -> Pattern a] -> Pattern a -> Pattern a
Chooses between a list of functions, using a pattern of integers.
squeeze :: Pattern Int -> [Pattern a] -> Pattern a
Chooses between a list of patterns, using a pattern of integers.
inhabit :: [(String, Pattern a)] -> Pattern String -> Pattern a
inhabit allows you to link patterns to some String, or in other words, to give patterns a name and then call them from within another pattern of Strings.
For example, we may make our own bassdrum, hi-hat and snaredrum kit using tidal:
let drum = inhabit [("bd",s "sine" |- accelerate 1.5),("hh",s "alphabet:7" # begin 0.7 # hpf 7000),("sd",s "invaders:3" # speed 12)]
d1 $ drum "[bd*8?, [~hh]*4, sd(6,16)]"
inhabit can be very useful when using MIDI controlled drum machines, since you can give understandable drum names to patterns of notes.
Boolean conditions
Type: struct :: Pattern Bool -> Pattern a -> Pattern a
struct places a rhythmic 'boolean' structure on the pattern you give it. For example:
d1 $ struct ("t ~ t*2 ~") $ sound "cp"
... is the same as ...
The structure comes from a boolean pattern, i.e. a binary one containing true or false values. Above we only used true values, denoted by t. It's also possible to include false values with f, which
struct will simply treat as silience. For example, this would have the same outcome as the above:
d1 $ struct ("t f t*2 f") $ sound "cp"
These true/false binary patterns become useful when you conditionally manipulate them, for example 'inverting' the values using every and inv:
d1 $ struct (every 3 inv "t f t*2 f") $ sound "cp"
In the above, the boolean values will be 'inverted' every third cycle, so that the structure comes from the fs rather than t. Note that euclidean patterns also create true/false values, for example:
d1 $ struct (every 3 inv "t(3,8)") $ sound "cp"
In the above, the euclidean pattern creates "t f t f t f f t" which gets inverted to "f t f t f t t f" every third cycle. Note that if you prefer you can use 1 and 0 instead of t and f.
Type: mask :: Pattern Bool -> Pattern a -> Pattern a
mask takes a boolean (aka binary) pattern and 'masks' another pattern with it. That is, events are only carried over if they match within a 'true' event in the binary pattern. For example consider
this kind of messy rhythm without any rests:
d1 $ sound (cat ["sn*8", "[cp*4 bd*4, hc*5]"]) # n (run 8)
If we apply a mask to it:
d1 $ mask "t t t ~ t t ~ t"
$ s (cat ["sn*8", "[cp*4 bd*4, bass*5]"])
# n (run 8)
Due to the use of cat here, the same mask is first applied to "sn*8" and in the next cycle to "[cp4 bd4, hc*5]".
You could achieve the same effect by adding rests within the cat patterns, but mask allows you to do this more easily. It kind of keeps the rhythmic structure and you can change the used samples
d1 $ mask "1 ~ 1 ~ 1 1 ~ 1"
$ s (cat ["can*8", "[cp*4 sn*4, jvbass*16]"])
# n (run 8)
Type: Pattern Bool -> Pattern a -> Pattern a -> Pattern a
sew uses a pattern of boolean (true or false) values to switch between two other patterns. For example the following will play the first pattern for the first half of a cycle, and the second pattern
for the other half.
d1 $ sound (sew "t f" "bd*8" "cp*8")
The above combines two patterns of strings, and passes the result to the sound function. It's also possible to sew together two control patterns, for example:
d1 $ sew "t <f t> <f [f t] t>" (n "0 .. 15" # s "future") (s "cp:3*16" # speed saw + 1.2)
You can also use Euclidean rhythm syntax in the boolean sequence:
d1 $ sew "t(11,16)" (n "0 .. 15" # s "future") (s "cp:3*16" # speed sine + 1.5)
Type: Pattern Bool -> Pattern a -> Pattern a -> Pattern a
stitch uses the first (binary) pattern to switch between the following two patterns. The resulting structure comes from the binary pattern, not the source patterns. This differs from sew where the
resulting structure comes from the source patterns. For example, the following uses a euclidean pattern to control CC0:
d1 $ ccv (stitch "t(7,16)" 127 0) # ccn 0 # "midi"
Type: euclid :: Pattern Int -> Pattern Int -> Pattern a -> Pattern a
euclid creates a Euclidean rhythmic structure. It produces the same output as the Euclidean pattern string. For example:
d1 $ euclid 3 8 $ sound "cp"
is the same as:
euclid accepts two parameters that can be patterns:
d1 $ euclid "<3 5>" "[8 16]/4" $ s "bd"
Type: euclidInv :: Pattern Int -> Pattern Int -> Pattern a -> Pattern a
Inverts the pattern given by euclid. For example:
d1 $ stack [euclid 5 8 $ s "bd",
euclidInv 5 8 $ s "hh27"]
to hear that the hi-hat event fires on every one of the eight even beats that the bass drum does not.
Type: euclidFull :: Pattern Int -> Pattern Int -> Pattern a -> Pattern a ->Pattern a
euclidFull is a convenience function for playing one pattern on the euclidean rhythm and a different pattern on the off-beat.
euclidFull 5 8 (s "bd") (s "hh27")
is equivalent to our above example.
ControlPattern conditions
Type: fix :: (ControlPattern -> ControlPattern) -> ControlPattern -> ControlPattern -> ControlPattern
With fix you can apply a ControlPattern as a condition and apply a function wich matches the controls. The first parameter is the function to apply and the second paramete is the condition.
d1 $ fix (ply 2) (s "hh") $ s "bd hh sn hh"
fixRange :: (ControlPattern -> Pattern ValueMap) -> Pattern (Map.Map String (Value, Value)) -> ControlPattern -> ControlPattern
The fixRange function isn't very user-friendly at the moment but you can create a fix variant with a range condition. Any value of a ControlPattern wich matches the values will apply the passed
d1 $ (fixRange ((# distort 1) . (# gain 0.8)) (pure $ Map.singleton "note" ((VN 0, VN 7)))) $ s "superpiano" <| note "1 12 7 11"
Type: ifp :: (Int -> Bool) -> (Pattern a -> Pattern a) -> (Pattern a -> Pattern a) -> Pattern a -> Pattern a
ifp decides whether to apply one or another function depending on the result of a test function, which is passed the current cycle as a number. For example:
d1 $ ifp ((== 0).(flip mod 2))
(striate 4)
(# coarse "24 48") $
sound "hh hc"
This will apply striate 4 for every even cycle, and # coarse "24 48" for every odd one.
The test function does not rely on anything Tidal-specific, it uses plain Haskell functionality for operating on numbers. That is, it calculates the modulo of 2 of the current cycle which is either 0
(for even cycles) or 1. It then compares this value against 0 and returns the result, which is either True or False. This is what the first part of ifp's type signature signifies (Int -> Bool), a
function that takes a whole number and returns either True or False. | {"url":"https://tidalcycles.org/docs/reference/conditions/","timestamp":"2024-11-04T10:13:17Z","content_type":"text/html","content_length":"213265","record_id":"<urn:uuid:6eaebe2b-0294-41b2-bc94-889e6acf98e3>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00225.warc.gz"} |
Click here to download the full example code
Neuron Types#
This tutorial will show you the different neuron types and how to work with them.
Depending your data/workflows, you will use different representations of neurons. If, for example, you work with light-level data you might end up extracting point clouds or neuron skeletons from
image stacks. If, on the other hand, you work with segmented EM data, you will typically work with meshes.
To cater for these different representations, neurons in NAVis come in four flavours:
Neuron type Description Core data
navis.TreeNeuron A hierarchical skeleton consisting - .nodes: the SWC node table
of nodes and edges.
navis.MeshNeuron A mesh with faces and vertices. - .vertices: (N, 3) array of x/y/z vertex coordinates
- .faces: (M, 3) array of faces
An image represented by either a - .voxels: (N, 3) array of voxels
navis.VoxelNeuron 2d array of voxels or a 3d voxel grid. - .values: (N, ) array of values (i.e. intensity)
- .grid: (N, M, K) 3D voxelgrid
navis.Dotprops A cloud of points, each with an - .points: (N, 3) array of point coordinates
associated local vector. - .vect: (N, 3) array of normalized vectors
Note that functions in NAVis may only work on a subset of neuron types: check out this table in the API reference for details. If necessary, NAVis can help you convert between the different neuron
types (see further below)!
In this guide we introduce the different neuron types using data bundled with NAVis. To learn how to load your own neurons into NAVis please see the tutorials on Import/Export.
TreeNeurons represent a neuron as a tree-like "skeleton" - effectively a directed acyclic graph, i.e. they consist of nodes and each node connects to at most one parent. This format is commonly used
to describe a neuron's topology and often shared using SWC files.
A navis.TreeNeuron is typically loaded from an SWC file via navis.read_swc but you can also constructed one yourself from e.g. pandas.DataFrame or a networkx.DiGraph. See the skeleton I/O tutorial
for details.
NAVis ships with a couple example Drosophila neurons from the Janelia hemibrain project published in Scheffer et al. (2020) and available at https://neuprint.janelia.org (see also the neuPrint
import navis
# Load one of the example neurons
sk = navis.example_neurons(n=1, kind="skeleton")
# Inspect the neuron
│ type │navis.TreeNeuron │
│ name │DA1_lPN_R │
│ id │1734350788 │
│ n_nodes │4465 │
│n_connectors│2705 │
│ n_branches │599 │
│ n_leafs │618 │
│cable_length│266476.875 │
│ soma │4177 │
│ units │8 nanometer │
│ created_at │2024-10-24 11:20:42.224436 │
│ origin │/home/runner/work/navis/navis/navis/data/swc/1... │
│ file │1734350788.swc │
navis.TreeNeuron stores nodes and other data as attached pandas.DataFrames:
│ │node_id│label│ x │ y │ z │ radius │parent_id │type│
│0│1 │0 │15784.0│37250.0│28062.0│10.000000 │-1 │root│
│1│2 │0 │15764.0│37230.0│28082.0│18.284300 │1 │slab│
│2│3 │0 │15744.0│37190.0│28122.0│34.721401 │2 │slab│
│3│4 │0 │15744.0│37150.0│28202.0│34.721401 │3 │slab│
│4│5 │0 │15704.0│37130.0│28242.0│34.721401 │4 │slab│
MeshNeurons consist of vertices and faces, and are a typical output of e.g. image segmentation.
A navis.MeshNeuron can be constructed from any object that has .vertices and .faces properties, a dictionary of vertices and faces or a file that can be parsed by trimesh.load. See the mesh I/O
tutorial for details.
Each of the example neurons in NAVis also comes as mesh representation:
m = navis.example_neurons(n=1, kind="mesh")
│ type │navis.MeshNeuron │
│ name │DA1_lPN_R │
│ id │1734350788 │
│ units │8 nanometer │
│n_vertices │6309 │
│ n_faces │13054 │
navis.MeshNeuron stores vertices and faces as attached numpy arrays:
(TrackedArray([[16384. , 34792.03125 , 24951.88085938],
[16384. , 36872.0625 , 25847.89453125],
[16384. , 36872.0625 , 25863.89453125],
[ 5328.08105469, 21400.07617188, 16039.99414062],
[ 6872.10498047, 19560.04882812, 13903.96191406],
[ 6872.10498047, 19488.046875 , 13927.96191406]]), TrackedArray([[3888, 3890, 3887],
[3890, 1508, 3887],
[1106, 1104, 1105],
[5394, 5426, 5548],
[5852, 5926, 6017],
[ 207, 217, 211]]))
Dotprops represent neurons as point clouds where each point is associated with a vector describing the local orientation. This simple representation often comes from e.g. light-level data or as
direvative of skeletons/meshes (see navis.make_dotprops).
Dotprops are used e.g. for NBLAST. See the dotprops I/O tutorial for details.
navis.Dotprops consist of .points and associated .vect (vectors). They are typically created from other types of neurons using navis.make_dotprops:
Turn our above skeleton into dotprops
dp = navis.make_dotprops(sk, k=5)
│ type │navis.Dotprops │
│ name │DA1_lPN_R │
│ id │1734350788 │
│ k │5 │
│ units │8 nanometer │
│n_points │4465 │
(array([[15784., 37250., 28062.],
[15764., 37230., 28082.],
[15744., 37190., 28122.],
[14544., 36430., 28422.],
[14944., 36510., 28282.],
[15264., 36870., 28282.]], dtype=float32), array([[-0.3002053 , -0.39364937, 0.8688596 ],
[-0.10845336, -0.2113751 , 0.9713694 ],
[-0.0435693 , -0.45593134, 0.8889479 ],
[-0.38446087, 0.44485292, -0.80888546],
[-0.9457323 , -0.1827982 , -0.26865458],
[-0.79947734, -0.5164282 , -0.30681902]], dtype=float32))
Check out the NBLAST tutorial for further details on dotprops!
VoxelNeurons represent neurons as either 3d image or x/y/z voxel coordinates typically obtained from e.g. light-level microscopy.
navis.VoxelNeuron consist of either a dense 3d (N, M, K) array (a "grid") or a sparse 2d (N, 3) array of voxel coordinates (COO format). You will probably find yourself loading these data from image
files (e.g. .nrrd via navis.read_nrrd()). That said we can also "voxelize" other neuron types to produce VoxelNeurons:
# Load an example mesh
m = navis.example_neurons(n=1, kind="mesh")
# Voxelize:
# - with a 0.5 micron voxel size
# - some Gaussian smoothing
# - use number of vertices (counts) for voxel values
vx = navis.voxelize(m, pitch="0.5 microns", smooth=2, counts=True)
│ type │navis.VoxelNeuron │
│ name │DA1_lPN_R │
│ id │1734350788 │
│units │500.0 nanometer │
│shape │(298, 392, 286) │
│dtype │float32 │
This is the grid representation of the neuron:
And this is the (N, 3) voxel coordinates + (N, ) values sparse representation of the neuron:
vx.voxels.shape, vx.values.shape
You may have noticed that all neurons share some properties irrespective of their type, for example .id, .name or .units. These properties are optional and can be set when you first create the
neuron, or at a later point.
In particular the .id property is important because many functions in NAVis will return results that are indexed by the neurons' IDs. If .id is not set explicitly, it will default to some rather
cryptic random UUID - you have been warned!
Neuron meta data#
NAVis was designed with connectivity data in mind! Therefore, each neuron - regardless of type - can have a .connectors table. Connectors are meant to bundle all kinds of connections: pre- &
postsynapses, electrical synapses, gap junctions and so on.
A connector table must minimally contain an x/y/z coordinate and a type for each connector. Here is an example of a connector table:
n = navis.example_neurons(1)
│ │connector_id │node_id│type│ x │ y │ z │ roi │confidence│
│0│0 │1436 │pre │6444│21608│14516│LH(R)│0.959 │
│1│1 │1436 │pre │6457│21634│14474│LH(R)│0.997 │
│2│2 │2638 │pre │4728│23538│14179│LH(R)│0.886 │
│3│3 │1441 │pre │5296│22059│16048│LH(R)│0.967 │
│4│4 │1872 │pre │4838│23114│15146│LH(R)│0.990 │
Connector tables aren't just passive meta data: certain functions in NAVis use or even require them. The most obvious example is probably for plotting:
# Plot neuron including its connectors
fig, ax = navis.plot2d(
n, # the neuron
connectors=True, # plot the neurons' connectors
color="k", # make the neuron black
cn_size=3, # slightly increase connector size
view=("x", "-z"), # set frontal view
method="2d" # connectors are better visible in 2d
In above plot, red dots are presynapses (outputs) and cyan dots are postsynapses (inputs).
Unless a neuron is truncated, it should have a soma somewhere. Knowing where the soma is can be very useful, e.g. as point of reference for distance calculations or for plotting. Therefore, {{ soma
}} neurons have a .soma property:
n = navis.example_neurons(1)
In case of this exemplary navis.TreeNeuron, the .soma points to an ID in the node table. We can also get the position:
array([[14957.1, 36540.7, 28432.4]], dtype=float32)
Other neuron types also support soma annotations but they may look slightly different. For a navis.MeshNeuron, annotating a node position makes little sense. Instead, we track the x/y/z position
m = navis.example_neurons(1, kind="mesh")
array([14957.1, 36540.7, 28432.4])
For the record: .soma / .soma_pos can be set manually like any other property (there are some checks and balances to avoid issues) and can also be None:
# Set the skeleton's soma on node with ID 1
n.soma = 1
Drop soma from MeshNeuron
NAVis supports assigning units to neurons. The neurons shipping with NAVis, for example, are in 8x8x8nm voxel space^1:
m = navis.example_neurons(1, kind="mesh")
To set the neuron's units simply use a descriptive string:
m.units = "10 micrometers"
Setting the units as we did above does not actually change the neuron's coordinates. It merely sets a property that can be used by other functions to interpret the neuron's coordinate space. See
below on how to convert the units of a neuron.
Tracking units is good practice in general but is also very useful in a variety of scenarios:
First, certain NAVis functions let you pass quantities as unit strings:
# Load example neuron which is in 8x8x8nm space
n = navis.example_neurons(1, kind="skeleton")
# Resample to 1 micrometer
rs = navis.resample_skeleton(n, resample_to="1 um")
Second, NAVis optionally uses the neuron's units to make certain properties more interpretable. By default, properties like cable length or volume are returned in the neuron's units, i.e. in 8x8x8nm
voxel space in our case:
You can tell NAVis to use the neuron's .units to make these properties more readable:
navis.config.add_units = True
navis.config.add_units = False # reset to default
2.1318150000000005 millimeter
Note that n.cable_length is now a pint.Quantity object. This may make certain operations a bit more cumbersome which is why this feature is optional. You can to a float by calling .magnitude:
Check out Pint's documentation to learn more.
To actually convert the neuron's coordinate space, you have two options:
You can multiply or divide any neuron or NeuronList by a number to change the units:
# Example neuron are in 8x8x8nm voxel space
n = navis.example_neurons(1)
# Multiply by 8 to get to nanometer space
n_nm = n * 8
# Divide by 1000 to get micrometers
n_um = n_nm / 1000
For non-isometric conversions you can pass a vector of scaling factors:
Note that for
, this is expected to be scaling factors for
(x, y, z, radius)
If your neuron has known units, you can let NAVis do the conversion for you:
n = navis.example_neurons(1)
# Convert to micrometers
n_um = n.convert_units("micrometers")
Addition & Subtraction
Multiplication and division will scale the neuro as you've seen above. Similarly, adding or subtracting to/from neurons will offset the neuron's coordinates:
n = navis.example_neurons(1)
# Convert to microns
n_um = n.convert_units("micrometers")
# Add 100 micrometers along all axes to the neuron
n_offset = n + 100
# Subtract 100 micrometers along just one axis
n_offset = n - [0, 0, 100]#
Operating on neurons#
Above we've already seen examples of passing neurons to functions - for example navis.plot2d(n).
For some NAVis functions, neurons offer have shortcut "methods":
import navis
sk = navis.example_neurons(1, kind='skeleton')
sk.reroot(sk.soma, inplace=True) # reroot the neuron to its soma
lh = navis.example_volume('LH')
sk.prune_by_volume(lh, inplace=True) # prune the neuron to a volume#
sk.plot3d(color='red') # plot the neuron in 3d
import navis
sk = navis.example_neurons(1, kind='skeleton')
navis.reroot_skeleton(sk, sk.soma, inplace=True) # reroot the neuron to its soma
lh = navis.example_volume('LH')
navis.in_volume(sk, lh, inplace=True) # prune the neuron to a volume
navis.plot3d(sk, color='red') # plot the neuron in 3d
In some cases the shorthand methods might offer only a subset of the full function's functionality.
The inplace parameter#
The inplace parameter is part of many NAVis functions and works like e.g. in the pandas library:
• if inplace=True operations are performed directly on the input neuron(s)
• if inplace=False (default) a modified copy of the input is returned and the input is left unchanged
If you know you don't need the original, you can use inplace=True to save memory (and a bit of time):
# Load a neuron
n = navis.example_neurons(1)
# Load an example neuropil
lh = navis.example_volume("LH")
# Prune neuron to neuropil but leave original intact
n_lh = n.prune_by_volume(lh, inplace=False)
print(f"{n.n_nodes} nodes before and {n_lh.n_nodes} nodes after pruning")
4465 nodes before and 344 nodes after pruning
All neurons are equal...#
... but some are more equal than others.
In Python the == operator compares two objects:
For NAVis neurons this is comparison done by looking at the neurons' attribues: morphologies (soma & root nodes, cable length, etc) and meta data (name).
n1, n2 = navis.example_neurons(n=2)
n1 == n1
To find out which attributes are compared, check out the neuron's .EQ_ATTRIBUTES property:
['n_nodes', 'n_connectors', 'soma', 'root', 'n_branches', 'n_leafs', 'cable_length', 'name']
Edit this list to establish your own criteria for equality.
Making custom changes#
Under the hood NAVis calculates certain properties when you load a neuron: e.g. it produces a graph representation (.graph or .igraph) and a list of linear segments (.segments) for TreeNeurons. These
data are attached to a neuron and are crucial for many functions. Therefore NAVis makes sure that any changes to a neuron automatically propagate into these derived properties. See this example:
n = navis.example_neurons(1, kind="skeleton")
print(f"Nodes in node table: {n.nodes.shape[0]}")
print(f"Nodes in graph: {len(n.graph.nodes)}")
Nodes in node table: 4465
Nodes in graph: 4465
Making changes will cause the graph representation to be regenerated:
n.prune_by_strahler(1, inplace=True)
print(f"Nodes in node table: {n.nodes.shape[0]}")
print(f"Nodes in graph: {len(n.graph.nodes)}")
Nodes in node table: 1761
Nodes in graph: 1761
If, however, you make changes to the neurons that do not use built-in functions there is a chance that NAVis won't realize that things have changed and properties need to be regenerated!
n = navis.example_neurons(1)
print(f"Nodes in node table before: {n.nodes.shape[0]}")
print(f"Nodes in graph before: {len(n.graph.nodes)}")
# Truncate the node table by 55 nodes
n.nodes = n.nodes.iloc[:-55].copy()
print(f"\nNodes in node table after: {n.nodes.shape[0]}")
print(f"Nodes in graph after: {len(n.graph.nodes)}")
Nodes in node table before: 4465
Nodes in graph before: 4465
Nodes in node table after: 4410
Nodes in graph after: 4410
Here, the changes to the node table automatically triggered a regeneration of the graph. This works because NAVis checks hash values of neurons and in this instance it detected that the node node
table - which represents the core data for TreeNeurons - had changed. It would not work the other way around: changing the graph does not trigger changes in the node table.
Again: as long as you are using built-in functions, you don't have to worry about this. If you do run some custom manipulation of neurons be aware that you might want to make sure that the data
structure remains intact. If you ever need to manually trigger a regeneration you can do so like this:
Clear temporary attributes of the neuron
Converting neuron types#
NAVis provides a couple functions to move between neuron types:
In particular meshing and skeletonizing are non-trivial and you might have to play around with the parameters to optimize results with your data! Let's demonstrate on some example:
# Start with a mesh neuron
m = navis.example_neurons(1, kind="mesh")
# Skeletonize the mesh
s = navis.skeletonize(m)
# Make dotprops (this works from any other neuron type
dp = navis.make_dotprops(s, k=5)
# Voxelize the mesh
vx = navis.voxelize(m, pitch="2 microns", smooth=1, counts=True)
# Mesh the voxels
mm = navis.mesh(vx.threshold(0.5))
/opt/hostedtoolcache/Python/3.10.15/x64/lib/python3.10/site-packages/skeletor/skeletonize/wave.py:198: DeprecationWarning:
Graph.clusters() is deprecated; use Graph.connected_components() instead
/opt/hostedtoolcache/Python/3.10.15/x64/lib/python3.10/site-packages/skeletor/skeletonize/wave.py:228: DeprecationWarning:
Graph.shortest_paths() is deprecated; use Graph.distances() instead
Inspect the results:
# Co-visualize the mesh and the skeleton
[m, s],
color=[(0.7, 0.7, 0.7, 0.2), "r"], # transparent mesh, skeleton in red
radius=False, # False so that skeleton is drawn as a line
Co-visualize the mesh and the dotprops
[m, dp],
color=[(0.7, 0.7, 0.7, 0.2), "r"], # transparent mesh, dotprops in red | {"url":"https://navis-org.github.io/navis/generated/gallery/tutorial_basic_01_neurons/","timestamp":"2024-11-03T12:13:40Z","content_type":"text/html","content_length":"1049366","record_id":"<urn:uuid:4b8a509a-f52b-4e0c-91e8-537d9508c0a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00886.warc.gz"} |
COUNT function: Description, Usage, Syntax, Examples and Explanation November 12, 2024 - Excel Office
COUNT function: Description, Usage, Syntax, Examples and Explanation
What is COUNT function in Excel?
COUNT function is one of Statistical functions in Microsoft Excel that is used counts the number of cells that contain numbers, and counts numbers within the list of arguments. Use
the COUNT function to get the number of entries in a number field that is in a range or array of numbers. For example, you can enter the following formula to count the numbers in the range A1:A20: =
COUNT(A1:A20). In this example, if five of the cells in the range contain numbers, the result is 5.
Syntax of COUNT function
COUNT(value1, [value2], …)
The COUNT function syntax has the following arguments:
• value1 Required. The first item, cell reference, or range within which you want to count numbers.
• value2, … Optional. Up to 255 additional items, cell references, or ranges within which you want to count numbers.
Note: The arguments can contain or refer to a variety of different types of data, but only numbers are counted.
COUNT formula explanation
• Arguments that are numbers, dates, or a text representation of numbers (for example, a number enclosed in quotation marks, such as “1”) are counted.
• Logical values and text representations of numbers that you type directly into the list of arguments are counted.
• Arguments that are error values or text that cannot be translated into numbers are not counted.
• If an argument is an array or reference, only numbers in that array or reference are counted. Empty cells, logical values, text, or error values in the array or reference are not counted.
• If you want to count logical values, text, or error values, use the COUNTA function.
• If you want to count only numbers that meet certain criteria, use the COUNTIF function or the COUNTIFSfunction.
Example of COUNT function
Steps to follow:
1. Open a new Excel worksheet.
2. Copy data in the following table below and paste it in cell A1
Note: For formulas to show results, select them, press F2 key on your keyboard and then press Enter.
You can adjust the column widths to see all the data, if need be.
Formula Description Result
=COUNT(A2:A7) Counts the number of cells that contain numbers in cells A2 through A7. 3
=COUNT(A5:A7) Counts the number of cells that contain numbers in cells A5 through A7. 2
=COUNT(A2:A7,2) Counts the number of cells that contain numbers in cells A2 through A7, and the value 2 4 | {"url":"https://www.xlsoffice.com/excel-functions/statistical-functions/count-function-description-usage-syntax-examples-and-explanation/","timestamp":"2024-11-12T07:29:35Z","content_type":"text/html","content_length":"63831","record_id":"<urn:uuid:a1cf696b-4a29-41d9-8b92-d2efe5c3d21f>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00442.warc.gz"} |
03696cam a2200505Ia 4500 ocn182559775 OCoLC 20210602115007.0 m o d cr cnu---unuuu 071128s1996 enka ob 001 0 eng d 96001679 N$T eng pn N$T YDXCP OCLCQ IDEBK ZCU E7B OCLCQ MERUC OCLCQ OCLCF NLGGC OPELS
DEBSZ UIU OCLCQ COO OCLCQ AZK LOA AGLDB MOR PIFAG OCLCQ U3W COCUF STF WRM D6H OCLCQ VTS NRAMU NLE INT REC VT2 OCLCQ UKMGB WYU TKN OCLCQ LEAUB M8D UKCRE 017584940 Uk 162589563 507575094 647671194
961626255 962628136 965994814 988465024 992039130 1035715889 1037796433 1038634104 1045559136 1055404142 1062907804 1081275831 1153540607 1228544768 9780080541716 (electronic bk.) 0080541712
(electronic bk.) 0750624698 (paper ; alk. paper) 9780750624695 (paper ; alk. paper) (OCoLC)182559775 (OCoLC)162589563 (OCoLC)507575094 (OCoLC)647671194 (OCoLC)961626255 (OCoLC)962628136 (OCoLC)
965994814 (OCoLC)988465024 (OCoLC)992039130 (OCoLC)1035715889 (OCoLC)1037796433 (OCoLC)1038634104 (OCoLC)1045559136 (OCoLC)1055404142 (OCoLC)1062907804 (OCoLC)1081275831 (OCoLC)1153540607 (OCoLC)
1228544768 QC174.8 .P38 1996eb SCI 055000 bisacsh 530.1/3 22 Pathria, R. K. Statistical mechanics / R.K. Pathria. 2nd ed. Oxford ; Boston : Butterworth-Heinemann, 1996. 1 online resource (xiv, 529
pages) : illustrations text txt rdacontent computer c rdamedia online resource cr rdacarrier Includes bibliographical references (pages 513-522) and index. Print version record. Front Cover;
Statistical Mechanics; Copyright Page; Contents; Preface to the Second Edition; Preface to the First Edition; Historical Introduction; Chapter 1. The Statistical Basis of Thermodynamics; Chapter 2.
Elements of Ensemble Theory; Chapter 3. The Canonical Ensemble; Chapter 4. The Grand Canonical Ensemble; Chapter 5. Formulation of Quantum Statistics; Chapter 6. The Theory of Simple Gases; Chapter
7. Ideal Bose Systems; Chapter 8. Ideal Fermi Systems; Chapter 9. Statistical Mechanics of Interacting Systems: The Method of Cluster Expansions. 'This is an excellent book from which to learn the
methods and results of statistical mechanics.' Nature 'A well written graduate-level text for scientists and engineers ... Highly recommended for graduate-level libraries.' Choice. This highly
successful text, which first appeared in the year 1972 and has continued to be popular ever since, has now been brought up-to-date by incorporating the remarkable developments in the field of 'phase
transitions and critical phenomena' that took place over the intervening years. This has been done by adding three new chapters (comprising over 150 pages and. Statistical mechanics. M�ecanique
statistique. SCIENCE Physics General. bisacsh Statistical mechanics. fast (OCoLC)fst01132070 Statistical mechanics Electronic books. Electronic books. Print version: Pathria, R.K. Statistical
mechanics. 2nd ed. Oxford ; Boston : Butterworth-Heinemann, 1996 0750624698 9780750624695 (DLC) 96001679 (OCoLC)34116798 ScienceDirect http://www.sciencedirect.com/science/book/9780750624695 0 0 0 0
MAIN MAIN 2021-06-02 0 2021-06-02 00:00:00 2021-06-02 EBK 9437 9437 | {"url":"https://webopac.iiserb.ac.in/cgi-bin/koha/opac-export.pl?op=export&bib=9437&format=marcxml","timestamp":"2024-11-11T14:04:39Z","content_type":"application/xml","content_length":"11332","record_id":"<urn:uuid:f9b82d32-a971-4520-a0a4-f21808a3e46c>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00151.warc.gz"} |
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where t 1; 2; . . . ; tmax . Thus, a feature vector x k s k ; s2 k ; r k; 1 ; r k; 2 ; . . . ; r k; tmax T of dimensionality 2 tmax is computed at every instant k, such that a set of G n-dimensional
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If this location is the same for both variables, then the variables are both referring to the same thing and the variables are therefore identical For example, if both C and D referred to our
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We assume the following has already been done by the user: 1. Both Microsoft Of ce 2003 and InfoSphere 9.5 are installed and the supporting ODBC drivers for Oracle and DB2 are in place in appropriate
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Since the content stored in the A variable is equal to the content stored in the A variable, this second mechanism would consider C and D to be identical for both the rst mechanism and the second
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As covered earlier in the chapter, video transmission requires special consideration. The following paragraphs summarize the special considerations a planner must take into account for video
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with equality iff A is diagonal. Since A is subject to a trace constraint, 1 n Aii P ,
This is far more complicated than the simple value type scenario depending on which mechanism we are using, we can consider two reference type variables to be identical to each other and different
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Impedance Factors Affecting PCB Board and Countermeasures
The annual growth rate of the electronics industry will exceed 20%, and the PCB board industry will also increase with the trend of the entire electronics industry. And more than a 20% growth rate.
The technological revolution and industrial structure changes in the world electronics industry are bringing new opportunities and challenges to the development of printed circuits. Printed circuits
develop with miniaturization, digitization, high frequency, and multi-function of electronic equipment. As the electrical interconnects in electronic devices - metal wires in PCB, it is not just a
matter of current flow or not. Instead, it acts as a signal transmission line. That is to say, electrical testing of PCBs for the transmission of high-frequency signals and high-speed digital
signals. It is not only necessary to measure whether the on, off, and short circuit of the circuit (or network) meets the requirements, but also whether the characteristic impedance value is within
the specified qualified range. Only when these two directions are qualified, is the printed board meets the requirements. The circuit performance provided by the printed circuit board must be able to
prevent reflection during signal transmission, keep the signal intact, reduce transmission loss, and play the role of matching impedance so that a complete, reliable, interference-free, noise-free
transmission signal can be obtained. This paper discusses the problem of characteristic impedance control of the surface microstrip line structure multilayer board commonly used in practice.
1. Surface microstrip line and characteristic impedance
The characteristic impedance of the surface microstrip line is high and widely used in practice. Its outer layer is the signal line surface of controlled impedance, and it is separated from the
adjacent reference surface by insulating materials.
For the surface microstrip line structure, the formula for calculating the characteristic impedance is:
Z0: Characteristic impedance of printed wire:
εr: Dielectric constant of insulating material:
h: The thickness of the medium between the printed wire and the reference plane:
w: width of printed wire:
t: Thickness of printed wire.
2. The dielectric constant of the material and its influence
The dielectric constant of the material is determined by the manufacturer of the material measured at a frequency of 1 MHz. The same material produced by different manufacturers is different due to
its different resin content. In this study, the relationship between the dielectric constant and the frequency change was studied by taking epoxy glass cloth as an example. The dielectric constant
decreases as the frequency increase, so in practical applications, the dielectric constant of the material should be determined according to the operating frequency. Generally, the average value can
be used to meet the requirements, and the transmission speed of the signal in the dielectric material will decrease with the increase of the dielectric constant. Therefore, in order to obtain a high
signal transmission speed, the dielectric constant of the material must be reduced, and at the same time, a high characteristic resistance must be used to obtain a high transmission speed, and a low
dielectric constant material must be selected for a high characteristic impedance.
3. The influence of wire width and thickness
Wire width is one of the main parameters that affect the variation of characteristic impedance. When the wire width changes by 0.025mm, the corresponding change in impedance value will be 5~6Ω. In
actual production, if 18um copper foil is used for the signal line surface of the control impedance, the allowable variation tolerance of the wire width is ±0.015mm. If the control impedance
variation tolerance is 35um copper foil, the allowable wire width variation tolerance is ±0.003 mm. It can be seen that the variation of the wire width allowed in the production will cause the
impedance value to change greatly. The width of the wire is determined by the designer according to various design requirements. It not only needs to meet the requirements of the current-carrying
capacity and temperature rise of the wire but also obtains the desired impedance value. This requires the manufacturer to ensure that the line width meets the design requirements and changes within
the tolerance range to meet the impedance requirements. The thickness of the wire is also determined according to the required current carrying capacity of the conductor and the allowable temperature
rise. In order to meet the requirements of use in production, the thickness of the coating is generally 25um on average. The wire thickness is equal to the copper foil thickness plus the plating
thickness. It should be noted that the surface of the wire should be clean before electroplating, and there should be no residues and trimming oil black so that the copper is not plated during
electroplating, which will change the thickness of the local wire and affect the characteristic impedance value. In addition, in the process of brushing the board, you must be careful not to change
the thickness of the wire and cause the impedance value to change.
4. Influence of dielectric thickness (h)
It can be seen from formula (1) that the characteristic impedance Z0 is proportional to the natural logarithm of the dielectric thickness, so it can be seen that the thicker the dielectric thickness,
the greater the Z0, so the dielectric thickness is another main factor affecting the characteristic resistance value. Because the wire width and the dielectric constant of the material have been
determined before production, the wire thickness process requirements can also be used as a fixed value, so controlling the laminate thickness (dielectric thickness) is the main means to control the
characteristic impedance in production. The relationship between the characteristic impedance value and the change in the thickness of the medium is obtained. When the thickness of the medium changes
by 0.025mm, it will cause a corresponding change in the impedance value of +5 to 8Ω. In the actual production process, the allowable variation in the thickness of each layer of the laminate will
result in a large change in the impedance value. In actual production, different types of prepregs are selected as the insulating medium, and the thickness of the insulating medium is determined
according to the number of prepregs. Take the surface microstrip line as an example, determine the dielectric constant of the insulating material at the corresponding operating frequency, then use
the formula to calculate the corresponding Z0, and then find out the corresponding dielectric thickness according to the wire width value and the calculated value Z0 proposed by the user. , and then
determine the type and number of prepregs according to the thickness of the selected copper-clad laminate and copper foil.
The effect of dielectric thickness of different structures on Z0
Compared with the stripline design, the design of the microstrip line structure has a higher characteristic impedance value under the same dielectric thickness and material, which is generally 20-40Ω
larger. Therefore, the design of the microstrip line structure is mostly used for high-frequency and high-speed digital signal transmission. At the same time, the characteristic impedance value will
increase with the increase of the dielectric thickness. Therefore, for high-frequency lines with strictly controlled characteristic impedance values, strict requirements should be placed on the error
of the dielectric thickness of the copper-clad laminate. Generally speaking, the change in the dielectric thickness should not exceed 10%. For multi-layer boards, the thickness of the media is also a
processing factor, especially when it is closely related to the multi-layer lamination process, so it should also be closely controlled.
5. Conclusion
In actual production, slight changes in the width and thickness of the wire, the dielectric constant of the insulating material, and the thickness of the insulating medium will cause the
characteristic impedance value to change, and the characteristic impedance value will also be related to other production factors. Therefore, in order to realize the control of the characteristic
impedance, the manufacturer must understand the factors affecting the change of the characteristic impedance value, master the actual production conditions, and adjust the various process parameters
according to the requirements of the designer to make the change within the allowable tolerance range. To get the desired impedance value on the PCB board. | {"url":"https://www.ipcb.com/pcb-blog/9676.html","timestamp":"2024-11-11T21:51:48Z","content_type":"application/xhtml+xml","content_length":"38345","record_id":"<urn:uuid:0b172e72-182e-430f-97eb-dfce2c39e2c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00489.warc.gz"} |
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Time From Now Calculator
The main purpose of the Time From Now Calculator is to get the exact time, date and number of days from the given number of hours, minutes and/or seconds. This simply means that if you want to know
what is 27 hours, 30 minutes and 40 seconds () from now, you just simply enter these values on the calculator and you'll get your answer right away.
How to Use the Time From Now Calculator
This calculator is helpful specially on cases that you want to know as quick as possible what is the time from now. Say for example, you are running late but your travel time takes usually about 25
minutes before arriving at the office. Sometimes you just want to know what is 25 minutes from now to get the exact time.
Another scenario is when you are cooking something for dinner, but it usually takes about 3 hours to prepare. Before cooking up, you may want to enter 3 hours on the calculator for you to make the
right estimates when you should begin cooking. This way, all will be prepared before dinner.
The usage of this Time From Now Calculator is quite not that hard, in fact, you just need to enter the time on the right input field, press Enter on your keyboard and that's it. However, here's a
step by step guide to its proper usage:
1. Step 1
There are input fields on the calculator labeled with Hours, Minutes and Seconds. Enter the time from now you want to figure out on its corresponding fields. For example, you want to know what is
7 hours and 45 minutes from now. Enter number 7 on Hours input field and 45 on Minutes input field.
2. Step 2
To get the result of the computation, you just need to press Enter key on your keyboard. You can also click the Calculate button on the calculator. Either way, it will trigger the calculator to
run the necessary computations.
Time From Now Calculator Inputs and Outputs
The calculator has three (3) input fields and two (2) buttons for calculation and clearing of inputs. The results are also displayed right below the buttons. Here are more details about them.
Time From Now Inputs
The inputs of the calculator include the hours, minutes and seconds. By clicking Calculate button or pressing Enter key, these actions will initialize the computation until the result is displayed.
There are three tiny boxes on the calculator labeled with Time, Date and Days. These are the exact results of the calculation with displays the information it is labeled with. A written paragraph
result is also displayed below it.
Time From Now Calculator Table
Here's a quick 1 to 50 hours from now in a table format.
Time From Now Date and Time
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What is the population of Wooster in Ohio? - Tiny Blue
What is the population of Wooster in Ohio?
If you ever wondered how many inhabitants Wooster in Ohio has, here is the answer:
Wooster, Ohio has a population of 24,811 residents.
With an area of 43.07 sq km (16.63 sq mi), that comes down to a population density of 576.06 inhabitants per square kilometer (1491.94 / sq mi).
As a reference: New York City has a population of 8,398,748 inhabitants and a population density of 6918 inhabitants per square kilometer (17918 / sq mi).
Wooster ( WUUS-tər) is a city in the U.S. state of Ohio and the county seat of Wayne County. The municipality is located in northeastern Ohio approximately 50 mi (80 km) SSW of Cleveland, 35 mi
(56 km) SW of Akron and 30 mi (48 km) W of Canton. The population was 24,811 at the 2000 census and 26,119 at the 2010 Census. The city is the largest in Wayne County, and the center of the
Wooster Micropolitan Statistical Area (as defined by the United States Census Bureau). Wooster has the main branch and administrative offices of the Wayne County Public Library.The College of
Wooster is located in Wooster.fDi magazine ranked Wooster among North America’s top 10 micro cities for business friendliness and strategy in 2013.
Cities with a similar population size as Wooster
Here a list of cities that have a similar number of inhabitants like Wooster, Ohio:
Cities with a similar size as Wooster
If you want to check which cities have a similar size as Wooster, Ohio, here you go:
Cities with a similar population density as Wooster
Other cities that have a similar population density as Wooster, Ohio are:
Cities from the same state: Ohio
Let’s see what other cities Ohio has to offer: | {"url":"https://tinyblue.info/what-is-the-population-of-wooster-in-ohio/","timestamp":"2024-11-10T11:05:13Z","content_type":"text/html","content_length":"40083","record_id":"<urn:uuid:68475ccb-1989-4da8-877c-2d64842059ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00185.warc.gz"} |
(PDF) Analysis and reduction of models using Persalys
Author content
All content in this area was uploaded by Claire-Eleuthèriane Gerrer on Sep 24, 2021
Content may be subject to copyright.
Analysis and reduction of models using Persalys
Claire-Eleuthèriane Gerrer1Hubert Blervaque2Julien Schueller1Daniel Bouskela3Sylvain
1Phimeca Engineering, France, {gerrer, schueller, girard}@phimeca.com
2Hubert Modelisation, France, hubertblervaque@gmail.com
3EDF Lab Chatou, France, daniel.bouskela@edf.fr
Advanced computer experiments enable to understand and
optimise a model. However, using these methods require
skills in programming and a deep understanding of their
underlying theory. Persalys is an open-source software,
based on OpenTURNS methods, which guides the user in
the analysis and makes computer experiments accessible
to non-programmers. This article aims to illustrate the use
of Persalys on an intelligible use case : a solar collector
from ThermoSysPro library.
We first performed a sensitivity analysis on the Model-
ica model exported as FMU. We then employed Persalys
capabilities to optimize and metamodel the solar collector.
We finally included the metamodel in the ThermoSysPro
concentrated solar power plant to observe its performance.
Persalys is developed for both Windows and Linux. We
succeeded in including metamodels on both OpenModel-
ica Connection Editor (OMEdit) and Dymola, in the form
of Modelica block or FMU.
Keywords: FMI, FMU, OtFMI, Persalys, ThermoSysPro,
sensitivity analysis, screening, metamodelling, model re-
duction, ExternalFunction.
1 Introduction
Analysing Modelica models is necessary to understand
their functioning and assert their reliability. Modellers do
it naturally when they modify inputs or parameters and
check the output variations. This verification step how-
ever demands lots of handwork when it is performed in
modelling GUIs, be it OMEdit or Dymola.
Persalys1is a user-friendly Python-based GUI dedi-
cated to the treatment of uncertainty and the management
of variabilities. Persalys enables the analysis of Modelica
models via the FMI standard. The software can be em-
ployed on all kinds of FMUs as well as with most finite
elements models. FMUs (Functional Mockup Units) are
models, written in Modelica or another 1D-modelling lan-
guage, exported under the FMI standard. This standard
provides an interface between different languages, such as
Amesim, Simulink, Python.
Persalys is based on the Python library for the treatment
of uncertainties, risks and statistics OpenTURNS2(Baudin
et al., 2015). The GUI is developed by part of the Open-
TURNS consortium, EDF and Phimeca Engineering, to
help and guide non-specialists in model analysis (see fig-
ure 1).
Aside from model analysis, Persalys enables to per-
form metamodelling, or model reduction, in a guided
way. Metamodelling consists in learning the behaviour of
the physical model using a mathematical function whose
computation time is negligible (usually in milliseconds
and below). Metamodels are thus a handy expedient to
prohibitive computation times when many model runs are
needed. Embedding metamodels in Modelica GUIs:
• permits to connect the metamodel with Modelica
blocks for larger-scale modelling,
• enables the inclusion of (metamodelled) external
models such as finite-element models.
Our purpose is to demonstrate how to, using only GUIs,
thoroughly analyse a FMU, metamodel it and employ the
metamodel in a Modelica environment.
The Modelica use case is detailed in section 2. The be-
haviour of a ThermoSysPro component, exported as FMU,
using Persalys is explored in section 3. Section 4 shows
how we computed the corresponding reduced model and
exported it from Persalys. We finally discuss the use of the
reduced model as FMU or Modelica component in sec-
tion 5.
2 The Modelica use case
We consider a Concentrated Solar Power Plant (CSPP)
transforming solar radiation to electricity. This industrial
installation provides up to 1 MWe.
2.1 The physics of a CSPP
The most visible part of a CSPP is its large amount of
surfaces reflecting the sun. The Parabolic Trough Solar
Collector (PTSC) contains a parabolic reflective surface
and a receiver tube, see figure 2. The reflected solar en-
ergy is transferred to the transparent receiver tube, which
contains an absorber tube coated with blackened nickel to
Figure 1. Persalys methods are presented as a tree. After defining the variables and outputs of interest, the user can evaluate the
model, perform screening, optimization, etc. The no-entry sign on a method signals if a prior step was not completed. For instance,
in the current analysis, the calibration method cannot be employed as observations must be provided first.
ensure high absorption. The absorber tubes heat up the
synthetic oil it contains to nearly 400 °C.
The synthetic oil circuit exchanges heat with a circuit
of water undergoing a Rankine cycle. The water is heated
to dry steam and passes through a set of turbines in rota-
tion, bringing an alternator in rotation to produce electric-
ity. The water is then condensed and sent back to the heat
exchangers, where it is heated again (Ferrière, 2008).
2.2 The modelling of a CSPP
The model ConcentratedSolarPowerPlant_PTSC3is an
emblematic example of the open-source ThermoSysPro li-
brary 4. ThermoSysPro provides components in the disci-
plines of thermal hydraulics and instrumentation and con-
trol for building nuclear, gas, coal or solar power plant
models. The modelling choices concerning the concen-
trated solar power plant is detailed in (El Hefni, 2014)
and (El Hefni and Bouskela, 2019). Figure 3 shows the
entire power plant model, using ThermoSysPro v3.2.
The component in the upper part, called SolarCollec-
tor5, models the PTSC. It is connected to the weather in-
puts: the sun radiation (direct normal incidence), sun in-
cidence angle and atmospheric temperature. The Solar-
Collector component is connected to the primary circuit
(whose fluid is oil) exchanging heat with the secondary
circuit. In this secondary circuit, the water is heated in
three steps (by the economizer, heater and super-heater)
before passing through the set of turbines.
2.3 A focus on the PTSC model
Before studying the entire solar power plant model, we fo-
cused on the PTSC model SolarCollector. Our purpose is
to better understand its behaviour using statistical analysis
and to replace it in the CSPP model by its metamodel.
We connected this component to ThermoSysPro inter-
face component HeatExchangerWall and to a HeatSource
setting the temperature value. We defined as output the
heat flow produced by the solar collector.
This model, named Env_PTSC, is steady-state. We thus
considered only the heat flow final value and constant
weather inputs. We exported this model as FMU to study
it using Persalys.
3 Understanding the PTSC’s be-
Modelica parameters and inputs have similar roles for sta-
tistical analysis as we consider a steady-state model. In
the following, we call “variables” the model inputs and
parameters under study.
We focus here on understanding the effect of the vari-
ables on the model output. We pre-selected 9 variables
of the SolarCollector, with causalities input or parameter,
3located in ThermoSysPro.Examples.Book.PowerPlant
5located in ThermoSysPro.Solar.Collectors
to study their effects. We chose their variation bounds in
accordance with the physics they represent, see table 1.
We loaded the PTSC model as FMU in Persalys. To
gain insights in the model behaviour, we first ran screen-
ing and Sobol’ sensitivity analyses (see subsection 3.1).
We then checked if the optimal values of important vari-
ables correspond to our understanding of the physics (see
subsection 3.2).
3.1 Screening and sensitivity analysis
Sensitivity analysis methods enable to determine the
model variables which affect most the output. Morris
method (also called “Morris screening”) and Sobol’ in-
dices are two global sensitivity analysis methods (Iooss,
2011). Morris’ qualifies the variables as important,with
nonlinear effect and/or interactions, or without effect. Its
results are rough, but the computation is rapid even with a
large number of variables (Morris, 1991). Sobol’ method
quantifies the fraction of the output variation for which
each variable is responsible. Sobol’ indices are computa-
tionally expensive, but very precise.
We screened the important variables using Morris
method, then studied more precisely their relative effect
on the heat flow. Figure 5 shows the results of Mor-
ris screening for the 9 variables (based on 80 trajectories
and 8 levels). The atmospheric temperature and the so-
lar collector heat transfer coefficient, glass transmittivity
and tube length variations have a negligible effect on the
model output. We thus set these 4 variables "without ef-
fect" to their value in the model ConcentratedSolarPow-
We refined the analysis of the effect of the 5 variables
categorized as important using Sobol’ sensitivity analysis.
The results of Morris analysis can be used by Persalys to
define the variables probabilistic model. Only the vari-
ables declared as significant are considered as uncertain:
the others are fixed to their default value.
Figure 6 shows the results of Sobol’ analysis in Per-
salys. The first-order index quantifies the effect of a vari-
able without interaction with any other variable on the
model output. The total index accounts for the effect of an
variable alone and in interaction with the other variables,
on the output. The variables with the greatest Sobol’ in-
dices are the most influential (with respect to their given
variation range). The heat flow produced by the solar
panel thus mainly depends on the sun radiation and the
incidence angle.
3.2 Variables optimal value
The value that maximises the model output is a comple-
mentary information to the relative influence of the vari-
ables. In theory, the heat flow produced is maximum when
the sun radiation and mirror reflectivity are maximal, the
angle incidence minimal. We checked the optimal com-
bined value of the 5 variables in Persalys.
Figure 7 shows that the maximal heat flow is reached
for maximal sun radiation and mirror reflectivity with
Figure 2. An example of PTSC. Source: (Tagle-Salazar et al., 2020)
Figure 3. Modelica model of a concentrated solar power plant, designed by El Hefni and Bouskela (El Hefni, 2014).
Variable group Name Unit Role Value Bounds
Geometric parameters
L m Length of the PTSC tube 450 [400, 500]
solarCollector.RimAngle ◦Rim angle 70 [65, 75]
solarCollector.f m Focal length 1.79 [1, 2]
solarCollector.h W/m2/KHeat transfer coefficient between
ambient air and glass envelope 3.06 [2.5, 5]
Cleanliness parameters solarCollector.R - Mirror reflectivity 0.93 [0.6, 0.95]
solarCollector.TauN - Glass transmittivity 0.95 [0.5, 0.95]
Weather inputs
T_atm.k ◦K Atmospheric temperature 300 [273, 303]
angle_incidence.k ◦Sun incidence angle (0° at zenith) 0 [0, 89]
radiation.k W/m2Direct Normal Irradiance 700 [0, 1000]
Table 1. Description of the 9 pre-selected variables of the PTSC models.
Figure 4. The Env_PTSC model relates the sun radiation, en-
vironment temperature and sun incidence to the heat flow pro-
duced by the SolarCollector.
minimal incidence angle as expected. The rim angle and
the focal length are maximal, meaning that the mirrors
have a large surface and “wrap” the tube. These results
must be considered with care, as the optimal point for the
5 variables may not be the same for each variable individ-
After gaining insights in the PTSC behaviour, we sta-
tistically learned this behaviour, i.e. we metamodelled the
Env_PTSC model. In existing solar power plants, the ge-
ometric parameters of the collectors cannot be modified:
only the weather inputs and the mirror cleanliness param-
eter evolve. Persalys enables us to reduce the PTSC model
with respect to these 3 variables.
4 Training and export of the meta-
Metamodelling, aka model reduction, consists in learn-
ing the model behaviour by a mathematical function. A
general purpose for metamodelling is to lower the compu-
tational cost, especially if many model runs are required
(e.g. to perform statistics or real-time simulations). We
reduced the PTSC model (a steady-state model) by learn-
ing the heatFlow final value with respect to the 3 afore-
mentioned variables with constant value.
Prior steps to metamodelling are the definition of a de-
sign of experiment and the simulation on these points, see
figure 1. We considered a Full Factorial Design (Friedman
et al., 2001) with 10 levels for each of the 3 variables, i.e.
1000 simulations. The metamodel is computed by Kriging
with constant mean and squared exponential covariance.
Figure 8 shows the metamodel validation by K-fold: the
cross-validation coefficient Q2is equal to 0.996, which is
very satisfying.
We saved the metamodel as a Python pickle object6
from the Persalys command line (see listing 1).
Listing 1. Save Persalys’ metamodel as Python Pickle object
import persalys as pls
import pickle
study = pls.Study.GetInstances()[0]
metamodel = study.getPhysicalModels()[1].
filename = "solar_collector.pkl"
with open(filename, ’wb’) as f:
pickle.dump(metamodel, f)
The pickled metamodel can be loaded in Python and ex-
ported as a Modelica graphical component and/or a FMU.
5 Using a Persalys metamodel in Mod-
elica GUI
Using the metamodel in a modelling environment enables
to connect it graphically to other components. We inserted
the metamodel as a power source in the concentrated solar
power plant model to show the validity of the interface be-
tween Persalys and Modelica GUIs. Following the same
methodology, metamodels reproducing the behaviour of
models from other environments (Excel, Amesim, Mat-
lab...) can be employed as blocks or FMUs in Modelica
The metamodel is a mathematical function, com-
puted by Persalys using OpenTURNS. The Python library
OTFMI interfaces FMUs with OpenTURNS objects, and
reciprocally (Girard and Yalamas, 2017)7.
We compare the advantages of exporting Persalys meta-
models as Modelica blocks or FMUs in subsection 5.1. We
show the insertion of the metamodel in a Modelica model
using a GUI in subsection 5.2.
5.1 Modelica block versus FMU
The metamodel can be used in Modelica as a block or as
a FMU. The FMU has the advantage to be standalone: it
embeds the contents required to run the metamodel, i.e.
a C function and C libraries. The “block” is a Modelica
wrapper of the C functions and XML file in which the
metamodel is stored. The simulation is faster, as unzip-
ping the FMU to access the underlying C is not necessary.
However, the C files need to be all stored and the wrapper
path must be updated manually if the user moves the C
The metamodel can be exported in both formats using
OTFMI, see listing 2. The function export_model exports
the given metamodel in a Modelica wrapper. If the gui op-
tion is prescribed, input and output connectors are gener-
ated, which enables to connect the wrapper to other Mod-
elica blocks. Otherwise inputs and outputs are defined us-
ing the Modelica language, which enables to simulate the
Figure 5. Top: Graph µ∗/σdrawn in Persalys GUI. The "no effect boundary" must be set by the user.
Bottom: Interpretation of the graphs µ∗/σand µ∗/µin Persalys GUI.
Figure 6. Top: the first-order and total Sobol’ indices of the 5 variables considered.
Bottom: the computed Sobol’ indices. Insufficient convergence of the estimates is signalled by warning panels.
Figure 7. Display of optimisation results in Persalys GUI. The heat flow is in watt ; the variable units are detailed in Table 1.
Figure 8. Metamodel validity using the K-fold method.
model via OMCompiler. The function export_fmu gener-
ates a FMU from a temporary Modelica wrapper. Both
ModelExchange or CoSimulation formats are supported.
Listing 2. Export the metamodel as component or FMU
fe = otfmi.FunctionExporter(metamodel,
fe.export_model(path_model, verbose=False,
fe.export_fmu(path_fmu, verbose=False)
These functionalities are supported on Windows and
Linux platforms. The generated component can be in-
cluded in OpenModelica Connection Editor (OMEdit) as
well as in Dymola.
5.2 Inserting the metamodel in a Modelica
We wrapped the metamodel in a Modelica block with
graphical input and output connectors. Figure 9 shows
the ThermoSysPro ConcentratedSolarPowerPlant_PTSC
model, whose solar panel has been replaced by the heat
source whose value is prescribed by the metamodel. We
call this model PartlyReducedCSPP. We successfully cali-
brated and ran the PartlyReducedCSPP model in Dymola.
At every time step of the simulation, the metamodel
value is queried by the solver (it is not co-simulation). The
initialization and integration steps last longer for the partly
reduced CSPP model than for the original CSPP, see ta-
ble 3. Table 2 shows that, even though the metamodel is
computationally less expensive than the Modelica model,
its inclusion in the Modelica environment increases the
simulation time.
Metamodel Env_PTSC
Simulation tool Python Modelica Modelica
Run time (s) 5 ×10−50.5 3 ×10−3
Table 2. The metamodel simulates in Python up to 100 times
faster than the corresponding model in Modelica.
We have two hypotheses to explain this difference.
First, the computation of the metamodel derivatives (nec-
essary to set the solver time steps) is possibly difficult.
Each derivative must be estimated by the Modelica solver
as the metamodel analytical function is not accessible.
This increase in simulation time could thus be due to a
larger number of calls to the metamodel.
Second, three different languages (Python, C and Mod-
elica) are employed for one computation of the metamodel
in the Modelica environment. This could be improved in
the following by using the C implementation of the Open-
TURNS library, instead of its Python implementation.
6 Conclusion and perspectives
Asserting the behaviour of a model and exploiting its re-
sults are very important tasks for a modeller. We presented
in this paper technological solutions to meet these needs
while staying (mostly) in graphical interface.
The Persalys software enables the statistical analysis of
Modelica (and others) models. Its strength resides in the
guidance for the analysis, performed in a very visual way.
We showed the use of Persalys for selecting the influen-
tial variables of the PTSC by Morris and Sobol’ sensi-
tivity analyses. We leveraged the results of these analy-
ses and our physical knowledge to successfully reduce the
model. We showed that the reduced model can be easily
imported in a Modelica GUI, be it OMEdit or Dymola,
and connected to other Modelica blocks. This technolog-
ical bridge can be used in a large scope of applications,
for instance to introduce reduced finite-elements models
in the Modelica environment.
The implementation in Persalys of statistical methods
for time-dependent outputs is underway. This will open
new possibilities to study models which output(s) evolve
over time without reaching a steady state. We also intend
to increase the simulation speed of the reduced model by
suppressing the underlying call to Python and directly us-
ing the C-implementation of OpenTURNS.
We thank warmly Arnaud Barthet, Pan Zhou and Li Xiao
(M4G team, EDF Chine) for sharing with us their experi-
ence of solar collectors.
This work was partially supported by the Paris region
through the FUI research project “Modeliscale”, a collab-
oration with Dassault Systèmes, Inria, EDF, Engie, CEA
INES, DPS, Eurobios and Phimeca Engineering.
Figure 9. The PartlyReducedCSPP model, based on ThermoSysPro emblematic ConcentratedSolarPowerPlant_PTSC.
PartlyReducedCSPP ConcentratedSolarPowerPlant_PTSC
CPU time for initialization (s) 0.345 0.179
CPU time for integration (s) 993 109
Table 3. The initialization and integration times of the partly reduced CSPP are larger than for the original ThermoSysPro model.
Michaël Baudin, Anne Dutfoy, Bertrand Iooss, and Anne-Laure
Popelin. Openturns: An industrial software for uncertainty
quantification in simulation. 2015.
Baligh El Hefni. Dynamic modeling of concentrated solar power
plants with the thermosyspro library (parabolic trough collec-
tors, fresnel reflector and solar-hybrid). Energy Procedia, 49:
1127–1137, 2014.
Baligh El Hefni and Daniel Bouskela. Modeling and Simula-
tion of Thermal Power Plants with ThermoSysPro. Springer,
Alain Ferrière. Centrales solaires thermodynamiques, 2008.
URL https://www.techniques-ingenieur.
Jerome Friedman, Trevor Hastie, and Robert Tibshirani. The
elements of statistical learning, volume 1. Springer series in
statistics New York, 2001.
Sylvain Girard and Thierry Yalamas. A probabilistic take
on system modeling with modelica and python. Tech-
nical report, Phimeca Engineering, 2017. https:
Bertrand Iooss. Revue sur l’analyse de sensibilité globale de
modèles numériques. Journal de la Société Française de
Statistique, 152(1):1–23, 2011.
Max D Morris. Factorial sampling plans for preliminary compu-
tational experiments. Technometrics, 33(2):161–174, 1991.
Pablo D Tagle-Salazar, Krishna DP Nigam, and Carlos I
Rivera-Solorio. Parabolic trough solar collectors: A general
overview of technology, industrial applications, energy mar-
ket, modeling, and standards. Green Processing and Synthe-
sis, 9(1):595–649, 2020. | {"url":"https://www.researchgate.net/publication/354810878_Analysis_and_reduction_of_models_using_Persalys","timestamp":"2024-11-14T01:28:15Z","content_type":"text/html","content_length":"461957","record_id":"<urn:uuid:251a82b6-a24e-419e-b591-c948b9724f5b>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00353.warc.gz"} |
Interval Subdivision
Dawood, Hend. InCLosure (Interval enCLosure): A Language and Environment for Reliable Scientific Computing
. 1.0 ed. Department of Mathematics, Faculty of Science, Cairo University, 2018.
InCLosure (Interval enCLosure) is a Language and Environment for Reliable Scientific Computing. InCLosure, provides rigorous and reliable results in arbitrary precision. From its name, InCLosure
(abbreviated as "InCL") focuses on "enclosing the exact real result in an interval". The interval result is reliable and can be as narrow as possible.
InCLosure supports arbitrary precision in both real and interval computations. In real arithmetic, the precision is arbitrary in the sense that it is governed only by the computational power of the
machine (default is 20 significant digits). The user can change the default precision according to the requirements of the application under consideration. Since interval arithmetic is defined in
terms of real arithmetic, interval computations inherit the arbitrary precision of real arithmetic with an added property that the interval subdivision method is provided with an arbitrary number of
subdivisions which is also governed only by the computational power of the machine. The user can get tighter and tighter guaranteed interval enclosures by setting the desired number of subdivisions
to cope with the problem at hand.
All the computations defined in terms of real and interval arithmetic (e.g., real and interval automatic differentiation) inherit the same arbitrary precision.
InCLosure is written in Lisp, the most powerful and fast language in scientific computations. InCLosure provides easy user interface, detailed documentation, clear and fast results. Anyone can
compute with InCLosure. | {"url":"https://scholar.cu.edu.eg/?q=henddawood/publications/term/7794","timestamp":"2024-11-03T20:17:15Z","content_type":"application/xhtml+xml","content_length":"48182","record_id":"<urn:uuid:8b7889df-c8bd-4201-ab90-b8e83e3d55c8>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00766.warc.gz"} |
gaussianValueAtRisk: Statistics tool for gaussian-assumption risk measures. - Linux Manuals (3)
gaussianValueAtRisk (3) - Linux Manuals
gaussianValueAtRisk: Statistics tool for gaussian-assumption risk measures.
