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Far Hills, We Have a Problem
Introduction – This paper explores problems with the United States Golf Association (USGA) Handicap System. These problems stem from a failure to understand the theory behind the Slope Handicap
System, and the dearth of USGA-sponsored research on handicap issues for the past 20 years. This in turn has led the USGA to make changes in the handicap system that reduce the equity of competition.
The device chosen to illustrate these problems is a case study involving Section 10-3, Reduction of Handicap Index Based on Exceptional Tournament Scores, of the Handicap System. The following
example and question were posed to the USGA:
Player A plays two tournaments at a course with a Slope Rating of 150. His two net scores are five strokes under the Course Rating.
Player B plays two tournaments at a course with a Slope Rating of 87. His two net scores are also five strokes under the Course Rating.
Assume these T-scores are the only ones in each player’s file. Under Section 10-3, Player A receives no reduction in index. Player B’s index is reduced by 4.8. Why is the player on the "easier"
course treated so harshly?
The problems are examined in three parts. First, the USGA’s response is analyzed for its understanding of the Slope Handicap System. The USGA’s answer is found to be unresponsive at best and invalid
at worst. Second, a flaw in the Slope Handicap System is demonstrated. This flaw is partially responsible for the counter-intuitive result of the above example, and its existence has been steadfastly
denied by various officials at the USGA and state golf associations. Third, the mistakes made in the development of the Handicap Reduction Table of Section 10.3 are documented. The USGA changed the
vocabulary used in the table to be consistent with the Slope Handicap System (e.g., “strokes under the course rating” became “index minus average tournament differential.” The USGA didn’t know or
didn’t care that these two concepts of exceptional performance are not synonymous. This mistake has led to inequity and the counter-intuitive result of the example. In a concluding section, the state
of research at the USGA is assessed and recommendations for improvement made.
The USGA Response – Scott Hovde, Manager of Course Rating and Handicap Education at the USGA, made the following response in an attempt to answer the questions posed in the example:[1]
As Slope Rating is a relative measurement of difficulty (not absolute), essentially an indication of how scores are spread out…a performance of 5 under the Course Rating on a Slope Rating of 87 is
more significant. On a course with an 87 Slope Rating, scores are expected to spread out at a rate of .77 (87/113…rounded) per difference of 1.0 in the Handicap Index. To beat a scratch golfer (who
is a very good player) by 5 strokes on a course where scores are very tight for players of all handicap levels, is more difficult than doing it on a course with a 150 Slope Rating, which spreads out
scores by 1.32 per 1.0 in the Index.
The advantage of a player with a lower Index grows as Slope Rating goes up and shrinks as Slope Rating goes down, regardless of which side of scratch a player is on.
Hovde argues scores are very tight on courses with low slope ratings and therefore performances of 5-under the Course Rating are more difficult to achieve. There is no empirical or theoretical basis
for such a claim. Papers proposing the Slope System only argue a player’s average score increases with the Slope rating.[2] These papers do not argue the Slope Rating affects the distribution of a
player’s score.
Hovde next argues the advantage of a player with a lower Index grows as Slope Rating goes up and shrinks as Slope Rating goes down. This is true by the definition, was never in question, and is
irrelevant to the question posed in the example. The question is whether indices are assigned properly, and not whether the handicap calculation based on those indices is correct. Why does the USGA
consider the player who scores 5-under the Course Rating at the PGA West Stadium Course (Slope Rating = 150) less skilled than a player scoring 5-under the Course Rating at the Oasis (Slope Rating =
87)? This question is examined next.
Flaw in the Slope Handicap System – To understand the flaw in the theory behind the Slope Handicap System, it is important to review its theoretical underpinnings. The Slope Handicap system was
introduced by the USGA in order to bring more equity to the game. The previous handicap system had computed handicaps by measuring the difference between a player's score and the course rating. It
was believed, however, the course rating did not properly measure the difficulty of the course for the higher handicapper.
The theory behind the Slope System is illustrated in the figure below:[3]
Figure - Player’s Score at Panther Mountain and Perfect Valley Versus Perfect Valley Handicap
The figure plots players' expected scores (defined here as 96 percent of the average of the ten best out of the last twenty rounds) on a tough course named Panther Mountain (slope rating = 155)
against their handicaps on a course of average difficulty named Perfect Valley (slope rating = 113). For example, a 10-handicap at Perfect Valley would be expected to have a score of 83.7 at Panther
Mountain or 13.7 strokes over the course rating of 70. Also shown in the figure is the score on Perfect Valley plotted against a player's Perfect Valley Handicap. For convenience the course rating at
each course is assumed to be the same (70), though that is not necessary to the argument.
Therefore, as illustrated in the figure, players from Perfect Valley with handicaps greater than scratch (i.e., 0 handicap), will score higher at Panther Mountain than they will at their home course.
The Slope Handicap System increases these players' handicaps at Panther Mountain reflecting the difficulty of the course. While the USGA has never presented any empirical evidence of this “slope
effect,” it does make intuitive sense. Obstacles at Panther Mountain such as length, water hazards, and rough may cause problems for bogey golfers that are not incorporated in the Course Rating.
Perfect Valley players with plus handicaps (i.e., less than scratch), however, are expected to score lower at Panther Mountain than at Perfect Valley. Therefore, these players will have their
handicaps reduced when they play at Panther Mountain. The Slope Handicap System assumes a course with a high slope rating is relatively easy for players with indices less than scratch. There is no
empirical evidence indicating this assumption is valid. Nor is there any theoretical reason to believe a player who scores 5-under the Course Rating at Perfect Valley is a much better player than one
who scores 5-under at Panther Mountain.
The problem arises because the USGA chose to set the scratch golfer as the standard. Had the standard been set at the “perfect golfer” (e.g., a +10) there would be no negative differentials and all
players at Perfect Valley would have their handicap stay the same or increase when traveling to Panther Mountain.[4]
Even if negative scoring differentials were eliminated, however, the peculiar result of the example would still exist. It is a mistake made in the development of the Handicap Reduction Table that
leads to the much harsher treatment of the player at the course with the low Slope Rating. We turn to this next.
Handicap Reduction Table – An abbreviated Handicap Reduction Table is shown below:
Handicap Reduction Table (Abbr.)
│Avg. of 2 Lowest T-Score Differentials Below Index │ Number of Eligible Tournament Scores │
│ ├──────┬──────┬──────┬──────┬──────────┤
│ │ 2 │ 3 │ 4 │ 5-9 │ 10-19 │
│ 4.0-4.4 │ 1.0 │ │ │ │ │
│ 4.5-4.9 │ 1.8 │ 1.0 │ │ │ │
│ 5.0-5.4 │ 2.6 │ 1.9 │ 1.0 │ │ │
│ 5.5-5.9 │ 3.4 │ 2.7 │ 1.9 │ 1.0 │ │
│ 6.0-6.4 │ 4.1 │ 3.5 │ 2.8 │ 1.9 │ 1.0 │
│ 6.5-6.9 │ 4.8 │ 4.3 │ 3.7 │ 2.9 │ 2.0 │
The table shows the higher the tournament differential (Index - Average of 2 lowest tournament differentials), the greater the possible reduction in index. The problem is there is no research
indicating the reduction should be a function of the tournament differential. The basic research was done by F.P. Engel who used Bogevold’s data in 1983 to determine the likelihood of a player
recording a low negative differential defined as net score minus the course rating.[5] Dean Knuth, former Senior Director of Handicapping at the USGA, developed a probability table for a player’s two
best scores to beat his handicap.[6] Even though Knuth is credited with inventing the Slope Handicap System, he used strokes below course rating and not index differentials in his analysis. In a
recent interview with the Wall Street Journal,[7] Knuth said “the odds of a mid-range player shooting eight strokes better than his handicap are 1 in 1,138.”[8] In other words, the handicap research
done under Knuth--which practically speaking is all of the research done by the USGA--viewed exceptional scores as strokes below the Course Rating. Knuth’s work estimates the probability of being
5-under the course rating is independent of the Slope Rating. This makes intuitive sense as well. A player’s score and handicaps are not affected by the Slope Rating if a player only plays at home.
If scores are normally distributed with a constant standard of deviation, a player has an equal chance of scoring n-strokes under the course rating whether the Slope Rating is 87 of 150.
So why did the USGA change “Strokes Under Course Rating” to “Index - Average Tournament Differential ” as the measure of exceptional performance? The two culprits were probably sloth and ignorance.
Sloth kept the USGA from doing any new research on the problem. When the current Appendix E, Exceptional Tournament Score Probability Table, of the Handicap System is based on 30-year old data, it is
a good indication no one in the Handicap Section of the USGA is working overtime.[9]
Ignorance probably guided someone at the USGA to make the change thinking it made little difference. According to Appendix E, a net differential 8.0 has a 1 in 1138 chance of occurring. A player
would have a net differential of 8.0 if his net score was 10.6 strokes under the Course Rating at a course with a Slope Rating of 150 or if his net score was 6.2 strokes under the Course Rating at a
course with a Slope Rating of 87. To believe those two events are equally likely, one has to suspend all judgment and disregard the bulk of USGA research. Apparently, that was no problem for some
USGA staffer and the Handicap Procedures Committee.
Conclusions and Recommendations – This paper has identified a flaw in the Slope Handicap System and problems with Sec. 10.3 of the Handicap System. The flaw is not a major impediment to equity since:
1. The flaw primarily affects players with plus indices who constitute a very small minority of all golfers,
2. Plus indices are typically not concerned about their handicap in inter-club net scoring events,
3. The flaw does not affect intra-club events where players’ handicaps are based primarily on scores from their home course.
Moreover, to correct the flaw would be very difficult, and only confuse the golfing community that is already sufficiently addled by the present Slope Handicap System.
It is recommended, however, that the USGA 1) update the research on exceptional performance, and 2) re-examine Sec. 10-3 for efficacy and effectiveness. Both tasks need to be completed to ensure an
equitable handicap system.
Updating Research on Exceptional Performance - Both Appendix E and the Handicap Reduction Table appear to be based on data and analysis that are 30 to 40-years old. Due to limits in computing power
back then, the sample size was probably small. With the advent of GHIN and modern computers, research no longer has such restraints. It would not be a difficult task, for example, to find the
percentage of golfers who have a score of 7-under the course rating in their last 20 scores. This empirical finding could be used to revise the probabilities presented in Appendix E.
Re-Examine sec. 10.3 - As discussed above, equivalent performances (i.e., n-strokes under the course rating) are treated differently depending on the Slope Rating. It is doubtful the USGA can
demonstrate Sec. 10-3 as presently constituted is equitable. Armed with new results on exceptional performance, the USGA would be in position to construct an equitable Handicap Reduction Table. This
re-examination should also review if Sec. 10-3 is serving its intended purpose (e.g., Is the lack of uniformity in the application of Section 10-3 affecting the equity of competition? Is the present
method of measuring tournament performance equitable?)[10]
Research on golf performance has undergone resurgence recently. Pope and Schweitzer have analyzed success rates when players are putting for par and birdie.[11] Minton has explored the effect of
randomness in golf and the mathematical strategies behind the sport.[12] Wells and Skowronski examined the evidence on whether players choke under pressure.[13] What all of these studies have in
common is that they were performed with data made available by the PGA Tour. There is no similar set of studies dealing with equity in amateur golf or using data from the USGA.
The dearth of research on handicapping is due to policies of the USGA. The USGA has not directed much attention or resources to research in years. The present policy of restricting public access to
its research has also not served the USGA well. Without external peer review, the probability of mistakes similar to those made in Sec. 10-3 and Appendix E will continue to be high. The USGA needs
something equivalent to perestroika to breakdown the insular barriers between it and the research community. If history is any judge, such an “opening” is decades off.
[1] E-mail from Scott Hovde to the author, April 16, 2012.
[2] R.C. Stroud and L.J. Riccio, Mathematical underpinnings of the slope handicap, in Science and Golf: The Proceedings of the First World Scientific Congress of Golf, Rutledge, Chapman and Hall,
London, 1990, pp. 135-140. D. Knuth, A two parameter golf course rating system, Ibid, pp. 141-146.
[3] The figure is a slight modification from that found in R.C. Stroud and L.J. Riccio, loc. cit. Stroud and Riccio wisely did not show or discuss the effect on plus handicaps at Perfect Valley.
[4]The USGA did not believe it had a problem even when the Slope Handicap was first adopted. Dean Knuth, Senior Director of Handicapping wrote “The hurdle of negative differentials could be bypassed
completely by setting the course rating at a low enough level to ensure their non-existence…The effect would be exactly the same as what we have and want now. What really is happening is that ability
differences are being magnified or contracted, as appropriate.” (Letter from Knuth to Russ Palmer, Executive Director of the Connecticut State Golf Association, Sept. 13, 1994.) The effect, of
course, would not be the same.
[5] D.L. Knuth, F.J. Scheid, and F.P. Engel, Outlier identification procedure for the reduction in handicap, Science and Golf II: Proceedings of the World Scientific Congress of Golf, E & F Spon,
London, 1994, p. 230.
[7] John Paul Newport, Fighting Back Against Sandbaggers, Wall Street Journal, July 2, 2011. Also see John Paul Newport, The Genius of Handicapping, Wall Street Journal, November 1, 2008 where
Newport cites the odds(sic) of beating your handicap by three strokes is 1 in 20.
[8] Kelly Neely, a member of the USGA Handicap Procedures Committee and Senior Director of Handicapping of the Oregon Golf Association wrote “Your chances of beating your handicap by 10 strokes in
one round is (one in) 37,000.” See www.oga.org/index.php/handicapping. Apparently, Ms. Neely relied on an old version of Appendix E.
[9] As an another example of sloth, Appendix E formerly showed the probability of a 13-21 handicap player having a net differential of 0 or better was 1 in 6. For every other index range, the
probability was 1 in 5. When the USGA was alerted to discrepancy, it changed this probability to 1 in 5 for all players in the 2012 Handicap Manual. No new research was done. Someone just replaced a
6 with a 5—problem solved. Equally disturbing, the 2012 Handicap System Manual contains typographical and editorial errors in Appendix E. When these were brought to the attention of the USGA, Mr.
Hovde blamed the errors on the printer. E-mail to author, February 9, 2012.
[10] Under the old Section 10-3(1994), the penalty for exceptional performance was a function of how well you played in tournaments. Under the current version of Section 10-3, the penalty is a
function of how well you played relative to your current index. This leads to some peculiar and perhaps inequitable results. See Dougharty, Laurence, Reduction in Index for Exceptional Tournament
Performance: Some Conceptual Problems, Measuring Golf Performance, Golf Economics, Manhattan Beach, CA, 2005.
[11] Pope, Devin and Schweitzer, Maurice, Is Tiger Woods Loss Averse? Persistent Bias in the Face of Experience, Competition, and High Stakes, American Economic Review, 101 (February 2011), pp.
[12] Minton, Roland, Golf by the Numbers, The Johns Hopkins University Press, Baltimore, MD, forthcoming June, 2012.
[13] Brett Wells & John Skowronski, Evidence of Choking Under Pressure on the PGA Tour, Basic and Applied Social Psychology, March/April 2012, Pages 175-182. | {"url":"http://www.ongolfhandicaps.com/2012/06/far-hills-we-have-problem.html","timestamp":"2024-11-02T17:37:02Z","content_type":"text/html","content_length":"151059","record_id":"<urn:uuid:2e1b626a-7503-49e0-86a9-dcb22f1cbba9>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00416.warc.gz"} |
How Many Slices in a 14-Inch Pizza? - Foods Guy
How Many Slices in a 14-Inch Pizza?
*This post may contain affiliate links. Please see my disclosure to learn more.
Ordering the right amount of pizza for a party or event is crucial. There is nothing worse than having starving guests or sitting with enough leftover pizza for a month!
But how many slices come in a 14-inch pizza? And how many people does that feed? Many factors influence the portion size and serving amount you should use. But on average, a 14-inch pizza has between
8-12 slices. This will feed between 3-5 people.
Okay, so how big are these slices then? This requires a little bit of math. But don’t worry. We’ve made it incredibly easy!
Today, we will show you everything you need to know about pizza slices. From slice amount to slice surface area, dimensions for packaging, and portion size for parties – we will cover it all!
These calculations can be used to calculate the dimensions and surface area of various-sized slices from different-sized pizzas.
How Big Is a 14-Inch Pizza?
When people refer to a “14-inch pizza,” it specifically references the diameter of a round pizza. That means a 14-inch pizza is 14 inches in diameter (across). Easy enough to understand, right?
But why is it important?
Understanding all of the size aspects of a pizza will help you better understand how many slices you will get from it and ultimately how many portions. This can help you calculate portion amounts for
events and even cost per portion, a crucial part of catering and party planning.
So, what do 14 inches mean?
A 14-inch (35.5cm) pizza is traditionally considered to be a large pizza. However, in some parts of the country and world, it’s classified as an extra-large pizza. The difference in classification
has to do with how much people eat in the area.
How Many Slices Are in a 14-Inch Pizza?
Now, the most important question we will answer today is: how many slices can you get from a 14-inch pizza? But the answer may not be as simple as you would like it to be.
On average, people divide a 14-inch pizza into 10 slices.
But again, depending on where you are and what that region’s portion sizing looks like, you may find 8 slices or 12 slices.
Plus, if you are catering for a specific event, it may even be smaller portions that will give you between 16-18 slices from that one pizza.
So the exact amount of slices per 14-inch large pizza depends. If you have to calculate numbers for an event, we’d recommend working on the traditionally recommended 10 slices.
How Big Are the Slices?
The size of the slice obviously depends on the number of slices you are cutting. To calculate the exact size, we will need to get into some maths!
Calculating the Surface Area of a Pizza
The formula for calculating the surface area of a circle (round pizza) is as follows: π x r²
• As we know, the diameter of a 14-inch pizza is 14 inches.
• The radius of a 14-inch pizza can be calculated by dividing the diameter by 2. In this case, the radius is 7 inches.
π x r²
= 3.1415 x 7²
= 153.94 in² (square inches)
Calculating the Surface Area of Each Slice
Working out the surface area of each slice is easy once you have the surface area of the entire pizza.
Again, the surface area of a 14-inch pizza is 153.94 in².
To calculate the surface area of the slices, divide this area by the number of slices you want to cut.
As you will see below, the more slices you cut, the smaller the surface area will be.
Calculating the Dimensions of the Slices
The main reason you would need the dimensions of pizza slices is for packaging or serving purposes. But it is an important point to discuss.
Step 1: Calculate the Circumference of the Pizza
The circumference of a circle can be calculated using the following formula: C = 2 x π x r
We already know the radius of a 14-inch pizza, which is 7 inches. Therefore;
C = 2 x π x r
C = 2 x 3.1415 x 7
C = 43.98 inches
Step 2: Divide the Circumference by Slices
This step will give you the size of the curved edge (crust) of the pizza. This is the most difficult measurement to get. But with our steps, it’s very easy.
Step 3: Dimensions of 1 Slice
Now, the dimensions of 1 slice consist of 3 measurements.
Two of these measurements will be the radius of the pizza. The two slices run straight from the center point to the edge.
The third dimension you just calculated is the curved edge.
So, here is the size of 1 slice of pizza from varying slice amounts.
How Many People Can You Feed From a 14-Inch Pizza?
Calculating the surface area and dimensions of pizza slices is much easier than calculating portion sizes.
Because how much do people eat? It’s a question whose answer has many determining factors.
Where are the people located? What is their background? Is pizza the only dish being served? What time of the day is it? What type of pizza is it?
Think of it this way:
• The elderly and children eat far less food than healthy teenagers or young adults.
• People tend to eat less food in the morning than they do at lunch or dinner.
• If many other appetizers or courses are being served, the portion size per dish will be smaller. People have more options to fill up.
• In some countries and even continents (like America for example) people eat far more than in others (like some parts of Europe). If portion sizes are bigger, people may need more.
• A plain Margarita pizza won’t fill guests up as quickly as a fully-loaded meat lovers pizza will. The same goes for the thick-crust versus thin-crust pizza types. One fills you up quicker than
the other.
So, how many people can you feed from a 14-inch pizza?
On average, we’d say between 3-5 people. The amount of slices you cut won’t change how much someone necessarily eats.
Whether they have 2 small slices or 1 massive slice, they have the same stomach capacity. | {"url":"https://foodsguy.com/slices-14-inch-pizza/","timestamp":"2024-11-11T17:13:01Z","content_type":"text/html","content_length":"295417","record_id":"<urn:uuid:0812e290-0a1f-4a60-86f0-d902e7796784>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00262.warc.gz"} |
AO for Groups is Here!
Announcements/RE: AO for Groups is Here! 07-25-2017, 01:40 AM
I have a question about placement. Prior to AO for Groups/2.0 going live, I was going to start my 7 year olds in Year 1 and my 9 year olds in Year 2. My 7 year olds are not reading yet and my 9 year
olds are not necessarily "on grade level" with their reading.
I'm considering using the AO 2.0 since I've got a group of my own here to simplify things a bit. I'm wondering A) Will it even simplify? B) Should I put my 9 yo's in Form 2A or do 1A with everyone?
If we do separate forms what (if anything) can we do together or would effectively I be running 2 separate homeschools? | {"url":"https://amblesideonline.org/forum/showthread.php?tid=31823&pid=474320&mode=threaded","timestamp":"2024-11-15T04:39:01Z","content_type":"application/xhtml+xml","content_length":"49619","record_id":"<urn:uuid:0b278eba-a761-41c4-aea7-cec3c33d922b>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00396.warc.gz"} |
Strong coupling physics, holography and cosmology
Dear all,
I am pleased to invite you to my PhD defense which will take place on November 15th at 14:00 in room 454A Luc Valentin, 4th floor.
I will present my work on strong coupling physics, holography and cosmology which has been done at APC under the supervision of Francesco Nitti.
The defense will be followed by a pot in the salle de convivialité.
Best regards,
Valentin Nourry
zoom link:
Strong coupling physics, holography and cosmology
This thesis studies the interplay between gravity and quantum effects of a conformal field theory (CFT).
In the first part of this thesis, the main goal is to investigate the stability of maximally symmetric spacetimes under small metric perturbations. We use AdS/CFT as a non-perturbative tool to
compute the 2-point function of the CFT stress-energy tensor in the presence of a dynamical metric. Interpreting the framework of semi-classical gravity as an effective field theory (EFT), we compute
the spectral functions of metric perturbations around maximally symmetric backgrounds. These spectral functions contain all the necessary information to settle the problem of stability in the
validity regime of the EFT. By studying the poles of these functions, we track the presence (or absence) of instabilities whether they are ghost-like or tachyonic. These instabilities are located on
the whole parameter space of the problem (the background curvature, the central charge of the CFT and the quadratic curvature coefficients of the gravity action). This project aims at distinguishing
the contributions of the CFT and the gravity action. We also compare the instabilities coming from different sectors of the scalar and tensor decomposition of the metric perturbations.
The second part of this thesis is dedicated to the study of holographic 2-point functions of a CFT defined in AdSd. The most generic system is made of two different CFTs which are connected through a
bulk AdSd+1 space whose boundary is made of two distinct AdS_d patches. The coupling between these two CFTs is responsible for breaking conformal invariance which is recovered by turning off one the
two CFTs. Alternatively, we can define boundary conditions on a tensionless end-of-the-world brane to impose conformal invari- ance at the level of holographic correlators. We find an expression for
the non-local action induced on such brane. This brane, as well as the AdS_d spaces, extend to the conformal boundary of AdS_{d+1}. We then investigate the possibility of defining sources there, but
leave the solution of this problem for future work.
Keywords: conformal field theory, stability, gravity, holography, AdS/CFT, effective field theory
Wednesday, 15 November, 2023 - 14:00 to 18:00
Nom/Prénom // Last name/First name:
Equipe(s) organisatrice(s) / Organizing team(s): | {"url":"https://www.apc.univ-paris-diderot.fr/APC_CS/en/node/5316","timestamp":"2024-11-07T12:13:05Z","content_type":"text/html","content_length":"36141","record_id":"<urn:uuid:698101fd-468f-462c-9709-696e11c0a8b8>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00312.warc.gz"} |
Barycentric dual grids
Some elements are defined on a barycentric dual grid. This grid is defined by taking a mesh of triangles:
Lines are then added connecting every vertex of each triangle to the midpoint of the opposite edge:
The cells in the dual grid are then defined as the union of all the triangles adjacent to one of the vertices in the original mesh:
In DefElement, regular polygons centred at the origin are used as reference elements of the dual grid. | {"url":"https://defelement.com/barycentric-dual-grid.html","timestamp":"2024-11-07T16:48:37Z","content_type":"text/html","content_length":"6304","record_id":"<urn:uuid:0dcaf031-67fc-4dcf-94e3-8afaf883d977>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00122.warc.gz"} |
Inductor is an electrical component that stores energy in magnetic field.
The inductor is made of a coil of conducting wire.
In an electrical circuit schematics, the inductor marked with the letter L.
The inductance is measured in units of Henry [L].
Inductor reduce current in AC circuits and short circuit in DC circuits.
Inductor picture
Inductor symbols
│Inductor ││
│Iron core inductor ││
│Variable inductor ││
Inductors in series
For several inductors in series the total equivalent inductance is:
L[Total] = L[1]+L[2]+L[3]+...
Inductors in parallel
For several inductors in parallel the total equivalent inductance is:
Inductor's voltage
Inductor's current
Energy of inductor
AC circuits
Inductor's reactance
X[L] = ωL
Inductor's impedance
Cartesian form:
Z[L] = jX[L] = jωL
Polar form:
Z[L] = X[L]∠90º
See also: | {"url":"https://jobsvacancy.in/electric/inductor.html","timestamp":"2024-11-03T21:57:32Z","content_type":"text/html","content_length":"6358","record_id":"<urn:uuid:2d8058bd-4f33-468e-bf7e-3d9c738b5a5a>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00494.warc.gz"} |
Relationships among Estimation Criteria
There is always some arbitrariness to classify the estimation methods according to certain mathematical or numerical properties. The discussion in this section is not meant to be a thorough
classification of the estimation methods available in PROC CALIS. Rather, classification is done here with the purpose of clarifying the uses of different estimation methods and the theoretical
relationships of estimation criteria.
Assumption of Multivariate Normality
GLS, ML, and FIML assume multivariate normality of the data, while ULS, WLS, and DWLS do not. Although the ML method with covariance structure analysis alone can also be based on the Wishart
distribution of the sample covariance matrix, for convenience GLS, ML, and FIML are usually classified as normal-theory based methods, while ULS, WLS, and DWLS are usually classified as
distribution-free methods.
An intuitive or even naive notion is usually that methods without distributional assumptions such as WLS and DWLS are preferred to normal theory methods such as ML and GLS in practical situations
where multivariate normality is doubt. This notion might need some qualifications because there are simply more factors to consider in judging the quality of estimation methods in practice. For
example, the WLS method might need a very large sample size to enjoy its purported asymptotic properties, while the ML might be robust against the violation of multi-normality assumption under
certain circumstances. No recommendations regarding which estimation criterion should be used are attempted here, but you should make your choice based more than the assumption of multivariate
Contribution of the Off-Diagonal Elements to the Estimation of Covariance or Correlation Structures
If only the covariance or correlation structures are considered, the six estimation functions,
• The functions
• The functions
The (Jöreskog and Sörbom; 1985, p. 23) shows that LISREL groups the
• Relationship between DWLS and WLS:
PROC CALIS: The
LISREL 7: This is not the case.
• Relationship between DWLS and ULS:
LISREL 7: The
PROC CALIS: To obtain the same results with
Because the reciprocal elements of the weight matrix are used in the discrepancy function, the off-diagonal residuals are weighted by a factor of 2.
ML and FIML Methods
Both the ML and FIML methods can be derived from the log-likelihood function for multivariate normal data. The preceding section Estimation Criteria mentions that
The two expressions differ only in the constant terms, which are independent of the model parameters, and in the formulas for computing the sample covariance matrix. While the FIML method assumes the
biased formula (with
The similarity (or dissimilarity) of the ML and FIML discrepancy functions leads to some useful conclusions here:
• Because the constant terms in the discrepancy functions play no part in parameter estimation (except for shifting the function values), overriding the default ML method with VARDEF=N (that is,
• Because the FIML function is evaluated at the level of individual observations, it is much more expensive to compute than the ML function. As compared with ML estimation, FIML estimation takes
longer and uses more computing resources. Hence, for data without missing values, the ML method should always be chosen over the FIML method.
• The advantage of the FIML method lies solely in its ability to handle data with random missing values. While the FIML method uses the information maximally from each observation, the ML method
(as implemented in PROC CALIS) simply throws away any observations with at least one missing value. If it is important to use the information from observations with random missing values, the
FIML method should be given consideration over the ML method.
See Example 25.13 for an application of the FIML method and Example 25.14 for an empirical comparison of the ML and FIML methods. | {"url":"http://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/statug_calis_sect074.htm","timestamp":"2024-11-13T01:27:10Z","content_type":"application/xhtml+xml","content_length":"27887","record_id":"<urn:uuid:67382f7f-db60-4cd8-8944-fdf6c3afa3a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00418.warc.gz"} |
The Bubble Factor and the Bubble Factor Table | ICMIZER Help Center
The display of bubble factors is a major feature of ICMIZER 3. Whenever you perform a Nash calculation you will see differently colored circles next to each opponent which will indicate their bubble
factor against you. The bigger the bubble factor, the more pressure you are putting on this opponent meaning that they will need more equity to call your pushes (as the most obvious example of
applying the bubble factor).
Here is the sample screenshot with bubble factors versus big stack on the bubble of a classic 9 max SNG tournament with 3 payouts:
As you can see, BTN has the smallest bubble factor and can actually call with the widest range (9.4%) versus our wide push.
SB, who has a larger stack, has a larger bubble factor and can only call with a 3.5% range, and BB who has the biggest stack and has a huge bubble factor of 2.7 versus us can only call with top 2.7%
hands against our almost any-two push.
The Bubble Factor concept is not very intuitive and can be hard to grasp for beginner players.
So let's start slow and build it up from the simpler bricks. Let's say we are in Chip EV mode, either playing cash or tournament with 1 prize, like Spin & Go.
In this case, the absolute change in our stack worth of doubling up or busting from the tournament is the same. So if we double up from 500 chips to 1000 or bust, technically both outcomes change our
stack worth by exactly 500 chips. Since both outcomes have the same absolute value (but a different sign, +500 or -500), winning 50% of the time would be enough to make us indifferent and justify the
potential all-in.
So, in this case, in order to calculate the bubble factor, we would take the absolute change of our stack if we lose (500) and divide it by absolute change if we win (500) and we get 500/500 = 1. So
in Chip EV mode or in tournaments with 1 prize the bubble factor is usually 1, you can play according to classic cash game pot odds.
Now in SNG or MTT tournaments where we have more than 1 payout, we have to use ICM to convert chips into their tournament prize equity value in order to compare decisions.
If he loses to UTG who has him covered, he gets nothing. If he wins, however, he doesn't even guarantee himself an ITM (although this win is big) and his new stack will be worth 40.16%: https://
Now its time to meet the Bubble Factor formula:
Bubble Factor = ChangeInOurTourneyEquityIfWeLose / ChangeInOurTourneyEquityIfWeWin or (EvBegin - EvLose)/(EvWin - EvBegin).
We can now calculate the bubble factor according to this formula: (29.23 - 0)/(40.16-29.23)= 29.23 / 10.93 = 2.67
So in this tournament situation, winning 4000 chips only adds 10.93% of the tournament prize pool to BB player. But losing costs him 29.23% of the prize pool, even though the chips that he wins or
loses are the same.
If you click only any bubble factor you will see the detailed bubble factor table which shows bubble factors of each player against each other player. As we see, the bubble factor for BB versus UTG
is indeed 2.67, so our math was correct.
How can we use Bubble Factors?
Great question! We can take the bubble factor and quickly estimate how much equity we need to call an all-in against a specific opponent. With bubble factor 1 we can say for simplicity that we will
need 50% equity versus their range.
If the bubble factor is bigger, we can use the following formula to estimate our required all-in equity:
Equity = BubbleFactor / (BubbleFactor + 1) * 100%
So with 2.67 bubble factor we can estimate Equity as 2.67 / (2.67 + 1) = 0.7275 or 73%.
We hit Calculate and see that our calling range is 99+.
With 99 being the worst calling hand, let's check the details for this calculation, namely how much equity 99 have versus CO's huge pushing range.
As we can see, it is pretty close to 72%, which we've calculated earlier. In fact, it is just slightly less because we weren't taking SB and antes into account with our bubble factor analysis. | {"url":"https://support.icmpoker.com/en/articles/3925388-the-bubble-factor-and-the-bubble-factor-table","timestamp":"2024-11-06T13:39:46Z","content_type":"text/html","content_length":"63791","record_id":"<urn:uuid:42e941e5-410d-4489-bc78-fb5499ea68cf>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00240.warc.gz"} |
Stresses with Gradients in an Unjointed Medium: Uniform Material
Variation in stress with depth cannot be ignored near the ground surface – the model gravity command is used to inform 3DEC that gravitational acceleration operates on the model. It is important to
understand that the model gravity command does not directly cause stresses to appear in the model; it simply causes body forces to act on all gridpoints of deformable blocks (or centroids of rigid
blocks). These body forces correspond to the weight of material surrounding each gridpoint. If no initial stresses are present, the forces will cause the material to move (during stepping) in the
direction of the forces until equal and opposite forces are generated by zone stresses. Given the appropriate boundary conditions (e.g., fixed bottom, roller side boundaries), the model will, in
fact, generate its own gravitational stresses compatible with the applied gravity. However, this process is inefficient, since many hundreds or thousands of steps may be necessary for equilibrium. It
is better to initialize the internal stresses such that they satisfy both equilibrium and the gravitational gradient. The block insitu topography command is usually the best approach. The internal
stresses must also match boundary stresses at stress boundaries.
Even though the boundary and in-situ stresses are specified to produce a force balance, some cycling of the model is normally required. This is because the boundary forces are only applied at the end
of a cycle; a small force imbalance is produced by the in-situ stresses. Usually, this imbalance is reduced within a few hundred cycles.
Consider, for example, a 20 m × 20 m × 20 m box of homogeneous unjointed material at a depth of 200 m underground, with fixed base and stress boundaries on the other five sides. The example below
produces an equilibrium system for this problem condition:
Initial stress state with gravitational gradient
model new
model large-strain on
block create brick 0,20 0,20 0,20
block zone generate edgelength 4.0
block zone cmodel assign mohr-coulomb
block zone property density 2500 bulk 5e9 shear 3e9 friction 35
model gravity 0 0 -10
; install initial stress forigin block with top 200 m below the surface
block insitu topography ratio-x 0.5 ratio-y 0.5 overburden [-200*10*2500]
; apply stresses to sides
block face apply stress -2.75e6,-2.75e6,-5.5e6 0,0,0 &
gradient-z 1.25e4, 1.25e4, 2.5e4 0,0,0 range position-x 0.0
block face apply stress -2.75e6,-2.75e6,-5.5e6 0,0,0 &
gradient-z 1.25e4, 1.25e4, 2.5e4 0,0,0 range position-x 20.0
block face apply stress -2.75e6,-2.75e6,-5.5e6 0,0,0 &
gradient-z 1.25e4, 1.25e4, 2.5e4 0,0,0 range position-y 0.0
block face apply stress -2.75e6,-2.75e6,-5.5e6 0,0,0 &
gradient-z 1.25e4, 1.25e4, 2.5e4 0,0,0 range position-y 20.0
; apply stress to top
block face apply stress 0,0,-5.0e6 0,0,0 range position-z 20.0
; fix bottom
block gridpoint apply velocity-z = 0.0 range position-z 0.0
model solve
In this example, horizontal stresses and gradients are equal to half the vertical stresses and gradients, but they may be set at any value that does not violate the yield criterion (Mohr-Coulomb, in
this case). After preparing a data file such as the one above, the model should be cycled to check that an equilibrium state is reached. If material failure does occur (e.g., reduce phi = 10º), this
will show as an unbalanced force magnitude that is roughly the same order of magnitude as the applied loading.