QuantLib::GenericGaussianStatistics - Statistics tool for gaussian-assumption risk measures.
#include <ql/math/statistics/gaussianstatistics.hpp>
Inherits Stat.
Public Types
typedef Stat::value_type value_type
Public Member Functions
GenericGaussianStatistics (const Stat &s)
Gaussian risk measures
Real gaussianDownsideVariance () const
Real gaussianDownsideDeviation () const
Real gaussianRegret (Real target) const
Real gaussianPercentile (Real percentile) const
Real gaussianTopPercentile (Real percentile) const
Real gaussianPotentialUpside (Real percentile) const
gaussian-assumption Potential-Upside at a given percentile
Real gaussianValueAtRisk (Real percentile) const
gaussian-assumption Value-At-Risk at a given percentile
Real gaussianExpectedShortfall (Real percentile) const
gaussian-assumption Expected Shortfall at a given percentile
Real gaussianShortfall (Real target) const
gaussian-assumption Shortfall (observations below target)
Real gaussianAverageShortfall (Real target) const
gaussian-assumption Average Shortfall (averaged shortfallness)
Detailed Description
template<class Stat> class QuantLib::GenericGaussianStatistics< Stat >
Statistics tool for gaussian-assumption risk measures.
This class wraps a somewhat generic statistic tool and adds a number of gaussian risk measures (e.g.: value-at-risk, expected shortfall, etc.) based on the mean and variance provided by the
underlying statistic tool.
Member Function Documentation
Real gaussianDownsideVariance () const
returns the downside variance, defined as [ ac{N}{N-1} imes ac{ um_{i=1}^{N} heta imes x_i^{2}}{ um_{i=1}^{N} w_i} ], where $ heta $ = 0 if x > 0 and $ heta $ =1 if x <0
Real gaussianDownsideDeviation () const
returns the downside deviation, defined as the square root of the downside variance.
Real gaussianRegret (Real target) const
returns the variance of observations below target [ ac{um w_i (min(0, x_i-target))^2 }{um w_i}. ]
See Dembo, Freeman 'The Rules Of Risk', Wiley (2001)
Real gaussianPercentile (Real percentile) const
gaussian-assumption y-th percentile, defined as the value x such that [ y = ac{1}{qrt{2
i}} int_{-infty}^{x} \xp (-u^2/2) du ]
percentile must be in range (0-100%) extremes excluded
Real gaussianTopPercentile (Real percentile) const
percentile must be in range (0-100%) extremes excluded
Real gaussianPotentialUpside (Real percentile) const
gaussian-assumption Potential-Upside at a given percentile
percentile must be in range [90-100%)
Real gaussianValueAtRisk (Real percentile) const
gaussian-assumption Value-At-Risk at a given percentile
percentile must be in range [90-100%)
Real gaussianExpectedShortfall (Real percentile) const
gaussian-assumption Expected Shortfall at a given percentile
Assuming a gaussian distribution it returns the expected loss in case that the loss exceeded a VaR threshold,
[ mathrm{E}
average of observations below the given percentile $ p $. Also know as conditional value-at-risk.
See Artzner, Delbaen, Eber and Heath, 'Coherent measures of risk', Mathematical Finance 9 (1999)
percentile must be in range [90-100%)
Generated automatically by Doxygen for QuantLib from the source code. | {"url":"https://www.systutorials.com/docs/linux/man/3-gaussianValueAtRisk/","timestamp":"2024-11-06T08:24:54Z","content_type":"text/html","content_length":"12134","record_id":"<urn:uuid:14274541-cc4a-4472-8dc3-d1af5531edb2>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00708.warc.gz"} |
Agreement and consensus problems in groups of autonomous agents with linear dynamics for ISCAS 2005
ISCAS 2005
Conference paper
Agreement and consensus problems in groups of autonomous agents with linear dynamics
View publication
We study two recent consensus problems in multi-agent coordination with linear dynamics. In Saber and Murray an agreement problem was studied which has linear continuous-time state equations and a
sufficient condition was given for the given protocol to solve the agreement problem; namely that the underlying graph is strongly connected. We give sufficient and necessary conditions which include
graphs that are not strongly connected. In addition, Saber and Murray show that the protocol solves the average consensus problem if and only if the graph is strongly connected and balanced. We show
how multi-rate integrators can solve the average consensus problem even if the graph is not balanced. We give lower bounds on the rate of convergence of these systems which are related to the
coupling topology. Saber and Murray also considered the case where the coupling topology changes with time but remain a balanced graph at all times. We relate this case of switching topology to
synchronization of nonlinear dynamical systems with timevarying coupling and give conditions for solving the consensus problem even when the graphs are not balanced. Jadbabaie et al. study a model of
leaderless and follow-the-leader coordination of autonomous agents using a discrete-time model with time-varying linear dynamics and show coordination if the underlying undirected graph is connected
across intervals. Mureau extended this to directed graphs which are strongly connected across intervals. We prove that coordination is possible even if the graph is not strongly connected. © 2005 | {"url":"https://research.ibm.com/publications/agreement-and-consensus-problems-in-groups-of-autonomous-agents-with-linear-dynamics","timestamp":"2024-11-04T15:35:01Z","content_type":"text/html","content_length":"68550","record_id":"<urn:uuid:94801872-2842-40bd-b077-69e62ba466aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00419.warc.gz"} |
Palantir’s Abundant Free Cash Flow Will Float PLTR Stock Higher - Prime Stock Profits
Palantir (NYSE:PLTR) reported stellar earnings on May 11 for the first quarter, showing massive revenue and free-cash-flow growth. I predicted that Palantir, the software company for the intelligence
community, would turn free-cash-flow positive late last year. Now it has come true. The company’s FCF will continue to spike higher with revenue growth. And that will push PLTR stock much higher.
Here is what happened.
Palantir reported that revenue rose 49% to $341 million. But here is what is astounding. Adjusted free cash flow came in at $151 million, up $441 million year-over-year. This also represents a very
large portion (44.3%) of revenue.
As a result, we can use this to value the company going forward.
Using FCF to Value Palantir
The 44.3% adjusted FCF margin can be applied to estimates for 2021 and 2021 revenue to derive FCF estimates. For example, analysts polled by Seeking Alpha forecast that revenue will be $1.48 billion
in 2021, up 35% from $1.093 billion last year. So using a 40% margin (slightly lower than Q1 to be conservative) adjusted FCF will hit $592 million this year.
And since revenue is forecast to climb 29% next year to $1.91 billion, adjusted FCF could be $764 million by the end of 2022. We can use this to value PLTR stock.
For example, using a 1% FCF yield measure, PLTR stock is worth $76.4 billion. This can be seen by dividing $764 million in the 2022 forecast adjusted FCF by 1%. This represents a potential gain of
53.7% over Palantir’s present $49.7 billion market value That makes it worth $40.82 per share (i.e., 53.9% above its price today $26.53).
Another way to value PLTR stock is to use a 1.5% FCF yield. This would make Palantir worth $50.933 billion, and the target price for PLTR stock would be 5% higher at $27.85 per share.
So, in effect, Palantir is worth somewhere between 5% and 53.9% higher — let’s call it 29.5% higher (about one-third). So that means one can reasonably expect PLTR stock to be at least 30% higher
within the next year, or $34.34 per share. That is based on the midpoint between a 1% and 1.5% FCF yield. But it could also be worth as much as 53.9% more at $40.82 per share using a 1.0% FCF yield.
What To Do With PLTR Stock
Analysts don’t agree with me. For example, Yahoo! Finance reports that the average of seven analysts’ price targets is $22.43, 15.5% below the current price. In addition, TipRanks.com reports that
eight analysts have an average target of $22, or 17.1% below the price on June 24. Marketbeat says $20.75, or 21.8% below today.
So, on average analysts say PLTR is worth 20% below today’s price. But my view is that the stock is worth at least 29.5% higher. Who is right?
Probability and Expected Returns
As you may surmise from reading my articles, I try to be objective about this by using probability analysis. For example, I put together three scenarios and weight them differently. I weigh the
possibility that analysts are right by 50%, and that I am right by 30%. The third scenario for 20% involves a market-based return of, say, 10% over the next year. All three scenarios add up to 100%
of the likely outcomes.
So here is how that works out. In scenario one, analysts’ average expectation of a 20% drop is multiplied by the probability weight of 50%. That produces an expected return (ER) of -10% (i.e., 0.50 x
0.2). The second scenario, where I’m right, is weighted by 30%. That results in an ER of +8.85% (i.e., 0.30 x 0.295). The third scenario results in an ER of +2.0%. Therefore, the total ER adds up to
1.1% (i.e., 8.85% +2% -9.75%).
This means that even if there is a 50% chance analysts are right, there is still a positive expected return for the stock. Palantir would be undervalued by at least 1.1%. But I have also shown that
PLTR stock could be worth as much as 54% more at $40.82 per share.
On the date of publication, Mark R. Hake did not hold a position in any security mentioned in the article. The opinions expressed in this article are those of the writer, subject to the
InvestorPlace.com Publishing Guidelines.
Mark Hake writes about personal finance on mrhake.medium.com and runs the Total Yield Value Guide which you can review here. | {"url":"https://primestockprofits.com/2021/06/27/palantirs-abundant-free-cash-flow-will-float-pltr-stock-higher/","timestamp":"2024-11-10T21:51:11Z","content_type":"text/html","content_length":"91976","record_id":"<urn:uuid:ea5f8e1e-73e9-4728-97b9-8ece17edd46c>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00174.warc.gz"} |
How Present Value Can Help You Save for Retirement - Age calculator
How Present Value Can Help You Save for Retirement
Retirement is a phase of life that everyone looks forward to. However, it requires careful planning and preparation to ensure that you have enough money to retire comfortably. One of the key concepts
in retirement planning is present value, which can help you save for retirement. This article will explain what present value is, why it is important for retirement, and how you can use it to plan
for your retirement.
What is Present Value?
Present value is the value of a future amount of money today. In other words, it is the amount of money you would need today to have a certain amount of money in the future. Present value takes into
account the time value of money, which is the idea that money today is worth more than money in the future due to factors such as inflation and the opportunity cost of not investing the money.
Why Is Present Value Important for Retirement?
Present value is important for retirement because it can help you determine how much money you need to save today to have enough money to retire comfortably in the future. It can also help you
compare different retirement savings options to determine which one is the most beneficial for you.
For example, if you want to retire in 20 years and you estimate that you will need $1 million to retire comfortably, you can use present value to determine how much you need to save today. If the
rate of return on your savings is 5%, you would need to save about $383,000 today to have $1 million in 20 years.
How Can You Use Present Value to Save for Retirement?
There are several ways you can use present value to save for retirement:
1. Estimate Your Retirement Expenses – To determine how much you need to save for retirement, you need to estimate your retirement expenses. This includes expenses such as housing, food,
transportation, healthcare, and leisure activities. Once you have estimated your retirement expenses, you can use present value to determine how much you need to save today to cover those expenses in
the future.
2. Compare Retirement Savings Options – There are several retirement savings options available, such as 401(k) plans, individual retirement accounts (IRAs), and annuities. You can use present value
to calculate the future value of each savings option and compare them to determine which one is the most beneficial for you.
3. Determine Your Required Rate of Return – Your required rate of return is the rate of return you need to earn on your savings to achieve your retirement goals. You can use present value to
determine your required rate of return by estimating your retirement expenses and the amount of money you need to save today.
4. Monitor Your Retirement Savings – Once you have calculated how much you need to save for retirement, you need to monitor your savings to ensure that you are on track to meet your retirement goals.
You can use present value to compare your actual savings to your projected savings and make adjustments as needed.
Q: How does present value affect my retirement savings?
A: Present value is an important concept in retirement savings because it helps you determine how much money you need to save today to have enough money to retire comfortably in the future. It also
helps you compare different retirement savings options to determine which one is the most beneficial for you.
Q: How can I calculate present value for my retirement savings?
A: You can calculate present value using a present value calculator or a financial calculator. The formula for present value is: present value = future value / (1 + (rate of return / number of
Q: What is the difference between present value and future value?
A: Present value is the value of a future amount of money today, while future value is the value of today’s money in the future. Both concepts are important in retirement savings because they help
you determine how much money you need to save today to have enough money in the future.
Q: What retirement savings options are available?
A: There are several retirement savings options available, such as 401(k) plans, individual retirement accounts (IRAs), and annuities. You should compare the benefits and drawbacks of each option to
determine which one is the most beneficial for you.
Q: What is a required rate of return?
A: A required rate of return is the rate of return you need to earn on your savings to achieve your retirement goals. You can use present value to determine your required rate of return by estimating
your retirement expenses and the amount of money you need to save today.
Recent comments | {"url":"https://age.calculator-seo.com/how-present-value-can-help-you-save-for-retirement/","timestamp":"2024-11-03T22:08:16Z","content_type":"text/html","content_length":"301951","record_id":"<urn:uuid:c68ca6cf-af58-4209-bbca-ab512fe3110b>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00849.warc.gz"} |
Volume GFX
After the long article about the derivation of the Dualgrid, the hard part of Dual Marching Cubes is done. Now, only the triangulation is left, which is now covered in a rather short article.
With that, some nice volume meshes can be created. But for terrain, a Level of Detail mechanism is still needed, this is what will come up next.
The last Datastructure before the Triangles appear in DMC: The Dualgrid
The longest article so far is online, it’s how to derive the Dualgrid from the Octree. This is the last tedious part, until the real Dual Marching Cubes triangles appear!
The first part of Dual Marching Cubes: Generating the Octree
After the introduction, the first article about the needed steps in Dual Marching Cubes is online, generating the Octree!
And beside that, a little donation button sneaked in at the front page as a possibility to directly support this page and this project.
Realtime Editing Support arrived in OGREs Volume Component!
I think, it’s also interesting on this site to post some news arround the ongoing development of the Volume Component of OGRE which is the implementation of all (currently described and upcoming)
articles on this site.
Since today, realtime editing of the terrain is possible!
This is the changelog since the last push to OGRE:
• Realtime editing is possible now!
• Refactored data shared all along the chunktree into an own class with the chunks pointing to it.
• Some more general code cleanup.
• Splitted the recursive tree loading function in the Chunk class into more readable subfunctions.
Of course, everything behind it will become some dedicated articles in the future.
Let’s start with Dual Marching Cubes
After having nicely textured Marching Cubes geometry, it’s time to start with something better, Dual Marching Cubes. But at some point, Marching Cubes will play a role again, hence the name Dual
Marching Cubes.
A small introduction to this algorithm is now online. | {"url":"https://www.volume-gfx.com/2013/03/","timestamp":"2024-11-05T13:17:39Z","content_type":"text/html","content_length":"34107","record_id":"<urn:uuid:3a0e363d-ffa7-470c-915e-4ed03d3f8074>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00804.warc.gz"} |
Machine Learning for the Masses: Regression Analysis | The Wiglaf Journal
Machine Learning for the Masses: Regression Analysis
Machine learning has become a very popular topic in recent years. However, I fear that most people hear “machine learning” and assume that it must be a very difficult topic to understand. After all,
machine learning is a subfield of artificial intelligence, so machine learning must be complex. On the contrary, some machine learning techniques are easy to understand, and they can be easily
implemented in data analysis.
Today, I am going to discuss a machine learning technique known as regression analysis, and I aim to show how it is accessible to the masses.
There are many types of regression analysis. To keep things simple, I am going to review an example of simple linear regression. Simple linear regression looks for the relationship between a single
independent variable and a single dependent variable.
Combining Exploratory Data Analysis and Regression Analysis
A friend recently asked me what exactly it is that I do for work. Wishing to spare her the technical details of a typical pricing workday, I summarized, “I look for patterns in data.” I realized that
my response is actually a good summary of exploratory data analysis: looking for patterns in your data and teasing out relationships. Exploratory data analysis is a way to glean insight from your
dataset, usually using visualizations.
A typical pricing project begins with exploratory data analysis – with visualizing the data. An important data visualization tool is the scatterplot. Typically, a scatterplot visualizes two variables
at a time, one on the x-axis and one on the y-axis.
A scatterplot can quickly show a relationship (or lack of relationship) between two separate variables in your dataset. Likewise, regression analysis allows you to model the relationship between
different variables in your dataset. I’m going to review aspects of both exploratory data analysis and regression analysis for this article.
Height vs. Weight
Determining the relationship of height and weight is a very common simple linear regression exercise, so I will use that for my example. If you wish to follow along, you can access the dataset I used
from Kaggle here.
First, I filtered the list to only males so that I can focus on the relationship between height and weight for a single sex. Second, I graphed the data in a scatterplot, with height on the x-axis and
weight on the y-axis. After shrinking the size of my markers and adjusting the axis ranges, I have the scatterplot shown below.
Generally, you want the independent variable on the x-axis and the dependent variable on the y-axis. But how do you know which variable is your independent and which is your dependent?
One way to think about it is to determine which variable changes in response to the other. Your independent variable is the input that you manipulate, and your dependent variable is what changes as a
result. For height and weight, you can see that adjusting someone’s height would result in a change in weight. However, the converse is not necessarily true: adjusting someone’s weight would not
result in a change in height. Thus, the independent variable is height, and the dependent variable is weight. Weight changes in response to height, but not the other way around.
Naturally, pricing professionals will more than likely want to see a price on the y-axis and some other marketing variable on the x-axis. For instance, you may want to see what happens to price as
the revenue of a transaction increases, or perhaps you are curious whether the price is impacted by the annual revenue of the customer purchasing.
Regression Analysis
Scatterplots help to give you an intuitive feel for your data and any relationships that may exist. It is very clear from the scatterplot above that there is a strong relationship between height and
weight. It appears that an increase in height results in an increase in weight, on average.
But by how many pounds does weight increase for each inch that height increases? Unfortunately, a scatterplot alone does not provide you with that level of detail. However, a regression analysis
Regression analysis is a common technique that is easily accessed in many statistical packages. The basic idea is that a regression analysis finds the line of best fit for the dataset. This
regression line shows how your two variables are related, on average.
In Microsoft Excel (which I used to create these charts), adding the regression line for a simple linear regression is as simple as selecting your scatterplot, going to “add chart element”, and
selecting “add trendline”. You can also choose to add the formula for the regression line, which I have.
(Microsoft Excel also allows you to complete a regression analysis using the Analysis ToolPak. This will give you additional statistics to help you determine how strong the fit is, in addition to the
regression line above. Detailing that option is beyond the scope of this article. However, you can find detailed instructions in the book here.)
The red line on the scatterplot is the regression line. This shows the relationship of weight to height, on average. The formula for the regression line provides us with the slope and y-intercept. We
can see that each 1-inch change in height is associated with a 6-pound change in weight.
Parting Thoughts
I should remind the reader that for this analysis, I only reviewed linear regression. You should note that this method will not work if you are dealing with a non-linear curve (i.e., if the
regression line is not best described by a straight line).
However, as you can see, simple linear regression is a machine learning technique that is very easy to implement, even if you are using Microsoft Excel. Combine it with a simple scatterplot, and you
have a simple and yet powerful tool for data analysis.
So, don’t fear machine learning. Embrace it!
Nathan L. Phipps
is a Senior Consultant at
Wiglaf Pricing
. His areas of focus include pricing transformations, marketing analysis, conjoint analysis, and commercial policy. Before joining Wiglaf Pricing, Nathan worked as a pricing analyst at Intermatic
Inc. (a manufacturer of energy control products) where he dealt with market pricing and the creation of price variance and minimum advertised price policies. His prior experience includes time in
aerosol valve manufacturing and online education. Nathan holds an MBA with distinction in Marketing Strategy and Planning & Entrepreneurship from the Kellstadt Graduate School of Business at DePaul
University and a BA in Biology & Philosophy from Greenville College. He is based in Chicago, Illinois. | {"url":"https://wiglafjournal.com/machine-learning-and-regression-analysis/","timestamp":"2024-11-05T10:19:51Z","content_type":"text/html","content_length":"55274","record_id":"<urn:uuid:996a54b8-1209-4927-bc43-5c6750640315>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00849.warc.gz"} |
--- title: Overview of the DRomics package author: Marie Laure Delignette-Muller, Aurélie Siberchicot, Elise Billoir, Floriane Larras date: '`r Sys.Date()`' output: html_vignette: # pdf_document:
toc: true number_sections: yes toc_depth: 4 urlcolor: blue linkcolor: blue toccolor: 'blue' header-includes: \renewcommand*\familydefault{\sfdefault} vignette: > %\VignetteEngine{knitr::rmarkdown} %\
VignetteIndexEntry{Overview of the DRomics package} %!\VignetteEncoding{UTF-8} \usepackage[utf8]{inputenc} --- ```{r setup, echo=FALSE, message=FALSE, warning=FALSE} require(DRomics) require(ggplot2)
set.seed(1234) options(digits = 3) knitr::opts_chunk$set(echo = TRUE, eval = TRUE, message=FALSE, warning=FALSE, cache=FALSE, fig.width = 7, fig.height = 5) ``` # Introduction {#introduction} DRomics
is a freely available tool for dose-response (or concentration-response) characterization from omics data. It is especially dedicated to **omics data** obtained using a **typical dose-response
design**, favoring a great number of tested doses (or concentrations) rather than a great number of replicates (no need of replicates to use DRomics). After a first step which consists in
**importing**, **checking** and if needed **normalizing/transforming** the data ([step 1](#step1)), the aim of the proposed workflow is to **select monotonic and/or biphasic significantly responsive
items** (e.g. probes, contigs, metabolites) ([step 2](#step2)), to **choose the best-fit model** among a predefined family of monotonic and biphasic models to describe the response of each selected
item ([step 3](#step3)), and to **derive a benchmark dose** or concentration from each fitted curve ([step 4](#step4)). Those steps can be performed in R using DRomics functions, or using the **shiny
application named DRomics-shiny**. In the available version, DRomics supports **single-channel microarray data** (in log2 scale), **RNAseq data** (in raw counts) and other **continuous omics data**
(in log scale), such as metabolomics data calculated from AUC values (area under the curve), or proteomics data when expressed as protein abundance from peak intensity values. Proteomics data
expressed in spectral counts could be analyzed as RNAseq data using raw counts after carefully checking the validity of the assumptions made in processing the RNAseq data. In order to link responses
across biological levels based on a common method, DRomics also handles **continuous apical data** as long as they meet the use conditions of **least squares regression** (homoscedastic Gaussian
regression, see the [section on least squares](#leastsquares) for a reminder if needed). As built in the environmental risk assessment context where omics data are more often collected on
non-sequenced species or species communities, DRomics does not provide an annotation pipeline. The **annotation of items selected by DRomics** may be complex in this context, and **must be done
outside DRomics** using databases such as KEGG or Gene Ontology. **DRomics functions** can then be used to help the interpretation of the workflow results in view of the **biological annotation**. It
enables a **multi-omics approach**, with the comparison of the responses at the different levels of organization (in view of a common biological annotation). It can also be used to **compare the
responses** at one organization level, but measured under **different experimental conditions** (e.g. different time points). This interpretation can be performed in R using DRomics functions, or
using a **second shiny application DRomicsInterpreter-shiny**. This vignette is intended to help users to start using the DRomics package. It is complementary to the reference manual where you can
find more details on each function of the package. The first part of this vignette ([Main workflow, steps 1 to 4](#mainworkflow)) could also help users of the first **shiny application
DRomics-shiny**. The second part ([Help for biological interpretation of DRomics outputs](#interpreter)) could also help users of the **second shiny application DRomicsInterpreter-shiny**. Both shiny
applications can be used locally in your R session after installation of the package and of some required shiny tools (see below and on the DRomics web page is you need some more help to install the
package: [https://lbbe.univ-lyon1.fr/fr/dromics](https://lbbe.univ-lyon1.fr/fr/dromics)) ```{r, eval = FALSE} # Installation of required shiny packages install.packages(c("shiny", "shinyBS",
"shinycssloaders", "shinyjs", "shinyWidgets", "sortable")) # Launch of the first shiny application DRomics-shiny shiny::runApp(system.file("DRomics-shiny", package = "DRomics")) # Launch of the
second shiny application DRomicsInterpreter-shiny shiny::runApp(system.file("DRomicsInterpreter-shiny", package = "DRomics")) ``` If you do not want to install the package on your computer, you can
also launch the two shiny applications from the shiny server of our lab, respectively at [https://lbbe-shiny.univ-lyon1.fr/DRomics/inst/DRomics-shiny/](https://lbbe-shiny.univ-lyon1.fr/DRomics/inst/
DRomics-shiny/) and [https://lbbe-shiny.univ-lyon1.fr/DRomics/inst/DRomicsInterpreter-shiny/](https://lbbe-shiny.univ-lyon1.fr/DRomics/inst/DRomicsInterpreter-shiny/). If you do not want to use R
functions and prefer to use the shiny applications, locally on your computer or from our shiny server, you can skip the pieces of code and focus on the explanations and on the outputs that are also
given in the shiny applications. And if one day you want go further using the R functions, we recommend you to start from the whole R code corresponding to your analysis that is provided on the last
page of each of the two shiny applications. # Main workflow {#mainworkflow} ## Step 1: importation, check and normalization / transformation of data if needed {#step1} ### General format of imported
data #### Importation of data from a unique text file {#textfile} Whatever the type of data imported in DRomics (e.g. RNAseq, microarray, metabolomic data), data can be imported from a .txt file
(e.g. "mydata.txt") organized with **one row per item** (e.g. transcript, probe, metabolite) and **one column per sample**. In **an additional first row**, after a name for the item identifier (e.g.
"id"), we must have the **tested doses or concentrations in a numeric format** for the corresponding sample (for example, if there are triplicates for each treatment, the first line could be "item",
0, 0, 0, 0.1, 0.1, 0.1, etc.). An **additional first column** must give the **identifier of each item** (identifier of the probe, transcript, metabolite, ..., or name of the endpoint for anchoring
data), and the other columns give the responses of the item for each sample. This file will be imported within DRomics using an internal call to the function read.table() with its default field
separator (sep argument) and its default decimal separator (dec argument at "."). So remember, if necessary, to **transform another decimal separator (e.g. ",") in "." before importing your data**.
Different examples of .txt files formatted for the DRomics workflow are available in the package, such as the one named "RNAseq_sample.txt". You can have a look of how the data are coded in this file
using the following code. **To use a local dataset formatted in the same way, such use a datafilename of type `"yourchosenname.txt"`.** ```{r ch1} # Import the text file just to see what will be
automatically imported datafilename <- system.file("extdata", "RNAseq_sample.txt", package = "DRomics") # datafilename <- "yourchosenname.txt" # for a local file # Have a look of what information is
coded in this file d <- read.table(file = datafilename, header = FALSE) nrow(d) head(d) ``` #### Importation of data as an R object {#Robject} **Alternatively an R object of class data.frame can be
directly given in input**, that corresponds to the output of read.table(file, header = FALSE) on a file described in the [previous section](#textfile). You can see below an example of an RNAseq data
set that is available in DRomics as an R object (named Zhou_kidney_pce) and which is an extended version (more rows) of the previous dataset coded in "RNAseq_sample.txt". ```{r ch2} # Load and look
at the dataset directly coded as an R object data(Zhou_kidney_pce) nrow(Zhou_kidney_pce) head(Zhou_kidney_pce) ``` If your data are already imported in R in a **different format as the one described
above**, you can use the **formatdata4DRomics() function to build an R object directly useable by the DRomics workflow**. formatdata4DRomics() needs two arguments in input: + the **matrix of the
data** with **one row for each item** and **one column for each sample**, + and the **numeric vector coding for the dose** for each sample. The names of the samples can be added in a third optional
argument (see ?formatdata4DRomics for details). Below is an example using a RNAseq dataset of the package coded as an R object named zebraf. ```{r ch3} # Load and look at the data as initially coded
data(zebraf) str(zebraf) (samples <- colnames(zebraf$counts)) # Formatting of data for use in DRomics # data4DRomics <- formatdata4DRomics(signalmatrix = zebraf$counts, dose = zebraf$dose,
samplenames = samples) # Look at the dataset coded as an R object nrow(data4DRomics) head(data4DRomics) ```
Whatever the way you format your data, **we strongly recommend you to carefully look at the following sections** to **check that you use the good scale for you data**, which **depends on the type of
measured signal ** (counts of reads, fluorescence signal, ...).
### What types of data can be analyzed using DRomics ? {#datatypes} #### Description of the classical types of data handled by DRomics DRomics offers the possibility to work on different types of
omics data (see following subsections for their description) but also on continuous anchoring data. **When working on omics data, all the lines of the data frame** (except the first one coding for
the doses or concentrations) **correspond to the same type of data** (e.g. raw counts for RNAseq data). **When working on anchoring data, the different lines** (except the first one coding for the
doses or concentrations) **correspond to different endpoints that may correspond to different types of data** (e.g. biomass, length,..), but all are assumed continuous data compatible with a Gaussian
(normal) error model (after transformation if needed, e.g. logarithmic transformation) for the selection and modeling steps (see the [section on least squares](#leastsquares) if you need a reminder
on the is condition). **Three types of omics data** may be imported in DRomics using the following functions: + **RNAseqdata()** should be used to import **RNAseq as counts of reads** (for details
look at the example with [RNAseq data](#RNAseqexample)), + **microarraydata()** should be used to import **single-channel microarray data in log2 scale** (for details look at the example with
[microarray data](#microarrayexample)), + **continuousomicdata()** should be used to import **other continuous omics data** such as metabolomics data, or proteomics data (only when expressed in
intensity),..., **in a scale that enables the use of a Gaussian error model** (for details look at the example with [metabolomic omics data](#metabolomicexample)). It is also possible to import in
DRomics **continuous anchoring data** measured at the apical level, especially for **sake of comparison of benchmark doses** (see [Step 4](#step4) for definition of BMD) estimated **at different
levels of organization** but using **the same metrics**. Nevertheless, one should keep in mind that the DRomics workflow was optimized for an automatic analysis of high throughput omics data
(especially implying a selection and modeling steps on high-dimensional data) and that **other tools may be better suited for the sole analysis of apical dose-response data** (for details look at the
example with [continuous apical data](#apicalexample)). In Steps 1 and 2 **count data** are internally analysed using functions of the Bioconductor package [DESeq2](https://bioconductor.org/packages/
release/bioc/html/DESeq2.html), continuous omics data (**microarray data and other continuous omics data**) are internally analysed using functions of the Bioconductor package [limma](https://
www.bioconductor.org/packages/release/bioc/html/limma.html) and **continuous anchoring data** are internally analysed using the classical lm() function. ##### An example with RNAseq data {#
RNAseqexample} For RNAseq data, **imperatively imported in raw counts** (if your counts come from Kallisto or Salmon put add the argument `round.counts = TRUE` in order to round them), you have to
choose the transformation method used to stabilize the variance ("rlog" or "vst"). In the example below "vst" was used to make this vignette quick to compile, but **"rlog" is recommended and chosen
by default even if more computer intensive than "vst" except when the number of samples is very large (> 30)** (as encountered for *in situ* data for example: see ?RNAseqdata and the [section
dedicated to *in situ* data](#insitudata) for details on this point). Whatever the chosen method, **data are automatically normalized with respect to library size and transformed in log2 scale**.
```{r ch4} RNAseqfilename <- system.file("extdata", "RNAseq_sample.txt", package = "DRomics") # RNAseqfilename <- "yourchosenname.txt" # for a local file ``` ```{r ch5} (o.RNAseq <- RNAseqdata
(RNAseqfilename, transfo.method = "vst")) plot(o.RNAseq, cex.main = 0.8, col = "green") ``` The plot of the output shows the distribution of the signal on all the contigs/genes, for each sample,
before and after the normalization and transformation of data. ##### An example with microarray data {#microarrayexample} For single-channel microarray data, **imperatively imported in log scale**
(classical and recommended log2 scale), you can choose between array normalization methods ("cyclicloess", "quantile", "scale" or "none"). In the example below, "quantile" was used to make this
vignette quick to compile, but **"cyclicloess" is recommended and chosen by default even if more computer intensive than the others** (see ?microarraydata for details). ```{r ch6} microarrayfilename
<- system.file("extdata", "transcripto_sample.txt", package = "DRomics") # microarrayfilename <- "yourchosenname.txt" # for a local file ``` ```{r ch7} (o.microarray <- microarraydata
(microarrayfilename, norm.method = "quantile")) plot(o.microarray, cex.main = 0.8, col = "green") ``` The plot of the output shows the distribution of the signal over all the probes, for each sample,
before and after the normalization of data. ##### An example with metabolomic data {#metabolomicexample} **Neither normalization nor transformation** is provided in the function continuousomicdata().
The **pre-treatment of metabolomic data must be done before importation of data**, and **data must be imported in log scale**, so that they can be directly modelled using a **Gaussian (normal) error
model**. This strong hypothesis is required both for selection of items and for dose-reponse modeling (see the [section on least squares](#leastsquares) for a reminder if needed). In the context of a
multi-omics approach we recommend the use of a log2 transformation, instead of the classical log10 for such data, so as to facilitate the comparison of results obtained with transcriptomics data
generally handled in a log2 scale. For instance, a basic procedure for the pre-treatment of metabolomic data could follow the three steps described thereafter: i) removing of metabolites for which
the proportion of missing data (non detected) across all the samples is too high (more than 20 to 50 percents according to your tolerance level); ii) retrieving of missing values data using half
minimum method (i.e. half of the minimum value found for a metabolite across all samples); iii) log-transformation of values. If a scaling to the total intensity (normalization by sum of signals in
each sample) or another normalization is necessary and pertinent, we recommend to do it before those three previously described steps. ```{r ch8} metabolofilename <- system.file("extdata",
"metabolo_sample.txt", package = "DRomics") # metabolofilename <- "yourchosenname.txt" # for a local file ``` ```{r ch9} (o.metabolo <- continuousomicdata(metabolofilename)) plot(o.metabolo, col =
"green") ``` The plot of the output shows the distribution of the signal over all the metabolites, for each sample. The deprecated metabolomicdata() function was renamed continuousomicdata() in the
recent versions of the package (while keeping the first name available) to **offer its use to other continuous omic data** such as **proteomics data** (when expressed in intensity) or **RT-qPCR
data**. As for metabolomic data, the **pre-treatment** of other continuous omic data must be done **before importation**, and **data must be imported in a scale that enables the use of a Gaussian
error model** as this strong hypothesis is required both for selection of items and for dose-response modeling. ##### An example with continuous anchoring apical data {#apicalexample} No
transformation is provided in the function continuousanchoringdata(). **If needed the pre-treatment of data must be done before importation of data**, so that they can be directly modelled using a
**Gaussian error model**. This strong hypothesis is required both for selection of responsive endpoints and for dose-reponse modeling (see the [section on least squares](#leastsquares) for a reminder
if needed). ```{r ch10} anchoringfilename <- system.file("extdata", "apical_anchoring.txt", package = "DRomics") # anchoringfilename <- "yourchosenname.txt" # for a local file ``` In the following
example the argument backgrounddose is used to specify that doses below or equal to 0.1 are considered as 0 in the DRomics workflow. Specifying this argument is necessary when there is no dose at 0
in the data (see [section on *in situ* data](#insitudata) for details on this point). ```{r ch11, fig.width = 7, fig.height = 3} (o.anchoring <- continuousanchoringdata(anchoringfilename,
backgrounddose = 0.1)) plot(o.anchoring) + theme_bw() ``` For such data the plot() function simply provides a dose-response plot for each endpoint. By default the dose is represented on a log scale,
it is why responses for the control (null dose, so minus infinity in log scale) appear as half points on the Y-axis. It can be changed using the argument dose_log_transfo as below. ```{r ch12,
fig.width = 7, fig.height = 3} plot(o.anchoring, dose_log_transfo = FALSE) + theme_bw() ``` #### Handling of data collected through specific designs {#specificdesigns} The DRomics workflow was first
developed for data collected through a typical dose-response experiment, with a reasonable number of tested doses (or concentrations - at least 4 in addition to the control and ideally 6 to 8) and a
small number of replicates per dose. Recently we made some modifications in the package to **make possible the use of designs with only 3 doses in addition to the control even if this type of design
is not recommended for dose-response modeling**. **We also extended our workflow to handle *in situ* (observational) data**, for which there is **no replication, as the dose (or concentration) is not
controlled** (see [an example with in situ data](#insitudata) for more details). It is also now possible to handle **experimental data collected using a design with a batch effect** using DRomics
together with functions from the Bioconductor package [sva](https://bioconductor.org/packages/release/bioc/html/sva.html) to correct for this batch effect before selection and modeling steps. We also
developed the **PCAplot() function to help visualizing this batch effect** and the impact of the **batch effect correction (BEC)** on data (see [an example with RNAseq data from an experiment with a
batch effect](#batcheffect) for more details on how to handle such a case and see ?PCAplot for details on this specific function **that could also be used to identify potential outlier samples**). ##
### An example with *in situ* (observational) RNAseq data {#insitudata} One of the problem that may occur in particular with *in situ* data, is the absence of real control samples, corresponding to a
strictly null exposure dose or concentration. **To prevent an hazardous calculation of the BMD (see [Step 4](#step4) for definition of BMD) by extrapolation in such a case, one should use the
argument backgrounddose to define the maximal measured dose that can be considered as a negligible dose.** All doses below or equal to the value given in backgrounddose will be fixed at 0, so as to
be considered at the **background level of exposition**. For *in situ* data (and more generally for data with a very large number of samples), the **use of the rlog transformation in RNAseqdata() is
not recommended**, both for speed reason and because you are **more likely to encounter a problem with the rlog transformation in case of outliers** in such a case (see [https://
support.bioconductor.org/p/105334/](https://support.bioconductor.org/p/105334/) for an explanation of the author of [DESeq2](https://bioconductor.org/packages/release/bioc/html/DESeq2.html) and if
you want to see an example of problems that may appear with outliers in that case, just force the `transfo.method` to `"rlog"` in the following example). ```{r ch13} datafilename <- system.file
("extdata", "insitu_RNAseq_sample.txt", package="DRomics") # Importation of data specifying that observed doses below the background dose # fixed here to 2e-2 will be considered as null dose to have
a control (o.insitu <- RNAseqdata(datafilename, backgrounddose = 2e-2, transfo.method = "vst")) plot(o.insitu) ``` The plot of the output shows the distribution of the signal on all the contigs, for
each sample (here the box plots are stuck to each other due to the large number of samples), before and after the normalization and transformation of data. ##### An example with RNAseq data from an
experiment with a batch effect {#batcheffect} When omics data are collected through a **design with a known potential batch effect**, the **DRomics function PCAplot()** can be used as in the example
below to **visualize the impact of this batch effect on the data**. If it seems necessary, **functions from specific packages** can then be used to perform **batch effect correction (BEC)**. We
recommend the use of functions **ComBat()** and **ComBat_seq()** from the Bioconductor **[sva](https://bioconductor.org/packages/release/bioc/html/sva.html)** package for this purpose, respectively
for **microarray** (or other continuous omic data) and **RNAseq data**. As **[sva](https://bioconductor.org/packages/release/bioc/html/sva.html)** is a Bioconductor package, it must be installed in
the same way as **[DESeq2](https://bioconductor.org/packages/release/bioc/html/DESeq2.html)** and **[limma](https://www.bioconductor.org/packages/release/bioc/html/limma.html)** previously to be
loaded. If needed look at the DRomics web page to get the good instruction to install Bioconductor packages: [https://lbbe.univ-lyon1.fr/fr/dromics](https://lbbe.univ-lyon1.fr/fr/dromics)). Below is
an example using ComBat-seq() on RNAseq data with batch effect. As the sva package does not import the RNAseq data in the same format as DRomics, it is necessary to use the DRomics function
formatdata4DRomics() to interoperate between ComBat-seq in DRomics functions (see the [section on importation as an R object](#Robject) for details on this function or ?formatdata4DRomics). ```{r
ch14} # Load of data data(zebraf) str(zebraf) # Look at the design of this dataset xtabs(~ zebraf$dose + zebraf$batch) ``` It appears in this design that the data were obtained using to batches, with
only the controlled condition (null dose) appearing in both batches. ```{r ch15} # Formating of data using the formatdata4DRomics() function data4DRomics <- formatdata4DRomics(signalmatrix =
zebraf$counts, dose = zebraf$dose) # Importation of data just to use DRomics functions # As only raw data will be given to ComBat_seq after (o <- RNAseqdata(data4DRomics)) # PCA plot with the sample
labels PCAdataplot(o, label = TRUE) + theme_bw() # PCA plot to visualize the batch effect PCAdataplot(o, batch = zebraf$batch) + theme_bw() ``` The PCA plot shows an impact of the batch effect, that
clearly appears on controls (red points) which were obtained on two different batches. ```{r ch16, results = "hide", message = FALSE} # Batch effect correction using ComBat_seq{sva} require(sva)
BECcounts <- ComBat_seq(as.matrix(o$raw.counts), batch = as.factor(zebraf$batch), group = as.factor(o$dose)) ``` ```{r ch17} # Formating of data after batch effect correction BECdata4DRomics <-
formatdata4DRomics(signalmatrix = BECcounts, dose = o$dose) o.BEC <- RNAseqdata(BECdata4DRomics) # PCA plot after batch effect correction PCAdataplot(o.BEC, batch = zebraf$batch) + theme_bw() ``` The
PCA plot after BEC shows the impact of the correction on the batch effect that is no more visible on controls (red points). ## Step 2: selection of significantly responding items {#step2} For the
second step of the workflow, the function **itemselect()** must be used simply **taking as input in a first argument the output of the function used in [step 1](#step1)** (output of RNAseqdata(),
microarraydata(), continuousomicdata() or continuousanchoringdata()). Below is an example with microarray data. ```{r ch18} (s_quad <- itemselect(o.microarray, select.method = "quadratic", FDR =
0.01)) ``` The **false discovery rate (FDR) corresponds to the expected proportion of items that will be falsely detected as responsive**. **With a very large data set** it is important to define a
selection step based on an FDR not only **to reduce the number of items to be further processed**, but also **to remove too noisy dose-response signals that may impair the quality of the results**.
We recommend to set a value between 0.001 and 0.1 depending of the initial number of items. When this number is very high (more than several tens of thousands), we recommend a FDR less than 0.05
(0.001 to 0.01) to increase the robustness of the results ([Larras et al. 2018](https://hal.science/hal-02309919/document)). Concerning the method used for selection, **we recommend the default
choice ("quadratic") for a typical omics dose-response design (many doses/concentrations with few replicates per condition)**. It enables the **selection of both monotonic and biphasic dose-response
relationships**. If you want to focus on monotonic dose-response relationships, the "linear" method could be chosen. For a design with a small number of doses/concentrations and many replicates (not
optimal for dose-response modeling), the "ANOVA" method could be preferable. **For *in situ* data** (observational data without replicates due to uncontrolled dose), **only trend tests will be
proposed** as the use of an ANOVA test in absence of replicates for some conditions is not reasonable. Each of the three methods proposed for this selection step is based on the use of a simple model
(quadratic,linear or ANOVA-type) linking the signal to the dose in a rank scale. This model is internally fitted to data by an **empirical Bayesian approach** using the respective packages **[DESeq2]
(https://bioconductor.org/packages/release/bioc/html/DESeq2.html)** and **[limma](https://www.bioconductor.org/packages/release/bioc/html/limma.html)** for **RNAseq data** and **microarray or
continuous omics data**, and by classical **linear regression** using the lm() function for **continuous anchoring data**. The adjustment of p-values according to the specified FDR is performed in
any case, even for continuous anchoring data, so as to ensure the unicity of the workflow independently of the type of data. See ?itemselect for more details and [Larras et al. 2018](https://
hal.science/hal-02309919/document) comparison of the three proposed methods on an example. It is easy, using for example the package **VennDiagram**, to compare the selection of items obtained using
two different methods, as in the following example. ```{r ch19, results = "hide"} require(VennDiagram) s_lin <- itemselect(o.microarray, select.method = "linear", FDR = 0.01) index_quad <-
s_quad$selectindex index_lin <- s_lin$selectindex plot(c(0,0), c(1,1), type = "n", xaxt = "n", yaxt = "n", bty = "n", xlab = "", ylab = "") draw.pairwise.venn(area1 = length(index_quad), area2 =
length(index_lin), cross.area = length(which(index_quad %in% index_lin)), category = c("quadratic trend test", "linear trend test"), cat.col=c("cyan3", "darkorange1"), col=c("black", "black"), fill =
c("cyan3", "darkorange1"), lty = "blank", cat.pos = c(1,11)) ``` ## Step 3: fit of dose-response models, choice of the best fit for each curve {#step3} ### Fit of the best model For Step 3 the
function drcfit() simply **takes as input in a first argument the output of itemselect()**. For each item selected in Step 2, the model that best fits the dose-response data is chosen among a
**family of five simple models built to describe a wide variety of monotonic and biphasic dose-response curves (DRC)** (and exclusively monotonic and biphasic curves : it is why more flexible models
such as polynomial third and fourth order classical polynomial models were deliberately not considered). For a complete description of those models see the [last section of Step 3](#models) or
[Larras et al. 2018](https://hal.science/hal-02309919/document). The procedure used to **select the best fit** is based on an **information criterion** as described in [Larras et al. 2018](https://
hal.science/hal-02309919/document) and in ?drcfit. The **classical** and **former default option** of the **AIC** (Akaike criterion - default information criterion used in DRomics versions < 2.2-0)
was **replaced by the default use of the AICc** (second-order Akaike criterion) in order to **prevent the overfitting** that may occur with dose-response designs with a small number of data points,
as recommended and now classically done in regression ([Hurvich and Tsai, 1989](https://www.stat.berkeley.edu/~binyu/summer08/Hurvich.AICc.pdf); Burnham and Anderson DR, 2004). As the call to the
drcfit() function may take time when the number of pre-selected items is large, by default a progressbar is provided. Some arguments of this function can be used to specify **parallel computing to
accelerate the computation** (see ?drcfit for details). ```{r ch20} (f <- drcfit(s_quad, progressbar = FALSE)) ``` In the following you can see the first lines of the output data frame on our example
(see ?drcfit for a complete description of the columns of the output data frame.) This output data frame provides information for each item, such as **the best-fit model**, **its parameter values**,
the **standard residual error (SDres)** (see the section on [least squares](#leastsquares) for his definition), **coordinates of particular points**, and the **trend of the curve** (among increasing,
decreasing, U-shaped, bell-shaped). An extensive description of the outputs of the complete DRomics workflow is provided in the [last section of the main workflow](#outputs). Note that the number of
items successfully fitted (output of Step 3) is often smaller that the number of items selected in Step 2, as for some of the selected items, all models may fail to converge or fail to significantly
better describe the data than the constant model. ```{r ch21} head(f$fitres) ``` ### Plot of fitted curves By default the plot() function used on the output of the drcfit() function provides the
first 20 fitted curves (or the ones you specify using the argument items) with observed points. **Fitted curves are represented in red, replicates are represented in open circles and means of
replicates at each dose/concentration are represented by solid circles**. All the fitted curves may be saved in a pdf file using the plotfit2pdf() function (see ?drcfit). ```{r ch22} plot(f) ``` The
fitted curves are by default **represented using a log scale** for dose/concentration, which is more suited in common cases where the range of observed doses/concentrations is very wide and/or where
tested doses/concentrations are obtained by dilutions. It is why the observations at the control appear differently as the other observations, as half circles on the y-axis, to remind that their true
value is minus infinity in a log scale. Use `dose_log_transfo = FALSE` to keep the raw scale of doses (see below). Another **specific plot function named targetplot() can be used to plot targeted
items, whether they were or not selected** in step 2 and fitted in step 3. See an example below and details in ?targetplot. In this example, the default arbitrary space between the y-axis (so the
points at control) and the points at the first non null doses was enlarged by fixing the limits of the x-axis as below : ```{r ch23} targetitems <- c("88.1", "1", "3", "15") targetplot(targetitems, f
= f) + scale_x_log10(limits = c(0.2, 10)) ``` ### Plot of residuals {#residuals} **To check the assumption of the Gaussian error model** (see the [section on least squares](#leastsquares)), two types
of residual plots can be used, `"dose_residuals"` for plot of residuals against the observed doses/concentrations, or `"fitted_residuals"` for plot of residuals against fitted values of the modeled
signal. The residual plots for all items may also be saved in a pdf file using the plotfit2pdf() function (see ?drcfit). ```{r ch24, fig.width = 7, fig.height = 5} plot(f, plot.type =
"dose_residuals") ``` ### Description of the family of dose-response models fitted in DRomics {#models} The best fit model is chosen among the five following models describing the observed signal $y$
as a function of $x$ the dose (or concentration): * the **linear model**: $$y = d + b \times x$$ with **2 parameters**, $b$ the slope and $d$ the mean signal at control. * the **exponential model**:
$$y = d + b \times \left(exp\left(\frac{x}{e}\right)-1\right)$$ with **3 parameters**, $b$ a shape parameter, $d$ the mean signal at control, $e$ a shape parameter. + When $e>0$ the dose response
curve - DRC - is increasing if $b>0$ and decreasing if $b<0$, with no asymptote for high doses. + When $e<0$ the DRC is increasing if $b<0$ and decreasing if $b>0$, with an asymptote at $d-b$ for
high doses. * the **Hill model**: $$y = c + \frac{d-c}{1+(\frac{x}{e})^b}$$ with **4 parameters**, $b$ ($>0$) a shape parameter, $c$ the asymtotic signal for high doses, $d$ the mean signal at
control, and $e$ ($>0$) the dose at the inflection point of the sigmoid. * the **Gauss-probit model** built as the sum of a Gauss and a probit part sharing the same parameters as defined below:$$y =
f \times exp\left(-0.5 \left(\frac{x-e}{b}\right)^2\right) +d+(c-d) \times \Phi\left(\frac{x-e}{b}\right)$$ with **5 parameters**, $b$ ($>0$) a shape parameter corresponding to the standard deviation
of the Gauss part of the model, $c$ the asymtotic signal for high doses, $d$ the asymptotic signal on the left of the DRC (generally corresponding to a fictive negative dose), $e$ ($>0$) a shape
parameter corresponding to the mean of the Gauss part of the model, and $f$ the amplitude and sign of the Gauss part of the model (the model is U-shaped if $f<0$ and bell-shaped if $f<0$). $\Phi$
represents the cumulative distribution function (CDF) of the standard Gauss (also named normal or Gaussian) distribution. This model encompasses **two simplifed versions with 4 parameters**, one
**monotonic** (for $f=0$) and one with **symmetrical asymptotes** (for $c=d$). * the **log-Gauss-probit model**, a variant of the previous one with on the log scale of the dose: $$y = f \times exp\
left(-0.5\left(\frac{ln(x)-ln(e)}{b}\right)^2\right) +d+(c-d) \times \Phi\left(\frac{ln(x)-ln(e)}{b}\right)$$ with **5 parameters**, $b$ ($>0$) a shape parameter corresponding to the standard
deviation of the Gauss part of the model, $c$ the asymtotic signal for high dose, $d$ the asymptotic signal on the left of the DRC, reached at the control (for $ln(x) = ln(0) = -\infty$), $ln(e)$ ($>
0$) a shape parameter corresponding to the mean of the Gauss part of the model, and $f$ the amplitude and sign of the Gauss part of the model (the model is U-shaped if $f<0$ and bell-shaped if $f
<0$). $\Phi$ represents the cumulative distribution function (CDF) of the standard Gauss distribution. As the previous one, this model encompasses **two simplifed versions with 4 parameters**, one
**monotonic** (for $f=0$) and one with **symmetrical asymptotes** (for $c=d$). This family of five models was built to be able to describe a wide range of monotonic and biphasic DRC. In the following
plot were represented typologies of curves that can be described using those models, depending of the definition of their parameters. In the following plot the curves are represented with the signal
in y-axis and the raw dose in x-axis. **As the range of tested or observed doses is often large**, we decided to plot model fits **by default using a log scale of doses**. The **shape of models will
be transformed in this log x-scale**, and **especially the linear model will no more appear as a straight line as below**. ```{r ch25, echo = FALSE, results = "hide", fig.height=8, fig.width = 8} par
(mfrow = c(4,4), mar = c(0,0,0,0), xaxt = "n", yaxt = "n") x <- seq(0,10, length.out = 50) # linear plot(x, DRomics:::flin(x, b = 1, d = 1), type = "l", lwd = 2, col = "red") legend("topleft", legend
= "linear, b > 0", bty = "n") plot(x, DRomics:::flin(x, b = -1, d = 1), type = "l", lwd = 2, col = "red") legend("bottomleft", legend = "linear, b < 0", bty = "n") # expo plot(x, DRomics:::fExpo(x, b
= 1, d = 1, e = 3), type = "l", lwd = 2, col = "red") legend("topleft", legend = "exponential, e > 0 and b > 0", bty = "n") plot(x, DRomics:::fExpo(x, b = -1, d = 1, e = 3), type = "l", lwd = 2, col
= "red") legend("bottomleft", legend = "exponential, e > 0 and b < 0", bty = "n") plot(x, DRomics:::fExpo(x, b = 1, d = 1, e = -3), type = "l", lwd = 2, col = "red") legend("topright", legend =
"exponential, e < 0 and b > 0", bty = "n") plot(x, DRomics:::fExpo(x, b = -1, d = 1, e = -3), type = "l", lwd = 2, col = "red") legend("bottomright", legend = "exponential, e < 0 and b < 0", bty =
"n") # Hill plot(x, DRomics:::fHill(x, b = 10, c = 3, d = 1, e = 3), type = "l", lwd = 2, col = "red") legend("bottomright", legend = "Hill, c > d", bty = "n") plot(x, DRomics:::fHill(x, b = 10, c =
1, d = 3, e = 3), type = "l", lwd = 2, col = "red") legend("topright", legend = "Hill, c < d", bty = "n") # Gauss-probit plot(x, DRomics:::fGauss5p(x, b = 2, c = 3, d = 1, e = 3, f = 2), type = "l",
lwd = 2, col = "red") legend("bottomright", legend = "Gauss-probit, c > d, f > 0", bty = "n") plot(x, DRomics:::fGauss5p(x, b = 2, c = 1, d = 3, e = 3, f = 2), type = "l", lwd = 2, col = "red")
legend("topright", legend = "Gauss-probit, c < d, f > 0", bty = "n") plot(x, DRomics:::fGauss5p(x, b = 2, c = 3, d = 1, e = 3, f = -2), type = "l", lwd = 2, col = "red") legend("bottomright", legend
= "Gauss-probit, c > d, f < 0", bty = "n") plot(x, DRomics:::fGauss5p(x, b = 2, c = 1, d = 3, e = 3, f = -2), type = "l", lwd = 2, col = "red") legend("topright", legend = "Gauss-probit, c < d, f <
0", bty = "n") # LGauss-probit x <- seq(0,100, length.out = 50) plot(x, DRomics:::fLGauss5p(x, b = 0.5, c = 3, d = 1, e = 20, f = 4), type = "l", lwd = 2, col = "red") legend("bottomright", legend =
"log-Gauss-probit, c > d, f > 0", bty = "n") plot(x, DRomics:::fLGauss5p(x, b = 0.5, c = 1, d = 3, e = 20, f = 4), type = "l", lwd = 2, col = "red") legend("topright", legend = "log-Gauss-probit, c <
d, f > 0", bty = "n") plot(x, DRomics:::fLGauss5p(x, b = 0.5, c = 3, d = 1, e = 20, f = -4), type = "l", lwd = 2, col = "red") legend("bottomright", legend = "log-Gauss-probit, c > d, f < 0", bty =
"n") plot(x, DRomics:::fLGauss5p(x, b = 0.5, c = 1, d = 3, e = 20, f = -4), type = "l", lwd = 2, col = "red") legend("topright", legend = "log-Gauss-probit, c < d, f < 0", bty = "n") ``` ### Reminder
on least squares regression {#leastsquares} It is important when using DRomics to have in mind that the dose-response models are fitted using the **least squares regression**, assuming an **additive
Gaussian (normal) error model** for the observed signal. It is why the scale under which you should import your data is very important: a log (or pseudo-log) transformation may be necessary to meet
the use conditions of the model for some types of data. Let us recall the formulation of the Gaussian model defining the signal (after transformation if needed) $y$ as a function of the dose (or
concentration) $x$, $f$ being one of the five models previously described, and $\theta$ the vector of its parameters (of length 2 to 5). $$y = f(x, \theta) + \epsilon$$ with $$\epsilon \sim N(0, \
sigma)$$ $N(0, \sigma)$ representing the Gaussian (normal) distribution of mean $0$ and standard deviation (SD) $\sigma$. In this model, the **residual standard deviation $\sigma$ is assumed
constant**. It is the classical "homoscedasticity" hypothesis (see the following figure for an illustration). The examination of residuals (see the section on [plot of residuals](#residuals)) is a
good way to check that the error model is not strongly violated on your data. ```{r ch26, echo = FALSE, fig.height = 4, fig.width = 7, results = "hide", out.width="80%"} par(mar = c(0.1, 0.1, 0.1,
0.1)) datafilename <- system.file("extdata", "apical_anchoring.txt", package = "DRomics") o_ls <- continuousanchoringdata(datafilename, check = TRUE, backgrounddose = 0.1) s_ls <- itemselect(o_ls)
f_ls <- drcfit(s_ls) growth <- f_ls$fitres[1,] #plot(f) plot(o_ls$dose, o_ls$data[1,], xlab = "dose", ylab = "signal", pch = 16, xlim = c(0, 30), ylim = c(-20, 80)) xfin <- seq(0, 80, length.out =
100) #plot(x, x+100, ylim = c(0, 7), xlab = "dose", ylab = "signal") valb <- growth$b; valc <- growth$c; vald <- growth$d vale <- growth$e; valf <- growth$f ytheo <- DRomics:::fGauss5p(xfin, valb,
valc, vald, vale, valf) lines(xfin, ytheo, col = "red", lwd = 2) # Ajout de lois normales en vertical doseu <- sort(unique(o_ls$dose)) ytheou <- DRomics:::fGauss5p(doseu, valb, valc, vald, vale,
valf) sy <- growth$SDres npts <- 50 # nb de points par normale coefsurx <- 12 tracenormale <- function(indice) { x <- doseu[indice] my <- ytheou[indice] yplot <- seq(my - 2*sy, my+2*sy, length.out =
npts) xplot <- dnorm(yplot, mean = my, sd = sy) lines(coefsurx*xplot+x, yplot, col = "blue", lwd = 2) segments(x, my - 2*sy, x, my + 2*sy, lty = 3, col = "blue") } sapply(1:7, tracenormale) ``` ##
Step 4: calculation of benchmark doses (BMD) {#step4} ### Calculation of BMD The **two types of benchmark doses (BMD-zSD and BMD-xfold) proposed by the [EFSA (2017)](https://
efsa.onlinelibrary.wiley.com/doi/full/10.2903/j.efsa.2017.4658) are systematically calculated** for each fitted dose-response curve using the function **bmdcalc()** with the output of the drcfit()
function as a first argument, **but we strongly recommend the use of the first one (BMD-zSD)** for reasons explained in [Larras et al. 2018](https://hal.science/hal-02309919/document) (see ?bmdcalc
for details on this function). ```{r ch27} (r <- bmdcalc(f, z = 1, x = 10)) ``` For the **recommended BMD-zSD**,the argument $z$, by default at 1, is used to define the **BMD-zSD** as the dose at
which the response is reaching the **BMR (benchmark response)** defined as $$BMR = y_0 \pm z \times SD$$ with $y_0$ the level at the control given by the dose-response fitted model and $SD$ the
residual standard deviation of the dose-response fitted model (also named $\sigma$ in the [previous mathematical definition of the Gaussian model](#leastsquares)). For the less recommended BMD-xfold,
the argument $x$, by default at 10 (for 10%), is used to define the BMD-xfold as the dose at which the response is reaching the BMR defined as $BMR = y_0 \pm \frac{x}{100} \times y_0$. So this second
BMD version does not take into account the residual standard deviation, and is strongly dependent of the magnitude of $y_0$, which may be a problem if the signal at the control is close to 0, which
is not rare on omics data that are classically handled on log scale. In the following you can see the first lines of the output data frame of the function bmdcalc() on our example. BMD values are
coded `NA` when the BMR stands within the range of response values defined by the model but outside the range of tested doses, and `NaN` when the BMR stands outside the range of response values
defined by the model due to asymptotes. Very low BMD values obtained by extrapolation between 0 and the smallest non null tested dose, that correspond to very sensitive items (that we do not want to
exclude), are thresholded at minBMD, an argument by default fixed at the smallest non null tested dose divided by 100, but that can be fixed by the user as what he considers to be a negligible dose.
An extensive description of the outputs of the complete DRomics workflow is provided in the [last section of the main workflow](#outputs). You can also see ?bmdcalc for a complete description of its
arguments and of the columns of its output data frame. ```{r ch28} head(r$res) ``` ### Plots of the BMD distribution The default plot of the output of the bmdcalc() function provides the distribution
of benchmark doses as an ECDF (Empirical Cumulative Density Function) plot for the chosen BMD ("zSD"" or "xfold"). See an example below. ```{r ch29} plot(r, BMDtype = "zSD", plottype = "ecdf") +
theme_bw() ``` Different alternative plots are proposed (see ?bmdcalc for details) that can be obtained using the argument plottype to choose the type of plot ("ecdf", "hist" or "density") and the
argument by to split the plot for example by "trend". You can also use the bmdplot() function to make an ECDF plot of BMDs and personalize it (see ?bmdplot for details). On a BMD ECDF plot one can
add a **color gradient for each item** coding for the **intensity of the signal** (after shift of the control signal at 0) as a function of the dose (see ?bmdplotwithgradient for details and an
example below). It is generally necessary to use the argument line.size to manually adjust the width of lines in that plot as the default value does not always give a satisfactory resut. It is also
recommended (but not mandatory but it is the default option for the argument `scaling`) to scale the signal in order to focus on the shape of the dose-reponse curves and not on the amplitude of the
signal change. ```{r ch30} bmdplotwithgradient(r$res, BMDtype = "zSD", facetby = "trend", shapeby = "model", line.size = 1.2, scaling = TRUE) ``` ### Calculation of confidence intervals on the BMDs
by bootstrap {#bootstrap} **Confidence intervals on BMD values** can be calculated by **bootstrap**. As the call to this function may take much time, by default a progressbar is provided and some
arguments can be used to specify parallel computing to accelerate the computation (see ?bmdboot for details). In the example below, a small number of iterations was used just to make this vignette
quick to compile, but **the default value of the argument niter (1000) should be considered as a minimal value to obtain stable results**. ```{r ch31} (b <- bmdboot(r, niter = 50, progressbar =
FALSE)) ``` This function gives an output corresponding to the output of the bmdcalc() function completed with bounds of BMD confidence intervals (by default 95% confidence intervals) and the number
of bootstrap iterations for which the model was successfully fitted to the data. An extensive description of the outputs of the complete DRomics workflow is provided in the [last section of the main
workflow](#outputs). ```{r ch32} head(b$res) ``` The plot() function applied on the output of the bmdboot() function gives an ECDF plot of the chosen BMD with the confidence interval of each BMD (see
?bmdcalc for examples). By default BMDs with an infinite confidence interval bound are not plotted. ### Filtering BMDs according to estimation quality {#bmdfilter} Using the bmdfilter() function, it
is possible to use one of the three filters proposed to retain only the items associated to the best estimated BMD values. + By default are retained only the items for which the BMD and its
confidence interval are defined (using `"CIdefined"`) (so excluding items for which the bootstrap procedure failed). + One can be even more restrictive by retaining items only if the BMD confidence
interval is within the range of tested/observed doses (using `"CIfinite"`), + or less restrictive (using `"BMDdefined"`) requiring that the BMD point estimate only must be defined within the range of
tested/observed doses. Let us recall that in the `bmdcalc()` output, if it is not the case the BMD is coded `NA` or `NaN`. Below is an example of application of the different filters based on
BMD-xfold values, chosen just to better illustrate the way filters work, as there far more bad BMD-xfold estimations than bad BMD-zSD estimations. ```{r ch33, fig.height = 3} # Plot of BMDs with no
filtering subres <- bmdfilter(b$res, BMDfilter = "none") bmdplot(subres, BMDtype = "xfold", point.size = 2, point.alpha = 0.4, add.CI = TRUE, line.size = 0.4) + theme_bw() # Plot of items with
defined BMD point estimate subres <- bmdfilter(b$res, BMDtype = "xfold", BMDfilter = "definedBMD") bmdplot(subres, BMDtype = "xfold", point.size = 2, point.alpha = 0.4, add.CI = TRUE, line.size =
0.4) + theme_bw() # Plot of items with defined BMD point estimate and CI bounds subres <- bmdfilter(b$res, BMDtype = "xfold", BMDfilter = "definedCI") bmdplot(subres, BMDtype = "xfold", point.size =
2, point.alpha = 0.4, add.CI = TRUE, line.size = 0.4) + theme_bw() # Plot of items with finite BMD point estimate and CI bounds subres <- bmdfilter(b$res, BMDtype = "xfold", BMDfilter = "finiteCI")
bmdplot(subres, BMDtype = "xfold", point.size = 2, point.alpha = 0.4, add.CI = TRUE, line.size = 0.4) + theme_bw() ``` ### Plot of fitted curves with BMD values and confidence intervals It is
possible to add the output of bmdcalc() (or of bmdboot()) in the argument BMDoutput of the plot() function of drcfit(), in order to add BMD values (when defined) as a **vertical line on each fitted
curve**, and **bounds of the confidence intervals** (when successfully calculated) as **two dashed lines**. **Horizontal dotted lines corresponding to the two BMR potential values will be also
added**. See an example below. ```{r ch34} # If you do not want to add the confidence intervals just replace b # the output of bmdboot() by r the output of bmdcalc() plot(f, BMDoutput = b) ``` All
the fitted curves may also be saved in the same way in a pdf file using the plotfit2pdf() function (see ?drcfit). ### Plot of all the fitted curves in one figure with points at BMD-BMR values It is
possible to use the curvesplot() function to plot all the fitted curves in one figure and to add The use of the curvesplot() function is more extensively described in the next parts and in the
corresponding help page. By default in this plot the curves are scaled to focus on the shape of the dose-response and not on their amplitude (add `scaling = FALSE` to see the curves without scaling)
and a log dose scale is used. In the following plot we also added vertical lines corresponding to tested doses on the plot and add transparency to visualize the density of curves when shapes are
similar (especially the case for linear shapes). ```{r ch35} tested.doses <- unique(f$omicdata$dose) g <- curvesplot(r$res, xmax = max(tested.doses), colorby = "trend", line.size = 0.8, line.alpha =
0.3, point.size = 2, point.alpha = 0.6) + geom_vline(xintercept = tested.doses, linetype = 2) + theme_bw() print(g) ``` The use of the package plotly to make such a plot interactive can be
interesting for example to get the identifiant of each curve or to choose what group of curves to eliminate or to focus on. You can try the following code to get an interactive version of the
previous figure. ```{r ch36, eval = FALSE} if (require(plotly)) { ggplotly(g) } ``` ### Description of the outputs of the complete DRomics workflow {#outputs} The **output of the complete DRomics
workflow**, given in `b$res` with `b` being the output of bmdboot(), or the output of `bmdfilter(b$res)` (see [previous section](#bmdfilter) for description of BMD filtering options) is a **data
frame** reporting the **results of the fit and BMD computation on each selected item** sorted in the ascending order of the adjusted p-values returned by the item selection step. ```{r ch37} str
(b$res) ``` The columns of this data frame are: * id: the item identifier * irow: the row number in the initial dataset * adjpvalue: the adjusted p-values returned by the item selection step * model:
the best model fitted * nbpar: the number of parameters of this best model (that may be smaller than the maximal number of parameters of the model if a simplified version of it was chosen) * b, c, d,
e, and f, the model parameter values * **SDres**: the **residual standard deviation of the best model** * typology: the typology of the curve depending of the chosen model and of its parameter values
with 16 classes, + "H.inc" for increasing Hill curves + "H.dec" for decreasing Hill curves + "L.inc" for increasing linear curves + "L.dec" for decreasing linear curves + "E.inc.convex" for
increasing convex exponential curves + "E.dec.concave" for decreasing concave exponential curves + "E.inc.concave" for increasing concave exponential curves + "E.dec.convex" for decreasing convex
exponential curves + "GP.U" for U-shape Gauss-probit curves + "GP.bell" for bell-shape Gauss-probit curves + "GP.inc" for increasing Gauss-probit curves + "GP.dec" for decreasing Gauss-probit curves
+ "lGP.U" for U-shape log-Gauss-probit curves + "lGP.bell" for bell-shape log-Gauss-probit curves + "lGP.inc" for increasing log-Gauss-probit curves + "lGP.decreasing" for decreasing log-Gauss-probit
curves * **trend**: the rough trend of the curve defined with four classes, + **U shape** + **bell shape** + **increasing** + **decreasing** * y0: the y theoretical value at the control * yatdosemax:
the theoretical y value at the maximal dose * yrange: the theoretical y range for x within the range of tested doses * maxychange: the maximal absolute y change (up or down) from the control *
xextrem: for biphasic curves, x value at which their extremum is reached * yextrem: the corresponding y value at this extremum * **BMD.zSD**: the BMD-zSD value * BMR.zSD: the corresponding BMR-zSD
value * BMD.xfold: the BMD-xfold value * BMR.xfold: the corresponding BMR-xfold value * nboot.successful: the number of bootstrap iterations for which the model was successfully fitted to the data An
incomplete version of this data frame is also given at the end of Step 3 (in `f$fitres` with `f` this output of drcfit()) and before bootstrap calculation on BMD values (in `r$res` with `r` is the
output of bmdcalc()). # Help for biological interpretation of DRomics outputs {#interpreter} This section illustrates functions that were developed in DRomics to help the biological interpretation of
outputs. The idea is to **augment the output data frame with a new column bringing biological information**, generally provided by **biological annotation** of the items (e.g. kegg pathway classes or
GO terms), and then to **use this information to organize the visualisation** of the DRomics output. The shiny application DRomicsInterpreter-shiny can be used to implement all the steps described in
this vignette without coding them in R. But in any case, **the biological annotation of items selected in the first DRomics workflow must be previously done outside DRomics using a database such as
the Gene Ontology (GO) of the kegg databases.** In this section we will **first present [a simple example from a metabolomic dataset](#monolevel)** and then **[an example with two molecular levels]
(#multilevels) using metabolomic and transcriptomic data** from the same experiment, to illustrate how to **compare the responses at different experimental levels** (in this example different
molecular levels). The different experimental levels could also be different time points, different experimental settings, different species, ... ## Interpretation of DRomics results in a simple case
with only one data set obtained in one experimental condition {#monolevel} ### Augmentation of the data frame of DRomics results with biological annotation {#augmentation} This augmentation is not
done using DRomics functions, but using simple R functions such as merge(). Nevertheless it is possible to perform this augmentation without coding in R, using the **shiny application
DRomicsInterpreter-shiny**. Report to the [introduction section](#introduction) to see how to launch the shiny application. Here is an example of how to proceed: 1. **Import the data frame with
DRomics results: the output `$res` of bmdcalc() or bmdboot() functions from Step 4 of the main DRomics workflow.** This step is not be necessary if previous steps were done directly in R, using the
DRomics package, as described previously in this vignette (see the [section describing this output data frame](#outputs)). We did it in this example, in order to take a real example that took a long
time to completely run, but from which results are stored in the package. ```{r ch38} # code to import the file for this example stored in our package resfilename <- system.file("extdata",
"triclosanSVmetabres.txt", package = "DRomics") res <- read.table(resfilename, header = TRUE, stringsAsFactors = TRUE) # to see the first lines of this data frame head(res) ``` 2. **Import the data
frame with biological annotation** (or any other descriptor/category you want to use), here KEGG pathway classes of each item present in the 'res' file. Examples are embedded in the DRomics package,
but be cautious, generally this file must be produced by the user. Each item may have more than one annotation (*i.e.* more than one line). If **items were annotated whatever they were selected by
the DRomics workflow or not**, you should **previously reduce the dimension of your annotation file by selecting only the items present in the DRomics output and that have at least one biological
annotation**. If each annotation stands in more than one word, you should surround each of them by quotes, or use tab as a column separator in your annotation file, and import it by adding `sep = "\
t"` in the arguments of `read.table()`. ```{r ch39} # code to import the file for this example in our package annotfilename <- system.file("extdata", "triclosanSVmetabannot.txt", package = "DRomics")
# annotfilename <- "yourchosenname.txt" # for a local file annot <- read.table(annotfilename, header = TRUE, stringsAsFactors = TRUE) # to see the first lines of this data frame head(annot) ``` 3.
**Merging of both previous data frames in order to obtain a so-called 'extendedres' data frame gathering, for each item, metrics derived from the DRomics workflow and biological annotation.**
Arguments by.x and by.y of the merge() function indicate the column name in res and annot data frames respectively, that must be used for the merging. ```{r ch40} # Merging extendedres <- merge(x =
res, y = annot, by.x = "id", by.y = "metab.code") # to see the first lines of the merged data frame head(extendedres) ``` ### Various plots of results by biological group #### BMD ECDF plots split by
group defined from biological annotation {#bmdplot} Using the function bmdplot() and its argument `facetby`, the BMD ECDF plot can be split by group (here by KEGG pathway class). Confidence intervals
can be added on this plot and color coding for trend in this example (See ?bmdplot for more options). ```{r ch41} bmdplot(extendedres, BMDtype = "zSD", add.CI = TRUE, facetby = "path_class", colorby
= "trend") + theme_bw() ``` The function ecdfplotwithCI() can also be used as an alternative as below to provide the same plot differing by the coloring of intervals only. (See ?ecdfplotwithCI for
more options.) ```{r ch42, eval = FALSE} ecdfplotwithCI(variable = extendedres$BMD.zSD, CI.lower = extendedres$BMD.zSD.lower, CI.upper = extendedres$BMD.zSD.upper, by = extendedres$path_class, CI.col
= extendedres$trend) + labs(col = "trend") ``` Using the function bmdplotwithgradient() and its argument `facetby`, the BMD plot with color gradient can be split here by KEGG pathway class. (See ?
bmdplotwithgradient for more options). ```{r ch43} bmdplotwithgradient(extendedres, BMDtype = "zSD", scaling = TRUE, facetby = "path_class", shapeby = "trend") ``` One can **focus on a group of
interest**, for instance the group "Lipid metabolism", and add the **labels of items** using argument `add.label` as below to display item identifiers instead of points. In that case in can be useful
to control the limits of the color gradient and the limits on the x-axis in order to use the same x-scale and signal-scale as in the global previous plot, as in the following example (see ?
bmdplotwithgradient for details). ```{r ch44} extendedres_lipid <- extendedres[extendedres$path_class == "Lipid metabolism",] bmdplotwithgradient(extendedres_lipid, BMDtype = "zSD", scaling = TRUE,
facetby = "path_class", add.label = TRUE, xmin = 0, xmax = 6, label.size = 3, line.size = 2) ``` #### Sensitivity plot of biological groups {#sensitivityplot} It is also possible to visualize the
sensitivity of **each biological group** using the sensitivityplot() function, choosing a **BMD summary** with the argument `BMDsummary` fixed at `"first.quartile"`, `"median"` or `"median.and.IQR"`
(for medians with the interquartile range as an interval). Moreover, this function will provide **information on the number of items** involved in each pathway/category (coding for the size of the
points). (See ?sensitivityplot for more options). As an example, below is an ECDF plot of 25th quantiles of BMD-zSD calculated here by pathway class. ```{r ch45} sensitivityplot(extendedres, BMDtype
= "zSD", group = "path_class", BMDsummary = "first.quartile") + theme_bw() ``` It is possible to use medians of BMD values represent and order the groups in a sensitivity plot and optionally to add
the interquartile range as a line on the plot, as below: ```{r ch46} sensitivityplot(extendedres, BMDtype = "zSD", group = "path_class", BMDsummary = "median.and.IQR") + theme_bw() ``` You can
customize the sensitivity plot and position the pathway class labels next to the point instead of on the y-axis, using ggplot2 functions. ```{r ch47} psens <- sensitivityplot(extendedres, BMDtype =
"zSD", group = "path_class", BMDsummary = "first.quartile") psens + theme_bw() + theme(axis.text.y = element_blank(), axis.ticks.y = element_blank()) + geom_text(aes(label = paste(" ",
psens$data$groupby, " ")), size = 3, hjust = "inward") ``` #### Trend plot per biological group {#trendplot} It is possible to represent the **repartition of trends in each biological group** using
the trendplot() function (see ?trendplot for details). ```{r ch48} trendplot(extendedres, group = "path_class") + theme_bw() ``` #### Plot of dose-response curves per biological group {#curvesplot}
The function curvesplot() can show the dose-response curves obtained for different groups (or one chosen group). As for the use of bmdplotwithgradient(), the **scaling of those curves can be used is
by default used to focus on the shape of them, and not on the amplitude of the signal change**. To use this function you have to define the dose range on which you want the computation of the
dose-response fitted curves, and **we strongly recommend you to choose a range corresponding to the range of tested/observed doses** in your dataset. Below is a code to plot the dose-response curves
split by biological group (argument facetby) and colored by trend (argument colorby). As below it is also possible to add a point at BMD-BMR values on each curve (See ?curvesplot for more options).
```{r ch49} # Plot of all the scaled dose-reponse curves split by path class curvesplot(extendedres, facetby = "path_class", scaling = TRUE, npoints = 100, colorby = "trend", xmax = 6.5) + theme_bw()
``` It is also possible using this function to visualize the modeled response of each item of one biological group, as below: ```{r ch50} # Plot of the unscaled dose-reponses for one chosen group,
split by metabolite LMres <- extendedres[extendedres$path_class == "Lipid metabolism", ] curvesplot(LMres, facetby = "id", npoints = 100, point.size = 1.5, line.size = 1, colorby = "trend", scaling =
FALSE, xmax = 6.5) + theme_bw() ``` ## Comparison of DRomics results obtained at different experimental levels, for example in a multi-omics approach {#multilevels} This section illustrates how to
use DRomics functions to help the interpretation of outputs from different data sets obtained at **different experimental levels (different molecular levels, different time points, different
experimental settings, ...)**. The idea is + to perform the augmentation of the DRomics output data frame obtained at each experimental level ([as previously described for one level](#augmentation)),
+ to bind the augmented data frames and add a column coding for the experimental level and + to use this column to organize the visualisation of the DRomics output and so make possible the
**comparison of the responses between experimental levels**. Below is used **an example corresponding to a multi-omics approach**, the experimental level corresponding to the molecular level, with a
transcriptomic (microarray) and a metabolomic data set issued from the same experiment. This example uses metabolomics and transcriptomics data for *Scenedesmus* and triclosan published by [Larras et
al. in 2020](https://doi.org/10.1016/j.jhazmat.2020.122727). It is possible to perform it without R coding within the **shiny application DRomicsInterpreter-shiny**. Report to the [introduction
section](#introduction) to see how to launch the shiny application. ### Augmentation of the data frames of DRomics results with biological annotation Following the same steps as described before [for
one level](#augmentation), below is an example of R code to **import the DRomics results for microarray data**, and to **merge them with the data frame giving on biological annotation of selected
items**. ```{r ch51} # 1. Import the data frame with DRomics results to be used contigresfilename <- system.file("extdata", "triclosanSVcontigres.txt", package = "DRomics") contigres <- read.table
(contigresfilename, header = TRUE, stringsAsFactors = TRUE) # 2. Import the data frame with biological annotation (or any other descriptor/category # you want to use, here KEGG pathway classes)
contigannotfilename <- system.file("extdata", "triclosanSVcontigannot.txt", package = "DRomics") # contigannotfilename <- "yourchosenname.txt" # for a local file contigannot <- read.table
(contigannotfilename, header = TRUE, stringsAsFactors = TRUE) # 3. Merging of both previous data frames contigextendedres <- merge(x = contigres, y = contigannot, by.x = "id", by.y = "contig") # to
see the first lines of the data frame head(contigextendedres) ``` The [previouly created](#augmentation) metabolomics data frame (extended results with biological annotation) is renamed for the sake
of homogeneity. ```{r ch52} metabextendedres <- extendedres ``` ### Binding of the data frames corresponding the results at each experimental level {#binding} The next step is the **bind of the
augmented data frames** of results obtained at the different levels (here transcriptomics and metabolomics data frames) and the **add of a variable** (here named level) **coding for the level** (here
a factor with two levels, metabolites and contigs). ```{r ch53} extendedres <- rbind(metabextendedres, contigextendedres) extendedres$explevel <- factor(c(rep("metabolites", nrow(metabextendedres)),
rep("contigs", nrow(contigextendedres)))) # to see the first lines of the data frame head(extendedres) ``` ### Comparison of results obtained at the different experimental levels using basic R
functions {#comparisonR} Below are examples of illustrations that can be made using basic R functions to globally compare the results obtained at the different experimental levels, for example to
compute and plot frequencies of pathways by molecular levels as below. ```{r ch54} (t.pathways <- table(extendedres$path_class, extendedres$explevel)) original.par <- par() par(las = 2, mar = c
(4,13,1,1)) barplot(t(t.pathways), beside = TRUE, horiz = TRUE, cex.names = 0.7, legend.text = TRUE, main = "Frequencies of pathways") par(original.par) ``` To do the same plot in proportions, just
apply the function prop.table() to the table of frequencies `t.pathways`. Here the ggplot2 grammar is used to plot the ECDF of BMD_zSD using different colors for the different molecular levels, after
removing the redundant lines corresponding to items corresponding to more than one pathway. ```{r ch55} unique.items <- unique(extendedres$id) ggplot(extendedres[match(unique.items, extendedres$id),
], aes(x = BMD.zSD, color = explevel)) + stat_ecdf(geom = "step") + ylab("ECDF") + theme_bw() ``` ### Comparison of results obtained at the different experimental levels using DRomics functions {#
comparisonDRomics} #### ECDF plot of BMD values per group and experimental level using DRomics functions Using the function bmdplot() the ECDF plot of the BMD-zSD values can be colored or split by
experimental level and/or split by group (here by KEGG pathway class) as below. (See ?bmdplot for more options, for example to add confidence intervals, ..., as in [the previous section presenting
bmdplot()](#bmdplot)). ```{r ch56} # BMD ECDF plot split by molecular level, after removing items redundancy bmdplot(extendedres[match(unique.items, extendedres$id), ], BMDtype = "zSD", facetby =
"explevel", point.alpha = 0.4) + theme_bw() # BMD ECDF plot colored by molecular level and split by path class bmdplot(extendedres, BMDtype = "zSD", facetby = "path_class", colorby = "explevel",
point.alpha = 0.4) + labs(col = "molecular level") + theme_bw() ``` #### Plot of the trend repartition per group and experimental level Using the function trendplot() and its arguments `facetby` it
is possible to show the repartition of trends of responses in each biological group for all experimental levels. ```{r ch57} # Preliminary optional alphabetic ordering of path_class groups
extendedres$path_class <- factor(extendedres$path_class, levels = sort(levels(extendedres$path_class), decreasing = TRUE)) # Trend plot trendplot(extendedres, group = "path_class", facetby =
"explevel") + theme_bw() ``` #### Sensitivity plot per group and experimental level Using the function sensitivityplot() and its arguments `group` and `colorby`, it is possible to show a summary of
BMD values with size of points coding for the number of items in each group as in the example, where the 25th quartiles of BMD values are represented per KEGG pathway class for each molecular level.
(See ?sensitivityplot for more options). ```{r ch58} sensitivityplot(extendedres, BMDtype = "zSD", group = "path_class", colorby = "explevel", BMDsummary = "first.quartile") + theme_bw() ``` ####
Selection of groups on which to focus using the selectgroups() function {#selectgroups} When the number of biological groups obtained after annotation of items is too high, it may be useful to
**select groups on which to focus**, to **enhance the visibility of plots**. This can be done for example using results of enrichment procedures in the case where enrichment is possible (e.g. for
sequenced organisms). One could also use selection criteria based **on the number of items in each biological group** (argument `nitems`, to select the **most represented groups**, represented by
more than `nitems`) and/or **on the BMD summary value** (argument `BMDmax`, to select the **most sensitive groups**, so those below `BMDmax`). The selectgroups() function can be used for this purpose
as in the example below (see ?selectgroups for details). When using this function you may optionally choose to keep the results of all the experimental levels (for comparison purpose) as soon as the
criteria are met for the group for at least one experimental level (as in the example below fixing the argument `keepallexplev` at `TRUE`). ```{r ch59} selectedres <- selectgroups(extendedres, group
= "path_class", explev = "explevel", BMDmax = 0.75, BMDtype = "zSD", BMDsummary = "first.quartile", nitems = 3, keepallexplev = TRUE) # BMDplot on this selection bmdplot(selectedres, BMDtype = "zSD",
add.CI = TRUE, facetby = "path_class", facetby2 = "explevel", colorby = "trend") + theme_bw() ``` #### BMD ECDF plot with color gradient split by group and experimental level Using the function
bmdplotwithgradient() and its arguments `facetby` and `facetby2`, the BMD plot with color gradient can be split here by group and experimental level, as on the example below on a manual selection of
pathway classes present at both molecular levels.(See ?bmdplotwithgradient for more options). ```{r ch60} # Manual selection of groups on which to focus chosen_path_class <- c("Nucleotide
metabolism", "Membrane transport", "Lipid metabolism", "Energy metabolism") selectedres2 <- extendedres[extendedres$path_class %in% chosen_path_class, ] bmdplotwithgradient(selectedres2, BMDtype =
"zSD", scaling = TRUE, facetby = "path_class", facetby2 = "explevel") ``` Especially as **metabolomic data and transcriptomic data were not imported in DRomics in the same scale** (in log2 for
transcriptomics and log10 for metabolomics), the **use of the scaling option of each dose-response curve is interesting here**. This option focuses on the shape of responses, skipping the amplitude
of changes to the control. #### Plot of the dose-response curves for a selection of groups Using the function curvesplot(), specific dose-response curves can be explored. In the following example,
only results related to the "lipid metabolism" pathway class are explored, using the argument `facetby` to split by experimental level. In the second example, the plot is split by biological group
using the argument `facetby` and by experimental level using the argument `facetby2` . (See ?curvesplot for more options). ```{r ch61} # Plot of the unscaled dose-response curves for the "lipid
metabolism" path class # using transparency to get an idea of density of curves with the shame shape LMres <- extendedres[extendedres$path_class == "Lipid metabolism", ] curvesplot(LMres, facetby =
"explevel", free.y.scales = TRUE, npoints = 100, line.alpha = 0.4, line.size = 1, colorby = "trend", xmax = 6.5) + labs(col = "DR trend") + theme_bw() # Plot of the scaled dose-response curves for
previously chosen path classes curvesplot(selectedres2, scaling = TRUE, facetby = "path_class", facetby2 = "explevel", npoints = 100, line.size = 1, line.alpha = 0.4, colorby = "trend", xmax = 6.5) +
labs(col = "DR trend") + theme_bw() ``` The scaling of the curves only used here in the second plot can be interesting to focus on the shapes of those curves, skipping the amplitude of the changes
from the control. This helps to evaluate the homogeneity of the shapes of the responses within each group. It may for example interesting to observe in this example, that some transcriptomics
responses (contigs) are gathering the same shape (when we use the scaling option - it is done by default) just differing by their sign (increasing / decreasing, or U-shape/bell-shape), which does not
clearly appear when dose-response curves are not scaled (as in the first plot). # References + Burnham, KP, Anderson DR (2004). Multimodel inference: understanding AIC and BIC in model selection.
Sociological methods & research, 33(2), 261-304. + Delignette-Muller ML, Siberchicot A, Larras F, Billoir E (2023). DRomics, a workflow to exploit dose-response omics data in ecotoxicology. Peer
Community Journal. doi : 10.24072/pcjournal.325. https://peercommunityjournal.org/articles/10.24072/pcjournal.325/ + EFSA Scientific Committee, Hardy A, Benford D, Halldorsson T, Jeger MJ, Knutsen
KH, ... & Schlatter JR (2017). Update: use of the benchmark dose approach in risk assessment. EFSA Journal, 15(1), e04658.[https://efsa.onlinelibrary.wiley.com/doi/full/10.2903/j.efsa.2017.4658]
(https://efsa.onlinelibrary.wiley.com/doi/full/10.2903/j.efsa.2017.4658) + Hurvich, CM, Tsai, CL (1989). Regression and time series model selection in small samples. Biometrika, 76(2), 297-307.
[https://www.stat.berkeley.edu/~binyu/summer08/Hurvich.AICc.pdf](https://www.stat.berkeley.edu/~binyu/summer08/Hurvich.AICc.pdf) + Larras F, Billoir E, Baillard V, Siberchicot A, Scholz S, Wubet T,
Tarkka M, Schmitt-Jansen M and Delignette-Muller ML (2018). DRomics : a turnkey tool to support the use of the dose-response framework for omics data in ecological risk assessment. Environmental
Science & Technology. [https://pubs.acs.org/doi/10.1021/acs.est.8b04752](https://pubs.acs.org/doi/10.1021/acs.est.8b04752). You can also find this article at : [https://hal.science/hal-02309919]
(https://hal.science/hal-02309919) + Larras F, Billoir E, Scholz S, Tarkka M, Wubet T, Delignette-Muller ML, Schmitt-Jansen M (2020). A multi-omics concentration-response framework uncovers novel
understanding of triclosan effects in the chlorophyte Scenedesmus vacuolatus. Journal of Hazardous Materials. [https://doi.org/10.1016/j.jhazmat.2020.122727](https://doi.org/10.1016/ | {"url":"https://cran.dcc.uchile.cl/web/packages/DRomics/vignettes/DRomics_vignette.Rmd","timestamp":"2024-11-07T06:47:36Z","content_type":"text/plain","content_length":"83771","record_id":"<urn:uuid:c694a0fd-2a6c-4f55-8e0d-7b2fd1a7c388>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00814.warc.gz"} |
Find The Measure Of Missing Angles - Angleworksheets.com
Finding The Measure Of Missing Angles Worksheets – If you have been struggling to learn how to find angles, there is no need to worry as there are many resources available for you to use. These
worksheets will help you understand the different concepts and build your understanding of these angles. Students will be able … Read more | {"url":"https://www.angleworksheets.com/tag/find-the-measure-of-missing-angles/","timestamp":"2024-11-15T02:49:17Z","content_type":"text/html","content_length":"46482","record_id":"<urn:uuid:1f7f494b-2fe9-4e47-a7b2-0951b1a0dc77>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00841.warc.gz"} |
Combining Imaginary Numbers Worksheet 2024 - NumbersWorksheets.com
Combining Imaginary Numbers Worksheet
Combining Imaginary Numbers Worksheet – A Realistic Amounts Worksheet can help your son or daughter be a little more informed about the methods powering this ratio of integers. In this particular
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numbers. Irrational figures are non-absolutely no, no-terminating decimals, and sq beginnings which are not best squares. They are often used in math applications, even though these types of numbers
are not used often in everyday life.
To define a rational number, you need to realize just what a rational number is. An integer is actually a entire amount, along with a rational variety is a proportion of two integers. The rate of two
integers may be the quantity on top split through the quantity at the base. If two integers are two and five, this would be an integer, for example. There are also many floating point numbers, such
as pi, which cannot be expressed as a fraction.
They are often created in a small fraction
A rational variety features a denominator and numerator that are not no. Consequently they are often depicted as a fraction. In addition to their integer numerators and denominators, logical numbers
can furthermore have a unfavorable importance. The bad worth must be placed left of and its particular complete importance is its distance from absolutely no. To simplify this illustration, we will
claim that .0333333 is actually a fraction that may be written as a 1/3.
As well as unfavorable integers, a reasonable variety can even be made into a small fraction. As an example, /18,572 is really a rational variety, when -1/ is not. Any small percentage made up of
integers is rational, so long as the denominator fails to contain a and can be written for an integer. Furthermore, a decimal that leads to a level is yet another rational amount.
They are sense
Even with their title, reasonable numbers don’t make very much sense. In mathematics, they may be solitary entities by using a exclusive duration around the quantity range. This means that when we
matter anything, we can easily get the size and style by its proportion to its original volume. This retains correct even if you can find unlimited reasonable numbers among two certain amounts. In
other words, numbers should make sense only if they are ordered. So, if you’re counting the length of an ant’s tail, a square root of pi is an integer.
If we want to know the length of a string of pearls, we can use a rational number, in real life. To get the time period of a pearl, as an example, we could add up its size. Just one pearl weighs ten
kilos, which is actually a realistic number. Additionally, a pound’s body weight is equal to twenty kilograms. Hence, we will be able to divide a pound by ten, with out be worried about the duration
of just one pearl.
They are often expressed like a decimal
If you’ve ever tried to convert a number to its decimal form, you’ve most likely seen a problem that involves a repeated fraction. A decimal amount can be written as a multiple of two integers, so
four times 5 various is the same as 8. A similar dilemma requires the repetitive fraction 2/1, and each side should be divided by 99 to have the appropriate response. But how do you make your
conversion process? Here are a few good examples.
A rational amount will also be developed in various forms, such as fractions and a decimal. A great way to represent a realistic number within a decimal is always to split it into its fractional
equivalent. There are 3 ways to divide a logical amount, and all these techniques results in its decimal counterpart. One of these brilliant methods is to break down it into its fractional equal, and
that’s what’s referred to as a terminating decimal.
Gallery of Combining Imaginary Numbers Worksheet
Quadratic Formulas With Imaginary Numbers Worksheet
Simplifying Complex Numbers Worksheet
Imaginary Numbers Worksheet
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The angles A,B and C of a △ABC are in AP and a:b=1:3. If c=4 c... | Filo
The angles and of a are in AP and . If , then the area (in ) of this triangle is
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Exp. (c)
It is given that angles of a are in . So,
[if and are in AP, then it taken as respectively, where is common difference of AP]
and [given]
by sine rule
From sine rule,
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Question Text The angles and of a are in AP and . If , then the area (in ) of this triangle is
Updated On Dec 5, 2022
Topic Sequences and Series
Subject Mathematics
Class Class 11
Answer Type Text solution:1 Video solution: 2
Upvotes 221
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3D Space
Introduction to Computer-aided Architectural Design 3D
To mail questions or comments please click here
3D Space - Concepts of modeling
1. The Concept of Three-dimensional Space
The concept of three-dimensional space is very old and very simple. By observation one can notice that the world around us is comprehensable by simply watching at it. Although there is a material
existence of objects (molecules, atoms, etc.) nonetheless it is enough to use our eyes to locate objects. We do not have to physically touch objects to locate them. The eye is a piece of glass where
rays from the real world are projected. Therefore we can distinguish two systems: a three-dimensional world, and a projection system. Historically, Euclid was able to specify the existence of such a
world, Descartes was able to specify a system of three-directions to locate objects within such a wolrd, and others were able to develop techniques to project that world on a canvas.
With the development of computer technology it became interesting and challenging to simulate the way humans see the world. The computer screen would be the surface of the eye, the 3-D world would be
defined as x,y,z numbers in the computer's memory, and the projection calculations would be done with the computer processor. Lets take an example: a cube has eight points, and each point has three
coordinates (x,y,z), therefore we need 8x3=24 numbers to describe the position of all points of the cube. Using simple mathematics we can calculate the projections of each point on the computer
screen and then draw the projected cube on the screen. If we change the coordinates (move, rotate, or scale the cube) we project the points from their new position on the screen.
In the following figures you can see the projection of an object with parallel lines and converging lines. The first projection system is called orthographic or parallel and the second is called
Lets go back to simple geometry. An object can be composed of points and connecting lines. A square is four points and four connections. A cube is eight points and 12 connecting lines (we can also
refer to the points as the geometry of the object and the connecting lines as the topology of the object). In the following example there are 10 points and 7 connecting line lists each for every
face. This is essentially how an object is represented internally as lists of numbers in the computer's memory. In addition to this information one can add attributes such as color, material, etc.
Three-dimensional objects can be created in many different ways:
• Extrusion. Here a 2D shape is extruded into a 3D object either by parallel lines of extrusion (such as a circle into a cylinder) or the extruded lines meet at a point (such as a circle into a
• Revolution. Here a 2D shape is revolved around an axis creating a 3D object of revolution (such as a hemicircle into a sphere)
• Result of set operations. Here an object is the result of union, intersection, and difference operations of elementary objects (such as cubes, spheres, cylinders, cones, etc.). We call these
elementary objects primitives. Set operations will be discussed later in these notes.
• Clouds of points and meshes. Here an object is represented as a set of points. When these points represent a surface in space and are connected as a grid we call it a mesh.
As mentioned earlier, three-dimensional objects are stored in the computer's memory as numbers representing points and connections. Then these points are projected to the surface of the screen. Any
change in the geometry (that is, the location of the points) is called a transformation. In the figure below each point of the object was multiplied by a series of sines and cosines resulting in
creating the impressionof rotation.
We distinguish many transformations, but the most important are:
Translation is another word for moving and object to a new position. It also sometimes called offsetting. Basically, it is the change of location of the object in space. By adding a constant number
(i.e. 5 units) to the x coordinate of all points the object is moved in the x direction (by 5 units).
A rotation specifies a pivoting, or angular displacement about an axis. Objects are rotated by specifying an angle of rotation and a pivot point. Then trigonometric functions are used to determine
the new position.
Scale (or sizing) either reduces or enlarges an object. Objects are scaled by specifying a percentage of scaling, the direction, and a pivot point. Then simple multiplication of each coordinate
determines the new position.
Set operations are logical combinations of objects. They behave as if the objects have mass. Imagine two object that are composed of molecules. The AND operator defines a new object that combines all
the molecules of object A AND object B. To be more specific:
For two objects A and B the union operator defines a new object that combines all the molecules of object A AND object B.
For two objects A and B the intersection operator defines a new object that combines all the molecules that are common to object A and B.
• Case 1 (A-B): For two objects A and B the difference (A-B) operator defines a new object that combines all the molecules that are in object A minus those in object B.
• Case 1 (B-A): For two objects A and B the difference (B-A) operator defines a new object that combines all the molecules that are in object B minus those in object A.
• Friedhoff, R., "Visualization", New York: Freeman and Co., 1989.
• Kerlow, I. and Rosebush, J., "Computer Graphics", New York: Van Nostrand Reinhold, 1994.
• Mitchell, W., "Computer-Aided Architectural Design", New York: Van Nostrand Reinhold, 1977. | {"url":"http://oldcda.design.ucla.edu/CAAD/Class_Notes/3DSpace_Folder/3DSpace.html","timestamp":"2024-11-09T04:48:37Z","content_type":"text/html","content_length":"8732","record_id":"<urn:uuid:95255797-6970-4a66-a6ce-7c30903bde0b>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00708.warc.gz"} |
Math Contest Repository
Euclid 2024 Question 2, CEMC UWaterloo
(Euclid 2024, Question 2, CEMC - UWaterloo)
(a) In a sequence with six terms, each term after the second is the sum of the previous two terms. If the fourth term is $13$ and the sixth term is $36$, what is the first term?
(b) For some real number $r \neq 0$, the sequence $5r$, $5r^2$, $5r^3$ has the property that the second term plus the third term equals the square of the first term. What is the value of $r$?
(c) Jimmy wrote four tests last week. The average of his marks on the first, second and third tests was $65$. The average of his marks on the second, third and fourth tests was $80$. His mark on the
fourth test was $2$ times his mark on the first test. Determine his mark on the fourth test.
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SolverCG (SolverControl &cn, VectorMemory< VectorType > &mem, const AdditionalData &data=AdditionalData())
SolverCG (SolverControl &cn, const AdditionalData &data=AdditionalData())
virtual ~SolverCG () override=default
template<typename MatrixType , typename PreconditionerType >
void solve (const MatrixType &A, VectorType &x, const VectorType &b, const PreconditionerType &preconditioner)
boost::signals2::connection connect_coefficients_slot (const std::function< void(typename VectorType::value_type, typename VectorType::value_type)> &slot)
boost::signals2::connection connect_condition_number_slot (const std::function< void(double)> &slot, const bool every_iteration=false)
boost::signals2::connection connect_eigenvalues_slot (const std::function< void(const std::vector< double > &)> &slot, const bool every_iteration=false)
boost::signals2::connection connect (const std::function< SolverControl::State(const unsigned int iteration, const double check_value, const VectorType ¤t_iterate)> &slot)
template<class Archive >
void serialize (Archive &ar, const unsigned int version)
Classes derived from EnableObserverPointer provide a facility to subscribe to this object. This is mostly used by the ObserverPointer class.
void subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const
void unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const
unsigned int n_subscriptions () const
template<typename StreamType >
void list_subscribers (StreamType &stream) const
void list_subscribers () const
template<typename VectorType = Vector<double>>
class SolverCG< VectorType >
This class implements the preconditioned Conjugate Gradients (CG) method that can be used to solve linear systems with a symmetric positive definite matrix. This class is used first in step-3 and
step-4, but is used in many other tutorial programs as well. Like all other solver classes, it can work on any kind of vector and matrix as long as they satisfy certain requirements (for the
requirements on matrices and vectors in order to work with this class, see the documentation of the Solver base class). The type of the solution vector must be passed as template argument, and
defaults to Vector<double>. The AdditionalData structure allows to control the type of residual for the stopping condition.
The CG method requires a symmetric preconditioner (i.e., for example, SOR is not a possible choice). There is a variant of the solver, SolverFlexibleCG, that allows to use a variable
preconditioner or a preconditioner with some slight non-symmetry (like weighted Schwarz methods), by using a different formula for the step length in the computation of the next search direction.
Eigenvalue computation
The cg-method performs an orthogonal projection of the original preconditioned linear system to another system of smaller dimension. Furthermore, the projected matrix T is tri-diagonal. Since the
projection is orthogonal, the eigenvalues of T approximate those of the original preconditioned matrix PA. In fact, after n steps, where n is the dimension of the original system, the eigenvalues of
both matrices are equal. But, even for small numbers of iteration steps, the condition number of T is a good estimate for the one of PA.
After m steps the matrix T_m can be written in terms of the coefficients alpha and beta as the tri-diagonal matrix with diagonal elements 1/alpha_0, 1/alpha_1 + beta_0/alpha_0, ..., 1/alpha_{m-1}
+beta_{m-2}/alpha_{m-2} and off-diagonal elements sqrt(beta_0)/alpha_0, ..., sqrt(beta_{m-2})/alpha_{m-2}. The eigenvalues of this matrix can be computed by postprocessing.
See also
Y. Saad: "Iterative methods for Sparse Linear Systems", section 6.7.3 for details.
The coefficients, eigenvalues and condition number (computed as the ratio of the largest over smallest eigenvalue) can be obtained by connecting a function as a slot to the solver using one of the
functions connect_coefficients_slot, connect_eigenvalues_slot and connect_condition_number_slot. These slots will then be called from the solver with the estimates as argument.
Observing the progress of linear solver iterations
The solve() function of this class uses the mechanism described in the Solver base class to determine convergence. This mechanism can also be used to observe the progress of the iteration.
Optimized operations with specific MatrixType argument
This class enables to embed the vector updates into the matrix-vector product in case the MatrixType and PreconditionerType support such a mode of operation. To this end, the VectorType needs to be
LinearAlgebra::distributed::Vector, the class MatrixType needs to provide a function with the signature
void MatrixType::vmult(
VectorType &,
const VectorType &,
const std::function<void(const unsigned int, const unsigned int)> &,
const std::function<void(const unsigned int, const unsigned int)> &) const
where the two given functions run before and after the matrix-vector product, respectively, and the PreconditionerType needs to provide a function either the signature
Number PreconditionerType::apply(unsigned int index, const Number src) const
to apply the action of the preconditioner on a single element (effectively being a diagonal preconditioner), or the signature
void PreconditionerType::apply_to_subrange(unsigned int start_range,
unsigned int end_range,
const Number* src_ptr_to_subrange,
Number* dst_ptr_to_subrange)
where the pointers src_ptr_to_subrange and dst_ptr_to_subrange point to the location in the vector where the operation should be applied to. If both functions are given, the more optimized apply path
is selected. The functions passed to MatrixType::vmult take as arguments a sub-range among the locally owned elements of the vector, defined as half-open intervals. The intervals are designed to be
scheduled close to the time the matrix-vector product touches those entries in the src and dst vectors, respectively, with the requirement that
• the matrix-vector product may only access an entry in src or dst once the operation_before_matrix_vector_product has been run on that vector entry;
• operation_after_matrix_vector_product may run on a range of entries [i,j) once the matrix-vector product does not access the entries [i,j) in src and dst any more.
The motivation for this function is to increase data locality and hence cache usage. For the example of a class similar to the one in the step-37 tutorial program, the implementation is
const std::function<void(const unsigned int, const unsigned int)>
const std::function<void(const unsigned int, const unsigned int)>
&operation_after_matrix_vector_product) const
In terms of the SolverCG implementation, the operation before the loop will run the updates on the vectors according to a variant presented in Algorithm 2.2 of [57] (but for a preconditioner),
whereas the operation after the loop performs a total of 7 reductions in parallel.
Preconditioned residual
AdditionalData allows you to choose between using the explicit or implicit residual as a stopping condition for the iterative solver. This behavior can be overridden by using the flag
AdditionalData::use_default_residual. A true value refers to the implicit residual, while false reverts it. The former uses the result of the matrix-vector product already computed in other algorithm
steps to derive the residual by a mere vector update, whereas the latter explicitly calculates the system residual with an additional matrix-vector product. More information on explicit and implicit
residual stopping criteria can be found link here.
Definition at line 196 of file solver_cg.h. | {"url":"https://dealii.org/developer/doxygen/deal.II/classSolverCG.html","timestamp":"2024-11-04T04:25:26Z","content_type":"application/xhtml+xml","content_length":"94549","record_id":"<urn:uuid:9c489de2-9540-4ff4-869d-14a792daae2c>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00724.warc.gz"} |
Kyber Post-Quantum KEM
Internet-Draft kyber January 2024
Schwabe & Westerbaan Expires 5 July 2024 [Page]
Intended Status:
Kyber Post-Quantum KEM
This memo specifies a preliminary version ("draft00", "v3.02") of Kyber, an IND-CCA2 secure Key Encapsulation Method.¶
This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79.¶
Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current
Internet-Drafts is at https://datatracker.ietf.org/drafts/current/.¶
Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as
reference material or to cite them other than as "work in progress."¶
This Internet-Draft will expire on 5 July 2024.¶
Copyright (c) 2024 IETF Trust and the persons identified as the document authors. All rights reserved.¶
This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/license-info) in effect on the date of publication of this document.
Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Revised BSD License
text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Revised BSD License.¶
Kyber is NIST's pick for a post-quantum key agreement [NISTR3].¶
Kyber is not a Diffie-Hellman (DH) style non-interactive key agreement, but instead, Kyber is a Key Encapsulation Method (KEM). A KEM is a three-tuple of algorithms (KeyGen, Encapsulate, Decapsulate
• KeyGen takes no inputs and generates a private key and a public key;¶
• Encapsulate takes as input a public key and produces as output a ciphertext and a shared secret;¶
• Decapsulate takes as input a ciphertext and a private key and produces a shared secret.¶
Like DH, a KEM can be used as an unauthenticated key-agreement protocol, for example in TLS [HYBRID] [XYBERTLS]. However, unlike DH, a KEM-based key agreement is interactive, because the party
executing Encapsulate can compute its protocol message (the ciphertext) only after having received the input (public key) from the party running KeyGen and Decapsulate.¶
A KEM can be transformed into a PKE scheme using HPKE [RFC9180] [XYBERHPKE].¶
NOTE This draft is not stable and does not (yet) match the final NIST standard ML-KEM (FIPS 203) expected in 2024. It also does not match the draft for ML-KEM published by NIST August 2023. [MLKEM]¶
Currently it matches Kyber as submitted to round 3 of the NIST PQC process [KYBERV302].¶
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described
in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.¶
Kyber is an IND-CCA2 secure KEM. It is constructed by applying a Fujisaki-Okamato style transformation on InnerPKE, which is the underlying IND-CPA secure Public Key Encryption scheme. We cannot use
InnerPKE directly, as its ciphertexts are malleable.¶
F.O. transform
InnerPKE ----------------------> Kyber
IND-CPA IND-CCA2
Kyber is a lattice-based scheme. More precisely, its security is based on the learning-with-errors-and-rounding problem in module lattices (MLWER). The underlying polynomial ring R (defined in
Section 5) is chosen such that multiplication is very fast using the number theoretic transform (NTT, see Section 5.1.3.1).¶
An InnerPKE private key is a vector s over R of length k which is small in a particular way. Here k is a security parameter akin to the size of a prime modulus. For Kyber512, which targets AES-128's
security level, the value of k is 2, for Kyber768 (AES-192 security level) k is 3, and for Kyber1024 (AES-256 security level) k is 4.¶
The public key consists of two values:¶
• A a k-by-k matrix over R sampled uniformly at random and¶
• t = A s + e, where e is a suitably small masking vector.¶
Distinguishing between such A s + e and a uniformly sampled t is the decision module learning-with-errors (MLWE) problem. If that is hard, then it is also hard to recover the private key from the
public key as that would allow you to distinguish between those two.¶
To save space in the public key, A is recomputed deterministically from a 256bit seed rho. Strictly speaking, A is not uniformly random anymore, but it's computationally indistuinguishable from it.¶
A ciphertext for a message m under this public key is a pair (c_1, c_2) computed roughly as follows:¶
c_1 = Compress(A^T r + e_1, d_u)
c_2 = Compress(t^T r + e_2 + Decompress(m, 1), d_v)
Distinguishing such a ciphertext and uniformly sampled (c_1, c_2) is an example of the full MLWER problem, see Section 4.4 of [KYBERV302].¶
To decrypt the ciphertext, one computes¶
m = Compress(Decompress(c_2, d_v) - s^T Decompress(c_1, d_u), 1).
It it not straight-forward to see that this formula is correct. In fact, there is negligable but non-zero probability that a ciphertext does not decrypt correctly given by the DFP column in Table 4.
This failure probability can be computed by a careful automated analysis of the probabilities involved, see kyber_failure.py of [SECEST].¶
To define all these operations precisely, we first define the field of coefficients for our polynomial ring; what it means to be small; and how to compress. Then we define the polynomial ring R; its
operations and in particular the NTT. We continue with the different methods of sampling and (de)serialization. Then, we first define InnerPKE and finally Kyber proper.¶
Kyber is defined over GF(q) = Z/qZ, the integers modulo q = 13*2^8+1 = 3329.¶
To define the size of a field element, we need a signed modulo. For any odd m, we write¶
for the unique integer b with -(m-1)/2 < b <= (m-1)/2 and b = a modulo m.¶
To avoid confusion, for the more familiar modulo we write umod; that is¶
is the unique integer b with 0 <= b < m and b = a modulo m.¶
Now we can define the norm of a field element:¶
|| a || = abs(a smod q)
3325 smod q = -4 || 3325 || = 4
-3320 smod q = 9 || -3320 || = 9
TODO (#23) Should we define smod and umod at all, since we don't use it.Bas¶
In several parts of the algorithm, we will need a method to compress fied elements down into d bits. To do this, we use the following method.¶
For any positive integer d, integer x and integer 0 <= y < 2^d, we define¶
Compress(x, d) = Round( (2^d / q) x ) umod 2^d
Decompress(y, d) = Round( (q / 2^d) y )
where Round(x) rounds any fraction to the nearest integer going up with ties.¶
Note that in Section 8.1 we extend Compress and Decompress to polynomials and vectors of polynomials.¶
These two operations have the following properties:¶
For implementation efficiency, these can be computed as follows.¶
Compress(x, d) = Div( (x << d) + q/2), q ) & ((1 << d) - 1)
Decompress(y, d) = (q*y + (1 << (d-1))) >> d
where Div(x, q) = Floor(x / q). TODO Do we want to include the proof that this is correct? Do we need to define >> and <<?Bas To prevent leaking the secret key, this must be computed in constant time
[KYBERSLASH]. On platforms where Div is not constant-time, the following equation is useful, which holds for those x that appear in the previous formula for 0 < d < 12.¶
Div(x, q) = (20642679 * x) >> 36
Kyber is defined over a polynomial ring Rq = GF(q)[x]/(x^n+1) where n=256 (and q=3329). Elements of Rq are tuples of 256 integers modulo q. We will call them polynomials or elements interchangeably.¶
A tuple a = (a_0, ..., a_255) represents the polynomial¶
a_0 + a_1 x + a_2 x^2 + ... + a_255 x^255.
With polynomial coefficients, vector and matrix indices, we will start counting at zero.¶
For a polynomial a = (a_0, ..., a_255) in R, we write:¶
|| a || = max_i || a_i ||
Thus a polynomial is considered large if one of its components is large.¶
Addition and subtraction of elements is componentwise. Thus¶
(a_0, ..., a_255) + (b_0, ..., b_255) = (a_0 + b_0, ..., a_255 + b_255),
(a_0, ..., a_255) - (b_0, ..., b_255) = (a_0 - b_0, ..., a_255 - b_255),
where addition/subtractoin in each component is computed modulo q.¶
Multiplication is that of polynomials (convolution) with the additional rule that x^256=-1. To wit¶
(a_0, ..., a_255) \* (b_0, ..., b_255)
= (a_0 * b_0 - a_255 * b_1 - ... - a_1 * b_255,
a_0 * b_1 + a_1 * b_0 - a_255 * b_2 - ... - a_2 * b_255,
a_0 * b_255 + ... + a_255 * b_0)
We will not use this schoolbook multiplication to compute the product. Instead we will use the more efficient, number theoretic transform (NTT), see Section 5.1.3.1.¶
The modulus q was chosen such that 256 divides into q-1. This means that there are zeta with¶
With such a zeta, we can almost completely split the polynomial x^256+1 used to define R over GF(q):¶
x^256 + 1 = x^256 - zeta^128
= (x^128 - zeta^64)(x^128 + zeta^64)
= (x^128 - zeta^64)(x^128 - zeta^192)
= (x^64 - zeta^32)(x^64 + zeta^32)
(x^64 - zeta^96)(x^64 + zeta^96)
= (x^2 - zeta)(x^2 + zeta)(x^2 - zeta^65)(x^2 + zeta^65)
... (x^2 - zeta^127)(x^2 + zeta^127)
Note that the powers of zeta that appear in the second, fourth, ..., and final lines are in binary:¶
0000001 1000001 0100001 1100001 0010001 1010001 0110001 ... 1111111
That is: brv(2), brv(3), brv(4), ..., where brv(x) denotes the 7-bit bitreversal of x. The final line is brv(64), brv(65), ..., brv(127).¶
These polynomials x^2 +- zeta^i are irreducible and coprime, hence by the Chinese Remainder Theorem for commutative rings, we know¶
R = GF(q)[x]/(x^256+1) -> GF(q)[x]/(x^2-zeta) x ... x GF(q)[x]/(x^2+zeta^127)
given by a |-> ( a mod x^2 - zeta, ..., a mod x^2 + zeta^127 ) is an isomorphism. This is the Number Theoretic Transform (NTT). Multiplication on the right is much easier: it's almost componentwise,
see Section 5.1.3.3.¶
A propos, the the constant factors that appear in the moduli in order can be computed efficiently as follows (all modulo q):¶
-zeta = -zeta^brv(64) = -zeta^{1 + 2 brv(0)}
zeta = zeta^brv(64) = -zeta^{1 + 2 brv(1)}
-zeta^65 = -zeta^brv(65) = -zeta^{1 + 2 brv(2)}
zeta^65 = zeta^brv(65) = -zeta^{1 + 2 brv(3)}
-zeta^33 = -zeta^brv(66) = -zeta^{1 + 2 brv(4)}
zeta^33 = zeta^brv(66) = -zeta^{1 + 2 brv(5)}
-zeta^127 = -zeta^brv(127) = -zeta^{1 + 2 brv(126)}
zeta^127 = zeta^brv(127) = -zeta^{1 + 2 brv(127)}
To compute a multiplication in R efficiently, one can first use the NTT, to go to the right "into the NTT domain"; compute the multiplication there and move back with the inverse NTT.¶
The NTT can be computed efficiently by performing each binary split of the polynomial separately as follows:¶
a |-> ( a mod x^128 - zeta^64, a mod x^128 + zeta^64 ),
|-> ( a mod x^64 - zeta^32, a mod x^64 + zeta^32,
a mod x^64 - zeta^96, a mod x^64 + zeta^96 ),
et cetera
If we concatenate the resulting coefficients, expanding the definitions, for the first step we get:¶
a |-> ( a_0 + zeta^64 a_128, a_1 + zeta^64 a_129,
a_126 + zeta^64 a_254, a_127 + zeta^64 a_255,
a_0 - zeta^64 a_128, a_1 - zeta^64 a_129,
a_126 - zeta^64 a_254, a_127 - zeta^64 a_255)
We can see this as 128 applications of the linear map CT_64, where¶
CT_i: (a, b) |-> (a + zeta^i b, a - zeta^i b) modulo q
for the appropriate i in the following order, pictured in the case of n=16:¶
For n=16 there are 3 levels with 1, 2 and 4 row groups respectively. For the full n=256, there are 7 levels with 1, 2, 4, 8, 16, 32 and 64 row groups respectively. The appropriate power of zeta in
the first level is brv(1)=64. The second level has brv(2) and brv(3) as powers of zeta for the top and bottom row group respectively, and so on.¶
The CT_i is known as a Cooley-Tukey butterfly. Its inverse is given by the Gentleman-Sande butterfly:¶
GS_i: (a, b) |-> ( (a+b)/2, zeta^-i (a-b)/2 ) modulo q
The inverse NTT can be computed by replacing CS_i by GS_i and flipping the diagram horizontally. TODO (#8) This section gives background not necessary for the implementation. Should we keep it?Bas¶
The modular divisions by two in the InvNTT can be collected into a single modular division by 128.¶
zeta^-i can be computed as -zeta^(128-i), which allows one to use the same precomputed table of powers of zeta for both the NTT and InvNTT.¶
TODO Add hints on lazy Montgomery reduction? Including https://eprint.iacr.org/2020/1377.pdfBas¶
As primitive 256th root of unity we use zeta=17.¶
As before, brv(i) denotes the 7-bit bitreversal of i, so brv(1)=64 and brv(91)=109.¶
The NTT is a linear bijection R -> R given by the matrix:¶
[ zeta^{ (2 brv(i>>1) + 1) (j>>1) } if i=j mod 2
(NTT)_{ij} = [
[ 0 otherwise
Recall that we start counting rows and columns at zero. The NTT can be computed more efficiently as described in section Section 5.1.3.1.¶
The inverse of the NTT is called InvNTT. It is given by the matrix:¶
[ zeta^{ 256 - (2 brv(j>>1) + 1) (i>>1) } if i=j mod 2
128 (InvNTT)_{ij} = [
[ 0 otherwise
NTT(1, 1, 0, ..., 0) = (1, 1, ..., 1, 1)
NTT(0, 1, 2, ..., 255) = (2429, 2845, 425, 1865, ..., 2502, 2134, 2717, 2303)
For elements a, b in R, we write a o b for multiplication in the NTT domain. That is: a * b = InvNTT(NTT(a) o NTT(b)). Concretely:¶
[ a_i b_i + zeta^{2 brv(i >> 1) + 1} a_{i+1} b_{i+1} if i even
(a o b)_i = [
[ a_{i-1} b_i + a_i b_{i-1} otherwise
Kyber makes use of various symmertic primitives PRF, XOF, KDF, H and G, where¶
XOF(seed) = SHAKE-128(seed)
PRF(seed, counter) = SHAKE-256(seed || counter)
KDF(prekey) = SHAKE-256(msg)[:32]
H(msg) = SHA3-256(msg)
G(msg) = (SHA3-512(msg)[:32], SHA3-512(msg)[32:])
Here counter is an octet; seed is 32 octets; prekey is 64 octets; and the length of msg varies.¶
On the surface, they look different, but they are all based on the same flexible Keccak XOF that uses the f1600 permutation, see [FIPS202]:¶
XOF(seed) = Keccak[256](seed || 1111, .)
PRF(seed, ctr) = Keccak[512](seed || ctr || 1111, .)
KDF(prekey) = Keccak[512](prekey || 1111, 256)
H(msg) = Keccak[512](msg || 01, 256)
G(msg) = (Keccak[1024](msg || 01, 512)[:32],
Keccak[1024](msg || 01, 512)[32:])
Keccak[c] = Sponge[Keccak-f[1600], pad10*1, 1600-c]
The reason five different primitives are used is to ensure domain separation, which is crucial for security, cf. [H2CURVE] §2.2.5. Additionally, a smaller sponge capacity is used for performance
where permissable by the security requirements.¶
The polynomials in the matrix A are sampled uniformly and deterministically from an octet stream (XOF) using rejection sampling as follows.¶
Three octets b_0, b_1, b_2 are read from the stream at a time. These are interpreted as two 12-bit unsigned integers d_1, d_2 via¶
d_1 + d_2 2^12 = b_0 + b_1 2^8 + b_2 2^16
This creates a stream of 12-bit ds. Of these, the elements >= q are ignored. From the resultant stream, the coefficients of the polynomial are taken in order. In Python:¶
def sampleUniform(stream):
cs = []
while True:
b = stream.read(3)
d1 = b[0] + 256*(b[1] % 16)
d2 = (b[1] >> 4) + 16*b[2]
for d in [d1, d2]:
if d >= q: continue
if len(cs) == n: return Poly(cs)
sampleUniform(SHAKE-128('')) = (3199, 697, 2212, 2302, ..., 255, 846, 1)
Now, the k by k matrix A over R is derived deterministically from a 32-octet seed rho using sampleUniform as follows.¶
sampleMatrix(rho)_{ij} = sampleUniform(XOF(rho || octet(j) || octet(i))
That is, to derive the polynomial at the ith row and jth column, sampleUniform is called with the 34-octet seed created by first appending the octet j and then the octet i to rho. Recall that we
start counting rows and columns from zero.¶
As the NTT is a bijection, it does not matter whether we interpret the polynomials of the sampled matrix in the NTT domain or not. For efficiency, we do interpret the sampled matrix in the NTT
Noise is sampled from a centered binomial distribution Binomial(2eta, 1/2) - eta deterministically as follows.¶
An octet array a of length 64*eta is converted to a polynomial CBD(a, eta)¶
CBD(a, eta)_i = b_{2i eta} + b_{2i eta + 1} + ... + b_{2i eta + eta-1}
- b_{2i eta + eta} - ... - b_{2i eta + 2eta - 1},
where b = OctetsToBits(a).¶
CBD((0, 1, 2, ..., 127), 2) = (0, 0, 1, 0, 1, 0, ..., 3328, 1, 0, 1)
CBD((0, 1, 2, ..., 191), 3) = (0, 1, 3328, 0, 2, ..., 3328, 3327, 3328, 1)
A k component small vector v is derived from a seed 32-octet seed sigma, an offset offset and size eta as follows:¶
sampleNoise(sigma, eta, offset)_i = CBD(PRF(sigma, octet(i+offset)), eta)
Recall that we start counting vector indices at zero.¶
Recall that Compress(x, d) maps a field element x into {0, ..., 2^d-1}. In Kyber d is at most 11 and so we can interpret Compress(x, d) as a field element again.¶
In this way, we can extend Compress(-, d) to polynomials by applying to each coefficient separately and in turn to vectors by applying to each polynomial. That is, for a vector v and polynomial p:¶
Compress(p, d)_i = Compress(p_i, d)
Compress(v, d)_i = Compress(v_i, d)
We will also use "o", from section Section 5.1.3.3, to denote the dot product and matrix multiplication in the NTT domain. Concretely:¶
1. For two length k vector v and w, we write¶
v o w = v_0 o w_0 + ... + v_{k-1} o w_{k-1}
2. For a k by k matrix A and a length k vector v, we have¶
where A_i denotes the (i+1)th row of the matrix A as we start counting at zero.¶
For a matrix A, we denote by A^T the tranposed matrix. To wit:¶
We define Decompress(-, d) for vectors and polynomials in the same way.¶
For any list of octets a_0, ..., a_{s-1}, we define OctetsToBits(a), which is a list of bits of length 8s, defined by¶
OctetsToBits(a)_i = ((a_(i>>3)) >> (i umod 8)) umod 2.
OctetsToBits(12,45) = (0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0)
For an integer 0 < w <= 12, we define Decode(a, w), which converts any list a of w*l/8 octets into a list of length l with values in {0, ..., 2^w-1} as follows.¶
Decode(a, w)_i = b_{wi} + b_{wi+1} 2 + b_{wi+2} 2^2 + ... + b_{wi+w-1} 2^{w-1},
where b = OctetsToBits(a).¶
Encode(-, w) is the unique inverse of Decode(-, w)¶
A polynomial p is encoded by passing its coefficients to Encode:¶
EncodePoly(p, w) = Encode(p_0, p_1, ..., p_{n-1}, w)
DecodePoly(-, w) is the unique inverse of EncodePoly(-, w).¶
A vector v of length k over R is encoded by concatenating the coefficients in the obvious way:¶
EncodeVec(v, w) = Encode((v_0)_0, ..., (v_0)_{n-1},
(v_1)_{0}, ..., (v_1)_{n-1},
..., (v_{k-1})_{n-1}, w)
DecodeVec(-, w) is the unique inverse of EncodeVec(-, w).¶
We are ready to define the IND-CPA secure Public-Key Encryption scheme that underlies Kyber. It is unsafe to use this underlying scheme directly as its ciphertexts are malleable. Instead, a
Public-Key Encryption scheme can be constructed on top of Kyber by using HPKE [RFC9180] [XYBERHPKE].¶
We have already been introduced to the following parameters:¶
XOF, H, G, PRF, KDF
Additionally, Kyber takes the following parameters¶
eta1, eta2
Size of small coefficients used in the private key and noise vectors.¶
d_u, d_v
How many bits to retain per coefficient of the u and v components of the ciphertext.¶
The values of these parameters are given in Section 12.¶
InnerKeyGen(seed) takes a 32 octet seed and deterministically produces a keypair as follows.¶
1. Set (rho, sigma) = G(seed).¶
2. Derive¶
1. AHat = sampleMatrix(rho).¶
2. s = sampleNoise(sigma, eta1, 0)¶
3. e = sampleNoise(sigma, eta1, k)¶
3. Compute¶
1. sHat = NTT(s)¶
2. tHat = AHat o sHat + NTT(e)¶
4. Return¶
1. publicKey = EncodeVec(tHat, 12) || rho¶
2. privateKey = EncodeVec(sHat, 12)¶
Note that in essence we're simply computing t = A s + e.¶
InnerEnc(msg, publicKey, seed) takes a 32-octet seed, and deterministically encrypts the 32-octet msg for the InnerPKE public key publicKey as follows.¶
InnerDec(cipherText, privateKey) takes an InnerPKE private key privateKey and decrypts a cipher text cipherText as follows.¶
Now we are ready to define Kyber itself.¶
A Kyber keypair is derived deterministically from a 64-octet seed as follows.¶
1. Split seed into¶
1. A 32-octet cpaSeed¶
2. A 32-octet z¶
2. Compute¶
1. (cpaPublicKey, cpaPrivateKey) = InnerKeyGen(cpaSeed)¶
2. h = H(cpaPublicKey)¶
3. Return¶
1. publicKey = cpaPublicKey¶
2. privateKey = cpaPrivateKey || cpaPublicKey || h || z¶
Kyber encapsulation takes a public key and generates a shared secret and ciphertext for the public key as follows.¶
1. Sample secret cryptographically-secure random 32-octet seed.¶
2. Compute¶
1. m = H(seed)¶
2. (Kbar, cpaSeed) = G(m || H(publicKey))¶
3. cpaCipherText = InnerEnc(m, publicKey, cpaSeed)¶
3. Return¶
1. cipherText = cpaCipherText¶
2. sharedSecret = KDF(KBar || H(cpaCipherText))¶
Kyber decapsulation takes a private key and a cipher text and returns a shared secret as follows.¶
For security, the implementation MUST NOT explicitly return or otherwise leak via a side-channel, decapsulation succeeded, viz cipherText == cipherText2.¶
Table 3: Description of kyber parameters
Name Description
k Dimension of module
eta1, eta2 Size of "small" coefficients used in the private key and noise vectors.
d_u How many bits to retain per coefficient of u, the private-key independent part of the ciphertext
d_v How many bits to retain per coefficient of v, the private-key dependent part of the ciphertext.