Was this helpful? ... Itasca Software © 2024, Itasca Updated: Aug 13, 2024 | {"url":"https://docs.itascacg.com/itasca910/3dec/docproject/source/modeling/problemsolving/stresses_unjointeduniform.html","timestamp":"2024-11-03T12:31:55Z","content_type":"application/xhtml+xml","content_length":"18784","record_id":"<urn:uuid:124e957e-5a2d-4de6-8c3a-995b1d2c1208>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00030.warc.gz"} |
cotangent in python
How to calculate cotangent in python
To calculate the cotangent in Python you can use the trigonometric function cot() of the mpmath module.
from mpmath import cot
The parameter x is the angle in radians.
The cot() function returns the cotangent of the angle.
What is the cotangent? In trigonometry the cotangent is the ratio between the cosine and the sine. The cotangent tends to infinity at an angle equal to zero. It is nothing in a 90 ° angle.
Alternative methods
The cotangent can also be calculated using the inverse of the tangent
from math import tan
Alternatively, using the ratio of the cosine to the sine.
from math import cos,sin
Example 1
To calculate the cotangent of an angle of 45 °
from math import radians
from mpmath import cot
The radians () function converts the angle measurement from sexagesimal to radians.
The cot() function calculates the cotangent.
The cotangent of an angle of 45 ° is equal to 1.
Example 2
The same result can be obtained by calculating the inverse of the tangent.
from math import tan
The output is as follows
Example 3
The cotangent can also be calculated as the ratio between the cosine and sine trigonometric functions of the same angle.
from math import cos,sin
The output is
Report an error or share a suggestion to enhance this page | {"url":"https://how.okpedia.org/en/python/how-to-calculate-cotangent-in-python","timestamp":"2024-11-07T13:15:24Z","content_type":"text/html","content_length":"13122","record_id":"<urn:uuid:a6680118-23cc-4f2c-994b-ec0dc5e1de0a>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00793.warc.gz"} |
Matrix Representation for SLR
Why matrices?
You might be wondering whether there are any other representations for linear regression that can be used to further simplify the solution. In the next video, Anjali will introduce you to the matrix
representation of linear regression and explain how it can be used to compute the model coefficients for linear regression models.
At 02:30, the dimensions are incorrectly written as (n,1) for the model coefficient vector. It should be (2,1).
So, in the video, you saw how the complex form of simple linear regression equations is converted to its matrix equivalent. Each observation can be represented with the following set of equations for
‘n’ observations:
The equations above have been converted to their matrix/vector equivalent as shown below:
The X matrix, i.e., the matrix with the predictors, is also known as the design matrix. Here, the first column in the design matrix is a column of 1’s for the intercept term β0 and the second column
contains the predictor values. There are n rows in the matrix for ‘n’ observations. The error vector consists of residuals for each observation, which is added to the product of the design matrix and
the parameter vector to obtain the response vector, Y.
You may want to refresh your memory of matrices by visiting the following link.
This matrix can be written in a very concise notation as:
Before delving further into the matrix representation, let’s try and understand some of the benefits of using matrices:
• Formulae become simpler, and more compact and readable.
• Code using matrices runs much faster than explicit ‘for’ loops.
• Python libraries, such as NumPy, help us build n-dimensional arrays, which occupy less memory than Python lists and computation is also faster.
Best parameter values through normal equations using matrices
In simple linear regression, we obtained the values of b0 and b1 by solving the normal equations using basic algebra. Let us see how can we use matrices for deriving the solution for β coefficients.
As we saw in the video, to get RSS, we multiply the error matrix by its transpose, as shown below:
Then, we differentiate RSS w.r.t. β and equate it to 0, as shown below:
Please note that we have skipped the differentiation step in the above derivation. It is not required to understand that here. The main objective is to understand that with simple matrix operations
you can find the value of the coefficients.
So, this is how we get our beta coefficients for the least RSS. The same expression can be used irrespective of the number of predictors or variables that are considered in the model. For example, we
can use this solution to find the coefficients for simple linear regression as:
The same equation can be used to find multiple coefficients present in Multiple Linear Regression. Now, in the forthcoming video, we will use this formula in our Python code and check whether or not
we get the same solution.
So, in the video, Anjali built a simple linear regression model on the marketing data set and verified the results obtained from the normal equations and from matrix calculations. So far, you have
built and verified a simple linear regression model. Now, as you are already aware, the next step is to build multiple linear regression models. In the next segment, you will learn about multiple
linear regression model.
Additional Reading: | {"url":"https://www.internetknowledgehub.com/matrix-representation-for-slr/","timestamp":"2024-11-09T22:46:57Z","content_type":"text/html","content_length":"82026","record_id":"<urn:uuid:ad4e142c-b527-4c1d-bcfb-3f8f92139255>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00167.warc.gz"} |
Free Fall Model
written by Andrew Duffy
The Free Fall model allows the user to examine the motion of an object in freefall. This is simply one-dimensional motion (vertical motion) under the influence of gravity.
The Free Fall model was created using the Easy Java Simulations (EJS) modeling tool. It is distributed as a ready-to-run (compiled) Java archive. Double clicking the ejs_bu_freefall.jar file will run
the program if Java is installed.
Please note that this resource requires at least version 1.5 of Java (JRE).
Subjects Levels Resource Types
Classical Mechanics
- Instructional Material
- Motion in One Dimension - Lower Undergraduate
= Curriculum support
= Acceleration - High School
= Interactive Simulation
= Gravitational Acceleration - Middle School
- Audio/Visual
= Position & Displacement - Upper Undergraduate
= Movie/Animation
= Velocity
Intended Users Formats Ratings
- Educators
- application/java
- Learners
Access Rights:
Free access
This material is released under a GNU General Public License Version 3 license.
Rights Holder:
Andrew Duffy, Boston University
EJS, Easy Java Simulations, acceleration, free fall, free fall simulation, gravity, position, position vs. time, velocity, velocity vs. time
Record Cloner:
Metadata instance created April 27, 2010 by Mario Belloni
Record Updated:
March 8, 2016 by wee lookang
Last Update
when Cataloged:
April 16, 2010
Other Collections:
Next Generation Science Standards
Crosscutting Concepts (K-12)
Patterns (K-12)
• Graphs and charts can be used to identify patterns in data. (6-8)
NGSS Science and Engineering Practices (K-12)
Analyzing and Interpreting Data (K-12)
• Analyzing data in 9–12 builds on K–8 and progresses to introducing more detailed statistical analysis, the comparison of data sets for consistency, and the use of models to generate and analyze
data. (9-12)
□ Analyze data using computational models in order to make valid and reliable scientific claims. (9-12)
Developing and Using Models (K-12)
• Modeling in 6–8 builds on K–5 and progresses to developing, using and revising models to describe, test, and predict more abstract phenomena and design systems. (6-8)
□ Develop and use a model to describe phenomena. (6-8)
• Modeling in 9–12 builds on K–8 and progresses to using, synthesizing, and developing models to predict and show relationships among variables between systems and their components in the natural
and designed worlds. (9-12)
□ Use a model to provide mechanistic accounts of phenomena. (9-12)
Using Mathematics and Computational Thinking (5-12)
• Mathematical and computational thinking at the 9–12 level builds on K–8 and progresses to using algebraic thinking and analysis, a range of linear and nonlinear functions including trigonometric
functions, exponentials and logarithms, and computational tools for statistical analysis to analyze, represent, and model data. Simple computational simulations are created and used based on
mathematical models of basic assumptions. (9-12)
□ Create or revise a simulation of a phenomenon, designed device, process, or system. (9-12)
□ Use mathematical or computational representations of phenomena to describe explanations. (9-12)
NGSS Nature of Science Standards (K-12)
Analyzing and Interpreting Data (K-12)
• Analyzing data in 9–12 builds on K–8 and progresses to introducing more detailed statistical analysis, the comparison of data sets for consistency, and the use of models to generate and analyze
data. (9-12)
Developing and Using Models (K-12)
• Modeling in 6–8 builds on K–5 and progresses to developing, using and revising models to describe, test, and predict more abstract phenomena and design systems. (6-8)
• Modeling in 9–12 builds on K–8 and progresses to using, synthesizing, and developing models to predict and show relationships among variables between systems and their components in the natural
and designed worlds. (9-12)
Using Mathematics and Computational Thinking (5-12)
• Mathematical and computational thinking at the 9–12 level builds on K–8 and progresses to using algebraic thinking and analysis, a range of linear and nonlinear functions including trigonometric
functions, exponentials and logarithms, and computational tools for statistical analysis to analyze, represent, and model data. Simple computational simulations are created and used based on
mathematical models of basic assumptions. (9-12)
AAAS Benchmark Alignments (2008 Version)
4. The Physical Setting
4B. The Earth
• 6-8: 4B/M3. Everything on or anywhere near the earth is pulled toward the earth's center by gravitational force.
4G. Forces of Nature
• 9-12: 4G/H1. Gravitational force is an attraction between masses. The strength of the force is proportional to the masses and weakens rapidly with increasing distance between them.
11. Common Themes
11B. Models
• 6-8: 11B/M1. Models are often used to think about processes that happen too slowly, too quickly, or on too small a scale to observe directly. They are also used for processes that are too vast,
too complex, or too dangerous to study.
• 6-8: 11B/M2. Mathematical models can be displayed on a computer and then modified to see what happens.
Common Core State Standards for Mathematics Alignments
Standards for Mathematical Practice (K-12)
MP.4 Model with mathematics.
High School — Algebra (9-12)
Creating Equations^? (9-12)
• A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential
Reasoning with Equations and Inequalities (9-12)
• A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
High School — Functions (9-12)
Linear, Quadratic, and Exponential Models^? (9-12)
• F-LE.1.b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
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A sofa is pushed with a force of 485 N in the of the sofa. If it is pushed 2.66 m, how much work is done on the sofa? | HIX Tutor
A sofa is pushed with a force of 485 N in the of the sofa. If it is pushed 2.66 m, how much work is done on the sofa?
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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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Distinct Elements in Streams and the Klee's Measure Problem | Department of CSE, IIT Hyderabad
Distinct Elements in Streams and the Klee's Measure Problem
Title Of the Talk: “ Distinct Elements in Streams and the Klee’s Measure Problem “
Speakers: Prof. Sourav Chakraborty (ISI, Kolkata) Host Faculty: Dr. Nitin Saurabh
date: : October 17 (Thursday); 11:30am – 12:30pm
Abstract: We will present a very simple streaming algorithm on F0 estimation that also caught the eye of Donald E. Knuth. In a recent article, Donald E. Knuth started with the following two
Sourav Chakraborty, N. V. Vinodchandran, and Kuldeep S. Meel have recently proposed an interesting algorithm for the following problem: A stream of elements ( (a_1, a_2, \ldots, a_m) ) is input, one
at a time, and we want to know how many of them are distinct. In other words, if ( A = { a_1, a_2, \ldots, a_m } ) is the set of elements in the stream, with multiplicities ignored, we want to know (
|A| ), the size of that set. But we don’t have much memory; in fact, ( |A| ) is probably a lot larger than the number of elements that we can hold in memory at any one time. What is a good strategy
for computing an unbiased estimate of ( |A| )?
Their algorithm is not only interesting, it is extremely simple. Furthermore, it’s wonderfully suited to teaching students who are learning the basics of computer science. (Indeed, ever since I saw
it, a few days ago, I’ve been unable to resist trying to explain the ideas to just about everybody I meet.) Therefore I’m pretty sure that something like this will eventually become a standard
textbook topic. This note is an initial approximation to what I might write about it, if I were preparing a textbook about data streams.”
This simple algorithm comes out of the first ever “efficient” streaming algorithm (from PODS 21) for the Klee’s Measure problem, which was a big open problem in the world of streaming for many years.
This work is based on joint works with N. V. Vinodchandran, and Kuldeep S. Meel across multiple articles, notable the following: Estimating the Size of Union of Sets in Streaming Models. PODS 2021
Estimation of the Size of Union of Delphic Sets: Achieving Independence from Stream Size. PODS 2022 Distinct Elements in Streams: An Algorithm for the (Text) Book. ESA 2022
Speaker Profile:
Sourav Chakraborty is a Professor in Computer Science at the Indian Statistical Institute, Kolkata, India. He finished his Bachelors in Mathematics from Chennai Mathematical Institute in 2003 and
then went on to do his Master’s (in 2005) and Ph.D. (in 2008) in Computer Science from the University of Chicago under the guidance of Prof. Laszlo Babai. After doing one year of postdoc at Technion,
Israel, and one year of postdoc at CWI Amsterdam, he joined Chennai Mathematical Institute in 2010. He joined the Indian Statistical Institute in 2018. His main area of interest is Theoretical
Computer Science with a particular focus on query complexity, property testing, Fourier Analysis of Boolean functions, streaming algorithms, and graph algorithms. Recently he has been using property
testing techniques in other more applied areas. Date:
October 17 (Thursday); 11:30am – 12:30pm
LHC-07 (Lecture Hall Complex) | {"url":"https://cse.iith.ac.in/talks/2024-10-14-Distinct-Elements-in-Streams.html","timestamp":"2024-11-08T15:46:37Z","content_type":"text/html","content_length":"15333","record_id":"<urn:uuid:3d0fde9a-5a3f-434a-a0a8-90cdea6c206f>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00287.warc.gz"} |
Alex and Heather each draw a triiangle with one side of 8cm, one angle of 45°and one angle of 80°. Alex says that their triangles are congruent. Explain why Alex might not be correct
Alex and Heather each draw a triiangle with one side of 8cm, one angle of 45°and one angle of 80°. Alex says that their triangles are congruent. Explain why Alex might not be correct
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Answer to a math question Alex and Heather each draw a triiangle with one side of 8cm, one angle of 45°and one angle of 80°. Alex says that their triangles are congruent. Explain why Alex might not
be correct
89 Answers
1. We know from the problem that there is a side of length 8\ \text{cm} and two angles: 45^\circ and 80^\circ.
2. The total sum of internal angles of a triangle is 180^\circ. Therefore, the remaining angle in each triangle is:
180^\circ - 45^\circ - 80^\circ = 55^\circ
3. This gives us a unique set of angles: 45^\circ, 80^\circ, 55^\circ in each triangle.
4. Since the corresponding angles in both triangles are equivalent, the triangles fall under the Angle-Angle-Angle (AAA) similarity criterion.
5. However, AAA similarity does not guarantee that the triangles are congruent. Congruence requires that the corresponding sides are congruent.
6. Because only one side of both triangles is given as 8\ \text{cm} and we have no information about the lengths of the other sides, the triangles might not be congruent.
Final Answer: The triangles are not necessarily congruent.
Frequently asked questions (FAQs)
Math question: Find the limit as x approaches infinity of [(x^2 + 4x + 3) / (x^2 - 2x + 1)] using L'Hospital's Rule.
What is the area of a triangle with base 6 units and height 9 units?
Question: What is the derivative of the function f(x) = sin(x) + ln(x) - e^x at x = π/4? | {"url":"https://math-master.org/general/alex-and-heather-each-draw-a-triiangle-with-one-side-of-8cm-one-angle-of-45-and-one-angle-of-80-alex-says-that-their-triangles-are-congruent-explain-why-alex-might-not-be-correct","timestamp":"2024-11-12T12:45:15Z","content_type":"text/html","content_length":"247759","record_id":"<urn:uuid:4a6819a1-2dd0-47b7-a0fd-3d19e345cc6d>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00805.warc.gz"} |
Planarity of Streamed Graphs | GIORDANO DA LOZZO
In this research we introduce a notion of planarity for graphs that are presented in a streaming fashion. A streamed graph is a stream of edges $e_1,e_2,\dots,e_m$ on a vertex set $V$. A streamed
graph is $\omega$-stream planar with respect to a positive integer window size $\omega$ if there exists a sequence of planar topological drawings $\Gamma_i$ of the graphs $G_i=(V,{e_j \mid i\leq j <
i+\omega})$ such that the common graph $G^{i}\cap=G_i\cap G{i+1}$ is drawn the same in $\Gamma_i$ and in $\Gamma_{i+1}$, for $1\leq i < m-\omega$. The Stream Planarity Problem with window size $\
omega$ asks whether a given streamed graph is $\omega$-stream planar. We also consider a generalization, where there is an additional backbone graph whose edges have to be present during each time
step. These problems are related to several well-studied planarity problems.
We show that the Stream Planarity Problem is $\mathcal{NP}$-complete even when the window size is a constant and that the variant with a backbone graph is $\mathcal{NP}$-complete for all $\omega \ge
2$. On the positive side, we provide $O(n+\omega{}m)$-time algorithms for (i) the case $\omega = 1$ and (ii) all values of $\omega$ provided the backbone graph consists of one $2$-connected component
plus isolated vertices and no stream edge connects two isolated vertices. Our results improve on the Hanani-Tutte-style $O((nm)^3)$-time algorithm proposed by Schaefer [GD'14] for $\omega=1$.
Theoretical Computer Science | {"url":"http://www.dia.uniroma3.it/~dalozzo/publication/journal-article/2019/tcs/","timestamp":"2024-11-05T15:55:38Z","content_type":"text/html","content_length":"32665","record_id":"<urn:uuid:20022110-ec0a-403b-806f-271903e00b9f>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00662.warc.gz"} |
2833 (number)
Interesting facts about the number 2833
• (2833) Radishchev is asteroid number 2833. It was discovered by L. I. Chernyj; N. S. Chernyj from Nauchni on 8/9/1978.
• There is a 2,833 miles (4,559 km) direct distance between Amritsar (India) and Changchun (China).
• There is a 1,761 miles (2,833 km) direct distance between Belgrade (Serbia) and Perm (Russia).
• There is a 1,761 miles (2,833 km) direct distance between Brasília (Brazil) and Mendoza (Argentina).
• There is a 2,833 miles (4,558 km) direct distance between Dalian (China) and Medan (Indonesia).
• More distances ...
• There is a 1,761 miles (2,833 km) direct distance between Dnipropetrovsk (Ukraine) and Riyadh (Saudi Arabia).
• There is a 1,761 miles (2,833 km) direct distance between Erbil (Iraq) and Lahore (Pakistan).
• There is a 1,761 miles (2,833 km) direct distance between Erbil (Iraq) and Tripoli (Libya).
• There is a 1,761 miles (2,833 km) direct distance between Guankou (China) and Patna (India).
• There is a 2,833 miles (4,559 km) direct distance between Hubli (India) and Omdurman (Sudan).
• There is a 2,833 miles (4,559 km) direct distance between Chandigarh (India) and Jilin (China).
• There is a 1,761 miles (2,833 km) direct distance between Chiba (Japan) and Xi’an (China).
• There is a 2,833 miles (4,559 km) direct distance between Jos (Nigeria) and Köln (Germany).
• There is a 2,833 miles (4,558 km) direct distance between Konya (Turkey) and Lagos (Nigeria).
• There is a 2,833 miles (4,558 km) direct distance between Łódź (Poland) and Peshawar (Pakistan).
• There is a 1,761 miles (2,833 km) direct distance between Manaus (Brazil) and Recife (Brazil).
• There is a 2,833 miles (4,559 km) direct distance between Munich (Germany) and Pikine (Senegal).
• There is a 2,833 miles (4,559 km) direct distance between Muzaffarābād (Pakistan) and Warsaw (Poland).
• There is a 2,833 miles (4,558 km) direct distance between Nagpur (India) and Rostov-na-Donu (Russia).
• There is a 2,833 miles (4,558 km) direct distance between Ningbo (China) and Rawalpindi (Pakistan).
• There is a 2,833 miles (4,558 km) direct distance between Oyo (Nigeria) and Paris (France).
• There is a 2,833 miles (4,558 km) direct distance between Peshawar (Pakistan) and Shenyang (China).
• There is a 2,833 miles (4,558 km) direct distance between Rio de Janeiro (Brazil) and Valencia (Venezuela).
What is 2,833 in other units
The decimal (Arabic) number
converted to a
Roman number
Roman and decimal number conversions
The number 2833 converted to a Mayan number is
Decimal and Mayan number conversions.
Length conversion
2833 kilometers (km) equals to
miles (mi).
2833 miles (mi) equals to
kilometers (km).
2833 meters (m) equals to
feet (ft).
2833 feet (ft) equals 863.509 meters (m).
Power conversion
2833 Horsepower (hp) equals to 2083.38 kilowatts (kW)
2833 kilowatts (kW) equals to 3852.33 horsepower (hp)
Time conversion
(hours, minutes, seconds, days, weeks)
2833 seconds equals to 47 minutes, 13 seconds
2833 minutes equals to 1 day, 23 hours, 13 minutes
Number 2833 morse code:
..--- ---.. ...-- ...--
Sign language for number 2833:
Number 2833 in braille:
Gregorian, Hebrew, Islamic, Persian and Buddhist Year (Calendar)
Gregorian year 2833 is Buddhist year 3376.
Buddhist year 2833 is Gregorian year 2290 .
Gregorian year 2833 is Islamic year 2279 or 2280.
Islamic year 2833 is Gregorian year 3370 or 3371.
Gregorian year 2833 is Persian year 2211 or 2212.
Persian year 2833 is Gregorian 3454 or 3455.
Gregorian year 2833 is Hebrew year 6593 or 6594.
Hebrew year 2833 is Gregorian year 927 a. C.
The Buddhist calendar is used in Sri Lanka, Cambodia, Laos, Thailand, and Burma. The Persian calendar is the official calendar in Iran and Afghanistan.
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Advanced math operations
Is Prime?
The number 2833 is a
prime number
. The closest prime numbers are
The 2833rd prime number in order is
Factorization and factors (dividers)
The prime factors of 2833
Prime numbers have no prime factors smaller than themselves. The factors of
2833 are
, 2833.
Total factors 2.
Sum of factors 2834 (1).
Prime factor tree
2833 is a prime number.
The second power of 2833
is 8.025.889.
The third power of 2833
is 22.737.343.537.
The square root √
is 53,225934.
The cube root of
is 14,149753.
The natural logarithm of No. ln 2833 = log
2833 = 7,949091.
The logarithm to base 10 of No. log
2833 = 3,452247.
The Napierian logarithm of No. log
2833 = -7,949091.
Trigonometric functions
The cosine of 2833 is 0,754061.
The sine of 2833 is -0,656805.
The tangent of 2833 is -0,871024.
Properties of the number 2833
Number 2833 in Computer Science
Code type Code value
PIN 2833 It's recommended that you use 2833 as your password or PIN.
2833 Number of bytes 2.8KB
Unix time Unix time 2833 is equal to Thursday Jan. 1, 1970, 12:47:13 a.m. GMT
IPv4, IPv6 Number 2833 internet address in dotted format v4 0.0.11.17, v6 ::b11
2833 Decimal = 101100010001 Binary
2833 Decimal = 10212221 Ternary
2833 Decimal = 5421 Octal
2833 Decimal = B11 Hexadecimal (0xb11 hex)
2833 BASE64 MjgzMw==
2833 MD5 ade55409d1224074754035a5a937d2e0
2833 SHA1 8aa80f7daf9fea98a5be5f6fe2e1c08a73e8474f
2833 SHA224 e842f5fa849b3136acbce982ac991f33b37a91b80d34f4d02ed0856e
2833 SHA256 99dbbc4aad5da980a3b5035ff28d2e9f828d973cf7be91cdf3b6886741879371
2833 SHA384 de3e35ce9144e2a958464f20b835dba9d37a9caae331595ff1695f98ed8f9202571a027dfb9721a8f90cb038f894e1af
More SHA codes related to the number 2833 ...
If you know something interesting about the 2833 number that you did not find on this page, do not hesitate to write us here.
Numerology 2833
The meaning of the number 3 (three), numerology 3
Character frequency 3: 2
The number three (3) came to share genuine expression and sensitivity with the world. People associated with this number need to connect with their deepest emotions. The number 3 is characterized by
its pragmatism, it is utilitarian, sagacious, dynamic, creative, it has objectives and it fulfills them. He/she is also self-expressive in many ways and with good communication skills.
More about the the number 3 (three), numerology 3 ...
The meaning of the number 8 (eight), numerology 8
Character frequency 8: 1
The number eight (8) is the sign of organization, perseverance and control of energy to produce material and spiritual achievements. It represents the power of realization, abundance in the spiritual
and material world. Sometimes it denotes a tendency to sacrifice but also to be unscrupulous.
More about the the number 8 (eight), numerology 8 ...
The meaning of the number 2 (two), numerology 2
Character frequency 2: 1
The number two (2) needs above all to feel and to be. It represents the couple, duality, family, private and social life. He/she really enjoys home life and family gatherings. The number 2 denotes a
sociable, hospitable, friendly, caring and affectionate person. It is the sign of empathy, cooperation, adaptability, consideration for others, super-sensitivity towards the needs of others.
The number 2 (two) is also the symbol of balance, togetherness and receptivity. He/she is a good partner, colleague or companion; he/she also plays a wonderful role as a referee or mediator. Number 2
person is modest, sincere, spiritually influenced and a good diplomat. It represents intuition and vulnerability.
More about the the number 2 (two), numerology 2 ...
№ 2,833 in other languages
How to say or write the number two thousand, eight hundred and thirty-three in Spanish, German, French and other languages. The character used as the thousands separator.
Spanish: 🔊 (número 2.833) dos mil ochocientos treinta y tres
German: 🔊 (Nummer 2.833) zweitausendachthundertdreiunddreißig
French: 🔊 (nombre 2 833) deux mille huit cent trente-trois
Portuguese: 🔊 (número 2 833) dois mil, oitocentos e trinta e três
Hindi: 🔊 (संख्या 2 833) दो हज़ार, आठ सौ, तैंतीस
Chinese: 🔊 (数 2 833) 二千八百三十三
Arabian: 🔊 (عدد 2,833) ألفان و ثمانمائة و ثلاثة و ثلاثون
Czech: 🔊 (číslo 2 833) dva tisíce osmset třicet tři
Korean: 🔊 (번호 2,833) 이천팔백삼십삼
Danish: 🔊 (nummer 2 833) totusinde og ottehundrede og treogtredive
Hebrew: (מספר 2,833) אלפיים שמונה מאות שלשים ושלש
Dutch: 🔊 (nummer 2 833) tweeduizendachthonderddrieëndertig
Japanese: 🔊 (数 2,833) 二千八百三十三
Indonesian: 🔊 (jumlah 2.833) dua ribu delapan ratus tiga puluh tiga
Italian: 🔊 (numero 2 833) duemilaottocentotrentatré
Norwegian: 🔊 (nummer 2 833) to tusen, åtte hundre og tretti-tre
Polish: 🔊 (liczba 2 833) dwa tysiące osiemset trzydzieści trzy
Russian: 🔊 (номер 2 833) две тысячи восемьсот тридцать три
Turkish: 🔊 (numara 2,833) ikibinsekizyüzotuzüç
Thai: 🔊 (จำนวน 2 833) สองพันแปดร้อยสามสิบสาม
Ukrainian: 🔊 (номер 2 833) дві тисячі вісімсот тридцять три
Vietnamese: 🔊 (con số 2.833) hai nghìn tám trăm ba mươi ba
Other languages ...
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If you know something interesting about the number 2833 or any other natural number (positive integer), please write to us here or on Facebook.
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Frequently asked questions about the number 2833
• Is 2833 prime and why?
The number 2833 is a prime number because its divisors are: 1, 2833.
• How do you write the number 2833 in words?
2833 can be written as "two thousand, eight hundred and thirty-three".
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Multiplication By 2 Worksheets
Mathematics, particularly multiplication, creates the foundation of numerous scholastic self-controls and real-world applications. Yet, for several learners, mastering multiplication can present a
challenge. To address this obstacle, instructors and moms and dads have welcomed an effective device: Multiplication By 2 Worksheets.
Intro to Multiplication By 2 Worksheets
Multiplication By 2 Worksheets
Multiplication By 2 Worksheets -
Multiplication by 2s This page is filled with worksheets of multiplying by 2s This is a quiz puzzles skip counting and more Multiplication by 3s Jump to this page if you re working on multiplying
numbers by 3 only Multiplication by 4s Here are some practice worksheets and activities for teaching only the 4s times tables Multiplication by 5s
Welcome to The Multiplying 2 Digit by 2 Digit Numbers A Math Worksheet from the Long Multiplication Worksheets Page at Math Drills This math worksheet was created or last revised on 2021 02 17 and
has been viewed 8 714 times this week and 10 384 times this month
Significance of Multiplication Practice Recognizing multiplication is pivotal, laying a solid foundation for advanced mathematical concepts. Multiplication By 2 Worksheets use structured and targeted
method, promoting a much deeper comprehension of this basic math procedure.
Advancement of Multiplication By 2 Worksheets
Multiplication Table Of 2 Repeated Addition by 2 s Read Write The Table Of 2
Multiplication Table Of 2 Repeated Addition by 2 s Read Write The Table Of 2
The worksheets below require students to multiply 2 digit numbers by 2 digit numbers Includes vertical and horizontal problems as well as math riddles task cards a picture puzzle a Scoot game and
word problems 2 Digit Times 2 Digit Worksheets Multiplication 2 digit by 2 digit FREE
Multiplication by 2 worksheets promotes an understanding of multiplication of numbers by 2 Multiplication is a basic arithmetic operation that is necessary to know if a student wants to be good at
mathematics These math worksheets consist of a variety of questions including multiplication of numbers by 2 word problems brain teasers etc
From standard pen-and-paper exercises to digitized interactive styles, Multiplication By 2 Worksheets have evolved, catering to diverse understanding styles and preferences.
Types of Multiplication By 2 Worksheets
Standard Multiplication Sheets Basic workouts concentrating on multiplication tables, helping learners construct a solid math base.
Word Trouble Worksheets
Real-life scenarios integrated into problems, improving crucial thinking and application abilities.
Timed Multiplication Drills Tests developed to boost speed and precision, aiding in quick psychological math.
Benefits of Using Multiplication By 2 Worksheets
Multiplication Worksheets 1 And 2 Multiplication Worksheets
Multiplication Worksheets 1 And 2 Multiplication Worksheets
Multiplying by Two Learners will practice multiplying by two in this fun math worksheet Children will discover that multiplying by two is like counting by two then use their math skills to solve 12
one and two digit multiplication problems Designed for second and third graders this worksheet supports students as they gain confidence in
Multiply by 2 s Lunita2 Member for 3 years 5 months Age 7 9 Level Grade 2 3 Language English en ID 149870 03 05 2020 Country code US Country United States School subject Math 1061955 Main content
Multiplication 2013181 One digit multiplication by 2 Share Print Worksheet Finish One digit multiplication by 2
Enhanced Mathematical Skills
Constant method sharpens multiplication efficiency, enhancing general math abilities.
Enhanced Problem-Solving Talents
Word problems in worksheets establish analytical reasoning and strategy application.
Self-Paced Learning Advantages
Worksheets suit specific understanding speeds, promoting a comfy and adaptable knowing atmosphere.
Exactly How to Produce Engaging Multiplication By 2 Worksheets
Incorporating Visuals and Shades Vibrant visuals and colors capture interest, making worksheets aesthetically appealing and involving.
Consisting Of Real-Life Scenarios
Associating multiplication to day-to-day scenarios adds importance and usefulness to exercises.
Customizing Worksheets to Different Skill Degrees Personalizing worksheets based upon varying proficiency degrees guarantees comprehensive understanding. Interactive and Online Multiplication
Resources Digital Multiplication Equipment and Games Technology-based sources use interactive knowing experiences, making multiplication appealing and delightful. Interactive Web Sites and Apps
Online systems offer varied and available multiplication method, supplementing conventional worksheets. Customizing Worksheets for Different Understanding Styles Visual Learners Visual help and
diagrams help comprehension for learners inclined toward aesthetic learning. Auditory Learners Verbal multiplication problems or mnemonics satisfy students who comprehend ideas with acoustic methods.
Kinesthetic Learners Hands-on activities and manipulatives sustain kinesthetic learners in understanding multiplication. Tips for Effective Application in Understanding Consistency in Practice
Regular practice strengthens multiplication skills, advertising retention and fluency. Stabilizing Repetition and Range A mix of repeated workouts and diverse trouble formats preserves interest and
understanding. Offering Useful Feedback Feedback aids in identifying areas of enhancement, urging ongoing development. Obstacles in Multiplication Method and Solutions Motivation and Engagement
Obstacles Monotonous drills can lead to disinterest; cutting-edge techniques can reignite inspiration. Getting Over Fear of Math Negative assumptions around mathematics can hinder progression;
creating a favorable learning setting is essential. Influence of Multiplication By 2 Worksheets on Academic Performance Research Studies and Research Study Searchings For Study shows a positive
relationship in between constant worksheet use and boosted mathematics efficiency.
Final thought
Multiplication By 2 Worksheets emerge as functional devices, cultivating mathematical proficiency in students while fitting varied knowing designs. From basic drills to interactive on-line resources,
these worksheets not only boost multiplication skills but additionally promote important thinking and analytical capacities.
2 Digit by 2 Digit multiplication Games And worksheets
Printable Multiplication Facts 0 12 PrintableMultiplication
Check more of Multiplication By 2 Worksheets below
4th Grade Two Digit Multiplication Worksheets Free Printable
Free Printable Multiplication Worksheets 2nd Grade
Multiplication Practice Sheets Printable Worksheets Multiplication Worksheets Pdf Grade 234
Worksheets On Multiplication For Grade 2 Printable Multiplication Flash Cards
Multiplication To 5x5 Worksheets For 2nd Grade
Multiplication Tables From 1 To 20 Printable Pdf Table Design Ideas
Multiplying 2 Digit by 2 Digit Numbers A Math Drills
Welcome to The Multiplying 2 Digit by 2 Digit Numbers A Math Worksheet from the Long Multiplication Worksheets Page at Math Drills This math worksheet was created or last revised on 2021 02 17 and
has been viewed 8 714 times this week and 10 384 times this month
Multiplication Worksheets K5 Learning
Our multiplication worksheets start with the basic multiplication facts and progress to multiplying large numbers in columns We emphasize mental multiplication exercises to improve numeracy skills
Choose your grade topic Grade 2 multiplication worksheets Grade 3 multiplication worksheets Grade 4 mental multiplication worksheets
Welcome to The Multiplying 2 Digit by 2 Digit Numbers A Math Worksheet from the Long Multiplication Worksheets Page at Math Drills This math worksheet was created or last revised on 2021 02 17 and
has been viewed 8 714 times this week and 10 384 times this month
Our multiplication worksheets start with the basic multiplication facts and progress to multiplying large numbers in columns We emphasize mental multiplication exercises to improve numeracy skills
Choose your grade topic Grade 2 multiplication worksheets Grade 3 multiplication worksheets Grade 4 mental multiplication worksheets
Worksheets On Multiplication For Grade 2 Printable Multiplication Flash Cards
Free Printable Multiplication Worksheets 2nd Grade
Multiplication To 5x5 Worksheets For 2nd Grade
Multiplication Tables From 1 To 20 Printable Pdf Table Design Ideas
Multiplication 2 worksheets Worksheets Multiplication Flashcards
Multiplication Times Tables Worksheets 2 3 4 5 Times Tables Four Worksheets FREE
Multiplication Times Tables Worksheets 2 3 4 5 Times Tables Four Worksheets FREE
A Worksheet On Multiplication 2 Teaching Resources
Frequently Asked Questions (Frequently Asked Questions).