# WARNING This is a specification of Kyber; not a production ready
# implementation. It is slow and does not run in constant time.
# Requires the CryptoDome for SHAKE. To install, run
# pip install pycryptodome pytest
from Crypto.Hash import SHAKE128, SHAKE256
import io
import hashlib
import functools
import collections
from math import floor
q = 3329
nBits = 8
zeta = 17
eta2 = 2
n = 2**nBits
inv2 = (q+1)//2 # inverse of 2
params = collections.namedtuple('params', ('k', 'du', 'dv', 'eta1'))
params512 = params(k = 2, du = 10, dv = 4, eta1 = 3)
params768 = params(k = 3, du = 10, dv = 4, eta1 = 2)
params1024 = params(k = 4, du = 11, dv = 5, eta1 = 2)
def smod(x):
r = x % q
if r > (q-1)//2:
r -= q
return r
# Rounds to nearest integer with ties going up
def Round(x):
return int(floor(x + 0.5))
def Compress(x, d):
return Round((2**d / q) * x) % (2**d)
def Decompress(y, d):
assert 0 <= y and y <= 2**d
return Round((q / 2**d) * y)
def BitsToWords(bs, w):
assert len(bs) % w == 0
return [sum(bs[i+j] * 2**j for j in range(w))
for i in range(0, len(bs), w)]
def WordsToBits(bs, w):
return sum([[(b >> i) % 2 for i in range(w)] for b in bs], [])
def Encode(a, w):
return bytes(BitsToWords(WordsToBits(a, w), 8))
def Decode(a, w):
return BitsToWords(WordsToBits(a, 8), w)
def brv(x):
""" Reverses a 7-bit number """
return int(''.join(reversed(bin(x)[2:].zfill(nBits-1))), 2)
class Poly:
def __init__(self, cs=None):
self.cs = (0,)*n if cs is None else tuple(cs)
assert len(self.cs) == n
def __add__(self, other):
return Poly((a+b) % q for a,b in zip(self.cs, other.cs))
def __neg__(self):
return Poly(q-a for a in self.cs)
def __sub__(self, other):
return self + -other
def __str__(self):
return f"Poly({self.cs}"
def __eq__(self, other):
return self.cs == other.cs
def NTT(self):
cs = list(self.cs)
layer = n // 2
zi = 0
while layer >= 2:
for offset in range(0, n-layer, 2*layer):
zi += 1
z = pow(zeta, brv(zi), q)
for j in range(offset, offset+layer):
t = (z * cs[j + layer]) % q
cs[j + layer] = (cs[j] - t) % q
cs[j] = (cs[j] + t) % q
layer //= 2
return Poly(cs)
def RefNTT(self):
# Slower, but simpler, version of the NTT.
cs = [0]*n
for i in range(0, n, 2):
for j in range(n // 2):
z = pow(zeta, (2*brv(i//2)+1)*j, q)
cs[i] = (cs[i] + self.cs[2*j] * z) % q
cs[i+1] = (cs[i+1] + self.cs[2*j+1] * z) % q
return Poly(cs)
def InvNTT(self):
cs = list(self.cs)
layer = 2
zi = n//2
while layer < n:
for offset in range(0, n-layer, 2*layer):
zi -= 1
z = pow(zeta, brv(zi), q)
for j in range(offset, offset+layer):
t = (cs[j+layer] - cs[j]) % q
cs[j] = (inv2*(cs[j] + cs[j+layer])) % q
cs[j+layer] = (inv2 * z * t) % q
layer *= 2
return Poly(cs)
def MulNTT(self, other):
""" Computes self o other, the multiplication of self and other
in the NTT domain. """
cs = [None]*n
for i in range(0, n, 2):
a1 = self.cs[i]
a2 = self.cs[i+1]
b1 = other.cs[i]
b2 = other.cs[i+1]
z = pow(zeta, 2*brv(i//2)+1, q)
cs[i] = (a1 * b1 + z * a2 * b2) % q
cs[i+1] = (a2 * b1 + a1 * b2) % q
return Poly(cs)
def Compress(self, d):
return Poly(Compress(c, d) for c in self.cs)
def Decompress(self, d):
return Poly(Decompress(c, d) for c in self.cs)
def Encode(self, d):
return Encode(self.cs, d)
def sampleUniform(stream):
cs = []
while True:
b = stream.read(3)
d1 = b[0] + 256*(b[1] % 16)
d2 = (b[1] >> 4) + 16*b[2]
assert d1 + 2**12 * d2 == b[0] + 2**8 * b[1] + 2**16*b[2]
for d in [d1, d2]:
if d >= q:
if len(cs) == n:
return Poly(cs)
def CBD(a, eta):
assert len(a) == 64*eta
b = WordsToBits(a, 8)
cs = []
for i in range(n):
cs.append((sum(b[:eta]) - sum(b[eta:2*eta])) % q)
b = b[2*eta:]
return Poly(cs)
def XOF(seed, j, i):
h = SHAKE128.new()
h.update(seed + bytes([j, i]))
return h
def PRF(seed, nonce):
assert len(seed) == 32
h = SHAKE256.new()
h.update(seed + bytes([nonce]))
return h
def G(seed):
h = hashlib.sha3_512(seed).digest()
return h[:32], h[32:]
def H(msg): return hashlib.sha3_256(msg).digest()
def KDF(msg): return hashlib.shake_256(msg).digest(length=32)
class Vec:
def __init__(self, ps):
self.ps = tuple(ps)
def NTT(self):
return Vec(p.NTT() for p in self.ps)
def InvNTT(self):
return Vec(p.InvNTT() for p in self.ps)
def DotNTT(self, other):
""" Computes the dot product <self, other> in NTT domain. """
return sum((a.MulNTT(b) for a, b in zip(self.ps, other.ps)),
def __add__(self, other):
return Vec(a+b for a,b in zip(self.ps, other.ps))
def Compress(self, d):
return Vec(p.Compress(d) for p in self.ps)
def Decompress(self, d):
return Vec(p.Decompress(d) for p in self.ps)
def Encode(self, d):
return Encode(sum((p.cs for p in self.ps), ()), d)
def __eq__(self, other):
return self.ps == other.ps
def EncodeVec(vec, w):
return Encode(sum([p.cs for p in vec.ps], ()), w)
def DecodeVec(bs, k, w):
cs = Decode(bs, w)
return Vec(Poly(cs[n*i:n*(i+1)]) for i in range(k))
def DecodePoly(bs, w):
return Poly(Decode(bs, w))
class Matrix:
def __init__(self, cs):
""" Samples the matrix uniformly from seed rho """
self.cs = tuple(tuple(row) for row in cs)
def MulNTT(self, vec):
""" Computes matrix multiplication A*vec in the NTT domain. """
return Vec(Vec(row).DotNTT(vec) for row in self.cs)
def T(self):
""" Returns transpose of matrix """
k = len(self.cs)
return Matrix((self.cs[j][i] for j in range(k))
for i in range(k))
def sampleMatrix(rho, k):
return Matrix([[sampleUniform(XOF(rho, j, i))
for j in range(k)] for i in range(k)])
def sampleNoise(sigma, eta, offset, k):
return Vec(CBD(PRF(sigma, i+offset).read(64*eta), eta)
for i in range(k))
def constantTimeSelectOnEquality(a, b, ifEq, ifNeq):
# WARNING! In production code this must be done in a
# data-independent constant-time manner, which this implementation
# is not. In fact, many more lines of code in this
# file are not constant-time.
return ifEq if a == b else ifNeq
def InnerKeyGen(seed, params):
assert len(seed) == 32
rho, sigma = G(seed)
A = sampleMatrix(rho, params.k)
s = sampleNoise(sigma, params.eta1, 0, params.k)
e = sampleNoise(sigma, params.eta1, params.k, params.k)
sHat = s.NTT()
eHat = e.NTT()
tHat = A.MulNTT(sHat) + eHat
pk = EncodeVec(tHat, 12) + rho
sk = EncodeVec(sHat, 12)
return (pk, sk)
def InnerEnc(pk, msg, seed, params):
assert len(msg) == 32
tHat = DecodeVec(pk[:-32], params.k, 12)
rho = pk[-32:]
A = sampleMatrix(rho, params.k)
r = sampleNoise(seed, params.eta1, 0, params.k)
e1 = sampleNoise(seed, eta2, params.k, params.k)
e2 = sampleNoise(seed, eta2, 2*params.k, 1).ps[0]
rHat = r.NTT()
u = A.T().MulNTT(rHat).InvNTT() + e1
m = Poly(Decode(msg, 1)).Decompress(1)
v = tHat.DotNTT(rHat).InvNTT() + e2 + m
c1 = u.Compress(params.du).Encode(params.du)
c2 = v.Compress(params.dv).Encode(params.dv)
return c1 + c2
def InnerDec(sk, ct, params):
split = params.du * params.k * n // 8
c1, c2 = ct[:split], ct[split:]
u = DecodeVec(c1, params.k, params.du).Decompress(params.du)
v = DecodePoly(c2, params.dv).Decompress(params.dv)
sHat = DecodeVec(sk, params.k, 12)
return (v - sHat.DotNTT(u.NTT()).InvNTT()).Compress(1).Encode(1)
def KeyGen(seed, params):
assert len(seed) == 64
z = seed[32:]
pk, sk2 = InnerKeyGen(seed[:32], params)
h = H(pk)
return (pk, sk2 + pk + h + z)
def Enc(pk, seed, params):
assert len(seed) == 32
m = H(seed)
Kbar, r = G(m + H(pk))
ct = InnerEnc(pk, m, r, params)
K = KDF(Kbar + H(ct))
return (ct, K)
def Dec(sk, ct, params):
sk2 = sk[:12 * params.k * n//8]
pk = sk[12 * params.k * n//8 : 24 * params.k * n//8 + 32]
h = sk[24 * params.k * n//8 + 32 : 24 * params.k * n//8 + 64]
z = sk[24 * params.k * n//8 + 64 : 24 * params.k * n//8 + 96]
m2 = InnerDec(sk, ct, params)
Kbar2, r2 = G(m2 + h)
ct2 = InnerEnc(pk, m2, r2, params)
return constantTimeSelectOnEquality(
ct2, ct,
KDF(Kbar2 + H(ct)), # if ct == ct2
KDF(z + H(ct)), # if ct != ct2
Kyber512, Kyber768 and Kyber1024 are designed to be post-quantum IND-CCA2 secure KEMs, at the security levels of AES-128, AES-192 and AES-256.¶
The designers of Kyber recommend Kyber768.¶
The inner public key encryption SHOULD NOT be used directly, as its ciphertexts are malleable. Instead, for public key encryption, HPKE can be used to turn Kyber into IND-CCA2 secure PKE [RFC9180] [
Any implementation MUST use implicit rejection as specified in Section 11.3.¶
National Institute of Standards and Technology, "FIPS PUB 202: SHA-3 Standard: Permutation-Based Hash and Extendable-Output Functions", n.d., <https://nvlpubs.nist.gov/nistpubs/fips/
Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, , <https://www.rfc-editor.org/rfc/rfc2119>.
Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, , <https://www.rfc-editor.org/rfc/rfc8174>.
The authors would like to thank C. Wood, Florence D., I. Liusvaara, J. Crawford, J. Schanck, M. Thomson, and N. Sullivan for their input and assistance.¶
• RFC Editor's Note: Please remove this section prior to publication of a final version of this document.¶
• Test specification against NIST test vectors.¶
• Fix two unintentional mismatches between this document and the reference implementation:¶
1. KDF uses SHAKE-256 instead of SHAKE-128.¶
2. Reverse order of seed. (z comes at the end.)¶
• Elaborate text in particular introduction, and symmetric key section.¶ | {"url":"https://datatracker.ietf.org/doc/html/draft-cfrg-schwabe-kyber","timestamp":"2024-11-06T07:51:00Z","content_type":"text/html","content_length":"159757","record_id":"<urn:uuid:e38e8d3b-2401-40d3-bcb5-c721091263d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00614.warc.gz"} |
[Solved] Also, What is the domain of f? and what i | SolutionInn
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Also, What is the domain of f? and what is the domain of f-1? Consider the graph of the one-to-one function shown in the figure
Also, What is the domain of f? and what is the domain of f-1?
Consider the graph of the one-to-one function shown in the figure below. y 10+ 00 8 6 4 2 Sketch the graph of f-1. -10 5 2 4 y 10 5 5 10 6 8 +x 10 5 X 10 -10 5 y 10+ 5 5 10 X 5 10
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Strong Column Condition Reports
A [c] = Cross-sectional area of the column
f [ck] = Characteristic compressive strength of concrete
f [cd] = Design compressive strength of concrete
f [yd] = Design yield strength of longitudinal reinforcement
M [ra] = Bearing strength calculated according to f [cd] and f [yd] at the lower end of the free height of the column or wall torque
M [r] = column or in the upper end of the free height of the curtain for [CDs] and f [yd] 'calculated according to the ultimate moment
M [r] = Moment of positive or negative bearing capacity calculated according to f [cd] and f [yd] on the column or wall face at the left end of the beam
M [rj] = Negative calculated according to f [cd] and f [yd] on the column or wall face at the right end of the beam j positive bearing strength moment
N [d] = Axial force calculated under the combined effect of vertical loads and earthquake loads multiplied by the load factors.
V [ik] = Sum of shear forces calculated in the direction of the earthquake in all columns on the i'th floor of the building
V [is] =The sum of shear forces calculated in the direction of the earthquake considered in the columns where (M [ra] + M [rü] ) ≥ 1.2 (M [ri] + M [rj] ) is provided at both the lower and upper
joints of the building. At the ends of the columns [meeting the] condition N [d] ≤ 0.10 A [c] f [ck] , Eq. Even if (7.3) is not provided, these columns are also not taken into account in the
calculation of V [is] .
Strong column controls are given as a subtitle in the column report.
• In structural systems consisting of only frames or a combination of walls and frames, it is checked whether the sum of the bearing strength moments of the columns joining each column-beam node is
at least 20% greater than the total bearing strength moments of the beams joining that node. (M [ra] + M [rü] ) ≥ 1.2 (M [ri] + M [rj] )
• In case of Nd≤ 0.10 A [c] f [ck in] both of the columns joining the node point , or at the nodal points of the top floor of single-storey buildings and multi-storey buildings, or if the curtain
where the beams are stuck working like a column in a weak direction, it is not considered that the above condition is not fulfilled. Information is written in the control column of the report
with the terms Nd≤ 0.10 A [c] f [ck] or Top floor .
• In case of N [d] ≥ 0.10 A [c] f [ck in] one of the columns joining the node point and there are columns that do not satisfy the condition (M [ra] + M [rü] ) ≥ 1.2 (M [ri] + M [rj] ) in the
system, in the earthquake direction considered, in any i'th floor of the building If the condition alpha [i] = V [is] / V [ik] is satisfied, it is allowed not to [meet] (M [ra] + M [rü] ) ≥ 1.2
(M [ri] + M [rj] ) in some columns above and / or below the relevant floor .
• Columns satisfying the condition N [d] ≤ 0.10 A [c] f [ck] can be used in the calculation of V [is] even if they do not satisfy the condition (M [ra] + M [rü] ) ≥ 1.2 (M [ri] + M [rj] ) .
• If the equation alpha [i] = V [is] / V [ik] is [met] 0.70 <1 / alpha [i] <1.00 in the interval (M [ra] + M [rü] ) ≥ 1.2 (M [ri] + M [rj] ), it is The bending moments and shear forces are
multiplied by the (1 / alpha [i] ) ratio.
• If any floor does not meet the condition, the user will be warned by the program.
• Information on strong column control in the program is displayed in the Weak Column Information tab in the Column Reinforced Concrete dialog. The columns that do not satisfy the (M [ra] + M [rü]
) ≥ 1.2 (M [ri] + M [rj] ) condition are marked in the ZK column in the Columns tab in the same dialog . | {"url":"https://help.idecad.com/ideCAD/strong-column-condition-reports","timestamp":"2024-11-10T17:49:50Z","content_type":"text/html","content_length":"37380","record_id":"<urn:uuid:121cf54e-24d4-40d3-a184-d9123f0a15b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00800.warc.gz"} |
reading time
3 minutes
Embedding of a watermak using DWT-SVD and a DWT-DCT tranform.
In my project, I utilized advanced watermarking techniques to embed a mark of size 512x512 into three different images. The primary focus was to ensure the highest possible image quality while
effectively embedding the watermark.
To achieve this, I combined cutting-edge watermarking techniques known for their ability to maintain image quality. These techniques involved optimizing the placement and strength of the watermark,
considering factors such as perceptual masking and robustness to attacks.
After successfully embedding the watermark in the images, another crucial aspect of the project was to develop an attacking technique. The objective of this technique was to attempt to destroy the
embedded mark created by another group while preserving the image’s best possible quality. Striking the right balance between attack effectiveness and its impact on overall image quality was
Throughout the project, special attention was given to the quality of the images to ensure seamless integration of the watermark without compromising visual appeal or clarity. By employing advanced
watermarking techniques and devising an effective attacking strategy, the aim was to demonstrate the robustness and resilience of the embedded mark against potential attacks while maintaining the
highest image quality.
Embedding Method Explanation
The watermarking method I employed utilizes two different mark embedding techniques:
• The first technique involves a two-level wavelet transform in the HL/HL quadrant, where a DCT transform is applied. The most significant bit is selected after sorting the bits, and an 8-bit long
code represents a single bit of the mark.
• The second technique utilizes a three-level wavelet transform in the LL/LL/LL quadrant, applying SVD. In this case, a 2-bit long code corresponds to a single bit of the mark.
This embedding method is designed to simultaneously embed the mark in three different images. A CSV file is utilized for the WPSNR function in the detection method. The detection method takes the
original image, the watermarked image, and the image after being attacked as inputs. It determines if the mark is still present in the attacked image and calculates the WPSNR of this image. This
method can be easily modified to detect the presence of the mark in any given image.
def mixed_embedding(....):
coefficient = pywt.dwt2(original, wavelet='haar')
quadrants = [coefficient[0],*coefficient[1]]
coefficient2_dct = pywt.dwt2(quadrants[loc_dct_lv1], wavelet='haar')
quadrants2_dct = [coefficient2_dct[0],*coefficient2_dct[1]]
coefficient2_svd = pywt.dwt2(quadrants[loc_svd_lv1], wavelet='haar')
quadrants2_svd = [coefficient2_svd[0],*coefficient2_svd[1]]
coefficient3_svd = pywt.dwt2(quadrants2_svd[loc_svd_lv1], wavelet='haar')
quadrants3_svd = [coefficient3_svd[0],*coefficient3_svd[1]
size = quadrants2_dct[1].shape[0]
size_svd = quadrants3_svd[1].shape[0]
#divisione in blocchi dei quadranti scelti
blocks_dct = quadrants2_dct[loc_dct_lv2]
blocks_dct = np.hsplit(blocks_dct, size//4)
blocks_svd = quadrants3_svd[loc_svd_lv2]
blocks_svd = np.hsplit(blocks_svd, size_svd//2)
for k in range(len(blocks_dct)):
blocks_dct[k] = np.vsplit(blocks_dct[k], size//4)
blocks_svd[k] = np.vsplit(blocks_svd[k], size_svd//2)
# dct, svd; embedding; idct, isvd
for i in range(len(blocks_dct)):
for j in range(len(blocks_dct)):
blocks_dct[i][j] = dct(dct(blocks_dct[i][j],axis=0, norm='ortho'),axis=1, norm='ortho')
U,S,VH = np.linalg.svd(blocks_svd[i][j])
if(mark[i][j] == 0):
blocks_dct[i][j] += alpha_dct*(np.array(padding(blocks_dct[i][j],seq_0_dct)).reshape(4,4))
S += alpha_svd*(np.array(seq_0_svd))
blocks_dct[i][j] += alpha_dct*(np.array(padding(blocks_dct[i][j],seq_1_dct)).reshape(4,4))
S += alpha_svd*(np.array(seq_1_svd))
blocks_svd[i][j] = np.dot(U*S,VH)
blocks_dct[i][j] = idct(idct(blocks_dct[i][j],axis=1, norm='ortho'),axis=0, norm='ortho')
for k in range(len(blocks_dct)):
blocks_dct[k] = np.array(np.vstack(blocks_dct[k])).reshape(128,4)
blocks_svd[k] = np.array(np.vstack(blocks_svd[k])).reshape(64,2)
quadrants2_dct[loc_dct_lv2] = np.array(np.hstack(blocks_dct)).reshape(128,128)
quadrants3_svd[loc_svd_lv2] = np.array(np.hstack(blocks_svd)).reshape(64,64)
coefficient3_svd = quadrants3_svd[0],(quadrants3_svd[1],quadrants3_svd[2],quadrants3_svd[3])
quadrants2_svd[loc_svd_lv1] = pywt.idwt2(coefficient3_svd, wavelet='haar')
coefficient2_dct = quadrants2_dct[0],(quadrants2_dct[1],quadrants2_dct[2],quadrants2_dct[3])
coefficient2_svd = quadrants2_svd[0],(quadrants2_svd[1],quadrants2_svd[2],quadrants2_svd[3])
#rimettiamo ogni quadrante al suo posto nel primo livello
quadrants[loc_dct_lv1] = pywt.idwt2(coefficient2_dct, wavelet='haar')
quadrants[loc_svd_lv1] = pywt.idwt2(coefficient2_svd, wavelet='haar')
coefficient = quadrants[0],(quadrants[1],quadrants[2],quadrants[3])
final = pywt.idwt2(coefficient, wavelet='haar')
return np.uint8(np.rint(np.clip(final, 0, 255)))
Attack Method Explanation
The “attacks” file contains various brute-force attacks for destroying the mark on the images:
• “Base-attacks” is a brute-force method in the spatial domain. It applies different single methods (AWGN, BLUR, SHARPENING, MEDIAN-FILTER, RESIZING, JPEG) to the given image. The best attack from
a list of successful attacks is selected, and attempts are made to attack the image again with combined attacks.
• “Wavelet_attack” follows a similar approach as “Base-attacks.” However, the attack is localized in the area of the DWT transform where the mark seems to be, based on a comparison between the
original and watermarked images.
• “Ftt_attack” and “Dct_attack” work similarly to the other attacks, but they are localized on the most significant bit of the DCT or FTT, where the mark seems to be, after comparing the original
and watermarked images.
You can find my full project on the Github Repository, where you can also check the code. | {"url":"https://niccoloparlanti.com/projects/watermarking/","timestamp":"2024-11-11T07:47:36Z","content_type":"text/html","content_length":"17579","record_id":"<urn:uuid:94be3890-49c1-4342-9cbd-5eeeffbd2b61>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00728.warc.gz"} |
False Positive Paradox
A disease that can be asymptomatic affects one percent of the population.
There’s a test for it that never produces false negatives but it has a five percent false positive rate.
You take the test before travelling, and it comes back positive.
What are the chances you have the disease?
A common answer is 95%, but it’s actually only 16.8%.
This is surprising to many people, but remember that only 1% of the population has the disease.
In a group of 10,000 people, we can expect that 100 people will have the disease and 9,900 will not.
Imagine we test everyone in the group.
All 100 people who have the disease will test positive since there are no false negatives.
Of the 9,900 people who don’t have the disease, 9,405 will test negative, and 495 will show false positives.
This means there are 595 positives: 100 of them are true positives and 495 are false positives.
100 true positives ÷ 595 total positives = 16.8% of the people who test positive are true positives
Note: If you took the test because you had symptoms of the disease, your probability judgment should account for this information.
The False Positive Paradox teaches us that when the prevalence of a disease is low, widespread testing of asymptomatic people leads to a high number of false positives.
Develop the skills to tackle logical fallacies through a series of 10 science-fiction videos with activities. Recommended for ages 8 and up.
Learn about common mistakes in data analysis with an interactive space adventure. Recommended for ages 12 and up.
Learn how to make sense of complicated arguments with 14 video lessons and activities. Recommended for ages 13 and up.
Learn to recognize, understand, and manage your emotions. Designed by child psychologist Ronald Crouch, Ph.D. Recommended for ages 5 and up.
Worksheets covering the basics of symbolic logic for children ages 12 and up.
These lesson plans and worksheets teach students in grades 2-5 about superstitions, different perspectives, facts and opinions, the false dilemma fallacy, and probability.
These lesson plans and worksheets teach students in grades 5-8 about false memories, confirmation bias, Occam's razor, the strawman fallacy, and pareidolia.
These lesson plans and worksheets teach students in grades 8-12 about critical thinking, the appeal to nature fallacy, correlation versus causation, the placebo effect, and weasel words.
These lesson plans and worksheets teach students in grades 9 and up the statistical principles they need to analyze data rationally. | {"url":"https://critikid.com/false-positive-paradox","timestamp":"2024-11-03T12:50:25Z","content_type":"text/html","content_length":"61833","record_id":"<urn:uuid:92ac58b1-1986-424f-93aa-1b9504dd4bbb>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00872.warc.gz"} |
Adjoints to "Enabling list-initialization for algorithms": find_last
1. Changelog
2. Motivation and scope
In the Tokyo 2024 meeting [P2248R8] was adopted. Due to an oversight (which is entirely our fault) std::ranges::find_last was accidentally excluded from the algorithms for which a default template
type parameter for the "value" argument was provided.
We propose to modify find_last’s specification, so that it matches the post-P2248 one for the rest of the algorithms (especially find).
3. Proposed Wording
All the proposed changes are relative to [N4971], assuming that [P2248R8]'s wording has been merged already.
Modify [alg.find.last] as shown:
template<forward_iterator I, sentinel_for<I> S,[DEL: class T,:DEL] class Proj = identity>
requires indirect_binary_predicate<ranges::equal_to, projected<I, Proj>, const T*>
constexpr subrange<I> ranges::find_last(I first, S last, const T& value, Proj proj = {});
template<forward_range R,[DEL: class T,:DEL] class Proj = identity>
requires indirect_binary_predicate<ranges::equal_to, projected<iterator_t<R>, Proj>, const T*>
constexpr borrowed_subrange_t<R> ranges::find_last(R&& r, const T& value, Proj proj = {});
4. Acknowledgements
Thanks to Jens Maurer for pointing out this oversight of [P2248R8].
Thanks to KDAB for supporting this work.
All remaining errors are ours and ours only. | {"url":"https://open-std.org/jtc1/sc22/wg21/docs/papers/2024/p3217r0.html","timestamp":"2024-11-07T13:44:47Z","content_type":"text/html","content_length":"72382","record_id":"<urn:uuid:ea396474-96ce-4a31-919a-93f13bcd10f2>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00194.warc.gz"} |
Data Structure
1. Introduction to Stack
A stack is a linear data structure that follows the Last In First Out (LIFO) principle, meaning the last element added to the stack is the first element to be removed. This data structure has various
applications in algorithms, computer memory management, and programming language implementation. In this article, we will delve into the intricacies of stacks, their relationship with other data
structures, and their practical applications in computer science.
1.1 Basic Stack Operations
The two primary operations that can be performed on a stack are push and pop. Push is the process of adding an element to the top of the stack, while pop is the process of removing the top element
from the stack. Additionally, the peek or top operation can be used to view the top element without removing it from the stack.
// Stack implementation in C++ using an array
#include <iostream>
#define MAX 10
class Stack {
int top;
int arr[MAX];
Stack() { top = -1; }
bool push(int x);
int pop();
int peek();
bool isEmpty();
bool Stack::push(int x) {
if (top >= MAX - 1) {
std::cout << "Stack Overflow";
return false;
} else {
arr[++top] = x;
return true;
int Stack::pop() {
if (top < 0) {
std::cout << "Stack Underflow";
return 0;
} else {
return arr[top--];
int Stack::peek() {
if (top < 0) {
std::cout << "Stack is Empty";
return 0;
} else {
return arr[top];
bool Stack::isEmpty() {
return (top < 0);
int main() {
Stack stack;
std::cout << "Stack top is: " << stack.peek() << std::endl;
std::cout << "Popping an element: " << stack.pop() << std::endl;
std::cout << "Stack top is: " << stack.peek() << std::endl;
if (stack.isEmpty()) {
std::cout << "Stack is empty" << std::endl;
} else {
std::cout << "Stack is not empty" << std::endl;
return 0;
In the above implementation, an array of fixed size MAX is used to store the stack elements. The push operation adds an element to the top of the stack, while the pop operation removes the top
element. The peek operation returns the top element without removing it, and the isEmpty operation checks if the stack is empty.
1.2 Stack Implementation using Linked List
Another common way to implement a stack is by using a singly-linked list. The advantage of using a linked list over an array is that it can grow and shrink dynamically according to the elements being
added or removed.
// Stack implementation in C++ using a linked list
#include <iostream>
class Node {
int data;
Node* next;
class Stack {
Node* top;
Stack() { top = nullptr; }
void push(int x);
int pop();
int peek();
bool isEmpty();
void Stack::push(int x) {
Node* newNode = new Node();
newNode->data = x;
newNode->next = top;
top = newNode;
int Stack::pop() {
if (isEmpty()) {
std::cout << "Stack Underflow";
return 0;
} else {
Node* temp = top;
top = top->next;
int poppedValue = temp->data;
delete temp;
return poppedValue;
int Stack::peek() {
if (isEmpty()) {
std::cout << "Stack is Empty";
return 0;
} else {
return top->data;
bool Stack::isEmpty() {
return (top == nullptr);
In this implementation, a linked list is used to store the stack elements. Each node in the linked list contains an integer data value and a pointer to the next node. The push operation adds a new
node at the beginning of the list, representing the top of the stack. The pop operation removes the top node and returns its data value. The peek operation returns the data value of the top node
without removing it, and the isEmpty operation checks if the stack is empty by checking if the top node is a nullptr.
2. Applications of Stack
Stacks have a wide range of applications in various domains of computer science, such as parsing expressions, managing function calls, and implementing algorithms. In this section, we will discuss
some of these applications in detail.
2.1 Expression Evaluation and Parsing
Stacks play a crucial role in evaluating and parsing mathematical expressions. They can be used to convert an infix expression to postfix or prefix notation and to evaluate postfix or prefix
expressions. The Shunting Yard Algorithm, which uses two stacks, is a popular method for parsing arithmetic expressions specified in infix notation.
2.1.1 Infix to Postfix Conversion
To convert an infix expression to postfix notation, we use a stack to store operators and parentheses. The algorithm scans the infix expression from left to right, and for each character encountered,
it performs one of the following actions:
1. If the character is an operand, append it to the output.
2. If the character is an operator, pop operators from the stack and append them to the output until an operator with a lower precedence is found or the stack is empty. Then push the new operator
onto the stack.
3. If the character is an open parenthesis, push it onto the stack.
4. If the character is a close parenthesis, pop operators from the stack and append them to the output until an open parenthesis is encountered. Pop and discard the open parenthesis.
After scanning the entire infix expression, pop any remaining operators from the stack and append them to the output. The resulting output string is the postfix notation of the given infix
2.1.2 Postfix Expression Evaluation
To evaluate a postfix expression, we use a stack to store intermediate results. The algorithm scans the postfix expression from left to right, and for each character encountered, it performs one of
the following actions:
1. If the character is an operand, push it onto the stack.
2. If the character is an operator, pop the top two operands from the stack, perform the operation, and push the result back onto the stack.
After scanning the entire postfix expression, the final result will be on the top of the stack.
2.2 Function Call Management
Stacks are used in programming languages to manage function calls and their associated memory. When a function is called, a new stack frame is created and pushed onto the call stack. The stack frame
contains local variables, function parameters, and return addresses. When the function returns, its stack frame is popped from the call stack, and control is transferred back to the calling function.
This mechanism allows for proper execution of nested and recursive function calls.
2.2.1 Activation Records and Stack Frames
An activation record, also known as a stack frame, is a data structure that contains information about a specific function call. It typically includes the following components:
• Function parameters: The values passed as arguments to the function.
• Local variables: Variables that are declared and used within the function.
• Return address: The memory address to which the control should be transferred when the function returns.
• Control link: A pointer to the previous stack frame, typically the calling function's stack frame.
• Space for intermediate calculations: Memory space for temporary results and intermediate values during the execution of the function.
When a function is called, a new activation record is created and pushed onto the call stack. This ensures proper nesting of function calls and allows for the correct execution of recursive
2.3 Depth-First Search (DFS) Algorithm
Stacks can be used to implement the depth-first search (DFS) algorithm, which is an important graph traversal technique. The DFS algorithm starts at a given vertex and explores as far as possible
along each branch before backtracking. The algorithm can be implemented using an explicit stack data structure or through recursion, which implicitly uses the call stack.
// DFS implementation using an explicit stack
#include <iostream>
#include <list>
#include <stack>
class Graph {
int V;
std::list<int>* adj;
Graph(int V);
void addEdge(int v, int w);
void DFS(int v);
Graph::Graph(int V) {
this->V = V;
adj = new std::list[V];
void Graph::addEdge(int v, int w) {
void Graph::DFS(int v) {
std::stack<int> stack;
bool* visited = new bool[V];
for (int i = 0; i < V; i++) visited[i] = false;
visited[v] = true;
while (!stack.empty()) {
int currentVertex = stack.top();
std::cout << currentVertex << " ";
for (auto i = adj[currentVertex].begin(); i != adj[currentVertex].end(); ++i) {
if (!visited[*i]) {
visited[*i] = true;
In this implementation, an explicit stack is used to store the vertices to be visited. The DFS algorithm starts at the given vertex, marks it as visited, and pushes it onto the stack. While the stack
is not empty, the algorithm pops the top vertex, processes it, and pushes its unvisited neighbors onto the stack. This process continues until all reachable vertices have been visited and the stack
is empty.
3. Relation between Stack and Other Data Structures
Stacks can be related to and used in conjunction with other data structures, such as queues, trees, and graphs. In this section, we will explore the relationships between stacks and these data
structures and their applications in various algorithms and problem-solving scenarios.
3.1 Stack and Queue
While both stacks and queues are linear data structures, they have different operational principles. A stack follows the LIFO (Last In First Out) principle, whereas a queue follows the FIFO (First In
First Out) principle. However, it is possible to simulate the behavior of one data structure using the other.
3.1.1 Implementing a Queue using Two Stacks
A queue can be implemented using two stacks (S1 and S2) with the following algorithm:
1. To enqueue an element, push it onto stack S1.
2. To dequeue an element, perform the following steps:
a. If both stacks are empty, the queue is empty, and there is no element to dequeue.
b. If stack S2 is empty, pop all elements from stack S1 and push them onto stack S2.
c. Pop the top element from stack S2, which is the dequeued element.
The intuition behind this approach is that elements enqueued onto stack S1 will be in reverse order when transferred to stack S2. Popping from stack S2 will then result in the original order being
preserved, as required by the queue's FIFO principle.
3.2 Stack and Trees
Stacks are often used in conjunction with tree data structures to perform various traversal algorithms. The most common tree traversal algorithms are depth-first traversals, such as inorder,
preorder, and postorder traversals. Stacks can be used in both recursive and iterative implementations of these algorithms.
3.2.1 Iterative Inorder Tree Traversal
To perform an iterative inorder tree traversal using a stack, follow these steps:
1. Initialize an empty stack.
2. Push the root node onto the stack and set the current node to the root.
3. While the stack is not empty or the current node is not null, perform the following steps:
a. If the current node is not null, push it onto the stack and move to its left child.
b. If the current node is null, pop a node from the stack, process it, and set the current node to its right child.
This algorithm simulates the inorder traversal of a binary tree without using recursion, which would implicitly use the call stack.
3.3 Stack and Graphs
As mentioned earlier, stacks can be used to implement graph traversal algorithms such as depth-first search (DFS). They can also be used in other graph algorithms, such as topological sorting and
finding strongly connected components in a directed graph.
3.3.1 Topological Sorting
Topological sorting is an algorithm that linearly orders the vertices of a directed acyclic graph (DAG) such that for every directed edge (u, v), vertex u comes before vertex v in the ordering. The
algorithm can be implemented using a modified depth-first search with a stack:
1. Perform a depth-first search on the graph and push vertices onto the stack as they finish their recursion (i.e., when all their descendants have been visited).
2. Pop vertices from the stack to obtain the topological order.
This algorithm works because a vertex is only pushed onto the stack after all its descendants have been visited, ensuring that the vertex comes before its descendants in the topological order.
3.3.2 Strongly Connected Components
Strongly connected components (SCCs) are subgraphs of a directed graph where each vertex is reachable from every other vertex within the same SCC. Tarjan's algorithm is a popular method for finding
SCCs, and it uses a stack and depth-first search to achieve this.
The algorithm maintains a depth-first search tree, assigning a unique index to each vertex, along with a low-link value, which is the smallest index reachable from the vertex. A stack is used to keep
track of vertices that have not yet been assigned to an SCC. When the low-link value of a vertex is equal to its index, it forms the root of an SCC, and all vertices on the stack up to and including
the root are part of the same SCC.
4. Tricky and Technical Aspects of Stack
In this section, we will discuss some of the more advanced and technical aspects of stacks, including memory management, stack overflow, and stack frame optimizations.
4.1 Memory Management
When using a stack, especially in programming languages that do not support automatic memory management, it is essential to allocate and deallocate memory appropriately. Memory leaks can occur if
memory is not freed when elements are popped from the stack. In languages with garbage collection, such as Java, this is less of a concern, as unused memory will be automatically reclaimed by the
garbage collector.
In C++, memory management for a stack implemented using a linked list can be handled using the destructor and copy constructor, ensuring proper allocation and deallocation of memory as elements are
added and removed from the stack.
4.2 Stack Overflow
Stack overflow occurs when the stack size exceeds its maximum capacity. This can happen in two scenarios:
1. When using a statically-allocated stack, an overflow occurs when the number of elements pushed onto the stack exceeds its fixed size. This can lead to data corruption and program crashes, as the
overflowed data may overwrite adjacent memory locations.
2. When using the call stack for recursion, an overflow occurs when the depth of recursion exceeds the available memory for stack frames. This typically results in a program crash or an error
message indicating that the maximum recursion depth has been reached.
To prevent stack overflow, it is essential to carefully manage stack size, ensure proper error handling, and avoid excessive recursion. Using dynamic data structures such as linked lists can help
avoid overflow in explicitly implemented stacks by allowing the stack to grow and shrink as needed.
4.3 Stack Frame Optimizations
Compiler optimizations can help reduce the overhead of stack frame management during function calls. Some of these optimizations include:
• Tail call optimization: When a function call is the last operation in another function, the compiler can optimize the call stack by reusing the current stack frame for the callee function,
effectively eliminating the need for additional stack frames.
• Inlining: The compiler may decide to inline small functions, replacing the function call with the function body itself. This eliminates the need for stack frame allocation and deallocation,
reducing the overhead associated with function calls.
• Stack frame elision: In some cases, the compiler can determine that certain stack frame elements, such as local variables or intermediate calculations, are not necessary and can be safely
eliminated, reducing the size of the stack frame and improving performance.
These optimizations can help improve the performance of programs that rely heavily on function calls and recursion. However, it is important to note that not all compilers or programming languages
support these optimizations, and the degree to which they are applied may vary depending on the specific implementation.
6. Advanced Stack Applications
In this section, we will explore some advanced applications of stacks, such as backtracking, memory management in programming languages, and parallel programming.
6.1 Backtracking
Backtracking is a general algorithm for finding all or some solutions to a problem that incrementally builds candidates to the solutions and abandons a candidate ("backtracks") as soon as it
determines that the candidate cannot be extended to a valid solution. Stacks are often used to store partial solutions in backtracking algorithms.
Some common problems that can be solved using backtracking with stacks include:
• Eight Queens Puzzle: Place eight chess queens on an 8x8 chessboard so that no two queens threaten each other.
• Hamiltonian Cycle: Find a cycle in a given graph that visits every vertex exactly once.
• Subset Sum: Given a set of integers and an integer value, determine if there is a subset of the given set with a sum equal to the given value.
In these problems, stacks can be used to store partial solutions, and the algorithm backtracks by popping elements from the stack whenever it reaches an invalid or incomplete solution.
6.2 Memory Management in Programming Languages
Many programming languages, such as C, C++, Java, and Python, use a combination of stack and heap memory to manage memory allocation and deallocation for variables and objects. The stack memory is
used for static memory allocation, such as function call frames and local variables, while the heap memory is used for dynamic memory allocation, such as objects and arrays created during runtime.
Understanding the role of the stack in memory management is crucial for writing efficient code and avoiding memory-related issues such as stack overflow and memory leaks. By managing the size of the
stack and the use of stack frames, programmers can optimize the performance and memory usage of their programs.
6.3 Parallel Programming
In parallel programming, multiple threads or processes work concurrently to execute tasks and solve problems. Stacks can be used in parallel programming for various purposes, such as managing the
call stack for each thread, storing intermediate results, or implementing thread-safe data structures.
When using stacks in parallel programming, it is essential to consider synchronization and contention issues. For example, when multiple threads access a shared stack, they may need to lock the stack
to avoid race conditions and ensure proper operation. This can be achieved using various synchronization techniques, such as mutexes, semaphores, or atomic operations.
Another consideration in parallel programming with stacks is the design of efficient, lock-free, or wait-free stack data structures that allow for high concurrency and low contention. These data
structures can improve the performance and scalability of parallel programs by minimizing the overhead of synchronization.
6.4 Coroutine and Stackful Coroutine
Coroutines are a general control structure that allows the flow of control to be cooperatively passed between two different routines without returning. Stackful coroutines, also known as asymmetric
coroutines, maintain their own stack for each coroutine, allowing for more advanced control flow and non-linear execution.
Stackful coroutines are particularly useful in implementing cooperative multitasking, where multiple tasks can run concurrently without the need for preemptive context switching. Stacks are essential
in stackful coroutines, as each coroutine has its own stack for managing function calls, local variables, and control flow.
Some programming languages, such as Lua and Python, provide support for stackful coroutines through libraries or language constructs. In these languages, stacks play a crucial role in enabling
advanced control flow and concurrency patterns.
7. Relation between Stack and Other Data Structures
In this section, we will explore the relationship between stacks and other data structures such as trees, deques, and queues, and how stacks can be used in conjunction with these data structures to
solve problems and implement algorithms.
7.1 Stack and Deque
A deque (short for double-ended queue) is a linear data structure that allows elements to be added or removed from both ends efficiently. Deques can be seen as a generalization of both stacks and
queues, as they support LIFO (Last In First Out) and FIFO (First In First Out) operations.
Stacks can be easily implemented using a deque by restricting insertions and deletions to one end only. Similarly, a deque can be simulated using two stacks with appropriate operations to maintain
the elements in the correct order. Understanding the relationship between stacks and deques is essential for designing efficient algorithms and data structures that require both LIFO and FIFO
7.2 Stack and Priority Queue
A priority queue is a data structure that supports efficiently inserting elements and retrieving the element with the highest priority. Unlike stacks, which follow the LIFO principle, priority queues
order elements based on their priority rather than their insertion order.
Priority queues can be implemented using various data structures, such as binary heaps, Fibonacci heaps, or even stacks. When using stacks to implement a priority queue, additional data structures
and algorithms may be required to maintain the elements in the correct order based on their priority. For example, two stacks can be used to implement a priority queue by maintaining a sorted order
of elements, with one stack holding the minimum elements and the other holding the maximum elements.
Understanding the relationship between stacks and priority queues is essential for designing efficient algorithms that require priority-based element retrieval, such as Dijkstra's shortest path
algorithm or the A* search algorithm.
7.3 Stack and Binary Search Tree
Binary search trees (BSTs) are tree data structures that maintain a sorted order of elements, allowing for efficient search, insert, and delete operations. Stacks can be used in conjunction with
binary search trees to implement various tree traversal algorithms, such as inorder, preorder, and postorder traversals.
As previously mentioned in section 3.2, stacks can be used in both recursive and iterative implementations of these traversal algorithms. By understanding the relationship between stacks and binary
search trees, one can design efficient algorithms for tree operations and problem-solving that involve sorted data structures.
8. Stack and Graph Algorithms
Stacks can be used in conjunction with graph algorithms to solve various problems and improve the efficiency of existing graph algorithms. In this section, we will discuss a few graph algorithms that
can benefit from the use of stacks.
8.1 Depth-First Search with Iterative Deepening
Depth-First Search (DFS) is a graph traversal algorithm that explores the vertices of a graph in a depthward motion, visiting a vertex's children before its siblings. DFS can be implemented both
recursively and iteratively using a stack. Iterative Deepening Depth-First Search (IDDFS) is a combination of DFS and Breadth-First Search (BFS) that performs DFS up to a certain depth, then
increases the depth limit and repeats the process until the desired node is found or the entire graph has been traversed.
IDDFS can be more efficient than regular DFS for searching large graphs with a small solution depth, as it combines the space efficiency of DFS with the optimal search properties of BFS. Using a
stack to implement IDDFS allows for efficient traversal of the graph while maintaining a low memory footprint.
8.2 All-Pairs Shortest Paths
The all-pairs shortest paths problem involves finding the shortest paths between all pairs of vertices in a graph. The Floyd-Warshall algorithm is a popular solution to this problem, with a time
complexity of O(n^3), where n is the number of vertices in the graph. However, an alternative solution using stacks can be employed to improve the efficiency of the algorithm for certain types of
The All-Pairs Shortest Paths with Stack (APSPS) algorithm uses a stack to keep track of intermediate vertices while traversing the graph. This allows the algorithm to avoid unnecessary calculations
and reduce the overall time complexity. Although the worst-case time complexity of APSPS is still O(n^3), it can be significantly faster for sparse graphs with a small number of edges.
8.3 Maximum Flow
The maximum flow problem involves finding the maximum amount of flow that can be sent from a source vertex to a sink vertex in a flow network. The Ford-Fulkerson algorithm is a well-known method for
solving the maximum flow problem, using augmenting paths to iteratively increase the flow until no more augmenting paths can be found.
Stacks can be used in the Ford-Fulkerson algorithm to implement DFS for finding augmenting paths in the residual graph. Using a stack for DFS in the Ford-Fulkerson algorithm can help reduce the
memory overhead and improve the overall efficiency of the algorithm.
9. Stack in Compiler Design and Parsing
Stacks play a crucial role in compiler design and parsing, particularly in the process of syntax analysis and semantic analysis. In this section, we will discuss how stacks are used in these aspects
of compiler design.
9.1 Stack in Syntax Analysis
Syntax analysis, also known as parsing, is the process of analyzing a sequence of tokens in a programming language to determine its grammatical structure. One of the most common parsing techniques is
the top-down parsing method called Recursive Descent Parsing, which uses a set of recursive procedures to process the input.
Stacks can be used to implement a non-recursive variant of Recursive Descent Parsing called Predictive Parsing. Predictive Parsing uses a stack to keep track of the production rules and input symbols
being processed. By using a stack, the parser can efficiently backtrack and try alternative production rules when necessary, allowing for a more efficient and flexible parsing process.
9.2 Stack in Semantic Analysis
Semantic analysis is the process of analyzing the meaning of a program's source code by checking for semantic errors, such as type mismatches and undeclared variables. Stacks are often used in
semantic analysis to manage the symbol table, a data structure that stores information about identifiers (such as variables, functions, and types) in the program's scope.
As the compiler processes the source code, it encounters various scopes (e.g., function or block scopes) that may introduce new identifiers or hide existing ones. By using a stack to manage the
symbol table, the compiler can efficiently handle the nesting of scopes and the visibility of identifiers, enabling accurate semantic analysis and error detection.
9.3 Stack in Intermediate Code Generation
Intermediate code generation is a stage in compiler design where the compiler generates an intermediate representation of the source code that is easier to optimize and translate to the target
machine code. One of the most common intermediate representations is the Three-Address Code (TAC), which represents the program as a sequence of instructions with at most three operands.
Stacks can be used in the generation of TAC to manage the evaluation of expressions and the allocation of temporary variables. By using a stack, the compiler can efficiently evaluate complex
expressions, generate intermediate code, and manage the lifetimes of temporary variables, leading to a more efficient and optimized intermediate representation.
10. Stack in Virtual Machines and Interpreters
Stacks play a vital role in the implementation of virtual machines and interpreters, which are used to execute programs written in high-level programming languages. In this section, we will discuss
how stacks are utilized in virtual machines and interpreters for various purposes.
10.1 Stack-based Virtual Machines
Stack-based virtual machines, such as the Java Virtual Machine (JVM) and the .NET Common Language Runtime (CLR), use stacks as the primary means of managing the execution state and data manipulation
during the execution of a program. In a stack-based virtual machine, operands are pushed onto the stack, and operations are performed by popping operands from the stack, processing them, and pushing
the result back onto the stack.
Using a stack as the primary data structure in a virtual machine simplifies the implementation and allows for more efficient execution of programs. Stack-based virtual machines can also take
advantage of various stack-based optimizations, such as constant folding and peephole optimization, to improve the performance of the executed code.
10.2 Stack in Interpreter Loop
Interpreters, such as those used for scripting languages like Python, JavaScript, and Lua, use a stack to manage the execution state and evaluate expressions during the interpretation of the program.
The interpreter loop, also known as the Read-Eval-Print Loop (REPL), reads a line of code, evaluates it, and prints the result before reading the next line.
Stacks are used in the interpreter loop to manage the call stack, store local variables and intermediate results, and evaluate expressions. By using a stack, the interpreter can efficiently manage
the execution state and evaluate complex expressions, allowing for the rapid execution of programs and interactive development in the REPL environment.
10.3 Stack in Bytecode Interpretation
Many interpreters and virtual machines use a bytecode representation of the program, which is a low-level, platform-independent representation of the source code. Bytecode interpreters, such as the
Python Virtual Machine (PVM) and the Lua Virtual Machine (LVM), use stacks to manage the execution state, evaluate expressions, and perform operations during the interpretation of the bytecode.
Stacks are used to store operands, local variables, and intermediate results during the execution of bytecode instructions. By using a stack, the bytecode interpreter can efficiently manage the
execution state and perform operations, leading to a more efficient and portable execution of the program across different platforms and environments.
11. Frequently Asked Questions
11.1 What is the difference between a stack and a heap?
A stack is a linear data structure that follows the LIFO principle, while a heap is a tree-based data structure that organizes elements based on their priorities or values. Stacks are used for
managing function calls, local variables, and control flow, while heaps are used for dynamic memory allocation and implementing priority queues.
11.2 Can a stack overflow?
Yes, a stack can overflow when it reaches its maximum capacity. Stack overflow occurs when too many function calls or local variables are pushed onto the stack, causing it to exceed its available
memory. Stack overflow can lead to program crashes or undefined behavior and is often the result of recursive function calls, infinite loops, or insufficient memory allocation.
11.3 Can stacks be implemented using other data structures?
Yes, stacks can be implemented using other data structures such as arrays, linked lists, or even other abstract data types like queues and deques. The choice of data structure for implementing a
stack depends on the specific requirements and performance characteristics of the application.
12. Further Exploration
The study of stacks in algorithms and data structures is an essential aspect of computer science and software engineering. This article has provided a comprehensive overview of the stack data
structure, its applications, and its relationship with other data structures. However, there are many more topics and techniques related to stacks that you can explore, including:
• Advanced stack-based optimization techniques
• Design and implementation of domain-specific stack languages
• Analysis of stack usage in various programming languages and paradigms
• Alternative stack implementations and data structures
• Stack-based parallel and concurrent algorithms
• Applications of stacks in artificial intelligence and machine learning
By delving deeper into these topics, you can gain a greater understanding of the intricacies of stacks, their applications, and their role in the broader field of computer science.
13. Conclusion
This article has provided an in-depth examination of the stack data structure and its various applications, from basic programming concepts to advanced algorithms and data structures. We have
discussed the implementation of stacks using arrays and linked lists, as well as the relationship between stacks and other data structures such as queues, trees, and deques. We have also explored the
use of stacks in compiler design, virtual machines, and interpreters, and their role in various graph algorithms. The knowledge and understanding gained from this article should serve as a solid
foundation for further exploration and study in the field of computer science. | {"url":"https://dmj.one/edu/su/course/csu1051/class/stack","timestamp":"2024-11-08T20:16:57Z","content_type":"text/html","content_length":"44465","record_id":"<urn:uuid:6e015ab1-3f98-44cf-ab0f-0475b8fab1f7>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00644.warc.gz"} |
On success, these functions return the natural logarithm of x.
If x is a NaN, a NaN is returned.
If x is 1, the result is +0.
If x is positive infinity, positive infinity is returned.
If x is zero, then a pole error occurs, and the functions return -HUGE_VAL, -HUGE_VALF, or -HUGE_VALL, respectively.
If x is negative (including negative infinity), then a domain error occurs, and a NaN (not a number) is returned. | {"url":"https://manual.cs50.io/3/logl","timestamp":"2024-11-11T14:07:50Z","content_type":"text/html","content_length":"13131","record_id":"<urn:uuid:a1cf8227-c094-4098-a6d1-2b59138e18fc>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00018.warc.gz"} |
LTV model for MPC, linearization at each prediction horizon step
This is my first post so please correct me if I’m doing something wrong.
I have the non-linear model of my system and successfully implemented a non-linear MPC using matlab as interface. However, the solving time is a bit to high on the target hardware. I would like to
linearize my model and use a different solver to achieve lower computational time. The main problem is that, to have a coherent model, I think that I should linearize at each step of the prediction
horizon using the previous step states and control actions. Is there a way to do so?
Thank you in advance for any help
1 Like
Hi Fabio,
I think this is exactly what an SQP method (or the RTI) does. It solvers at every MPC step an QP (you can think of it as local linear MPC) - where the QP is obtained by evaluating the linearization’s
at the previous iterate. For more technical details on this you can read https://publications.syscop.de/Gros2016.pdf which exactly explains the connection between LTV MPC and nonlinear MPC.
To help you with your problem in practice, you probably have tried different horizon lengths, samplings time.
You may use also multiphase OCP (lookt at the acados examplse) and have a finer discretization at the beginning and coarser in the reminder of the MPC prediction horizon, and this can save you lot of
computation time without harming the closed loop performance much.
Do you have a least squres objective? What kind of numerical integration do you use in your MPC?
Changing these things can speed up quite a bit.
Hi, thank you for the reply!
I set cost_type to ‘linear_ls’ and sim_method to ‘erk’.
I’m indeed using as nlp_solver ‘sqp_rti’, I got what you say but shouldn’t it be faster if it wouldn’t need to lineatize but just replace the previous iteration states in a matrix? If I linearize the
system I would get the matrices which would be function of the linearization point, for the solver it would be just a plug in and matrices operations.
Thank you for the insights, I’m a student and I still have to learn a lot about optimization | {"url":"https://discourse.acados.org/t/ltv-model-for-mpc-linearization-at-each-prediction-horizon-step/1845","timestamp":"2024-11-03T18:41:53Z","content_type":"text/html","content_length":"20220","record_id":"<urn:uuid:a3e52729-cd84-4ab7-bdd9-c1c902edf51e>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00649.warc.gz"} |
Friedman Test Calculator | Online Tutorials Library List | Tutoraspire.com
Friedman Test Calculator
by Tutor Aspire
The Friedman Test is the non-parametric alternative to the one-way ANOVA with repeated measures. It is used to test for differences between groups when the dependent variable is ordinal.
To perform a Friedman Test for a given dataset, simply enter the values for up to five samples into the cells below, then press the “Calculate” button.
The calculator will output the test statistic Q, the p-value of the test, and the calculations that were used to derive the test statistic Q.
Group 1 Group 2 Group 3 Group 4 Group 5
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Stata Tips, Fourth Edition
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Volume I: Tips 1-119
Volume II: Tips 120-152
Authors: Nicholas J. Cox (editor)
Publisher: Stata Press
Copyright: 2024
ISBN-13: 978-1-59718-405-2
Pages: 510; paperback
Volume I: Tips 1-119
ISBN-13: 978-1-59718-407-6
Pages: 327; paperback
Volume II: Tips 120-152
ISBN-13: 978-1-59718-409-0
Pages: 183; paperback
Stata Tips provides concise and insightful notes about commands, features, and tricks that will help you obtain a deeper understanding of Stata.
The book comprises the contributions of the Stata community that have appeared in the Stata Journal since 2003. Each tip is a brief article that provides practical advice on using Stata. With tips
covering a breadth of topics in statistics, graphics, data management, and programming, both new and experienced Stata users are sure to find tips that will be useful in their research.
Nicholas Cox is a statistically minded geographer at Durham University. He contributes talks, postings, FAQ, and programs to the Stata user community. He has also coauthored 16 commands in official
Stata. He was an author of several inserts in the Stata Technical Bulletin and is Editor-at-Large of the Stata Journal.
Introducing Stata tips
Stata tip 1: The eform() option of regress, R. Newson
Stata tip 2: Building with floors and ceilings, N. J. Cox
Stata tip 3: How to be assertive, W. Gould
Stata tip 4: Using display as an online calculator, P. Ryan
Stata tip 5: Ensuring programs preserve dataset order, R. Newson
Stata tip 6: Inserting awkward characters in the plot, N. J. Cox
Stata tip 7: Copying and pasting under Windows, S. Driver and P. Royston
Stata tip 8: Splitting time-span records with categorical time-varying covariates, B. Jann
Stata tip 9: Following special sequences, N. J. Cox
Stata tip 10: Fine control of axis title positions, P. Ryan and N. Winter
Stata tip 11: The nolog option with maximum-likelihood modeling commands, P. Royston
Stata tip 12: Tuning the plot region aspect ratio, N. J. Cox
Stata tip 13: generate and replace use the current sort order, R. Newson
Stata tip 14: Using value labels in expressions, K. Higbee
Stata tip 15: Function graphs on the fly, N. J. Cox
Stata tip 16: Using input to generate variables, U. Kohler
Stata tip 17: Filling in the gaps, N. J. Cox
Stata tip 18: Making keys functional, S. Driver
Stata tip 19: A way to leaner, faster graphs, P. Royston
Stata tip 20: Generating histogram bin variables, D. A. Harrison
Stata tip 21: The arrows of outrageous fortune, N. J. Cox
Stata tip 22: Variable name abbreviation, P. Ryan
Stata tip 23: Regaining control over axis ranges, N. Winter
Stata tip 24: Axis labels on two or more levels, N. J. Cox
Stata tip 25: Sequence index plots, U. Kohler and C. Brzinsky-Fay
Stata tip 26: Maximizing compatibility between Macintosh and Windows, M. S. Hanson
Stata tip 27: Classifying data points on scatter plots, N. J. Cox
Stata tip 28: Precise control of dataset sort order, P. Schumm
Stata tip 29: For all times and all places, C. H. Franklin
Stata tip 30: May the source be with you, N. J. Cox
Stata tip 31: Scalar or variable? The problem of ambiguous names, G. I. Kolev
Stata tip 32: Do not stop, S. P. Jenkins
Stata tip 33: Sweet sixteen: Hexadecimal formats and precision problems, N. J. Cox
Stata tip 34: Tabulation by listing, D. A. Harrison
Stata tip 35: Detecting whether data have changed, W. Gould
Stata tip 36: Which observations?, N. J. Cox
Stata tip 37: And the last shall be first, C. F. Baum
Stata tip 38: Testing for groupwise heteroskedasticity, C. F. Baum
Stata tip 39: In a list or out? In a range or out?, N. J. Cox
Stata tip 40: Taking care of business, C. F. Baum
Stata tip 41: Monitoring loop iterations, D. A. Harrison
Stata tip 42: The overlay problem: Offset for clarity, J. Cui
Stata tip 43: Remainders, selections, sequences, extractions: Uses of the modulus, N. J. Cox
Stata tip 44: Get a handle on your sample, B. Jann
Stata tip 45: Getting those data into shape, C. F. Baum and N. J. Cox
Stata tip 46: Step we gaily, on we go, R. Williams
Stata tip 47: Quantile–quantile plots without programming, N. J. Cox
Stata tip 48: Discrete uses for uniform(), M. L. Buis
Stata tip 49: Range frame plots, S. Merryman
Stata tip 50: Efficient use of summarize, N. J. Cox
Stata tip 51: Events in intervals, N. J. Cox
Stata tip 52: Generating composite categorical variables, N. J. Cox
Stata tip 53: Where did my p-values go?, M. L. Buis
Stata tip 54: Post your results, P. Van Kerm
Stata tip 55: Better axis labeling for time points and time intervals, N. J. Cox
Stata tip 56: Writing parameterized text files, R. Gini
Stata tip 57: How to reinstall Stata, W. Gould
Stata tip 58: nl is not just for nonlinear models, B. P. Poi
Stata tip 59: Plotting on any transformed scale, N. J. Cox
Stata tip 60: Making fast and easy changes to files with filefilter, A. R. Riley
Stata tip 61: Decimal commas in results output and data input, N. J. Cox
Stata tip 62: Plotting on reversed scales, N. J. Cox and N. L. M. Barlow
Stata tip 63: Modeling proportions, C. F. Baum
Stata tip 64: Cleaning up user-entered string variables, J. Herrin and E. Poen
Stata tip 65: Beware the backstabbing backslash, N. J. Cox
Stata tip 66: ds—A hidden gem, M. Weiss
Stata tip 67: J() now has greater replicating powers, N. J. Cox
Stata tip 68: Week assumptions, N. J. Cox
Stata tip 69: Producing log files based on successful interactive commands, A. R. Riley
Stata tip 70: Beware the evaluating equal sign, N. J. Cox
Stata tip 71: The problem of split identity, or how to group dyads, N. J. Cox
Stata tip 72: Using the Graph Recorder to create a pseudograph scheme, K. Crow
Stata tip 73: append with care!, C. F. Baum
Stata tip 74: firstonly, a new option for tab2, R. G. Gutierrez and P. A. Lachenbruch
Stata tip 75: Setting up Stata for a presentation, K. Crow
Stata tip 76: Separating seasonal time series, N. J. Cox
Stata tip 77: (Re)using macros in multiple do-files, J. Herrin
Stata tip 78: Going gray gracefully: Highlighting subsets and downplaying substrates, N. J. Cox
Stata tip 79: Optional arguments to options, N. J. Cox
Stata tip 80: Constructing a group variable with specified group sizes, M. Weiss
Stata tip 81: A table of graphs, M. L. Buis and M. Weiss
Stata tip 82: Grounds for grids on graphs, N. J. Cox
Stata tip 83: Merging multilingual datasets, D. L. Golbe
Stata tip 84: Summing missings, N. J. Cox
Stata tip 85: Looping over nonintegers, N. J. Cox
Stata tip 86: The missing() function, B. Rising
Stata tip 87: Interpretation of interactions in nonlinear models, M. L. Buis
Stata tip 88: Efficiently evaluating elastics with the margins command, C. F. Baum
Stata tip 89: Estimating means and percentiles following multiple imputation, P. A. Lachenbruch
Stata tip 90: Displaying partial results, M. Weiss
Stata tip 91: Putting unabbreviated varlists into local macros, N. J. Cox
Stata tip 92: Manual implementation of permutations and bootstraps, L. Ãngquist
Stata tip 93: Handling multiple y axes on twoway graphs, V. Wiggins
Stata tip 94: Manipulation of prediction parameters for parametric survival regression models, T. Boswell and R. G. Gutierrez
Stata tip 95: Estimation of error covariances in a linear model, N. J. Horton
Stata tip 96: Cube roots, N. J. Cox
Stata tip 97: Getting at ρ's and σ's, M. L. Buis
Stata tip 98: Counting substrings within strings, N. J. Cox
Stata tip 99: Taking extra care with encode, C. Schechter
Stata tip 100: Mata and the case of the missing macros, W. Gould and N. J. Cox
Stata tip 101: Previous but different, N. J. Cox
Stata tip 102: Highlighting specific bars, N. J. Cox
Stata tip 103: Expressing confidence with gradations, U. Kohler and S. Eckman
Stata tip 104: Added text and title options, N. J. Cox
Stata tip 105: Daily dates with missing days, S. J. Samuels and N. J. Cox
Stata tip 106: With or without reference, M. L. Buis
Stata tip 107: The baseline is now reported, M. L. Buis
Stata tip 108: On adding and constraining, M. L. Buis
Stata tip 109: How to combine variables with missing values, P. A. Lachenbruch
Stata tip 110: How to get the optimal k-means cluster solution, A. Makles
Stata tip 111: More on working with weeks, N. J. Cox
Stata tip 112: Where did my p-values go? (Part 2), M. L. Buis
Stata tip 113: Changing a variable's format: What it does and does not mean, N. J. Cox
Stata tip 114: Expand paired dates to pairs of dates, N. J. Cox
Stata tip 115: How to properly estimate the multinomial probit model with heteroskedastic errors, M. Herrmann
Stata tip 116: Where did my p-values go? (Part 3), M. L. Buis
Stata tip 117: graph combine—Combining graphs, L. Ãngquist
Stata tip 118: Orthogonalizing powered and product terms using residual centering, C. Sauer
Stata tip 119: Expanding datasets for graphical ends, N. J. Cox
Introducing Stata tips
Stata tip 120: Certifying subroutines, M. L. Buis
Stata tip 121: Box plots side by side, N. J. Cox
Stata tip 122: Variable bar widths in two-way graphs, B. Jann
Stata tip 123: Spell boundaries, N. J. Cox
Stata tip 124: Passing temporary variables to subprograms, M. L. Buis
Stata tip 125: Binned residual plots for assessing the fit of regression models for binary outcomes, J. Kasza
Stata tip 126: Handling irregularly spaced high-frequency transactions data, C. F. Baum and S. Bibo
Stata tip 127: Use capture noisily groups, R. B. Newson
Stata tip 128: Marginal effects in log-transformed models: A trade application, L. J. Uberti
Stata tip 129: Efficiently processing textual data with Stata’s new Unicode features, A. Koplenig
Stata tip 130: 106610 and all that: Date variables that need to be fixed., N. J. Cox
Stata tip 131: Custom legends for graphs that use translucency, T. P. Morris
Stata tip 132: Tiny tricks and tips on ticks, N. J. Cox and V. Wiggins
Stata tip 133: Box plots that show median and quartiles only, N. J. Cox
Stata tip 134: Multiplicative and marginal interaction effects in nonlinear models, W. H. Dow, E. C. Norton, and J. T. Donahoe
Stata tip 135: Leaps and bounds, M. L. Buis
Stata tip 136: Between-group comparisons in a scatterplot with weighted markers, A. Musau
Stata tip 137: Interpreting constraints on slopes of rank-deficient design matrices, D. Christodoulou
Stata tip 138: Local macros have local scope, N. J. Cox
Stata tip 139: The by() option of graph can work better than graph combine, N. J. Cox
Stata tip 140: Shorter or fewer category labels with graph bar, N. J. Cox
Stata tip 141: Adding marginal spike histograms to quantile and cumulative distribution plots, N. J. Cox
Stata tip 142: joinby is the real merge m:m, D. Mazrekaj and J. Wursten
Stata tip 143: Creating donut charts in Stata, A. Musau
Stata tip 144: Adding variable text to graphs that use a by() option, N. J. Cox
Stata tip 145: Numbering weeks within months, N. J. Cox
Stata tip 146: Using margins after a Poisson regression model to estimate the number of events prevented by an intervention, M. Falcaro, R. B. Newson, and P. Sasieni
Erratum: Stata tip 145: Numbering weeks within months, N. J. Cox
Stata tip 147: Porting downloaded packages between machines, R. B. Newson
Stata tip 148: Searching for words within strings, N. J. Cox
Stata tip 149: Weighted estimation of fixed-effects and first-differences models, J. Gardner
Stata tip 150: When is it appropriate to xtset a panel dataset with panelvar only? C. Lazzaro
Stata tip 151: Puzzling out some logical operators, N. J. Cox
Stata tip 152: if and if: When to use the if qualifier and when to use the if command, N. J. Cox and C. B. Schechter | {"url":"https://jat.co.kr/shopb/62","timestamp":"2024-11-08T20:18:52Z","content_type":"text/html","content_length":"98517","record_id":"<urn:uuid:e47d781f-8934-4357-b722-920661861ebd>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00058.warc.gz"} |
Solve a tridiagonal matrix
tridiagmatrix {cmna} R Documentation
Solve a tridiagonal matrix
use the tridiagonal matrix algorithm to solve a tridiagonal matrix
tridiagmatrix(L, D, U, b)
L vector of entries below the main diagonal
D vector of entries on the main diagonal
U vector of entries above the main diagonal
b vector of the right-hand side of the linear system
tridiagmatrix uses the tridiagonal matrix algorithm to solve a tridiagonal matrix.
the solution vector
See Also
Other linear: choleskymatrix(), detmatrix(), gdls(), invmatrix(), iterativematrix, lumatrix(), refmatrix(), rowops, vecnorm()
version 1.0.5 | {"url":"https://search.r-project.org/CRAN/refmans/cmna/html/tridiagmatrix.html","timestamp":"2024-11-13T09:12:11Z","content_type":"text/html","content_length":"2993","record_id":"<urn:uuid:45d1976b-8548-4922-a9af-6ed4f316811c>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00677.warc.gz"} |
Annual interest rate example
Annual Percentage Rate (APR) Calculator. Loan Amount. $. Interest Rate. %. Term. Yr. Finance Charges (Added to loan amount). $. Prepaid Finance Charges examples of the types of confusion that we
found is presented and evaluated. Keywords: Annual Percentage Rate; APR, Annual Effective Rate; AER; Nominal To calculate the effective annual interest rate, when the nominal rate and compounding
periods are given, you can use the EFFECT function. In the example
13 Jul 2017 The daily periodic interest rate generally can be calculated by dividing the annual percentage rate, or APR, by either 360 or 365, depending on An Effective Interest Rate is also known
as effective annual interest rate, AER - annual equivalent rate and expressed as a percentage in yearly basis. This is the 21 May 2015 The interest rate reflects the base cost of borrowing money
and is a percentage of the principal loan amount. For example, the average interest Simply put, interest rates determine the amount paid by borrowers (debtors) for to compare the annual interest
rates with different compounding terms (daily,
The annual percentage rate (APR) that you are charged on a loan may not be the amount of interest you actually pay. The amount of interest you effectively pay
The annual percentage rate (APR) on a mortgage is a better indication of the true cost By changing any value in the following form fields, calculated values are For example, what if you want to
compare a 30-year fixed-rate mortgage at 7 The annual percentage rate (APR) that you hear so much about allows you to 18 Jun 2019 Calculation of interest amount and interest rate. Loan interest
amount is being calculated based on the nominal rate, whereas the annual 13 Jul 2017 The daily periodic interest rate generally can be calculated by dividing the annual percentage rate, or APR, by
either 360 or 365, depending on An Effective Interest Rate is also known as effective annual interest rate, AER - annual equivalent rate and expressed as a percentage in yearly basis. This is the
21 Feb 2020 The effective annual interest rate is calculated by taking the nominal interest rate and adjusting it for the number of compounding periods the
On the contrary, APR or Annual Percentage Rate is the amount that includes the nominal interest rate, processing fees, penalties and all other charges that are The Windows-based version of the
Annual Percentage Rate program (APRWIN v 6.2 - Released 5/2008) is an efficient tool for verifying annual percentage rates The nominal rate is the interest rate as stated, usually compounded more
than once per year. The effective rate (or effective annual rate) is a rate that, compounded annually, gives the same interest as Example 1. Suppose we want to find 4.2 Calculation of the EAIR. •
EAIR – “the Effective Annual Interest Rate”. • The EAIR is the true, annual rate given a frequency of compounding within the year.
18 Jun 2019 Calculation of interest amount and interest rate. Loan interest amount is being calculated based on the nominal rate, whereas the annual
Example 3: In a bank, an amount of Rs. 20,000 is deposited for one year. The rate of interest is 8% per annum and is compounded semi-annually. What is the If you are shopping around for a personal
loan, you have no doubt seen banks advertise two different interest rates: Annual Flat Rate and Effective Interest Rate Annual Percentage Rate (APR) Calculator. Loan Amount. $. Interest Rate. %.
Term. Yr. Finance Charges (Added to loan amount). $. Prepaid Finance Charges examples of the types of confusion that we found is presented and evaluated. Keywords: Annual Percentage Rate; APR, Annual
Effective Rate; AER; Nominal
Effective Period Rate = Nominal Annual Rate / n. Effective annual interest rate calculation. The effective interest rate is equal to 1 plus the nominal interest rate in percent divided by the number
of compounding persiods per year n, to the power of n, minus 1. Effective Rate = (1 + Nominal Rate / n) n - 1 . Effective interest rate calculation
These fees are considered, however, in the calculation of the annual percentage rate. Covers the compound-interest formula, and gives an example of how to use it. amount, "P" is the beginning amount
(or "principal"), "r" is the interest rate ( expressed If interest is compounded yearly, then n = 1; if semi-annually, then n = 2; APR is used for comparing credit cards and unsecured loans, and is
expressed as a percentage of the amount you've borrowed. For example, a personal loan 23 Jul 2013 Annual Interest Rate Equation. If the lender offers a loan at 1% per month and it compounds monthly,
then the annual percentage rate (APR) on 22 Aug 2019 The Annual Percentage Rate (APR) is a calculation of the overall cost of your loan. It is expressed as an annual rate that represents the actual
«Nominal rate» - is the annual rate of interest on the credit, which is designated in the agreement with the Bank. In this example – is 18% (0, 18). «Number of It is calculated on a daily basis, so
your APR must be converted to a daily rate. The math equation for that is annual percentage rate (APR) ÷ 365 (number of
To calculate the effective annual interest rate, when the nominal rate and compounding periods are given, you can use the EFFECT function. In the example The most common and comparable interest rate
is the APR (annual percentage The EIR calculation is used in cases where interest is compounded, i.e. when These fees are considered, however, in the calculation of the annual percentage rate.
Covers the compound-interest formula, and gives an example of how to use it. amount, "P" is the beginning amount (or "principal"), "r" is the interest rate ( expressed If interest is compounded
yearly, then n = 1; if semi-annually, then n = 2; APR is used for comparing credit cards and unsecured loans, and is expressed as a percentage of the amount you've borrowed. For example, a personal | {"url":"https://topbinhpaupmb.netlify.app/riston61514xyf/annual-interest-rate-example-boga.html","timestamp":"2024-11-14T20:16:46Z","content_type":"text/html","content_length":"33275","record_id":"<urn:uuid:4e1b6bb5-a206-4b44-a13f-64b0fbd313a3>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00397.warc.gz"} |
E-mail List Archives
Thread: Question - Presenting simple mathematic expression
Number of posts in this thread: 4 (In chronological order)
From: Rabab Gomaa
Date: Tue, Apr 29 2014 12:20PM
Subject: Question - Presenting simple mathematic expression
No previous message | Next message →
Jaws does not recognize <sup> for simple mathematics expressions such as =
"10 <sup>4</sup>".=20
e.g. it reads "One hundred four" instead of "10 to the power of 4" .
I am looking for a way to present simple mathematics expression instead of =
using MathML.=20
Thank you,
Rabab Gomaa
From: Joe Chidzik
Date: Wed, Apr 30 2014 7:54AM
Subject: Re: Question - Presenting simple mathematic expression
← Previous message | Next message →
MathJax provides a way of using TeX notation to display equations. There is also pretty good accessibility support provided by MathJax. http://www.mathjax.org/resources/articles-and-presentations/
Without using some sort of plugin, or markup, I can't think of a simple way of conveying mathematical semantics short of writing out equations long hand as you have done e.g. "10 to the power 4"
From: Noble,Stephen L.
Date: Wed, Apr 30 2014 8:20AM
Subject: Re: Question - Presenting simple mathematic expression
← Previous message | Next message →
If the original question--" I am looking for a way to present simple mathematics expression instead of using MathML"--is centered on an easier way of hand-coding simple math expression within a web
page, then I would also suggest using MathJax to render the math, and perhaps making the hand-coding issue even simpler by using AsciiMath which MathJax can also pick up assuming you set it up
properly. See http://docs.mathjax.org/en/latest/asciimath.html for more details.
Of course, if one is constructing a webpage, then one *should* use MathML, even for such simple expressions. If the concern is for cross-browser support, then that is where MathJax comes in, since
MathJax can serve MathML to browsers that can use it, even if TeX or AsciiMath is used as the original markup.
To get JAWS to read the math, though, you'll have to use IE+MathPlayer.
--Steve Noble
= EMAIL ADDRESS REMOVED =
From: = EMAIL ADDRESS REMOVED = [ = EMAIL ADDRESS REMOVED = ] on behalf of Joe Chidzik [ = EMAIL ADDRESS REMOVED = ]
Sent: Wednesday, April 30, 2014 9:54 AM
To: WebAIM Discussion List
Subject: Re: [WebAIM] Question - Presenting simple mathematic expression
MathJax provides a way of using TeX notation to display equations. There is also pretty good accessibility support provided by MathJax. http://www.mathjax.org/resources/articles-and-presentations/
Without using some sort of plugin, or markup, I can't think of a simple way of conveying mathematical semantics short of writing out equations long hand as you have done e.g. "10 to the power 4"
From: Jonathan C. Cohn
Date: Wed, Apr 30 2014 8:23AM
Subject: Re: Question - Presenting simple mathematic expression
← Previous message | No next message
I understand that the latest version of box for Google Chrome supports math checks.
Sent from my iPhone
> On Apr 30, 2014, at 9:54 AM, Joe Chidzik < = EMAIL ADDRESS REMOVED = > wrote:
> MathJax provides a way of using TeX notation to display equations. There is also pretty good accessibility support provided by MathJax. http://www.mathjax.org/resources/articles-and-presentations/
> Without using some sort of plugin, or markup, I can't think of a simple way of conveying mathematical semantics short of writing out equations long hand as you have done e.g. "10 to the power 4"
> Joe | {"url":"https://webaim.org/discussion/mail_thread?thread=6404","timestamp":"2024-11-10T12:45:27Z","content_type":"text/html","content_length":"11224","record_id":"<urn:uuid:17580497-f2b6-4f6f-abec-95eb7608af00>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00797.warc.gz"} |
... The main goal of the present study is therefore to investigate the understanding of the generality of mathematical statements and the relation to the reading and construction of proofs in
first-year university students. Previous studies have shown differences in students' understanding and evaluation of different types of arguments (e.g., Healy & Hoyles, 2000;Kempen, 2018Kempen, ,
2021. Thus, I focused on the influence of reading different types of arguments (no argument, empirical argument, generic proof, and ordinary proof) on students' understanding of generality and other
proof-related activities. ...
... While ratings regarding students' conviction were higher than for verification, the generic proofs still received lower ratings than the ordinary proofs. In comparison to empirical arguments,
Kempen (2021) found that students gave generic proofs higher ratings regarding both familiar and unfamiliar statements, which may indicate that students "do not mix up the idea of generic proofs with
purely empirical verifications" (p. 4). ...
... A positive effect was also expected regarding familiar statements, because the role of being familiar with statements and modes of argumentations for the acceptance of proof has been highlighted
in the literature (e.g., Hanna, 1989). However, so far, no such influence on proof evaluation has been shown (e.g., Kempen, 2021;Martin & Harel, 1989). As research findings suggest that the
comprehension of the argument is important regarding students' conviction of it (e.g., Sommerhoff & Ufer, 2019;Weber, 2010), a positive effect was expected, i.e., students' with higher levels of
comprehension also have higher levels of conviction. ... | {"url":"https://www.researchgate.net/publication/353243516_INVESTIGATING_THE_DIFFERENCE_BETWEEN_GENERIC_PROOFS_AND_PURELY_EMPIRICAL_VERIFICATIONS","timestamp":"2024-11-08T00:12:20Z","content_type":"text/html","content_length":"377333","record_id":"<urn:uuid:d010dcc1-99d9-44b1-a6f3-35e4f3e12e5b>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00568.warc.gz"} |
Types of Return on Principal: Gross vs Net ROP in context of return on principal
31 Aug 2024
Title: An Examination of Gross and Net Return on Principal (ROP) in the Context of Investment Returns
Abstract: Return on Principal (ROP) is a fundamental concept in finance, representing the return earned by an investor on their principal investment. However, two distinct types of ROP exist: gross
and net ROP. This article delves into the definitions, formulas, and implications of these two types of ROP, providing a comprehensive understanding of their differences.
Introduction: Return on Principal (ROP) is a crucial metric in finance, measuring the return earned by an investor on their principal investment. In the context of investments, ROP can be calculated
using various methods, resulting in different types of returns. This article focuses on two primary types of ROP: gross and net ROP.
Gross Return on Principal (ROP): The gross ROP represents the total return earned by an investor, including all income generated from the investment, such as interest, dividends, or capital gains.
The formula for calculating gross ROP is:
Gross ROP = (Principal + Income) / Principal
In ASCII format:
• P represents the principal investment
• I represents the income generated from the investment
Net Return on Principal (ROP): The net ROP, on the other hand, represents the return earned by an investor after deducting all expenses associated with the investment. This includes fees, taxes, and
other charges. The formula for calculating net ROP is:
Net ROP = (Principal + Income - Expenses) / Principal
In ASCII format:
Net ROP = (P + I - E) / P
• E represents the expenses associated with the investment
Conclusion: In conclusion, gross and net ROP are two distinct types of returns that investors can earn on their principal investments. While gross ROP represents the total return earned by an
investor, including all income generated from the investment, net ROP represents the return earned after deducting all expenses associated with the investment. Understanding the differences between
these two types of ROP is essential for making informed investment decisions.
• [Insert relevant references here]
Note: This article provides a general overview of gross and net ROP without numerical examples. The formulas provided are in ASCII format, as requested.
Related articles for ‘return on principal’ :
• Reading: Types of Return on Principal: Gross vs Net ROP in context of return on principal
Calculators for ‘return on principal’ | {"url":"https://blog.truegeometry.com/tutorials/education/393d47a5a3ca824afa839534be04a24c/JSON_TO_ARTCL_Types_of_Return_on_Principal_Gross_vs_Net_ROP_in_context_of_retur.html","timestamp":"2024-11-03T15:37:40Z","content_type":"text/html","content_length":"17088","record_id":"<urn:uuid:b94944d7-951d-4325-a570-eec7819ab515>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00189.warc.gz"} |
Punnett square practice and examples
Punnett square definition
As is well known make a punnett square is widely used for solving genetics problems in mendelian genetics. An ability to make a punnett squares will be useful for middle and high school students
in biology classes. But professional geneticists use these skills in their work too. So what is punnett square?
Punnett Square - is a graphical method proposed by the British geneticist R. Punnett in 1906 to visualize all the possible combinations of different types of gametes in particular crosses or
breeding experiments (each gamete is combination of one maternal allele with one paternal allele for each gene being studied in the cross).
Punnett Square looks like a two-dimensional table, where over the square horizontally fit the gametes of one parent, and the left edge of the square in the vertical - the gametes of the other
parent. Within the square, at the intersection of rows and columns, write genotypes making from the gametes combinations. Thus, it becomes very easy to determine the probability of each genotype in a
particular cross.
Monohybrid punnett square
For monohybrid cross we study the inheritance of a single gene. In the classical monohybrid cross each gene has two alleles. For example, to make our punnett square, we take the maternal and
paternal organisms with the same genotype - "Gg". For dominant allele in genetics we use upper-case letters and for recessive allele lower-case letters. This genotype can produce only two types of
gametes that contain either the "G" or "g" allele.
So this Punnett square look like that:
From punnett square in the offspring we have genotype ratio and probability: 1(25%)GG : 2(50%)Gg : 1(25%)gg - this typical genotypes ratio (1:2:1) for a monohybrid cross.Dominant allele will mask
the recessive allele that means, that the organisms with the genotypes "GG" and "Gg" have the same phenotype.
For example, if allele "G" is yellow color and allele "g" is green color, then "gg" have green phenotype, "GG" and "Gg" have yellow phenotype. From punnett square in the amount we have 3G-(yellow
phenotype) and 1gg(green phenotype) - this typical phenotypes ratio (3:1) for a monohybrid cross. And probability of the offspring will be 75%G- : 25%gg.
Mendelian genetics
At first these results were obtained in the experiments of Gregor Mendel to the plant - garden pea (Pisum sativum). To expound the results Gregor Mendel made the following conclusions:
• Every trate of this organism is controlled by a pair of alleles.
• If the organism contains two different alleles for this trait, then one of them (the dominant) can manifest itself, completely suppressed the manifestation of the other (recessive).
• During meiosis, each pair of alleles split(segregate), and each gamete receives one allele from each pair of alleles the principle of segregation.
Without this basic genetics lows we can't solve any punnett square problems. Determine possibility to predict the results of one pair of alternative traits, Mendel went on to study the inheritance of
two pairs of such traits.
Dihybrid punnett square
For dihybrid cross we study the inheritance of two genes. For dihybrid cross the Punnett squares only works if the genes are independent of each other, which meomans when form a maternal and
paternal gametes - each of them can get any allele of one pair, along with any allele of the another pair. This principle of independent assortment was discovered by Mendel in experiments on dihybrid
and polyhybrid crosses.
The following example illustrates Punnett square for a dihybrid cross between two heterozygous pea plants. We have two genes Shape and Color. For shape: "R" is dominant allele with round phenotype
and "w" is recessive allele with wrinkled phenotype. For color: "Y" is dominant allele with yelloy phenotype and "g" is recessive allele whith green phenotype. Maternal and paternal organisms have
some genotype- "RwYg".
First you need to determine all possible combinations of gametes, for this you can also use Punnett squares:
Then they can produce four types of gametes with all possible combinations: RY, Rg, wY, wg. And now form the Punnett square for genotypes:
RY Rg wY wg
RY RRYY RRYg RwYY RwYg
Rg RRYg RRgg RwYg Rwgg
wY RwYY RwYg wwYY wwYg
wg RwYg Rwgg wwYg wwgg
From punnett square in the offspring we have genotype ratio and probability: 1(6,25%)RRYY : 2(12,5%)RwYY : 1(6,25%)wwYY : 2(12,5%)RRYg : 4(25%)RwYg : 2(12,5%)wwYg : 1(6,25%)RRgg : 2(12,5%)Rwgg : 1
Since dominant traits mask recessive traits, from punnett square we have phenotypes combinations whith ratio and probability: 9(56,25%)R-Y-(round, yellow) : 3(18,75%)R-gg(round,green) : 3(18,75%)
wwY-(wrinkled, yellow) : 1(6,25%)wwgg(wrinkled, green). The ratio 9:3:3:1 is typical for a dihybrid cross.
Trihybrid punnett square
Make punnett square for trihybrid cross between two heterozygous plants is more complicated. To solve this problem, we can use our knowledge of mathematics. To determine all possible combinations
of gametes for trihybrid cross we have to remember the solution of polynomials:
• Let make polynomial for this cross: (A + a) X (B + b) X (C + c).
• We multiply the expression in the first bracket on the expression of a second and we get : (AB + Ab + aB + ab) X (C + c).
• Now multiply this expression by the expression in the third bracket and we get: ABC + ABc + AbC + Abc + aBC + aBc + abC + abc.
Then they can produce eight types of gametes with all possible combinations. This solution can be illustrated by the Punnett square:
C c
AB ABC ABc
Ab AbC Abc
aB aBC aBc
ab abC abc
Now form the Punnett square for genotypes (we get 64 punnett square):
ABC aBC AbC abC ABc aBc Abc abc
ABC AABBCC AaBBCC AABbCC AaBbCC AABBCc AaBBCc AABbCc AaBbCc
aBC AaBBCC aaBBCC AaBbCC aaBbCC AaBBCc aaBBCc AaBbCc aaBbCc
AbC AABbCC AaBbCC AAbbCC AabbCC AABbCc AaBbCc AAbbCc AabbCc
abC AaBbCC aaBbCC AabbCC aabbCC AaBbCc aaBbCc AabbCc aabbCc
ABc AABBCc AaBBCc AABbCc AaBbCc AABBcc AaBBcc AABbcc AaBbcc
aBc AaBBCc aaBBCc AaBbCc aaBbCc AaBBcc aaBBcc AaBbcc aaBbcc
Abc AABbCc AaBbCc AAbbCc AabbCc AABbcc AaBbcc AAbbcc Aabbcc
abc AaBbCc aaBbCc AabbCc aabbCc AaBbcc aaBbcc Aabbcc aabbcc
Genotypes ratio and probability for Trihybrid cross
But how do we calculate the ratio of genotypes from this punnett square. Again use the polynomials.
• We know the genotype ratio for monohybrid cross: 1AA: 2Aa: 1aa.
• Now we form a polynomial for our case: (1AA + 2Aa + 1aa) X (1BB + 2Bb + 1bb) X (1CC + 2Cc + 1cc).
• Multiply the first two expressions we get : (1AABB + 2AABb + 1AAbb + 2AaBB + 4AaBb + 2Aabb + 1aaBB + 2aaBb + 1aabb) X (1CC + 2Cc + 1cc).
• Multiplying this expression on the third we get the results for genotypes : 1AABBCC : 2AABbCC : 1AAbbCC : 2AaBBCC : 4AaBbCC : 2AabbCC : 1aaBBCC : 2aaBbCC : 1aabbCC : 2AABBCc : 4AABbCc : 2AAbbCc :
4AaBBCc : 8AaBbCc : 4AabbCc : 2aaBBCc : 4aaBbCc : 2aabbCc : 1AABBcc : 2AABbcc : 1AAbbcc : 2AaBBcc : 4AaBbcc : 2Aabbcc : 1aaBBcc : 2aaBbcc : 1aabbcc
Phenotypes ratio and probability for Trihybrid cross
And as well we can calculate from our punnett square the ratio of phenotypes.
• We know the phenotype ratio for monohybrid cross: 3A-: 1aa.
• Now we form a polynomial for our case: (3A- + 1aa) X (3B- + 1bb) X (3C- + 1cc).
• Multiply the first two expressions we get : (9A-B-+ 3A-bb + 3aaB-+ 1aabb) X (3C-+ 1cc).
• Multiplying this expression on the third we get the results for phenotypes : 27A-B-C- : 9A-bbC- : 9aaB-C- :3aabbC- : 9A-B-cc : 3A-bbcc : 3aaB-cc : 1aabbcc
All this results we can get from polinomial method, without making punnett squares.
How to solve a large and complicated punnett square examples
But what should we do if we need to solve the problem with a large number of genes. Even using the polynomials will be difficult to avoid mistakes and get the right results. In addition it may be
time-consuming. And if you do not want to do all this job manually, then you can use our professional Punnett Square Calculator.
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What is the slope of any line perpendicular to the line passing through (14,12) and (12,5)? | HIX Tutor
What is the slope of any line perpendicular to the line passing through #(14,12)# and #(12,5)#?
Answer 1
Slope of the perpendicular line is #(y_2-y_1)/(x_2-x_1) = (5-12)/(12-14)=7/2# We know the condition of perpendicularity of two lines is the product of their slopes will be equal to be #-1#.i.e #
m_1*m_2=-1 ; m_1=7/2 :. m_2= -2/7#[answer]
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Answer 2
The slope of any line perpendicular to the line passing through the points (14,12) and (12,5) is the negative reciprocal of the slope of the original line. To find the slope of the original line,
calculate the change in y divided by the change in x between the two points. Then, take the negative reciprocal of that slope to find the slope of any line perpendicular to it.
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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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Algorithm Design Techniques
Now, with all the components of the algorithmic problem solving in place, how do you design an algorithm to solve a given problem? This is the main question this book seeks to answer by teaching you
several general design techniques.
What is an algorithm design technique?
An algorithm design technique (or “strategy” or “paradigm”) is a general approach to solving problems algorithmically that is applicable to a variety of problems from different areas of computing.
Check this book’s table of contents and you will see that a majority of its chapters are devoted to individual design techniques. They distill a few key ideas that have proven to be useful in
designing algorithms. Learning these techniques is of utmost importance for the following reasons.
First, they provide guidance for designing algorithms for new problems, i.e., problems for which there is no known satisfactory algorithm. Therefore—to use the language of a famous proverb—learning
such techniques is akin to learning to fish as opposed to being given a fish caught by somebody else. It is not true, of course, that each of these general techniques will be necessarily applicable
to every problem you may encounter. But taken together, they do constitute a powerful collection of tools that you will find quite handy in your studies and work.
Second, algorithms are the cornerstone of computer science. Every science is interested in classifying its principal subject, and computer science is no exception. Algorithm design techniques make it
possible to classify algorithms according to an underlying design idea; therefore, they can serve as a natural way to both categorize and study algorithms.
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
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VITEEE 2017 Physics full syllabus | ENTRANCE INDIA
VITEEE 2017 Physics Syllabus
PART – I – PHYSICS
1.Laws of Motion & Work, Energy and Power
Law of conservation of linear momentum and its applications. Static and kinetic friction – laws of friction – rolling friction – lubrication.
Work done by a constant force and a variable force; kinetic energy – work-energy theorem – power.
Conservative forces: conservation of mechanical energy (kinetic and potential energies) – non-conservative forces: motion in a vertical circle – elastic and inelastic collisions in one and
two dimensions.
2.Properties of Matter
Elastic behaviour – Stress-strain relationship – Hooke’s law – Young’s modulus – bulk modulus – shear modulus of rigidity – Poisson’s ratio – elastic energy. Viscosity – Stokes’ law –
terminal velocity – streamline and turbulent flow – critical velocity. Bernoulli’s theorem and its applications.
Heat – temperature – thermal expansion: thermal expansion of solids – specific heat capacity: Cp, Cv – latent heat capacity. Qualitative ideas of Blackbody radiation: Wein’s displacement Law –
Stefan’s law.
Charges and their conservation; Coulomb’s law-forces between two point electric charges – Forces between multiple electric charges-superposition principle. Electric field – electric field due to a
point charge, electric field lines; electric dipole, electric field intensity due to a dipole – behaviour of a dipole in a uniform electric field. Electric potential – potential
difference-electric potential due to a point charge and dipole-equipotential surfaces – electrical potential energy of a system of two point charges.
Electric flux-Gauss’s theorem and its applications. Electrostatic induction-capacitor and capacitance – dielectric and electric polarisation – parallel plate capacitor with and without
dielectric medium – applications of capacitor – energy stored in a capacitor – Capacitors in series and in parallel – action of points – Van de Graaff generator.
4.Current Electricity
Electric Current – flow of charges in a metallic conductor – drift velocity and mobility and their relation with electric current. Ohm’s law, electrical resistance – V-I characteristics –
electrical resistivity and conductivity-classification of materials in terms of conductivity – Carbon resistors – colour code for carbon resistors – combination of resistors – series and parallel
– temperature dependence of resistance – internal resistance of a cell – potential difference and emf of a cell – combinations of cells in series and in parallel.
Kirchoff ’s law – Wheatstone’s Bridge and its application for temperature coefficient of resistance measurement – Me t re br i dge – sp ec i a l cas e o f Wheats to n e br i dge – Potentiometer
principle – comparing the emf of two cells.
5.Magnetic Effects of Electric Current
Magnetic effect of electric current – Concept of magnetic field – Oersted’s experiment – Biot – Savart law – Magnetic field due to an infinitely long current carrying straight wire and circular coil
– Tangent galvanometer – construction and working – Bar magnet as an equivalent solenoid – magnetic field lines.
Ampere’s circuital law and its application. Force on a moving charge in uniform magnetic field and electric field – cyclotron – Force on current carrying conductor in a uniform magnetic field –
Forces between two parallel current carrying conductors – definition of ampere.
Torque experienced by a current loop in a uniform magnetic field – moving coil galvanometer – conversion to ammeter and voltmeter – current loop as a magnetic dipole and its magnetic dipole moment –
Magnetic dipole moment of a revolving electron.
6.Electromagnetic Induction and Alternating Current
Electromagnetic induction – Faraday’s law – induced emf and current – Lenz’s law. Self induction – Mutual induction – self inductance of a long solenoid – mutual inductance of two long solenoids.
Methods of inducing emf – (i) by changing magnetic induction (ii) by changing area enclosed by the coil and (iii) by changing the orientation of the coil (quantitative treatment).
AC generator – commercial generator. (Single phase, three phase). Eddy current – applications – transformer – long distance transmission. Alternating current – measurement of AC – AC circuit with
resistance – AC circuit with inductor – AC circuit with capacitor – LCR series circuit – Resonance and Q – factor – power in AC circuits.
Reflection of light, spherical mirrors, mirror formula. Refraction of light, total internal reflection and its applications, optical fibers, refraction at spherical surfaces, lenses, thin lens
formula, lens maker ’s formula. Magnification, power of a lens, combination of thin lenses in contact, combination of a lens and a mirror. Refraction and dispersion of light through a
prism. Scattering of light-blue colour of sky and reddish appearances of the sun at sunrise and sunset.
Wavefront and Huygens’s principle – Reflection, total internal reflection and refraction of plane wave at a plane surface using wavefronts. Interference – Young’s double slit experiment
and expression for fringe width – coherent source – interference of light – Formation of colours in thin films – Newton’s rings. Diffraction – differences between interference and diffraction of
light- diffraction grating. Polarisation of light waves – polarisation by reflection – Brewster’s law – double refraction – nicol prism – uses of plane polarised light and Polaroids –
rotatory polarisation – polarimeter.
8.Dual Nature of Radiation and Atomic Physics
Electromagnetic waves and their characteristics – Electromagnetic spectrum – Photoelectric effect – Light waves and photons- Einstein’s photoelectric equation – laws of photoelectric emission –
particle nature of light – photo cells and their applications.
Atomic structure – discovery of the electron – specific charge (Thomson’s method) and charge of the electron (Millikan’s oil drop method) – alpha scattering – Rutherford’s atom model.
9.Nuclear Physics
Nuclear properties – nuclear radii, masses, binding energy, density, charge – isotopes, isobars and isotones – nuclear mass defect – binding energy – stability of nuclei – Bainbridge
mass spectrometer.
Nature of nuclear forces – Neutron – discovery – properties – artificial transmutation – particle accelerator. Radioactivity – alpha, beta and gamma radiations and their properties – Radioactive
decay law – half life – mean life – artificial radioactivity – radio isotopes – effects and uses – Geiger – Muller counter. Radio carbon dating. Nuclear fission – chain reaction – atom bomb – nuclear
reactor – nuclear fusion – Hydrogen bomb – cosmic rays – elementary particles.
10.Semiconductor Devices and their Applications
Semiconductor basics – energy band in solids: difference between metals, insulators and semiconductors – semiconductor doping – Intrinsic and Extrinsic semiconductors. Formation of P-N Junction –
Barrier potential and depletion layer-P-N Junction diode – Forward and reverse bias characteristics – diode as a rectifier – Zener diode-Zener diode as a voltage regulator – LED. Junction transistors
– characteristics – transistor as a switch – transistor as an amplifier – transistor as an oscillator.
Logic gates – NOT, OR, AND, EXOR using discrete components – NAND and NOR gates as universal gates – De Morgan’s theorem – Laws and theorems of Boolean algebra.
Latest Govt Job & Exam Updates: | {"url":"https://entranceindia.com/viteee/viteee-2017-physics-syllabus/","timestamp":"2024-11-04T23:52:39Z","content_type":"text/html","content_length":"75481","record_id":"<urn:uuid:a0521839-f4fd-4338-8c30-f5c24b05ff0a>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00296.warc.gz"} |
Expected value of a random divisor selector | Thien Hoang
Consider a sequence , in which let and . Apparently there are a lot of sequences like that. For each , is selected randomly with equal possibilities among the divisors of . Let's denote the expected
value of . What is ? Read the
problem statement
For example, we have and need to calculate . All possible sequences of are:
Can we simply calculate the average value of ?, in this case:
No we can't, because the possibilities for the sequences to happen are not all the same. For example, the possibility of is , while that of is only .
Consider (prime factorization). The process of choosing (which is a divisor of ) can be interpreted as independent steps (events). In step , we choose as a random number from 0 to . After all steps,
we have .
This means, instead of dealing with the problem above with any number , we can break it down into (the number of prime factors of ) smaller problems. Each small problem requires finding the expected
value of given , in other word, finding . Suppose that , then:
The smaller problem
How do we calculate ? This should be done easily with dynamic programming or recursion. We consider the probability for the -th term of the sequence to equal :
Then the rest is straightforward:
Complexity analysis
Factorizing takes . Calculating takes . Since , the overall complexity is .
my solution
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Problemi Attuali di Fisica Teorica 06
Posted by Urs Schreiber
This year’s conference in the annual series Problemi Attuali di Fisica Teorica will be held April 7 - April 13 in Vietri, Italy (like last year).
The HEP program is
Monday 10 - Quantum Gravity
Tuesday 11 - Strings and Fields
Wednesday 12 - Duality and Branes
Thursday 13 - Geometric and topological aspects of strings and branes
$\;\;\;\;\;$ 1) Non Abelian gerbes and brane theory.
$\;\;\;\;\;$ 2) Manifolds, supermanifolds, special holonomy and superstring compactifications.
$\;\;\;\;\;$ 3) Generalized complex geometry and supersymmetric sigma models.
$\;\;\;\;\;$ 4) Poisson geometry and non commutative geometry.
Francesco Bonechi
(I.N.F.N., Firenze)
The Poisson sigma model and the quantization of Poisson manifolds
Abstract: The Poisson sigma model is a bidimensional field theory having as target manifold a Poisson manifold. Kontsevich formula for the deformation quantization of the target manifold is
interpretable as the perturbative expansion of a particular correlator of the model. The non perturbative dynamics of the model is instead still largely unexplored. In this seminar, we clarify
the meaning of the inegrality condition of the Poisson tensor which appears both in the integration of the gauge transformations of the model and in the geometric quantization of the target
Francesco D’Andrea (S.I.S.S.A., Trieste)
Local index formulas on quantum spheres
Abstract: A general introduction to the basic ideas of index theory in noncommutative geometry is presented, clarified through the q-sphere example. One of the main motivation of this work is the
classification of deformations of instantons, whose charge can be computed using the local formulae of Connes-Moscovici. After a brief introduction of the main notions, some results concerning
the geometrical properties of the quantum SU(2) group and of Podles spheres, which are deformation of the Lie group SU(2) and of Riemann sphere, respectively, will be discussed. Finally, the
outlook of the subject and in particular the connection with the theory of modular forms will be illustrated.
Jarah Evslin (Free University, Bruxelles)
Twisted K-Theory as a BRST Cohomology
Abstract: We argue that twisted K-theory is a BRST cohomology. The original Hilbert space is the integral cohomology of a spatial slice, corresponding to the lattice of quantized Ramond-Ramond
field strengths. The gauge symmetry consists of large gauge transformations that correspond geometrically to choices of trivializations of gerbes. The BRST operator is identified with the
differential of the Atiyah-Hirzebruch spectral sequence.
Branislav Jurco (Munich University, Munich)
Nonabelian gerbes, differential geometry and stringy applications
Abstract: We will discuss nonabelian gerbes and their twistings as well as the corresponding differential geometry. We describe the classifying space, the corresponding universal gerbe and their
relation to string group and string bundles. Finally we show the relevance of twisted nonabelian gebres in the study and resolution of global anomalies of multiple coinciding M5-branes.
Urs Schreiber (Hamburg University, Hamburg)
Surface transport, gerbes, TFT and CFT
Abstract: Segal’s conception of a 2D QFT as a functor on cobordisms may be refined to that of a 2-functor on surface elements. Surface transport in gerbes, as well as 2D TFTs and CFTs provide
Alessandro Tanzini (SISSA Trieste)
Recent developments in topological brane theories
Abstract: We will discuss the formulation of topological theories for branes and its relevance for the recent conjectures about S-duality in topological string and topological M theory.
Alessandro Torrielli (Humboldt University, Berlin)
D-brane decay in electric fields and noncommutative geometry
Abstract: We study tachyon condensation in the presence of overcritical electric fluxes, by means of a toy model based on the noncommutative deformation of the one proposed by Minahan and
Zwiebach. We discuss the relation with Sen’s standard picture of D-brane decay, and the connection with the S-brane paradigma.
Maxim Zabzine (Upsala University, Upsala, and University of California, Santa Barbara)
New results in generalized Kahler geometry
Abstract: I will review the different decsriptions of generalized Kahler geometry and its relation with $N=(2,2)$ supersymmetric sigma model. I will sketch the proof of the existence of
generalized Kahler potential and will explain the relation to off-shell supersymmetry.
Posted at February 9, 2006 12:27 PM UTC | {"url":"https://golem.ph.utexas.edu/string/archives/000749.html","timestamp":"2024-11-03T18:22:18Z","content_type":"application/xhtml+xml","content_length":"19206","record_id":"<urn:uuid:12abe123-3538-496a-ae5b-70c30872147d>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00254.warc.gz"} |
Dynamic Model for Division
The model presented is valid for the interpretation of the whole-number quotient of a whole number. For every case of a division “whole-number quotient of a whole number” must be designed a model
which helps to interpret the case. Here is chosen the case of partitioning the set of 12 objects into: a) 2 groups of equal shares in each one, symbolically denoted as 12÷2 b) 3 groups of equal
shares in each one, symbolically denoted as 12÷3 c) 4 groups of equal shares in each one, symbolically denoted as 12÷4 The 12 objects are represented by 12 blue circles. The respective model for the
division interpretation is constructed using GeoGebra software with its effective virtual tools and the excellent periodic properties of the trigonometric functions. The model of this case serves as
a demonstrative model for the teachers to use in their classroom, equipped with computers, or in computer laboratories. *** Play with the slider and observe the dynamic of partitioning of the set of
12 objects into 2, 3 and 4 groups, respectively, simultaneously are shown the respective equations of the division. | {"url":"https://www.geogebra.org/m/p2tf4DwW","timestamp":"2024-11-08T01:56:47Z","content_type":"text/html","content_length":"91836","record_id":"<urn:uuid:2488148b-ff95-478e-b94d-0201b237c57b>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00666.warc.gz"} |
How do you graph (x-1)^2+(y-4)^2=9? | Socratic
How do you graph #(x-1)^2+(y-4)^2=9#?
1 Answer
You find the centre, the vertices, and the endpoints of the function. Then you plot the graph.
${\left(x - 1\right)}^{2} + {\left(y - 4\right)}^{2} = 9$
This is the standard form for the equation of a circle with centre at ($1 , 4$) and radius $\sqrt{9} = 3$.
This means that, to find the vertices, you go 3 units up from the centre and 3 units down.
Thus, the vertices are at ($1 , 7$) and ($1 , 1$).
To find the endpoints, you go 3 units left of the centre and 3 to the right.
Thus, the endpoints are at ($- 2 , 4$) and ($4 , 4$).
Plot these points on a graph.
Now draw a smooth circle through these four points.
And you have your graph.
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Understanding Mathematical Functions: How To Calculate Gradient Of A F
Introduction: Unveiling the Concept of Gradients in Mathematics
Mathematical functions are a fundamental concept in the field of mathematics, playing a crucial role in various branches of science, engineering, and economics. They are used to represent the
relationship between two or more variables, and they provide a way to understand and analyze real-world phenomena.
One of the key aspects of understanding functions is grasping the concept of gradient. The gradient is a measure of the steepness of a function at any given point and provides valuable information
about the rate of change of the function. In this chapter, we will delve into the definition of mathematical functions, the significance of gradients, and the process for calculating them.
A Definition of mathematical functions and their importance in various fields
A mathematical function is a rule that assigns to each input value exactly one output value. It is denoted by a symbol f(x) and can be represented graphically as a curve or a line. Functions are used
in a wide range of fields, including physics, engineering, biology, and economics, to model and analyze various phenomena.
Functions allow us to understand the relationship between different variables and make predictions about how one variable might change as another variable changes. They are essential for solving
problems and making decisions in both theoretical and practical contexts.
Explanation of the gradient and its significance in understanding functions
The gradient of a function at a given point measures how the function changes as you move along the curve. It provides information about the steepness, or slope, of the function at that point. In
simpler terms, the gradient tells us how rapidly the function is increasing or decreasing at a specific point.
Understanding the gradient is crucial for analyzing the behavior of a function, identifying maximum and minimum points, and predicting the direction of change. It allows us to gain insights into the
behavior of real-world phenomena and make informed decisions based on those insights.
Overview of the process for calculating the gradient and its applications
Calculating the gradient of a function involves finding the rate of change of the function with respect to the independent variable. This can be done using various mathematical techniques such as
differentiation, which is a fundamental concept in calculus.
The gradient has numerous applications in various fields, such as physics, engineering, economics, and machine learning. For example, in physics, the gradient is used to calculate the force acting on
a particle, while in machine learning, it is used to optimize algorithms and make predictions based on data.
Key Takeaways
• Understanding the concept of gradient in mathematics.
• Calculating the gradient of a function using first principles.
• Using the derivative to find the gradient of a function.
• Applying the gradient to solve real-world problems.
• Understanding the relationship between the gradient and the rate of change.
Understanding Mathematical Functions: How to Calculate Gradient of a Function
When it comes to understanding mathematical functions, one of the key concepts to grasp is the calculation of gradients. Gradients play a crucial role in determining the steepness of curves and the
rate of change in functions. In this chapter, we will delve into the basic principles behind gradients, the concept of derivatives, and the relationship between gradients and the steepness of curves.
Explanation of the Slope as the Rate of Change in Functions
Slope refers to the steepness of a line or a curve. In the context of functions, slope represents the rate of change. When we talk about the slope of a function, we are essentially referring to how
the function's output (y-value) changes with respect to its input (x-value). A positive slope indicates an increasing function, while a negative slope indicates a decreasing function. Understanding
the concept of slope is fundamental to grasping the idea of gradients in mathematical functions.
Introduction to the Concept of Derivatives as a Foundational Tool for Gradients
Derivatives are a foundational tool for calculating gradients. In calculus, the derivative of a function at a certain point gives the slope of the tangent line to the curve at that point. This means
that derivatives provide us with a way to measure the rate of change of a function at any given point. By finding the derivative of a function, we can determine the gradient at a specific point,
which is essential for understanding the behavior of the function.
The Relationship Between Gradients and the Steepness of Curves
The relationship between gradients and the steepness of curves is crucial in understanding the behavior of functions. The gradient of a function at a particular point gives us information about how
steep the curve is at that point. A higher gradient indicates a steeper curve, while a lower gradient indicates a gentler slope. By analyzing the gradients of a function at different points, we can
gain insights into the overall shape and behavior of the function.
Steps to Calculate the Gradient of a Function
Understanding how to calculate the gradient of a function is an essential skill in mathematics, particularly in the field of calculus. The gradient of a function represents the rate of change of the
function with respect to its variables. In this chapter, we will explore the steps to calculate the gradient of a function, including a clarification of the difference between partial and total
derivatives, a detailed example of computing the gradient for a simple linear function, and a step-by-step walkthrough of calculating the gradient of a non-linear function.
A Clarification of the difference between partial and total derivatives
Before delving into the calculation of the gradient of a function, it is important to understand the difference between partial and total derivatives. In the context of multivariable calculus, a
partial derivative measures how a function changes with respect to one of its variables, while holding the other variables constant. On the other hand, a total derivative measures how a function
changes with respect to all of its variables simultaneously. Understanding this distinction is crucial when calculating the gradient of a function, as it determines the approach to be taken.
Detailed example of computing gradient for a simple linear function
Let's consider a simple linear function, f(x, y) = 2x + 3y. To calculate the gradient of this function, we can use the concept of partial derivatives. The gradient of the function is represented by
the vector (∂f/∂x, ∂f/∂y), where ∂f/∂x and ∂f/∂y denote the partial derivatives of f with respect to x and y, respectively. In this case, the partial derivatives are ∂f/∂x = 2 and ∂f/∂y = 3.
Therefore, the gradient of the function is (2, 3).
Step-by-step walkthrough of calculating the gradient of a non-linear function
Now, let's consider a non-linear function, f(x, y) = x^2 + y^2. Calculating the gradient of a non-linear function involves finding the partial derivatives of the function with respect to each of its
variables. In this example, the partial derivatives are ∂f/∂x = 2x and ∂f/∂y = 2y. Therefore, the gradient of the function is represented by the vector (2x, 2y). This step-by-step approach allows us
to determine the rate of change of the function with respect to each of its variables, providing valuable insights into its behavior.
Applying Gradient Calculations in Real-World Scenarios
Understanding mathematical functions and their gradients is not just a theoretical exercise, but it has practical applications in various real-world scenarios. Let's take a closer look at how
gradient calculations are applied in different fields.
A. Examination of gradient's role in physics, particularly in force fields
In physics, the concept of gradient plays a crucial role in understanding force fields. The gradient of a function represents the direction of the steepest ascent of the function. In the context of
force fields, the gradient of a scalar function helps in determining the direction and magnitude of the force acting on a particle at any given point in space. This is particularly important in
fields such as electromagnetism and fluid dynamics, where the understanding of force fields is essential for various applications.
B. Usage of gradient in economics for cost functions and optimization
In economics, gradient calculations are used in the analysis of cost functions and optimization problems. Cost functions, which represent the relationship between the cost of production and the level
of output, often involve the calculation of gradients to determine the rate of change of costs with respect to different input variables. Additionally, optimization problems in economics, such as
maximizing profit or minimizing cost, rely on gradient calculations to find the optimal solutions. This application of gradients is fundamental in decision-making processes for businesses and
C. How gradients aid in creating machine learning algorithms and navigating topographical maps
In the realm of machine learning, gradients are extensively used in training algorithms and optimizing models. The process of gradient descent, which involves iteratively adjusting the parameters of
a model to minimize a cost function, relies on the calculation of gradients. This enables machine learning algorithms to learn from data and make predictions with high accuracy. Moreover, in the
context of topographical maps, gradients are essential for understanding the terrain and navigating through different elevations. By calculating gradients, one can determine the slope and direction
of the terrain, which is valuable for various applications such as urban planning, environmental studies, and outdoor recreation.
Advanced Techniques in Gradient Calculation
When it comes to understanding mathematical functions, calculating the gradient of a function is a crucial skill. In this chapter, we will explore advanced techniques in gradient calculation,
including multivariable functions, incorporating constraints using Lagrange multipliers, and utilizing software tools and calculators for complex gradient calculations.
A Introduction to multivariable functions and multiple partial derivatives
When dealing with multivariable functions, the concept of gradient extends to multiple partial derivatives. The gradient of a function of several variables is a vector that points in the direction of
the greatest rate of increase of the function. To calculate the gradient, we take the partial derivatives of the function with respect to each variable and form a vector with these derivatives. This
allows us to understand how the function changes in different directions in the multivariable space.
For example, in a function f(x, y, z), the gradient is represented as ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z). Understanding how to calculate and interpret the gradient of multivariable functions is essential for
various applications in mathematics, physics, and engineering.
B Incorporating constraints using Lagrange multipliers to find gradient under conditions
When dealing with optimization problems involving constraints, the use of Lagrange multipliers is a powerful technique to find the gradient of a function under specific conditions. Lagrange
multipliers allow us to incorporate constraints into the optimization problem and find the critical points of the function while satisfying these constraints.
The method involves setting up a system of equations using the gradient of the function and the gradient of the constraint, and then solving for the critical points. This technique is widely used in
economics, engineering, and other fields to optimize functions subject to certain constraints.
C Utilizing software tools and calculators for complex gradient calculations
With the advancement of technology, there are various software tools and calculators available to assist in complex gradient calculations. These tools can handle functions with multiple variables,
constraints, and intricate mathematical operations, making it easier to compute gradients for complex functions.
Software packages such as MATLAB, Mathematica, and Python libraries like NumPy provide built-in functions for gradient calculation, allowing users to focus on the problem at hand rather than the
intricacies of the computation. Additionally, online calculators and graphing tools offer user-friendly interfaces for visualizing and computing gradients of functions.
By leveraging these software tools, mathematicians, scientists, and engineers can efficiently analyze and optimize multivariable functions with ease, leading to advancements in various fields of
Troubleshooting Common Challenges with Gradients
Understanding and calculating gradients is an essential skill in mathematics and is crucial for various applications in fields such as physics, engineering, and computer science. However, there are
common challenges that arise when working with gradients, and knowing how to troubleshoot these challenges is important for mastering this concept.
A. Handling undefined gradients and points of non-differentiability
One common challenge when dealing with gradients is encountering undefined gradients or points of non-differentiability. This often occurs when working with functions that have sharp corners, cusps,
or vertical tangents. In such cases, it is important to understand that the gradient does not exist at these points.
To handle undefined gradients and points of non-differentiability:
• Identify the points where the gradient is undefined or the function is non-differentiable.
• Use alternative methods such as directional derivatives or subgradients to analyze the behavior of the function at these points.
• Consider the context of the problem to determine if the undefined gradient has any physical significance.
B. Strategies for dealing with the complexities of higher-dimensional gradients
Working with higher-dimensional gradients can introduce additional complexities, especially when visualizing and interpreting the behavior of the function in multiple dimensions. Understanding how to
navigate these complexities is essential for effectively working with higher-dimensional gradients.
Strategies for dealing with the complexities of higher-dimensional gradients:
• Utilize visualization tools such as contour plots, surface plots, and vector fields to gain insights into the behavior of the function in higher dimensions.
• Break down the problem into one-dimensional or two-dimensional slices to analyze the behavior of the function along specific directions.
• Consider the geometric interpretation of gradients in higher dimensions, such as the direction of steepest ascent and descent.
C. Tips for visualizing gradients to better understand their behavior
Visualizing gradients can provide valuable insights into the behavior of a function and how it changes across different points in its domain. However, effectively visualizing gradients requires a
good understanding of the underlying concepts and techniques for interpreting the visual representations.
Tips for visualizing gradients to better understand their behavior:
• Use color mapping or contour lines to represent the magnitude of the gradient at different points in the domain.
• Experiment with different visualization techniques such as streamlines or gradient vector plots to gain a comprehensive understanding of the function's behavior.
• Consider the relationship between the gradient and level sets of the function to visualize how the function changes across its domain.
Conclusion & Best Practices for Mastering Gradients
Understanding how to accurately calculate gradients is essential for mastering mathematical functions. It not only helps in solving complex problems but also provides a deeper insight into the
behavior of functions. In this final section, we will summarize the importance of accurately calculating gradients, highlight best practices, and encourage the application of gradient knowledge to
diverse problems for mastery.
A Summarization of the importance of accurately calculating gradients
• Insight into Function Behavior: Calculating gradients allows us to understand how a function changes at any given point, providing valuable information about its behavior.
• Optimization: Gradients are crucial in optimization problems, helping us find the maximum or minimum values of a function.
• Applications in Various Fields: From physics to economics, accurately calculating gradients is fundamental in various fields for modeling and problem-solving.
Highlight best practices, including consistent practice and leveraging visualization tools
• Consistent Practice: Mastering gradients requires consistent practice and application of different techniques to solve problems.
• Leveraging Visualization Tools: Using visualization tools such as graphs and diagrams can aid in understanding the concept of gradients and their impact on functions.
• Exploring Real-World Examples: Applying gradients to real-world problems can enhance understanding and provide practical experience in calculating gradients.
Encouragement to apply gradient knowledge to new and diverse problems for mastery
• Seeking Diverse Problems: To master gradients, it is important to apply the knowledge to a wide range of problems, including those from different domains and complexities.
• Continuous Learning: Embracing new challenges and seeking to solve diverse problems will contribute to the mastery of calculating gradients.
• Collaboration and Discussion: Engaging in discussions and collaborating with peers can provide new perspectives and insights into calculating gradients for various functions. | {"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-calculate-gradient-function","timestamp":"2024-11-09T04:15:59Z","content_type":"text/html","content_length":"230253","record_id":"<urn:uuid:71fdefe7-3d01-43b2-a4b4-be77fc801323>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00787.warc.gz"} |
CBSE Class 4 Maths Metric Measures Question bank
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Capillary Waves
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Understanding Capillary Waves
The subject of capillary waves is a fascinating topic in physics—these waves, ubiquitous yet underappreciated, hold intriguing depth and significance. You'll discover how these waves are everywhere,
from the raindrop hitting the surface of a pond to the ripple effect in your morning coffee.
Capillary Waves Definition
A capillary wave is a wave propagating along the interface between two fluid mediums, driven predominantly by the effects of surface tension.
Surface tension pulls the liquid's surface towards the centre of mass, which eventually results in a circular cross-section around the disturbance. As the wave propagates, this effect combined with
gravity causes the wave to 'drape' slightly, creating a sinusoidal form.
For an imagined scenario, picture dropping a pebble into a still pond—the circular ripples that spread outwards are nothing but capillary waves.
Under the guidance of concept 'dispersion relation', capillary waves behave differently depending on their wavelength. Specifically, \[ \Omega(k) = \sqrt { (\rho g k + \gamma k^3)/\rho } \] where \(
\Omega \) is the angular frequency, \( k \) is the wave number, \( \rho \) is the fluid density, \( g \) is the gravitational acceleration and \( \gamma \) is the surface tension. It depicts how
capillary waves with shorter wavelengths are primarily influenced by surface tension, whereas those with longer wavelengths lean more on gravity.
Significance of Capillary Waves in Physics
The study of capillary waves opens the door to understanding the diverse effects of surface tension itself.
• Aiding meteorologists in evaluating wind speeds over oceans and seas.
• Facilitating satellite imaging by revealing wave spectra on planetary surfaces.
• In physics, capillary waves often serve as a useful analogy for quantum mechanical concepts.
Common Examples of Capillary Waves
In your everyday life, you might have noticed a droplet of water creating a ripple effect when it falls onto a calm water surface; those small ripples are capillary waves.
Interestingly, the wind that blows over a water body can give rise to capillary waves; a phenomenon that has evolved as a vital tool for weather prediction. They also exist in large-scale occurrences
like tsunamis, making these waves relevant to both micro and macro events.
Observing Capillary Waves in Everyday Life
Capillary waves aren't solely restricted to vast bodies of water; their presence extends to your daily household activities too.
Small scale examples include the ripples created in a cup of coffee when stirred, or the waves created in a bathtub when a toddler splashes water. Another interesting instance is the 'tears' that
form on the inside of a wine glass, which is also a manifestation of a type of capillary wave.
Exploring Capillary Wave Theory
The theory behind capillary waves forms the backbone of our understanding of various natural and earth sciences, ranging from meteorology to oceanography.
Origins of Capillary Wave Theory
Often referenced as ripples, capillary waves have been a part of scientific study for centuries. The theory around these was first conceptualised during the 19th century. The scientist Thomas Young
made significant strides towards our understanding of this phenomenon.
He described capillary waves as the oscillation of fluid under the influence of surface tension, in the absence of other external forces. Diving deeper into these oscillatory movements, Young
discovered the strong relationship between the wave’s wavelength, velocity, and the surface tension acting upon it.
Expanding upon Young’s work, James Clerk Maxwell and Lord Rayleigh made significant contributions to what we now understand as the modern theory of capillary waves. Adapting their predecessors'
findings, they developed the dispersion relation for waves:
\[ \Omega (k) = \sqrt{ (\frac{\gamma k^3}{\rho} + gk) } \]
where \( \Omega \) represents the frequency of the waves, \( k \) is the wave number, \( \gamma \) is the surface tension of the liquid, \( \rho \) is the fluid density, and \( g \) is the
acceleration due to gravity. This equation beautifully captures the dual influence of both gravity and surface tension on capillary waves.
Key Contributions to Capillary Wave Theory
Holistically capillary wave theory is the amalgamation of work from pioneering physicists. Many made further advancements like the contributions of Sommerfeld and Lamb. Among these, a few stand out
due to their profundity and widespread influence on the understanding of capillary waves.
Scientist's Name Their Contribution
Thomas Young First to espouse and explain the impact of surface tension on fluid oscillation
James Clerk Maxwell Revised Young’s work on capillary wave theory introducing the influence of other factors like wavelength
Lord Rayleigh Built on Maxwell’s work to formulate a comprehensive dispersion relationship for capillary waves
Insights from Capillary Wave Theory
Capillary wave theory provides key insights into the interaction of surface tension and gravity, shaping our understanding of oceans, weather, and even quantum mechanics. Because these waves result
from the potentially morphic interface of two fluids, they offer unparalleled detail into the mechanics of fluid dynamics.
Applications of capillary wave theory are abundant. In meteorology, changes of capillary waves in the oceans provide critical data to predict weather patterns. The theory is also central to
oceanography by providing insights into wave dynamics in oceans and seas. Interestingly, a significant parallel is drawn between the behaviour of capillary waves and quantum mechanical waves, notably
in their mutual obeisance to the uncertainty principle.
Decoding the Dynamics of Capillary Waves
Decoding the dynamics of capillary waves warrants a comprehensive understanding of the dispersion relation, which encapsulates how these waves react to fluctuating variables such as surface tension,
fluid density, and wave number.
Capillary waves exhibit two unique states dictated by their wavelength. When the wavelength is tiny—typically under 1.7 cm in water—surface tension dominates, driving the ripples we commonly refer to
as capillary waves. If the wavelength is longer, gravity rules, and the phenomena are often referred to as gravity waves.
This duality has led scientists and mathematicians to derive two separate forms of the dispersion relation equation, each catering to the dominant contributor—surface tension or gravity.
• When surface tension is in control (\(k < \sqrt{\frac{\rho g}{\gamma}} \), where \(k\) is the wavenumber), the relation simplifies to \( \Omega^2 = \frac{\gamma k^3}{\rho} \), a behaviour at the
core of capillary waves.
• For gravity-dominant scenarios (\(k > \sqrt{\frac{\rho g}{\gamma}} \)), the equation becomes \( \Omega^2 = gk \), aligning with gravity waves.
These simplified forms allow a deeper delve into the dynamics of capillary waves under different conditions, paving the way for imaginative experiments, practical applications, and academic growth in
the field of physics.
Deconstructing Gravity Capillary Waves
To begin, it is important to understand that both surface tension and gravity play vital roles in the formation of waves on a fluid's surface. Depending on their relative importance, different wave
behaviors emerge. The arena where these forces strike a balance, giving rise to a unique form of waves, is what we explore under gravity capillary waves. Your detailed understanding of this science
begins here.
Differences between Capillary Waves and Gravity Waves
At the heart of the differences between capillary waves and gravity waves are the driving forces behind each - surface tension and gravity, respectively. But, there's another factor that makes a big
difference too: the length of the waves.
Let's begin by examining capillary waves. Capillary waves, or ripples, are ubiquitous—they can be seen when you toss a stone into a pond, or even when you stir your coffee. The leading force behind
such waves is surface tension. These waves are characterised by a shorter wavelength, typically less than about 1.7 cm for water at room temperature, and a higher frequency.