Are Multiplication By 2 Worksheets appropriate for every age groups?
Yes, worksheets can be customized to various age and ability levels, making them versatile for different students.
How usually should trainees exercise utilizing Multiplication By 2 Worksheets?
Constant practice is vital. Regular sessions, preferably a couple of times a week, can produce considerable renovation.
Can worksheets alone improve mathematics abilities?
Worksheets are a valuable tool but needs to be supplemented with varied knowing techniques for comprehensive skill growth.
Are there on-line systems offering cost-free Multiplication By 2 Worksheets?
Yes, numerous instructional websites provide open door to a large range of Multiplication By 2 Worksheets.
Exactly how can parents sustain their children's multiplication practice at home?
Motivating constant technique, giving support, and developing a favorable learning setting are advantageous actions. | {"url":"https://crown-darts.com/en/multiplication-by-2-worksheets.html","timestamp":"2024-11-12T23:51:26Z","content_type":"text/html","content_length":"28931","record_id":"<urn:uuid:3eda5767-7b55-4a8b-b5f4-c9d2f85f196c>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00042.warc.gz"} |
We are now going to move to computing the derivative of the regression cost function. Recall that the cost function is the sum over the data points of the squared difference between an observed
output and a predicted output, plus the L2 penalty term.
= SUM[ (prediction - output)^2 ]
+ l2_penalty*(w[0]^2 + w[1]^2 + ... + w[k]^2).
Since the derivative of a sum is the sum of the derivatives, we can take the derivative of the first part (the RSS) as we did in the notebook for the unregularized case in Week 2 and add the
derivative of the regularization part. As we saw, the derivative of the RSS with respect to w[i] can be written as:
2*SUM[ error*[feature_i] ].
The derivative of the regularization term with respect to w[i] is:
Summing both, we get
2*SUM[ error*[feature_i] ] + 2*l2_penalty*w[i].
That is, the derivative for the weight for feature i is the sum (over data points) of 2 times the product of the error and the feature itself, plus 2*l2_penalty*w[i].
We will not regularize the constant. Thus, in the case of the constant, the derivative is just twice the sum of the errors (without the 2*l2_penalty*w[0] term).
Recall that twice the sum of the product of two vectors is just twice the dot product of the two vectors. Therefore the derivative for the weight for feature_i is just two times the dot product
between the values of feature_i and the current errors, plus 2*l2_penalty*w[i].
With this in mind complete the following derivative function which computes the derivative of the weight given the value of the feature (over all data points) and the errors (over all data points).
To decide when to we are dealing with the constant (so we don't regularize it) we added the extra parameter to the call feature_is_constant which you should set to True when computing the derivative
of the constant and False otherwise. | {"url":"https://notebook.community/Benedicto/ML-Learning/Linear_Regression_4_ridge_regression_assignment_2","timestamp":"2024-11-09T14:26:29Z","content_type":"text/html","content_length":"99069","record_id":"<urn:uuid:05e8b4ea-172f-4287-bca1-ba1e80e721c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00535.warc.gz"} |
Linear Regression for Data Science
A Complete Guide on Linear Regression for Data Science
Linear Regression is a basic Machine learning Algorithm, which tries to build a relationship between two variables through a linear equation.
y\quad =\quad \alpha \quad +\quad \beta x
One variable in the linear equation called the independent variable and the other one called the dependent variable.
First should determine the strength of the relationship between the desired variable before trying to fit a linear equation to any given data.
Fitting a linear equation to the given data will not serve the purpose If there is no significant relationship between the two variables.
Scatter Plot and Correlation coefficient are the most common ways to determine the strength of the relationship between the desired variable.
Scatter Plot gives us a visual idea about the relationship between the two or more variables.
The correlation coefficient gives us a numerical measure of the relationship between the two variables.
A Linear Regression model contains the following form of the equation where X is the independent variable and Y is the dependent variable (because Y depends on X)
Y\quad =\quad a\quad +\quad bX\quad +\quad \varepsilon
Whereas b is the slope of the Straight line, X is the intercept(i.e. the value of Y when X=0) and e is the random error.
Assumption of the Linear Regression Model:
1. Linearity: There is a linear relationship between the dependent(X) and independent(Y) variables.
2. Homoscedasticity: The Variance of the Residual can be the same for any value of X (i.e Random Error which is in the form of normal independent distribution)
3. Independence: The successive observations of the independent variable X are independent of each other.
4. Normality: It can be used for any value of X, Y which is the form of Normal Distribution.
Least Square Estimation:
The Simple Linear Regression uses a single independent variable (X) and dependent variable(Y), which has a general form of equation Y = a + bX + e.
The Least Square Estimation method applied to estimate the regression coefficient a & b which are intercept and slope of the straight line respectively and based on both the estimated coefficient, it
decides on the best-fitted line.
The simple principle on which it works that try to decrease the vertical deviation of each point from the best-fitted line.
The next step is to square the deviation and adds it all up so the risk of canceling the positive and negative values can be ruled out.
Adequacy of the Regression Model:
Residual Analysis:
Once a regression model gets fitted it is always helpful to plot the Residuals (the difference between the fitted line to observed values) in time sequence, against Y values, or against X Values.
It helps to get an idea of the distribution of the residuals (to know whether it is normally distributed or not)
At the start of our study of simple linear regression, we assume that the Random error is the Normal Independent distribution.
If the assumption does not hold a model then fitting a linear regression model to the said data will not be fruitful.
R Code for Simple Linear Regression
#Load the data like Train and Test datasets
#The values must be numeric and NumPy arrays for the linear regression model
# Read the csv dataset file
train <- read.csv("data.csv")
# Sample the data In Train and Test Dataset
train_data = train[sample(seq(1,200),0.8*nrow(train)),]
test_data = train[sample(seq(1,200),0.2*nrow(train)),]
# Train the model using the training sets and check the score
Linear_model <- lm(y_variable_train_data ~ ., data = train_data)
# Get The Statistical Summary of data
#Predict Output
prediction = predict(linear_model, test_data)
# Print the predicted output
# find the root mean square error of the liner model
RMSE_of_model = rmse(test$y_variable_test_data, prediction)
Python Code for Simple Linear Regression
# Import the Python Libraries for linear regression
import numpy as np
from sklearn.linear_model import LinearRegression
# Create a numpy array data
x = np.array([2, 10, 20, 30, 42, 65, 72]).reshape((-1, 1))
y = np.array([7, 14, 18, 27, 33, 42, 55])
# Build the regression model on x and y array matrix
model = LinearRegression().fit(x, y)
# After the model fitted, you can check the efficiency of the model and it's working capability.
# You can get the coefficient of determination (𝑅²), intercept, and Slope with specific functions.
# Get the 𝑅² of model
squ_of_r = model.score(x, y)
print('coefficient of determination:', squ_of_r)
# Get the intercept of line
intercept = model.intercept_
print('intercept of model:', intercept)
# get the slop of line
Slope = model.coef_
print('slope of model:', Slope)
# To get the prediction of data based on test data
Prediction = model.predict(x)
print('prediction of responce variable:', Prediction, sep='\n')
In the end, Linear regression is a very efficient model which contain simple and multiple approaches for numerical data prediction.
Simple linear regression makes the linear modeling very easy to understand and apply for a very wide variety of data.
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Leave a Reply Cancel reply | {"url":"https://analyticslearn.com/a-complete-guide-on-linear-regression-for-data-science","timestamp":"2024-11-07T02:42:57Z","content_type":"text/html","content_length":"194650","record_id":"<urn:uuid:cab5bd12-10bd-4c77-9df4-1726bc1323bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00124.warc.gz"} |
Although the sample size for simple logistic regression can be readily
Although the sample size for simple logistic regression can be readily determined using currently available methods the sample size calculation for multiple logistic regression requires some
additional information such as the coefficient of determination (for multiple logistic regression (ii) available interim or group-sequential designs and (iii) much smaller required sample size. of a
logistic regression follows a logit-normal (LN) distribution which is generated from your logistic transformation of a normal distribution [7]. These properties of logistic regression have inspired
us to develop a transformation-based approach to determine the sample size. Our approach is 1st to transform a logistic end result measure into a normal distribution and then the sample size is
determined by the t-test. The Sapacitabine (CYC682) sample size determination using a Sapacitabine (CYC682) transformed end result measure offers three major advantages over the existing methods: (i)
no need for in the case of multiple logistic regression; (ii) straightforward implementation of interim or group-sequential designs based on a transformed end result measure; and (iii) much smaller
required sample size. It should be noted that our approach would be applied when a logistic end result measure is continuous and comes from a logistic regression model. When a logistic end result
measure is definitely either binary or ordinal the proposed approach would not be used. 2 Motivating example Prostate malignancy (PrC) is the second leading cause of cancer-related death in males.
Although prostate specific antigen (PSA) blood testing remains the most widely used tool for PrC detection important efforts have been conducted to determine alternate biomarkers to conquer its lack
of specificity. Recently it has been discovered that sarcosine alanine glutamate and glycine are metabolic biomarkers of PrC progression [8 9 Using these metabolic biomarkers a PrC diagnostic
algorithm (a logistic regression model) was developed which also required into account medical information such as PSA and prostate volume. The outcome measure of this logistic algorithm will be
called the M-score. A new study was planned to validate the M-score by comparing with the PrC biopsy result in African American (AA) males who were referred Sapacitabine (CYC682) for prostate biopsy
for any clinical indication. The primary hypothesis was that the M-score which has not been extensively analyzed in AA males would have related test characteristics in AA males as it did in Western
American males. The question that we were asked as statisticians was: how many AA males need to be included in the study? In the previous study the M-score was elevated in AA PrC individuals compared
to those with benign prostate disease using a small sample size of 18. Based on this earlier result the study was designed to detect a difference of 10 points in the imply M-score of AA males with
vs. those without PrC based on biopsy results. Since the M-score was generated from a logistic regression model a sample size could be determined based on a logistic regression model. However
covariates included in the logistic model were blinded and the biopsy result positive or bad was the only covariate available to us. 3 Methods A logit-normal (LN) distribution is a probability
distribution of a random variable whose logit follows a normal distribution. If a random variable follows a normal distribution then its logistic = = stands for a logit-normal distribution with imply
of ?i Sapacitabine (CYC682) and standard deviation (SD) of ψi we = 0 1 and j = 1 2 ? ni. An investigator desires to test the null Rabbit polyclonal to AGMAT. hypothesis that the two human population
means are equivalent stands for a normal distribution. The pdf depends on the mean and SD of are the only available information and no analytical remedy exists to recover the mean and SD of is
definitely developed for the power and sample size dedication for an LN distribution and it is freely available at http://cansur.sourceforge.net. The brief instruction on how to use the R package can
be found in the Supplementary Info. 4 Simulation studies Because there is no analytical remedy of the imply and SD of an LN distribution Monte Carlo simulation was performed to find the true imply
and true SD of an LN distribution related to the people of a normal distribution. In particular the normal distributions were chosen corresponding to the LN distributions with the difference in imply
(Δ) of 0.1 and 0.2 and the same SD (i.e. ψ = 0.10. | {"url":"http://healthyguide.info/although-the-sample-size-for-simple-logistic-regression-can-be-readily/","timestamp":"2024-11-05T09:41:51Z","content_type":"text/html","content_length":"34938","record_id":"<urn:uuid:5d9d256a-5edb-41f2-a3e1-0be85c12a320>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00263.warc.gz"} |
Families of Solutions for ODEs
This Demonstration explores families of solutions of an ordinary differential equation (ODE) of the form , which are plotted using the vector field . For each choice of you can see how the solutions
depend on the value of the parameters; in some cases going from negative to positive values causes a significant change in the behavior of the solutions. | {"url":"https://www.wolframcloud.com/obj/73021e17-ac43-4903-a9eb-52342e6220f7","timestamp":"2024-11-12T03:32:42Z","content_type":"text/html","content_length":"169796","record_id":"<urn:uuid:a2b8fbff-4a66-4088-b86a-18902d14ee87>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00086.warc.gz"} |
Effect of suction chamber baffles on pressure fluctuations in a low specific speed centrifugal pump
In order to study the effect of suction chamber baffles on hydraulic performance and unsteady characteristics in a low specific speed centrifugal pump, a model pump was design with enlarging flow
mothed and four schemes of suction baffle, including no baffle (scheme 0), only one baffle in the suction (scheme 1), two baffles (scheme 2) and three baffles in the suction (scheme 3), were
considered. Commercial code FLUENT was applied to simulate the flow of the pump. RNG $k$-$\epsilon$ turbulence model was adopted to handle with the turbulent flows in the pump. The sliding mesh
technique was applied to take into account the impeller-volute interaction. Based on the simulation results, the hydraulic performance and pressure fluctuations were obtained and analyzed in detail.
The head value of no baffle in the suction (scheme 0) is lower than that with baffles in the suction (scheme 1, scheme 2, scheme 3) at each condition point. Hump point in scheme 0 is at $\phi =$
0.00596 (1.2 times ${Q}_{o}$). The hump point in scheme 1, scheme 2, scheme 3 is at 0.8${Q}_{o}$, 1.0${Q}_{o}$, 1.0${Q}_{o}$, respectively. The ε value of scheme 1 is the smallest and that of scheme
0 is the largest in the four schemes. Six wave troughs are observed clearly at each monitoring point as the impeller rotates in a circle. Each time the impeller is turned 10 degrees, there are six
obvious troughs around the impeller. With the rotation of the impeller, peak value of pressure fluctuations at blade passing frequency (BPF) is gradually decrease. At low flow ($\varphi =$0.002383),
the main frequency of pressure fluctuation at P36 and P1 under scheme 0, scheme 2 and scheme 3 is 295 Hz, which is corresponding to BPF. The pressure fluctuation levels are decreased by –2.72 %,
–2.13 %, and –2.21 % respectively when the number of baffle in the suction is one, two, three, respectively. And decrease rate of pressure fluctuation (${∆C}_{p}$) on scheme 1 is maximum. It
indicates that Adding baffles to the suction chamber is beneficial to reduce the amplitude of pressure pulsation at BPF in the volute. The best number of baffles in the suction is one. Based on
scheme 1 simulation results, the prototype was manufactured and the performance experiments were carried out. A good agreement of the head and efficiency between numerical results and experimental
results are observed.
1. Introduction
Low specific speed centrifugal pumps (30 $\le {n}_{s}\le$ 80) have comprehensive applications on the field of aviation, aerospace, petrochemical industry, agricultural irrigation and light industry
due to its characteristics of low flow rate and high head [1, 2]. On the basis of the traditional design method, the passages of impeller are long and narrow, which give rise to lots of unavoidable
defects for low speed centrifugal pumps. For example, lower efficiency, easily having a camp on the performance curve, easily leading to overload at the large flow rate and so on. Hence, lots of
researches on characteristics of low specific speed centrifugal pumps have been performed.
At present, experimental test and numerical simulation are the main research methods of characteristic in centrifugal pumps [3]. Although experimental test is reliable, high cost and more time will
be wasted in it. With the rapid development of CFD technology, numerical simulation method, which is very useful and convenient to predict the unsteady dynamic characteristics, has gradually become
the main method to research the inner flow in pumps.
Gao [4] researched the influence of five typical blade trailing edge profiles on the performance and unsteady pressure fluctuations in a low specific centrifugal pump (${n}_{s}=$51.68) and pointed
out that the blade trailing edge profiles had a significantly effect on the pump performances and well-designed blade trailing edge profile could reduce vortex intensity. Shi [5] investigated the
effect of the blade wrap angle and outlet angle on the hydraulic performance of the low-specific speed sewage pump (${n}_{s}=$60) and reminded of that changing the blade outlet angle had much more
influence on the performance of the pump than changing the wrap angle. Zhu [6] performed experiments on transient performance of a low specific speed centrifugal pump (${n}_{s}=$45) during starting
and stopping periods with open impeller and captured the appearance of the slowly rising of the flow rate at the beginning of starting period and suddenly dropping at the end of stopping process.
Zhang [7] carried out experimental research on internal flow in impeller of a specific speed centrifugal pump (${n}_{s}=$47) with PIV and pointed out that the absolute velocity value increases with
radius and from pressure side to suction side at the same radius gradually. [8, 9] investigated the rotor-stator interaction and flow unsteadiness in a low specific speed centrifugal pump with the
large eddy simulation (LES) method and stated briefly that the pressure fluctuation amplitude is determined by the corresponding vorticity magnitude. Gao, et al. [10-13] researched unsteady pressure
pulsation measurements and analysis of a low specific speed centrifugal pump.
Besides inner flow in a low specific speed centrifugal pump was researched, some scholars researched the influence of volute geometry on the characteristics of low specific speed centrifugal pump.
Hamed Alemi, et al. [14-16] investigated effects of volute curvature on performance of a low specific-speed centrifugal pump at design and off-design conditions and presented that constant velocity
design method of volute casing give more satisfactory performance.
In general, the most researches of characteristics in low specific speed centrifugal pumps are concentrated on impeller parameters and volute parameters. As is known to all, the suction chamber is in
front of the impeller. The outlet velocity field distribution of suction chamber has a great influence on the hydraulic efficiency of impeller. there is little study about the influence of suction on
pressure pulsation in a low specific speed centrifugal pump. In this paper, the centrifugal pump with specific speed ${n}_{s}=$25 was designed, four schemes of suction were presented, and unsteady
dynamic characteristics were discussed.
2. Research model and scheme
2.1. Research model
The parameter of research model in this paper is a super low specific centrifugal pump with a diffusion suction chamber, a spiral volute and a shrouded impeller. In order to obtain higher efficiency
value at operation point in the super lower centrifugal, enlarged flow design method was adopted. Basic principle of enlarged flow design method is operation flow rate (OFR) and specific speed value
of original pump are amplified in reason to obtain a new centrifugal pump. Higher efficiency value will be gained when the new design centrifugal pump runs at OFR. Due to range of efficiency curve in
the new centrifugal pump cover that of the original model, efficiency at operation flow rate is increased, and average efficiency in the planned range is enhanced [1]. The specific speed value of the
research pump is amplified from 25 at the designed point to 39, and the nominal flow rate is enlarged from 12.5 m^3/h to 30 m^3/h. The impeller with six blades is designed at 2950 r/min. Main
geometric parameters of the pump are presented in Table 1. The specification of design parameters is defined in details in [17].
Table 1Design parameters
Nominal flow rate ${Q}_{n}=$30 m^3/h
Operation flow rate (OFR) ${Q}_{o}=$12.5 m^3/h
Specific speed ${n}_{s}=$39
Rotational speed $n=$2950 r/min
Head at OFR $H=$74 m
Blade number $Z=$ 6
Impeller inlet diameter ${D}_{1}=$68 mm
Impeller outlet diameter ${D}_{2}=$228 mm
Impeller outlet width ${b}_{2}=$7 mm
Impeller wrap angle $\alpha =$149°
Volute inlet diameter ${D}_{3}=$245 mm
Volute inlet width ${b}_{3}=$18 mm
Diffuser outer diameter ${D}_{4}=$32 mm
2.2. Scheme
Due to preswirl and backflow coexisting in the suction at small flow rate, centrifugal pump performance is in a sharp decline. It is beneficial to reduce the impact loss at the entrance of the
impeller and improve efficiency that there is a preswirl before impeller entrance. But if the preswirl strength is too high, Head value at zero flow will be reduce and hump phenomenon will be
occurrence on the performance curve. Therefore, it is necessary to reduce the preswirl strength properly in the suction. In addition, backflow will cause loss of energy and centrifugal pump
performance will be in a sharp decline. Measure must be taken to restrain the backflow generation.
In general, setting a baffle in the suction can reduce the preswirl strength before the impeller entrance and, at the same time, can restrain the backflow generation. In order to investigate the
effect of the suction baffles on hydraulic performance in the low specific centrifugal, four research schemes with no baffle (scheme 0), only one baffle (scheme 1), two baffles (scheme 2) and three
baffles in the suction (scheme 3) were designed. The thickness of baffle is 7 mm. The distances between the top of baffle and axis of the suction are 13 mm, and the length size of the suction is 48
mm. Geometry parameters of the baffle in the suction are shown in Fig. 1.
Fig. 1Sketch of research schemes
a) Baffle program of scheme 1
b) Baffle program of scheme 2
c) Baffle program of scheme 3
3. Mathematical models and numerical simulations
3.1. Mathematical models
For unsteady flow of incompressible fluid, the equation of continuity equation and moment conservation equations can be written as follows:
$\frac{\partial \rho }{\partial t}+\frac{\partial }{\partial {x}_{j}}\left(\rho {u}_{j}\right)=0,$
$\frac{\partial }{\partial t}\left(\rho {u}_{i}\right)+\frac{\partial }{\partial {x}_{j}}\left(\rho {u}_{i}{u}_{j}\right)=-\frac{\partial p}{\partial {x}_{i}}+\frac{\partial }{\partial {x}_{j}}\left
[\mu \left(\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}}-\frac{2}{3}{\delta }_{ij}\frac{\partial {u}_{k}}{\partial {x}_{k}}\right)\right]+\frac{\partial }{\
partial {x}_{j}}\left(-\rho \stackrel{-}{{u}_{i}^{"}{u}_{j}^{\text{'}}}\right),$
where $p$ is the static pressure, $\rho$ is the density and ${u}_{i}$ ($i=$1, 2, 3) is fluid velocity. $\mu$ is the molecular viscosity. $-\rho \stackrel{-}{{u}_{i}^{"}{u}_{j}^{\text{'}}}$ is
Reynolds stress.
Two-equation turbulence models allow the determination of both, a turbulent length and time scale by solving two separate transport equations. The RNG $k$-$\epsilon$ was adopted in the research. The
turbulence kinetic energy, $k$, and its rate of dissipation, $\epsilon$, are obtained from the following transport equations:
$\frac{\partial }{\partial t}\left(\rho k\right)+\frac{\partial }{\partial {x}_{i}}\left(\rho k{u}_{i}\right)=\frac{\partial }{\partial {x}_{j}}\left({\alpha }_{k}{u}_{eff}\frac{\partial k}{\partial
{x}_{j}}\right)+{G}_{k}+{G}_{b}-\rho \epsilon -{Y}_{M}+{S}_{k},$
$\frac{\partial }{\partial t}\left(\rho \epsilon \right)+\frac{\partial }{\partial {x}_{i}}\left(\rho {\epsilon u}_{i}\right)=\frac{\partial }{\partial {x}_{j}}\left[{\alpha }_{\epsilon }{u}_{eff}\
frac{\partial \epsilon }{\partial {x}_{j}}\right]+{C}_{1\epsilon }\frac{\epsilon }{k}\left({G}_{k}+{C}_{3\epsilon }{G}_{b}\right)-{C}_{2\epsilon p}\frac{{\epsilon }^{2}}{k}-{R}_{\epsilon }+{S}_{\
epsilon }.$
In these equations, ${G}_{k}$ represents the generation of turbulence kinetic energy due to the mean velocity gradients. This term may be defined as ${G}_{k}=-\rho \stackrel{-}{{u}_{i}^{"}{u}_{j}^{\
text{'}}}\frac{\partial {u}_{j}}{\partial {x}_{i}}$. ${G}_{b}$ is the generation of turbulence kinetic energy due to buoyancy and is given by ${G}_{b}=\beta g\frac{{\mu }_{t}}{{\mathrm{P}\mathrm{r}}_
{t}}\frac{\partial T}{\partial {x}_{i}}$. ${Y}_{M}$ represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate, which is defined as ${Y}_{M}=
2\rho \epsilon {M}_{t}^{2}$, where ${M}_{t}$ is the turbulent Mach number, defined as ${M}_{t}=\sqrt{k/{a}^{2}}$, and $a=\sqrt{\gamma RT}$ is the speed of sound. The quantities ${\alpha }_{k}$ and $
{\alpha }_{\epsilon }$ are the inverse effective Prandtl numbers for $k$ and $\epsilon$, respectively. ${S}_{k}$ and ${S}_{\epsilon }$ are user-defined source terms. ${C}_{1\epsilon }=\text{1.42}$, $
{C}_{2\epsilon }=\text{1.68}$.
3.2. Geometry and grid
Fluid domains of physical model were built by 3D modeling commercial software Creo 3.0. The grids of the geometry model were generated by the manufacturing code ANSYS ICEM. In order to guarantee
fluid development enough, the length of inlet extension is 5 times inlet diameter of the centrifugal pump, and the outlet extension length is as 10 times as outlet diameter of the volute. The whole
flow passage model with inlet extension, suction chamber, clearance of wear-ring, impeller, volute casing and outlet extension was employed to obtain precise calculation data. The thickness of wear
ring clearance in the calculation is 0.2 mm. Hexahedral cells were generated in the inlet, outlet, impeller and wear-ring clearance zones. Unstructured tetrahedral cells were used to define the
volute casing because of its complex geometric structure. The grids near tongue zone of volute were refined so as to gain precise unsteady flow structures. Computational model and grids of low
specific centrifugal pump are shown in Fig. 2.
Fig. 2Computational model and grids of low specific centrifugal pump
a) Computational model of low specific centrifugal pump
A detailed mesh sensitivity analysis of computational domain was performed so as to eliminate the influence of mesh factor. The same topological structure of computational domain was employed for the
research model. Five schemes of grids were gained at the same premise of mesh quantity. The numbers of grids for five cases were 213 000, 642 000, 1 046 000, 1 435 000, 1 973 000, respectively.
The value of head was for the qualitative dimensions of mesh independent check, and the computational head value of five cases were 75.4 m, 74.3 m, 73.9 m, 73.6 m, 73.7 m, respectively. As can be
observed in Fig. 3, the head value was gradual decline with the number of girds increasing. Apparent, the head value of case 4 and case 5 were almost the same. It indicates that it will be little
impact on numerical results to increase the number of grids from those of case 4. Taking the computer resources in account and keeping the cost under control, the girds numbers of case 4 were chosen
to calculate the computational domain.
Fig. 3Mesh sensitivity analysis
3.3. Boundary condition and solution method
The inlet boundary condition of the model pump was defined as velocity-inlet while the outlet boundary condition was set up as pressure-outlet. The Hydraulic diameter $D$ of the low specific
centrifugal is 50 mm and the value of turbulence intensity $I$ was calculated directly [18]. Sliding mesh technology was employed to deal with the information transfer between the rotating impeller
and the stationary volute. The walls of the model pump were defined as no-slip condition, and the standard wall function was used to solve the low Reynolds number flow in the near wall region.
Commercial software Fluent based on the finite volume method was employed to simulate unsteady flow field in the low specific centrifugal. SIMPLEC algorithm was adopted to calculate pressure-velocity
In order to obtain detailed resolution of unsteady flow results of the low specific centrifugal pump, time step of the unsteady calculation should satisfy the Courant number criterion [19]:
${C}_{O}=\frac{\left|\stackrel{-}{v}\right|∆t}{l}\le 50,$
where $\left|\stackrel{-}{v}\right|$ is the absolute value of the estimated average velocity. $l$ is the smallest size of the grid. Considering the scale of grid and impeller speed of the model pump,
time step was set as $∆t=$5.65×10^-5, which indicate that each impeller revolution will be calculated in a time sequence of 360 times steps corresponding to 1° of the impeller rotation speed. The
numerical residual convergence criterion was set as 10^-5 so as to ensure the result to be converged. To obtain a rapid convergence process, steady simulation was first carried out in advance, which
was set as the initial condition for transient calculation.
4. Results and discussions
4.1. Performance characteristics analysis
Comparison of head value for four schemes are shown in Fig. 4. Flow at the dashed is operation flow rate ($\varphi =$ 0.00596). As can be seen in Fig. 4, with the increase of flow rate, the head
value of four schemes increases first and then decreases. Namely, there is a hump phenomenon in the low specific speed centrifugal pump.
It is easily found the head value of no baffle in the suction (scheme 0) is lower than that with baffles in the suction (scheme 1, scheme 2, scheme 3) at each condition point. Hump point in scheme 0
is at$\varphi =$ 0.00596 (1.2 times ${Q}_{o}$). The hump point in scheme 1, scheme 2, scheme 3 is at 0.8${Q}_{o}$, 1.0${Q}_{o}$, 1.0${Q}_{o}$, respectively. It indicates that the baffle in the
suction can affect the inlet flow pattern of the impeller and further impact the energy distribution in the impeller.
Fig. 4External characteristics curves for four schemes
In order to measure the hump degree in each scheme, the dimensionless formula is introduced as follows:
${\epsilon }_{n}=\frac{{H}_{{h}_{n}}-{H}_{{l}_{n}}}{{H}_{{h}_{n}}}×100%,$
${\pi }_{i}=\frac{{\epsilon }_{i}}{{\epsilon }_{1}},$
where ${H}_{hn}$ is the head coefficient at hump point for scheme $n$. ${H}_{ln}$ is the head coefficient at $\varphi =$ 0.002383 for four schemes. $n=$0, 1, 2.3; $i=$0, 1, 2, 3. The comparison of $
{\epsilon }_{n}$ and ${\pi }_{i}$ for four schemes are is list in Table 2.
Table 2The comparison of ε for four schemes
Scheme ${H}_{h}$ ${H}_{l}$ ${\epsilon }_{n}$ ${\pi }_{i}$
0 0.57287 0.55738 2.704 % 159
1 0.58729 0.58719 0.017 % 1
2 0.58691 0.57791 1.533 % 90
3 0.59022 0.57916 1.873 % 110
As can be observed in Table 2, the $\epsilon$ value of scheme 1 is the smallest and scheme 0 one is the largest in the four schemes (159 times scheme 1). The scheme 2 and scheme 3 are 90 times and
110 times scheme 1, respectively. It shows that the number of baffle affect centrifugal pump hump location and intensity, and the best number of baffles in the suction is one.
4.2. Analysis of time domain on monitoring points
In order to explore the pressure pulsation characteristics of the low specific speed centrifugal pump under various schemes in details, thirty-six monitoring points were set up evenly in the volute
calculation domains. As shown in Fig. 5, thirty-six monitoring points were all placed in the symmetry plane of the volute, and they were arranged as counter clockwise, which was in the same direction
with the impeller rotating. The diameter of base circle the monitoring points on is 230 mm. Angle of monitoring point P1 was defined as $\theta =$0°.
The static pressure value of numerical simulation on each monitoring point is given at different times. In order to accurately compare the diffidence of pressure on each monitoring point, the
pressure coefficient is introduced:
${C}_{p}=\frac{2\left(p-\overline{P}\right)}{\rho {u}^{2}},$
where $p$ is the static pressure on the monitoring point. $\overline{P}$ is the average static pressure for the impeller rotating a circle and u is the circumferential velocity of the impeller.
Fig. 5Distribution of pressure pulsation monitoring points
Fig. 6 shows the distribution of the pressure pulsation time domain of thirty-six monitoring points around the impeller at one impeller rotation period. Abscissa represents different monitoring
points, and ordinate is angle around volute inlet.
From the Fig. 6 It can be seen that six wave troughs are observed clearly at each monitoring point as the impeller rotates in a circle, which corresponds to the six blades. The appearance of the
former trough is earlier than that of the latter one at one impeller channel period. Each time the impeller is turned 10 degrees, there are six obvious troughs around the impeller. It shows that the
rotor-stator interaction between the impeller and the volute is the main cause of the pressure pulsation.
With the increase of the flow rate, the pressure fluctuation of each scheme tends to be gradual. The pressure fluctuation in time domain is severe at$\varphi =$ 0.002383 in four schemes. It is
because that the flow rate is much small and there are lots of secondary flow phenomenon in impeller runner.
4.3. Frequency domain analysis on monitoring point
Due to space limitations and taking into account the causes of static and dynamic interference, frequency analysis of three points P36, P1, P2 under different flow condition, which are near the
volute tongue, is performed in this section. The frequency distribution of the pressure pulsations coefficients at 3 monitoring points near the tongue is shown in Fig. 7. It can be found that with
the rotation of the impeller, peak value of pressure fluctuation at blade passing frequency (BPF) is gradually decrease, which means amplitude on P36 > P1 > P2 at BPF under all schemes. At low flow (
$\varphi =$0.002383), the main frequency of pressure fluctuation at P36 and P1 under scheme 0, scheme 2 and scheme 3 is 295 Hz, which is corresponding to BPF. The main frequency of P2 under scheme 0,
scheme 2 and scheme 3 is 49.17 Hz. At scheme 1, the fluctuation main frequency is 49.17 Hz at P1 and P2, and the main frequency at P36 is 295 Hz, which are shaft frequency and BPF, respectively. It
indicates that fluctuation frequency distribution around impeller is affected by the suction baffles distribution. Under other flow rates, the main frequency of pressure fluctuation at three points
is 295 Hz.
Fig. 8 shows the comparison of fluctuation amplitude at BPF for each point. It can be found that with the change of angle, the peak of pressure fluctuation at BPF in the volute varies periodically.
The maximum value of amplitude is all at 360° of the volute (at P36 location). The minimum value of amplitude is all near 150°. Seven peaks and troughs of amplitude are captured. It indicates that
amplitude of frequency at BPF is in periodic fluctuations.
Fig. 6Time domain diagram of monitoring point: a) time domain diagram of scheme 0 at ϕ= 0.002383; b) time domain diagram of scheme 0 at ϕ= 0.005958; c) time domain diagram of scheme 0 at ϕ= 0.008342;
d) time domain diagram of scheme 1 at ϕ= 0.002383; e) time domain diagram of scheme 1 at ϕ= 0.005958; f) time domain diagram of scheme 1 at ϕ= 0.008342; g) time domain diagram of scheme 2 at ϕ=
0.002383; h) time domain diagram of scheme 2 at ϕ= 0.005958, i) time domain diagram of scheme 2 at ϕ= 0.008342; j) time domain diagram of scheme 3 at ϕ= 0.002383; k) time domain diagram of scheme 3
at ϕ= 0.005958; l) time domain diagram of scheme 3 at ϕ= 0.008342
In order to evaluate the influence of baffle on the pressure fluctuation level of the low specific speed pump, formula of decrease rate for pressure fluctuation is introduced as follows:
where $\stackrel{-}{{C}_{pn}}$ is the average value of pressure fluctuation coefficient at BPF, $n=$ 0, 1, 2, 3.
Fig. 7Frequency distribution of three points at different scheme
a) Frequency distribution of three points at scheme 0
b) Frequency distribution of three points at scheme 1
c) Frequency distribution of three points at scheme 2
d) Frequency distribution of three points at scheme 3
Fig. 8Comparison of fluctuation amplitude at BPF for each point
As can be observed in Table 3, the pressure fluctuation levels were decreased by –2.72 %, –2.13 %, and –2.21 % respectively when the number of baffle in the suction is one, two, three. And ${∆C}_{p}$
of scheme 1 is maximum. It indicates that Adding a baffle to the suction chamber is beneficial to reduce the amplitude of pressure pulsation at BPF in the volute. The best number of baffles in the
suction is one.
Table 3Mean pressure fluctuation amplitude at BPF
Scheme Frequency (Hz) $\stackrel{-}{{C}_{p}}$ ${∆C}_{p}$
0 295 0.01779 0
1 295 0.01732 –2.72 %
2 295 0.01742 –2.13 %
3 295 0.01741 –2.21 %
5. Test verification
The suction of prototype was designed based on the scheme 1 simulation results, and the performance test was performed. Fig. 9 shows the model pump geometry profile and the physical drawing of the
prototype is shown in Fig. 10.
Fig. 9Model pump geometry profile: 1 – suspension, 2 – rear pump cover, 3 – impeller, 4 – inlet flange, 5 – baffle, 6 – pump shaft, 7 – mechanical seal, 8 – bracket
Fig. 10Physical drawing of model pump
Performance experiments of the low centrifugal pump were performed in a closed-loop test platform schematized in Fig. 11(a). The model pump was driven by a variable speed electric AC motor controlled
by a frequency converter. The water was pumped from and returned to reservoir. The shaft torque and rotational speed were monitored by a torque and speed sensor with errors under ±0.10 %. Static
pressure values were measured at the inlet and outlet of the pump by a differential pressure transfer, and the uncertainty was within ±0.10 %. The flow rate was measured by a magnetic flow meter with
the uncertainty less than ±0.14 %. The measurement accuracy of pump efficiency was quantified as ±0.30 %. Scheme and physical map of the test rig are shown in Fig. 11.
Fig. 12. shows the Contrast of head and efficiency between simulation and experiment. The numerical results are compared with experimental results of the super low specific speed centrifugal pump
designed by increasing flow method so as to validate the reliability of CFD. Q-H and Q-$\eta$ curve of the model pump is presented in Fig. 12. A good agreement of the head between numerical results
and experimental results are observed. The head value of simulation and experiment at operational flow rate (labeled at dotted line) are 73.74 m, 75.57 m, respectively. The relative head error
between calculation and experiment values is less than 5 % under all flow rate. The maximum absolute error value of efficiency is no more than 2.3 %. It is worth mentioning that in order to ensure
the repeatability of the test results, the experiment was performed 3 times, and the experimental data were averaged over 3 times. No hump was captured from the H-Q curve of the experiment.
Fig. 11Scheme and physical map of the test rig
a) Scheme of the test rig
b) Physical map of the test rig
Fig. 12Contrast of head and efficiency between simulation and experiment
The efficiency curves between numerical results and experimental results are monotone increasing with flow rate increasing. This is because that the model pump was designed by enlarging flow method
and the maximum flow point presented in Fig. 12 don’t reach the nominal flow rate (30 m^3/h) Efficiency value of the simulation and test at operational flow rate are 38.66 %, 37.88 %, respectively.
It indicts that the calculation is feasible for the model pump and can be used to perform detail analysis.
6. Conclusions
In this paper, a low specific speed centrifugal pump was designed based on the enlarged flow method. Four schemes were designed to research the effect of baffles on performance and unsteady
characteristics of pressure fluctuations of centrifugal pump, and some conclusions are taken as follows:
1) The head value of no baffle in the suction is lower than that with baffles in the suction at each condition point. Hamp point in scheme 0 is at $\phi =$ 0.00596 (1.2 times ${Q}_{o}$). The hump
point in scheme 1, scheme 2, scheme 3 is at 0.8${Q}_{o}$, 1.0${Q}_{o}$, 1.0${Q}_{o}$, respectively. The $\epsilon$ value of scheme 1 is the smallest and the scheme 0 one is the largest in the four
schemes (159 times scheme 1). The scheme 2 and scheme 3 are 90 times and 110 times scheme 1, respectively. The best number of baffles in the suction is one.
2) Six wave troughs are observed clearly at each monitoring point as the impeller rotates in a circle, which corresponds to the 6 blades. The appearance of the former trough is earlier than that of
the latter one at an impeller channel period. Each time the impeller is turned 10 degrees, there are six obvious troughs around the impeller. It shows that the rotor-stator interaction between the
impeller and the volute is the main cause of the pressure pulsation.
3) With the rotation of the impeller, peak value of pressure fluctuations at blade passing frequency (BPF) is gradually decrease, which means amplitude on P36 > P1 > P2 at BPF under all schemes. At
low flow ($\varphi =$0.002383), the main frequency of pressure fluctuation at P36 and P1 under scheme 0, scheme 2 and scheme 3 is 295 Hz, which is corresponding to BPF. the main frequency of P2 under
scheme 0, scheme 2 and scheme 3 is 49.17 Hz. At scheme 1, the fluctuation main frequency is 49.17 Hz at P1 and P2, and the main frequency at P36 is 295 Hz, which are shaft frequency and BPF,
4) The pressure fluctuation levels were decreased by –2.72 %, –2.13 %, and 2.21 % respectively when the number of baffle in the suction is one, two, three, respectively. And ${∆C}_{p}$ of scheme 1 is
maximum. It indicates that Adding a baffle to the suction chamber is beneficial to reduce the amplitude of pressure pulsation at BPF in the volute. The best number of baffles in the suction is one.
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About this article
Flow induced structural vibrations
low specific speed
centrifugal pump
pressure pulsation
numerical simulation
The authors would like to thank the support by the National Natural Science Foundation of China (51239005, 51779106), Jiangsu Industry University Research Cooperation Innovation Fund – Forward Joint
Research Project (BY2015064-10), Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), Open subject of Key Laboratory of Fluid and Power Machinery, Ministry of
Education, Xihua University. (szjj2016-068), Jiangsu Top Six Talent Summit Project (GDZB-017).
Author Contributions
Kaikai Luo performed the data analyses and wrote the manuscript. Yong Wang conceived and designed the experiments. Houlin Liu contributed to the conception of the study. Jie Chen helped perform the
analysis with constructive discussions. Yu Li helped carry out experiments. Jun Yan contributed significantly to analysis and manuscript preparation.
Copyright © 2019 Kaikai Luo, et al.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. | {"url":"https://www.extrica.com/article/18943","timestamp":"2024-11-05T12:06:51Z","content_type":"text/html","content_length":"186570","record_id":"<urn:uuid:ba0d681b-922e-449b-87a8-4dcc54a4a9c4>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00515.warc.gz"} |
Classification Of Numbers (Exciting Facts To Secure 100%) - Logicxonomy
The Classification of Numbers is vast and one of the most important topics asked in competitive examinations. It is loaded with many facts, which are important to keep in mind because even a single
mistake in language-based math problems, is capable of knocking you out from the competition.
Types of Numbers
Number Theory
In the broader sense, numbers are primarily divided into three parts (Classification of Numbers):
(a) Real Numbers
(b) Imaginary Numbers
(c) Complex Numbers
(a) Real Numbers
Real numbers are the set of positive, negative, decimal, fraction, n[th ]roots of any numbers, etc. That can be marked on an infinitely long number line.
Example: $-5, 0, \frac{1}{3}, \sqrt[3]{2}, 1.\overline{26}, 1.543...., 2\frac{3}{5}, etc.$
(b) Imaginary Numbers
When a real number is multiplied by an imaginary unit ‘i (iota)’ (where i=$\sqrt{-1}$), the product is called an Imaginary Number.
Imaginary Numbers are the square root of non-positive real numbers, According to this definition ‘Zero (0)’ is considered the Real and Imaginary Number ($0=\sqrt{0}$).
Note: Zero (0) is neither a positive nor a negative number. It’s a non-positive as well as a non-negative real number. (Classification of Numbers)
Example: $2i, -5i, \sqrt{3}i$ ,etc.
(c) Complex Numbers
A combination of real and imaginary numbers, in the form of ‘a+ib’, is called a Complex Number. Where a & b are Real Numbers and iota (i) is an imaginary unit.
Example: 2+3i (Real part=2, Imaginary Part=3i)
Note: All Real numbers are complex numbers with their imaginary part equal to Zero (0). (Classification of Numbers)
Complex Number Real Number Imaginary Number
-2+3i -2 3i
5-2i 5 -2i
-8i 0 -8i (Purely Imaginary)
4 4 (Purely Real) 0
Real and Imaginary Numbers
Types of Real Numbers
Real Numbers can be classified into two major parts (Classification of Numbers):
(a) Rational Numbers
(b) Irrational Numbers
(a) Rational Numbers
Any number that can be expressed in the form of $\frac{p}{q}$, where p & q are integers (non-decimal, positive, or negative numbers) and q≠0. Rational Numbers include recurring (repeating) and finite
decimal numbers.
According to this definition Zero (0) is a rational number because it can be expressed in the form of $\frac{p}{q}$, where p=0.
Example: $\frac{1}{3} , 1, 0, 7,1.2\overline{3} ,1.333..., 2\frac{3}{4}$, etc.
Note: $\pi=\frac{22}{7}$ is not actually true, it is just an approximate value of (pi), commonly used in calculations. (Classification of Numbers)
$\frac{22}{7}=3.1428571428.....=3.\overline{142857}$ (Recurring Decimal)
$\pi=3.141592653589793238.......$ (Non-Repeating and Non-Terminating)
Since the value of pi($\pi$) is non-terminating and non-repeating, $\pi$ is an irrational number.
Definition of Pi
Let, the length of the circumference of a circle be ‘C’ and the length of its diameter be ‘d’ then
The earliest use of this Greek letter ($\pi$) for the ratio of a Circle’s circumference to its diameter is introduced by British Mathematician William Jones in 1706.
(b) Irrational Numbers
Irrational numbers can’t be expressed in the form of $\frac{p}{q}$, the decimal expansion of an irrational number neither terminates nor repeats.
$\sqrt[n]{m}$ or $m^{\frac{1}{n}}$ , is an irrational number if n and m are positive real numbers but its decimal expansion has an irrational value.
Example: $\pi(pi), e , \varphi, \sqrt{2},\sqrt[3]{9}$, etc.
$e (Euler's Number)=2.71828182......=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+.......$.
$\varphi(Golden Ratio)=1.61803398......=\frac{1+\sqrt{5}}{2}$.
Rational and Irrational Numbers
Numbers Fractional Form Rational?
0 $\frac{0}{q}$ Yes
1.225 $\frac{49}{40}$ Yes
-0.1 $-\frac{1}{10}$ Yes
0.33333….=$0.\overline{3}$ $\frac{1}{3}$ Yes
$\sqrt{2}$ – No
$\sqrt{4}$ $\frac{2}{1}$ Yes
Difference between Rational and Irrational Numbers
Types of Irrational Numbers
Irrational numbers are divided into two parts (Classification of Numbers):
(a) Surds
(b) Transcendental Numbers
(a) Surds
If the value of $\sqrt[n]{x}$ is an irrational number then it’s a Surd. Here x is a rational number.
Click Here for detailed concepts and tricks of Surds and Indices. (Classification of Numbers)
(b) Transcendental Numbers
A number that is not algebraic, is called a Transcendental Number.
Example: $\pi, e, ln(2), 2^{\sqrt{2}}, e^{\pi}, \pi^{e}, i^{i}$, etc.
Note: (Classification of Numbers)
• The natural logarithm (base e) of any natural number other than 1, is a Transcendental Number. Example: ln(2)
• If a and b are algebraic numbers (a≠0,1) but b is irrational then $a^{b}$ is a transcendental number. Example: $2^{\sqrt{2}}$
Concepts of Algebraic Numbers:
Before starting the concept of Algebraic numbers, we have to know about Polynomials and Polynomial Equations.
An expression, composed of variables, constants, and exponents which is combined using arithmetic operators like +, -, ÷, and ×, is called a polynomial.
Here the exponents of the variables must be in whole numbers (0,1,2,3….) means fractions and negative exponents are not allowed.
Operators: +,-
Exponents: 2,1 ($x^{2}, y^{1}$)
Coefficient of variables: 2,3
Variables: x,y
Constant: 7
Expression Polynomial?
5 Yes ($5x^{0}$)
3x Yes
$2y^{2}-\frac{5}{3}x$ Yes
$3xy^{-1}$ No (Negative Exponent)
$\frac{3}{x+2}$ No (Negative Exponent)
$\frac{x}{2}$ Yes
$\sqrt{2}$ Yes
$\sqrt{x}$ No (Fractional Exponent)
Identify Polynomials
Polynomial Equation: When a polynomial expression equals Zero, called Polynomial Equation.
Example: $3x^{2}+5x-2=0$
Algebraic Numbers: It is the solution (root) of a non-zero polynomial equation in one variable, with integer coefficients.
Examples: (i) $x^{2}-x-1=0$ then $x=\frac{1+\sqrt{5}}{2}$
(ii) $x^{4}+4=0$ then $x=(1+i)$
Here both the values of x are algebraic numbers.
Note: $\pi$ can not be the root of a polynomial that has integer coefficients, so it’s a non-algebraic number.
Types of Rational Numbers
The Rational Numbers are divided into three parts: (Classification of Numbers)
(a) Decimal numbers
(b) Fractions
(c) Integers
Types of Decimal Numbers
(a) Decimal Numbers
A decimal number consists of two parts, a whole and a fractional part separated by a ‘.’ (Decimal Point) sign.
Example: 1.25, 2.3, 0.33, 1.41356……, etc.
Decimals can be further divided into two parts: (Classification of Numbers)
(i) Terminating (Finite) Decimals: This type of decimal number has an end digit after a decimal point or has a finite number of decimal places.
Example: 1.5, 2.4172, etc.
(ii) Non-Terminating Decimals: This type of decimal number doesn’t have an end digit or it goes on indefinitely. Non-Terminating decimal numbers can be further divided into two types: (Classification
of Numbers)
1. Repeating (Recurring Decimals): The numbers which are uniformly repeated after the decimal point, are called Recurring decimals. It can be represented by drawing an over-line (Bar Sign) on
repeating digits.
Example: $1.3333....=1.\overline{3}$.
2. Non-Repeating Decimals: This type of number is non-terminating and non-repeating in nature. So, it falls in the category of Irrational Numbers.
Example: 1.2357821….., 0.5732182……, etc.
(b) Fractions
If a whole object is divided into ‘n’ equal parts then each part is termed as a Portion. If we select ‘k’ portions out of these then in fractions it is represented as $\frac{k}{n}$.
k, the number of parts we have
n, the Total number of parts
Example: 3 out of 5 girls can be represented in fractions as $\frac{3}{5}$.
Fractions can be further classified into three parts: (Classification of Numbers)
(i) Proper Fractions: If the numerator value of a fraction is less than its denominator.
Example: $\frac{5}{7}, \frac{1}{2}, \frac{3}{5}$, etc.
(ii) Improper Fractions: If the numerator value of a fraction is greater than or equal to its denominator.
Example: $\frac{5}{4}, \frac{7}{3}, \frac{11}{11}$, etc.
(iii) Mixed fractions: When a fraction consists of two parts, a whole number, and a fractional part then it is called a Mixed Fraction.
Example: $2\frac{5}{7}, 8\frac{1}{2}$, etc.
# How to convert an Improper Fraction into a Mixed Fraction (Classification of Numbers)
Que: Convert $\frac{13}{5}$ into a Mixed fraction?
Step 1: Divide 13 by 5.
Step 2: Mixed fraction= $Quotient\frac{Remainder}{Divisor}=2\frac{3}{5}$.
# How to convert a Mixed Fraction into an improper Fraction (Classification of Numbers)
(c) Integers
Integer means whole or intact. The positive or Negative numbers including Zero (0), which have no decimal or fractional part are called Integers.
It is expressed as Z={….., -2, -1, 0, 1, 2, …….}
Integers are divided into two parts:
1. Whole Numbers: A set of all positive integers and Zero (0) is called Whole Numbers.
W={0,1,2,3,4,5…… }
Whole numbers are further divided into two parts:
(i) Zero(0)
(ii) Natural Numbers: A set of all positive integers is called Natural Number. N={1,2,3,4,…… }
2. Negative Numbers: A set of all negative integers is called Negative Numbers (-1,-2,-3,-4,….. ).
Numeral Symbols
R, Real Numbers
Z, Integers
N, Natural Numbers
Q, Rational Numbers
P, Irrational Numbers
C, Complex Numbers
W, Whole Numbers
Discrete and Continuous Numbers
Natural numbers, Whole numbers, Integers, and Rational Numbers are Discrete Numbers. Discrete numbers are finite, numeric, countable, and non-negative Integers.
Real Numbers are continuous Numbers because it is impossible to count real numbers existing between any two real numbers (it’s infinite).
Very Special Numbers
1. Even and Odd Numbers
2. Prime Numbers
3. Co-Prime Numbers
4. Composite Numbers
5. Perfect Numbers
6. Deficient Numbers
7. Abundant Numbers
8. Happy Numbers
9. Narcissistic Numbers
10. Palindromic Numbers
11. Fermat Number
12. Kaprekar Number
13. Sierpinski Numbers
14. Riesel Numbers
15. Sociable Numbers
16. Harshad/Niven Numbers
17. Mersenne Numbers
18. Armstrong Numbers
19. Wieferich Prime Numbers ……. and many more
Click the link for details: Types of Numbers
Frequently Asked Questions
1. What are the types of natural numbers?
The Classification of Numbers: Prime numbers, Composite numbers, Co-Prime Numbers, Perfect numbers, Abundant numbers, Palindromic numbers, even-odd numbers, etc.
2. What are the symbols of different types of numbers?
The Classification of Numbers: Real Numbers (R), Integers (Z), Natural Numbers (N), Rational Numbers (Q), Irrational Numbers (P), Complex Numbers (C), and Whole Numbers (W).
3. What is the difference between rational numbers and irrational numbers?
Rational numbers can be expressed in fractional form but irrational numbers can’t.
4. Zero (0) is a real number or imaginary?
As we know, the square root of a non-positive real number is called an Imaginary number and Zero is neither a negative nor positive, real number. Thus Zero is the Real and Imaginary number.
5. What are complex numbers?
The numbers which can be expressed in the form of a+ib where a & b are Real Numbers, imaginary unit i=$\sqrt{-1}$. All real numbers are complex numbers with their imaginary parts equal to zero.
6. Zero is a rational or irrational number?
Zero can be expressed in the fractional form where the numerator is equal to zero so Zero is a rational number.
7. 𝜋 (pi) is a rational or irrational number?
𝜋 (pi) is an irrational number because its decimal expansion is non-terminating and non-repeating.
8. What are the properties of 𝜋 (pi)?
𝜋 (pi) is an irrational number, a Real number, a Transcendental number, and a non-algebraic number.
9. What is a Polynomial?
An expression composed of variables, constants, and exponents, combined with arithmetic operators (+,-,÷,×) is called a Polynomial. The exponents of variables should be in whole numbers.
Classification of Numbers (MCQ)
20 multiple choice questions with solutions on the concepts of Classification of numbers and Simplification: | {"url":"https://logicxonomy.com/classification-of-numbers/","timestamp":"2024-11-08T14:32:47Z","content_type":"text/html","content_length":"254831","record_id":"<urn:uuid:6def1356-9ece-416b-aa12-9bab2942055d>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00574.warc.gz"} |
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• answers prentice hall | {"url":"https://softmath.com/math-com-calculator/function-range/how-to-divide-radicals.html","timestamp":"2024-11-10T19:21:16Z","content_type":"text/html","content_length":"190468","record_id":"<urn:uuid:912026a4-3e59-4d3e-a988-265cdd599f50>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00468.warc.gz"} |
Physics Problem Set: Tensions and Moments of Inertia | Assignments Physics | Docsity
Download Physics Problem Set: Tensions and Moments of Inertia and more Assignments Physics in PDF only on Docsity! Name Phys 2010 (NSCC), Fall 2005 Problem Set #11 Ye Olde Shoppe 30o T 1. A 500-N
sign is suspended from a horizontal 6.00-meter long uniform 100-N rod as indicated at the right. The sign is attached to the rod at a point which is 4.0 m from the wall; The left end of the rod is
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N is supported by two wires as shown at the right. What is the tension in each of the cables supporting the rod? 1 T1 T3 40oT2 3. The rod from Problem 2 is now supported (horizontally) by three
ropes, as shown to the right. Find the tensions in all the ropes. 35o x 4. A uniform beam of weight 500 N, 6.00 m in length is supported by a cable at the far end as shown at the right. The cable
will break if its tension exceeds 1300 N A man of weight 700 N goes walking out on the beam. How away from the wall can he get before the cable breaks? 2 | {"url":"https://www.docsity.com/en/docs/problems-set-11-for-algebra-based-physics-i-phys-2010/6349768/","timestamp":"2024-11-14T17:07:55Z","content_type":"text/html","content_length":"229225","record_id":"<urn:uuid:fc1963c7-d768-4e2b-ad23-d3454c0749a4>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00359.warc.gz"} |
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2023.66
URN: urn:nbn:de:0030-drops-186007
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18600/
Mayr, Peter
On the Complexity Dichotomy for the Satisfiability of Systems of Term Equations over Finite Algebras
For a fixed finite algebra ?, we consider the decision problem SysTerm(?): does a given system of term equations have a solution in ?? This is equivalent to a constraint satisfaction problem (CSP)
for a relational structure whose relations are the graphs of the basic operations of ?. From the complexity dichotomy for CSP over fixed finite templates due to Bulatov [Bulatov, 2017] and Zhuk
[Zhuk, 2017], it follows that SysTerm(?) for a finite algebra ? is in P if ? has a not necessarily idempotent Taylor polymorphism and is NP-complete otherwise. More explicitly, we show that for a
finite algebra ? in a congruence modular variety (e.g. for a quasigroup), SysTerm(?) is in P if the core of ? is abelian and is NP-complete otherwise. Given ? by the graphs of its basic operations,
we show that this condition for tractability can be decided in quasi-polynomial time.
BibTeX - Entry
author = {Mayr, Peter},
title = {{On the Complexity Dichotomy for the Satisfiability of Systems of Term Equations over Finite Algebras}},
booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
pages = {66:1--66:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-292-1},
ISSN = {1868-8969},
year = {2023},
volume = {272},
editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18600},
URN = {urn:nbn:de:0030-drops-186007},
doi = {10.4230/LIPIcs.MFCS.2023.66},
annote = {Keywords: systems of equations, general algebras, constraint satisfaction}
Keywords: systems of equations, general algebras, constraint satisfaction
Collection: 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)
Issue Date: 2023
Date of publication: 21.08.2023
DROPS-Home | Fulltext Search | Imprint | Privacy | {"url":"http://dagstuhl.sunsite.rwth-aachen.de/opus/frontdoor.php?source_opus=18600","timestamp":"2024-11-06T20:10:57Z","content_type":"text/html","content_length":"5968","record_id":"<urn:uuid:e9a8a721-9c77-4df3-8823-049d85c937c1>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00759.warc.gz"} |
Risk Analysis of Portfolios Under Uncertainty: Minimizing Average Rates of Falling
Yuji Yoshida
Faculty of Economics and Business Administration, University of Kitakyushu, 4-2-1 Kitagata, Kokuraminami, Kitakyushu 802-8577, Japan
February 12, 2010
April 22, 2010
January 20, 2011
average value-at-risk, risk-sensitive portfolio, fuzzy random variable, perception-based extension, probability of bankruptcy
A portfolio model to minimize the risk of falling under uncertainty is discussed. The risk of falling is represented by the value-at-risk of rate of return. Introducing the perception-based
extension of the average value-at-risk, this paper formulates a portfolio problem to minimize the risk of falling with fuzzy random variables. In the proposed model, randomness and fuzziness are
evaluated respectively by the probabilistic expectation and the mean with evaluation weights and λ-mean functions. The analytical solutions of the portfolio problem regarding the risk of falling
are given. This paper gives formulae to show the explicit relations among the following important parameters in portfolio: the expected rate of return, the risk probability of falling and
bankruptcy, and the average rate of falling regarding the asset prices. A numerical example is given to explain how to obtain the optimal portfolio and these parameters from the asset prices in
the stock market.
Cite this article as:
Y. Yoshida, “Risk Analysis of Portfolios Under Uncertainty: Minimizing Average Rates of Falling,” J. Adv. Comput. Intell. Intell. Inform., Vol.15 No.1, pp. 56-62, 2011.
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21. [21] Y. Yoshida, M. Yasuda, J. Nakagami, and M. Kurano, “A new evaluation of mean value for fuzzy numbers and its application to American put option under uncertainty,” Fuzzy Sets and
Systems, Vol.157, pp. 2614-2626, 2006.
22. [22] M. Inuiguchi and T. Tanino, “Portfolio selection under independent possibilistic information,” Fuzzy Sets and Systems, Vol.115, pp. 83-92, 2000. | {"url":"https://www.fujipress.jp/jaciii/jc/jacii001500010056/?lang=ja","timestamp":"2024-11-14T18:32:30Z","content_type":"text/html","content_length":"48216","record_id":"<urn:uuid:4ce93b99-46f1-4873-8ecc-0eb0c89fcb3b>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00409.warc.gz"} |
How do you explain Bland Altman plot?
The Bland–Altman plot is a method for comparing two measurements of the same variable. The concept is that X-axis is the mean of your two measurements, and the Y-axis is the difference between the
two measurements.
How do you interpret contract limits?
The limits of agreement estimate the interval that a given proportion of differences between measurements is likely to lie within. The limits can be used to determine if the methods can be used
interchangeably, or if a new method can replace an old method without changing the interpretation of the results.
Why is Bland Altman better than correlation?
Correlation analysis may lead to incorrect or debated results in comparison of two measurement methods. The Bland-Altman analysis is a simple and accurate way to quantify agreement between two
variables and may help clinicians to compare a new measurement method against another one or a reference standard.
When would you use a Bland Altman plot?
Bland–Altman plots are extensively used to evaluate the agreement among two different instruments or two measurements techniques. Bland–Altman plots allow identification of any systematic difference
between the measurements (i.e., fixed bias) or possible outliers.
Why use a Bland Altman plot?
A Bland-Altman plot is a useful display of the relationship between two paired variables using the same scale. It allows you to perceive a phenomenon but does not test it, that is, does not give a
probability of error on a decision about the variables as would a test.
What is Bland Altman test used for?
How do you calculate Bland-Altman limits of agreement?
The Bland–Altman method calculates the mean difference between two methods of measurement (the ‘bias’), and 95% limits of agreement as the mean difference (2 sd) [or more precisely (1.96 sd)]. It is
expected that the 95% limits include 95% of differences between the two measurement methods.
How do you do a Bland-Altman plot in R?
How to Create a Bland-Altman Plot in R (Step-by-Step)
1. Step 1: Create the Data.
2. Step 2: Calculate the Difference in Measurements.
3. Step 3: Calculate the Average Difference & Confidence Interval.
4. Step 4: Create the Bland-Altman Plot.
Which is an example of a Bland Altman plot?
What is a Bland-Altman Plot? (Definition & Example) A Bland-Altman plot is used to visualize the differences in measurements between two different instruments or two different measurement techniques.
How is the bias determined in Bland Altman?
The first page of Bland-Altman results shows the difference and average values and is used to create the plot. The second results page shows the average bias, or the average of the differences. The
bias is computed as the value determined by one method minus the value determined by the other method.
How is the B and a plot analysis used?
The B&A plot analysis is a simple way to evaluate a bias between the mean differences, and to estimate an agreement interval, within which 95% of the differences of the second method, compared to the
first one, fall. Data can be analyzed both as unit differences plot and as percentage differences plot.
What does a B plot for no CAEP mean?
A B-A plot for No CAEP paired with CAEP1. In this type, the cluster of points may lie above or below the mean, indicating an offset, a bias, a systematic error. It might be wise to test means. | {"url":"https://mysqlpreacher.com/how-do-you-explain-bland-altman-plot/","timestamp":"2024-11-03T22:31:59Z","content_type":"text/html","content_length":"107227","record_id":"<urn:uuid:2467d265-7a6b-4ed0-b241-5435e93eb84e>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00749.warc.gz"} |
Statistical Interpretation of the Wave Function
About Articles Books Lectures Presentations Glossary Cite page Help? Translate
Philosophers Statistical Interpretation of the Wave Function
Mortimer Adler It is often said that Max Born gave us the “statistical interpretation” of quantum
Rogers Albritton mechanics that lies at the heart of Niels Bohr’s and Werner Heisenberg’s principle of
Alexander of complementarity and so was included in their “Copenhagen Interpretation” of quantum
Aphrodisias mechanics. But Born himself said many times he had only applied an idea of Albert Einstein
Samuel Alexander that had circulated privately for many years. To be sure, Born and Einstein quarreled for
William Alston years over determinism and causality, but as we have seen, it was Einstein who in 1916
Anaximander discovered “chance” in the statistical or "chance" interaction of matter and radiation,
G.E.M.Anscombe even if he considered chance a “weakness in the theory.” Probability and statistics were
Anselm very important in the two centuries before Born’s work, but most physicists and
Louise Antony philosophers saw the implied randomness to be “epistemic,” the consequence of human
Thomas Aquinas ignorance. Random distributions of all kinds were thought to be completely deterministic at
Aristotle the particle level, with collisions between atoms following Newton’s time-reversible
David Armstrong dynamical laws. Ludwig Boltzmann’s transport equation and H-Theorem showed that the
Harald increase of entropy is statistically irreversible at the macroscopic level, even if the
Atmanspacher motions of individual particles were time reversible. Boltzmann did speculate that there
Robert Audi might be some kind of molecular “chaos” or “disorder” that could cause particles traveling
Augustine between collisions to lose the “correlations” or information about their past paths that
J.L.Austin would be needed for the paths to be time reversible and deterministic, but nothing came of
A.J.Ayer this idea. In his early career, Erwin Schrödinger was a great exponent of fundamental
Alexander Bain chance in the universe. He followed his mentor Franz S. Exner, who as a colleague of
Mark Balaguer Boltzmann at the University of Vienna was a great promoter of statistical thinking. In his
Jeffrey Barrett inaugural lecture at Zurich in 1922, Schrödinger argued that available evidence can not
William Barrett justify our assumptions that physical laws are deterministic and strictly causal. His
William Belsham inaugural lecture was modeled on that of Exner in 1908.
Henri Bergson
George Berkeley Exner’s assertion amounts to this: It is quite possible that Nature’s laws are of
Isaiah Berlin thoroughly statistical character. The demand for an absolute law in the background of
Richard J. the statistical law — a demand which at the present day almost everybody considers
Bernstein imperative — goes beyond the reach of experience.
Bernard Berofsky
Robert Bishop Exner and Boltzmann both said that determinism goes beyond experience
Max Black
Susanne Bobzien Such a dual foundation for the orderly course of events in Nature is in itself
Emil du improbable. The burden of proof falls on those who champion absolute causality, and not
Bois-Reymond on those who question it. For a doubtful attitude in this respect is to-day by far the
Hilary Bok more natural.
Laurence BonJour
George Boole Several years later, Schrödinger presented a paper on “Indeterminism in Physics” to the
Émile Boutroux June, 1931 Congress of A Society for Philosophical Instruction in Berlin. He supported the
Daniel Boyd idea of Boltzmann that “an actual continuum must consist of an infinite number of parts;
F.H.Bradley but an infinite number is undefinable..”
Michael Burke If nature is more complicated than a game of chess, a belief to which one tends to
Lawrence Cahoone incline, then a physical system cannot be determined by a finite number of
C.A.Campbell observations. But in practice a finite number of observations is all that we can make.
Joseph Keim All that is left to determinism is to believe that an infinite accumulation of
Campbell observations would in principle enable it completely to determine the system. Such was
Rudolf Carnap the standpoint and view of classical physics, which latter certainly had a right to see
Carneades what it could make of it. But the opposite standpoint has an equal justification: we
Nancy Cartwright are not compelled to assume that an infinite number of observations, which cannot in
Gregg Caruso any case be carried out in practice, would suffice to give us a complete determination.
Ernst Cassirer
David Chalmers In the history of science it is hard to find ears more likely to be sympathetic to a new
Roderick Chisholm idea than Schrödinger should have been to Max Born’s suggestion that the absolute modulus
Chrysippus of the amplitude of Schrödinger’s wave function |ψ|^2 should be interpreted statistically
Cicero as the likelihood of finding a particle. And Schrödinger should have known Einstein thought
Tom Clark quantum mechanics is statistical. Yet Schrödinger objected strenuously, not so much to the
Randolph Clarke probability and statistics as to the conviction of Born and his brilliant student
Samuel Clarke Heisenberg that quantum phenomena, like quantum jumps between atomic energy levels, were
Anthony Collins only predictable statistically, and that there is a fundamental indeterminacy in the
Antonella classical idea that particles have simultaneously knowable exact positions and velocities
Corradini (momenta). Born, Heisenberg, and Bohr had declared classical determinism and causality
Diodorus Cronus untrue of the physical world. It is likely that Schrödinger was ecstatic that his wave
Jonathan Dancy equation implied a deterministic physical theory. His wave function ψ evolves in time to
Donald Davidson give exact values for itself for all times and places. Perhaps Schrödinger thought that the
Mario De Caro waves themselves could provide a field theory of physics, much as fields in Newton's
Democritus gravitational theory and in Maxwell's electromagnetic theory provide complete descriptions
Daniel Dennett of nature. Schrödinger began to wonder whether nature might be only waves, no particles. In
Jacques Derrida July of 1926, Born used Louis de Broglie’s matter waves for electrons, as described by
René Descartes Schrödinger’s wave equation, but he interpreted the wave as the probability of finding an
Richard Double electron going off in a specific collision direction, proportional to the square of the
Fred Dretske wave function ψ, with ψnow seen as a "probability amplitude." Born's interpretation of the
John Dupré quantum mechanical wave function of a material particle as the probability (amplitude) of
John Earman finding the material particle was a direct extension of Einstein's interpretation of light
Laura Waddell waves giving probability of finding photons. To be sure, Einstein's interpretation may be
Ekstrom considered only qualitative, where Born's was quantitative, since the new quantum mechanics
Epictetus now allowed exact calculations and measurements to test its predictions. Born initially
Epicurus gave Einstein full credit for the statistical interpretation as Einstein had described his
Austin Farrer "ghost field" idea. Although the original idea is pure Einstein, it is widely referred to
Herbert Feigl today as “Born’s statistical interpretation” or "Born's Rule," another example of others
Arthur Fine getting credit for a concept first seen by Einstein. Born described his insights in 1926,
John Martin
Fischer Collision processes not only yield the most convincing experimental proof of the basic
Frederic Fitch assumptions of quantum theory, but also seem suitable for explaining the physical
Owen Flanagan meaning of the formal laws of the so-called “quantum mechanics.”... The matrix form of
Luciano Floridi quantum mechanics that was founded by Heisenberg and developed by him and the author of
Philippa Foot this article starts from the thought that an exact representation of processes in space
Alfred Fouilleé and time is quite impossible and that one must then content oneself with presenting the
Harry Frankfurt relations between the observed quantities, which can only be interpreted as properties
Richard L. of the motions in the limiting classical cases. On the other hand, Schrödinger (3)
Franklin seems to have ascribed a reality of the same kind that light waves possessed to the
Bas van Fraassen waves that he regards as the carriers of atomic processes by using the de Broglie
Michael Frede procedure; he attempts “to construct wave packets that have relatively small dimensions
Gottlob Frege in all directions,” and which can obviously represent the moving corpuscle directly.
Peter Geach Neither of these viewpoints seems satisfactory to me. Here, I would like to try to give
Edmund Gettier a third interpretation and probe its utility in collision processes. I shall recall a
Carl Ginet remark that Einstein made about the behavior of the wave field and light quanta. He
Alvin Goldman said that perhaps the waves only have to be wherever one needs to know the path of the
Gorgias corpuscular light quanta, and in that sense, he spoke of a “ghost field.” It determines
Nicholas St. John the probability that a light quantum - viz., the carrier of energy and impulse –
Green follows a certain path; however, the field itself is ascribed no energy and no impulse.