On the other hand, gravity waves, also known as surface gravity waves, are largely determined by the force of gravity. Unlike capillary waves, these waves occur when the wavelength is longer—greater
than 1.7 cm for water—are of lower frequency and hence take longer to occur.
Here's a simple breakdown:
• Capillary waves: Shorter wavelength (< 1.7 cm), higher frequency, driven by surface tension.
• Gravity waves: Longer wavelength (> 1.7 cm), lower frequency, driven by gravity.
Delineating Characteristics of Gravity Capillary Waves
Gravity capillary waves sit at the intersection of the two aforementioned types of waves—they reflect the tug of war between the forces of surface tension and gravity.
If you were to graph the phase speed of a wave against its wavelength, you would find a valley around the point marking a 1.7 cm wavelength (for pure water at room temperature). This is the realm of
gravity capillary waves. At these wavelengths, the competing effects of gravity and surface tension balance each other out giving rise to wave phenomena that can't simply be classified as either
'capillary' or 'gravity' waves.
Remarkably, these waves exhibit wave speeds that are slower than both capillary and gravity waves of comparable wavelengths. Such waves find their application in various scientific and technological
fields, including weather forecasting, remote sensing of the sea surface, and even studies of quantum wave-particle duality.
Gravity Capillary Waves Interaction with Surroundings
Gravity capillary waves don't exist in isolation — they are influenced in significant ways by their surroundings. The nature of the fluid, the ambient temperature, the presence of impurities or
surfactants, wind speed, pressure variations, and many more variables can dramatically impact the characteristics of these waves.
For instance, pressure variations can add nontrivial alterations to wave dynamics. At higher altitudes, the atmospheric pressure is lower, resulting in a decrease in wave speeds. Impurities or
surfactants, in turn, can lower the surface tension of the fluid, leading to concomitant alterations in wave behaviours.
Such depth in understanding allows us to infer a host of environmental information by studying these waves. For instance, changes in the pattern of gravity capillary waves on the ocean could reveal
the start of a gusty wind or an underwater seismic event.
Tracking the Impact of Gravity on Capillary Waves
Believe it or not, the Earth's gravity has a substantial impact on the life of capillary waves. As the force simultaneously helping to form and also to slow down these waves, gravity's influences
lend interesting contours to the dynamics of capillary waves.
To examine gravity's effects, let's consider a scenario of dropping a pebble into a calm body of water. As the disturbance travels outward, the wave's inherent surface tension tries to restore the
water surface to its flat state. This effect, aided by gravity, then pulls the water surface back downwards, raising other parts of the water surface, effectively forming another wave, and this cycle
keeps repeating.
This cyclical event clarifies why surface capillary waves aren't permanent, and dissolve after a point of time. But it isn't a unidirectional impact. Waves influence gravity as well. Scientists
routinely measure variations in the Earth's gravity field due to changes in ocean surface waves, a testament to the intricate connections between these waves and their surroundings.
While the explanation might seem intricate now, its comprehension can pave the way for a much deeper and intuitive understanding of the fascinating world of fluid dynamics.
Are Capillary Waves Dispersive?
In the world of physics, the short and straightforward answer is: yes, capillary waves are indeed dispersive. The dispersive nature of capillary waves arises from the wavelength-dependent phase
speeds, where waves of different wavelengths propagate at different velocities. To grasp the essence of this characteristic, one must delve into the realm of fluid dynamics, surface tension and
dispersion relations.
Analyzing the Dispersive Nature of Capillary Waves
A key feature of oscillatory movements in fluids is their dispersive nature—different frequencies, or wavelengths, progress at different speeds. As previously stated, capillary waves don't stray from
this rule. By definition, the dispersion relation for capillary waves is:
\[ \Omega(k) = \sqrt{(\frac{\gamma k^3}{\rho} + g k)} \]
where \( \Omega \) represents the frequency of waves, \( k \) is the wave number, \( \gamma \) is surface tension, \( \rho \) is fluid density, and \( g \) is gravitational acceleration. This
equation showcases how both gravity and surface tension contribute to the dispersion of capillary waves.
But it doesn't stop here. For shorter wavelengths or lighter densities, the influence of surface tension becomes more prominent than that of gravity. This is where the dispersion relation simplifies
to \( \Omega^2 = \frac{\gamma k^3}{\rho} \), indicating that the wave speed now depends solely upon the wavelength and surface tension of the fluid.
On the other hand, when it comes to longer wavelengths or higher densities, gravity takes the higher ground. As the most significant force, the dispersion relation then alters itself to \( \Omega^2 =
gk \). Here, the speed of the wave now only depends on the gravitational acceleration and the wavelength.
To unravel this wavelength dependency even further, one may qualitatively observe that shorter wavelength waves travel at a slower speed as compared to their longer wavelength counterparts. This
variation in speed with wavelength roots from the differences in the impact of forces (surface tension and gravity), which epitomises the dispersive nature of capillary waves.
Factors Affecting the Dispersion of Capillary Waves
Now at this stage, you might be curious about the various factors that affect the dispersion of capillary waves. Let's take a deeper look into some of them:
• Surface tension (\(\gamma\)): Surface tension plays a pivotal role in the formation of capillary waves, especially at shorter wavelengths where its effect tends to dominate. An increase in
surface tension enhances the dispersive effect, slowing down the waves of shorter wavelengths while leaving the longer ones relatively unaffected.
• Fluid density (\(\rho\)): Density, being inversely proportional to the phase speed, has a discernible effect on wave propagation. Higher fluid density means slower waves, resulting in an enhanced
dispersion of different wavelengths.
• Gravitational acceleration (\(g\)): Gravitational acceleration impacts the wave speed directly. A higher acceleration due to gravity increases the speed of waves, thereby modifying the dispersion
These factors, in different combinations, can create an array of varying wave behaviours that not only affect the dispersive nature of capillary waves, but also their shape and propagation.
Mathematical and experimental inquiries into the effects of surface tension, fluid density, and gravitational acceleration have led to a deeper understanding of wave dispersion in capillary waves.
This knowledge doesn't only add to the fundamental understanding of wave behavior but also finds application in other areas of study such as meteorological forecasting, marine exploration, and
environmental science to name a few.
Determining Causes of Capillary Waves
Capillary waves, also commonly known as ripples, are formed due to the fascinating equilibrium between the force of gravity and the surface tension of a fluid. To begin with a fair understanding of
ripple creation, it is crucial to grasp the delicate interaction between the external factors and the inherent properties of the fluid itself.
Physical Conditions Leading to Capillary Waves
At the heart of the mechanism that leads to the formation of capillary waves are two key forces: surface tension and gravity. The primary force responsible for the creation of capillary waves,
however, is surface tension. This property, inherent to all fluids, arises from the cohesive forces between liquid molecules. Because molecules on the surface do not have similar molecules all around
them, they are pulled inwards, causing this phenomena.
When a disturbance occurs on the surface — say a pebble is thrown into a pond or a gust of wind blows over the water surface — the surface tension acts to restore the fluid back to its unmoved shape.
This disturbance moves outwards as waves, the wavelengths of which are determined by the balance between the gravitational force and the surface tension.
However, there are more intricate details which govern the formation of these waves. Explained through the lens of the wave particle duality from quantum mechanics, a particle (in this case the
disrupting element such as a pebble or the gust of wind) imparts quantised 'packets' of energy, called quanta, to the fluid surface. The amount of these energy packets determines the wavelength and
frequency of the resulting wave. It's a wonderful instance of quantum effects playing out in our observable world!
This leads us to wavelength, one of the factors that distinguishes capillary waves. When the wavelength is less than approximately 1.7 cm (for water at room temperature), the wave is defined as a
capillary wave. For larger wavelengths, the wave is influenced more by gravity and is classified as a gravity wave. This '1.7 cm' is not a hard and fast rule, as it depends on the surface tension,
density, and temperature of the fluid.
Environmental Effects on Capillary Waves Formation
It's not just the internal conditions of a fluid and the interplay of forces that orchestrate the creation and shape of capillary waves. The environment housing the fluid acts as a major determinant
too. So, allow us to explore how environmental factors can influence the formation of capillary waves from several perspectives.
The presence of surfactants can dramatically alter the behaviour of capillary waves. Surfactants are compounds that reduce the fluid's surface tension. For example, soap added to water decreases its
surface tension, which subsequently alters the formation, propagation, and characteristics of the capillary waves – smaller wavelengths are dominant-
Next, the temperature of the fluid also significantly affects wave formation. Higher temperatures decrease both the surface tension and density of the fluid, which can increase the wavelength
threshold at which surface tension becomes the dominant force. Therefore, the capillary wave formations at higher temperatures are influenced differently than those at lower temperatures.
Fittingly, wind speed heavily influences the generation of capillary waves, especially on large bodies of water like oceans and seas. A stronger wind can input a larger amount of energy into the
fluid surface, generating waves of higher amplitudes and longer wavelengths. Likewise, randomness in wind direction and speed can give capillary waves a wide range of sizes, shapes, and propagation
Lastly, even underwater disturbances like seismic activities can play their role. Oscillations created by a submerged earthquake, for instance, propagate to the surface, leading to capillary wave
formation. Interestingly, such underwater disturbances can sometimes propagate over huge distances before surfacing as capillary waves, and hence can act as an early warning system for tsunamis or
other disruptive events.
To conclude, capillary waves provide an intriguing gateway into the world of fluid dynamics, wave behaviour, and much more. They depict a delicate balance of forces and unfold some intricate aspects
of our natural world, all at the same time!
Capillary Waves - Key takeaways
• Capillary waves, often referred to as ripples, oscillate under the influence of surface tension in the absence of other external forces.
• Capillary wave theory involves significant contributions from scientists Thomas Young, James Clerk Maxwell and Lord Rayleigh, among others.
• Applications of capillary wave theory are varied, including meteorology, oceanography, and quantum mechanics.
• Capillary waves and gravity waves are differentiated by their dominant forces (surface tension and gravity, respectively) and their wavelength; capillary waves have a shorter wavelength and
higher frequency, while gravity waves have a longer wavelength and lower frequency.
• Capillary waves are indeed dispersive, implying that waves of different wavelengths propagate at different velocities.
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Frequently Asked Questions about Capillary Waves
What causes capillary waves to form on the surface of water?
Capillary waves, also known as ripple waves, are caused by the disturbance of a fluid surface, such as water, due to the effects of surface tension and gravity. Forces like wind, nearby vibrations,
or objects entering the water can trigger these waves.
What factors can influence the speed and direction of capillary waves?
The speed and direction of capillary waves can be influenced by factors such as the water's surface tension, the density of the fluid, the force of gravity, and external forces like wind or
disturbances in the water surface.
How are capillary waves linked to weather forecasting and prediction?
Capillary waves, or ripples, on the ocean surface can indicate wind speed and direction, crucial factors in weather forecasting. Their characteristics, studied via remote sensing technologies, help
to develop accurate meteorological models for weather prediction.
Can capillary waves occur on liquids other than water?
Yes, capillary waves can occur on any liquid surface, not just water. They occur due to the effects of surface tension and gravity on the liquid.
How does surface tension contribute to the formation of capillary waves?
Surface tension is the driving force behind capillary waves. It causes the surface of a liquid to behave like a stretched elastic sheet. When disturbed, this 'sheet' forms tiny waves—capillary
waves—which quickly flatten out again due to the surface tension.
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K-Nearest Neighbor (KNN) Algorithm in Python • datagy
In this tutorial, you’ll learn how all you need to know about the K-Nearest Neighbor algorithm and how it works using Scikit-Learn in Python. The K-Nearest Neighbor algorithm in this tutorial will
focus on classification problems, though many of the principles will work for regression as well.
The tutorial assumes no prior knowledge of the K-Nearest Neighbor (or KNN) algorithm. By the end of this tutorial, you’ll have learned:
• How the algorithm works to predict classes of data
• How the algorithm can be tweaked to use different types of distances
• How the algorithm works with multiple dimensions
• How to work with categorical or non-numeric data in KNN classification
• How to validate your algorithm and test its effectiveness
• How to improve your algorithm using hyper-parameter turning in Python
Let’s get started!
What is the K-Nearest Neighbor Algorithm?
The K-Nearest Neighbor Algorithm (or KNN) is a popular supervised machine learning algorithm that can solve both classification and regression problems. The algorithm is quite intuitive and uses
distance measures to find k closest neighbours to a new, unlabelled data point to make a prediction. Because of this, the name refers to finding the k nearest neighbors to make a prediction for
unknown data.
• In classification problems, the KNN algorithm will attempt to infer a new data point’s class by looking at the classes of the majority of its k neighbours. For example, if five of a new data
point’s neighbors had a class of “Large”, while only two had a class of “Medium”, then the algorithm will predict that the class of the new data point is “Large”.
• In regression problems, the KNN algorithm will predict a new data point’s continuous value by returning the average of the k neighbours’ values. For example, if the five closest neighbours had
values of [100, 105, 95, 100, 110], then the algorithm would return a value of 102, which is the average of those five values.
In an upcoming section, we’ll walk through step-by-step how this algorithm works.
Why is the K-Nearest Neighbor Algorithm a Good Algorithm to Learn?
The K-Nearest Neighbor Algorithm is a great machine learning algorithm for beginners to learn. This isn’t to discount the immense value that a machine learning practitioner can get from the
Let’s look at three reasons why the KNN algorithm is great to learn:
1. It’s an intuitive algorithm, that is easy to visualize. This makes it great for beginners to learn and easy to explain to non-technical audiences.
2. It’s very versatile, since it can be applied to both regression and classification problems.
3. The algorithm can work with relatively small datasets and can run quite quickly. It can also be very easily tuned (as you’ll later learn) to improve its accuracy.
Now, let’s dive into how the algorithm actually works!
How does the K-Nearest Neighbor Algorithm Work?
The K-Nearest Neighbor algorithm works by calculating a new data points class (in the case of classification) or value (in the case of regression) by looking at its most similar neighbors. How does
it determine which data points are the most similar? Generally, this is done by using a distance calculation, such as the Euclidian distance or the Manhattan distance.
As a machine learning scientist, it’s your job to determine the following:
1. Which similarity measure to use,
2. How many neighbours (k) to look at, and
3. Which features (or dimensions) of your data are most important
Let’s take a look at a sample dataset. Take a look at the data below? Do you see any patterns in the data?
It looks like there are three clusters in our data
Upon first inspection, it looks like there are two clusters of data. Thankfully, our dataset is pre-labelled and we can actually colour the different labels differently. Let’s take a look at our
graph now.
There are actually categories in our data
Now we can see there are actually three classes of data. We have “Small”, “Medium”, and “Large” classes of data. We can see that as the values of x and y increase, that the data tends to move closer
to a larger class.
Knowing this, let’s introduce a new data point. We don’t know what class the data point belongs to, but we do know its values for x and y, which are stored in the tuple (6, 3).
Let’s see where this new data point fits on our graph:
Adding a new data point to our dataset
We can see that our data point is near some Large points (in green) and some Medium points (in orange). It’s a little hard to tell at this point whether our data point should be labelled as Medium or
This is where the KNN algorithm comes into play. Let’s take a look at our first hyper-parameter of the algorithm: k. The value of k determines the number of neighbors to look at. In classification
problems, it can be helpful to use odd values of k, since it requires a majority vote (which can be more difficult with an even number).
To start, let’s use the value of k=5, meaning that we’ll look at the new data point’s five closest neighbours. One very useful measure of distance is the Euclidian distance, which represents the
shortest distance between two points. Imagine the distance as representing the way a crow would fly between two points, completely unobstructed.
Let’s draw the distances to only the nearest five data points and see which classes are most connected to our new data.
Identifying the five closest neighbors
By looking at this, we can see that the majority of the nearest points are classed as Large, with only a single nearest neighbor being classed as Medium.
Using this simple calculation, we’re able to say with some degree of confidence that the label of this data point should be Large.
Later in the tutorial, you’ll learn how to calculate the accuracy of your model, as well as how to improve it.
Using the K-Nearest Neighbor Algorithm in Python’s Scikit-Learn
In this section, you’ll learn how to use the popular Scikit-Learn (sklearn) library to make use of the KNN algorithm. To start, let’s begin by importing some critical libraries: sklearn and pandas:
import pandas as pd
from sklearn.neighbors import KNeighborsClassifier
from seaborn import load_dataset
For this tutorial, we’ll focus on the Penguins dataset that comes bundled with Seaborn. The dataset covers information on different species of penguins, including the island the sample was taken
from, as well as their bill length and depth.
The dataset focuses on predicting the species of a penguin based on its physical characteristics. There are three types of Penguins that the dataset has data on: the Adelie, Chinstrap, and Gentoo
penguins, as shown below:
Artwork by @allison_horst
We can load the dataset as a Pandas DataFrame to explore the data a bit using the load_dataset() function:
# Loading the penguins dataset
df = load_dataset('penguins')
# Returns:
# species island bill_length_mm bill_depth_mm flipper_length_mm body_mass_g sex
# 0 Adelie Torgersen 39.1 18.7 181.0 3750.0 Male
# 1 Adelie Torgersen 39.5 17.4 186.0 3800.0 Female
# 2 Adelie Torgersen 40.3 18.0 195.0 3250.0 Female
# 3 Adelie Torgersen NaN NaN NaN NaN NaN
# 4 Adelie Torgersen 36.7 19.3 193.0 3450.0 Female
We can see that our dataset has six features and one target. Let’s break this down a little bit:
Column Type Description Data Type # of Unique Observations
species Target The species of the penguin String 3
island Feature The island on which the penguin’s data was taken String 3
bill_length_mm Feature The length of the penguin’s bill, measured in millimetres Float N/A (continuous)
bill_depth_mm Feature The depth of the penguin’s bill, measured in millimetres Float N/A (continuous)
flipper_length_mm Feature The length of the penguin’s flipper, measured in millimetres Float N/A (continuous)
body_mass_g Feature The mass of the penguin, measured in grams Float N/A (continuous)
sex Feature The sex of the penguin String 2
Describing the Penguins dataset from Seaborn
We can see how the measurement’s of the penguins are taken by taking a look at this helpful image below:
Artwork by @allison_horst
Splitting our Data into Training and Testing Data
We’ll need to split our data into both features and target arrays.
• The features array, commonly referred to as X, is expected to be multi-dimensional array.
• Meanwhile, the target array, commonly noted as y, is expected to be of a single dimension.
Lets focus only one a single dimension for now: bill length. We’ll extract that column as a DataFrame (rather than as a Series), so that sklearn can load it properly.
# Splitting our DataFrame into features and target
df = df.dropna()
X = df[['bill_length_mm']]
y = df['species']
One important piece to note above is that we’ve dropped any missing records. Technically it may be a good idea to try and impute these values. However, this is a bit out of the scope of this
We can also split our data into training and testing data to prevent overfitting our analysis and to help evaluate the accuracy of our model. This can be done using the train_test_split() function in
sklearn. To learn more about this function, check out my in-depth tutorial here.
For this, we’ll need to import the function first. We’ll then set a random_state= value so that our results are reproducible. This, of course, is optional. However, it lets you reproduce your results
consistently, so it’s a good practice.
# Splitting data into training and testing data
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state = 100)
Now that we have our dataset lined up, let’s take a look at how the KNeighborsClassifier class works in sklearn.
Understanding KNeighborsClassifier in Sklearn
Before diving further into using sklearn to calculate the KNN algorithm, let’s take a look at the KNeighborsClasifier class:
n_neighbors=5, # The number of neighbours to consider
weights='uniform', # How to weight distances
algorithm='auto', # Algorithm to compute the neighbours
leaf_size=30, # The leaf size to speed up searches
p=2, # The power parameter for the Minkowski metric
metric='minkowski', # The type of distance to use
metric_params=None, # Keyword arguments for the metric function
n_jobs=None # How many parallel jobs to run
In this tutorial, we’ll focus (in time) on the n_neighbors=, weights=, p=, and n_jobs= parameters.
To kick things off though, let’s focus on what we’ve learned so far: measuring distances using the Euclidian distance, and finding the five nearest neighbors.
In order to use the Euclidian distance, we can either modify the metric= parameter to 'euclidian', or we can change the p= parameter to 1. We’ll get more into this in a later section, but for now,
let’s simply change the p= value.
Conventionally, the classifier object is assigned to a variable clf. Let’s load the class with the parameters discussed above:
# Creating a classifier object in sklearn
clf = KNeighborsClassifier(p=1)
In the object above, we’ve instantiated a classifier object that uses the Euclidian distance (p=1) and looks for five neighbours (default n_neighbors=5).
Now that we have our classifier set up, we can pass in our training data to fit the algorithm. This will handle the steps we visually undertook earlier in the tutorial by finding the nearest
neighbours’s class for each penguin:
# Fitting our model
clf.fit(X_train, y_train)
At this point, we’ve made our algorithm! Sklearn has abstracted a lot of the complexities of the calculation behind the scenes.
We can now use our model to make predictions on the data. To do this, we can use the .predict() method and pass in our testing feature:
# Making predictions
predictions = clf.predict(X_test)
# Returns:
# ['Adelie' 'Gentoo' 'Chinstrap' 'Adelie' 'Gentoo' 'Gentoo' 'Gentoo'
# 'Chinstrap' 'Gentoo' 'Gentoo' 'Gentoo' 'Adelie' 'Adelie' 'Gentoo'
# 'Gentoo' 'Chinstrap' 'Chinstrap' 'Adelie' 'Gentoo' 'Gentoo' 'Adelie'
# 'Gentoo' 'Adelie' 'Adelie' 'Adelie' 'Adelie' 'Gentoo' 'Chinstrap'
# 'Adelie' 'Adelie' 'Adelie' 'Adelie' 'Gentoo' 'Adelie' 'Chinstrap'
# 'Gentoo' 'Adelie' 'Gentoo' 'Gentoo' 'Gentoo' 'Adelie' 'Gentoo' 'Adelie'
# 'Adelie' 'Chinstrap' 'Chinstrap' 'Chinstrap' 'Adelie' 'Gentoo' 'Gentoo'
# 'Gentoo' 'Gentoo' 'Adelie' 'Adelie' 'Gentoo' 'Gentoo' 'Adelie' 'Gentoo'
# 'Gentoo' 'Adelie' 'Gentoo' 'Gentoo' 'Gentoo' 'Adelie' 'Adelie' 'Adelie'
# 'Chinstrap' 'Adelie' 'Gentoo' 'Gentoo' 'Chinstrap' 'Chinstrap' 'Adelie'
# 'Chinstrap' 'Gentoo' 'Gentoo' 'Gentoo' 'Chinstrap' 'Adelie' 'Gentoo'
# 'Adelie' 'Adelie' 'Adelie' 'Chinstrap']
Similarly, if we wanted to simply pass in a single mock-penguins data, we could pass in a list containing that one value. Say we measured our own pet penguin’s bill length and found that it was 45.2
mm. We could simply write:
# Making your own predictions
predictions = clf.predict([[44.2]])
# Returns
# 'Gentoo'
At this point, you may be wondering: “Great! We’ve made a prediction. But, how accurate is that prediction?” Let’s dive into the next section to learn more about evaluating our model’s performance.
Validating a K-Nearest Neighbor Algorithm in Python’s Scikit-Learn
At this point, you’ve created a K-Nearest Neighbor classifier using a single feature. You’re probably also curious to know how accurate your model is. We can measure the model’s accuracy because our
original dataset was pre-labelled.
Because we split our data into training and testing data, it can be helpful to evaluate the model’s performance using the testing data. This is because this is data that the model hasn’t yet seen.
Because of this, we can be confident that the model’s effectiveness to new data can be accurately tested.
In classification problems, one helpful measurement for a model’s effectiveness is the accuracy score. This looks at the proportion of accurate predictions out of the total of all predictions.
When we made predictions using the X_test array, sklearn returned an array of predictions. We already know the true values for these: they’re stored in y_test.
We can use the sklearn function, accuracy_score() to return a proportion out of 1 that measures the algorithms effectiveness. Let’s see how we can do this:
# Measuring the accuracy of our model
from sklearn.metrics import accuracy_score
print(accuracy_score(y_test, predictions))
# Returns: 0.7023809523809523
We can see that the model’s accuracy is 70%, meaning that it returned true positives 7/10 times. This isn’t bad, but we can definitely tweak our algorithm to improve its effectiveness. In order to do
this, let’s take a quick look at how else we can measure distances using a K-Nearest Neighbor algorithm.
How Do You Calculate Distances in K-Nearest Neighbor?
So far in this tutorial, we’ve explored how to measure the distance between two points using the Euclidian distance. Another distance metric that can be used is the Manhattan distance, which is often
referred to as the tax-cab distance. While the Euclidian distance measures the shortest distance between two points, the Manhattan distance measures the distance that a taxi-cab would have to take if
it could only make right-angle turns.
Take a look at the image below to see how these distances differ:
The differences between the Euclidian and Manhattan distances
When the metric= parameter was first introduced, the default value was specified to 'minkowski'. The Minkowski distance is the generalization of both the Euclidian distance and the Manhattan
Let’s take a look at the formula:
The formula for the Minkowski Distance
When p is equal to 2, the distance represents the Euclidian distance. When p is equal to 1, the Minkowski distance represents the Manhattan distance.
Because of this, we can toggle between the two distances by setting the values of p to either 1 or 2. Later on in the tutorial, you’ll learn to test which of the distances is a better for the context
of your data. Neither is inherently better and will depend on the particulars of your dataset.
When would you use either distance? There is no hard and fast rule. However, the Manhattan distance has been suggested to work better for high-dimensional datasets and when the influence of outliers
can have a dramatic effect.
Measuring the Influence of a Neighbours’ Distance
One other piece to consider is how the neighbours themselves are weighted. By default, sklearn will weight each neighbour in a uniform manner. This means that regardless of how far each neighbour is
away from the new data point, it will have the same weight.
This is controlled by the weights= parameter. If we wanted to change this to instead weigh each data point based on its distance from the new data point, we could set the argument to 'distance'. In
this case, the weight of each neighbour is the inverse of its distance to the data point.
Again, neither of these approaches is inherently right or wrong, or better or worse. This is where the “art” of machine learning comes in, as each hyper-parameter is tuned using the specifics of your
Using Multiple Dimensions in K-Nearest Neighbor in Python
When we first loaded this dataset, we knew that we had access to seven features. Perhaps our model’s accuracy was only 70% because we used only one feature. Perhaps this feature didn’t have as much
influence as we’d have hoped.
In this section, you’ll learn how to build your KNN algorithm using multiple dimensions. We’ll focus on only numeric features so far, but cover off categorical data in the next section. By necessity,
sklearn requires that each value passed into the algorithm is numeric and not missing.
Let’s change our X variable to represent all the numeric columns in our dataset. We can accomplish this by using the select_dtypes() function and asking Pandas to only include the type of numbers:
df = load_dataset('penguins')
df = df.dropna()
X = df.select_dtypes(include='number')
y = df['species']
In this case, Pandas included the following columns:
• bill_length_mm
• bill_depth_mm
• flipper_length_mm
• body_mass_g
We now have four dimensions to work with. What’s great about this is that it introduces significantly more data, which will hopefully improve our predictions!
In order to modify our algorithm, we don’t actually need to do anything differently. Let’s repeat the steps from above by first splitting our data and then passing in our training data:
# Loading a multi-dimensional dataset into KNN
X = df.select_dtypes(include='number')
y = df['species']
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state = 100)
clf = KNeighborsClassifier(p=1)
clf.fit(X_train, y_train)
predictions = clf.predict(X_test)
Let’s see how much our accuracy score changed in this case:
# Checking our new accuracy score
print(accuracy_score(y_test, predictions))
# Returns: 0.7738095238095238
We can see that our model’s accuracy improved by 7%! Let’s take a look at how we can encode our categorical data to make it work with our algorithm.
Working with Categorical Data in K-Nearest Neighbor in Python
Machine learning models work with numerical data. Because of this, we need to transform the data in our categorical columns into numbers in order for our algorithm to work successfully.
There are a number of different ways in which we can encode our categorical data. One of these methods is known as one-hot encoding. This process converts each unique value in a categorical column
into its own binary column. The image below demonstrates how this can be done:
One-hot encoding in Scikit-Learn
In the image above, each unique value is turned into its own column. This means that for the three unique values we now have three distinct column. The values represent either a 1 (for that value
being represented) or a 0 (for the value not being represented).
You may be wondering why we didn’t encode the data as 0, 1, and 2. The reason for this is because the data isn’t ordinal or interval data, where the order means anything. By assigning a value of 0 to
one island and 2 to another would imply the difference between these two islands is greater than between one island and another.
Let’s see how we can one-hot encode the two categorical columns, 'sex' and 'island':
# One-hot encoding categorical variables in Sklearn
from sklearn.preprocessing import OneHotEncoder
from sklearn.compose import make_column_transformer
X = df.drop(columns = ['species'])
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state = 100)
column_transformer = make_column_transformer(
(OneHotEncoder(), ['sex', 'island']),
X_train = column_transformer.fit_transform(X_train)
X_train = pd.DataFrame(data=X_train, columns=column_transformer.get_feature_names())
Let’s break down what we did here:
1. We imported both the OneHotEncoder class and the make_column_transformer class
2. We created a make_column_transformer class and passed in a tuple containing the transformation we wanted to happen. We wanted to apply a OneHotEncoder() class with default arguments to both the
'sex' and 'island' columns. The argument of remainder = 'passthrough' simply instructs sklearn to ignore the other columns.
3. We used the .fit_transform() method to transform X_train
4. Finally, we created a new DataFrame out of the transformed data.
Scaling Data for K-Neatest Neighbor
One of the things you may have noticed in our DataFrame is that some of our features have significantly larger ranges than others. For example, all of our one-hot encoded variables have either a 0 or
a 1. However, our body_mass_g variable has a range of 2700 through 6300.
One of the things that may occur is that the variables with larger ranges dominate the algorithm. This can be mitigated if we scale the data. By using a Min-Max normalization method, all the values
will exist on a range from 0 to 1, though the original distribution of the data will be maintained.
Let’s see how we can use sklearn to apply Min-Max scaling to our data. Because we’re applying a transformation to our columns, we can actually build this into the make_column_transformer class from
earlier! This will save us a bit of trouble to fit and transform two sets of data.
# Adding Min-max Scaling to our Preprocessing
import pandas as pd
from sklearn.neighbors import KNeighborsClassifier
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import OneHotEncoder
from sklearn.compose import make_column_transformer
from sklearn.metrics import accuracy_score
from sklearn.preprocessing import MinMaxScaler
from seaborn import load_dataset
df = load_dataset('penguins')
df = df.dropna()
X = df.drop(columns = ['species'])
y = df['species']
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state = 100)
column_transformer = make_column_transformer(
(OneHotEncoder(), ['sex', 'island']),
(MinMaxScaler(), ['bill_depth_mm', 'bill_length_mm', 'flipper_length_mm', 'body_mass_g']),
X_train = column_transformer.fit_transform(X_train)
X_train = pd.DataFrame(data=X_train, columns=column_transformer.get_feature_names())
Now that we’ve preprocessed our data both by one-hot encoding and scaling our data, let’s take a look at how the accuracy has changed.
We’ll need to apply the fitted transformations of our data to the testing data, so there’ll be a bit of pre-work to do before we can make predictions.
# Making predictions with our new algorithm
X_test = column_transformer.transform(X_test)
X_test = pd.DataFrame(data=X_test, columns=column_transformer.get_feature_names())
clf = KNeighborsClassifier(p=1)
clf.fit(X_train, y_train)
predictions = clf.predict(X_test)
print(accuracy_score(y_test, predictions))
# Returns: 1.0
Let’s break down what we did above:
1. We used our column_transformer to transform the testing features in the same way as our training data. Note, in particular, we’re using only the .transform() method – not the .train_transform()
method we previously used.
2. We used our training data to train our classifier and passed in our transformed training data to make predictions.
3. Finally, we used our testing labels to test our accuracy
Note that the accuracy we returned was 100%! This may have been a fluke based on the random state that we used. Changing the random state will result in slightly lower accuracy (though still in the
high 90%s).
It’s important to note that accuracy is just a single criterion for evaluating the performance of a classification problem. If you want to learn more about this, check out my in-depth post on
calculating and visualizing a confusion matrix in Python.
Hyper-Parameter Tuning for K-Nearest Neighbor in Python’s Scikit-Learn
To close out this tutorial, let’s take a look at how we can improve our model’s accuracy by tuning some of its hyper-parameters. Hyper-parameters are the variables that you specify while building a
machine learning model. This includes, for example, the number of neighbours to consider or the type of distance to use.
Hyper-parameter tuning, then, refers to the process of tuning these values to ensure a higher accuracy score. One way to do this is, simply, to plug in different values and see which hyper-parameters
return the highest score.
This, however, is quite time-consuming. Scikit-Learn comes with a class GridSearchCV which makes the process simpler. You simply provide a dictionary of values to run through and sklearn returns the
values that worked best.
What’s more, is that the class also completes a process of cross-validation. When we picked our random-state, we trained based on a single selection of training data. GridSearchCV will cycle through
the different combinations of training and testing splits that can be created.
Want to learn about a more efficient way to optimize hyperparameters? You can optimize and speed up your hyperparameter tuning using the Optuna library.
Let’s consider some of the hyper-parameters that we may want to tune to improve the accuracy of our KNN model:
• n_neighbors measures the number of neighbours to use to determine what classification to make. Since we’re working in classification and need a majority vote, it can be helpful to consider only
odd numbers.
• p determines what type of distance to use. We can test both Euclidian and Manhattan distances.
• weights determines whether to weigh all neighbours equally or to take their distances into consideration.
Let’s pass in a dictionary of these parameters into our GridSearchCV class:
# Creating a dictionary of parameters to use in GridSearchCV
from sklearn.model_selection import GridSearchCV
params = {
'n_neighbors': range(1, 15, 2),
'p': [1,2],
'weights': ['uniform', 'distance']
clf = GridSearchCV(
clf.fit(X_train, y_train)
# Returns: {'n_neighbors': 5, 'p': 1, 'weights': 'uniform'}
Let’s break this code down a bit:
1. We defined a dictionary of parameters to test.
2. We created a classifier using the GridSearchCV class, asking for a cross-validation of 5
3. Finally, we fitted the data and printed the best values for our hyper-parameters.
In this case, sklearn returned the values of {'n_neighbors': 5, 'p': 1, 'weights': 'uniform'} for our best parameters. Note that this is only the best parameters of the options we asked to be tested.
It could very well be that the value for n_neighbors=21 would have produced the best results – but we didn’t ask sklearn to test that!
Now that you have these parameters, simply pass them into your model.
What to Learn After This Tutorial
Ok! You’ve made it to the end of the tutorial. At this point, you may be thinking, “Ok Nik – what now?” The answer to that question is simply to play around with this model and see if you can make it
work with other datasets! Try and learn about different parameters in the KNeighborsClassifier class and see how you can tweak them to make your model run better or more efficiently.
Take a look at the sklearn documentation for the KNeighborsClassifier. While it can often seem dense, hopefully this tutorial makes them a little clearer.
In this tutorial, you learned how to use the KNeighborsClassifier to take on the K-Nearest Neighbors classifier algorithm in sklearn. You first learned why the algorithm is a good choice, especially
for beginners or for those who need to explain what’s going on. You then learned, visually, how the algorithm works. Then you walked through an end-to-end exercise of preparing your data for the
algorithm, evaluating its performance, and tweaking it to make it perform better.
Additional Resources
To learn more about related topics, check out the articles below:
2 thoughts on “K-Nearest Neighbor (KNN) Algorithm in Python”
1. Hi,
Your coding content is very useful . But due to heavy ads, site is very slow and boring. Can you publish some plotting technique using matplotlib library?
1. Thanks Bruce! I appreciate the feedback. I’m definitely planning on adding more on Matplotlin the future. | {"url":"http://datagy.io/python-knn/","timestamp":"2024-11-14T16:54:47Z","content_type":"text/html","content_length":"210552","record_id":"<urn:uuid:89c418b2-b73e-4485-bf0c-bf308c4dda5b>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00581.warc.gz"} |
Jump to navigation Jump to search
A predicate P is a hyponym of another predicate Q iff P is a special case of Q:
For any pair of predicates P,Q:
for all x,
P(x) → Q(x)
not (Q(x) → P(x)))
The term 'hyponym' is a converse of the term 'hyperonym'.
• 'Dog' is a hyponym of 'animal'.
Other languages
REF This article has no reference(s) or source(s).
Please remove this block only when the problem is solved. | {"url":"http://glottopedia.org/index.php?title=Hyponym&oldid=15802","timestamp":"2024-11-14T02:27:30Z","content_type":"text/html","content_length":"16491","record_id":"<urn:uuid:7d17188d-1ca4-449f-a01b-698b003f92b6>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00131.warc.gz"} |
Geometry Can Help Make Clear What Algebra and Multiplication Mean
Many people who have learned arithmetic successfully have some difficulty moving to algebra which is more abstract and symbolic. In fact algebra is just arithmetic but using letters to be more
general. They say a picture is worth a thousand … Continue reading
Posted in Algebra, Geometry, Multiplication, Numbers Leave a comment
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The Complete Data Science & Machine Learning Bootcamp in Python 3 - reason.townThe Complete Data Science & Machine Learning Bootcamp in Python 3
The Complete Data Science & Machine Learning Bootcamp in Python 3
Are you looking to learn Data Science and Machine Learning? Then this bootcamp is for you! This 12-week course will teach you everything you need to know about Python 3, from the basics all the way
to advanced concepts. By the end of the course, you’ll be able to confidently tackle any data science or machine learning problem. So what are you waiting for? Sign up today!
Checkout this video:
Introduction to Data Science and Machine Learning
Data science is a quickly growing field that is all about extracting knowledge and insights from data. Machine learning is a subset of data science that focus on building algorithms that can
automatically learn and improve from experience.
Python is a programming language that is particularly well suited for data science and machine learning. In this bootcamp, we will be using Python 3 to cover all the material.
This bootcamp will cover all the essential topics in data science and machine learning, including:
– Exploratory data analysis
– Manipulating and cleaning data
– Visualization
– Linear regression
– Logistic regression
– Decision trees and random forests
– Support vector machines
– Neural networks
The Python Programming Language
Python is a widely used high-level interpreted language. Python’s design philosophy emphasizes code readability with its notable use of significant whitespace. Its language constructs and
object-oriented approach aim to help programmers write clear, logical code for small and large-scale projects.
Data Manipulation with Pandas
As a data analyst or scientist, you’ll almost always need to work with data in a tabular format. Luckily, the pandas Python library makes working with tabular data much easier. In this article, we’ll
go over the basics of pandas so you can start working with tabular data in Python.
Pandas is a Python library for working with tabular data that is built on top of the NumPy library. It provides powerful tools for dealing with missing data, working with time series data, and
performing other advanced operations on numerical data.
One of the most important pieces of Pandas is the DataFrame, which is similar to a Excel spreadsheet or SQL table. DataFrames allow you to manipulate and analyze tabular data in a variety of ways.
In this article, we’ll cover the following topics:
– Creating DataFrames
– Manipulating DataFrames
– Slicing and Indexing DataFrames
– Transforming DataFrames
Data Visualization with Matplotlib
In this section of the bootcamp, we’ll learn how to visualize data using Matplotlib. Matplotlib is a powerful Python library that can be used to create a variety of different types of plots and
visualizations. We’ll learn how to use Matplotlib to create line graphs, bar charts, scatter plots, and more.
Machine Learning with Scikit-Learn
If you want to learn about machine learning, there’s no better library to use than Scikit-learn. In this section of the course, we’ll show you how to use Scikit-learn to build machine learning models
in Python.
We’ll start by covering some of the basics of machine learning, including what it is, how it works, and some of the development tasks that are involved in building machine learning models. Then we’ll
move on to using Scikit-learn to build and evaluate a wide variety of supervised and unsupervised learning models. By the end of this section, you will have a strong understanding of how to use
Scikit-learn to build machine learning models that can make predictions on data.
Deep Learning with TensorFlow
Deep learning is a subset of machine learning that uses algorithms to model high-level abstractions in data. Using deep learning, a computer can learn to recognize objects, identify voices, and make
predictions. Deep learning is the key to creating self-driving cars and understanding human behavior.
TensorFlow is a powerful tool for deep learning. It was created by Google Brain and allows you to create complex neural networks. TensorFlow is open source and can be used on any platform.
In this course, you will learn how to use TensorFlow to build deep learning models. You will also learn how to train and optimize your models. By the end of this course, you will be able to build
your own deep learning models with TensorFlow.
Natural Language Processing with NLTK
Natural Language Processing (NLP) is a subfield of Artificial Intelligence (AI) that is concerned with helping computers to understand human language. In other words, NLP is all about teaching
computers how to read and write like humans.
NLTK is a powerful Python library that makes it easy to work with human language data. NLTK includes many different algorithms for performing various tasks such as tokenization, part-of-speech
tagging, stemmer, and lemmatization.
Big Data with Apache Spark
Spark is a powerful tool for working with Big Data, and it’s becoming increasingly popular in the world of data science and machine learning. In this bootcamp, we’ll give you a crash course in Spark,
covering the basics of working with Spark dataframes, performing common data manipulation tasks, and running simple machine learning algorithms on Spark data.
Project: Building a Machine Learning Model
In this project, we are going to use a dataset of Kickstarter projects to predict whether or not a project will be successfully funded. We will use a variety of different machine learning models and
compare their accuracy in order to find the best model for this particular dataset.
The dataset can be found here: https://www.kaggle.com/kemical/kickstarter-projects
This project is ideal for beginner to intermediate level data science and machine learning students.
In conclusion, we have covered a lot of ground regarding data science and machine learning in Python 3. We have seen how to pre-process data, how to build and train models, and how to evaluate their
performance. We have also seen how to use a variety of tools and libraries for data science and machine learning, including NumPy, pandas, matplotlib, seaborn, Scikit-learn, TensorFlow, and Keras.
We hope that this bootcamp has given you a good foundation on which to build your data science and machine learning skills. There is a lot more to learn, but we believe that this bootcamp has
equipped you with the basics that you need to get started on your journey. Thank you for taking the time to learn with us! | {"url":"https://reason.town/complete-data-science-machine-learning-bootcamp-python-3/","timestamp":"2024-11-11T18:04:05Z","content_type":"text/html","content_length":"96962","record_id":"<urn:uuid:5d1a7c25-0612-4f2c-9fcc-70946238315b>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00613.warc.gz"} |
Counting in 2s Missing Numbers Number Line | Learn and Solve Questions
Introduction to Counting in 2s
Learning to add numbers by 2 is done through skip counting by twos. We begin by skip counting and adding when we first gain the concept of addition. Skip counting by 2 is adding 2 to each previous
number to obtain the next number in the series. This method helps us in learning the concept of the addition of 2. Starting with 0, skipping counting by two results in a series that looks like $0, 2,
4, 6, 8$ and so on.
Skip counting is a mathematical strategy taught as a sort of multiplication. Earlier textbooks referred to this method as "counting by twos" (threes, fours, etc.). A person can count to ten using
only the other even numbers $2,4,6,8$ and 10 when skip counting by 2s.
Counting in 2s
Counting in 2s
To skip count by two, add two to each previous number to get the subsequent number in the sequence. This skill helps in our understanding of the addition of two concepts. When learning addition by
counting, skip counting by two is a highly helpful skill that advances our understanding of addition by adding two to each number. The table of two is easier to memorise when we skip count by two.
Skip Counting by 2 on Number Line
In this section, we will learn the concept of skip counting by 2 on a number line. We will make two jumps to reach one number to another. Let us have a look at the number line below. Starting from
the number 0, when skip counting by 2, the next number will be 0 + 2 = 2.
So, we have landed on the number 2. Next, when we take another jump, we land on 2 + 2 = 4. Continuing in the same manner, we skip a number and land on the next number. So, the series continues like
0, 2, 4, 6, 8, 10, and so on.
Skip Counting by 2 on Number Line
Find the missing numbers in the series 12, ___ , ___, 18, 20, 22, ___ using skip counting by 2.
Ans: To find the missing numbers by skip counting by 2, we add to each number in the series to obtain the next number.
12 + 2 = 14
14 + 2 = 16
22 + 2 = 24
So, the series is 12, 14, 16, 18, 20, 22, 24
The missing numbers are 14, 16 and 24.
Example: Write the multiples of 2 up to 20.
Ans: We will use the method of skip counting by 2 to write the table of 2. The first multiple is 2, so we obtain the remaining multiples by adding 2 to each multiple.
2 + 2 = 4
4 + 2 = 6
6 + 2 = 8
8 + 2 = 10
10 + 2 = 12
12 + 2 = 14
14 + 2 = 16
16 + 2 = 18
18 + 2 = 20
So, the multiples of 2 up to 20 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.
Counting in 2s Activities
Counting in 2s activities
Counting in 2s Game
Write all the multiples of two which are not circled.
Counting in 2s game
Ans: Multiple of 2s, which are not circled, are as follows:
2, 8, 24 and 30.
Counting in 2s Worksheet
Counting in 2s worksheet
To conclude all the conceptual understanding regarding Counting in 2s in this article, we can say the idea of skipping two numbers on a number line. To get from one number to another, we have to make
the jump of two. Then, later on, we saw some examples based on skip counting, like the missing number between the multiple of 2s. Also, to get more clarity on the topic, we have included the activity
worksheet and games through which students will get more engaged with counting in 2s.
FAQs on Counting in 2s Missing Numbers Number Line
1. Why is it necessary to count in 2s?
Children's ability to recognise patterns and solve problems is boosted by doing this. It helps in learning the tables. They can better comprehend the idea of even and odd when they count in twos.
2. Kids should count by 2s when?
Your child's developing mathematical knowledge heavily relies on numbers and counting. These fundamental mathematical ideas lay the groundwork for later, more sophisticated mathematical operations.
Eight-year-olds frequently count up to 1,000 and have mastered the art of skip counting (counting by 2s, 5s, 10s).
3. When should children start counting by 2s?
A youngster can first recognise when there is one and more than one, but not whether there are two or six. By the time a youngster is two years old, he can count to two ("one, two"), and by the age
of three, he can count to three. However, if he can count to ten, he is probably reciting from memory. | {"url":"https://www.vedantu.com/maths/counting-in-2s-missing-numbers-number-line","timestamp":"2024-11-07T14:08:44Z","content_type":"text/html","content_length":"220408","record_id":"<urn:uuid:ee0252df-2347-41de-86a5-e2f34bf19d50>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00768.warc.gz"} |
Homework 4 ISyE 6420 The Bounded Normal Mean solution
1. Metropolis: The Bounded Normal Mean.
Suppose that we have information
that the normal mean θ is bounded between −m and m, for some known number m. In this
case it is natural to elicit a prior on θ with the support on interval [−m, m].
A prior with interesting theoretical properties supported on [−m, m] is Bickel-Levit prior1
π(θ) = 1
, − m ≤ θ ≤ m.
Assume that a sample [−2, −3, 4, −7, 0, 4] is observed from normal distribution
f(y|θ) ∝
τ exp
(y − θ)
with a known precision τ = 1/4. Assume also that the prior on θ is Bickel-Levit, with m = 2.
This combination likelihood/prior does not result in an explicit posterior (in terms of
elementary functions). Construct a Metropolis algorithm that will sample from the posterior
of θ.
(a) Simulate 10,000 observations from the posterior, after discarding first 500 observations
(burn-in), and plot the histogram of the posterior.
(b) Find Bayes estimator of θ, and 95% equitailed Credible Set based on the simulated
(i) Take uniform distribution on [−m, m] as a proposal distribution since it is easy to
sample from. This is an independence proposal, the proposed θ
′ does not depend on the
current value of the chain, θ.
1When m is large, this prior is an approximation of the least favorable distribution in a Bayes-Minimax
problem with class of priors limited to symmetric and unimodal distributions.
(ii) You will need to calculate Pn
i=1(yi − θ)
for current θ and Pn
i=1(yi − θ
for the
proposed θ
, prior to calculating the Metropolis ratio.
Gibbs Sampler and High/Low Protein Diet in Rats. Armitage and Berry (1994,
p. 111)2
report data on the weight gain of 19 female rats between 28 and 84 days after birth.
The rats were placed in randomized manner on diets with high (12 animals) and low (7
animals) protein content.
High protein Low protein
We want to test the hypothesis on dietary effect: Did a low protein diet result in a
significantly lower weight gain?
The classical t test against the one sided alternative will be significant at 5% significance
level, but we will not go in there. We will do the test Bayesian way using Gibbs sampler.
Assume that high-protein diet measurements y1i
, i = 1, . . . , 12 are coming from normal
distribution N (θ1, 1/τ1), where τ1 is the precision parameter,
|θ1, τ1) ∝ τ
(y1i − θ1)
, i = 1, . . . , 12.
The low-protein diet measurements y2i
, i = 1, . . . , 7 are coming from normal distribution
N (θ2, 1/τ2),
|θ2, τ2) ∝ τ
(y2i − θ2)
, i = 1, . . . , 7.
Assume that θ1 and θ2 have normal priors N (θ10, 1/τ10) and N (θ20, 1/τ20), respectively. Take
prior means as θ10 = θ20 = 110 (apriori no preference) and precisions as τ10 = τ20 = 1/100.
Assume that τ1 and τ2 have the gamma Ga(a1, b1) and Ga(a2, b2) priors with shapes
a1 = a2 = 0.01 and rates b1 = b2 = 4.
(a) Construct Gibbs sampler that will sample θ1, τ1, θ2, and τ2 from their posteriors.
2Armitage, P. and Berry, G. (1994). Statistical Methods in Medical Research (3rd edition). Blackwell
(b) Find sample differences θ1 − θ2 . Proportion of positive differences approximates the
posterior probability of hypothesis H0 : θ1 > θ2. What is this proportion if the number of
simulations is 10,000, with burn-in of 500?
(c) Using sample quantiles find the 95% equitailed credible set for θ1 − θ2. Does this set
contain 0?
Hint: No WinBUGS should be used (except maybe to check your results). Use Octave
(MATLAB), or R, or Python here. You may want to consult Handout GIBBS.pdf from the
course web repository. | {"url":"https://jarviscodinghub.com/product/homework-4-isye-6420-the-bounded-normal-mean-solution/","timestamp":"2024-11-03T09:53:05Z","content_type":"text/html","content_length":"107601","record_id":"<urn:uuid:d843c504-56f5-46ab-a7ad-829bfae05470>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00750.warc.gz"} |
J. Whitbeck and M. Dias de Amorim and V. Conan and J.-L. Guillaume
The 18th Annual International Conference on Mobile Computing and Networking, Mobicom’12, pp. 377-388
While a natural fit for modeling and understanding mobile networks, time-varying graphs remain poorly understood. Indeed, many of the usual concepts of static graphs have no obvious counterpart in
time-varying ones. In this paper, we introduce the notion of temporal reachability graphs. A (tau, sigma)-reachability graph is a time-varying directed graph derived from an existing connectivity
graph. An edge exists from one node to another in the reachability graph at time t if there exists a journey (i.e., a spatiotemporal path) in the connectivity graph from the first node to the second,
leaving after t, with a positive edge traversal time tau, and arriving within a maximum delay sigma. We make three contributions. First, we develop the theoretical framework around
temporal reachability graphs. Second, we harness our theoretical findings to propose an algorithm for their efficient computation. Finally, we demonstrate the analytic power of the temporal
reachability graph concept by applying it to synthetic and real-life data sets. On top of defining clear upper bounds on communication capabilities, reachability graphs highlight asymmetric
communication opportunities and offloading potential. | {"url":"https://www.complexnetworks.fr/tag/reachability/","timestamp":"2024-11-10T15:41:20Z","content_type":"text/html","content_length":"74198","record_id":"<urn:uuid:c6d4774d-e4d1-4118-8208-2896270c5a53>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00812.warc.gz"} |
VTU Applied Thermodynamics - December 2013 Exam Question Paper | Stupidsid
Total marks: --
Total time: --
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary
1 (a) Define the therms: i) Stoichiometric air; ii) Enthalpy of combustion iv) Enthalpy of formation; v) Adiabatic flame temperature.
10 M
1 (b) The volumetric composition of dry fuel gases obtained by the combustion of an unknown hydrocarbon is 12.7% CO[2], 0.9% CO, 3.9% O[2] and 82.5% N[2] Determine: i) Composition of the fuel; ii)
Theoretical air required for complete combustion ; iii) Percentage of excess air required.
10 M
2 (a) With the help of P-V and T-S diagrams, derive an equation for theoretical air standard efficiency of a semi diesel (dual) cycle in terms of compression ratio, cut off ratio and explosion ratio,
with suitable assumptions.
10 M
2 (b) In an air standard diesel cycle the compression ratio is 15 and the fluid properties at the beginning of compression are 100kPa and 300K. For a peak temperature of 1600K calculate i) the
percentage of stroke at which cut off occurs; ii) the cycle efficiency and iii) the work output/kg.
10 M
3 (a) Describe the following as applied to I.C engine i) Morse test ii) Heat balance sheet.
8 M
3 (b) During a test on a single cylinder 4 stroke oil engine the following observation were made Bore=30cm, stroke=45 cm, duration of trail =1hr , total fuel consumption=7.6 kg calorific value of
fuel=45,000kJ/kg. Total revolution made=12000, mean effective pressure 6 bar, net brake load=1.47 kN. Brake drum diameter 1.8m rope diameter 3cm. Mass of jacket cooling water circulated=550kg water
enters at 15°C water leaves at 60°C. Total air consumption 360kh room temperature 20°C, exhaust gas temperature 300°C, calculate: i) Indicated and break power ii) Indicated thermal efficiency iii)
Mechanical efficiency ; iv) Draw the heat balance sheet on minute basis
12 M
4 (a) Why Carnot cycle is not used as a reference cycle for steam power plant?
3 M
4 (b) Sketch the flow diagram and corresponding T-S diagram,of a reheat vapour cycle and derive an expression for reheat cycle efficiency
7 M
4 (c) A 40MW steam power plant working on Rankine cycle operates between boiler pressure of 4MPa and condenser pressure of 10 kPa. The steam leaves the boiler and enters the steam turbine at 400°C.
The isentropic efficiency of steam turbine is 85%. determine : i) The cycle efficiency ii)The quality of exhaust steam turbine and iii) Steam flow rate in kg/hr considering pump work.
Properties of steam
│ │ │Specific volume m^3/kg│Specific enthalpy kJ/kg │Specific entropy kJ/kgK │
│Pressure bar│ts c ├────────────┬─────────┼──────┬────────┬────────┼───────┬────────┬────────┤
│ │ │V[f] │V[g] │h[f] │h[f] │h[g] │s[f] │s[f] │s[g] │
│40 │250.3│0.00125 │0.049 │1087.4│1712.9 │2800.3 │2.797 │3.272 │6.069 │
│0.1 │45.83│0.0010 │14.675 │191.8 │2392.9 │2584.7 │0.649 │7.502 │8.151 │
10 M
5 (a) State the advantages of multi-stage compression.
4 M
5 (b) Derive the relation among volumetric efficiency with clearance for various pressure ratio.
8 M
5 (c) Atmospheric air at 1 bar and 27°C is taken into a single stage reciprocating compressor. It is compressed according to the law PV^13=C, to the delivery pressure of 6 bar. The compressor takes
1m^3 of air/min. The speed of the compressor is 300rpm. Stroke to diameter ratio is equal to 1.5:1 mechanical efficiency of the compressor 0.85 motor transmission efficiency 0.9 calculate:
i)The indicated power and isothermal efficiency.
ii) The cylinder dimensions and power of motor required to drive the compressor.
8 M
6 (a) Discuss with help of T-S diagram the three methods of improving the thermal efficiency of an open cycle gas turbine plant.
12 M
6 (b) A gas turbine unit has pressure ratio 6.1 and maximum cycle temperature of 610°C. The isentropic efficiency of the compressor and turbine are 0.80 and 0.82 respectively. Calculate the power
output in kilo wants of an electric generator geared to the turbine when the air enters the compressor at 15°C and rate of 16kg/s. Assume C_{\rho}=1.005 for C_{\rho}=1.11 for r=1.4 compression=
expansion =n=1.33
8 M
7 (a) Write a short note on air cycle refrigeration.
4 M
7 (b) With the help of neat flow diagram explain the working of steam jet refrigeration system.
8 M
7 (c) An ammonia vapour compression refrigeration machine works between 25°C and -20°C. The ammonia leaves the compressor in dry and saturated condition. Liquid ammonia is under cooled to 21.5°C
before passing through throttle valve. The average specific heat of liquid ammonia is 4.75 kJ/kg°C. Find the theoretical cop of machine. The following properties of NH[3] are given. If the net
refrigeration required is 400 × 10^3 kj/hr. Find the mass of ammonia circulated /min. Assume cop actual is 75% of cop theoretical:
│ │Liquid kJ/kg │Vapour kJ/kg K │
│Temp C├──────────┬─────┼────────┬──────┤
│ │h[f]kJ/kg │s[f] │h[g] │s[g] │
│25 │537.6 │4.612│1708.5 │8.534 │
│-20 │328.4 │3.854│1661.0 │9.118 │
8 M
8 (a) Derive the equation for relative humidity and specific humidity of moist air.
4 M
8 (b) With neat sketch explain the working of air conditioning system for hot and dry weather. Present the process involved on a psychometric chart.
6 M
8 (c) Calculate i) relative humidity; ii) humidity ratio; iii) dev point temperature; iv)density and v)Enthalpy of atmospheric air when the DBT is 35°C, WBT=23°C and the barometer reads 750mm Hg.
10 M
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Let f (x) = √x − 1 and g(x) = 1x − 2 .(i) State the domains for f and g, and the range of f .
Let f (x) = √x − 1 and g(x) = 1x − 2 .(i) State the domains for f and g, and the range of f .
Solution 1
The domain of a function is the set of all possible input values (x-values) which will output real numbers.
For the function f(x) = √x - 1, the input x must be greater than or equal to 1. This is because the square root of a negative number is not a real number. So, the domain of f is [1, ∞) Knowee AI is a
powerful AI-powered study tool designed to help you to solve study problem.
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5 Best Free Factoring Cubics Calculator For Windows
Here is a list of Free Factoring Cubics Calculator For Windows. Using these free calculators, you can find out the roots of a cubic polynomial easily. All these factoring cubics calculator work on
the same principle. You have to provide the values of all coefficients of a cubic equation in order to get all its roots.
Most of these freeware also tell the nature of the roots thus obtained. I have also added a cubics equation calculator in this list which lets you create your own macros to solve different
mathematical problems.
My favorite factoring cubics calculator:
Out of all these factoring cubics calculator, I like Microsoft Mathematics the most. It is a fully-featured calculator, the use of which is not limited to solve cubic equations. You can use it to
solve both maths and science problems. It comes with a worksheet. You can write any type of problem on the worksheet to get its solution. This feature makes it very easy to use. Besides this, it
offers many advanced features. Read the article to know more about it.
You may also like some best free Math Equation Editor, Geometry Calculator, and Matrix Calculator software for Windows.
Microsoft Mathematics
Microsoft Mathematics is a free factoring cubics calculator for Windows. It is a featured scientific calculator that is designed to solve both maths and science problems. You can solve equations,
plot graphs, convert units, and solve many other problems with this advanced calculator.
Using Microsoft Mathematics is a piece of cake. Simply write a problem on the worksheet provided on its interface and click on Enter button. You can enter any type of problem on the worksheet. You
will then get the solution to the entered problem. Apart from this, you can also use the Equation Solver to solve the system of linear equations, quadratic equations, cubic equations, quartic
equations, etc. After solving an equation, you can also plot the graph of that equation in the Graphing section.
Following are some of the problems which you can solve with the help of Microsoft Mathematics:
Precise Calculator
Precise Calculator is a free and open source scientific calculator which can be used as roots of cubic equation calculator.
How to solve a cubic equation using Precise Calculator:
Go to Macro menu and select Cubic Equation. This will write a code in the upper empty box of the calculator with default values of coefficients of a cubic equation. You just have to change the values
of these coefficients (a, b, c, and d) and press Enter button. All three roots of a cubic equation are displayed in the output box. The time taken to solve a problem is also displayed on its
interface. It only solves the roots of a cubic equation but does not tell the nature of roots.
Apart from finding the roots of a cubic equation, it is also capable to find the roots of a quadratic equation, find prime numbers, solve trigonometric equations, solve hyperbolic functions, find
area, perimeter, volume, etc. of 2D and 3D geometric shapes. Some of the 2D and 3D geometric shapes include circle, kite, parallelogram, square, trapezoid, cube, cone, trapezium, cylinder, pyramid,
It has four modes of calculation, namely, Norm, Fix, Sci, and Eng. You can switch the result to any of these modes. Moreover, it also features decimal, hexadecimal, and binary conversions. For
scientific calculations, it comes with some predefined constants, like Speed of Light, Permittivity of Vacuum, Electron Mass, Proton Mass, Neutron Mass, Boltzmann’s constant, Stefan-Boltzmann’s
constant, etc. You can use these constants directly in your calculation.
General Features of Precise Calculator:
• It supports more than 5 languages: English, Catalan, Espanol, French, Italiano, etc.
• You can change font style and font size.
• It lets you save the result in .txt format.
• You can create your own macros and save them.
• It automatically saves calculation history. You can clear it anytime.
Precise Calculator is a portable factoring cubics calculator.
NH Mathematical Tools
NH Mathematical Tools is another free factoring cubic polynomials calculator for Windows. The cubic equation is displayed in its standard form, i.e., AX^3+BX^2+CX+D=0 with empty boxes in place of
coefficients. You have to fill the value of the coefficients in the required places and click on Solve this Equation button. The final answer is displayed along with the nature of the roots.
It comes with five categories of tools:
• Algebra: In this section, you will find factoring quadratics calculator, factoring cubics calculator, solving systems of equations calculator with 2 and 3 unknowns, and HCF and LCM finder.
• Arithmetic-Discrete: This section includes base converter, factorial calculator, and random number generator.
• Sequence: Fibonacci and Fermat number calculators are provided here.
• Probability: You can find out the probability by using probability calculator.
• Quiz: Quiz is available in three levels (easy, medium, and hard). You can also set time for playing the quiz.
Cubique - Cubic Equation Solver
Cubique – Cubic Equation Solver is another free factoring cubics calculator for Windows. The calculator has a very simple interface. The general format of the third order (cubic) equation is shown.
You have to enter the values of all four coefficients in the empty spaces. You can enter the coefficients either in decimal, fraction, or exponential form.
It also displays the nature of the roots. The good part of the software is it displays the complex conjugate roots of the equation with red color. On the other hand, all real roots are displayed with
blue color. So, you can easily identify them.
A copy to clipboard feature is also available by which you can easily copy the solved roots of the equation. But, this feature did not work while testing.
Root Calculator
Root Calculator is another free factoring cubics calculator for Windows. Using this free calculator, you can find the roots of a cubic equation. The structure of a cubic equation (AX^3+BX^2+CX+D=0)
is displayed on its interface. You have to fill up the values of coefficients in the required fields of the cubic equation thus displayed and click on Solve button. The roots of a cubic equation are
displayed in the Solutions section along with the nature of roots.
Other advantages of this freeware include interpolation, extrapolation, quadratic roots calculator, quartic equation roots calculator (only for paid users), and complex multiplication.
NOTE: The free version of this software displays only real roots and not the imaginary one. If the cubic equation has any imaginary roots, it displays a message: Complex Roots are shown on Upgrade.
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MATH 106
Designed for future K-6 teachers. Focus is on mathematical concepts, including counting, number sense, operations, algorithms, fractions, ratio, and proportion. Method topics include teaching
strategies, assessment methods, and processes of doing mathematics as related to elementary mathematics. This course does not fulfill the quantitative skills requirement for the AA-DTA degree. This
class may include students from multiple sections. (Elective)
1. Understand and apply foundations of current pedagogical theories of the learning mathematics by elementary students, particularly with respect to the mathematical concepts in the K-8 curriculum.