H.Paul Grice One would do better to postpone these thoughts, when coupled directly to quantum
Ian Hacking mechanics, until the place of the electromagnetic field in the formalism has been
Ishtiyaque Haji established. However, from the complete analogy between light quanta and electrons, one
Stuart Hampshire might consider formulating the laws of electron motion in a similar manner. This is
W.F.R.Hardie closely related to regarding the de Broglie-Schrödinger waves as “ghost fields,” or
Sam Harris better yet, “guiding fields.” I would then like to pursue the following idea
William Hasker heuristically: The guiding field, which is represented by a scalar function ψ of the
R.M.Hare coordinates of all particles that are involved and time, propagates according to
Georg W.F. Hegel Schrödinger’s differential equation. However, impulse and energy will be carried along
Martin Heidegger as when corpuscles (i.e., electrons) are actually flying around. The paths of these
Heraclitus corpuscles are determined only to the extent that they are constrained by the law of
R.E.Hobart energy and impulse; moreover, only a probability that a certain path will be followed
Thomas Hobbes will be determined by the function ψ. One can perhaps summarize this, somewhat
David Hodgson paradoxically, as: The motion of the particle follows the laws of probability, but the
Shadsworth probability itself propagates in accord with causal laws.
Baron d'Holbach This last sentence is a remarkably concise description of the dualism in quantum mechanics,
Ted Honderich a strange mixture of indeterminism and determinism, of chance and necessity. In his 1948
Pamela Huby Waynflete lectures, Born elaborated on his understanding of chance,
David Hume
Ferenc Huoranszki There is no doubt that the formalism of quantum mechanics and its statistical
Frank Jackson interpretation are extremely successful in ordering and predicting physical
William James experiences. But can our desire of understanding, our wish to explain things, be
Lord Kames satisfied by a theory which is frankly and shamelessly statistical and indeterministic?
Robert Kane Can we be content with accepting chance, not cause, as the supreme law of the physical
Immanuel Kant world? To this last question I answer that not causality, properly understood, is
Tomis Kapitan eliminated, but only a traditional interpretation of it, consisting in its
Walter Kaufmann identification with determinism. I have taken pains to show that these two concepts are
Jaegwon Kim not identical. Causality in my definition is the postulate that one physical situation
William King depends on the other, and causal research means the discovery of such dependence. This
Hilary Kornblith is still true in quantum physics, though the objects of observation for which a
Christine dependence is claimed are different: they are the probabilities of elementary events,
Korsgaard not those single events themselves.
Saul Kripke
Thomas Kuhn Ever since 1930, when Born's young graduate student Heisenberg had been selected for the
Andrea Lavazza Nobel Prize in physics although much of the theory was his own work, Born felt he had been
Christoph Lehner treated unfairly. He finally received recognition, with the Nobel Prize for physics in
Keith Lehrer 1954, for his "statistical interpretation." But Born's voluminous correspondence with
Gottfried Leibniz Einstein reveals that he had perhaps come to think that Einstein's supposed determinism
Jules Lequyer meant Einstein did not believe in the statistical nature of quantum physics, so this idea
Leucippus may now rightfully belong to Born. He called it "his own" in the 1950's.
Michael Levin
Joseph Levine Normal | Teacher | Scholar
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Free Will
Mental Causation
James Symposium
Exner’s assertion amounts to this: It is quite possible that Nature’s laws are of thoroughly statistical character. The demand for an absolute law in the background of the statistical law — a demand
which at the present day almost everybody considers imperative — goes beyond the reach of experience. Exner and Boltzmann both said that determinism goes beyond experience Such a dual foundation for
the orderly course of events in Nature is in itself improbable. The burden of proof falls on those who champion absolute causality, and not on those who question it. For a doubtful attitude in this
respect is to-day by far the more natural.
Exner and Boltzmann both said that determinism goes beyond experience
If nature is more complicated than a game of chess, a belief to which one tends to incline, then a physical system cannot be determined by a finite number of observations. But in practice a finite
number of observations is all that we can make. All that is left to determinism is to believe that an infinite accumulation of observations would in principle enable it completely to determine the
system. Such was the standpoint and view of classical physics, which latter certainly had a right to see what it could make of it. But the opposite standpoint has an equal justification: we are not
compelled to assume that an infinite number of observations, which cannot in any case be carried out in practice, would suffice to give us a complete determination.
Collision processes not only yield the most convincing experimental proof of the basic assumptions of quantum theory, but also seem suitable for explaining the physical meaning of the formal laws of
the so-called “quantum mechanics.”... The matrix form of quantum mechanics that was founded by Heisenberg and developed by him and the author of this article starts from the thought that an exact
representation of processes in space and time is quite impossible and that one must then content oneself with presenting the relations between the observed quantities, which can only be interpreted
as properties of the motions in the limiting classical cases. On the other hand, Schrödinger (3) seems to have ascribed a reality of the same kind that light waves possessed to the waves that he
regards as the carriers of atomic processes by using the de Broglie procedure; he attempts “to construct wave packets that have relatively small dimensions in all directions,” and which can obviously
represent the moving corpuscle directly. Neither of these viewpoints seems satisfactory to me. Here, I would like to try to give a third interpretation and probe its utility in collision processes. I
shall recall a remark that Einstein made about the behavior of the wave field and light quanta. He said that perhaps the waves only have to be wherever one needs to know the path of the corpuscular
light quanta, and in that sense, he spoke of a “ghost field.” It determines the probability that a light quantum - viz., the carrier of energy and impulse – follows a certain path; however, the field
itself is ascribed no energy and no impulse. One would do better to postpone these thoughts, when coupled directly to quantum mechanics, until the place of the electromagnetic field in the formalism
has been established. However, from the complete analogy between light quanta and electrons, one might consider formulating the laws of electron motion in a similar manner. This is closely related to
regarding the de Broglie-Schrödinger waves as “ghost fields,” or better yet, “guiding fields.” I would then like to pursue the following idea heuristically: The guiding field, which is represented by
a scalar function ψ of the coordinates of all particles that are involved and time, propagates according to Schrödinger’s differential equation. However, impulse and energy will be carried along as
when corpuscles (i.e., electrons) are actually flying around. The paths of these corpuscles are determined only to the extent that they are constrained by the law of energy and impulse; moreover,
only a probability that a certain path will be followed will be determined by the function ψ. One can perhaps summarize this, somewhat paradoxically, as: The motion of the particle follows the laws
of probability, but the probability itself propagates in accord with causal laws.
There is no doubt that the formalism of quantum mechanics and its statistical interpretation are extremely successful in ordering and predicting physical experiences. But can our desire of
understanding, our wish to explain things, be satisfied by a theory which is frankly and shamelessly statistical and indeterministic? Can we be content with accepting chance, not cause, as the
supreme law of the physical world? To this last question I answer that not causality, properly understood, is eliminated, but only a traditional interpretation of it, consisting in its identification
with determinism. I have taken pains to show that these two concepts are not identical. Causality in my definition is the postulate that one physical situation depends on the other, and causal
research means the discovery of such dependence. This is still true in quantum physics, though the objects of observation for which a dependence is claimed are different: they are the probabilities
of elementary events, not those single events themselves. | {"url":"https://www.informationphilosopher.com/quantum/statistical/","timestamp":"2024-11-10T09:14:06Z","content_type":"text/html","content_length":"106912","record_id":"<urn:uuid:d1453294-2c83-4b8e-ad80-f42fd03b7750>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00144.warc.gz"} |
The class Triangulation_cw_ccw_2 offer two functions int cw(int i) and int ccw(int i) which given the index of a vertex in a face compute the index of the next vertex of the same face in clockwise or
counterclockwise order. This works also for neighbor indexes. Thus, for example the neighbor neighbor(cw(i)) of a face f is the neighbor which is next to neighbor(i) turning clockwise around f. The
face neighbor(cw(i)) is also the first face encountered after f when turning clockwise around vertex i of f.
Many of the classes in the triangulation package inherit from Triangulation_cw_ccw_2. This is for instance the case for CGAL::Triangulation_2<Traits,Tds>::Face. Thus, for example the neighbor
neighbor(cw(i)) of a face f is the neighbor which is next to neighbor(i) turning clockwise around f. The face neighbor(cw(i)) is also the first face encountered after f when turning clockwise around
vertex i of f.
Figure 37.17: Vertices and neighbors.
#include <CGAL/Triangulation_2.h>
Triangulation_cw_ccw_2 a;
default constructor.
int a.ccw ( const int i) const returns the index of the neighbor or vertex that is next to the neighbor or vertex with index i in counterclockwise order around a face.
int a.cw ( const int i) const returns the index of the neighbor or vertex that is next to the neighbor or vertex with index i in counterclockwise order around a face.
See Also | {"url":"https://doc.cgal.org/Manual/latest/doc_html/cgal_manual/Triangulation_2_ref/Class_Triangulation_cw_ccw_2.html","timestamp":"2024-11-03T22:39:00Z","content_type":"text/html","content_length":"6492","record_id":"<urn:uuid:f3ba5c58-ced4-4924-8f72-545413d6a8ce>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00439.warc.gz"} |
Time Value of Money
Time Value of Money Calculator
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Included are:
- amortizations
- annuities
- loans
- leases
- mortgages
- savings accounts
In addition to the standard 30/360 calendar used in standard TVM calculations, it also offers the option of calculating payment based on a 30/365 and 30/365.25 calendar.
Cash Inflows and Outflows
In time value of money problems, positive and negative numbers have different meanings: positive numbers are inflows of cash (cash received) while negative numbers are outflows (cash paid). See
Understanding Cash Flows below for more details.
To calculate, enter the data you know then select ? button on the row you don't.
Time Value of Money
- Present Value: Current value of the annuity.
- Future Value: Future value of the annuity.
- Payment: Periodic payment for the annuity.
- Interest/Year: Interest per year as a percentage.
- Periods: Number of total periods. This number is the number of years and months times the periods per year. For example, if the loan is 4 years with 12 payments per year (monthly payments), periods
should be 48 (4 x 12).
- Periods/Year: Number of payment periods per year. For example, if payments are made quarterly, periods per year should be 4.
- Compounds/Year: Number of interest compounding periods per year. Most of the time, compounding periods per year should equal payment periods per year. For example, if payments are made monthly and
interest is compounded monthly, compounding periods per year and periods per year should both be 12.
- Payment Timing: Payments occur at the beginning or end of the period. Payments made at the beginning of the period are called Annuity Due. Most leases are this kind. A payment made at the end of
the period is called an Ordinary Annuity. Most loans are this kind.
- Begin Period: Starting period to calculate the amortization information.
- End Period: Ending period to calculate the amortization information.
- Interest Paid: Total interest paid over the amortization period.
- Principal Paid: Total principal paid over the amortization period.
- Payment Paid: Total payments made over the amortization period.
- End Balance: Balance at the end of the amortization period.
Amortizations always round to 2 decimal places.
- Years: Number of years (Periods / Periods per Year)
- Total Interest: Total interest paid during the entire amortization period
- Total PI: Total principal and interest paid during the entire amortization period
- 30/365 Payment: Calculates the payment based on a 30 days per month/365 days per year (30/365) calendar basis. Standard TVM calculates on a 30/360 basic.
- 30/365¼ Payment: Calculates the payment based on a 30 days per month/365½ days per year (30/365½) calendar basis. Standard TVM calculates on a 30/360 basic.
Understanding Cash Flows
To further understand the cash flow model, here is an example of a timeline. Note that inflows of cash are treated as positive amounts (designated by no sign or a [+] sign) and outflows of cash as
negative amounts (designated by a [–] sign).
[https://poweronecalc.s3.amazonaws.com/templates/tvm_loan.png | loan example]
In Time Value of Money problems the interval between cash flows are always the same and the payment amounts are always the same. In this example, the borrower receives an initial, Present Value (PV)
amount followed by subsequent payments (PMT) made back to the lending institution, each an equal distance of time apart (say, one month). This is a typical loan or mortgage scenario.
Lease Example:
[https://poweronecalc.s3.amazonaws.com/templates/tvm_lease.png | lease example]
Investment Example [DEP=deposit, FV=future value]:
[https://poweronecalc.s3.amazonaws.com/templates/tvm_investment.png | investment example]
Balloon Payment Example:
[https://poweronecalc.s3.amazonaws.com/templates/tvm_balloon.png | balloon payment example]
Car Loan
When purchasing a new car, the auto dealer has offered a 12.5% interest rate over 36 months on a $7,500 loan. What will be the monthly payment?
- Payment Timing: End
- Present Value: 7,500
- Future Value: 0
- Interest/Year: 12.5%
- Periods: 36 [3 years * 12 periods/yr]
- Periods/Year: 12
- Compounds/Year: 12
Select ? on Payment row. The payment is -$250.90 per month. It is negative because it's a cash outflow.
Car Loan, Amortization
Continued from 'Car Loan' example, how much principal was paid in the first 12 periods?
- Beg Period: 1
- End Period: 12
The principal paid is -$2,196.29. It is negative because it's a cash outflow.
Retirement Annuity
With 35 years until retirement and $15,000 in the bank, how much would have to be put aside at the beginning of each month to reach $2.5 million if an interest rate of 10% can be expected?
- Payment Timing: Begin
- Present Value: -15,000
- Future Value: 2,500,000
- Interest/Year: 10.0%
- Periods: 420 [35 years * 12 periods/yr]
- Periods/Year: 12
- Compounds/Year: 12
Select ? on Payment row. The payment is -$525.15 per month. It is negative because it's a cash outflow.
Savings Account
With $3,000 in a savings account and 3.75% interest, how many months does it take to reach $4,000?
- Payment Timing: End
- Present Value: -3,000
- Future Value: 4,000
- Payment: 0
- Interest/Year: 3.75%
- Periods/Year: 12
- Compounds/Year: 12
Select ? on Periods row. To reach $4,000, it will take 92.2 periods (or 92.20 / 12 = 7.68 years).
You have decided to buy a house but you only have $900 to spend each month on a 30-year mortgage. The bank has quoted an interest rate of 8.75%. What is the maximum purchase price?
- Payment Timing: End
- Future Value: 0
- Payment: -900
- Interest/Year: 8.75%
- Periods: 360 [30 years * 12 periods/yr]
- Periods/Year: 12
- Compounds/Year: 12
Select ? on Present Value row. You can afford a home with a price of $114,401.87.
Mortgage w/Balloon Payment
Continued from 'Mortgage' example, you realize that you will only own the house for about 5 years and then sell it. How much will the balloon payment (the repayment to the bank) be?
- Periods: 60 [5 years * 12 periods/yr]
Select ? on Future Value row. The balloon payment will be $109,469.92 after five years.
Canadian Mortgages
Canadian mortgages compound interest twice per year instead of monthly. What is the monthly payment to fully amortize a 30-year, CA$80,000 Canadian mortgage if the interest rate is 12%?
- Payment Timing: End
- Present Value: 80,000
- Future Value: 0
- Interest/Year: 12.0%
- Periods: 360 [30 years * 12 periods/yr]
- Periods/Year: 12
- Compounds/Year: 2
Select ? on Payment row. The payment is -CA$805.11. It is negative because it is a cash outflow.
Bi-Weekly Mortgage Payments
A buyer is considering a $100,000 home loan with monthly payments, an annual interest rate of 9% and a term of 30 years. Instead of making monthly payments, the buyer realizes that he can build
equity faster by making bi-weekly payments (every two weeks). How long will it take to pay off the loan?
Part 1: Calculate Monthly Payment
- Payment Timing: End
- Present Value: 100,000
- Future Value: 0
- Interest/Year: 9.0%
- Periods: 360 [30 years * 12 periods/yr]
- Periods/Year: 12
- Compounds/Year: 12
Select ? on Payment row. The monthly payment is -$804.62. It is negative because it is a cash outflow.
Part 2: Periods/Bi-Weekly Payments
- Payment: -402.31 (Payment / 2)
- Periods/Year: 26
Select ? on Periods row. Calculating shows periods equal to 567.4 periods (567.40 ¸ 26 = 21.82 years).
APR of Loan with Fees
The Annual Percentage Rate (APR) is the interest rate when fees are included with the mortgage amount. Because the fees increase the cost of the loan, the effective interest rate on the borrowed
amount is higher. For example, a borrower is charged two points for the issuance of a mortgage (one point is equal to 1% of the mortgage amount). If the mortgage amount is $60,000 for 30 years with
an interest rate of 11.5%, what is the APR?
Part 1: Calculate Monthly Payment
- Payment Timing: End
- Present Value: 60,000
- Future Value: 0
- Interest/Year: 11.5%
- Periods: 360 [30 years * 12 periods/yr]
- Periods/Year: 12
- Compounds/Year: 12
Select ? on Payment row. The payment is -$594.17. It is negative because it is a cash outflow.
Part 2: Calculate APR
- Present Value: 58,800 [$60,000 loan -2% fees]
Select ? on Interest/Year row. The APR is 11.764%.
Present Value
Future Value
Payment Timing
Begin Period
End Period
Interest Paid
Principal Paid
Payment Paid
End Balance
Amortization Table
Total Interest
Total PI
30/365 Payment
30/365¼ Payment
30/365 Payment
savings accounts | {"url":"https://power.one/t/2a7e0fef321c3ce81ae7/time-value-of-money-calculator","timestamp":"2024-11-08T09:04:23Z","content_type":"text/html","content_length":"24100","record_id":"<urn:uuid:0b8161e9-5957-4511-94c2-c8c0cd077a3a>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00057.warc.gz"} |
Exploring Graphs
This function can be thought of as a text ‘Visual Inspection’ of the graph. It has enough information that you might be able to draw some conclusions however it mostly will tell you about the layout
of the graph. This function can also be used to investigate non graph objects like lm and aov etc. Only the graph application will be discussed here.
There is full support here for many layers as well as faceted displays (multiple plots in one graph object).
For every graph there will be a few lines at the start telling you the title subtitle what the axes are and there tick marks. This is the same across all of the graph printouts. It is the individual
layer sections can be quite different.
The VI function is built into the print function for ggplot objects. This means anytime you have BrailleR loaded and try to display a ggplot it will give you the text output.
This effect can be seen below
## This is an untitled chart with no subtitle or caption.
## It has x-axis 'Sepal.Length' with labels 5, 6, 7 and 8.
## It has y-axis 'Sepal.Width' with labels 2.0, 2.5, 3.0, 3.5, 4.0 and 4.5.
## The chart is a set of 150 big solid circle points of which about 78% can be seen.
However you can also easily call it explicitly without having to print the graph.
## This is an untitled chart with no subtitle or caption.
## It has x-axis 'Sepal.Length' with labels 5, 6, 7 and 8.
## It has y-axis 'Sepal.Width' with labels 2.0, 2.5, 3.0, 3.5, 4.0 and 4.5.
## The chart is a set of 150 big solid circle points of which about 78% can be seen.
Below is not only a list of the supported Geom but also a little explanation about there printouts.
## This is an untitled chart with no subtitle or caption.
## It has x-axis 'mpg' with labels .
## It has y-axis 'cyl' with labels 4.950, 4.975, 5.000, 5.025 and 5.050.
## The chart is 1 horizontal line as follows:
## Line at y position 5.
As this is quite a simple geom it is quite a simple printout.
It will tell you how many lines and at what height they are at.
## This is an untitled chart with no subtitle or caption.
## It has x-axis 'mpg' with labels 10, 15, 20, 25, 30 and 35.
## It has y-axis 'cyl' with labels 4, 5, 6, 7 and 8.
## The chart is a set of 32 big solid circle points of which about 81% can be seen.
Information will include here the number of points and the shape. There is also a percentage of approximately how many points can be seen. This is used to help you understand the over plotting. This
is only effective for when the points sizes are not too big (size < 18). Don’t worry it will only show you the percentage if it is accurate.
GeomBar / GeomCol
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## This is an untitled chart with no subtitle or caption.
## It has x-axis 'mpg' with labels 10, 15, 20, 25, 30 and 35.
## It has y-axis 'count' with labels 0, 1, 2, 3, 4 and 5.
## The chart is a bar chart with 30 vertical bars.
Even though there is a geom_histogram, geom_bar and geom_col in the backend of ggplot it is the same. So for the VI they are treated all pretty much the same.
There will be information just on the number of bars displayed.
## This is an untitled chart with no subtitle or caption.
## It has x-axis 'mpg' with labels 10, 15, 20, 25, 30 and 35.
## It has y-axis 'wt' with labels 2, 3, 4 and 5.
## The chart is a set of 1 line.
## Line 1 connects 32 points.
Very similar to the hline it will tell you about how many lines there are. Then for each line it will tell you the number of points that make up the line.
## This is an untitled chart with no subtitle or caption.
## It has x-axis 'mpg' with labels 10, 15, 20, 25, 30 and 35.
## It has y-axis 'as.factor(cyl)' with labels 4, 6 and 8.
## The chart is a boxplot comprised of 3 horizontal boxes with whiskers.
## There is a box at y=1.
## It has median 26. The box goes from 22.8 to 30.4, and the whiskers extend to 21.4 and 33.9.
## There are 0 outliers for this boxplot.
## There is a box at y=2.
## It has median 19.7. The box goes from 18.65 to 21, and the whiskers extend to 17.8 and 21.4.
## There are 0 outliers for this boxplot.
## There is a box at y=3.
## It has median 15.2. The box goes from 14.4 to 16.25, and the whiskers extend to 13.3 and 18.7.
## There are 1 max and 2 min outliers for this boxplot.
As a boxplot is effectively a 5 number summary with outliers this printout will include all of that information.
It shall include this summary for each boxplot in the layer.
## `geom_smooth()` using method = 'loess' and formula = 'y ~ x'
## This is an untitled chart with no subtitle or caption.
## It has x-axis 'mpg' with labels 10, 15, 20, 25, 30 and 35.
## It has y-axis 'wt' with labels 1, 2, 3, 4, 5 and 6.
## The chart is a 'lowess' smoothed curve with 95% confidence intervals covering 15% of the graph.
This output tell you the method used to get the smoothed curve and the confidence interval level which is set.
It also tells you what percentage of the graph is covered by the CI. This information can be used to quickly gauge how confident the graph is
GeomRibbon / GeomArea
## This is an untitled chart with no subtitle or caption.
## It has x-axis 'carat' with labels 0, 1, 2, 3, 4 and 5.
## It has y-axis 'price' with labels 0, 5000, 10000 and 15000.
## The chart is a ribbon which is bound on the y axis, with non constant widths: 345, 2882, 5766, 10913 and 18018 with centers at: 172.5, 1441, 2883, 5456.5 and 9009. The ribbon takes up 65% of the graph.
Once again this layer gives you some information about the layers data. It will tell you whether it is bound on the y or x axis and then if it is constant or non constant width. For it to be bound on
the y axis means that it covers the whole range of x and vice versa for y.
It includes some widths and centers throughout 5 points in the layer. The width will either be bottom to top for y bound or left to right for x bound.
It also includes the area covered statistic found in the smooth layer.
blank <- ggplot(BOD, aes(x = demand, y = Time)) +
geom_line() +
expand_limits(x = c(15, 23, 6), y = c(30))
## This is an untitled chart with no subtitle or caption.
## It has x-axis 'demand' with labels 10, 15 and 20.
## It has y-axis 'Time' with labels 0, 10, 20 and 30.
## It has 2 layers.
## Layer 1 is a set of 1 line.
## Line 1 connects 6 points, at (8.3, 1), (10.3, 2), (15.6, 5), (16, 4), (19, 3) and (19.8, 7).
## Layer 2 is a expand limits, increasing x axis by 48%, and increasing y axis by 383%.
GeomBlank is made by the expand_limits function. It is a normal layer like all the others however it just doesn’t have any points to be displayed.
So what this VI does is explain the effect that this layer had on the limits. It shall tell you how much larger the x and y axis are. | {"url":"https://cran.mirror.garr.it/CRAN/web/packages/BrailleR/vignettes/ExploringGraphs.html","timestamp":"2024-11-08T01:02:23Z","content_type":"text/html","content_length":"61234","record_id":"<urn:uuid:316eb0d8-a8f8-465e-adce-fd8c48cfc1b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00807.warc.gz"} |
Collection of Solved Problems
Nitrogen in a Vessel
Task number: 2173
One mole of nitrogen is enclosed in a vessel. Average translational kinetic energy of the nitrogen molecules is 3.5 kJ. Determine the most probable speed of nitrogen molecules after increasing
temperature in the vessel by 200 K.
• Hint 1
Using the formula for average molecular kinetic energy of a gas \(\bar{E}_k\), express the temperature T[1] of nitrogen before heating.
• Hint 2: The Most Probable Speed
To calculate the most probable speed v[p] of the heated nitrogen molecules, use the following relation:
where M[m] is the molar mass of nitrogen, T[2] is the temperature of the heated nitrogen and R is the molar gas constant.
• Notation
\(\bar{E}_k=3.5\,\mathrm{kJ}=3500\,\mathrm{J}\) average translational kinetic energy of nitrogen molecules
n = 1 mol amount of nitrogen
ΔT = 200 K increase in temperature
v[p] = ? the most probable speed
From the Tables:
M[m] = 28 gmol^−1 = 0.028 kgmol^−1 molar mass of nitrogen
R = 8.31 Jmol^−1K^−1 molar gas constant
• Analysis
First we express the temperature of nitrogen before heating from the formula for the average kinetic energy of a gas, according to which the energy is directly proportional to the thermodynamic
temperature. After that we add the temperature change to the calculated initial temperature and obtain the temperature of the heated nitrogen, which we subsequently substitute in the formula for
the most probable molecular speed.
• Solution
The average kinetic energy \(\bar{E}_k\) of any gas is directly proportional to its thermodynamic temperature T. The following relation applies
where n is the amount of the gas and R is the molar gas constant.
From this formula we can easily determine temperature T[1] of the nitrogen before heating:
For temperature T[2] of the heated nitrogen it holds true
Finally, to calculate the most probable speed v[p] of the nitrogen molecules, we use the known formula
where M[m] is the molar mass of the nitrogen, and substitute for the obtained expression of temperature T[2]:
\[v_p=\sqrt{\frac{2R}{M_m}\left(\frac{2\bar{E}_k}{3nR}+\mathrm{\Delta} T\right)}.\]
• Numerical Substitution
\[v_p=\sqrt{\frac{2R}{M_m}\left(\frac{2\bar{E}_k}{3nR}+\mathrm{\Delta} T\right)}=\sqrt{\frac{2\cdot{8.31}}{0.028}\cdot\left(\frac{2\cdot{3500}}{3\cdot{1}\cdot{8.31}}+200\right)}\,\mathrm{m\,s}^
{-1}\] \[v_p\dot{=}530\,\mathrm{m\,s}^{-1}\]
Just out of interest let's also try to determine the most probable speed of the nitrogen molecules before heating:
\[v_p=\sqrt{\frac{4\bar{E}_k}{3nM_m}}=\sqrt{\frac{4\cdot{3500}}{3\cdot{1}\cdot{0.028}}}\,\mathrm{m\,s}^{-1}\] \[v_p\dot{=}410\,\mathrm{m\,s}^{-1}\]
• Answer
The most probable speed of the nitrogen molecules after its temperature been increased is approximately 530 ms^−1.
• Comment
Try to compare the determined speed of the nitrogen molecules with the speeds we encounter in everyday life.
By comparing the calculated value with the speed of the air under normal conditions, we obtain the approximate value of the Mach number 1.6. So the speed is supersonic. | {"url":"https://physicstasks.eu/2173/nitrogen-in-a-vessel","timestamp":"2024-11-11T18:13:01Z","content_type":"text/html","content_length":"29986","record_id":"<urn:uuid:b77cf77f-cab9-44bc-b31d-ddb0d3fb512a>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00043.warc.gz"} |
Undirected Graph using Tarjan's Algorithm
If you haven't already gone through the chapter on
Articulation Points
and watched the video, then please do so before reading this chapter as we will be using all the concepts introduced in
Articulation Points
chapter while discussing the algorithm to find
in an
undirected graph
Video Explanation of finding Bridges using Tarjan's Algorithm:
Let G = (V, E) be a connected, undirected graph. A
of G is an edge whose removal disconnects G.
In the diagram below all the
are the edges colored in red.
Two Important Observations:
1. If we apply the same concept of keeping track of discovery time and earliest discovered node reachable and constructing a DFS Spanning Tree as discussed in the chapter Articulation Points, then
for every bridges U-V there is something interesting about the earliest discovered node reachable from the subtree (subgraph with all the descendants of V) rooted at V . If edge U-V is a bridge
and U is discovered before V in the DFS, i.e, discoveryTime of U is less than discoveryTime of V then the earliest discovered vertex reachable from the subgraph rooted at V will be a vertex with
discoveryTime GREATER than the discoveryTime of U.
And why is it so? It is because an edge U-V happens to be a bridge only when none of the vertices which were discovered before V in the DFS is reachable from the subtree (subgraph with all the
descendants of V, in original given graph) rooted at vertex V in DFS Spanning Tree. Look at the connected undirected graph in the diagram below: the subgraph rooted at V which consists of V, W
and X is NOT connected to any of vertices which were discovered before the root of this subgraph, and the root of this subgraph is V. To say in a different way, this subtree has no backedge that
takes it to a vertex discovered before the root of this DFS subtree. So if the edge connecting the root (V) of this subgraph is to the root's parent (U) is removed the subgraph becomes
disconnected from the rest of the graph.
So for all the descendents of V, the earliest discovered vertex happens to be V, and no other earlier discovered point. Which means the subgraph rooted at V is totally dependent on the back-edge
U-V to be connected to the rest of the graph due to absence of any other backedge leading to a vertex discovered before discovering V.
So, in our Depth First Search (DFS) after we just backtracked from vertex V to vertex U if we see that the earliest discovered node reachable from V through its descendents and at most one back
edge but not using the back edge V-U is LESS THAN OR EQUAL TO the discovery time of vertex U then the edge U-V is definitely not a Bridge.
The edge U-V is a Bridge when the earliest discovered node reachable from V through the descendants of V using at most one back edge but not using the back edge V-U is GREATER THAN the discovery
time of vertex U.
NOTE: When I say subtree I refer to the subtree in the DFS Spanning Tree, but when I say subgraph I refer to the corresponding subgraph in the given original graph. The subtree and the
corresponding subgraph will have same root, and have same nodes in it. For example, the subtree and its corresponding subgraph rooted at V: both of them have V, W and X as nodes.
Please click on the image below to enlarge it in a new tab. The whole algorithm we discussed above is depicted in the diagram below:
2. Since we can do DFS from any vertex, we need to do a check to see if the logic discussed above ALWAYSholds true if any of the adjacent edge of the start vertex happens to be a bridge. A quick
check by doing a DFS starting from either U and V in the graph in the diagram above shows that the logic discussed above in (1) applies as is for the start node of the DFS as well.
The code below implements the algorithm we just came up with:
Java code:
Login to Access Content
Python code:
Login to Access Content
Time Complexity:
The time complexity will be the time taken by the Depth First Search (DFS) : O(V + E), where V = total number of vertices in the connected undirected graph, and E = total number of edges in the
graph. We are visiting all the edges and all the vertices while finding the bridges.
Problem Solving:
Problem Statment:
There are n servers numbered from 0 to n-1 connected by undirected server-to-server connections forming a network where connections[i] = [a, b] represents a connection between servers a and b. Any
server can reach any other server directly or indirectly through the network.
A critical connection is a connection that, if removed, will make some server unable to reach some other server.
Return all critical connections in the network in any order.
/ |
/ |
1 |
| \ |
| \|
| 0
Input: n = 4, connections = [[0,1],[1,2],[2,0],[1,3]]
Output: [[1,3]]
Explanation: [[3,1]] is also accepted.
This problem is about nothing but finding bridges in the given connected undirected graph. Take a look at the implementation below to see how we use the knowledge we gained in this chapter so far to
solve real-world problems:
Java code:
Login to Access Content
Python code:
Login to Access Content
More on Tarjan's Algorithm: | {"url":"https://systemsdesign.cloud/Algo/GraphTheory/Tarjan/Bridges","timestamp":"2024-11-07T16:16:11Z","content_type":"text/html","content_length":"50816","record_id":"<urn:uuid:53297409-4a53-4f32-a266-53142bc396d1>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00151.warc.gz"} |
Discussion for October 19th
Well I will have to take a look and see what I will play for this next draw. That 23 has come up 3 times now. I think we are heading into the 30's and 40's.
The 46 has repeated 2 times now, that might be in my picks for Saturday's draw. The 17 hasn't played in the picks along with the 10 in either the bonus draws or regular draws so I might play them
along with the 32 and 36. Still thinking though.
Any thoughts on these?
Last edited:
The 20s 30s and 40s all look pretty good Renata I see 24 is hot, 25 is pretty much due and as a prior bonus # it may pull through. 02 could also repeat. 05 looks good with 08 & 10 looking strong.
09 is a possible cold #. 32 has got to come out and play soon(unless they forgot to put it in the machine) and 37 also looks good. 44 & 49 look due for a hit. 40 41 & 43 although cold look pretty
good. As for the 17 my feeling is you would have greater sucess with the 01 or 07.
I will play some of the same numbers that I have been playing for a while now along with some others. The numbers that I have been playing regularly now for about a month are:
11,27,32,40,47. They have to come up sometime.
I am not sure about that 2 repeating but I do think that the 14 will or maybe the 25 along with the 26? Although the 20's may skip this one they have been playing now for the last 4 draws.
Who knows... just to be on the safe side I will play 1 or 2 of them but then play HIGH, like the 30's and 40's
The high 40's are lurking....and 49 is not so far way here but we might see the 01 coming to the surface along with the 15...setting up you know what....
As you see above I play a total of 22 different numbes and all I seem to get is 1 hit!
So my sugestion is too all players is to pick other numbers
You will have better luck..
30 & 32
Surely, now I see 30 and 32 showing themselves so I will play
them this time. No 27 still. I will take a gamble and insert the 22
and 43(cold numbers) pulling both of them from ON49. Any
comments on this?
Re: 30 & 32
Florie said:
Surely, now I see 30 and 32 showing themselves so I will play
them this time. No 27 still. I will take a gamble and insert the 22
and 43(cold numbers) pulling both of them from ON49. Any
comments on this?
Looking at that
Dennis Bassboss said:
The high 40's are lurking....and 49 is not so far way here but we might see the 01 coming to the surface along with the 15...setting up you know what....
I see those high 40's also -
These long-shots are going to strike hard
I am not sure about the 2 repaeting but I think the 14 will. I will also play that 22 and 32 even though that 32 is as cold as ice.
Beaker what do you think about the chance of 25 repeating?
That one I am not sure about, in fact I am not sure about the teens or the 20's because they have been playing together in the last 4 or 5 draws. I think one of them will skip but which one?
renata said:
I am not sure about the 2 repaeting but I think the 14 will. I will also play that 22 and 32 even though that 32 is as cold as ice.
Beaker what do you think about the chance of 25 repeating?
That one I am not sure about, in fact I am not sure about the teens or the 20's because they have been playing together in the last 4 or 5 draws. I think one of them will skip but which one?
All of those numbers from the last draw are good repeaters - including 25. I wouldn't be surprised if two or three hit again
You can play that 32 and I'll try the 23 4peat
I don't think too many 20's or 30's will hit - somenthing from there should be quiet this time. Looks like we are moving to some more balanced draws so consider covering all decades.
Last edited:
Many good repeaters indeed....got to double check these ones...Be back later I have to go to work now!
Dennis Bassboss said:
Many good repeaters indeed....got to double check these ones...Be back later I have to go to work now!
I'm working my other job today
Jackpot Hunter
I got that feeling this morning...
HINTS: 19 and 27
Good for 649 and Super7
Florie said:
First set - 03 04 10 13 15 18 28 30 31 32 37 43 44 48
Second set - 06 08 10 12 13 15 17 22 28 32 37 42
Omitted all probable repeaters in this draw to make my wheel smaller.
Am taking a gamble on 32 and 10 and 22.
Good Luck to us all.
They are some good long-shots Florie
Good luck with them
Florie, what is your sure-fire best bet??
Last edited:
Florie said:
OCTOBER 18TH SUPER 7 WINNING NUMBERS ARE-
02 16 17 31 34 36 38 BONUS 05
This is the biggest Hint ever!!
I am PRETTY!!!!!I like a few from there too....that 31 is becoming a hanging ghost as we speak! That 38 is also scary and so is the 34...and what a weird combo that 16-17 consecutive...
Dennis Bassboss said:
I am PRETTY!!!!!I like a few from there too....that 31 is becoming a hanging ghost as we speak! That 38 is also scary and so is the 34...and what a weird combo that 16-17 consecutive...
My picks
Here goes...
I'm going with the following 18 number wheel:
Wish me luck and as always good luck to all!
I think that there are going to be few in the teen's and 20's,
I am going to play the 17, and 19 because like my reasoning for playing the 2 in the last draw, it hasn't played much if at all in the bonus draws along with the 10. I also think the 29 is going to
play in here somewhere. The 4-14-44 also look good to me.
I remember someone saying that the SUPER 7 actually follows the 649 not the other way around.
My feeling is that the 40's are going to hit hard tonight. There might be one or two low numbers and then maybe one or two 30's and the rest in the 40's. I also think more ODD numbers and few EVEN.
I haven't narrowed my numbers down yet.
I HAVE TO AGREE WITH YOU ON THIS RENATA
17,19 HAVE NOT BEEN HITTING HARD BUT NEITHER HAVE THE 30'S OR 40'S EVEN IN THE BONUS DRAW | {"url":"https://lottoforums.com/threads/discussion-for-october-19th.795/","timestamp":"2024-11-06T12:14:11Z","content_type":"text/html","content_length":"112996","record_id":"<urn:uuid:6bef4ef1-ac92-40a1-900e-c7523ce43eaf>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00796.warc.gz"} |
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Converting int to float in C - One Step! Code
In this post, we’ll see how to manipulate an integer value to get its single-precision floating-point bit representation. Specifically, we’ll convert an int data type to a float data type in C
assuming a word size of 32 bits. An example can be seen below.
value int float
12345 0x00003039 0x4640e400
Most of the work will be done using bit-level operations, addition, and subtraction. We’ll help ourselves with the log function at one point, though.
As a first approach, this “converter” will not handle rounding effects. This means that it will yield the same results as casting int to float only in the range -10M to +10M. Larger values need to be
rounded due to the limited precision of floating-point. This converter won’t handle those cases. For a converter that handles rounding effects as well, see this post.
The IEEE single-precision floating-point
Floating-point representation approximates a real value x as:
Each of these variables is encoded with a field of bits. For single-precision floating-point (32 bits), we have:
• The sign field s with 1 bit.
• The magnificand M with 23 bits.
• The exponent E with 8 bits.
The sign field is just encoded to 0 for positive values and 1 for negative ones.
The magnificand M is a binary fraction in the form:
Where each x is a binary digit, 0 or 1. For whole numbers such as integers, the range of floating-point values are said to be normalized. In this case, we encode only the bits following the binary
point (a leading 1 is implied). The precision of the fractionary part xxx...x is limited to 23 bits. This is known as the frac field.
Finally, we have the exponent field. The value of E is linked to the 8 bits used to encode it as follows:
Where exp is an unsigned 8-bits value (range 0-255) and Bias = 127. The possible range of E values is between -126 to 127 (the exponent field can’t be all zeros or all ones for normalized values).
These three fields together form the 32 bit single-precision floating-point bit representation of a number as follows:
With s being 1 bit, exp 8 bits and frac 23 bits long.
Floating-point encoding example
As an example, let’s see how is the number 24800 encoded in floating-point representation.
Since it’s a positive value, we’ll have s=0.
To decode the values of M and E we’ll transform 24800 to a binary value:
Now, we’ll express this binary value in the form M * 2^E:
We get E=14 which means that exp = E + Bias = 141. The value 141 (in decimals) needs to be encoded as an unsigned 8-bit number: exp = 10001101.
We also get M=1.10000011100000. The digits after the binary point are 10000011100000. These are encoded in the frac field. We pad to the right with zeros (they don’t change the binary fraction value)
to get 23 bits:
Finally, we’ll get the following 32-bit encoding for the number 24800 as a single-precision floating-point:
Integer to floating-point converter
Now, we’ll see how to program the converter in C. The steps that we’ll follow are pretty much those of the example above. We’ll assume int encodes a signed number in two’s complement representation
using 32 bits.
We’ll reproduce the floating-point bit representation using theunsiged data type. We’ll call this data type float_bits.
typedef unsigned float_bits;
First, we’ll determine the sign bit s. The value we are converting to a float is int i.
unsigned s = i>>31;
To get s, we’re just shifting to the leftmost bit of the 32 bit integer. In two’s complement representation, this value is 1 for negative values and 0 for positive ones, just like for floats.
Next, we’ll get the exponent.
unsigned E = (int) (log(i<0 ? -i : i)/log(2));
unsigned exp = E + 127;
With the first line, we get the highest power of two E such that i >= 2^E. We do that with the logarithm base 2 operator:
With the second line, we account for the Bias = 127 to get exp.
Finally, we’ll calculate the frac field. We start by calculating M. We can drop the negative sign and deal only with the absolute value of i since the sign is already encoded by s:
unsigned M= i>0 ? i : -i;
The frac field is obtained by dropping the most siginificant 1:
unsigned frac = M ^ (1<<E);
At this point, frac contains all of the bits after the leading one.
Next, we push the start of the frac field to the 23th bit position:
if (E>=23)
frac >>= E-23;
frac <<= 23-E;
The frac field is 23 bits long. The exponent E says how many of these places are already used. We truncate to only the most significant 23 bits if the field is too long or pad with zeros to the right
if the field is too short.
See that what we do for large numbers is to truncate. This is equivalent to rounding towards zero. The default C behavior is to round-to-even (it will round to the closest value). For this reason,
the result of our converter will differ from C’s casting for large values (those with more than 24 significant bits) .
Finally, we accommodate all of the fields together:
s<<31 | exp<<23 | frac;
Putting it all together in the float_i2f function, we’ll get:
float_bits float_i2f(int i) {
/* Special case : 0 is not a normalized value */
if (i==0)
return 0;
/* sign bit */
unsigned s = i>>31;
/* Exponent */
unsigned E = (int) (log(i<0 ? -i : i)/log(2));
unsigned exp = E + 127;
/* Magnificand*/
unsigned M= i>0? i : -i;
unsigned frac = M ^ (1<<E);
/* Move frac to start at bit postion 23 */
if (E>=23)
/* Too long: Truncate to first 23 bits */
frac >>= E-23;
/* Too short: Pad to the right with zeros */
frac <<= 23-E;
return s<<31 | exp<<23 | frac;
We included a special clause for i=0 that is not a normalized value.
To test the validity of our converter, we’ll compare the results of the float_i2f function with the casting operation (float) i.
int main() {
int i;
float *fp;
float_bits f;
for (i=-1e7; i<1e7; i++) {
f = float_i2f(i);
fp = &f;
if (*fp != (float) i)
printf("Casting not equal for value: %d\nConverted value is %.0f\nCasted value is %.0f\n", i, *fp, (float) i);
Under the hood, the float_bits data type is just the unsigned data type. In order to make C recognize those bits as a float, we declare a float pointer float *fp and pass it the address of the
float_bits result: &f. This way, the bits at *fp will be recognized as floats by C.
After compiling and running this code, we’ll get no output. This means that the values of the converter and the casting operation were all equal for the tested range of integer values. For a
converter that handles the whole range of int values, you can this post.
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seminars - Rational liftings and gRSK
Kashiwara's theory of crystal bases has been fruitful to many different areas of mathematics. A classical application has been to representation theory (where it originates), which leads to another
combinatorial proof of the Cauchy identity, where the famous Robinson-Schensted-Knuth (RSK) bijection becomes an isomorphism of crystals. However, its discrete (and combinatorial) nature makes it
hard to work with sometimes, so we transform its description into piecewise-linear functions. This description allows us to construct a rational lifting, enabling a more continuous theory of crystals
that can lead to further connections. This was formalized in the work of Berestein and Kazhdan with their theory of geometric and unipotent crystals. The RSK bijection can also be formulated into
piecewise-linear functions, which has a rational lifting to an isomorphism of algebraic varieties called geometric RSK (gRSK) with nice properties. In this talk, we discuss the background on
geometric crystals and gRSK, including the two different descriptions due to Noumi-Yamada and O'Connell-Seppäläinen-Zygouras.
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Keras Plot Model | How to Plot Model Architecture in Keras?
Introduction to Keras Plot Model
Keras plot model is defined in tensorflow, a machine learning framework provided by Google. Basically, it is an open source that was used in the Tensorflow framework in conjunction with Python to
implement the deep learning algorithm. It contains the optimization technique that was used to perform the complicated and mathematical operations. It is more scalable and will support multiple
Key Takeaways
• The keras plot model is plotted as layers of the graph. We can achieve the keras graph by using the function name as plot_model in keras.
• We can define it by using the keras model and keras model sequential function. We need to import the plot_model library.
What is Keras Plot Model?
Keras API is a deep learning API written in Python. It is a high-level API with a productive interface that helps us in solving machine learning problems. Keras plot is built on top of the Tensorflow
framework. It is designed to experiment in a specific way. It will provide the building blocks and abstractions that will allow us to encapsulate the machine learning solution.
The keras function API will help us in creating models that are more flexible than models created using the sequential API. The functional API will work with a model that has a nonlinear topology,
sharing layers and handling multiple outputs and inputs. The deep learning model has multiple layers and is directed. The keras functional API helps us in the creation of graph layers.
How to Use the Keras Plot Model?
The below steps show how we can use the keras plot model as follows. For using the plot model we need to import the required models as follows:
1. In this step we are importing the required model to use the keras plot model as follows. We are importing the same by using keywords as an import.
import pandas as pd
from keras.utils import plot_model
import numpy as np
from keras.datasets import mnist
from keras.layers import Dense
2. After importing the module now in this step we are loading the dataset. We are loading the dataset name mnist as follows.
(X_train, y_train), (X_test, y_test) = mnist.load_data()
3. After loading the dataset now in this step we are defining the model. We are defining the model name as sequential.
mod = Sequential()
mod.add(layers.Dense(128, kernel_initializer = 'uniform', input_shape = (10,)))
4. While defining the model now in this step we are defining the activation function by using the created model as follows.
mod.add (layers.Activation ('relu'))
mod.add (Dense(256))
mod.add (Dropout(0.2))
mod.add (Dense(256, activation = 'relu'))
mod.add (Dropout(0.1))
5. After defining the function of activation now in this step we are setting up the configuration of the image as follows.
img_width, img_height = 22, 22
batch_size = 64
no_epochs = 15
no_classes = 5
validation_split = 0.1
verbosity = 1
6. After the setup of image configuration now in this step we are adding the optimizer.
mod.compile (optimizer = 'adamax')
7. After adding the optimizer now in this step we are plotting the model by using keras plot_model.
ifile = 'cat.jpg'
tf.keras.utils.plot_model (mod, to_file = ifile, show_shapes = True, show_layer_names = True)
How to Plot Model in Architecture?
For understanding the model of keras we have an idea of model layers virtual representation. We can display the architechture of the keras model and save the same into a file. The tf.keras.utils
provide the plot_model function for plotting and saving the architechture of the model into the file. For plotting the architechture of the plot model we are importing the below model and creating a
model snippet.
import tensorflow as tf
ip = tf.keras.Input (shape = (100,), dtype = 'int32', name = 'ip')
x_ax = tf.keras.layers.Embedding (
output_dim=512, input_dim = 1000, input_length = 100)(ip)
x_ax = tf.keras.layers.LSTM(32)(x_ax)
x_ax = tf.keras.layers.Dense(64, activation = 'relu')(x_ax)
op = tf.keras.layers.Dense (1, activation = 'sigmoid', name = 'op')(x_ax)
mod = tf.keras.Model(inputs = [ip], outputs = [op])
After defining the model snippet now in this step, we are defining the plot model architechture.
ifile = 'dog.jpg'
tf.keras.utils.plot_model (mod, to_file = ifile, show_shapes = True, show_layer_names = True)
Keras Plot Model Function
The function will convert the keras model in the dot format and save the file. Below is the keras plot_model function and its argument as follows. It will contain multiple arguments.
to_file = "cat.jpg",
show_shapes = False,
show_dtype = False,
rankdir = "GB",
expand_nested = False,
dpi = 64,
layer_range = None,
show_layer_activations = False,
Below are the arguments of the plot_model function as follows. Keras plot_model contains multiple arguments.
• Model – This is an instance of the keras model.
• To_file – This is defined as the file name of the plot image.
• Show_shapes – It is defined as whether we display the shape information.
• Show_dtype – It is defined as whether we display layer dtype.
• Show_layer_name – It is defined as whether we display layer names.
• Rankdir – We are passing this argument with pydot.
• Expand_nested – This is defined as whether we are expanding the models which were nested.
• Dpi – This is defined as per inch dot.
• Layer_range – This is defined as a list which was containing the items of two strings.
• Show_layer_activations – This is defined as display layer activations.
Keras Plot Model Graph
The summary is very useful when using simple models, but it becomes very confusing when we have multiple inputs and outputs. Keras provides a function for creating a plot for the graph of a network
neural network, which makes the model more complex, but it is very simple to understand. The plot model function in Keras will generate a plot of our network. When we use this in our code, it will
accept multiple arguments.
from keras.layers import Dense
from keras.models import Sequential
from keras.utils.vis_utils import plot_model
mod = Sequential()
mod.add (Dense (2, input_dim = 1, activation = 'relu'))
mod.add (Dense(1, activation = 'sigmoid'))
plot_model (mod, to_file = 'cat.jpg', show_shapes = True, show_layer_names = True)
Given below are the FAQs mentioned:
Q1. What is the use of keras plot model?
Answer: It is used to convert the keras model into the dot format and after converting it will save in a specified image file.
Q2. Which libraries module do we need to import at the time of using keras plot model?
Answer: We need to import the keras, pandas, numpy, plot_model, and tensorflow module at the time of using the keras plot model.
Q3. How we can use the keras to plot the model by using python?
Answer: We can use the keras plot_model to plot the graph in keras. We need to use the function name as plot_model and need to import the library of the keras module.
The functional API will work with a model that has a nonlinear topology, sharing layers and working with multiple outputs and inputs. Basically, it is an open source that was used in conjunction with
the Tensorflow framework in Python to implement the deep learning algorithm.
Recommended Articles
This is a guide to Keras Plot Model. Here we discuss the introduction, and how to use the keras plot model. how to plot a model in architecture? and FAQ. You may also have a look at the following
articles to learn more – | {"url":"https://www.educba.com/keras-plot-model/","timestamp":"2024-11-08T05:29:36Z","content_type":"text/html","content_length":"321345","record_id":"<urn:uuid:7db93cf1-0368-4a58-8479-29d89218e9b5>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00514.warc.gz"} |
On Grading Algorithms
On Grading Algorithms
I finished grading the finals (graduate algorithms) before I left on vacation. Grading algorithms finals has its
• challenges: grading is not just checking "equality". A student might propose a solution different from the one you have in mind but equally valid. So you have to understand their idea, think
about if it will work, check proof of correctness and make sure there are no bugs, etc. All time consuming tasks.
• surprises: a silent student in the classroom may show a spark in the finals. A regular class room participant may fumble solutions to easy problems.
• and, in this case, self-inflicted quirks: borrowing from someone (who? I don't now recall), I award 1/4th of the points on a question if a student simply writes "I Do Not Know".This rewards the
students for knowing they do not know the solution, as well as cutting down on fluff a grader has to read through. How do students react to this? Some are professional, fill in "I Do Not Know"
just before the time runs out, some are principled and never use this route, and yet others forget to use this route and regret it later.
Labels: aggregator
12 Comments:
I also use the policy of 1/4 credit for the answer "I don't know". I like the policy well enough to keep it, but it is not perfect. I find that most students are not so good at estimating the
probability that they can give a correct answer.
Kirk Pruhs
I first heard the 1/4 policy (for some value of 1/4) from Martin F-C. Not sure if he was the original.
I've been giving 25% for "I don't know" (or "no idea", or "WTF", but definitely not " ") for about ten years. I don't remember who gave me the idea.
Most students seem to appreciate the WTF policy. (Yay, free points!) But a few (always undergrads) think it's just a sadistic mind game, because they never know whether their partial answers are
worth more or less than "WTF". Encouraging those students to spend more time thinking about solving the problem instead of strategizing about partial credit just pisses them off.
Ciao Dude,
it might be the case that you heard the the policy of 1/4 credit for the answer "I don't know" from Allan Borodin in Rome?
It is my case, and maybe we talked about it...
I heard about it from Allan Borodin in UofT.
Hi, Well probably someone else used
this type of scheme before me, but I have been using it for many many years at Toronto. (I probably told Muthu when we were both enjoying great academic visits in Rome, La Sapienza.) I give 20%
for any question for which a student writes "I do not know how to answer the question". I also allow students to say "I do not know how to do this problem set/text/exam" and get 20% for the
entire set/test/exam. In recent years I have been giving 10% for those who write nothing. This goes against the pedagogy of "giving credit for knowing what you dont know" but it also cuts down on
those who get flustered at the end (esp in a test), forget to write it, and then beg. I have found that students love this system as they feel you are being generous and good students recognize
the value in not bullshitting. I rarely get nonsense since using this 20% rule.
I first heard about it from Naomi Nishimura in 1991 when co-teaching a course with her at Waterloo. We wanted to ask "essay" like questions but we were worried about lengthy responses from the
students. At some point the concept of giving students 20% for a blank question came up.
I don't know if she got this idea from from her grad years at U of T where Allan is or if the flow of information was in the opposite direction.
Alex Lopez-Ortiz
I now distinctly remember Allan telling me about the nuances of this scheme, as Rome, the eternal city, blinked, in the evening. Thank you, La Sapienza!
The scheme -- its variants upto O(1) factors --- cuts down bullshit, discourages strategizing for partial credits, but seems to need the students to get used to the idea some, and estimate the
quality of their solutions a little better.
I also liked the tweak of awarding 20% for no answers: to avoid students kicking themselves for not writing IDNTK. Have to see how that works, next semester.
-- metoo.
In my grad classes I combine this with the policy of asking the students to answer any X of the Y questions on the test. In general I would rather that students have a full understanding of 1/2
of the material than have a 1/2-***** understanding of all of the material.
Kirk Pruhs
To followup on my previous message, we found that it was important to warn the students about this marking scheme well ahead of time. We allowed them to ask questions in class about how would
this work. Overall it was very effective in stopping students from fishing for marks.
Alex Lopez-Ortiz
p.s. I asked Naomi and she can't remember if this is something that occurred to her on the spot or if she had heard about it from someone else.
My last word on the subject. Prabhakar Ragde who did a postdoc in Toronto remembers hearing this idea as far back as 1987 from Charlie Rackoff, who might well have been quoting Al.
In sum, all sources lead to Rome, i.e. Al Borodin.
Muthu, is this new for your class? And specific to algorithms? I took your databases course once, and no such policy! Or is it that databases doesn't deserve such grace :-) | {"url":"https://mysliceofpizza.blogspot.com/2008/12/on-grading-algorithms.html","timestamp":"2024-11-11T13:49:33Z","content_type":"application/xhtml+xml","content_length":"31419","record_id":"<urn:uuid:123ca1bb-78a4-49c9-8372-b14f3a5849bf>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00364.warc.gz"} |
A3T4 Path Tracing
Up to this point, your renderer has only computed object visibility using ray tracing. Now, we will simulate the complicated paths that light can take throughout the scene, bouncing off many surfaces
before eventually reaching the camera. Simulating this multi-bounce light is referred to as global illumination, and it is critical for producing realistic images, especially when specular surfaces
are present. Note that all functions in src/scene/material.cpp are in local space to the surface with respect to the ray intersection point, while functions in src/pathtracer/pathtracer.cpp are
generally in world space.
Step 1: Pathtracer::trace
Pathtracer::trace is the function responsible for coordinating the path tracing procedure. We've given you code to intersect a ray with the scene and collect information about the surface
intersection necessary for computing the lighting at that point. You should read this function and understand where/why functions of the bsdf are called.
Step 2: Lambertian
Implement Lambertian::scatter, Lambertian::evaluate, and Lambertian::pdf. Note that their interfaces are defined in src/scene/material.h. Task 5 will further discuss sampling BSDFs, so reading ahead
may help your understanding.
• Lambertian::albedo is a texture giving the ratio of incoming light to reflected light, also known as the base color of the Lambertian material. Call albedo.lock()->evaluate(uv) to get the albedo
at the current point. Note that an albedo of $1$ should correspond to perfect energy conservation. (I.e., this value has not been pre-divided by $\pi$.)
• Lambertian::scatter returns a Scatter object, with direction and attenuation components. You can use a Samplers::Hemisphere::Cosine sampler to randomly sample a direction from a cosine-weighted
hemisphere distribution and you can compute the attenuation component via Lambertian::evaluate.
• Lambertian::evaluate computes the ratio of incoming to outgoing radiance given a pair of directions. Traditionally, BSDFs are specified as the ratio of incoming radiance to outgoing irradiance,
which necessitates the extra cos(theta) factor in the rendering equation. In Scotty3D, however, we expect the BSDF to operate only on radiance, so you must scale the evaluation accordingly.
• Lambertian::pdf computes the PDF for sampling some incoming direction given some outgoing direction. However, the Lambertian BSDF in particular does not depend on the outgoing direction. Since we
sampled the incoming direction from a cosine-weighted hemisphere distribution, what is its PDF?
Note: a variety of sampling functions are provided in src/pathtracer/samplers.h.
Note: for testing, notice that sample_direct_lighting_task4 already samples "delta lights" (i.e., non-area lights). So a scene with point or directional lights should show your material working
without requiring Step 3 to be complete.
Step 3: Pathtracer::sample_indirect_lighting
In this function, you will estimate light that bounced off at least one other surface before reaching our shading point. This is called indirect lighting.
• (1) Randomly sample a new ray direction from the BSDF distribution using Material::scatter().
• (2) Create a new world-space ray and call Pathtracer::trace() to get incoming light. You should modify Ray::dist_bounds so that the ray does not intersect at time = 0. Remember to set the new
depth value to avoid infinite recursion.
• (3) Compute a Monte Carlo estimate of incoming indirect light scaled by BSDF attenuation.
NOTE: you may wish to add some ray logging to help debug. See, for example, the code in sample_direct_lighting_task6 for and example of such code. Guarding it with a constant (in the example:
LOG_AREA_LIGHT_RAYS) is useful so it is easy to turn off for increased performance.
Step 4: Pathtracer::sample_direct_lighting_task4
Finally, you will estimate light that hit our shading point after being emitted from a light source without any bounces in between. For now, you should use the same sampling procedure as
Pathtracer::sample_indirect_lighting, except for using the direct component of incoming light. Note that since we are only interested in light emitted from the first intersection, we can trace a ray
with depth = 0.
Note: separately sampling direct lighting might seem silly, as we could have just gotten both direct and indirect lighting from tracing a single BSDF sample. However, separating the components will
allow us to improve our direct light sampling algorithm in task 6.
Reference Results
After correctly implementing task 4, your renderer should be able to make a beautifully lit picture of the Cornell Box with Lambertian spheres (A3-cbox-lambertian-spheres.s3d). Below is a render
using 1024 samples per pixel (spp):
cbox lambertian
Note the time-quality tradeoff here. This image was rendered with a sample rate of 1024 camera rays per pixel and a max ray depth of 8. This will produce a relatively high quality result, but will
take quite some time to render. Rendering a fully converged image may take a even longer, so start testing your path tracer early!
Thankfully, runtime will scale (roughly) linearly with the number of samples. Below are the results and runtime of rendering the Lambertian cornell box at 240x240 on an Intel Core i7-8086K (max ray
depth 8):
cbox-lambertian timing
Extra Credit
• Instead of setting a maximum ray depth, implement un-biased russian roulette for path termination. Though russian roulette will increase variance, use of a good heuristic (such as overall path
throughput) should improve performance enough to show better convergence in an equal-time comparison. Refer to Physically Based Rendering chapter 13.7. (You may need to add a throughput member to
Ray to support this change.)
• (Advanced) Implement homogeneous volumetric scattering. Refer to Physically Based Rendering chapters 11 and 15. | {"url":"https://kokecacao.me/page/Course/F22/15-462/code/15462/Scotty3D/assignments/A3/T4-path-tracing.md","timestamp":"2024-11-08T15:54:13Z","content_type":"text/html","content_length":"12968","record_id":"<urn:uuid:3cd79569-fb85-4108-bfd1-b20547c1590b>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00846.warc.gz"} |
Interpolation with Curve Fitting Toolbox
Interpolation is a process for estimating values that lie between known data points.
Interpolation involves creating of a function f that matches given data values y[i] at given data sites x[i] where f(x[i]) = y[i], for all i.
Most interpolation methods create the interpolant f as the unique function of the formula
$f\left(x\right)=\sum _{j}{f}_{j}\left(x\right){a}_{j},$
where the form of the functions f[j] depends on the interpolation method.
For spline interpolation, the f[j] are the n consecutive B-splines B[j](x) = B(x|t[j],...,t[j+k]), j = 1:n, of order k for a knot sequence t[1] ≤ t[2] ≤ ... ≤ t[n + k].
About Interpolation Methods
Curve Fitting Toolbox™ supports the interpolation methods described in the following table.
Method Description
Nearest neighbor Nearest neighbor interpolation. This method sets the value of an interpolated point to the value of the nearest data point.
Linear Linear interpolation. This method fits a different linear polynomial between each pair of data points for curves, or between sets of three points for surfaces.
Natural neighbor Natural neighbor interpolation. This method sets the value of an interpolated point to a weighted average of the nearest data points. The interpolating surface is C1 continuous,
except at the sample points.
Shape-preserving Piecewise cubic Hermite interpolation (PCHIP). This method preserves monotonicity and the shape of the data (for curves only).
Cubic spline Cubic spline interpolation. This method fits a different cubic polynomial between each pair of data points for curves, or between sets of three points for surfaces.
Biharmonic (v4) MATLAB^® 4 griddata method. This method fits surfaces that also extrapolate well (for surfaces only).
Thin-plate spline Thin-plate spline interpolation. This method fits smooth surfaces that also extrapolate well (for surfaces only).
Interpolant surface fits use the MATLAB function scatteredInterpolant for the linear, nearest neighbor, and natural neighbor methods, and the MATLAB function griddata for the cubic spline and
biharmonic methods. The thin-plate spline method uses the tpaps function.
The interpolant method you use depends on several factors, including the characteristics of the data being fit, the required smoothness of the curve, speed considerations, and post-fit analysis
requirements. The linear and nearest neighbor methods fit models efficiently, and the resulting curves are not very smooth. The natural neighbor, cubic spline, shape-preserving, and biharmonic
methods take longer to fit models, and the resulting curves are very smooth.
For example, the following plot shows a nearest neighbor interpolant fit and a shape-preserving (PCHIP) interpolant fit for the nuclear reaction data from the carbon12alpha.mat sample data set. The
nearest neighbor interpolant is not as smooth as the shape-preserving interpolant.
Goodness-of-fit statistics, prediction bounds, and weights are not defined for interpolants. Additionally, the fit residuals are always 0 (within computer precision) because interpolants pass through
the data points.
Biharmonic interpolant fits consist of radial basis function interpolants. All other interpolants supported by Curve Fitting Toolbox are piecewise polynomials and consist of multiple polynomials
defined between data points. For cubic spline and PCHIP interpolation, four coefficients describe each piece. Curve Fitting Toolbox uses a cubic (third-degree) polynomial to calculate the four
coefficients. Refer to the following for more information:
• spline for cubic spline interpolation
• pchip for shape-preserving (PCHIP) interpolation, and for a comparison of PCHIP and cubic spline interpolation
• scatteredInterpolant, griddata, and tpaps for surface interpolation
It is possible to fit a single polynomial interpolant to data, with a degree one less than the number of data points. However, the behavior of such fits is unpredictable between data points.
Piecewise polynomials with lower-order segments do not diverge significantly from the fitting data domain, so they are useful for analyzing a wider range of data sets.
Selecting an Interpolant Fit
Select Interpolant Fit Interactively
Open the Curve Fitter app by entering curveFitter at the MATLAB command line. Alternatively, on the Apps tab, in the Math, Statistics and Optimization group, click Curve Fitter.
On the Curve Fitter tab, in the Fit Type section, select an Interpolant fit. The app fits an interpolating curve or surface that passes through every data point.
In the Fit Options pane, you can specify the Interpolation method value.
For curve data, you can set Interpolation method to Linear, Nearest neighbor, Cubic spline, or Shape-preserving (PCHIP). For surface data, you can set Interpolation method to Linear, Nearest
neighbor, Natural neighbor, Cubic spline, Biharmonic (v4), or Thin-plate spline.
For surfaces, the Interpolant fit uses the scatteredInterpolant function for the Linear, Nearest neighbor, and Natural neighbor methods, the griddata function for the Cubic Spline and Biharmonic (v4)
methods, and the tpaps function for the Thin-plate spline method. Try the Thin-plate spline method when you require both smooth surface interpolation and good extrapolation properties.
If your data variables have very different scales, clear the Center and scale check box to see the difference in the fit. Normalizing the inputs might influence the results of the piecewise Linear
and Cubic Spline interpolation methods, and the Nearest neighbor and Natural neighbor surface interpolation methods.
Fit Linear Interpolant Model Using the fit Function
Load the census sample data set.
The variables pop and cdate contain data for the population size and the year the census was taken, respectively.
You can use the fit function to fit any of the interpolant models described in Interpolant Model Names. In this case, fit a linear interpolant model using the 'linearinterp' option, and then plot the
f = fit(cdate,pop,'linearinterp');
Compare Linear Interpolant Models
Load the carbon12alpha sample data set. Create both nearest neighbor and PCHIP interpolant fits using the 'nearestinterp' and 'pchip' options.
load carbon12alpha
f1 = fit(angle,counts,'nearestinterp');
f2 = fit(angle,counts,'pchip');
Compare the fitted curves f1 and f2 by plotting them in the same figure.
p1 = plot(f1,angle,counts);
hold on
p2 = plot(f2,'b');
hold off
legend([p1;p2],'Counts per Angle','Nearest Neighbor','PCHIP',...
See Also
Related Topics | {"url":"https://nl.mathworks.com/help/curvefit/interpolation-methods.html","timestamp":"2024-11-10T02:10:14Z","content_type":"text/html","content_length":"84510","record_id":"<urn:uuid:b33b928d-43e3-45c8-9beb-9a80d354bd94>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00465.warc.gz"} |
Wirklich witzige Google-Anfragen kann ich aus letzter Zeit leider nicht vermelden. Festzustellen bleibt lediglich, daß sich immer wieder Leute hierhin verirren, die auf ihre Fragen fast eine Antwort
bekommen -- dann aber keinen Kommentar hinterlassen, sondern wortlos wieder abziehen. Wer nach povray mandelbulb sucht, will doch sicher den Code für das Fraktal in einer Form haben, die von PoV-Ray
gelesen werden kann? Naja, vielleicht hilft es ja doch noch jemandem, wenn auch der ursprüngliche Besucher längst weg ist (siehe unten).
Außerdem: mein Hauptblog läßt sich wunderbar über Google finden; das Nibelungen-Projekt aber nur über die Blog-Suche. Das finde ich ein bißchen frustrierend, weil es für die Besucherzahlen alles
andere als förderlich ist. Naja, vielleicht sollte ich trotzdem noch ein paar Seiten ablichten und vor allem auch transkribieren, damit die Suchmaschinen ein bißchen mehr Text zum Zerkauen haben.
So, das war's auch schon für heute. Ach ja, der Code:
#declare iteratex = function (x, y, z, a) {
a + pow(f_sphere(x,y,z,0),4)*sin(f_ph(y,z,x)*8)
#declare iteratey = function(x, y, z, b) {
b + pow(f_sphere(x,y,z,0),4)*sin(f_ph(y,z,x)*8)
#declare iteratez = function(x, y, z, c) {
c + pow(f_sphere(x,y,z,0),4)*cos(f_ph(y,z,x)*8)
isosurface {
function {
f_sphere(iteratex(iteratex(iteratex(x,y,z,x), iteratey(x,y,z,y), iteratez(x,y,z,z), x),
iteratey(iteratex(x,y,z,x), iteratey(x,y,z,y), iteratez(x,y,z,z), y),
iteratez(iteratex(x,y,z,x), iteratey(x,y,z,y), iteratez(x,y,z,z), z), x),
iteratey(iteratex(iteratex(x,y,z,x), iteratey(x,y,z,y), iteratez(x,y,z,z), x),
iteratey(iteratex(x,y,z,x), iteratey(x,y,z,y), iteratez(x,y,z,z), y),
iteratez(iteratex(x,y,z,x), iteratey(x,y,z,y), iteratez(x,y,z,z), z), y),
iteratez(iteratex(iteratex(x,y,z,x), iteratey(x,y,z,y), iteratez(x,y,z,z), x),
iteratey(iteratex(x,y,z,x), iteratey(x,y,z,y), iteratez(x,y,z,z), y),
iteratez(iteratex(x,y,z,x), iteratey(x,y,z,y), iteratez(x,y,z,z), z), z), 2)
[Edit: Typo in den Metadaten]
Kommentare deaktiviert für Gefindet
Das Wort des Tages: Unbill. Außerhalb alter Texte (oder der Fantasy-Literatur) hört man es äußerst selten. Heute aber bat unser IT-Dienstleister, diese zu entschuldigen: ein zentraler Dienst ist vor
Weihnachten ausgefallen, der Hersteller arbeitet immer noch an der Behebung des Problems.
Außerdem: Ich habe mich in einem Irrgarten aus Wenn-Dann-Abfragen verlaufen und ein Softwarepaket an zwei Stellen repariert, aber gleichzeitig an drei anderen kaputtgemacht. Bis die Reparatur dann
getestet war, ging es schon stark auf acht zu. Naja, wer früh nach Hause will, sollte im Fenster mit den Build-Resultaten vielleicht ganz bis nach rechts scrollen. Dann sähe man nämlich die roten
Und: Während ich auf den Buildservice warte, bereite ich diesen Blogeintrag in einem TextEdit-Fenster vor. Das Programm mag aber den Satzanfang Naja nicht und ersetzt ihn automatisch durch Anja. Zum
Glück kann man das abschalten.
Und zu guter Letzt: Der neue Mandelbulb-Film ist fertig. Es hat ein paar Wochen gedauert, ihn zu berechnen; aber für glatte Kanten und eine höhere Framerate hat ich der Aufwand gelohnt.
Kommentare deaktiviert für Vom Tage
Today, another piece of the Mandelbulb Puzzle fell into place (actually, the lower right corner). The mosaic series will continue for a while, but most important parts are there by now. One could
start thinking about the next step, say zooming in. However, a higher order is hardly possible without software optimization.