2. Analyze, understand, and apply the four fundamental operations of arithmetic.
3. Analyze, understand, and apply number theory, including divisibility and factorization.
4. Analyze, understand, and extend the number system to include fractions and rational numbers, decimals, exponents, and real numbers. | {"url":"https://catalog.pencol.edu/mathematics-mathmath/math-106","timestamp":"2024-11-07T20:08:52Z","content_type":"text/html","content_length":"17606","record_id":"<urn:uuid:26541602-2e20-4f43-af13-576032208d1c>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00139.warc.gz"} |
how to get the beam size and divergence at different FW for as non Gaussian beam | Zemax Community
Dear all,
I noticed that zemax gives the beam size and divergence at 1/e^2 FW (full width) (13.5% FW) but, I would like to have it at 10%. Does anyone know whether there is a way in zemax to get the beam size
and divergence at other FW. My beam is not Gaussian so, I cannot simply convert the 13.5%, the zemax output, to 10% using the intensity distribution.
Thanks in advance, | {"url":"https://community.zemax.com/got-a-question-7/how-to-get-the-beam-size-and-divergence-at-different-fw-for-as-non-gaussian-beam-3809?postid=12161","timestamp":"2024-11-07T15:54:28Z","content_type":"text/html","content_length":"212754","record_id":"<urn:uuid:a28d0721-472b-48a4-8af2-d3cd5fd1a826>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00256.warc.gz"} |
Probability Calculations | Lexique de mathématique
Probability Calculations
To calculate probabilities, we apply the rules and principles that govern probability theory.
When rolling a six-sided fair die, the possible outcomes are: Ω = {1, 2, 3, 4, 5, 6}.
In this situation, the probability of getting a 3 is given by: P(3) \(=\frac{1}{6}\). | {"url":"https://lexique.netmath.ca/en/probability-calculations/","timestamp":"2024-11-05T04:34:50Z","content_type":"text/html","content_length":"63315","record_id":"<urn:uuid:bd8793c9-598d-4b4b-82e5-a78a0b8818e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00760.warc.gz"} |
Nguyen-Thi Dang - Dynamics of the Weyl chamber flow (over a higher rank locally symmetric space)
Qiongling Li - Higgs bundle and minimal surfaces in non-compact symmetric space
A Higgs bundle over a Riemann surface X equipped with a harmonic metric is called a harmonic bundle. Conformal harmonic bundles over a Riemann surface X correspond to equivariant minimal branched
immersion from the universal cover of X to the symmetric space associated to GL(n,C). Our plan of the mini-course is as follows: Part I explains the explicit correspondence between conformal harmonic
bundles with minimal surfaces in the symmetric space associated to GL(n,C). Part II provides various interesting examples of minimal surfaces in the product of symmetric space of non-compact type and
the Euclidean space in terms of harmonic bundles. Part III discusses further developments on the topics like Labourie conjecture, Morse index, total curvature and so on.
James Farre - Convex pleated surfaces
A quasi-Fuchsian surface group is a discrete, convex co-compact surface subgroup of PSL(2,C), which acts isometrically on hyperbolic 3-space. The boundary of the convex core of a quasi-Fuschian
surface group has the intrinsic structure of a hyperbolic surface. This hyperbolic surface is bent along a family of geodesic lines that form a geodesic lamination, and the bending angle defines a
transverse measure on this lamination. These convex surfaces, embedded in a complete hyperbolic 3-manifold, are examples of Thurston’s pleated surfaces.
Assuming a bit of familiarity with basic hyperbolic geometry, will give a crash course on (measured) geodesic laminations on closed hyperbolic surfaces and then build quasi-Fuchsian surface groups by
bending a totally geodesic plane in hyperbolic space along a lamination in a group equivariant way.
If time permits, we will also explain how to bend convex projective structures on closed surfaces along a geodesic lamination in 3-dimensional real projective space to obtain certain convex
co-compact surface subgroups of SL(4,R).
Bram Petri - Probabilistic methods in hyperbolic geometry | {"url":"https://submanifolds-autrans.sciencesconf.org/resource/page/id/1","timestamp":"2024-11-02T02:09:01Z","content_type":"application/xhtml+xml","content_length":"12272","record_id":"<urn:uuid:ea5ba0c6-e94b-4cb0-9bdc-011e49c6ae8e>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00094.warc.gz"} |
Sveriges lantbruksuniversitet - Primo - SLU-biblioteket
Alexander Forsman - Project Engineer - FMV - Försvarets
The WTI are like layered 2D QSH states, but are destroyed by disorder. The STI are robust and lead to novel ‘‘topological metal’’ surface states. face states of a 3D topological insulator do strongly
resemble the edge states of a 2D topological insulator. As in the 2D case, the direction of electron motion along the surface of a 3D topological insulator is deter-mined by the spin direction, which
now varies continu-ously as a function of propagation direction (figure 1d).
Better your understanding of this method with this roundup of tips, materials and more. There are a number of reasons to know all you should about insulation. Insulation, when used properly, will
Insulators work as protectors. They may protect heat, sound and the passage of electricity.
Therefore, the half Hall conductance is a unique property of the surface states of 3D topological insulators which is determined by the bulk topology.
Sveriges lantbruksuniversitet - Primo - SLU-biblioteket
Two-dimensional (2D) topological insulators are … 2 days ago Topological materials with exotic quantum properties are promising candidates for quantum spin electronics. Different classes of
topological materials, including Weyl semimetal, topological superconductor, topological insulator and Axion insulator, etc., can be connected to each other via quantum phase transition.
TWO-DIMENSIONAL MATERIALS - Dissertations.se
This new class of materials Recently, topological material has been a hot topic in condensed matter physics, but I don't know what is topological material and how to distinguish topological material
from band diagram. And how does it divide the types of topological materials according to band diagram, such as Topological Insulators,Topological Semimetal. 2020-03-05 · Topological materials
include topological insulators (TI) [3, 4], Weyl and Dirac semimetals (WSM, DSM) , and topological superconductors (TSC) . The distinction is not always clear cut, since some categories overlap.
Ruck’s bismuth-rhodium-iodine material stays a topological insulator at temperatures up to roughly 2,000 K (Nat. Mater. 2013, DOI: 10.1038/nmat3570).
From the adiabatic theorem of quantum mechanics to topological states of matter [Review] Electronic structure of the 3D topologic Insulator Bi 2 Te 3.
Fragor att stalla pa anstallningsintervjun
Topological insulators represent a new quantum state of matter which is characterized by peculiar edge or surface states that show up due to a topological character of the bulk wave functions. This
review presents a pedagogical account on topological insulator materials with an emphasis on basic theory and materials properties. Topological Materials Topological insulators are a new state of
quantum matter with a bulk gap and odd number of relativistic Dirac fermions on the surface. The bulk of such materials is insulating but the surface can conduct electric current with well-defined
spin texture. Topological insulators are materials that are electrically insulating in the bulk but can conduct electricity due to topologically protected electronic edge or surface states.
Topological Materials Topological insulators are a new state of quantum matter with a bulk gap and odd number of relativistic Dirac fermions on the surface. The bulk of such materials is insulating
but the surface can conduct electric current with well-defined spin texture. Topological insulators are materials that are electrically insulating in the bulk but can conduct electricity due to
topologically protected electronic edge or surface states. Topological insulators represent a new quantum state of matter which is characterized by peculiar edge or surface states that show up due to
a topological character of the bulk wave functions. This review presents a pedagogical account on topological insulator materials with an emphasis on basic theory and materials properties.
Topological insulators represent a new quantum state of matter which is characterized by peculiar edge or surface states that show up due to a topological character of the bulk wave functions. This
review presents a pedagogical account on topological insulator materials with an emphasis on basic theory and materials properties.
Vårdcentralen bokskogen sjukgymnast
The study of topological materials has been an important task in condensed Abstract [en]. Topological insulators and topological crystalline insulators are materials that have a bulk band structure
that is gapped, but that also have av J Schmidt · 2020 — frequency superconductivity in the doped topological insulator Bi2Se3. The. Kitaev materials are characterized by the interplay of
electronic Physical Review B. Condensed Matter and Materials Physics. 102.
These materials have emerged as exceptionally fertile ground for materials science research.
Arkeologiska termer
Majorana and Weyl Modes in Designer Materials - Aaltodoc
In two-dimensional tungsten ditelluride, two different states of matter — topological insulator and Chris Hooley explains this phenomenon in 100 seconds. Visit physicsworld.com for more videos and
podcasts. http://physicsworld.com/cws/channel/multimedia Thus far, the field of topological insulators has been focused on bismuth and antimony chalcogenide based materials such as Bi 2 Se 3, Bi 2 Te
3, Sb 2 Te 3 or Bi 1 − x Sb x, Bi 1.1 Sb 0.9 Te 2 S. The choice of chalcogenides is related to the Van der Waals relaxation of the lattice matching strength which restricts the number of materials
and substrates. 2013-04-21 · Topological insulators represent a new quantum state of matter which is characterized by peculiar edge or surface states that show up due to a topological character of
the bulk wave functions. This review presents a pedagogical account on topological insulator materials with an emphasis on basic theory and materials properties.
Gta 5 franklin
体育即时比分app - Valt projekt - Uppsala universitet
This review presents a pedagogical account on topological insulator materials with an emphasis on basic theory and materials 2013-01-01 · Another correlated material α-Fe 2 O 3 with corundum
structure is predicted to be a possible topological magnetic insulator .
Ph.D. position in experimental nanoelectronics and
M. Malard | Extern. Postdoc, Department of Physics and Astronomy, Materials Theory, Uppsala Effect of the Rashba splitting on the RKKY interaction in topological-insulator thin topological insulator
by introducing the spin-orbit coupling （SOC） or mass term. In this review paper, we introduce theoretical materials that show the nodal Aharonov-Bohm interference in topological insulator
nanoribbons-article. ISSN: 1476-1122 , 1476-4660 ,. , Nature Materials , Vol.9(3), p.225-229 ,.
Topological insulators represent a new quantum state of matter which is characterized by peculiar edge or surface states that show up due to a topological character of the bulk wave functions. This
review presents a pedagogical account on topological insulator materials with an emphasis on basic theory and materials properties. Topological insulators have attracted much interest recently from
Condensed Matter Physics as well as the wider scientific community. They are characterized by having bulk electronic states like that of a standard band gap insulator but with spin-momentum locked
conduction channels only at the surface of the material. Chris Hooley explains this phenomenon in 100 seconds. Visit physicsworld.com for more videos and podcasts. | {"url":"https://hurmanblirrikfqthlu.netlify.app/48722/31520","timestamp":"2024-11-14T08:37:15Z","content_type":"text/html","content_length":"18544","record_id":"<urn:uuid:91a90d00-4287-49f3-bb08-8dbf7bcc3643>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00740.warc.gz"} |
s in Johannesburg
Mathematics tutors in Johannesburg
Personalized Tutoring Near You Mathematics lessons for online or at home learning in Johannesburg
Extra Math Lessons Johannesburg, Gauteng by Top Math Tutors
University and School classes in JHB are fast paced and can be tough to achieve the results you are looking for. We help in all levels of Primary, High School and University levels up until PHD. We
cover all parts of Johannesburg including Johannesburg North, JHB South, Johannesburg East, Joburg West or Joburg Central. Our private tutors come from the most trusted Universities in the area
including the University of Johannesburg and the University of Witwaterstrand (WITS). Some popular topics include Advanced Mathematics, Additional Mathematics, Applied Mathematics, Financial
Mathematics, Trigonometry, Probability & Statistics, Number Theory, Number System, Logic, Geometry, Game Theory, Dynamical Systems, Differential Equations, Coordinate Geometry, Computation,
Combinatorics, Calculus, Arithmetic, Algebra and Mathematics IEB.
Mathematics tutors in Johannesburg near you
Rivaldo Messia M
Auckland Park
Josh G
Cheltondale, Johannesburg
During my four and a half years of experience, I taught Maths to students ranging from grades 8 to 12 and by final exams, each of my students significantly improved their marks with some even
achieving distinction. Moreover, having studied a Bachelor of Commerce with Law, my subjects throughout my degree included Computational and Applied Maths, Business Statistics and Financial Maths.
Teaches: English as a foreign Language, Mathematics, History, Accounting, Science, Biology, English
Available for Mathematics lessons in Johannesburg
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Highlands North
Uhuru R
Parktown, Johannesburg
I have achieved brilliant results in Mathematics. I have the ability to explain anything to its simplest form. My students have achieved great results after been tutored by me. I come highly
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Teaches: Law, Mathematics, Humanities, History, Politics, Economics, Accounting, English Language and Literature
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Waverley, Johannesburg
I am maths graduate and have gained the tools for mathematical analysis and best course/s of action in the application of mathematics fundamentals while working in a research facility. I am able to
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Subjects related to Mathematics in Johannesburg
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How do you solve abs(2x+9)=abs(7x-2)? | HIX Tutor
How do you solve #abs(2x+9)=abs(7x-2)#?
Answer 1
Solution: $x = \frac{11}{5} , x = - \frac{7}{9}$
#|2 x+ 9| =|7 x- 2| # . Squaring both sides we get,
#(2 x+ 9)^2 =(7 x- 2)^2 # or
#4 x^2+36 x +81 =49 x^2-28 x +4 # or
#45 x^2- 64 x -77= 0 ; a=45 ,b=-64 ,c=-77#
Discriminant # D= b^2-4a c = 64^2-4*77*45=17956#
Quadratic formula: #x= (-b+-sqrtD)/(2a) #
#:. x= (64+-sqrt 17956)/(2*45)= 32/45 +- 134/90 # or
#x = 32/45 +- 67/45:. x = 99/45=11/5 and x= -35/45=-7/9 #
Solution: #x=11/5, x= -7/9# [Ans]
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Answer 2
To solve the equation ( |2x + 9| = |7x - 2| ), you need to consider two cases: when ( 2x + 9 ) is positive or zero, and when ( 2x + 9 ) is negative. Then, solve each case separately to find the
values of ( x ).
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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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Find Index of an Element in a Tuple in Python - Data Science Parichay
In this tutorial, we will look at how to find the index of an element in a tuple in Python with the help of some examples.
How to get the index of an element inside a tuple?
You can use the Python tuple index() function to find the index of an element in a tuple. The following is the syntax.
# find index of element e in tuple t
It returns the index of the first occurrence of the value inside the tuple. If the value is not present in the tuple, it gives an error.
Let’s look at some examples.
Index of an element in a tuple
Let’s find the index of the element 3 inside the tuple (1, 3, 2, 5, 3, 7).
# create a tuple
t = (1, 3, 2, 5, 3, 7)
# find index of 3
We get the index of 3 as 1. Note that the element 3 is present twice inside the tuple, at indexes 1 and 4 respectively but the index function only returned the index of its first occurrence, that is,
All indexes of an element in a tuple
If you want all the indexes of occurrences of a value, you can iterate over the values in a loop. For example, let’s find all the indexes of 3 in the above tuple.
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# create a tuple
t = (1, 3, 2, 5, 3, 7)
# find all indexes of 3
result = []
for i in range(len(t)):
if t[i] == 3:
[1, 4]
We get both its indexes. Here we iterate over each value in the tuple and compare it with 3, if it’s equal, we add its index to our result list.
You can use a list comprehension to reduce the above code to a single line.
# create a tuple
t = (1, 3, 2, 5, 3, 7)
# find all indexes of 3
result = [i for i in range(len(t)) if t[i]==3]
[1, 4]
We get the same result as above.
What if the element is not present?
If the element is not present inside the tuple, the index() function will give an error. For example, let’s try to find the index of 4, an element that is not present in the above tuple.
# create a tuple
t = (1, 3, 2, 5, 3, 7)
# find index of 4
ValueError Traceback (most recent call last)
Input In [4], in <module>
2 t = (1, 3, 2, 5, 3, 7)
3 # find index of 4
----> 4 print(t.index(4))
ValueError: tuple.index(x): x not in tuple
We get a ValueError indicating that the element for which we are trying to find the index is not present in the tuple.
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Stacking Data
Data stacking involves splitting a data set up into smaller data sets, and stacking the values for each of the variables into a single column. It is a type of data wrangling, which is used when
preparing data for further analysis. Common applications of stacking are: to unloop data, to allow multiple outcome variables to be used in regression, and to simplify reporting.
Example of stacking
In the image below, each row shows the data for one of four respondents in a survey. The data file contains a looped structure, where three sets of information appear for three different brands. In
total, there are four observations and 10 variables.
The same data is shown below, in stacked form. It has been reshaped to contain 12 observations and five variables. The last three variables (columns) show data that has been stacked. The first column
contains the ID variable, which has been stretched. The second column contains the unique variable names from the original data and is also stretched to line up with the other data.
Stacking can occur multiple times
You can perform data stacking multiple times. For example, you could stack the data set on the right, to contain two variables, where the first variable contained all the values in the table, stacked
on top of each other, and the second variable contained the variable names, stretched to line up with the appropriate values.
Stacking to unloop data
Often, people create data files where each row reflects how the data has been collected, rather than how it should be analyzed. For example, surveys often have data on a whole household of people in
a single row, but analysis may require each person in the household to be treated as a separate analysis unit (and thus to have their own row in the data file). By unlooping the data, calculating
summary statistics, such as averages and percentages, become more straightforward because the data to be included is in a single column vs spread over multiple columns. In the example above, most
statistical programs would not readily be able to compute an average answer for "Likelihood to Recommend" in the original data file, but can easily do so using the stacked data file.
Stacking for regression analysis
Most software for regression assumes that there is a single outcome variable. However, this is commonly not the case. For example, in the data set above, there are three potential outcome variables
(the three variables measuring the likelihood to recommend). Stacking the data means you can analyze it using standard regression analysis.
Simplifying reporting
When you stack data it becomes possible to update calculations by applying a filter. If the data is not stacked, you'll need to update your analysis by changing variables or recreating the analysis
from scratch. This takes more time and increases the risk of error.
Data stacking in software
For small data files, stacking is often performed using spreadsheets. With larger files, specialist software is required. For example:
• R has various functions that perform stacking (e.g., reshape).
• SPSS has: Data > Restructure > Restructure select variables into cases
• Q has: Tools > Stack SPSS .sav Data File
• Displayr has Anything > Data > Data Sets > Stack
Stacked data and statistical significance
Stacking the data from variables in a data set has the consequence of inflating the sample size (e.g., 100 respondents with 10 rows of data become 1,000 observations). This can cause problems with
statistical tests. This can be partially ameliorated by using a weight (e.g., in this example, assigning each observation a weight of 0.1), although this is a hack. It is better to either treat the
data as hierarchical in a modeling sense (e.g., fitting some kind of Bayesian model), or, treat the data as being from a cluster sample.
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Complex Arithmetic
17.2 Complex Arithmetic
In the descriptions of the following functions, z is the complex number x + iy, where i is defined as sqrt (-1).
Compute the magnitude of z.
The magnitude is defined as |z| = sqrt (x^2 + y^2).
For example:
See also: arg.
Compute the argument, i.e., angle of z.
This is defined as, theta = atan2 (y, x), in radians.
For example:
See also: abs.
Return the complex conjugate of z.
The complex conjugate is defined as conj (z) = x - iy.
See also: real, imag.
Sort the numbers z into complex conjugate pairs ordered by increasing real part.
The negative imaginary complex numbers are placed first within each pair. All real numbers (those with abs (imag (z) / z) < tol) are placed after the complex pairs.
tol is a weighting factor which determines the tolerance of matching. The default value is 100 and the resulting tolerance for a given complex pair is 100 * eps (abs (z(i))).
By default the complex pairs are sorted along the first non-singleton dimension of z. If dim is specified, then the complex pairs are sorted along this dimension.
Signal an error if some complex numbers could not be paired. Signal an error if all complex numbers are not exact conjugates (to within tol). Note that there is no defined order for pairs with
identical real parts but differing imaginary parts.
cplxpair (exp (2i*pi*[0:4]'/5)) == exp (2i*pi*[3; 2; 4; 1; 0]/5)
Return the imaginary part of z as a real number.
See also: real, conj.
Return the real part of z.
See also: imag, conj. | {"url":"https://docs.octave.org/v4.2.0/Complex-Arithmetic.html","timestamp":"2024-11-14T17:13:01Z","content_type":"text/html","content_length":"7459","record_id":"<urn:uuid:a11454af-dac5-4184-9210-d689d7a1bf5e>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00749.warc.gz"} |
Chemistry Examples
Step 1
To find the mass of mole of look up the atomic mass of each element and multiply it by the number of atoms contained in each element in the molecule.
mass of mole of +
Step 2
Fill in the atomic masses from the periodic table.
mass of mole of +
Step 3
Step 3.1
Remove parentheses.
mass of mole of
Step 3.2
Step 4
One mole of any gas occupies at STP (standard temperature and pressure).
Step 5
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Untangling the Calculus Mysteries
Muhammad Moazzam Amjad
Calculus originally known as infinitesimal calculus, is a mathematical branch of study that focuses on limits, continuity, derivatives, integrals and infinite series. Many counting elements appeared
in ancient Greece, then in China and the Middle East, and then again in medieval Europe and India. Infinitesimal calculus was developed independently by Isaac Newton and Gottfried Wilhelm Leibniz at
the end of the 17th century.
Calculus is a word that frequently conjures a combination of energy and fear in the hearts of students. This branch of mathematics has empowered mankind to understand and display the complex cycles
of progress and movement in our real world. Whether you're an eager numerical devotee or somebody who shudders at the prospect of calculus, we should set out on an excursion to demystify math and
investigate its principal ideas. It has two significant branches; differential calculus and integral calculus.
Calculus is been narrowed down to further subdivisions by stepwise revelation;
At the core of calculus lies the idea of limits. Envision following the movement of an article as it speeds up or dials back. To comprehend its momentary speed, you really want to look at its
movement at increasingly small time spans. This is where limit become an integral factor - they permit us to grasp how a point acts as it moves toward a specific point of destination. This idea is
vital for laying out the foundations of calculus and examining persistent functions, where no abrupt leaps or breaks exist.
Differentiation is in many cases thought about the lead idea of calculus. It empowers us to work out paces of progress and comprehend how a function behaves. Consider a winding street - its incline
changes at each point. Essentially, in math, differentiation estimates the pace of progress of a function at some random point. This is especially valuable in fields like physics, designing, and
financial aspects, where understanding how variables change is fundamental.
Integration, the kin of differentiation, fills an alternate need. It assists us with tracking down the gathered amount or region under a curve. Envision plotting the speed of a moving vehicle on a
chart. Coordinating this curve gives you the distance the vehicle has voyaged. In certifiable situations, coordination helps with working out regions, volumes, and even probabilities.
• Fundamental Theory of Calculus:
The major hypothesis of Calculus flawlessly integrates differentiation and integration. It expresses that differentiation and integration are inverse operations of each other. In less complex terms,
while differentiation gives the pace of progress of a function, integration uncovers the first function itself. This hypothesis supports many high level ideas in calculus and exhibits its binding
together power.
Calculus could appear to be conceptual, however its certifiable applications are immense and various. Physics depends on calculus to portray the movement of articles, anticipate divine peculiarities,
and plan the laws of nature. Engineering utilizes calculus to configurate structures, advance cycles, and investigate frameworks. Financial and economic matters benefits from calculus to demonstrate
monetary development, anticipate market drifts, and upgrade asset portion.
• Edition; Multivariable Calculus:
As we adventure into a multi-layered universe, we experience multivariable calculus. In this domain, we investigate functions with various inputs and outputs, urgent for figuring out complex
frameworks. It finds applications in fields like PC designs, fluid dynamics, and even machine learning algorithms.
Calculus isn't bound to the limited world. It digs into the endless domain with arrangements and series. Consider including a limitless number of numbers - appears stunning, isn't that so? Calculus
prepares us to handle these limitless aggregates, offering insights into numerical examples and approximations that support present day innovation.
Generally, Calculus is undeniably beyond the set of equations to be solved; a language empowers us to unravel the powerful texture of our reality. From understanding the universe's crucial laws to
making state of the art innovation, calculus engages us to reveal the secrets of change and motion. In this way, whether you're an aspiring mathematician or basically inquisitive about the powers
molding our world, embrace the excursion of unraveling calculus - it's a fare to the journey of exploring heart of the mathematics. | {"url":"https://www.mathnonprofittutoring.org/post/untangling-the-calculus-mysteries","timestamp":"2024-11-06T05:52:03Z","content_type":"text/html","content_length":"1050493","record_id":"<urn:uuid:c651e543-1289-4101-bed1-48c68b1e8414>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00338.warc.gz"} |
Archive for March
Caution: This math may be wrong.
In episode 6 of Neon Genesis Evangelion, NERV borrows a positron cannon from a government research lab to deal with the latest threat. This cannon takes all of Japan’s power output, for 37 seconds,
to charge the rifle for one shot. According to the CIA World Factbook, Japan produced 1.017 trillion kWh for the whole year in 2003.
1.017 trillion kWh per year/365 days/24 hours = 116 million kWh per hour
So, Japan produces 116 million kWh per hour. However, Makoto Hyuga states that it will take atleast 180 million watts to pierce the AT field of the angel Ramiel. Seeing as it takes 37 seconds to
charge the rifle*…
180 million kW per 37 seconds * (3600 seconds / 37 seconds) = 17460 million kWh per hour
So, if my calculations are correct, the rifle would need 17460 million kWh to charge for an hour; unfortunately, this is quite a lot higher than the 116 million kWh per hour Japan produces, infact,
it is 150 times more. So, if Evangelion’s Japan in the year 2015 is to be able to power the rifle, it needs to produce 150 times more power than 2003 Japan does.
[* Counting from the ejection of the first shot’s fuse, to the pulling of the trigger for the second shot.] | {"url":"https://adterrasperaspera.com/blog/2006/03/","timestamp":"2024-11-13T01:15:27Z","content_type":"text/html","content_length":"17997","record_id":"<urn:uuid:956702e6-72b0-4dd8-8e37-7b53e3c5c448>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00179.warc.gz"} |
How to calculate the monthly interest on a loan 2023 - INFINANZAS
How to calculate the monthly interest on a loan 2023
Calculating interest each month is an essential skill. Interest rates are often expressed as an annual percentage yield (APY), although it can also be expressed as an annual percentage rate (APR).
Still, it doesn’t hurt to really know how many dollars and cents you are dealing with, and that gives you the ability to budget and calculate on a monthly basis. This is important, for example, for
calculating monthly utility bills, groceries or car costs.
Interest also accrues monthly (if not every day), and these periodic interest calculations accumulate over the course of a year. The process for taking an annual rate to a monthly rate is the same
regardless of whether interest is paid or rather interest is earned.
How can I do the monthly interest calculation myself?
Very simple to calculate any monthly interest rate you have to divide the annual interest by 12, which will account for the 12 months of the year. You will need to convert the percentage format to
decimal to complete each of these steps.
Example: Suppose you have an APR or APY of 10%. What is your particular monthly interest rate and how much would you pay or otherwise earn on $2,000?
Convert the annual interest from a percentage expression to a decimal expression by dividing by 100: 10/100 = 0.10.
Now divide that number by 12, this allows you to get the monthly interest rate in decimal form, being10/12 = 0.0083
To calculate the monthly interest on $2,000, multiply that number by the total amount: 0.0083 x $2,000 = $16.60 per month.
Convert the monthly rate in decimal format back to a percentage (multiplying by 100): 0.0083 x 100 = 0.83%.
The monthly interest rate is 0.83%.
The above example serves to illustrate the simplest way in which you can calculate interest rates and individual costs for each month. You can calculate interest by months, also by days, years or any
required period. Regardless of the period you choose, the rate you use in the calculation is called the term.
Most of the time you will see the interest rate expressed as an annual rate, so you will usually need to convert it to a periodic rate that suits your financial problem or product. You can use the
same concept to calculate interest as for other time periods:
For a daily interest rate, divide the annual by 360 (or 365, depending on your bank).
For a quarterly one, divide the annual by four.
For a weekly rate, divide the annual rate by 52.
On many loans, the balance changes each month. For example, in auto, home and personal loans, the balance gradually decreases over time and usually ends up decreasing each month.
Over time, these monthly interest costs decrease and the amount that goes toward the loan balance increases.
Home loans and credit cards
Home loans can be complicated. It is advisable to use an amortization schedule to understand the interest costs, but you may have to do extra work to figure out the actual interest rate. You can use
our mortgage calculator (below) to see how the principal payment, interest charges, taxes and insurance add up to your monthly mortgage payment.
You may know the annual percentage rate (APR) of your mortgage, and keep in mind that the APR may contain additional costs on top of interest charges (such as closing costs). Also, the interest rate
on variable rate mortgages is subject to change.
In the case of credit cards, you may add new charges and pay off debts numerous times throughout the month. All this activity makes the calculations more complicated, but it’s still worth knowing how
monthly interest accrues. In many cases, you can use an average daily balance, which is the sum of each day’s balance divided by the number of days in each month (and the finance charge can be
calculated using the average daily balance). In other cases, the card issuer charges interest on a daily basis (so the daily interest rate, not the monthly rate, must be calculated).
Interest rates and APY
Be sure to use the specific interest rate in your calculations, not the annual percentage yield. The APY takes into account compounding, which is the interest you earn as your account grows due to
interest payments. The APY will be higher than the actual APY unless the interest is compounded annually, so it has to be taken into consideration that the APY may provide an inaccurate result. That
said, APY makes it easy to quickly find out how much you will earn annually on a savings account with no additions or withdrawals.
What is a good interest rate for a credit card?
The average credit card interest rate was 16.17% in February 2022. You can expect to pay a few points more for store credit cards. Business and student credit cards will help you minimize the
interest rate.
What is the prime rate?
The prime rate is the rate the bank charges its best customers. In other words, it is the lowest interest rate for that particular day. This rate is generally only available to institutional
customers. The average consumer pays a prime rate and a different rate depending on their risk as a borrower.
How can the credit card interest rate be reduced?
Credit card rates are negotiable, but it depends on the card issuer. Your card issuer is more likely to offer you a lower interest rate if you have good credit habits, such as keeping up with your
monthly payments.
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Word Sudoku Under Bergdorfbib Co Printable Sudoku Challenger | Sudoku Printables
Word Sudoku Under Bergdorfbib Co Printable Sudoku Challenger
Word Sudoku Under Bergdorfbib Co Printable Sudoku Challenger – If you’ve ever had trouble with sudoku, you’re aware that there are many different kinds of sudoku puzzles which is why it’s difficult
to choose which ones you’ll need to solve. But there are also many options to solve them. In fact, it is likely that a printable version can be an excellent method to get started. The rules to solve
sudoku are similar to the rules for solving other puzzles however the way they are presented differs slightly.
What Does the Word ‘Sudoku’ Mean?
The word “Sudoku” is an abbreviation of the Japanese words suji and dokushin meaning “number” as well as “unmarried’, respectively. The objective of the puzzle is to fill all the boxes with numbers
such that each numeral from one to nine appears just once on every horizontal line. The word Sudoku is an emblem associated with the Japanese puzzle maker Nikoli, which originated in Kyoto.
The name Sudoku originates in the Japanese word”shuji wa Dokushin Ni Kagiru meaning ‘numbers have to not be separated’. The game is composed of nine 3×3 squares that have nine smaller squares.
Originally called Number Place, Sudoku was a mathematical puzzle that stimulated development. Although the origins of the game are unknown, Sudoku is known to have roots that go back to the earliest
number puzzles.
Why is Sudoku So Addicting?
If you’ve played Sudoku you’ll realize how addictive the game can be. A Sudoku addict will never be able to put down the thought of the next problem they’ll solve. They’re constantly planning their
next fix, while various aspects of their life are slipping to the sidelines. Sudoku is a highly addictive game It’s crucial in order to hold the addictive potential of the game in check. If you’ve
developed a craving for Sudoku here are some methods to reduce your dependence.
One of the most popular methods to determine if the addict you are to Sudoku is to watch your behavior. Most people carry books and magazines with them as well as scroll through social media updates.
Sudoku addicts, however, carry newspapers, books exercise books and smartphones wherever they go. They are constantly working on puzzles and aren’t able to stop! Many people discover it is easier to
complete Sudoku puzzles than normal crosswords, so they can’t quit.
Super Challenger Sudoku Printable Puzzles
What is the Key to Solving a Sudoku Puzzle?
A great strategy to solve a printable sudoku problem is to try and practice with different approaches. The top Sudoku puzzle solvers do not use the same method for every puzzle. The most important
thing is to practice and experiment with various approaches until you find the one that is effective for you. After some time, you’ll be able to solve puzzles without difficulty! But how do you learn
to solve an printable Sudoku game?
The first step is to understand the basics of suduko. It’s a game that requires logic and deduction and requires you to examine the puzzle from many different angles to see patterns and then solve
it. When solving a suduko puzzle, do not try to figure out the numbers; instead, you should scan the grid for ways to recognize patterns. It is also possible to apply this technique to rows and
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Watt conversions
The watt conversion selector on this page selects the power measurement unit to convert to starting from watts (W). To make a conversion starting from a unit of power other than watt, simply click on
the "Reset" button.
About watt
The watt is the derived unit of power in the International System of Units (SI).
One watt (W) is equal to 1,000,000 microwatts (μW), or 1,000 milliwatts (mW), or 0.001 kilowatts (kW), or 10^-6 megawatts (MW) or 10^-9 gigawatts (GW).
The watt measures the rate at which the energy is transformed. One watt is equal to one joule per second. The joule is a derived unit of energy in the SI while the second is a unit of time in the SI.
The watt unit is named after the Scottish engineer James Watt who lived between 1736 and 1819.
1 W = 1 J/s = 1 N×m/s = 1 kg×m^2/s^3
Symbol: W
Plural: watts
Subdivisions and multiples of the watt using SI prefixes
Subdivisions of the watt
Name Symbol Value
deciwatt dW 10^-1 W
centiwatt cW 10^-2 W
milliwatt mW 10^-3 W
microwatt μW 10^-6 W
nanowatt nW 10^-9 W
picowatt pW 10^-12 W
femtowatt fW 10^-15 W
attowatt aW 10^-18 W
zeptowatt zW 10^-21 W
yoctowatt yW 10^-24 W
Multiples of the watt
Name Symbol Value
decawatt daW 10^1 W
hectowatt hW 10^2 W
kilowatt kW 10^3 W
megawatt MW 10^6 W
gigawatt GW 10^9 W
terawatt TW 10^12 W
petawatt PW 10^15 W
exawatt EW 10^18 W
zettawatt ZW 10^21 W
yottawatt YW 10^24 W
Watt conversions: a compehensive list with conversions from watts to other (metric, imperial, or customary) power measurement units is given below. | {"url":"https://conversion-website.com/power/from-watt.html","timestamp":"2024-11-08T15:50:16Z","content_type":"text/html","content_length":"18973","record_id":"<urn:uuid:cf168ba0-0eb4-4acf-a725-b19862e7dde4>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00290.warc.gz"} |
optimSPAN: Optimization of sample configurations for variogram and... in Laboratorio-de-Pedometria/spsann-package: Optimization of Spatial Samples via Simulated Annealing
Optimize a sample configuration for variogram and spatial trend identification and estimation, and for spatial interpolation. An utility function U is defined so that the sample points cover, extend
over, spread over, SPAN the feature, variogram and geographic spaces. The utility function is obtained aggregating four objective functions: CORR, DIST, PPL, and MSSD.
optimSPAN( points, candi, covars, strata.type = "area", use.coords = FALSE, lags = 7, lags.type = "exponential", lags.base = 2, cutoff, criterion = "distribution", distri, pairs = FALSE, schedule,
plotit = FALSE, track = FALSE, boundary, progress = "txt", verbose = FALSE, weights, nadir = list(sim = NULL, seeds = NULL, user = NULL, abs = NULL), utopia = list(user = NULL, abs = NULL) ) objSPAN(
points, candi, covars, strata.type = "area", use.coords = FALSE, lags = 7, lags.type = "exponential", lags.base = 2, cutoff, criterion = "distribution", distri, pairs = FALSE, x.max, x.min, y.max,
y.min, weights, nadir = list(sim = NULL, seeds = NULL, user = NULL, abs = NULL), utopia = list(user = NULL, abs = NULL) )
Integer value, integer vector, data frame (or matrix), or list. The number of sampling points (sample size) or the starting sample configuration. Four options are available:
• Integer value. The required number of sampling points (sample size). The sample configuration used to start the optimization will consist of grid cell centres of candi selected using
simple random sampling, i.e. base::sample() with x = 1:nrow(candi) and size = points.
• Integer vector. A set of row indexes between one (1) and nrow(candi). These row indexes identify the grid cell centres of candi that will form the starting sample configuration for
the optimization. The length of the integer vector, length(points), is the sample size.
• Data frame (or matrix). The Cartesian x- and y-coordinates (in this order) of the starting sample configuration.
points • List. An object with two named sub-arguments:
□ fixed An integer vector or data frame (or matrix) specifying an existing sample configuration (see options above). This sample configuration is kept as-is (fixed) during the
optimization and is used only to compute the objective function values.
□ free An integer value, integer vector, data frame or matrix (see options above) specifying the (number of) sampling points to add to the existing sample configuration. These new
sampling points are free to be moved around (jittered) during the optimization.
Most users will want to set an integer value simply specifying the required sample size. Using an integer vector or data frame (or matrix) will generally be helpful to users willing to
evaluate starting sample configurations, test strategies to speed up the optimization, and fine-tune or thin an existing sample configuration. Users interested in augmenting a possibly
existing real-world sample configuration or fine-tuning only a subset of the existing sampling points will want to use a list.
Data frame (or matrix). The Cartesian x- and y-coordinates (in this order) of the cell centres of a spatially exhaustive, rectangular grid covering the entire spatial sampling domain. The
spatial sampling domain can be contiguous or composed of disjoint areas and contain holes and islands. candi provides the set of (finite) candidate locations inside the spatial sampling
candi domain for a point jittered during the optimization. Usually, candi will match the geometry of the spatial grid containing the prediction locations, e.g. newdata in gstat::krige(), object
in raster::predict(), and locations in geoR::krige.conv().
covars Data frame or matrix with the spatially exhaustive covariates in the columns.
(Optional) Character value setting the type of stratification that should be used to create the marginal sampling strata (or factor levels) for the numerical covariates. Two options are
• "area" (Default) Equal-area marginal sampling strata.
strata.type • "range" Equal-range marginal sampling strata.
The first option ("area") is equivalent to drawing the frequency histogram of the numerical covariates with bins of variable width but equal area. The second, however, would result in a
frequency histogram with bins of equal width but variable area such as when using graphics::hist() with its default options. Strata of equal area will include virtually the same number of
individual covariate grid cells per stratum, while equal-range strata aim for the same number of unique covariate values in each stratum.
use.coords (Optional) Logical value. Should the projected spatial x- and y-coordinates be used as spatially exhaustive covariates? Defaults to use.coords = FALSE.
Integer value, the number of lag-distance classes. Alternatively, a vector of numeric values with the lower and upper bounds of each lag-distance class, the lowest value being larger than
lags zero (e.g. 0.0001). Defaults to lags = 7.
lags.type Character value, the type of lag-distance classes, with options "equidistant" and "exponential". Defaults to lags.type = "exponential".
lags.base Numeric value, base of the exponential expression used to create exponentially spaced lag-distance classes. Used only when lags.type = "exponential". Defaults to lags.base = 2.
Numeric value, the maximum distance up to which lag-distance classes are created. Used only when lags is an integer value. If missing, it is set to be equal to the length of the diagonal
cutoff of the rectangle with sides x.max and y.max as defined in scheduleSPSANN().
criterion Character value, the feature used to describe the energy state of the system configuration, with options "minimum" and "distribution". Defaults to objective = "distribution".
Numeric vector, the distribution of points or point-pairs per lag-distance class that should be attained at the end of the optimization. Used only when criterion = "distribution".
distri Defaults to a uniform distribution.
pairs Logical value. Should the sample configuration be optimized regarding the number of point-pairs per lag-distance class? Defaults to pairs = FALSE.
schedule List with named sub-arguments setting the control parameters of the annealing schedule. See scheduleSPSANN().
(Optional) Logical for plotting the evolution of the optimization. Plot updates occur at each ten (10) spatial jitters. Defaults to plotit = FALSE. The plot includes two panels:
1. The first panel depicts the changes in the objective function value (y-axis) with the annealing schedule (x-axis). The objective function values should be high and variable at the
beginning of the optimization (panel's top left). As the optimization proceeds, the objective function values should gradually transition to a monotone decreasing behaviour till they
plotit become virtually constant. The objective function values constancy suggests the end of the optimization (panel's bottom right).
2. The second panel shows the starting (grey circles) and current spatial sample configuration (black dots). Black crosses indicate the fixed (existing) sampling points when a spatial
sample configuration is augmented. The plot shows the starting sample configuration to assess the effects on the optimized spatial sample configuration: the latter generally should be
independent of the first. The second panel also shows the maximum possible spatial jitter applied to a sampling point in the Cartesian x- (x-axis) and y-coordinates (y-axis).
(Optional) Logical value. Should the evolution of the energy state be recorded and returned along with the result? If track = FALSE (the default), only the starting and ending energy
track states return along with the results.
(Optional) An object of class SpatialPolygons (see sp::SpatialPolygons()) with the outer and inner limits of the spatial sampling domain (see candi). These SpatialPolygons help depict the
boundary spatial distribution of the (starting and current) sample configuration inside the spatial sampling domain. The outer limits of candi serve as a rough boundary when plotit = TRUE, but the
SpatialPolygons are missing.
(Optional) Type of progress bar that should be used, with options "txt", for a text progress bar in the R console, "tk", to put up a Tk progress bar widget, and NULL to omit the progress
progress bar. A Tk progress bar widget is useful when using parallel processors. Defaults to progress = "txt".
verbose (Optional) Logical for printing messages about the progress of the optimization. Defaults to verbose = FALSE.
List with named sub-arguments. The weights assigned to each one of the objective functions that form the multi-objective combinatorial optimization problem. They must be named after the
weights respective objective function to which they apply. The weights must be equal to or larger than 0 and sum to 1.
List with named sub-arguments. Three options are available:
• sim: the number of simulations that should be used to estimate the nadir point, and seeds vector defining the random seeds for each simulation;
• user: a list of user-defined nadir values named after the respective objective functions to which they apply;
• abs: logical for calculating the nadir point internally (experimental).
List with named sub-arguments. Two options are available:
utopia • user: a list of user-defined values named after the respective objective functions to which they apply;
• abs: logical for calculating the utopia point internally (experimental).
x.min, Numeric value defining the minimum and maximum quantity of random noise to be added to the projected x- and y-coordinates. The minimum quantity should be equal to, at least, the minimum
y.max, distance between two neighbouring candidate locations. The units are the same as of the projected x- and y-coordinates. If missing, they are estimated from candi.
Integer value, integer vector, data frame (or matrix), or list. The number of sampling points (sample size) or the starting sample configuration. Four options are available:
Integer value. The required number of sampling points (sample size). The sample configuration used to start the optimization will consist of grid cell centres of candi selected using simple random
sampling, i.e. base::sample() with x = 1:nrow(candi) and size = points.
Integer vector. A set of row indexes between one (1) and nrow(candi). These row indexes identify the grid cell centres of candi that will form the starting sample configuration for the optimization.
The length of the integer vector, length(points), is the sample size.
Data frame (or matrix). The Cartesian x- and y-coordinates (in this order) of the starting sample configuration.
fixed An integer vector or data frame (or matrix) specifying an existing sample configuration (see options above). This sample configuration is kept as-is (fixed) during the optimization and is used
only to compute the objective function values.
free An integer value, integer vector, data frame or matrix (see options above) specifying the (number of) sampling points to add to the existing sample configuration. These new sampling points are
free to be moved around (jittered) during the optimization.
Most users will want to set an integer value simply specifying the required sample size. Using an integer vector or data frame (or matrix) will generally be helpful to users willing to evaluate
starting sample configurations, test strategies to speed up the optimization, and fine-tune or thin an existing sample configuration. Users interested in augmenting a possibly existing real-world
sample configuration or fine-tuning only a subset of the existing sampling points will want to use a list.
Data frame (or matrix). The Cartesian x- and y-coordinates (in this order) of the cell centres of a spatially exhaustive, rectangular grid covering the entire spatial sampling domain. The spatial
sampling domain can be contiguous or composed of disjoint areas and contain holes and islands. candi provides the set of (finite) candidate locations inside the spatial sampling domain for a point
jittered during the optimization. Usually, candi will match the geometry of the spatial grid containing the prediction locations, e.g. newdata in gstat::krige(), object in raster::predict(), and
locations in geoR::krige.conv().
Data frame or matrix with the spatially exhaustive covariates in the columns.
(Optional) Character value setting the type of stratification that should be used to create the marginal sampling strata (or factor levels) for the numerical covariates. Two options are available:
The first option ("area") is equivalent to drawing the frequency histogram of the numerical covariates with bins of variable width but equal area. The second, however, would result in a frequency
histogram with bins of equal width but variable area such as when using graphics::hist() with its default options. Strata of equal area will include virtually the same number of individual covariate
grid cells per stratum, while equal-range strata aim for the same number of unique covariate values in each stratum.
(Optional) Logical value. Should the projected spatial x- and y-coordinates be used as spatially exhaustive covariates? Defaults to use.coords = FALSE.
Integer value, the number of lag-distance classes. Alternatively, a vector of numeric values with the lower and upper bounds of each lag-distance class, the lowest value being larger than zero (e.g.
0.0001). Defaults to lags = 7.
Character value, the type of lag-distance classes, with options "equidistant" and "exponential". Defaults to lags.type = "exponential".
Numeric value, base of the exponential expression used to create exponentially spaced lag-distance classes. Used only when lags.type = "exponential". Defaults to lags.base = 2.
Numeric value, the maximum distance up to which lag-distance classes are created. Used only when lags is an integer value. If missing, it is set to be equal to the length of the diagonal of the
rectangle with sides x.max and y.max as defined in scheduleSPSANN().
Character value, the feature used to describe the energy state of the system configuration, with options "minimum" and "distribution". Defaults to objective = "distribution".
Numeric vector, the distribution of points or point-pairs per lag-distance class that should be attained at the end of the optimization. Used only when criterion = "distribution". Defaults to a
uniform distribution.
Logical value. Should the sample configuration be optimized regarding the number of point-pairs per lag-distance class? Defaults to pairs = FALSE.
List with named sub-arguments setting the control parameters of the annealing schedule. See scheduleSPSANN().
(Optional) Logical for plotting the evolution of the optimization. Plot updates occur at each ten (10) spatial jitters. Defaults to plotit = FALSE. The plot includes two panels:
The first panel depicts the changes in the objective function value (y-axis) with the annealing schedule (x-axis). The objective function values should be high and variable at the beginning of the
optimization (panel's top left). As the optimization proceeds, the objective function values should gradually transition to a monotone decreasing behaviour till they become virtually constant. The
objective function values constancy suggests the end of the optimization (panel's bottom right).
The second panel shows the starting (grey circles) and current spatial sample configuration (black dots). Black crosses indicate the fixed (existing) sampling points when a spatial sample
configuration is augmented. The plot shows the starting sample configuration to assess the effects on the optimized spatial sample configuration: the latter generally should be independent of the
first. The second panel also shows the maximum possible spatial jitter applied to a sampling point in the Cartesian x- (x-axis) and y-coordinates (y-axis).
(Optional) Logical value. Should the evolution of the energy state be recorded and returned along with the result? If track = FALSE (the default), only the starting and ending energy states return
along with the results.
(Optional) An object of class SpatialPolygons (see sp::SpatialPolygons()) with the outer and inner limits of the spatial sampling domain (see candi). These SpatialPolygons help depict the spatial
distribution of the (starting and current) sample configuration inside the spatial sampling domain. The outer limits of candi serve as a rough boundary when plotit = TRUE, but the SpatialPolygons are
(Optional) Type of progress bar that should be used, with options "txt", for a text progress bar in the R console, "tk", to put up a Tk progress bar widget, and NULL to omit the progress bar. A Tk
progress bar widget is useful when using parallel processors. Defaults to progress = "txt".
(Optional) Logical for printing messages about the progress of the optimization. Defaults to verbose = FALSE.
List with named sub-arguments. The weights assigned to each one of the objective functions that form the multi-objective combinatorial optimization problem. They must be named after the respective
objective function to which they apply. The weights must be equal to or larger than 0 and sum to 1.
sim: the number of simulations that should be used to estimate the nadir point, and seeds vector defining the random seeds for each simulation;
user: a list of user-defined nadir values named after the respective objective functions to which they apply;
user: a list of user-defined values named after the respective objective functions to which they apply;
Numeric value defining the minimum and maximum quantity of random noise to be added to the projected x- and y-coordinates. The minimum quantity should be equal to, at least, the minimum distance
between two neighbouring candidate locations. The units are the same as of the projected x- and y-coordinates. If missing, they are estimated from candi.
The help page of minmaxPareto() contains details on how spsann solves the multi-objective combinatorial optimization problem of finding a globally optimum sample configuration that meets multiple,
possibly conflicting, sampling objectives.
There are multiple mechanism to generate a new sample configuration out of an existing one. The main step consists of randomly perturbing the coordinates of a single sample, a process known as
‘jittering’. These mechanisms can be classified based on how the set of candidate locations for the samples is defined. For example, one could use an infinite set of candidate locations, that is, any
location in the spatial domain can be selected as a new sample location after a sample is jittered. All that is needed is a polygon indicating the boundary of the spatial domain. This method is more
computationally demanding because every time an existing sample is jittered, it is necessary to check if the new sample location falls in spatial domain.
Another approach consists of using a finite set of candidate locations for the samples. A finite set of candidate locations is created by discretising the spatial domain, that is, creating a fine
(regular) grid of points that serve as candidate locations for the jittered sample. This is a less computationally demanding jittering method because, by definition, the new sample location will
always fall in the spatial domain.
Using a finite set of candidate locations has two important inconveniences. First, not all locations in the spatial domain can be selected as the new location for a jittered sample. Second, when a
sample is jittered, it may be that the new location already is occupied by another sample. If this happens, another location has to be iteratively sought for, say, as many times as the size of the
sample configuration. In general, the larger the size of the sample configuration, the more likely it is that the new location already is occupied by another sample. If a solution is not found in a
reasonable time, the the sample selected to be jittered is kept in its original location. Such a procedure clearly is suboptimal.
spsann uses a more elegant method which is based on using a finite set of candidate locations coupled with a form of two-stage random sampling as implemented in spcosa::spsample(). Because the
candidate locations are placed on a finite regular grid, they can be taken as the centre nodes of a finite set of grid cells (or pixels of a raster image). In the first stage, one of the “grid cells”
is selected with replacement, i.e. independently of already being occupied by another sample. The new location for the sample chosen to be jittered is selected within that “grid cell” by simple
random sampling. This method guarantees that virtually any location in the spatial domain can be selected. It also discards the need to check if the new location already is occupied by another
sample, speeding up the computations when compared to the first two approaches.
Visit the help pages of optimCORR, optimDIST, optimPPL, and optimMSSD to see the details of the objective functions that compose SPAN.
optimSPAN returns an object of class OptimizedSampleConfiguration: the optimized sample configuration with details about the optimization.
objSPAN returns a numeric value: the energy state of the sample configuration – the objective function value.
spsann always computes the distance between two locations (points) as the Euclidean distance between them. This computation requires the optimization to operate in the two-dimensional Euclidean
space, i.e. the coordinates of the sample, candidate and evaluation locations must be Cartesian coordinates, generally in metres or kilometres. spsann has no mechanism to check if the coordinates are
Cartesian: you are the sole responsible for making sure that this requirement is attained.
##################################################################### # NOTE: The settings below are unlikely to meet your needs. # ###################################################################
## ## Not run: # This example takes more than 5 seconds to run! require(sp) data(meuse.grid) candi <- meuse.grid[, 1:2] nadir <- list(sim = 10, seeds = 1:10) utopia <- list(user = list(DIST = 0, CORR
= 0, PPL = 0, MSSD = 0)) covars <- meuse.grid[, 5] schedule <- scheduleSPSANN(chains = 1, initial.temperature = 1, x.max = 1540, y.max = 2060, x.min = 0, y.min = 0, cellsize = 40) weights <- list
(CORR = 1/6, DIST = 1/6, PPL = 1/3, MSSD = 1/3) set.seed(2001) res <- optimSPAN( points = 10, candi = candi, covars = covars, nadir = nadir, weights = weights, use.coords = TRUE, utopia = utopia,
schedule = schedule) objSPSANN(res) - objSPAN(points = res, candi = candi, covars = covars, nadir = nadir, use.coords = TRUE, utopia = utopia, weights = weights) ## End(Not run)
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/github/Laboratorio-de-Pedometria/spsann-package/man/optimSPAN.html","timestamp":"2024-11-08T22:33:23Z","content_type":"text/html","content_length":"42789","record_id":"<urn:uuid:41c39d8d-fb95-4af9-bb1b-c1866bd3762a>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00471.warc.gz"} |
Newton-Raphson Method for Solving Nonlinear Equations
1 Introduction
Nonlinear equations, unlike their linear counterparts, can’t be solved with simple algebra. But fear not, for Newton’s method comes to the rescue! This iterative technique is a powerful tool for
finding the roots (also called zeros) of these nonlinear equations.
Imagine a curvy graph and you’re looking for the x-value where the graph intersects the x-axis (i.e., the root). Newton’s method works by taking an initial guess for this x-value. Then, it cleverly
uses the concept of the tangent line: it draws a line tangent to the curve at your guess, and finds where this tangent line meets the x-axis. This new x-axis intercept becomes a better approximation
for the root.
The beauty lies in the repetition. You keep using the same idea, drawing a tangent line at your new guess, and iteratively getting closer and closer to the actual root. With each step, the
approximation gets refined, leading to a more accurate solution.
Newton’s method or Newton-Raphson method is a numeric method for finding a numerical solution of an equation of the form f(x) = 0 where f(x) is continuous and differentiable and also it is know that
the equation have a solution approximate to a given point. The method is shown in figure below.
2 Algorithm for Newton-Raphson Method
Algorithm for Newton-Raphson method is simple:
1. Choose a point x 1 as an initial guess of the solution.
2. For each iteration until the error is smaller than a specified value, calculate the next solution using the formula in the last section.
3 When Should Iterations Stop?
Sample Problem: Finding the Square Root
Let's use Newton's method to find the square root of a number (say, 2). The square root of a number x satisfies the equation x^2 - y = 0, where y is the desired square root. We can rewrite this as f
(y) = y^2 - 2 = 0.
Here's the Python program to solve it:
def f(y):
"""Function representing the equation (y^2 - 2 = 0)."""
return y**2 - 2
def df(y):
"""Derivative of the function f(y) (2*y)."""
return 2*y
def newton_method(f, df, x0, tol, max_iter):
Implements Newton's method for finding the root of a function.
f: The function for which we want to find the root.
df: The derivative of the function f.
x0: The initial guess for the root.
tol: The tolerance level for convergence.
max_iter: The maximum number of iterations allowed.
The root (if found) or None.
for i in range(max_iter):
x_new = x0 - f(x0) / df(x0)
if abs(x_new - x0) < tol:
return x_new
x0 = x_new
return None
# Define initial guess, tolerance, and max iterations
x_guess = 1.0 # Initial guess (can be adjusted)
tolerance = 1e-6
max_iterations = 10
# Solve for the square root using Newton's method
root = newton_method(f, df, x_guess, tolerance, max_iterations)
if root:
print(f"Square root of 2 (approximately): {root:.6f}")
print(f"No root found within tolerance or {max_iterations} iterations.")
This program defines the function f(y) and its derivative df(y) representing the equation y^2 - 2 = 0. The newton_method function remains the same. It then defines the initial guess, tolerance, and
maximum iterations. Finally, it calls the newton_method to find the root and prints the result.
This example demonstrates how to apply Newton's method to solve a simple nonlinear equation. You can modify the f(y) and df(y) functions to solve other nonlinear equations following the same
4 Consideration When Using Newton-Raphson Method | {"url":"https://bastakiss.com/blog/numerical-methods-16/newton-raphson-method-for-solving-nonlinear-equations-145","timestamp":"2024-11-05T15:35:27Z","content_type":"text/html","content_length":"57427","record_id":"<urn:uuid:8c0efbf0-ade0-41f9-b630-038fbc63d857>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00366.warc.gz"} |
Question ID - 103098 | SaraNextGen Top Answer
The rate of diffusion of methane is twice that of X. The molecular mass of X is divided by 32. What is value of
Question ID - 103098 | SaraNextGen Top Answer
The rate of diffusion of methane is twice that of X. The molecular mass of X is divided by 32. What is value of
1 Answer
127 votes
Answer Key / Explanation : (2)
Given Molecular mass is divided by 32 therefore,
127 votes
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An Introduction to Geospatial Processing with Spark | HyperCodeLab
In this new era of information, geospatial data is becoming more and more relevant in the Data Engineering and Data Analytics work… but what do we mean when we talk about Geospatial Data?
In this post, I would like to introduce some basic concepts about Geospatial Processing using Spark, one of the most popular data processing frameworks of 2020.
Let’s start with the basics, what is geospatial data?
Geospatial data is about objects, events or phenomena that have a location on the surface of the Earth (let’s not worry about other planets yet… Sorry, Mars Rover)
And what does this mean in terms of data? Well… We can represent Geospatial data with just a String, and the two most popular formats for doing so are:
• GeoJson: { “geometry”: { “type”: “Polygon”, “coordinates”: [ [ [100.0, 0.0], [101.0, 0.0],[101.0, 1.0],[100.0, 1.0],[100.0, 0.0]] ]} }
• Well-Known Text (WKT): POLYGON ((100.0 0.0, 101.0 0.0, 101.0 1.0, 100.0 1.0, 100.0 0.0))
The examples above are representing this polygon:
But data as a String is not very useful because we would like to use some spatial operators on them, and doing that with a String can be really hard… hence why we use something called a Geometry Type
. This data type, which is not available in every system, allows to use spatial operators like Containts, Insersect, and many more:
Of course, we can do that without any Geometry Type, but it’s much easier to do Intersection of (Polygon A and Polygon B), than to code all the math behind that operation.
So… How can I do all of this in Spark?
Spark doesn’t have a Geometry Type embedded, but luckily there are some people out there that have done, and continue doing, that work for us (Thanks open source community!)… So there are a few
choices on how to do this:
• Use a third-party library: there are a few options available like GeoSpark, GeoMesa… this is suitable if you can find the transformations that you need there… So don’t reinvent the wheel and use
what other people have already developed and tested.
• Wrap an existing core library: if the available Spark libraries don’t fit your requirements, you can go one level deeper and wrap one of the existing geospatial implementations in Spark. Normally
JTS is the most low-level library that you would find. All the others use this one as a starting point. It only provides the data types and some basic operations between geometries. It’s a java
library, so if you want to use it with Spark you need to adapt it yourself.
• Implement everything for scratch: this will probably require years of work…
In this introduction I’m going to show the basics of GeoSpark …and we might go deeper in an upcoming post :)
Assuming that you’re working on a Databricks environment, using GeoSpark is straightforward. You just need to add two Maven libraries to your cluster and that’s it:
Once you have the libraries installed, you can open a notebook and start using those geospatial functions! It’s seriously that simple.
Let’s start by defining a table with some geospatial data: (I will be copying the code of the cells here and adding a screenshot of the result)
// Create a new DataFrame with a column representing geospatial data as strings
val data = Seq(("ID 1", "POLYGON ((0.0 0.0, 10.0 0.0, 10.0 10.0, 0.0 10.0, 0.0 0.0))"),
("ID 2", "POLYGON ((5.0 5.0, 15.0 5.0, 15.0 15.0, 5.0 15.0, 5.0 5.0))"),
("ID 3", "POLYGON ((16.0 0.0, 20.0 0.0, 20.0 5.0, 16.0 5.0, 16.0 0.0))")).toDF("ID", "geometry")
These rows represent some polygons:
In order to use the UDFs that GeoSpark provides, we can run the following:
import org.datasyslab.geosparksql.utils.GeoSparkSQLRegistrator
We need to convert our strings in a proper geometry type, for that we need to do the following:
val data2 = data.selectExpr("ID", "ST_GeomFromWKT(geometry) as geometry")
// Checking the schema
So now our geometry column is of type geometry:
We are ready for running some spatial operations… But to make this easier, let’s create a temporary table so we can run SQL queries.
Let’s get the area of each polygon:
SELECT *, ST_Area(geometry) AS area FROM GeoTable
Let’s get the Centroid of each polygon:
SELECT *, ST_Centroid (geometry) AS centroid FROM GeoTable
Those functions, the ones that you apply directly to the geometry column are called just Functions.
There are other types, like Predicates, that allows you to check conditions between different polygons.
To use them, you need to have at least two geometries, so let’s get to that situation by creating a new table:
val data3 = Seq(("ID 1", "POLYGON ((0.0 0.0, 10.0 0.0, 10.0 10.0, 0.0 10.0, 0.0 0.0))", "ID 2", "POLYGON ((5.0 5.0, 15.0 5.0, 15.0 15.0, 5.0 15.0, 5.0 5.0))"),
("ID 1", "POLYGON ((0.0 0.0, 10.0 0.0, 10.0 10.0, 0.0 10.0, 0.0 0.0))", "ID 3", "POLYGON ((16.0 0.0, 20.0 0.0, 20.0 5.0, 16.0 5.0, 16.0 0.0))"),
("ID 2", "POLYGON ((5.0 5.0, 15.0 5.0, 15.0 15.0, 5.0 15.0, 5.0 5.0))", "ID 1", "POLYGON ((0.0 0.0, 10.0 0.0, 10.0 10.0, 0.0 10.0, 0.0 0.0))"),
("ID 2", "POLYGON ((5.0 5.0, 15.0 5.0, 15.0 15.0, 5.0 15.0, 5.0 5.0))", "ID 3", "POLYGON ((16.0 0.0, 20.0 0.0, 20.0 5.0, 16.0 5.0, 16.0 0.0))")
).toDF("ID1", "geometry1", "ID2", "geometry2")
val data4 = data3.selectExpr("ID1", "ST_GeomFromWKT(geometry1) as geometry1", "ID2", "ST_GeomFromWKT(geometry2) as geometry2")
With this new DataFrame we can run a few Spatial Queries to check for some conditions, like for example:
• Checking if two geometries intersect each other.
• Checking if the geometry is within other.
SELECT ID1,
ST_Intersects(geometry1, geometry2)AS Intersects,
ST_Within(geometry1, geometry2)AS Within
FROM GeoTable2
You can see that using Geospark can be really easy thanks to its SQL API.
In future posts, I will go deeper into more geospatial processing.
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Sharpness is arguably the most important single image quality factor: it determines the amount of detail an image can convey. The image on the upper right illustrates the effects of reduced sharpness
(from running Image Processing with one of the Gaussian filters set to 0.7 sigma).
Device or system sharpness is measured as a Spatial Frequency Response (SFR), also called Modulation Transfer Function (MTF). MTF is the contrast at a given spatial frequency (measured in cycles or
line pairs per distance) relative to low frequencies. The 50% MTF frequency correlates well with perceived sharpness— much better than the old vanishing resolution measurement, which indicated where
the detail wasn’t.
Sharpness and MTF are introduced in Sharpness: What is it and how is it measured?
The perceived sharpness of a print or display is measured by Subjective Quality Factor (SQF) or Acutance, which are derived from MTF, the Contrast Sensitivity Function of the human eye, and viewing
Imatest‘s primary sharpness measurement uses slanted-edge patterns analyzed by SFRplus, eSFR ISO , SFRreg, or Checkerboard which have automated region detection. SFR, can be used for manually
selected SFR regions. Analysis can be performed on targets you can purchase or print with the Imatest Test Charts module. Concise instructions are found in How to test lenses with Imatest.