Kommentare deaktiviert für Ecke • deutsch
The latest addition has taken quite some time, but it is also a larger piece of the image: the upper right corner is complete. (Of the fractal, that is; I will supply the remaining background at a
later date.)
You can find the entire series by tag. Its beginning is here.
Kommentare deaktiviert für Eckstein • deutsch
Today, I only have a quick little thumbnail to add to the mosaic.
Kommentare deaktiviert für Briefmarke • deutsch
Today, the jigsaw puzzle has grown a little bit.
Kommentare deaktiviert für Mandelpuzzle 2 • deutsch
Some time ago, I announced my intentions to calculate a new picture of Mandelbulb with one more iteration step. Unfortunately, PoV-Ray proved to be even slower than I feared. Watching my computer
computing on my own is not much fun, so I show you a preview with three iteration steps -- after more than a week, one can get an impression of what it will look like once it is finished. Try
watching the large picture to see full details.
Admittedly, a few more iteration steps would be nice: the detailed structure seen on Daniel White's images is much better (he seems to use ten steps).
2 Kommentare • deutsch
A few days ago, I promised to write more about Mandelbulb, a three-dimensional fractal related to the Mandelbrot set. It has taken several days on a two-processor machine, but now the first movie is
finally ready.
However, I would like to tell you a bit about sequences first. Sequences are, well, just numbers following one another; exactly what numbers make up the sequence is determined by certain rules.
Let us try to square numbers: 2, 4, 16, 256, 32768, ... -- these numbers will grow very quickly. We might try to start with a smaller number instead: 1, 1, 1, ... -- I will admit this to be a rather
boring sequence: 1^2=1, so there is little change. Maybe we could try this: 0.1, 0.01, 0.0001, ... -- now the numbers are actually shrinking. So: if we keep on squaring a number less than one in this
way, it will shrink to nothing; numbers larger than one are growing, and one itself remains unchanged. By the way, trying negative number will not change this pattern -- the number will become
positive during the first step, and the sequence proceeds as before. If we draw the number line, it might look a bit like this:
Everything between -1 and +1 remains finite. All other numbers will grow ever larger when squared. We could try to make our formula (i.e. keep on squaring) more complicated, but we would not get
funny pictures, anyway: the number line is, well, just a single line -- what can we make of it? Having an area to draw on would be much nicer:
How shall we assign numbers to different spots on this plane? Luckily, this is no problem in mathematics: we simply invent new numbers. We shall do this in such a way that in addition to zero left of
one and to to the right, we will have additional number above and below. Now we have to think of a way to square such numbers:
We imagine a clock's hand pointing to each number. When trying to square such a number, we will first square its length. Then, we shall turn its hand a bit: one o'clock gets two o'clock, three
o'clock becomes six o'clock, and four o'clock becomes eight o'clock. And, just in case you are wondering: mathematicians' clocks turn the wrong way -- and in this case, twelve o'clock (zero) is to
the right, not on top.
This definition might look a tad strange, but it has a distinct advantage: all those ordinary numbers on the number line are at twelve o'clock (the positive ones), or six o'clock (the negative ones).
When squaring them according to our new rule, the positive numbers will remain positive numbers, and the negative ones will be turned from six to twelve o'clock. That is, when squaring a number from
the number line, we obtain a number from the positive half of the number line. Since we are squaring each hand's length, the result is just the same as when following the ordinary rules from school.
Going back to sequences, we square away. The result is a bunch of hands rotating madly -- but all hands longer than on grow indefinitely, while the shorter ones shrink away to nothing.
This lets us draw at least a little picture, if not a very interesting one: a disc with radius one noting all numbers shrinking when they are squared.
It seems obvious that a more complex formula can create more complex pictures. However, if you think the formula has to be much more complex, you are mistaken: try to square and add the number
alternately, like this: 1^2=1, 1+1=2, 2^2=4, 4+1=5, 5^2=25; thus: 1, 2, 4, 5, 25, ... and the numbers are getting ever larger. Starting with a negative number, we encounter a small surprise: -2^2=4,
4+(-2)=2, 2^2=4, 4+(-2)=2... we are getting an infinite sequence of the number two! Therefore, when painting every converging sequence black (i.e. every sequence remaining finite), we will not get a
circle: +1 would remain white, but -2 would become black. Maybe an ellipse? It is hard to tell intuitively what our clock-hand numbers[1] do, so we have to use a computer to calculate the picture.
This is the result:
It is surprising and quite wonderful that such a simple formula creates such a complex image; and this simplicity is, to a large extent, the beauty of the Mandelbrot set: one gets so much more than
one has put in.
Finally, I would like to show my movie of the three-dimensional variant. An explanation of the corresponding formula will have to wait until next time. The fractal looks pretty smooth here, whereas
the two-dimensional Mandelbrot set is quite hairy. This is not a property of the Mandelbulb, but of my alculations: in order to get a result in no more than a few days, I have calculated only the
first two steps of the sequence. To get a more realistic picture, I would have had to use ten or so. Moreover, I will do something about the rough, pixelled structure.
[1]Those of you who have paid very close attention might notice that while we have defined how to square a clock-hand number, we have neglected to define the summation. However, doing so is quite
simple: to add to hands, we shift one of them so that its base coincides with the tip of the other hand. During that shift, neither hand must change the direction in which it points. The sum is then
defined as a new hand that points from the base of the first to the tip of the second hand. This will look familiar if you know vectors[2].
[2]Moreover, the clock-hand numbers will look quite familiar to those who know complex numbers.
1 Kommentar • deutsch
Earlier this week, Slashdot published the news that a three-dimensional equivalent of the Mandelbrot set has been found. Visiting Daniel White's page, not only did I find an explanation, but also
unprecedented images -- and I knew immediately that I would like to create such images myself.
The class of objects to which the Mandelbrot and the new Mandelbulb belong is called fractals. What are fractals, anyway? Well, a fractals in an object with a dimension that is not an integer. At
first glance, it is not quite clear what this means: an object is either one-dimensional, such as a line; or a sheet of paper, which has two dimensions (ignoring its thickness); or, for example, a
cube with three dimensions. But two and a half?
If we look at the dimension of a vector space, this is, indeed, impossible: there, one would simply count the (independent) directions that are to be found in a given object, and name there number
the object's dimension: if there is just length (such as on a piece of string), the dimension is one. The living room has length, width, and height, thus it is three-dimensional.
But there is a different way: let us take a ball that is large enough to hide the piece of string. For a one-meter string, we would need a ball one meter in diameter. Then, try with a ball half that
size: we would need two to cover the string. If we tried with ten-centimeter balls, we would need ten of them, and so on.
That does not work for my living-room, however: I could put the whole room into a large ball of, say, five meters; but I would need many more than just five balls of one meter each -- more than a
hundred, in fact. That is, the dimension of an object tells us how quickly the number of balls grows when the balls themselves shrink. For a one-dimensional object, 2 balls of size 1/2 are
sufficient, for something three-dimensional, we would need 8, or 2^3.
Now, let us turn around and use this counting as a definition: a chessboard is two-dimensional precisely because we need 64=8^2 squares of size 1/8 to cover it.
What would an object need to look like to make a fractional number drop out of this computation? Let us make a little experiment. You only need a sheet of paper, a pencil, an eraser, and maybe a
ruler (I did without). First, draw a straight line, like this:
Then, take the eraser to remove the middle third of the line:
Erase the middle third of the two remaining lines as well:
And so on -- in principle, you can continue indefinitely. That is, as long as your eraser is fine enough.
What happens if you try to cover your creation with balls? Start with one that is large enough to cover all those little dashes. If you then try balls a third of its size, you do not need three of
them -- instead, two are sufficient: one for the left side, one for the right. You have removed everything in the middle with your eraser, after all. Using balls of diameter 1/3, only 2=3^0.63 of
them are enough to cover the drawing. So you could say your little piece of art has a dimension of 0.63. This is not as weird as it may look at first glance: There is an enormous number of dots, most
of which are very close together. Isolated dots would have a dimension of 0; and dots that are infinitely close together form a line of dimension 1. Our result is somewhere inbetween, which should
hardly come as a surprise.
The Mandelbrot set is similar: regardless of how far we magnify parts of it, there are ever finer structures appearing.In principle, this is true for the Mandelbulb above; however, the picture has
been rendered with simplified equation, so only the largest knobs are visible: it is like throwing away the erasor after the first two steps.
Appearing soon
During the next few days, I would like to show how to calculate the Mandelbrot set, and the new Mandelbulb. Hopefully, there will also be more pictures, and perhaps a movie or two.
[Edit: Typos]
3 Kommentare • deutsch | {"url":"http://kirjoittaessani.de/tag/fraktal/","timestamp":"2024-11-09T01:43:30Z","content_type":"application/xhtml+xml","content_length":"80031","record_id":"<urn:uuid:6f84e8c8-ee92-48e1-be2b-d3db28ee36d7>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00066.warc.gz"} |
Main Content
Set axis limits and aspect ratios
axis(limits) specifies the limits for the current axes. Specify the limits as vector of four, six, or eight elements.
axis style uses a predefined style to set the limits and scaling. For example, specify the style as equal to use equal data unit lengths along each axis.
axis mode sets whether MATLAB^® automatically chooses the limits or not. Specify the mode as manual, auto, or one of the semiautomatic options, such as 'auto x'.
axis ydirection, where ydirection is ij, places the origin at the upper left corner of the axes. The y values increase from top to bottom. The default for ydirection is xy, which places the origin at
the lower left corner. The y values increase from bottom to top.
axis visibility, where visibility is off, turns off the display of the axes background. Plots in the axes still display. The default for visibility is on, which displays the axes background.
lim = axis returns the x-axis and y-axis limits for the current axes. For 3-D axes, it also returns the z-axis limits. For polar axes, it returns the theta-axis and r-axis limits.
___ = axis(ax,___) uses the axes or polar axes specified by ax instead of the current axes. Specify ax as the first input argument for any of the previous syntaxes. Use single quotes around input
arguments that are character vectors, such as axis(ax,'equal').
Set Axis Limits
Plot the sine function.
x = linspace(0,2*pi);
y = sin(x);
Change the axis limits so that the x-axis ranges from $0$ to $2\pi$ and the y-axis ranges from -1.5 to 1.5.
Add Padding Around Stairstep Plot
Create a stairstep plot, and use the axis padded command to add a margin of padding between the plot and the plot box.
x = 0:12;
y = sin(x);
axis padded
Use Semiautomatic Axis Limits
Create a plot. Set the limits for the x-axis and set the minimum y-axis limit. Use an automatically calculated value for the maximum y-axis limit.
x = linspace(-10,10,200);
y = sin(4*x)./exp(.1*x);
axis([-10 10 0 inf])
Set Axis Limits for Multiple Axes
Call the tiledlayout function to create a 2-by-1 tiled chart layout. Call the nexttile function to create the axes objects ax1 and ax2. Plot data in each axes. Then set the axis limits for both axes
to the same values.
x1 = linspace(0,10,100);
y1 = sin(x1);
ax1 = nexttile;
x2 = linspace(0,5,100);
y2 = sin(x2);
ax2 = nexttile;
axis([ax1 ax2],[0 10 -1 1])
Display Plot Without Axes Background
Plot a surface without displaying the axes lines and background.
Use Tight Axis Limits and Return Values
Plot a surface. Set the axis limits to equal the range of the data so that the plot extends to the edges of the axes.
Return the values of the current axis limits.
l = 1×6
1.0000 49.0000 1.0000 49.0000 -6.5466 8.0752
Change Direction of Coordinate System
Create a checkerboard plot and change the direction of the coordinate system.
First, create the plot using the summer colormap. By default, the x values increase from left to right and the y values increase from bottom to top.
C = eye(10);
colormap summer
Reverse the coordinate system so that the y values increase from top to bottom.
Retain Current Axis Limits When Adding New Plots
Plot a sine wave.
x = linspace(0,10);
y = sin(x);
Add another sine wave to the axes using hold on. Keep the current axis limits by setting the limits mode to manual.
y2 = 2*sin(x);
hold on
axis manual
hold off
If you want the axes to choose the appropriate limits, set the limits mode back to automatic.
Input Arguments
limits — Axis limits
four-element vector | six-element vector | eight-element vector
Axis limits, specified as a vector of four, six, or eight elements.
For Cartesian axes, specify the limits in one of these forms:
• [xmin xmax ymin ymax] — Set the x-axis limits to range from xmin to xmax. Set the y-axis limits to range from ymin to ymax.
• [xmin xmax ymin ymax zmin zmax] — Also set the z-axis limits to range from zmin to zmax.
• [xmin xmax ymin ymax zmin zmax cmin cmax] — Also set the color limits. cmin is the data value that corresponds to the first color in the colormap. cmax is the data value that corresponds to the
last color in the colormap.
The XLim, YLim, ZLim, and CLim properties for the Axes object store the limit values.
For polar axes, specify the limits in this form:
• [thetamin thetamax rmin rmax] — Set the theta-axis limits to range from thetamin to thetamax. Set the r-axis limits to range from rmin to rmax.
The ThetaLim and RLim properties for the PolarAxes object store the limit values.
For partially automatic limits, use inf or -inf for the limits you want the axes to choose automatically. For example, axis([-inf 10 0 inf]) lets the axes choose the appropriate minimum x-axis limit
and maximum y-axis limit. It uses the specified values for the maximum x-axis limit and minimum y-axis limit.
If the x-axis, y-axis, or z-axis displays categorical, datetime, or duration values, then use the xlim, ylim, and zlim functions to set the limits instead.
Example: axis([0 1 0 1])
Example: axis([0 1 0 1 0 1])
Example: axis([0 inf 0 inf])
mode — Manual, automatic, or semiautomatic selection of axis limits
manual | auto | 'auto x' | 'auto y' | 'auto z' | 'auto xy' | 'auto xz' | 'auto yz'
Manual, automatic, or semiautomatic selection of axis limits, specified as one of the values in this table. All of the auto mode values use the tickaligned style to calculate the limits for the
particular axis or set of axes you specify.
Value Description Axes Properties That Change
manual Freeze all axis limits at their current values. Sets XLimMode, YLimMode, and ZLimMode to 'manual'. If you are working with polar axes, then this option sets ThetaLimMode and RLimMode to
auto Automatically choose all axis limits. Sets XLimMode, YLimMode, and ZLimMode to 'auto'. If you are working with polar axes, then this option sets ThetaLimMode and RLimMode to
'auto x' Automatically choose the x-axis limits. Sets XLimMode to 'auto'.
'auto y' Automatically choose the y-axis limits. Sets YLimMode to 'auto'.
'auto z' Automatically choose the z-axis limits. Sets ZLimMode to 'auto'.
'auto xy' Automatically choose the x-axis and y-axis Sets XLimMode and YLimMode to 'auto'.
'auto xz' Automatically choose the x-axis and z-axis Sets XLimMode and ZLimMode to 'auto'.
'auto yz' Automatically choose the y-axis and z-axis Sets YLimMode and ZLimMode to 'auto'.
You cannot use these options with polar axes.
style — Axis limits and scaling
tight | padded | fill | equal | image | square | vis3d | normal
Axis limits and scaling, specified as one of these values.
Value Description Axes Properties That Change
XLimMode, YLimMode, and ZLimMode change to 'auto'.
In general, align the edges of the axes box with the tick marks that are XLimitMethod, YLimitMethod, and ZLimitMethod change to 'tickaligned'.
tickaligned closest to your data without excluding any data. The appearance might vary
depending on the type of data you plot and the type of chart you create. XLim, YLim, and ZLim automatically update to incorporate new data added to the axes. To keep the limits from
changing when using hold on, use axis tickaligned manual.
XLimMode, YLimMode, and ZLimMode change to 'auto'. If you are working with polar axes, then ThetaLimMode and
RLimMode change.
tight Fit the axes box tightly around the data by setting the axis limits equal XLimitMethod, YLimitMethod, and ZLimitMethod change to 'tight'.
to the range of the data.
XLim, YLim, and ZLim automatically update to incorporate new data added to the axes. To keep the limits from
changing when using hold on, use axis tight manual.
XLimMode, YLimMode, and ZLimMode change to 'auto'.
Fit the axes box around the data with a thin margin of padding on all XLimitMethod, YLimitMethod, and ZLimitMethod change to 'padded'.
padded sides. The width of the margin is approximately 7% of your data range.
XLim, YLim, and ZLim automatically update to incorporate new data added to the axes. To keep the limits from
changing when using hold on, use axis padded manual.
Sets DataAspectRatio to [1 1 1] and sets DataAspectRatioMode and PlotBoxAspectRatioMode to 'manual'. For 2-D
views, it also sets the XLimMode and YLimMode appropriately so that the axes fills its allotted space within the
parent figure or other container. For 3-D Views, XLimMode, YLimMode, and ZLimMode are set to 'auto' and
equal Use the same length for the data units along each axis. XLimitMethod, YlimitMethod, and ZLimitMethod are set to 'tight'.
This style disables the default “stretch-to-fill” behavior.
Sets DataAspectRatio to [1 1 1], DataAspectRatioMode to 'manual', and PlotBoxAspectRatioMode to 'auto'. It also
Use the same length for the data units along each axis and fit the axes sets XLimMode, YLimMode, and ZLimMode to 'auto' and XLimitMethod, YlimitMethod, and ZLimitMethod to 'tight'.
image box tightly around the data.
This style disables the default “stretch-to-fill” behavior.
Sets PlotBoxAspectRatio to [1 1 1] and sets the associated mode property to manual.
square Use axis lines with equal lengths. Adjust the increments between data
units accordingly. This style disables the default “stretch-to-fill” behavior.
Enable the “stretch-to-fill” behavior (the default). The lengths of each Sets DataAspectRatioMode and PlotBoxAspectRatioMode to 'auto'.
fill axis line fill the position rectangle defined in the Position property of
the axes.
vis3d Freeze the aspect ratio properties. Sets DataAspectRatioMode, PlotBoxAspectRatioMode, and CameraViewAngleMode to 'manual'.
normal Restore the default behavior. Sets DataAspectRatioMode and PlotBoxAspectRatioMode to 'auto'.
For more information on the plot box aspect ratio and the data aspect ratio, see the PlotBoxAspectRatio and DataAspectRatio properties.
You cannot use these options with polar axes, except for the axis tight and axis normal commands.
ydirection — y-axis direction
xy (default) | ij
y-axis direction, specified as one of these values:
• xy — Default direction. For axes in a 2-D view, the y-axis is vertical with values increasing from bottom to top.
• ij — Reverse direction. For axes in a 2-D view, the y-axis is vertical with values increasing from top to bottom.
You cannot use these options with polar axes.
visibility — Axes lines and background visibility
"on" | "off" | true or 1 | false or 0 | OnOffSwitchState value
Axes lines and background visibility, specified as any of these values:
• "on" or "off" — A value of "on" displays the axes lines and background, and "off" hides them. You can also specify the character vectors 'on' or 'off'.
• Numeric or logical 1 (true) or 0 (false) — A value of 1 or true displays the axes lines and background, and 0 or false hides them. (since R2024a)
• A matlab.lang.OnOffSwitchState value — A value of matlab.lang.OnOffSwitchState.on displays the axes lines and background, and matlab.lang.OnOffSwitchState.off hides them. (since R2024a)
Use parentheses to specify 1, 0, true, false, or an OnOffSwitchState value. For example, axis(0) and axis(false) hide the axes.
Parentheses are optional for the values "on" and "off". For example, axis off hides the axes.
Specifying the visibility sets the Visible property of the Axes object or PolarAxes object to the specified value.
ax — Target axes
one or more axes
Target axes, specified as one or more axes. You can specify Axes objects or PolarAxes objects. If you do not specify the axes, then axis sets the limits for the current axes (gca).
When you specify the axes, use single quotes around other input arguments that are character vectors.
Example: axis(ax,'tight')
Example: axis(ax,limits)
Example: axis(ax,'manual')
Output Arguments
lim — Current limit values
four-element vector | six-element vector
Current limit values, returned as a four-element or six-element vector.
• For Cartesian axes in a 2-D view, lim is of the form [xmin xmax ymin ymax]. For axes in a 3-D view, lim is of the form [xmin xmax ymin ymax zmin zmax]. The XLim, YLim, and ZLim properties for the
Axes object store the limit values.
• For polar axes, lim is of the form [thetamin thetamax rmin rmax]. The ThetaLim and RLim properties for the PolarAxes object store the limit values.
• You can combine multiple input arguments together, for example, axis image ij. The options are evaluated from left to right. Subsequent options can overwrite properties set by prior ones.
• If axes do not exist, the axis function creates them.
• Use hold on to keep plotting functions from overriding preset axis limits.
Version History
Introduced before R2006a
R2024b: Querying limit selection mode, visibility, and y-axis direction is not supported
This syntax, which returns the axis limit selection mode (m), visibility (v), and y-axis direction (d) is no longer supported and returns an error.
[m,v,d] = axis('state')
You can get the same information by querying these Axes properties.
This change was announced in R2015a.
R2024a: Display or hide axes by specifying logical or OnOffSwitchState value
Display or hide the axes by specifying the visibility input argument as a logical value or as a matlab.lang.OnOffSwitchState value. The values 1 and true are equivalent to "on", and 0 and false are
equivalent to "off".
The values "on" and "off" are still supported.
See Also | {"url":"https://uk.mathworks.com/help/matlab/ref/axis.html","timestamp":"2024-11-14T22:08:31Z","content_type":"text/html","content_length":"138383","record_id":"<urn:uuid:bd01903a-c9e0-41fa-bdf3-bb1d02ce6949>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00090.warc.gz"} |
imdev(imagename, outfile='', region='', box='', chans='', stokes='', mask='', overwrite=False, stretch=False, grid=[1, 1], anchor='ref', xlength='1pix', ylength='1pix', interp='cubic', stattype=
'sigma', statalg='classic', zscore=-1, maxiter=-1)[source]
Create an image that can represent the statistical deviations of the input image.
[Description] [Examples] [Development] [Details]
☆ imagename (path) - Input image name
☆ outfile (string=’’) - Output image file name. If left blank (the default), no image is written but a new image tool referencing the collapsed image is returned.
☆ region ({string, record}=’’) - Region selection. Default is to use the full image.
☆ box (string=’’) - Rectangular region(s) to select in direction plane. Default is to use the entire direction plane.
☆ chans (string=’’) - Channels to use. Default is to use all channels.
☆ stokes (string=’’) - Stokes planes to use. Default is to use all Stokes planes.
☆ mask (string=’’) - Mask to use. Default setting is none.
mask != ''
■ stretch (bool=False) - Stretch the mask if necessary and possible? Default value is False.
☆ overwrite (bool=False) - Overwrite (unprompted) pre-existing output file? Ignored if “outfile” is left blank.
☆ grid (intVec=[1, 1]) - x,y grid spacing. Array of exactly two positive integers.
☆ anchor ({string, intVec}=’ref’) - x,y anchor pixel location. Either “ref” to use the image reference pixel, or an array of exactly two integers.
☆ xlength ({string, int}=’1pix’) - Either x coordinate length of box, or diameter of circle. Circle is used if ylength is empty string.
☆ ylength ({string, int}=’1pix’) - y coordinate length of box. Use a circle if ylength is empty string.
☆ interp (string=’cubic’) - Interpolation algorithm to use. One of “nearest”, “linear”, or “cubic”. Minimum match supported.
☆ stattype (string=’sigma’) - Statistic to compute. See full description for supported statistics.
☆ statalg (string=’classic’) - Statistics computation algorithm to use. Supported values are “chauvenet” and “classic”, Minimum match is supported.
statalg = chauvenet
■ zscore (double=-1) - For chauvenet, this is the target maximum number of standard deviations data may have to be included. If negative, use Chauvenet”s criterion. Ignored if
algorithm is not “chauvenet”.
■ maxiter (int=-1) - For chauvenet, this is the maximum number of iterations to attempt. Iterating will stop when either this limit is reached, or the zscore criterion is met. If
negative, iterate until the zscore criterion is met. Ignored if algortihm is not “chauvenet”.
Create an image that can represent the statistical deviations of the input image.
This application creates an image that reflects the statistics around specified grid points of the input image. The output image has the same dimensions and coordinate system as the (selected
region in the) input image. The grid parameter describes how many pixels apart the grid pixels are from one another and the statistics are computed around each grid pixel. Grid pixels are
limited to the direction plane only (typically RA and dec); independent statistics are computed for each direction plane (i.e., at each frequency/stokes pixel should the input image happen to
have such additional axes).
Using the xlength and ylength parameters, one may specify either a rectangular or circular region around each grid point that defines which surrounding pixels are used in the statistic
computation for individual grid points. If the ylength parameter is the empty string, then a circle of diameter provided by xlength centered on the grid point is used. If ylength is not
empty, then a rectangular box of dimensions xlength x ylength centered on the grid pixel is used. These two parameters may be specified in pixels, using either numerical values or valid
quantities with “pix” as the unit (e.g., “4pix”). Otherwise, they must be specified as valid angular quantities, with recognized units (e.g., “4arcsec”). As with other region selections in
CASA, full pixels are included in the computation even if the specified region includes only a fraction of that pixel.
WARNING: Beware of machine precision issues because you may get a smaller number of pixels included in a region than you expect if you specify, e.g., an integer number of pixels. In such
cases, you probably want to specify that number plus a small epsilon value (e.g., “2.0001pix” rather than “2pix”) to mitigate machine precision issues when computing region extents.
The output image is formed by putting the statistics calculated at each grid point at the corresponding grid point in the output image. Interpolation of these output values is then used to
compute values at non-grid-point pixels. The user may specify which of the standard interpolation algorithms to use for this computation using the interp parameter. The input image pixel mask
is copied to the output image. If interpolation is performed, output pixels are masked where the interpolation fails.
ANCHORING THE GRID
The user may choose at which pixel to anchor the grid. For example, if one specifies grid=[4,4] and anchor=[0,0], grid points will be located at pixels [0,0], [0,4], [0,8] … [4,0], [4,4],
etc. This is exactly the same grid that would be produced if the user specified anchor=[4,4] or anchor=[20,44]. The value “ref”, which is the default, indicates that the reference pixel of
the input image should be used to anchor the grid. The x and y values of this pixel will be rounded to the nearest integer if necessary.
SUPPORTED STATISTICS AND STATISTICS ALGORITHMS
One may specify which statistic should be represented using the stattype parameter. The following values are recognized (minimum match supported):
☆ ‘iqr’ - inner quartile range (q3 - q1)
☆ ‘max’ - maximum
☆ ‘mean’ - mean
☆ ‘medabsdevmed’ or ‘madm’ - median absolute deviation from the median
☆ ‘median’ - median
☆ ‘min’ - minimum
☆ ‘npts’ - number of points
☆ ‘q1’ - first quartile
☆ ‘q3’ - third quartile
☆ ‘rms’ - rms
☆ ‘sigma’ or ‘std’ - standard deviation
☆ ‘sumsq’ - sum of squares
☆ ‘sum’ - sum
☆ ‘var’ - variance
☆ ‘xmadm’ - median absolute deviation from the median converted to an RMS-equivalent value. Result is MADM multipied by x, where x is the reciprocal of \(Phi^{-1}*(3/4)\) and \(Phi^{-1}\)
is the reciprocal of the quantile function. Numerically, x = 1.482602218505602. See here for an example.
Using the statalg parameter, one may also select whether to use the default Classical (“classic”, which uses the “framework” statistics method) or Chauvenet/ZScore (“chauvenet”) statistics
algorithm to compute the desired statistic (see the help for ia.statistics or imstat for a full description of these algorithms; see this page for further information on Operations and
Statistics on CASA images).
Parameter descriptions
The name of the input image that imdev will use.
Output image file name. If left blank (the default), no image is written but a new image tool referencing the collapsed image is returned.
Region selection. Default is to use the full image.
Rectangular region(s) to select in direction plane. Default is to use the entire direction plane.
Channels from the input image to use. Default is to use all channels.
Stokes planes to use. Default is to use all Stokes planes.
Mask to use. Default setting is none.
mask expandable parameters
Stretch the mask if necessary and possible. Default value is False.
Overwrite (unprompted) pre-existing output file. Ignored if outfile is left blank.
x,y grid spacing. Array of exactly two positive integers.
x,y anchor pixel location. Either “ref” to use the image reference pixel, or an array of exactly two integers.
Either x coordinate length of box, or diameter of circle. Circle is used if ylength is empty string.
y coordinate length of box. Use a circle if ylength is empty string.
Interpolation algorithm to use. One of “nearest”, “linear”, or “cubic”. Minimum match supported.
Statistic to compute. Accepted values discussed in the section above.
Statistics computation algorithm to use. Supported values are “chauvenet” and “classic”, Minimum match is supported.
statalg=’chauvenet’ expandable parameters
This is the target maximum number of standard deviations data may have to be included. If negative, use Chauvenet”s criterion.
This is the maximum number of iterations to attempt. Iterating will stop when either this limit is reached, or the zscore criterion is met. If negative, iterate until the zscore criterion is
Compute standard deviations in circles of diameter 10arcsec around grid pixels spaced every 4 x 5 pixels and anchored at pixel [30, 40], and use linear interpolation to compute values at
imdev("my.im", "sigma.im", grid=[4, 5], anchor=[30, 40],
xlength="10arcsec", stattype="sigma", interp="lin",
Compute median of the absolute deviations from the median values using the z-score/Chauvenet algorithm, by fixing the maximum z-score to determine outliers to 5. Use cubic interpolation to
compute values for non-grid-point pixels. Use a rectangular region of dimensions 5arcsec x 20arcsec centered on each grid point as the region in which to include pixels for the computation of
stats for that grid point.
imdev("my.im", "madm.im", grid=[4, 5], anchor=[30, 40],
xlength="5arcsec", ylength="20arcsec, stattype="madm",
interp="cub", statalg="ch", zscore=5)
No additional development details
Parameter Details
Detailed descriptions of each function parameter
imagename (path) - Input image name
outfile (string='') - Output image file name. If left blank (the default), no image is written but a new image tool referencing the collapsed image is returned.
region ({string, record}='') - Region selection. Default is to use the full image.
box (string='') - Rectangular region(s) to select in direction plane. Default is to use the entire direction plane.
chans (string='') - Channels to use. Default is to use all channels.
stokes (string='') - Stokes planes to use. Default is to use all Stokes planes.
mask (string='') - Mask to use. Default setting is none.
overwrite (bool=False) - Overwrite (unprompted) pre-existing output file? Ignored if “outfile” is left blank.
stretch (bool=False) - Stretch the mask if necessary and possible? Default value is False.
grid (intVec=[1, 1]) - x,y grid spacing. Array of exactly two positive integers.
anchor ({string, intVec}='ref') - x,y anchor pixel location. Either “ref” to use the image reference pixel, or an array of exactly two integers.
xlength ({string, int}='1pix') - Either x coordinate length of box, or diameter of circle. Circle is used if ylength is empty string.
ylength ({string, int}='1pix') - y coordinate length of box. Use a circle if ylength is empty string.
interp (string='cubic') - Interpolation algorithm to use. One of “nearest”, “linear”, or “cubic”. Minimum match supported.
stattype (string='sigma') - Statistic to compute. See full description for supported statistics.
statalg (string='classic') - Statistics computation algorithm to use. Supported values are “chauvenet” and “classic”, Minimum match is supported.
zscore (double=-1) - For chauvenet, this is the target maximum number of standard deviations data may have to be included. If negative, use Chauvenet”s criterion. Ignored if algorithm is not
maxiter (int=-1) - For chauvenet, this is the maximum number of iterations to attempt. Iterating will stop when either this limit is reached, or the zscore criterion is met. If negative, iterate
until the zscore criterion is met. Ignored if algortihm is not “chauvenet”. | {"url":"https://casadocs.readthedocs.io/en/latest/api/tt/casatasks.analysis.imdev.html","timestamp":"2024-11-10T21:45:21Z","content_type":"text/html","content_length":"37134","record_id":"<urn:uuid:69d66bf8-8d20-43fc-b29a-ac60fe5cde6f>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00780.warc.gz"} |
Received: 16 August 2018; Published on-line: 26 October 2018
Improving the technique for controlled cryogenic destruction of conjunctival tumors located in the projection of the ciliary body onto the sclera: a preliminary report
O.S. Zadorozhnyy1, Cand Sc (Med); N.V. Savin2, Cand Sc (Techn); A.S. Buiko1, Dr Sc (Med), Prof
1 Filatov Institute of Eye Diseases and Tissue Therapy, NAMS of Ukraine; Odessa (Ukraine)
2 Municipal Clinical Hospital No.1; Odessa (Ukraine)
E-mail: laserfilatova@gmail.com
TO CITE THIS ARTICLE: Zadorozhnyy OS, Savin NV, Buiko AS. Improving the technique for controlled cryogenic destruction of conjunctival tumors located in the projection of the ciliary body onto the
sclera: a preliminary report. J.ophthalmol.(Ukraine).2018;5:60-65. https://doi.org/10.31288/oftalmolzh201856065
Background: In case of controlled cryogenic destruction of an epibulbar tumor, making direct temperature measurements in the tumor and surrounding tissues during the treatment process is difficult
and dangerous. Infrared thermography (IRT) is, however, capable of imaging and recording only superficial-tissue temperature changes. Mathematical modeling can be used to assess the temperature
distribution in the underlying structures based on their thermal physical characteristics.
Purpose: To develop a model for temperature distribution in the ocular tunics in cryogenic destruction of conjunctival tumors located within a zone involving the ciliary body, in order to determine
the tunic freezing parameters enabling reduced risk of complications while meeting the principles of ablastics.
Materials and Methods: Twenty-five patients (25 eyes) underwent real-time IRT during controlled cryogenic destruction of benign and malignant epibulbar lesions located in the projection of the
ciliary body onto the sclera. A model for temperature distribution in the ocular tunics in cryogenic destruction of conjunctival tumors located within a ciliary body zone was implemented using
Microsoft Quick BASIC Version 4.5.
Results: IRT-based analysis of temperature distribution found that, in cryogenic destruction of epibulbar tumors located in the projection of the ciliary body onto the sclera, initially, the sclera
surrounding the centre of exposure gradually cooled down, while the cornea started cooling down rapidly only in 30 to 60 seconds, depending on tumor size and cryogenic unit parameters. The model (a)
was developed while taking into account differences in heat conduction and heat capacity between the sclera, ciliary body and cornea, and (b) was also in agreement with the above finding.
Conclusion: The method for IRT-based temperature field monitoring in cryogenic destruction of epibulbar tumors, and the thermal physical model developed will allow determining the individual
cryogenic exposure parameters that would prevent excessive freezing of the surrounding structures (in particular, the ciliary body and corneal endothelium). IRT allows for the assessment of
post-freezing increases in ocular tissue temperature to the baseline temperature, thus contributing to reduced complication rate, should a repeat cycle of cryogenic destruction of the tumor be
required. Further studies should match individual cryogenic exposure durations computed by the model to the data relating to long-term clinical outcomes.
Keywords: epibulbar tumors, cryogenic destruction, infrared thermography, model
Cryogenic destruction of conjunctival tumors is an effective treatment for the condition, associated with a low recurrence rate [1-3], and successfully used both as a stand-alone treatment (Fig.1)
and in combination with other modalities [4-7].
Cryotherapy for the tumor located within a zone involving the ciliary body, limbus, and peripheral cornea (Fig. 2) in some cases may be followed by complications (keratitis, iritis, iridocyclitis, or
uveitis with deformation of the pupil) owing to the duration of cryogenic exposure.