Several alternative patterns, which cause cameras to apply differing amounts of sharpening and noise reduction, can be used for measuring MTF. All require more real estate than the slanted-edge. They
The MTF Measurement Matrix compares the different methods.
System sharpness is affected by the lens (design and manufacturing quality, position in the image field, aperture, and (for zoom lenses) focal length), sensor (pixel count and anti-aliasing filter),
and signal processing (especially sharpening and noise reduction). In the field, sharpness is affected by camera shake (a good tripod can be helpful), focus accuracy, and atmospheric disturbances
(thermal effects and aerosols).
Some lost sharpness can be restored by sharpening, but sharpening has limits. It can’t restore detail where MTF is very low (under about 10%). Oversharpening, illustrated on the right, can also
degrade image quality (especially at large magnifications) by causing “halos” to appear near contrast boundaries. Images from many compact digital cameras and phones are oversharpened.
Sharpness: What is it and how is it measured?
Visualizing Sharpness | Rise Distance and Frequency Domain | Modulation Transfer Function | Spatial Frequency Units
Summary metrics | MTF measurement Matrix| Slanted-Edge measurement | Edge angles | Slanted-Edge modules
Edge contrast and clipping | Slanted-Edge algorithm | Differences with ISO | Noise reduction
Related sharpness techniques | Key takeaways | Additional resources
Visualizing Sharpness
Sharpness determines the amount of detail an imaging system can reproduce. It is defined by the boundaries between zones of different tones or colors.
In Figure 1, sharpness is illustrated with a bar pattern of increasing spatial frequency. The top portion of the figure is sharp and its boundaries are crisp; the lower portion is blurred and
illustrates how the bar pattern is degraded after passing through a simulated lens.
Note: All lenses blur images to some degree.
Sharpness is most visible on features like image edges (Figure 2) and can be measured by the edge (step) response.
Several methods are used for measuring sharpness that include the 10-90% rise distance technique, modulation transfer function (MTF), special and frequency domains, and slanted-edge algorithm.
Rise Distance and Frequency Domain
Image sharpness can be measured by the “rise distance” of an edge within the image.
With this technique, sharpness can be determined by the distance of a pixel level between 10% to 90% of its final value (also called 10-90% rise distance; see Figure 3).
Rise distance is not widely used because there is no convenient way of calculating the rise distance of an imaging system from the rise distances of its individual components (i.e., lens, digital
sensor, and software sharpening).
To overcome this issue, measurements are made in the frequency domain where frequency is measured in cycles or line pairs per distance (millimeters, inches, pixels, image height, or sometimes angle
[degrees or milliradians]).
In the frequency domain, a complex signal (audio or image) can be created by combining signals consisting of pure tones (sine waves), which are characterized by a period or frequency (Figure 4). The
response of a complete system is the product of the responses of each component.
Frequency and spatial domains are related by the Fourier transform.
\(\displaystyle F(x)=\int_{-\infty}^{\infty}f(t)e^{-i\omega t}dt\)
\(\displaystyle f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)e^{i \omega t}d\omega\)
f = Frequency = 1/Period (a shorter period corresponds to a higher frequency);
t = time; ω = 2πf
The greater the system response at high frequencies (short periods), the more detail the system can convey (Figure 5). System response can be characterized by a frequency response curve, F(f).
Note: High frequencies correspond to fine detail.
Modulation Transfer Function
The relative contrast at a given spatial frequency (output contrast/input contrast) is called Modulation Transfer Function (MTF), which is similar to the Spatial Frequency Response (SFR), and is a
key to measuring sharpness. In Figure 6, MTF is illustrated with sine and bar patterns, an amplitude plot, and a contrast plot—each of which has spatial frequencies that increase continuously from
left to right.
Note: Imatest uses SFR and MTF interchangeably. SFR is more commonly associated with complete system response, where MTF is commonly associated with the individual effects of a particular component.
In other words, system SFR is equivalent to the product of the MTF of each component in the imaging system.
High spatial frequencies (on the right) correspond to fine image detail. The response of photographic components (film, lenses, scanners, etc.) tends to “roll off” at high spatial frequencies. These
components can be thought of as low-pass filters that pass low frequencies and attenuate high frequencies.
Figure 6 consists of upper, middle, and lower plots and are described as follows:
• Bar pattern, Sine pattern (upper plot) —The upper plot illustrates (1) the original sine patterns, (2) the sine pattern with lens blur, (3) the original bar pattern, and (4) the bar pattern with
lens blur. Note that lens blur causes contrast to drop at high spatial frequencies.
• Amplitude (middle plot)—The middle plot displays the luminance (“modulation” V in the MTF Equation section) of the bar pattern with lens blur (see red curve in Figure 6). The modulation of the
sine pattern, which consists of pure frequencies, is used to calculate MTF. (Note that contrast decreases at high spatial frequencies.)
• MTF % (lower plot)—The lower plot shows the corresponding sine pattern contrast (see blue curve; represents MTF), which also is defined in the MTF Equation section. By definition, the low
frequency MTF limit is always 1 (100%). In Figure 6, the MTF is 50% at 61 lp/mm (line pairs per millimeter) and 10% at 183 lp/mm. Note that both frequency and MTF are displayed on logarithmic
scales with exponential notation [10^0 = 1%; 10^1 = 10%; 10^2 = 100%, etc.]; amplitude (middle plot) is displayed on a linear scale. The MTF of a complete imaging system is the product of the MTF
of its individual components.
MTF Equation
The equation for MTF is derived from the sine pattern contrast C(f) at spatial frequency f, where
\(\displaystyle C(f)=\frac{V_{max}-V_{min}}{V_{max}+V_{min}}\) for luminance (“modulation”) V.
\(\displaystyle MTF(f)=100\% \times\frac{C(f)}{C(0)}\) Note: this normalizes MTF to 100% at low spatial frequencies.
To correctly normalize MTF at low spatial frequencies, a test chart must have some low-frequency energy. This is supplied by large light and dark areas in slanted edges and by features in most
patterns used by Imatest, but is not present in lines and grids. For systems where sharpening can be controlled, the recommended primary MTF calculation is the slanted-edge, which is calculated from
the Fourier transform of the impulse response (i.e., response to a narrow line), which is the derivative (d/dx or d/dy) of the edge response. Fortunately, you don’t need an understanding of Fourier
transforms to understand MTF.
Traditional Resolution Measurements
Traditional resolution measurements involve observing an image of bar patterns, most frequently the USAF 1951 chart (Figure 7) and estimating the highest spatial frequency (lp/mm) where bar patterns
are visibly distinct. This observation (also called vanishing resolution) corresponds to an MTF of roughly 10-20%. Because the vanishing resolution is the spatial frequency where image information
disappears— where it isn’t visible, it is strongly dependent on observer bias and is a poor indicator of image sharpness. (It’s Where the Woozle Wasn’t in Winnie the Pooh.)
Note: The USAF 1951 chart (long-since abandoned by the Air Force) is poorly suited for computer analysis because it uses space inefficiently and its bar triplets lack a low frequency reference.
Furthermore, small changes in chart position (sampling phase) can cause the appearance of its bars to change as they shift from being in phase to out of phase with the pixel array. This adversely
affects the vanishing resolution estimate.
Better indicators of image sharpness are spatial frequencies where MTF is 50% of its low frequency value (MTF50) or 50% of its peak value (MTF50P). MTF50 and MTF50P are recommended for comparing the
sharpness of different cameras and lenses because
1. Image contrast is half its low frequency or peak value thus detail is still quite visible. (The eye is insensitive to detail at spatial frequencies where MTF is 10% or less.)
2. The response of most cameras falls off rapidly in the vicinity of MTF50 and MTF50P. MTF50P is a better metric for strongly sharpened cameras (explained in our paper from Electronic Imaging 2020).
3. The 50% level is related to the image’s information capacity.
Note: Additional sharpness indicators are discussed in Summary metrics, below.
Although MTF can be estimated directly from images of sine patterns (using Rescharts,Log Frequency, Log F-Contrast, and Star Chart), the ISO 12233 slanted-edge technique provides more accurate and
repeatable results and uses space more efficiently. Slanted-edge images can be analyzed by one of the modules listed in the MTF Measurement Matrix, below.
Spatial Frequency Units
Most readers will be familiar with temporal frequency. For example, the frequency of a sound—measured in Cycles/Second or Hertz—is closely related to its perceived pitch. The frequencies of radio
transmissions (measured in kilohertz, megahertz, and gigahertz) are also familiar.
Spatial frequency is measured in cycles (or line pairs) per distance instead of time. As with temporal (e.g., audio) frequency response, the more extended the response, the more detail can be
Note: In imaging systems, one cycle (C) is equivalent to one line pair (LP).
The two nomenclatures are used interchangeably.
Spatial frequency units can be selected from the Settings or More settings windows of SFR and Rescharts modules (SFRplus, eSFR ISO, Star, etc. Figure 8) and is the measurement intended to determine
how much detail a camera can reproduce or how well the pixels are utilized.
Past film camera lens tests used line pairs per millimeter (lp/mm), which worked well for comparing lenses because most 35mm film cameras have the same 24 x 36mm picture size. But digital sensor
sizes vary widely—from under 5mm diagonal in camera phones to 43mm diagonal for full-frame cameras to an even larger diagonal for medium format. For this reason, line widths per picture height (LW/
PH) is recommended for measuring the total detail a camera can reproduce. Note that LW/PH is equal to 2 × lp/mm × (picture height in mm).
Another useful spatial frequency unit is cycles per pixel (C/P), which gives an indication of how well individual pixels are utilized. The choice of units is also influenced by whether performance at
the image (sensor) or on the object has primary importance: see Comparing sharpness in different cameras. There is no need to use actual distances (millimeters or inches) with digital cameras,
although such measurements are available (Table 1).
Table 1. Summary of spatial frequency units with equations that refer to MTF in selected frequency units.
│ MTF Unit │ Application │ Equation │
│ Cycles/Pixel (C/P) │ Shows how well pixels are utilized. Nyquist frequency f[Nyq] is always 0.5 C/P. │ │
│ Cycles/Distance │ Cycles per distance on the sensor. Pixel spacing or pitch must be entered. Popular for comparing resolution in the old days of │ \(\frac{MTF(C/P)}{\text{pixel │
│ │ standard film formats (e.g., 24x36mm for 35mm film). │ pitch}}\) │
│ (cycles/mm or cycles/inch) │ │ │
│ Line Widths/Picture Height (LW/ │ │ │
│ PH) │ Measures overall image sharpness. This is the best unit for comparing the performance of cameras with different sensor sizes │ \(2 \times MTF\bigl(\frac{LP} │
│ │ and pixel counts. Line Widths is traditional for TV measurements. Recommended for image-centric applications in Comparing │ {PH}\bigr)\) ; │
│ note: for cropped images enter │ sharpness in different cameras. │ \(2 \times MTF\bigl(\frac{C} │
│ the original picture height into │ Note that 1 Cycle = 1 Line Pair (LP) = 2 Line Widths (LW). │ {P}\bigr) \times PH\) │
│ the more settings dimensions │ │ │
│ input │ │ │
│ Line Pairs/Picture Height (LP/ │ │ │
│ PH) │ │ \(MTF\bigl(\frac{LW}{PH}\bigr) │
│ │ │ / 2\) ; │
│ note: for cropped images enter │ Measures overall image sharpness. Differs from LW/PH by a factor of 2. Used by dpreview.com. │ \(MTF\bigl(\frac{C}{P}\bigr) \ │
│ the original picture height into │ │ times PH\) │
│ the more settings dimensions │ │ │
│ input │ │ │
│ │ Angular frequencies. Pixel spacing or pitch must be entered. Focal length (FL) in mm is usually included in EXIF data in │ │
│ Angular frequency │ commercial image files. If it isn’t available it must be entered manually, typically in the EXIF parameters region at the │ \(0.001 \times MTF\bigl(\frac │
│ │ bottom of the settings window. If pixel spacing or focal length is missing, units will default to Cycles/Pixel. │ {\text{cycles}}{\text{mm}}\ │
│ Cycles/milliradian │ Cycles/degree is useful for comparing camera systems to the human eye, which has an MTF50 of roughly 20 Cycles/Degree │ bigr) \times FL(\text{mm})\) │
│ │ (depending on the individual’s eyesight and illumination). │ │
│ │ │ │
├──────────────────────────────────┤FL can be calculated from the simple lens equation*, \(1/FL = 1/s_1 + 1/s_2\), where s[1] is the lens-to-chart distance (easy ├────────────────────────────────┤
│ │ to measure), s[2] is the lens-to-sensor distance, and magnification \(M = s_2/s_1\). │ │
│ │ │ \(\frac{\pi}{180} \times MTF\ │
│ │ \(FL = s_1/(1+1/|M|)\ =\ s_2/(1+|M|)\) │ bigl(\frac{\text{cycles}}{\ │
│ Cycles/degree │ │ text{mm}}\bigr) \times FL(\ │
│ │ *Unless s[1] >> s[2], (by 100× or more), lens geometry (s[1], s[2], and FL) is not reliable for calculating M because lenses │ text{mm})\) │
│ │ can deviate significantly from the simple lens equation. │ │
│ │ │ │
│ │ Cycles per distance on the object being photographed (what some people think of as the “subject”). Pixel spacing and │ \(MTF\bigl( \frac{\text │
│ │ magnification must be entered with an important exception*. Should be used when the system specification references the object │ {Cycles}}{\text{Distance}} \ │
│ │ being photographed (for example, if features of a certain width need to be detected). Recommended for object-centric │ bigr) \times |\text │
│ Cycles/object mm │ applications in Comparing sharpness in different cameras. │ {Magnification}|\) │
│ Cycles/object in │ │ │
│ │ *For SFRplus when bar-to-bar spacing is entered, eSFR ISO when the registration mark vertical spacing is entered, or │ Cycles/distance is Cycles/mm │
│ │ Checkerboard when the square length is entered, Cycles per object distance is calculated directly without using pixel spacing │ or Cycles/in on the image │
│ │ or entering magnification, which is calculated from the geometry. Before Imatest 2021.2 you had to enter a number in the Pixel │ sensor. │
│ │ spacing field, but this number is not used for the actual calculation. We apologize for the confusion. │ │
│ Line Widths/Crop Height │ Primarily used for testing when the active chart height (rather than the total image height) is significant. No longer │ │
│ Line Pairs/Crop Height │ recommended because it’s dependent on the crop size, which is not standardized. │ │
│ Line Widths/ │ │ \(2 \times MTF\bigl(\frac{C} │
│ Feature Ht (Px) │ When either of these is selected, a Feature Ht pixels box appears to the right of the MTF plot units (sometimes used for │ {P}\bigr) \times \text{Feature │
│ Line Pairs/ │ Magnification) that lets you enter a feature height in pixels, which could be the height of a monitor under test, a test chart, │ Height}\) │
│ Feature Ht (Px) │ or the active field of view in an image that has an inactive area. This unit selection is useful for comparing the resolution │ │
│ │ of specific objects for cameras with different image or pixel sizes. Recommended for object-centric applications in Comparing │ \(MTF\bigl(\frac{C}{P}\bigr) \ │
│ (formerly Line Widths or Line │ sharpness in different cameras. │ times \text{Feature Height}\) │
│ Pairs/N Pixels (PH)) │ │ │
│ PH = Picture Height in pixels. FL(mm) = Lens focal length in mm. Pixel pitch = distance per pixel = 1/(pixels per distance). │
│ Note: Different units scale differently with image sensor and pixel size. │
Comparing sharpness in different cameras recommends spatial frequency units based on one of two broad types of application:
□ Image-centric (such as landscape photography, where detail on the image sensor is important): Line Widths (or Pairs) per Picture Height is recommended.
□ Object-centric (for medical, machine vision, etc., where details of the object are important): Cycles/object distance or LW (or LP) per Feature Height are recommended.
Summary Metrics
Several summary metrics are derived from MTF curves to characterize overall performance. These metrics are used in a number of displays, including secondary readouts in the SFR/SFRplus/eSFR ISO Edge/
MTF plot (see Imatest Slanted-Edge Results) and in the SFRplus 3D maps.
│ Summary │ Description │ Comments │
│ Metric │ │ │
│ MTF50 │ Spatial frequency where MTF is 50% (nn%) of the low (0) frequency MTF. │ The most common summary metric; correlates well with perceived sharpness. Increases with increasing │
│ MTFnn │ MTF50 (nn = 50) is widely used because it corresponds to bandwidth (the │ software sharpening; may be misleading because it “rewards” excessive sharpening, which results in visible │
│ │ half-power frequency) in electrical engineering. │ and possibly annoying “halos” at edges. │
│ MTF50P │ │ Identical to MTF50 for low to moderate software sharpening, but lower than MTF50 when there is a software │
│ MTFnnP │ Spatial frequency where MTF is 50% (nn%) of the peak MTF. │ sharpening peak (maximum MTF > 1). Much less sensitive to software sharpening than MTF50 (as shown in a │
│ │ │ paper we presented at Electronic Imaging 2020). All in all, a better metric. │
│ MTF area │ Area under an MTF curve (below the Nyquist frequency), normalized to │ A particularly interesting new metric because it closely tracks MTF50 for little or no sharpening, but does │
│ normalized │ its peak value (1 at f = 0 when there is little or no sharpening, but │ not increase for strong oversharpening; i.e., it does not reward excessive sharpening. Still relatively │
│ │ the peak may be » 1 for strong sharpening). │ unfamiliar. Described in Slanted-Edge MTF measurement consistency. │
│ MTF10, │ │ These numbers are of interest because they are comparable to the “vanishing resolution” (Rayleigh limit). │
│ MTF10P, │ Spatial frequencies where MTF is 10 or 20% of the zero frequency or │ Noise can strongly affect results at the 10% levels or lower. MTF20 (or MTF20P) in Line Widths per Picture │
│ MTF20, │ peak MTF │ Height (LW/PH) is closest to analog TV Lines. Details on measuring monitor TV lines are found here. │
│ MTF20P │ │ │
│ Shannon │ Not strictly a sharpness metric! Combines MTF, noise, and contrast loss │ Can be measured from Siemens Star or Slanted-edge patterns. Slanted-edges are faster and more convenient. │
│ information │ to give a figure of merit for imaging systems.. │ Sensitive to exposure level. New and still unfamiliar: we are making an effort to educate the industry │
│ capacity │ │ about its merits. Details and news on Image Information Metrics: Information Capacity and more. │
MTF Measurement Matrix — comparing different charts and measurement techniques
Imatest has many patterns for measuring MTF— slanted-edge, Log frequency, Log f-contrast, Siemens Star, Dead Leaves (Spilled Coins), Random 1/f, and Hyperbolic wedge— each of which tends to give
different results in consumer cameras, most of which have nonuniform image processing— commonly bilateral filtering— that depends on local scene content. Sharpening (high frequency boost) tends to be
maximum near contrasty features (larger near higher contrast edges), while noise reduction (high frequency cut, which can obscure fine texture) tends to be maximum in their absence. For this reason
MTF measurements can be very different with different test charts.
In principle, MTF measurements should be the same when no nonuniform or nonlinear image processing (bilateral filtering) is applied, for example when the image is demosaiced with dcraw or LibRaw with
no sharpening and noise reduction. But this does not exactly happen because demosaicing, which is present in all cameras that use Color Filter Arrays (CFAs) involves some nonlinear processing. The
sensitivity of different patterns to image processing is summarized in the image below.
Comparison of the effects of image processing (bilateral filtering) on MTF measurements:
Slanted-edges and wedges tend to be sharpened the most.
The random 1/f pattern has the least sharpening and the most noise reduction.
The MTF Matrix table below lists the attributes, advantages, and disadvantages of Imatest’s methods for measuring MTF.
MTF Measurement Matrix
Note that most of these patterns benefit from averaging multiple (identically-registered) images to reduce the effects of noise.
│ Measurement │ Advantages / Disadvantages / Sensitivity │ Primary use & comments │
│ pattern │ │ │
│ │ Most efficient use of space: enables creation of a detailed map │ │
│ │ of MTF response. │ │
│ │ Fast, automated region detection in SFRplus, eSFR ISO, SFRreg, │ │
│ │ and Checkerboard. │ │
│ │ Fast calculations. │ │
│ │ Relatively insensitive to noise (more immune if noise reduction │ This is the primary MTF measurement in Imatest. │
│ │ is applied). │ │
│ Slanted-edge │ Compliant with the ISO 12233 standard, using a “binning” │ The most efficient pattern for lens and camera testing, especially where an MTF response map is required. │
│ (SFR │ (super-resolution) algorithm that allows MTF to be measured │ │
│ SFRplus │ above the Nyquist frequency (0.5 C/P). │ The high contrast (≥40:1) recommended in the old ISO 12233:2000 standard produced unreliable results │
│ eSFRiso │ The best pattern for manufacturing testing. │ (clipping, gamma issues, excessive sharpening with bilateral filters). The new ISO 12233:2014 standard │
│ SFRreg │ May give optimistic results in systems with strong sharpening │ recommends 4:1 contrast. This is our recommendation (with SFRplus or eSFR ISO) for all new work. │
│ Checkerboard) │ and noise reduction (i.e., it can be fooled by signal │ │
│ │ processing, especially with high contrast (≥ 10:1) edges. │ Compared favorably with the Siemens star in Slanted-edge versus Siemens Star. │
│ │ Gives inconsistent results in systems with extreme aliasing │ │
│ │ (strong energy above the Nyquist frequency), especially with │ │
│ │ small regions. │ │
│ │ Most sensitive to sharpening, especially for high contrast │ │
│ │ (≥10:1) edges; │ │
│ │ Least sensitive to software noise reduction. │ │
│ Log frequency │ Calculated from first principles. Displays color moire. │ Primarily used as a check on other methods, which are not calculated from first principles. │
│ │ Sensitive to noise. Inefficient use of space. │ │
│ │ Best pattern for illustrating the effects of nonuniform image │ │
│ │ processing. │ The sensitivity to sharpening/noise reduction is an advantage for this chart, which is designed to Illustrate │
│ Log f-Contrast │ Sensitive to noise. │ how signal processing varies with image content (feature contrast). Shows loss of fine detail due to software │
│ │ Strong sensitivity to sharpening near the (high contrast) top │ noise reduction. │
│ │ of the image and noise reduction near the (low contrast) │ │
│ │ bottom, with a gradual transition in-between. │ │
│ │ Included in the ISO 12233:2014 standard. Relatively insensitive │ │
│ │ to noise. Provides directional MTF information. │ Promoted for general testing by Image Engineering, but spatial detail is limited to a 3×3 or 4×3 grid. │
│ Siemens star │ Slow, inefficient use of space. Limited low frequency │ Compared with the slanted-edge in Slanted edge vs. Siemens star MTF calculations: 2024 white paper. │
│ │ information at outer radius makes MTF normalization difficult. │ │
│ │ Moderate sensitivity to sharpening and noise reduction. │ │
│ │ Measures texture blur / sharpness / acutance. Pattern │ │
│ │ statistics are similar to typical images. │ Consists of stacked randomly-sized circles. Strong industry interest, particularly from the Camera Phone │
│ │ Inefficient use of space. A tricky noise power subtraction │ Image Quality (CPIQ) group. │
│ Dead Leaves │ algorithm* can reduce very high sensitivity to noise, but │ │
│ (Spilled Coins) │ signal-averaging of multiple identical images works better. │ Both Dead Leaves (Spilled Coins) and Random charts are analyzed with the Random (Dead Leaves) module. Strong │
│ │ Moderate sensitivity to sharpening and strong sensitivity to │ bilateral filtering can cause misleading results. │
│ │ noise reduction make it usable for an overall texture sharpness │ │
│ │ metric that correlates well with subjective observations. │ │
│ │ Reveals how well fine detail (texture) is rendered: system │ Measures a camera’s ability to render fine detail (texture), i.e., low contrast, high spatial frequency image │
│ Random │ response to software noise reduction. │ content. *Noise power can be removed from the measurement in Imatest using the gray patches adjacent to the │
│ (scale-invariant) │ Lest sensitive to sharpening, │ pattern. │
│ │ Most sensitive to Software noise reduction │ │
│ │ Makes use of wedge patterns on the ISO 12233:2000 or eSFR ISO │ │
│ │ chart. │ Measures “vanishing resolution”: where lines start disappearing in wedge patterns, frequently in the ISO │
│ │ MTF is not accurate around Nyquist and half-Nyquist frequencies │ 12233 chart, where three regions (including a square region for a low-frequency reference) are required to │
│ Wedge │ (it’s very sensitive to sampling phase variations). Not │ get a reasonable MTF measurement (which is less accurate than other methods due to sampling phase │
│ │ suitable as a primary MTF measurement. │ sensitivity). More convenient with eSFR ISO. │
│ │ Sensitive to sharpening. │ │
│ │ Sensitive to noise. Inefficient use of space. │ │
│ The effects of noise (and low Signal-to-Noise Ratio – SNR) can be greatly reduced by acquiring and signal-averaging multiple images. │
Slanted-Edge Measurement for Spatial Frequency Response
Several Imatest modules measure MTF using the slanted-edge technique and include:
• Slanted-edge test charts that may be purchased from Imatest or created with Imatest Test Charts. Charts that employ automatic detection (SFRplus, eSFR ISO, SFRreg, or Checkerboard) are
• Briefly, the ISO 12233 slanted-edge method calculates MTF by finding the average edge (4X oversampled using a clever binning algorithm), differentiating it (to obtain the Line Spread Function
(LSF)), and taking the absolute value of the Fourier transform of the LSF. The algorithm is described in detail here.
The key output of slanted edge analysis is the Edge/MTF plot, which can be viewed by clicking the button below. Many additional results are available, including summary and 3D plots, showing Lateral
Chromatic Aberration and other results as well as MTF.
The Edge/MTF plot: Imatest’s primary slanted-edge result
An Edge/MTF plot from Imatest SFR (for an SFRplus chart image) is shown on the right. SFRplus, eSFR ISO, SFRreg, and Checkerboard produce similar results and much more.
(Upper-left) A narrow image illustrating the tones of the averaged edge. It is aligned with the average edge profile (spatial domain) plot, immediately below.
(Middle-left) Average Edge (Spatial domain): The average edge profile shown here linearized (the default). A key result is the edge rise distance (10-90%), shown in pixels and in the number of rise
distances per Picture Height. Other parameters include overshoot and undershoot (if applicable). This plot can optionally display the line spread function (LSF: the derivative of the edge).
(Bottom-left) MTF (Frequency domain): The Spatial Frequency Response (MTF), shown to twice the Nyquist frequency. Key summary results include MTF50, the frequency where contrast falls to 50% of its
low frequency value, and MTF50P, the frequency where contrast falls to 50% of its peak value, which corresponds well with perceived image sharpness. Units are cycles per pixel (C/P) and Line Widths
per Picture Height (LW/PH). Other results include MTF at Nyquist (0.5 cycles/pixel; sampling rate/2), which indicates the probable severity of aliasing and user-selected secondary readouts, and
Secondary readouts. The Nyquist frequency is displayed as a vertical blue line. The diffraction-limited MTF response is shown as a pale brown dashed line when the pixel spacing is entered (manually)
and the lens focal length is entered (usually from EXIF data, but can be manually entered).
This image is strongly (but not excessively) sharpened.
SFR Results: MTF (sharpness) plot describes this Figure in more detail.
MTF curves and Image appearance contains several examples illustrating the correlation between MTF curves and perceived sharpness.
Why is the edge slanted?
MTF results for pure vertical or horizontal edges are highly dependent on sampling phase (the relationship between the edge and the pixel locations), and hence can vary from one run to the next
depending on the precise (sub-pixel) edge position. The edge is slanted so MTF is calculated from the average of many sampling phases, which makes results much more stable and robust (Figure 9).
What edge angles work best?
Where possible, edge angles should be greater than ±2 degrees from the closest Vertical (V), Horizontal (H), or 45 degree orientation. The reason is that results from vertical, horizontal, and 45°
edges are very sensitive to the relationship between the edge and the pixels (i.e., they are phase-sensitive). Tilting the edges by more than 2 or 3 degrees avoids this issue.
The ISO 12233 standard recommends an angle of either 5 or 5.71 degrees (arctan(0.1)). This angle is not sacred— MTF is not strongly dependent on edge angle. Angles from 3 to 7 degrees work fine. For
nonzero edge angles θ relative to the closest V or H orientation, a cosine correction is applied, as illustrated on the right. The correction is significant when θ is greater than about 8 degrees
(cos(8º) = 0.99). The initial MTF and corresponding frequency f are calculated from a Vertical or Horizontal line (shown in blue), based on the region selection. The true MTF is defined normal to the
edge— along the red line. Since the length of the actual transition along the red line (normal to the edge) is shorter than the measured transition along the blue (V or H) line, and since the
frequency f used to measure MTF is inversely proportional to the actual transition length,
\(f=f_{initial} / cos(\theta)\)
Corresponding summary metrics MTFnn (MTF50, MTF50P, etc.), which have units of frequency, are increased over the initial values.
\(MTFnn = MTFnn(\text{initial}) / cos(\theta)\)
The primary disadvantage of large edge angles is that the available region area may be reduced, especially for SFRreg patterns.
Edge Contrast and clipping
Edge Contrast should be limited to 10:1 at the most, and a 4:1 edge contrast is generally recommended. The reason is that high contrast edges (>10:1, such as found in the old ISO 12233:2000 chart)
can cause saturation or clipping, resulting in edges with sharp corners that exaggerate MTF measurements. For more details, see Using Rescharts slanted-edge modules, Part 2: Warnings – clipping.
Advantages and Disadvantages of Slanted Edge
• Most efficient use of space, which makes it possible to create a detailed map of MTF response
• Fast, automated region detection in SFRplus, eSFR ISO, SFRreg, and Checkerboard
• Fast calculations
• Relatively insensitive to noise (highly immune if noise reduction is applied)
• Compliant with the ISO 12233 standard, whose “binning” (super-resolution) algorithm allows MTF to be measured above the Nyquist frequency (0.5 C/P)
• The best pattern for manufacturing testing
• May give optimistic results in systems with strong image-dependent sharpening (i.e., where the amount of sharpening increases with edge contrast). This type of image processing (bilateral
filtering) is almost universal in consumer cameras.
• Gives inconsistent results in systems with extreme aliasing (strong energy above the Nyquist frequency), especially with small regions.
• Not suitable for measuring fine texture, where the Log Frequency-Contrast or Spilled Coins (Dead Leaves) patterns are recommended.
Note: Imatest Master can calculate MTF for edges of virtually any angle, though exact vertical, horizontal, and 45° should be avoided because of sampling phase sensitivity.
Slanted-Edge Modules
Imatest Slanted-Edge Modules include SFR, SFRplus, eSFR ISO, Checkerboard, and SFRreg (see Table 2 and Sharpness Modules for details).
Note: See “How to test lenses with Imatest” for a good summary of how to measure MTF using SFRplus or eSFR ISO.
Table 2. Brief summary of Imatest slanted-edge modules.
Module Description Examples
• Measures MTF from slanted edges in a variety of charts and wherever there is a clean edge; region selection is manual.
SFR • Two useful regions from the old ISO-12233:2000 chart are indicated by red and blue arrows (see top, far-right image in the Examples column).
• A typical region (a crop of a vertical edge slanted about 5.7 degrees) is used to calculate horizontal MTF response (see lower, far-right image in the Examples column).
• Measures MTF and other image quality parameters from Imatest SFRplus chart (recommended) or created using Imatest Test Charts (a wide-body printer, advanced printing skills,
and knowledge of color management required).
SFRplus • Offers numerous advantages over the old ISO 12233:2000 test chart: automatic feature detection, lower contrast for improved accuracy, more edges (less wasted space) for a
detailed map of MTF over the image surface.
• Measurements are ISO-compliant; includes automatic region detection.
• Measures MTF and other image quality parameters using an enhanced version of the ISO 12233:2014 and 2017 Edge SFR (E-SFR) test chart
eSFR ISO • Has slightly less spatial detail than SFRplus, but much more noise detail.
• Includes automatic region detection.
• Sensitive to framing, making it ideal for through-focus tests.
Checkerboard • Provides precise distortion calculations.
• Includes automatic region detection.
Several individual charts are typically placed around the image field; works with:
• Extreme fisheye lenses (>180º)
SFRreg • Charts at different distances to test focus and depth of field.
• Extreme high resolution (>36MP) cameras, large fields of view, and large distances.
• Includes automatic region detection.
Slanted-Edge Algorithm
The MTF calculation is derived from ISO standard 12233. The Imatest calculation contains a number of enhancements, listed below. The original ISO calculation is performed when the ISO standard
SFR checkbox in the SFR input dialog box is checked (we recommended leaving it unchecked unless it’s specifically required).
• The cropped image is linearized; i.e., the pixel levels are adjusted to remove the gamma encoding applied by the camera. (Gamma is adjustable with a default of 0.5).
• The edge locations for the red, green, blue, and luminance (Y) channels:
Y = 0.2125R + 0.7154G + 0.0721B (default) or 0.3R + 0.59G + 0.11B or (selectable in Options III)
are determined for each scan line (horizontal lines in the above image).
• A second order fit to the edge is calculated for each channel using polynomial regression. The second order fit removes the effects of lens distortion. In the above image, the equation would have
the form:
\(x = a_0 + a_1y + a_2y^2\)
• Depending on the value of the fractional part of scan line i,
fp = x[i] – int(x[i ])
of the second order fit at each scan line, the shifted edge is added to one of four bins:
bin 1 if 0 ≤ fp < 0.25
bin 2 if 0.25 ≤ fp < 0.5
bin 3 if 0.5 ≤ fp < 0.75
bin 4 if 0.75 ≤ fp < 1
Note: The bin mentioned in the previous equation does not depend on the detected edge location.
□ The four bins are combined to calculate an averaged 4x oversampled edge. This allows analysis of spatial frequencies beyond the normal Nyquist frequency.
☆ The derivative (d/dx) of the averaged 4x oversampled edge is calculated. A centered Hamming window is applied to force the derivative to zero at its limits.
☆ MTF is the absolute value of the Fourier transform (FFT) of the windowed derivative.
Note: Origins of Imatest slanted-edge SFR calculations were adapted from a Matlab program, sfrmat, which was written by Peter Burns to implement the ISO 12233:2000 standard. Imatest’s SFR calculation
incorporates numerous improvements, including improved edge detection, better handling of lens distortion, and better noise immunity. The original Matlab code is available here. In comparing sfrmat
results with Imatest, tonal response is assumed to be linear; i.e., gamma = 1 if no OECF (tonal response curve) file is entered into sfrmat. Since the default value of gamma in Imatest is 0.5, which
is typical of digital cameras in standard color spaces such as sRGB, you must set gamma to 1 to obtain good agreement with sfrmat.
Imatest and ISO Standard 12233 calculation options
New in Version 22.1 – an Imatest/ISO Standard SFR dropdown menu is located on the lower left of slanted-edge More Settings window. (This option was formerly a checkbox for “ISO compatible”
This new dropdown allows you to choose between Imatest and ISO-compliant calculations. There are now four options that can be used for SFR Settings to control the Edge SFR Algorithm.
1. Imatest pre-22.1 – this option maintains the pre-22.1 Imatest SFR calculations, which are based on ISO 12233:2017, but have an additional correction factor.
2. ISO 12233:2017 & earlier – implements the 12233:2017 algorithm with Hamming window and linear edge fitting.
3. Imatest 22.1 (recommended) – Our recommended calculation – uses the Tukey window (alpha=1), and 5th order polynomial edge fitting, for most accurate results. It is based on the ISO 12233:2022
standard, but has an additional correction factor
4. ISO 12233:2024 – implements the current 12233:2024 algorithm, with Tukey window (alpha=1) and 5th order polynomial edge fitting.
Differences Between the Imatest calculations and standard ISO 12233 Calculations
Setting #2 (ISO 12233 2017 & earlier) is not recommended because the Imatest and newer ISO calculations are more accurate— definitely superior in the presence of noise and optical distortion. More
information on calculations can be found below:
• The center of each scan line is calculated from the peak of the lowpass-filtered edge derivative in the Imatestcalculations. The ISO calculation uses a centroid, which is optimum in the absence
of noise. But noise is always present to some degree, and the centroid is extremely sensitive to noise because noise at large distances from the edge has the same weight as the edge itself. The
lowpass-filter is closer to a matched filter, which optimally detects the edge derivative peak.
• Gamma(used to linearize the data) is entered as an input value or derived from known chart contrast. In ISO-standard implementations it is assumed to be 1 unless an OECF file is entered.
• Imatest assumes that the edge may have some (second-order) curvature due to optical distortion. ISO-standard calculation through 2017 assumes a straight line, which can result in degraded MTF
measurements in the presence of optical distortion. Curved edges (5th order fit) are included in the ISO 12233:2022+ calculation. There is very little difference with the 2nd order fit of the
older Imatest calculation.
• Imatest’s “modified apodization” noise reduction (on by default) results in more consistent measurements (no systematic difference). It is turned off when the ISO standard calculation is checked
• The Line Spread Function (LSF) correction factor, introduced in ISO 12233:2014 has been implemented in Imatestsince 2015. The correction factor D(j) (or D(k)), which compensates for the high
frequency loss from numerical differentiation when calculating LSF from the Edge Spread Function (ESF), is incorrect in both the ISO 12233:2014 and 2017 standards. The Imatest and newer ISO
implementation, which is based on first principles (we have the full set of equations), complies with the intent of both versions of the standard.
• A Tukey window is used in the newer Imatest and ISO calculations. It seems to make very little difference.
Note that Additional calculation details can be found in the Peter Burns links (below)
Key Takeaways
• Frequency and spatial domain plots convey similar information, but in a different form. A narrow edge in spatial domain corresponds to a broad spectrum in frequency domain (extended frequency
response) and vice-versa.
• Imatest measures the system response, which includes image processing: not just the lens response.
• Sensor response above the Nyquist frequency can cause aliasing that is visible as Moiré patterns of low spatial frequency. In Bayer sensors (all sensors except Foveon), Moiré patterns appear as
color fringes. Moiré in Foveon sensors is far less bothersome because it is monochrome and the effective Nyquist frequency of the Red and Blue channels is lower than with Bayer sensors.
• MTF at and above the Nyquist frequency is not an unambiguous indicator of aliasing problems. MTF is the product of the lens and sensor response, demosaicing algorithm, and sharpening that
frequently boosts MTF at the Nyquist frequency. MTF should be interpreted as a warning that there could be problems.
• Results are calculated for the R, G, B, and Luminance (Y) channels, (by default, Y = 0.2125R + 0.7154G + 0.0721B, but can be set to the older (NTSC) value, 0.3R+0.59G+0.11B, in the Options III
window). The Y channel is normally displayed in the foreground, but any of the other channels can selected. All are included in the .CSV output file.
• Horizontal and vertical resolution can be different for CCD sensors and should be measured separately. They’re nearly identical for CMOS sensors. Recall, horizontal resolution is measured with a
vertical edge and vertical resolution is measured with a horizontal edge. Resolution is only one of many criteria that contributes to image quality.
• MTF can vary throughout the image, and it doesn’t always follow the expected pattern of sharpest near the center and less sharp near the corners. There are any number of reasons: lens
misalignment, curvature of field, misfocus, etc. That is why measurements are important.
See Also
Additional Resources | {"url":"https://www.imatest.com/imaging/sharpness/","timestamp":"2024-11-14T01:13:27Z","content_type":"text/html","content_length":"392966","record_id":"<urn:uuid:006c2db8-4a98-4dca-b0be-1665ab56fe72>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00441.warc.gz"} |
76,985 research outputs found
In the standard flat cosmological constant ($\Lambda$) cold dark matter (CDM) cosmology, a model of two populations of lens halos for strong gravitational lensing can reproduce the results of the
Jodrell-Bank VLA Astrometric Survey (JVAS) and the Cosmic Lens All-Sky Survey (CLASS) radio survey. In such a model, lensing probabilities are sensitive to three parameters: the concentration
parameter $c_1$, the cooling mass scale $M_\mathrm{c}$ and the value of the CDM power spectrum normalization parameter $\sigma_8$. The value ranges of these parameters are constrained by various
observations. However, we found that predicted lensing probabilities are also quite sensitive to the flux density (brightness) ratio $q_{\mathrm{r}}$ of the multiple lensing images, which has been,
in fact, a very important selection criterion of a sample in any lensing survey experiments. We re-examine the above mentioned model by considering the flux ratio and galactic central Super Massive
Black Holes (SMBHs), in flat, low-density cosmological models with different cosmic equations of state $\omega$, and find that the predicted lensing probabilities without considering $q_{\mathrm{r}}$
are over-estimated. A low value of $q_\mathrm{r}$ can be compensated by raising the cooling mass scale $M_\mathrm{c}$ in fitting the predicted lensing probabilities to JVAS/CLASS observations. In
order to determine the cosmic equation of state $\omega$, the uncertainty in $M_\mathrm{c}$ must be resolved. The effects of SMBHs cannot be detected by strong gravitational lensing method when $q_{\
mathrm{r}}\leq 10$.Comment: 7 pages, 2 figures, corrected to match published version in A& | {"url":"https://core.ac.uk/search/?q=author%3A(Chen%2C%20Da)","timestamp":"2024-11-03T07:08:09Z","content_type":"text/html","content_length":"77803","record_id":"<urn:uuid:b944a338-a951-41aa-a08d-1ac9930d67f4>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00389.warc.gz"} |
Probing the core of the strong nuclear interaction
The strong nuclear interaction between nucleons (protons and neutrons) is the effective force that holds the atomic nucleus together. This force stems from fundamental interactions between quarks and
gluons (the constituents of nucleons) that are described by the equations of quantum chromodynamics. However, as these equations cannot be solved directly, nuclear interactions are described using
simplified models, which are well constrained at typical inter-nucleon distances^1–5 but not at shorter distances. This limits our ability to describe high-density nuclear matter such as that in the
cores of neutron stars^6. Here we use high-energy electron scattering measurements that isolate nucleon pairs in short-distance, high-momentum configurations^7–9, accessing a kinematical regime that
has not been previously explored by experiments, corresponding to relative momenta between the pair above 400 megaelectronvolts per c (c, speed of light in vacuum). As the relative momentum between
two nucleons increases and their separation thereby decreases, we observe a transition from a spin-dependent tensor force to a predominantly spin-independent scalar force. These results demonstrate
the usefulness of using such measurements to study the nuclear interaction at short distances and also support the use of point-like nucleon models with two- and three-body effective interactions to
describe nuclear systems up to densities several times higher than the central density of the nucleus. | {"url":"https://pure.york.ac.uk/portal/en/publications/probing-the-core-of-the-strong-nuclear-interaction","timestamp":"2024-11-04T10:57:28Z","content_type":"text/html","content_length":"67067","record_id":"<urn:uuid:5c910bdf-d49a-4e4f-9867-5bdf35bf87b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00524.warc.gz"} |
Financial Modelling Workshop
Financial Modelling Workshop - Analysis, Measurement and Valuation
Duration: 2 days
• Review of Basic Financial Mathematics
• Probability Analysis and Statistical Methods
• Basic Risk Measures
• Performance Measurement
• Total Return Analysis
• The Economics of Project Finance
The objective of this highly practical PC-based workshop is to give you a good understanding and hands-on experience with tools and techniques used in financial modelling and investment analysis.
First, we give a review of the basic mathematical and statistical concepts that are widely used in the field of finance. We explain concepts such as “Time Value of Money”, variance and standard
deviation and other descriptive statistics. We also present and explain a variety of probability measures and statistical tools such as regression analysis and random number generation.
The remaining part of the workshop is divided into four practical “sessions”.
In session one, we present and explain a number basic risk measures that are used in the risk assessment of equity and fixed income investments. The measures include sensitivity measures such as beta
for stocks and modified duration and convexity for bonds, as well as statistical risk measures derived from estimated loss distributions, such as “Value-at-Risk”.
In session two, we look at measures of investment performance, including “time-weighted” and “money-weighted” return and various measures of risk-adjusted and relative performance.
In session three, we explain how “total return analysis” can be used to rank alternative investments, based upon explicit assumptions about horizon yields and prices, reinvestment rates, dividend
growth and other inputs.
Finally, in session four, we look at the economics of “project finance”. We explain how a project is set up and organized, and we explain and present a number of methods and key ratios for evaluating
the costs, benefits and risk of projects.
The individual sessions will be designed as a mix of theoretical presentations, practical examples and cases, and “hands-on” exercises. Participants are requested to bring their own laptop.
Day One
09.00 - 09.15 Welcome and Introduction
09.15 - 12.00 Review of Basic Financial Mathematics
• Time Value of Money and Cash Flow Analysis
• Descriptive Statistics
□ Mean, variance, standard deviation, skewness, kurtosis,…
• Probability Analysis
• Statistical Tools
□ Regression analysis, random number generation,…
• Small Exercises
Session 1: Basic Risk Measures
• Building Blocks in Risk Analysis
• Sensitivity Measures for Equity Investments
□ Alpha, Beta and factor sensitivities
• Sensitivity Measures for Fixed-Income Investments
□ Duration, modified duration, convexity, key-rate duration
• Statistical Measures of Risk
□ Value-at-Risk, Earnings-at-Risk, Capital-at-Risk,..
• Practical Examples
• Workshop
12.00 - 13.00 Lunch
13.00 - 16.30 Session 2: Performance Measurement
• Measuring Return
□ Arithmetic and geometric returns
• One-Dimensional Measures of Performance
□ Internal rate of return
□ Money-weighted return
□ Time-weighted return
• Risk-Adjusted Measures of Performance
□ Treynor index
□ Sharpe index
□ Sortino ratio
□ Lower partial moments
• Relative Measures of Performance
□ Tracking error
□ Information ratio
• Workshop
Day Two
09.00 - 09.15 Recap
09.15 - 12.00 Session 3: Total Return Analysis (Horizon Analysis)
• Purpose of Total Return Analysis
• “Total Return” Concept
• Assessing Reinvestment Risk
• Assessing Principal Risk
• Calculating Expected Return
• Sensitivity Analysis
• Switch Analysis (Bonds)
□ Arbitrage switch
□ Duration switch
□ Convexity switch
□ Spread trades
• Using Total Return Analysis in Constructing Optimal Portfolios
• Workshop
• Total Return Analysis for Equity Investments
□ Assessing residual value
□ Estimating total return and risk
□ Constructing optimal portfolios
• Workshop
12.00 - 13.00 Lunch
13.00 - 16.30 Session 4: The Economics of Project Finance
• Introduction to Project Finance
□ Definition
□ Rationale and motives
□ Characteristics
• Structuring Projects
• Typical Financing Structure
• Typical Structure for Allocation of Risk
• Valuing Projects
• Analyzing Debt Capacity
• Analyzing Debt Sensitivities
• Managing Project Risk
• Case Study: The A2 Motorway
• Workshop
Evaluation and Termination of the Workshop | {"url":"http://riskrepair.com/Financial-Modelling-Workshop.html","timestamp":"2024-11-03T15:11:09Z","content_type":"text/html","content_length":"58961","record_id":"<urn:uuid:c45f457d-ded5-4d00-9869-bf277cf686cb>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00033.warc.gz"} |
Math Contest Repository
POTW #21 - July 8, 2024
MCR Problem of the Week, July 8, 2024 Edition
In the Pascal's Triangle, all outside numbers are 1 and all inside numbers are found by adding the two numbers above it:
What is the $19^{\text{th}}$ number in the $29^{\text{th}}$ row of the Pascal's Triangle?
Please login or sign up to submit and check if your answer is correct.
flag Report Content
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• It may be offensive.
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Thanks for keeping the Math Contest Repository a clean and safe environment! | {"url":"https://mathcontestrepository.pythonanywhere.com/problem/potw21/","timestamp":"2024-11-04T06:10:53Z","content_type":"text/html","content_length":"9702","record_id":"<urn:uuid:b4b48952-9c67-467b-81fb-e8b52da3f968>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00553.warc.gz"} |
PROOF OF {ART}WORK - Collection | OpenSea
Interactive, software-based art and one of the very first NFT collections to deviate from simple image and video formats.
PROOF OF {ART}WORK pieces are mathematical entities resulting from a simple equation: z_n -> {z_n}^2 + z_0
These NFTs aren't images, but explorable galaxies of millions of points plotted onto 2D space. PO{A}W uses bespoke software to explore these on-demand, inspired by the logic underpinning Google Maps.
PO{A}W is an early example of an on-chain NFT collection, with token IDs representing unique z_0 values for the equation above. As aesthetically pleasing IDs are rare and must be found at random,
their discovery can be considered akin to blockchain mining—with a reward of an artistic proof.
Interactive, software-based art and one of the very first NFT collections to deviate from simple image and video formats.
PROOF OF {ART}WORK pieces are mathematical entities resulting from a simple equation: z_n -> {z_n}^2 + z_0
These NFTs aren't images, but explorable galaxies of millions of points plotted onto 2D space. PO{A}W uses bespoke software to explore these on-demand, inspired by the logic underpinning Google Maps.
PO{A}W is an early example of an on-chain NFT collection, with token IDs representing unique z_0 values for the equation above. As aesthetically pleasing IDs are rare and must be found at random,
their discovery can be considered akin to blockchain mining—with a reward of an artistic proof. | {"url":"https://opensea.io/collection/proof-of-artwork/activity?tab=items","timestamp":"2024-11-11T11:19:37Z","content_type":"text/html","content_length":"403890","record_id":"<urn:uuid:7cdaf239-4af7-4c22-84ef-b454f13b2ef9>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00479.warc.gz"} |
Entropy Distribution Setup
Finally moving on from the Held Karp relaxation, we arrive at the second step of the Asadpour asymmetric traveling salesman problem algorithm. Referencing the Algorithm 1 from the Asadpour paper, we
are now finally on step two.
Algorithm 1 An $O(\log n / \log \log n)$-approximation algorithm for the ATSP
Input: A set $V$ consisting of $n$ points and a cost function $c\ :\ V \times V \rightarrow \mathbb{R}^+$ satisfying the triangle inequality.
Output: $O(\log n / \log \log n)$-approximation of the asymmetric traveling salesman problem instance described by $V$ and $c$.
1. Solve the Held-Karp LP relaxation of the ATSP instance to get an optimum extreme point solution $x^*$. Define $z^*$ as in (5), making it a symmetrized and scaled down version of $x^*$. Vector
$z^*$ can be viewed as a point in the spanning tree polytope of the undirected graph on the support of $x^*$ that one obtains after disregarding the directions of arcs (See Section 3.)
2. Let $E$ be the support graph of $z^*$ when the direction of the arcs are disregarded. Find weights ${\tilde{\gamma}}_{e \in E}$ such that the exponential distribution on the spanning trees, $
\tilde{p}(T) \propto \exp(\sum_{e \in T} \tilde{\gamma}_e)$ (approximately) preserves the marginals imposed by $z^*$, i.e. for any edge $e \in E$, $$\sum_{T \in \mathcal{T} : T \ni e} \tilde
{p}(T) \leq (1 + \epsilon) z^*_e$$ for a small enough value of $\epsilon$. (In this paper we show that $\epsilon = 0.2$ suffices for our purpose. See Section 7 and 8 for a description of how
to compute such a distribution.)
3. Sample $2\lceil \log n \rceil$ spanning trees $T_1, \dots, T_{2\lceil \log n \rceil}$ from $\tilde{p}(.)$. For each of these trees, orient all its edges so as to minimize its cost with
respect to our (asymmetric) cost function $c$. Let $T^*$ be the tree whose resulting cost is minimal among all of the sampled trees.
4. Find a minimum cost integral circulation that contains the oriented tree $\vec{T}^*$. Shortcut this circulation to a tour and output it. (See Section 4.)
Sections 7 and 8 provide two different methods to find the desired probability distribution, with section 7 using a combinatorial approach and section 8 the ellipsoid method. Considering that there
is no ellipsoid solver in the scientific python ecosystem, and my mentors and I have already decided not to implement one within this project, I will be using the method in section 7.
The algorithm given in section 7 is as follows:
1. Set $\gamma = \vec{0}$.
2. While there exists an edge $e$ with $q_e(\gamma) > (1 + \epsilon) z_e$:
☆ Compute $\delta$ such that if we define $\gamma’$ as $\gamma_e’ = \gamma_e - \delta$, and $\gamma_f’ = \gamma_f$ for all $f \in E\ \backslash {e}$, then $q_e(\gamma’) = (1 + \epsilon/2)
☆ Set $\gamma \leftarrow \gamma’$.
3. Output $\tilde{\gamma} := \gamma$.
This structure is fairly straightforward, but we need to know what $q_e(\gamma)$ is and how to calculate $\delta$.
Finding $\delta$ is very easy, the formula is given in the Asadpour paper (Although I did not realize this at the time that I wrote my GSoC proposal and re-derived the equation for delta. Fortunately
my formula matches the one in the paper.)
$$ \delta = \ln \frac{q_e(\gamma)(1 - (1 + \epsilon / 2)z_e)}{(1 - q_e(\gamma))(1 + \epsilon / 2) z_e} $$
Notice that the formula for $\delta$ is reliant on $q_e(\gamma)$. The paper defines $q_e(\gamma)$ as
$$ q_e(\gamma) = \frac{\sum_{T \ni e} \exp(\gamma(T))}{\sum_{T \in \mathcal{T}} \exp(\gamma(T))} $$
where $\gamma(T) = \sum_{f \in T} \gamma_f$.
The first thing that I noticed is that in the denominator the summation is over all spanning trees for in the graph, which for the complete graphs we will be working with is exponential so a `brute
force’ approach here is useless. Fortunately, Asadpour and team realized we can use Kirchhoff’s matrix tree theorem to our advantage.
As an aside about Kirchhoff’s matrix tree theorem, I was not familiar with this theorem before this project so I had to do a bit of reading about it. Basically, if you have a laplacian matrix (the
adjacency matrix minus the degree matrix), the absolute value of any cofactor is the number of spanning trees in the graph. This was something completely unexpected to me, and I think that it is very
cool that this type of connection exists.
The details of using Kirchhoff’s theorem are given in section 5.3. We will be using a weighted laplacian $L$ defined by
$$ L_{i, j} = \left\{ \begin{array}{l l} -\lambda_e & e = (i, j) \in E \\\ \sum_{e \in \delta({i})} \lambda_e & i = j \\\ 0 & \text{otherwise} \end{array} \right. $$
where $\lambda_e = \exp(\gamma_e)$.
Now, we know that applying Krichhoff’s theorem to $L$ will return
$$ \sum_{t \in \mathcal{T}} \prod_{e \in T} \lambda_e $$
but which part of $q_e(\gamma)$ is that?
If we apply $\lambda_e = \exp(\gamma_e)$, we find that
$$ \begin{array}{r c l} \sum_{T \in \mathcal{T}} \prod_{e \in T} \lambda_e &=& \sum_{T \in \mathcal{T}} \prod_{e \in T} \exp(\gamma_e) \\\ && \sum_{T \in \mathcal{T}} \exp\left(\sum_{e \in T} \
gamma_e\right) \\\ && \sum_{T \in \mathcal{T}} \exp(\gamma(T)) \\\ \end{array} $$
So moving from the first row to the second row is a confusing step, but essentially we are exploiting the properties of exponents. Recall that $\exp(x) = e^x$, so could have written it as $\prod_{e \
in T} e^{\gamma_e}$ but this introduces ambiguity as we would have multiple meanings of $e$. Now, for all values of $e$, $e_1, e_2, \dots, e_{n-1}$ in the spanning tree $T$ that product can be
expanded as
$$ \prod_{e \in T} e^{\gamma_e} = e^{\gamma_{e_1}} \times e^{\gamma_{e_2}} \times \dots \times e^{\gamma_{e_{n-1}}} $$
Each exponential factor has the same base, so we can collapse that into
$$ e^{\gamma_{e_1} + \gamma_{e_2} + \dots + \gamma_{e_{n-1}}} $$
which is also
$$ e^{\sum_{e \in T} \gamma_e} $$
but we know that $\sum_{e \in T} \gamma_e$ is $\gamma(T)$, so it becomes
$$ e^{\gamma(T)} = \exp(\gamma(T)) $$
Once we put that back into the summation we arrive at the denominator in $q_e(\gamma)$, $\sum_{T \in \mathcal{T}} \exp(\gamma(T))$.
Next, we need to find the numerator for $q_e(\gamma)$. Just as before, a `brute force’ approach would be exponential in complexity, so we have to find a better way. Well, the only difference between
the numerator and denominator is the condition on the outer summation, which the $T \in \mathcal{T}$ being changed to $T \ni e$ or every tree containing edge $e$.
There is a way to use Krichhoff’s matrix tree theorem here as well. If we had a graph in which every spanning tree could be mapped in a one-to-one fashion onto every spanning tree in the original
graph which contains the desired edge $e$. In order for a spanning tree to contain edge $e$, we know that the endpoints of $e$, $(u, v)$ will be directly connected to each other. So we are then
interested in every spanning tree in which we reach vertex $u$ and then leave from vertex $v$. (As opposed to the spanning trees where we reach vertex $u$ and then leave from that same vertex). In a
sense, we are treating vertices $u$ and $v$ is the same vertex. We can apply this literally by contracting $e$ from the graph, creating $G / {e}$. Every spanning tree in this graph can be uniquely
mapped from $G / {e}$ onto a spanning tree in $G$ which contains the edge $e$.
From here, the logic to show that a cofactor from $L$ is actually the numerator of $q_e(\gamma)$ parallels the logic for the denominator.
At this point, we have all of the needed information to create some pseudo code for the next function in the Asadpour method, spanning_tree_distribution(). Here I will use an inner function q() to
find $q_e$.
def spanning_tree_distribution
input: z, the symmetrized and scaled output of the Held Karp relaxation.
output: gamma, the maximum entropy exponential distribution for sampling spanning trees
from the graph.
def q
input: e, the edge of interest
# Create the laplacian matrices
write lambda = exp(gamma) into the edges of G
G_laplace = laplacian(G, lambda)
G_e = nx.contracted_edge(G, e)
G_e_laplace = laplacian(G, lambda)
# Delete a row and column from each matrix to made a cofactor matrix
G_laplace.delete((0, 0))
G_e_laplace.delete((0, 0))
# Calculate the determinant of the cofactor matrices
det_G_laplace = G_laplace.det
det_G_e_laplace = G_e_laplace.det
# return q_e
return det_G_e_laplace / det_G_laplace
# initialize the gamma vector
gamma = 0 vector of length G.size
while true
# We will iterate over the edges in z until we complete the
# for loop without changing a value in gamma. This will mean
# that there is not an edge with q_e > 1.2 * z_e
valid_count = 0
# Search for an edge with q_e > 1.2 * z_e
for e in z
q_e = q(e)
z_e = z[e]
if q_e > 1.2 * z_e
delta = ln(q_e * (1 - 1.1 * z_e) / (1 - q_e) * 1.1 * z_e)
gamma[e] -= delta
valid_count += 1
if valid_count == number of edges in z
return gamma
Next Steps#
The clear next step is to implement the function spanning_tree_distribution using the pseudo code above as an outline. I will start by writing q and testing it with the same graphs which I am using
to test the Held Karp relaxation. Once q is complete, the rest of the function seems fairly straight forward.
One thing that I am concerned about is my ability to test spanning_tree_distribution. There are no examples given in the Asadpour research paper and no other easy resources which I could turn to in
order to find an oracle.
The only method that I can think of right now would be to complete this function, then complete sample_spanning_tree. Once both functions are complete, I can sample a large number of spanning trees
to find an experimental probability for each tree, then run a statistical test (such as an h-test) to see if the probability of each tree is near $\exp(\gamma(T))$ which is the desired distribution.
An alternative test would be to use the marginals in the distribution and have to manually check that
$$ \sum_{T \in \mathcal{T} : T \ni e} p(T) \leq (1 + \epsilon) z^*_e,\ \forall\ e \in E $$
where $p(T)$ is the experimental data from the sampled trees.
Both methods seem very computationally intensive and because they are sampling from a probability distribution they may fail randomly due to an unlikely sample.
A. Asadpour, M. X. Goemans, A. Mardry, S. O. Ghran, and A. Saberi, An o(log n / log log n)-approximation algorithm for the asymmetric traveling salesman problem, Operations Research, 65 (2017), pp. | {"url":"https://blog.scientific-python.org/networkx/atsp/entropy-distribution-setup/","timestamp":"2024-11-02T18:51:03Z","content_type":"text/html","content_length":"30001","record_id":"<urn:uuid:d1d6f9ec-5315-4d6e-a41f-c754dbd76960>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00700.warc.gz"} |
Arrange the following fraction in descending order: 121,231,... | Filo
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Arrange the following fraction in descending order:
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Sure, here are the step by step solutions: We are given the following fractions: We need to compare them and arrange them in descending order. Step 1: Compare and Since 9 is greater than 7, we know
that is smaller than . Therefore: Step 2: Compare and Since 12 is greater than 9, we know that is smaller than . Therefore: Step 3: Compare and Since 17 is greater than 12, we know that is smaller
than . Therefore: Step 4: Compare and Since 23 is greater than 17, we know that is smaller than . Therefore: Step 5: Compare and Since 50 is greater than 23, we know that is smaller than . Therefore:
Putting it all together, we have: Therefore, the fractions in descending order are:
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Question Text Arrange the following fraction in descending order:
Updated On Feb 13, 2024
Topic All topics
Subject Mathematics
Class Class 12
Answer Type Text solution:1 | {"url":"https://askfilo.com/user-question-answers-mathematics/arrange-the-following-fraction-in-descending-order-36393033393037","timestamp":"2024-11-14T14:21:41Z","content_type":"text/html","content_length":"414137","record_id":"<urn:uuid:9517766b-0e5c-4819-af6a-fe1c4e21d30d>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00492.warc.gz"} |
Faster parallel string matching via larger deterministic samples
Building on previous results of Breslauer, Galil, and Vishkin, we obtain for every p(m) = Ω(log log m) an optimal speedup parallel string matching algorithm that can preprocess a pattern P of length
m in time O(p(m)) and can then find all occurrences of P in a text of an arbitrary length in time O(log log m/log p(m)). Letting p(m) = (log m)^ε[lunate], for any ε > 0, we obtain a constant O(1/ε)
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Dive into the research topics of 'Faster parallel string matching via larger deterministic samples'. Together they form a unique fingerprint. | {"url":"https://cris.tau.ac.il/en/publications/faster-parallel-string-matching-via-larger-deterministic-samples","timestamp":"2024-11-11T00:18:09Z","content_type":"text/html","content_length":"49002","record_id":"<urn:uuid:2cc64d26-a51f-4390-8329-179dbb8b0661>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00246.warc.gz"} |
a box of 60 tissues has a length of 11 cm, a width of 10.5 cm and a height of 13.5 cm. find the volume of the box of tissues
Find an answer to your question 👍 “a box of 60 tissues has a length of 11 cm, a width of 10.5 cm and a height of 13.5 cm. find the volume of the box of tissues ...” in 📗 Mathematics if the answers
seem to be not correct or there’s no answer. Try a smart search to find answers to similar questions.
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Misc 2 - State converse and contrapositive of statement - p: A positiv
Misc 2 (i) - Mathematical Reasoning
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is prime Contrapositive It a positive integer has divisor other than 1 and itself , then it is not prime | {"url":"https://www.teachoo.com/2959/654/Misc-2---State-converse-and-contrapositive-of-each/category/Miscellaneous/","timestamp":"2024-11-09T20:50:43Z","content_type":"text/html","content_length":"120898","record_id":"<urn:uuid:b8275524-296b-4fca-a435-14a1c603b0a1>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00543.warc.gz"} |