The importance of knowing the duration of cryogenic exposure for cryogenic destruction of tumors of any location has been acknowledged [8]. The duration depends on multiple factors, but mainly on the
nature, size and location of the tumor. In order to determine the duration of cryogenic exposure while treating for eyelid skin tumors, we have used a model developed previously [9] which takes into
account direct temperature measurements both of tumor tissues and adjacent healthy tissues during the process of cryogenic destruction.
However, in case of cryogenic destruction of an epibulbar tumor, making direct temperature measurements in the tumor tissue and especially surrounding tissues during the treatment process is
difficult and dangerous, and a non-invasive approach is required. Such an approach was rather surprisingly found while recording scleral-surface infrared thermography (IRT) patterns during cryogenic
destruction of epibulbar tumors located in the projection of the ciliary body onto the sclera. Interest of medical practitioners (including ophthalmologists) in applications of IRT has been renewed
in recent years [10-12]. The modality is, however, capable of imaging and recording only superficial-tissue temperature changes during cryogenic destruction of the tumor. We found it reasonable to
use mathematical modeling for the assessment of temperature distribution in the underlying structures (sclera, vitreous, and ciliary body) [13, 14]. Knowing temperature gradients of freeze zones for
the epibulbar tumor located in the projection of the ciliary body onto the sclera will contribute to reduced rate of complications with improvement in treatment outcomes, while meeting the principles
of ablastics.
The purpose of the study was to develop a model for temperature distribution in the ocular tunics in cryogenic destruction of conjunctival tumors located within a zone involving the ciliary body, in
order to determine the tunic freezing parameters enabling reduced risk of complications while meeting the principles of ablastics.
Materials and Methods
Since October 2016, 25 patients (25 eyes) have undergone real-time IRT of the freeze zone during controlled cryogenic destruction of benign and malignant epibulbar lesions (melanoma, 10 cases;
carcinoma, 8; dysplasia, 2; papilloma, 3; hemangioendothelioma, 1; and hemangiopericytoma, 1) located in the projection of the ciliary body onto the sclera.
A microcryogenic cylinder-and-throttle system capable of producing low temperatures within the range of -120…-90 °С depending on gas pressure in the cylinder was used to perform the destruction [15].
The cryogenic destruction methodology provided for stable freezing, with the duration of cryogenic exposure depending on the amount and location of tumor tissue, dimensions of cryogenic tip, and
cryogen pressure. Double freeze-thaw cycles were performed in case of conjunctival melanoma.
A smartphone attached IRT system, FLIR ONE (FLIR® Systems, Inc., USA; spectral range, 8-14 μm) [16], was used for displaying the images of thermal patterns of the freeze zone. Although the
temperature characteristics of FLIR ONE (temperature range, 20°С to +120°С) were, a priori, too low for enabling quantitative assessment of temperature distribution within the freeze zone, the
pattern of temperature distribution was clearly seen.
Thermal parameters of the eye (scleral and corneal conjunctiva) were monitored before and after surgery. Images of thermal patterns were obtained. In addition, time of freezing and time of thawing
with regard to external ocular surface structures (including the cornea) surrounding the center of freezing were recorded.
Examination and treatment of patients were performed under stable environmental conditions (air temperature and humidity control and minimal indoor air velocity). Patients spent 20 minutes indoors
with their eyes closed for adaptation to the indoor environment prior to IRT.
This study followed the ethical standards stated in the Declaration of Helsinki and was approved by the Local Ethics Committee of the Filatov Institute. Written informed consent was obtained from all
individual participants included in the study.
A model for temperature distribution in the ocular tunics in cryogenic destruction of conjunctival tumors located within a zone involving the ciliary body was implemented using Microsoft Quick BASIC
Version 4.5 (Microsoft, Redmond, WA).
Clinical Stage Results
In a number of cases, the pattern of temperature distribution in the freeze zone was surprisingly found to change over time. Initially, the pattern took the shape of regular concentric circles
spreading outward from the freeze center, but over time the circles started becoming irregular, with their circumferences extending towards the cornea. However, there were no substantial visual
changes in the shape of the freeze zone.
Thus, initially, the sclera surrounding the centre of freezing was the first to gradually cool down, while the cornea started cooling down only in 30 to 60 seconds. In addition, the temperature of
the central cornea decreased from 35.1±0.8 °C to 22.4±0.9 °C, whereas that of the corneal site adjacent to the tumor decreased to less than 10 °C. In each patient, individual gradual thawing-induced
increases in tissue temperature in the exposure zone over time were recorded. The ocular surface temperature returned to baseline 10 minutes on average after the cryogenic destruction process was
stopped and the eyelid retractor was removed.
Mathematical formulation
Choroidal circulation is known to be a key source of heat to the human eye. Blood entering the eye (a) has a temperature that is practically equal to that of the body, and (b) produces a temperature
gradient between the choroid and cornea which induces a heat transfer from blood to ocular tissues [13, 17]. We hypothesize that the irregularities found in scleral surface thermal patterns in the
course of cryogenic destruction are caused by the fact that the ciliary body becomes occluded from the blood circulation, and the subsequent heat extraction becomes more active in structures of
minimal circulation (the cornea). Therefore, the task to be accomplished is to determine a time point at which the initial freeze zone deformation that cannot be visually identified occurs in each
specific case.
Knowing the anatomical structure of the human eye, thermophysical properties of ocular structures (Table 1), temperature characteristics of cryogenic tip, and the size of the tumor would allow to
accomplish the task by thermophysical modeling, in particular, for the cases of a tumor located within the ciliary body and adjacent choroid [13, 17-19].
The shape of the human eye is nearly spherical, with deviations from sphericity being small enough to be considered negligible in modeling. The internal structure of the eye is symmetrical with
respect to the axis passing through the center of the cornea, lens and vitreous [13]. Therefore, it would be appropriate to consider a model for temperature processes in such structures using
spherical polar coordinates.
The heat conduction equation in spherical polar coordinates is as follows [20]:
Mathematical expressions for temperature function T ( r, φ,t) can be expressed as analytical formulas only in the simplest case, when the function depends on a single variable, radius (r). In this
case, the problem to be solved becomes a centrally symmetrical problem [21].
The finite element method[22] was used to solve equation (3), with curvilinear quadrilateral elements in polar coordinates employed in order to simplify algorithms for mesh generation and numbering
the nodes of finite-element meshes.
Quadrilateral elements can be approximated with hyperboloidal surfaces, and the temperature function T ( r, φ,t) relevant to each element can be approximated by a quadratic function. A quadrilateral
element is described in space by 8 nodes located at the element corners and midsides, and quadratic approximation is performed with quadratic parabolas located at these nodes. Quadratic functions
show much similarity with a function of temperature T ( r, φ) in a physical system with regard to spatial distribution.
The so called coordinate functions in relative coordinates (ξ,η) are used for implementation of approximation, and in case of a square expressed coordinates (ξ,η), these functions are expressed as
follows [23]: for the nodes located at the corner points (i = 1,3,5,7),
for the nodes located at the middle points of the horizontal sides of the square (i = 2,6),
and for the nodes located at the middle points of the vertical sides of the square (i = 4,8),
Coordinate functions equal 1 at eigennodes and equal 0 at any other nodes. In addition, an algebraic sum of coordinate functions for all the nodes of a finite element equals 1:
Coordinate functions perform mapping of a square, expressed in relative coordinates (ξ ,η), onto a curvilinear quadrilateral, expressed in absolute coordinates (r, φ). Coordinate transformation is
performed as follows:
where r[i ]and φ[i] are coordinates of finite element nodes expressed in the (r,φ) system.
One may use coordinate functions in order to transform the heat conduction equation (1) in the unknown function T(r,φ) into the system of linear algebraic equations in the unknown values of this
function at approximation nodes:
Components of the matrices of finite-element equation system (3) expressed in spherical polar coordinates are calculated through coordinate functions:
for the coefficient matrix,
for the free-term matrix,
and for the heat capacity matrix,
The components taking into account convective cooling of the ocular surface are integrated only over the angular coordinate, with the eye-globe radius R being constant. Therefore,
In addition, matrix equation (3) is written for the condition that the cornea and the exposed scleral surface are in contact with ambient air, the unexposed sclera surface is surrounded by orbital
tissues, and these media differ in temperature. In this equation T[0]
is the ambient temperature matrix for the relevant finite-element nodes on the integration surface.
The Crank-Nicolson scheme [23] in a matrix form is used to solve non-stationary problems:
The model was implemented as a BASIC language computer program that provides calculation of thermal conditions in the human eye based on the following input parameters:
tn, baseline temperature in various ocular compartments;
t0, ambient temperature;
dt, integration step for the cryogenic destruction process (1 to 3 seconds);
mL, number of integration steps;
Rg, average eye globe radius based on ultrasound measurements (mm);
dsc, average scleral thickness based on ultrasound measurements (mm);
dblu, average choroidal thickness based on ultrasound measurements (mm);
dse, average retinal thickness based on ultrasound measurements (mm);
f1, distance from the centre of the cornea to the limbus (mm);
f2, distance from the limbus to the (mm);
f2, distance from the tip of the cryoprobe to the conjunctival fornix (mm).
The software will split the domain under investigation into finite elements, and assign thermal physical characteristics to each finite element of the mesh (as per Table 1) automatically. This makes
operating the software easy; however, the location of the target tumor and type of the cryogenic unit are limitations of the software. While the software is running, it will display plots for radial
isotherms from the probe tip and across the ocular surface. In addition, it will display temperature values at points specified by the user on completion of computations.
Figure 4 shows plots for radial isotherms from the probe tip and across the ocular surface in cryogenic destruction of the conjunctival tumor at second 60 of intensive freezing to illustrate the
results of software execution (isotherms are plotted at 8.8 °C intervals).
Currently, there are no criteria for dosing the cryogenic exposure in the treatment of epibulbar tumors, which may cause complications in some cases. Peksayar et al [3, 7] have reported that
excessive freezing of the tumors located on the corneoscleral limbus and ciliary body resulted in sectorial iris atrophy, ocular hypotony, iritis, and corneal scarring.
Cryotherapy is thought to act by destroying cells initially by its thermal effect and later by obliteration of the microcirculation, resuiting in ischemic infarction of; this is
relevant both to superficially located tumors and to deep infiltrating tumors [24].
Currently, a degree of cryogenic exposure for the tumor during cryogenic destruction is determined visually by the surgeon. Peksayar et al [7] have applied cryotherapy on the ocular tissues until the
ice ball formed 0.5 to 2 mm around the probe. The intended extension of the ice ball was 2 mm for the conjunctiva, 1 mm for the episcleral tissues and the corneoscleral limbus, and 0.5 mm for the
In our current study, we used infrared thermography and found that, in cryogenic destruction of epibulbar tumors located in the projection of the ciliary body onto the sclera, initially, the sclera
surrounding the centre of freezing gradually cooled down, while the cornea started cooling down rapidly only in 30 to 60 seconds. This phenomenon may indicate (a) the ciliary body in the exposure
area is deeply frozen, and (b) the microcirculation is possibly obliterated and stopped functioning as a heat sink, which is consistent with the hypotheses of others [7, 24].
The model (a) was developed while taking into account differences in heat conduction and heat capacity between the sclera, ciliary body and cornea, and (b) also confirmed the above finding. The
cornea has no microcirculation of its own, and rapidly loses heat when ciliary body structures undergo deep freezing. However, as expected, corneal regions differed in the rate of cooling down. The
corneal site adjacent to the freeze center was the first to begin cooling down, followed by the central cornea.
In addition, our current study demonstrated the potential for real-time imaging and prediction of individual gradual increases in tissue temperature at the freezing centre and in the surrounding
tissues. Recording gradual increases in tissue temperature with time at the freezing centre may be useful in cryogenic destruction of large tumors that necessitate a number of sequential freeze-thaw
cycles with a control of the time when the ocular temperature returns to baseline.
IRT with FLIR ONE (FLIR® Systems) system allows real-time heat-pattern imaging for the freeze zone during controlled cryogenic destruction of epibulbar tumors located in the projection of the ciliary
body onto the sclera. A thermophysical model was developed to determine temperature characteristics of the thermal fields. The developed model allows for the assessment of distribution of temperature
gradients in the superficial and underlying tissues in cryogenic destruction of tumors varying in size. We believe that the developed model will allow determining individual cryogenic exposure
durations that would prevent excessive freezing of the surrounding structures (in particular, the ciliary body and cornea), thus contributing to reduced post-operative complication rate. In addition,
IRT allows for the assessment of gradual post-freezing increases in ocular tissue temperature to the baseline temperature, should a repeat cycle of cryogenic destruction of the tumor be required.
Further studies should match individual cryogenic exposure durations computed by the model to the data relating to (a) visual estimates of the freeze zone and (b) clinical outcomes, in order to
determine the details of the developed “method for controlled cryogenic destruction of epibulbar tumors” which require correction.
1. Divine RD, Anderson RL. Nitrous oxide cryotherapy for intraepithelial epithelioma of the conjunctiva. Arch Ophthalmol. 1983;101(5):782–6.
2. Fraunfelder FT, Wingfield D. Management of intraepithelial conjunctival tumors and squamous cell carcinomas. Am J Ophthalmol. 1983 Mar;95(3):359–63.
3. Peksayar G, Soyturk MK, Demiryont M. Long-term results of cryotherapy on malignant epithelial tumors of the conjunctiva. Am J Ophthalmol. 1989 Apr 15;107(4):337-40.
4. Buiko AS, Safronenkova IA, Elagina VA. [Methodological guidelines for cryogenic and radiocryogenic surgical treatment for malignant epithelial eyelid tumors]. Odessa: Astroprint; 2015. Russian
5. Basti S, Macsai MS. Ocular surface squamous neoplasia: a review. Cornea. 2003;22:687–704.
6. Kenawy N, Lake SL, Coupland SE, et al. Conjunctival melanoma and melanocytic intraepithelial neoplasia. Eye (Lond). 2013;27(2):142–52.
7. Peksayar G, Altan-Yaycioglu R, Onal S. Excision and cryosurgery in the treatment of conjunctival malignant epithelial tumours. Eye (Lond). 2003;17(2): 228–32.
8. Buschmann W. Kryochirurgie von Tumoren in der Augenregion. Stuttgart, New York: Thieme; 1999. pp.56–104.
9. Buĭko AS, Karpovskii EYa, Safronenkova IA et al. [Epithelial tumors of the eyelids: cryosurgery or scalpel?]. Oftalmol Zh. 1991;(6):338–44. Russian
10. Kaczmarek M, Nowakowski A, Suchowirski M, et al. Active dynamic thermography in cardiosurgery. Quant Infr Therm J. 2007;4(1):107–23
11. Kawasaki S, Mizoue S, Yamaguchi M, Shiraishi A, et al. Evaluation of filtering bleb function by thermography. Br J Ophthalmol. 2009 Oct;93(10):1331-6.
12. Tan JH, Ng EYK, Rajendra Acharya U. Infrared thermography on ocular surface temperature: a review. Infrared Phys Technol. 2009;52:97–108.
13. Anatychuk LI, Pasechnikova NV, Kobylianskyi RR, et al. Computer simulation of thermal processes in human eye. Journal of Thermoelectricity. 2017;(5):41–58. Russian
14. Ooi EH, Ng EYK. Ocular temperature distribution: a mathematical perspective. J Mech Med Biol. 2009;9(2):199–227.
15. Buĭko AS, Elagina VA, Landa Iu. [Potentials for increasing the effectiveness of cryogenic treatment of eyelid tumors using a device based on an adjustable balloon throttle microcryogenic system].
Oftalmol Zh. 1987;(5):272–6. Russian
16. Zadorozhnyy OS, Guzun OV, Brarishko AIu, et al. Infrared thermography of external ocular surface in patients with absolute glaucoma in transscleral cyclophotocoagulation: a pilot study. J
Ophthalmol (Ukraine). 2018; (2):23–8.
17. Mapstone R. Determinants of ocular temperature. Br J Ophthalmol. 1968;52:729–41.
18. Vit VV. [The structure of the human visual system]. Odessa: Astroprint; 2003. Russian
19. Scott JA. A finite element model of heat transport in the human eye. Phys Med Biol. 1988; 33: 227–41.
20. Karlslow HS, Jaeger JC. [Conduction of Heat in Solids]. Moscow: Nauka; 1964. Russian
21. Savin SN. [Modeling the epoxy cure processes in spherical layers]. Visnyk ONU. Khimiia. 2013;18(4):38–45. Russian
22. Segerlind LJ. [Applied Finite Element Analysis]. Moscow: Mir; 1979. Russian
23. Altoiz BA, Savin NV, Shatagina EA. [Effect of heat release in a microinterlayer of a liquid on the measurement of its viscosity]. Zhurnal Tekhnicheskoi Fiziki. 2014; 59 (5):21–7. Russian
24. Lee GA, Hirst LW. Ocular surface squamous neoplasia. Surv Ophthalmol. 1995 May-Jun;39(6):429–50. | {"url":"https://www.ozhurnal.com/en/archive/2018/5/10-fulltext","timestamp":"2024-11-13T13:09:28Z","content_type":"text/html","content_length":"88120","record_id":"<urn:uuid:e96f2895-26ff-4db4-a099-fbf93d6187e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00071.warc.gz"} |
Trigonometric Substitution Calculator | Get Solution in Seconds!
Introduction to Trigonometric Substitution Calculator
The trig substitution calculator with steps is a free tool that helps to find the solution of integral problems that cannot be solved by any other method of integration.
The trig sub calculator determines the antiderivative functions that have square roots or rational powers of quadratic expressions in a fraction of a second. Trigonometric substitution is a technique
commonly used in calculus to simplify such integrals by substituting trigonometric functions for certain variables.
Our calculator can help you solve integrals more efficiently and accurately. It serves as a valuable tool for students and professionals working in calculus and related fields where such integrals
are encountered frequently. Additionally, if you would like to find the solution of complex integral questions, you can utilize our U substitution calculator. Our calculator also determines the
integral function who do not solve by any other method of integration in the run of time.
What is Trigonometric Substitution?
Trigonometric substitution is a process in which trigonometric functions are substitution into an original problem that has a square root or radical expression for the solution of integral questions
Trigonometric substitution is a technique commonly used in calculus to simplify integrals that involve expressions with square roots or the sum/difference of squares. It is particularly useful for
integrating functions where algebraic manipulation alone is insufficient.
Our calculator is a powerful technique in calculus, particularly for handling integrals involving square roots or expressions that resemble trigonometric identities. Further, if you eould like to
find the antiderivative for the given function, you can utilize our integration by parts calculator. Our calculator also helps you to easily determine the special type of integration which is a
product of two functions.
Formula Used by Trig Substitution Calculator
There are three expressions that take trigonometric functions to solve these types of integral problems. According to the expression, different trigonometric substitutions are used. The expression
used by our integration by trigonometric substitution calculator is as follows,
$$ \sqrt{a^2 - x^2} \; x \;=\; asin \theta \; dx \;=\; acos \theta d \theta $$
$$ \sqrt{a^2 + x^2} or \sqrt{a^2 + x^2} \; x \;=\; atan \theta \; dx \;=\; asec^2 \theta d \theta $$
$$ \sqrt{x^2 - a^2} \; x\;=\; asec \theta \;dx\;=\; asec \theta tan \theta d \theta $$
Working Method of Trig Sub Calculator
The trig sub integral calculator takes specific types of antiderivative expressions and substitutes trigonometric functions in these integral problems to give instant solutions.
When you give your integral question as an input in this tool.
First, our calculator identifies the problem that whether the given integral has one of the same expressions mentioned above. After recognizing the nature of the function, change x and dx into
cross-pounding trigonometric substitution.
Then simplify the function according to the rules of integration after substitution, two case finds which are:
• In case of definite integral apply upper and lower limit and get result. For further clarification, you can utilize the bounded integral calculator to solve definite integrals with specified
limits, providing a comprehensive approach to integral calculus problems.
• In the case of an indefinite integral function replace trigonometric substitution from the original function which is in terms of x.
Let's take an example to see the trigonometric substitution process for an integral problem to know how this online trig substitution calculator with steps works
Example of Trigonometric Substitution:
An example of the trigonometric substitution problem is given to know how to solve such problems manually. These problems can be solved by using the trigonometric substitution calculator but it is
also important to know each step so, here’s an example,
Find the following:
$$ \int_{-3}^{3} \sqrt{9 - x^2} dx $$
$$ 9 sin^2 \theta + 9cos^2 \theta \;=\; 9 $$
$$ 9cos^2 \theta \;=\; 9 - 9 sin^2 \theta $$
If we let,
$$ x \;=\; 3sin \theta $$
$$ 9 - x^2 \;=\; 9 - 9 sin^2 \theta \;=\; 9cos^2 \theta $$
$$ \int_{-3}^{3} \sqrt{9 - x^2} dx \;=\; \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \sqrt{9 - 9sin^2 \theta} (3cos \theta) d \theta $$
$$ \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \sqrt[3]{9 cos^2 \theta} cos \theta d \theta $$
$$ \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} 3|3 cos \theta| cos \theta d \theta $$
$$ \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} 9cos^2 \theta d \theta $$
$$ \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{9}{2} (1+cos(2 \theta)) d \theta $$
$$ -\frac{9}{2} \biggr( \theta + \frac{1}{2} sin(2 \theta) \biggr) \biggr|_{\frac{-\pi}{2}}^{\frac{\pi}{2}} -\frac{9}{2} \pi $$
Thus it is the final solution of our function with specific limits. If you would like to solve this problem further, you can use our integral indefinite calculator. Additionally, our calculator also
provides a comprehensive tool for further analysis and exploration of functions without specific limits.
How to Use Trigonometric Substitution Calculator with Steps
The trig substitution calculator with steps has a simple design that you use for solving integral problems easily if you follow the given guidelines. These are:
• Enter your integral function(definite or indefinite) in the trig sub calculator.
• Select a variable from the given list(x,y,z) that you want to evaluate for integral questions.
• If the problem is related to a definite integral then add the upper and lower limits in our calculator.
• If a given question is related to an indefinite then you do not need to add limit values for the solution
• Press the “Calculate” button to get the solution of the integral problem
• Recalculate button brings you back to the new page for the new calculation
Result Obtained from Trig Sub Integral Calculator
You will obtain the solution of the integral question after you give input in the integration by trigonometric substitution calculator. It may contain as:
Our calculator serves as an educational tool for learning and practicing trigonometric substitution techniques in calculus. It helps reinforce understanding of concepts and provides a platform for
exploration and experimentation.
Additionally, For more advanced calculations involving integrals, you can use our integral evaluator that provides accurate results with step-by-step explanations
• Result section provides the solution of the given antiderivative problem
• Possible steps section provides you with solutions in a step-wise process
• Plot section sketch a graph according to the result of the given integral problem
Benefits of Using Our Calculator
The trig substitution calculator gives you a lot of benefits while using this tool for solving integral questions. These benefits are:
Our Calculator also offers efficiency, accuracy, comprehensiveness, and user-friendliness, making it a valuable tool for anyone working with integration problems involving trigonometric functions.
Additionally, if you're interested in exploring more mathematical tools and calculators for various applications, you can access our comprehensive collection of calculators by visiting the All
Calculators page.
• It saves our time and effort in doing complex calculations of integrals
• It gives results in a fraction of a second.
• This trig sub calculator provides accurate results with steps.
• You can use this calculator to practice more and more integral examples.
• It will improve your learning experience about the trigonometric substitution method.
The trig sub integral calculator is a valuable tool for such integral problem who has a specific expression with radical and do not solve any other rule of integration.
The trigonometric substitution calculator with steps is a reliable tool as it provides a precise result of your given problem every time whenever you give input into it.
Explore other calculus techniques with our integral calculator partial fraction. Delve deeper into solving complex integrals by employing both trigonometric substitution and integration by parts | {"url":"https://integral-calculators.net/trigonometric-substitution-calculator","timestamp":"2024-11-04T20:16:27Z","content_type":"text/html","content_length":"83578","record_id":"<urn:uuid:fbae66cd-5860-4de8-aab2-9378f483b890>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00565.warc.gz"} |
Systems formed by translates of one element in L p ( R ) L_{p}(mathbb R)
• Let 1 p < , f Lp(R), and R. We consider the closed subspace of Lp(R), Xp(f, ), generated by the set of translations f() of f by . If p = 1 and {f(): } is a bounded minimal system in L1(R), we
prove that X1(f, ) embeds almost isometrically into 1. If {f(): } is an unconditional basic sequence in Lp(R), then {f(): } is equivalent to the unit vector basis of p for 1 p 2 and Xp(f, )
embeds into p if 2 < p 4. If p > 4, there exists f Lp(R) and Z so that {f(): } is unconditional basic and Lp(R) embeds isomorphically into Xp(f, ). 2011 American Mathematical Society. | {"url":"https://vivo.library.tamu.edu/vivo/display/n223251SE","timestamp":"2024-11-03T13:21:11Z","content_type":"text/html","content_length":"21801","record_id":"<urn:uuid:57c43af4-e146-467e-bf97-5014be549dc0>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00780.warc.gz"} |
Astronomie I
Our group is responsible for the Astronomie I lecture, held in German, which is a general introduction to astronomy for (mainly) 1st semester students.
The content includes:
• Kapitel 1: Geschichte der Astronomie
• Kapitel 2: Unser Sonnensystem
• Kapitel 3: Die Sonne
• Kapitel 4: Space Weather
• Kapitel 5: Spektroskopie, Strahlungstransport
• Kapitel 6: Gravitationswellen, Interferometrie
• Kapitel 7: Sternatmosphären
• Kapitel 8: Sterne und ihre Entwicklung
• Kapitel 9: Teleskope und Beobachtungen
The lecture includes exercises of building a telescope, visiting an inflatable planetarium, and fabricating star charts.
Modeling Techniques in Physics
The Modeling Techniques in Physics lecture is held during the 5th semester. It consists of 4 parts: ordinary differential equations, partial differential equations, Monte Carlo simulations and
machine learning. Our group is responsible for the machine learning part.
The content includes:
• Introduction to ML, Unsupervised techniques
• Clustering, k-means, PCA, outlier detection
• Supervised techniques
• Linear and logistic regression, regularization
• k nearest neighbors, SVM
• Perceptron, neural networks
• Optimization (gradient descent, backpropagation)
• Binary classification - multiclass classification
• Convolutional neural networks
• Encoder, VAE, GAN, ChatGPT
The lecture includes programming exercises. Interested students are welcome to deepen their knowledge in MSc theses. | {"url":"https://sml.unige.ch/teaching.html","timestamp":"2024-11-13T05:22:45Z","content_type":"text/html","content_length":"4406","record_id":"<urn:uuid:748495af-c2d3-417f-9352-8572e18f319e>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00378.warc.gz"} |
Dilated Convolution [explained]
Open-Source Internship opportunity by OpenGenus for programmers. Apply now.
Dilated convolution, also known as Atrous Convolution or convolution with holes, first came into light by the paper "Semantic Image Segmentation with Deep Convolutional Nets and Fully Connected
CRFs". The idea behind dilated convolution is to "inflate" the kernel which in turn skips some of the points. We can see the difference in the general formula and some visualization.
Table of content:
1. Introduction to Dilated Convolution
2. Dilated convolution in Tensorflow
3. Dilated convolution in action
4. Dilated Convolution: Results of the context module
5. Complexity analysis of dilated convolutions
6. Usefulness of Dilated Convolutions
We will dive into Dilated Convolution now.
Introduction to Dilated Convolution
The following convolution represents the standard discrete convolution:
We see that the kernel looks up all the points of the input. The mathematical formula of discrete convolution is:
This represents dilated convolution:
As you can clearly see, the kernel is skipping some of the points in our input. The mathematical formula of dilated convolution is:
We can see that the summation is different from discrete convolution. The l in the summation s+lt=p tells us that we will skip some points during convolution. When l = 1, we end up with normal
discrete convolution. The convolution is a dilated convolution when l > 1. The parameter l is known as the dilation rate which tells us how much we want to widen the kernel. As we increase the value
of l, there are l-1 gaps between the kernel elements. The following image shows us three different dilated convolutions where the value of l are 1, 2 and 3 respectively.
The red dots represent that the image we get after convolution is with 3 x 3 pixels. We see that the output of all three dilated convolutions have equal dimensions but the receptive field observed by
the model is entirely different. Receptive field simply tells us how far the red dot can "see through". It is 3 x 3 when l=1, 5 x 5 when l=2, and 7 x 7 when l=3. The increase in receptive field means
that we are able to observe more without any additional costs!
Dilated convolution in Tensorflow
Tensorflow has a built-in function for dilated convolution (or atrous convolution). The syntax for the dilated convolution function is:
value, filters, rate, padding, name=None
This computes a 2-D atrous convolution, with a given 4-D value and filters tensors.
The rate parameters defines the dilation rate (l). If the rate parameter is equal to one, it performs regular 2-D convolution. If the rate parameter is greater than one, it performs atrous
convolution, sampling the input values every rate pixels in the height and width dimensions.
Dilated convolution is similar if we are convolving the input with a set of upsampled filters, produced by inserting rate - 1 zeros between two consecutive values of the filters along the height and
width dimensions.
The convolution is the most efficient when they are stacked on top of one another. A sequence of atrous_conv2d operations with identical rate parameters, 'SAME' padding, and filters with odd heights/
net = atrous_conv2d(net, filters1, rate, padding="SAME")
net = atrous_conv2d(net, filters2, rate, padding="SAME")
net = atrous_conv2d(net, filtersK, rate, padding="SAME")
can be equivalently performed cheaper in terms of computation and memory as:
pad = ... # padding so that the input dims are multiples of rate
net = space_to_batch(net, paddings=pad, block_size=rate)
net = conv2d(net, filters1, strides=[1, 1, 1, 1], padding="SAME")
net = conv2d(net, filters2, strides=[1, 1, 1, 1], padding="SAME")
net = conv2d(net, filtersK, strides=[1, 1, 1, 1], padding="SAME")
net = batch_to_space(net, crops=pad, block_size=rate)
because each pair of consecutive space_to_batch and batch_to_space have the same block_size. Thus, they cancel each other out provided that their respective paddings and crops inputs are identical.
Dilated convolution in action
In the paper “Multi-scale context aggregation by dilated convolutions”, the authors build a network out of multiple layers of dilated convolutions. They increase the dilation rate l exponentially
at each layer. As a result, while the number of parameters grows only linearly with layers, the effective receptive field grows exponentially with layers!
In the paper, a context module was made comprising of 7 layers that apply 3 x 3 convolutions with varying values of the dilation rate.
The dilation rates were 1, 1, 2, 4, 8, 16 and 1. The final convolution is a 1 x 1 convolution to make sure that the number of channels are the same as in the input one. This implies that the input
and output has an equal number of channels.
At the bottom of the table, we can see two different types of output channels: Basic and Advanced. The Basic context module has only 1 channel (1C) in the whole module whereas the Advanced context
module has increased the number of channels from 1C in the first layer to 32C at the penultimate layer (7th layer).
Dilated Convolution: Results of the context module
The model was tested on PASCAL VOC 2012 dataset. VGG-16 is used as the front-end module. Following was the setup of the model:
• The last two pooling and striding layers of VGG-16 were removed entirely, and the context module (discussed above) was plugged in.
• Padding of the intermediate feature maps was also removed.
• Padding of the input feature maps by a width of 33.
• A weight initialization considering the number of channels of input and output is used instead of standard random initialization.
We can see that the dilated convolutions performed better than the previous FCN-8s and DeepLabV1 by about 5 percent on the test set. Also, a mean IoU (Intersection over Union) of 67.6% is also
Complexity analysis of dilated convolutions
For any dilated convolution,
• There is a time complexity of O(d) for one dot product, which is simply d multiplications and d-1 addition.
• Since we perform k number of dot products, this amounts to O(k.d)
• Next, at the layer level, we apply our kernel n - k + 1 times over the input. If n >> k, this amounts to O(n.k.d)
• Finally, if we assume to have d number of kernels, our final time complexity of dilated convolution would be O(n.k.d^2).
Usefulness of Dilated Convolutions
Following are the advantages of using Dilated convolutions:
1. Since dilated convolutions support exponential expansion in the context of the receptive field, there is no loss of resolution.
2. Dilated convolutions use 'l' as the parameter for the dilation rate. As we increase the value of 'l' it allows one to have a larger receptive field which is really helpful as we are able to view
more data points thus saving computation and memory costs.
While dilated convolutions provide a cheap way to increase the receptive field and helps in the saving computation costs, the main drawback of such methods
is the requirement for learning a large amount of extra parameters.
With this article at OpenGenus, you must have the complete idea of Dilated Convolution. Enjoy.
Read these Research papers: | {"url":"https://iq.opengenus.org/dilated-convolution/","timestamp":"2024-11-07T22:18:50Z","content_type":"text/html","content_length":"65319","record_id":"<urn:uuid:84e6360c-8f23-48fc-8945-9757ac8ae62f>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00238.warc.gz"} |
What is the apothem of a polygon? Thank you very much.... | Socratic
What is the apothem of a polygon? Thank you very much....
1 Answer
Defined only for regular polygons, apothema is a segment from a center of a polygon to a midpoint of any side.
Here is the apothema of a regular octagon:
Apothema is also a radius of an inscribed circle and the shortest segment that connects a center of a regular polygon with points on its sides.
Impact of this question
2140 views around the world | {"url":"https://api-project-1022638073839.appspot.com/questions/what-is-the-apothem-of-a-polygon-thank-you-very-much","timestamp":"2024-11-07T00:48:39Z","content_type":"text/html","content_length":"33460","record_id":"<urn:uuid:b143ff75-38c5-4804-9a65-6caf3765c8d9>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00259.warc.gz"} |
Re: [Bug] math.mod behaves differently than the modulus operator.
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• Subject: Re: [Bug] math.mod behaves differently than the modulus operator.
• From: Mike Pall <mikelu-0801@...>
• Date: Mon, 28 Jan 2008 13:20:12 +0100
Grellier, Thierry wrote:
> Well, I find it weak defense to say that manual doesn't claim
> consistency (one may then consider this a manual issue).
Umm, I guess you've not been on the list back then ... math.mod
was there before %. It's also clearly marked as a compatibility
function (renamed to math.fmod). Which btw is the main reason why
the behaviour is different and why two variants are required.
The manual is pretty consistent with the implementation:
"[The] Modulo [operator] is defined as a % b == a - math.floor(a/b)*b
That is, it is the remainder of a division that rounds the quotient
towards minus infinity."
"math.fmod (x, y) -- Returns the remainder of the division of x by y." | {"url":"http://lua-users.org/lists/lua-l/2008-01/msg00613.html","timestamp":"2024-11-06T10:21:34Z","content_type":"text/html","content_length":"5065","record_id":"<urn:uuid:2c017afc-20b4-48b7-af6a-64951dcfbf50>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00482.warc.gz"} |
MathSciDoc: An Archive for Mathematician
We consider a general class of $N\times N$ random matrices whose entries $h_{ij}$ are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local
semicircle law which improves previous results [14] both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, $\max_{i,j} \E \abs{h_{ij}}^2$. As
a consequence, we prove the universality of the local $n$-point correlation functions in the bulk spectrum for a class of matrices whose entries do not have comparable variances, including random
band matrices with band width $W\gg N^{1-\e_n}$ with some $\e_n>0$ and with a negligible mean-field component. In addition, we provide a coherent and pedagogical proof of the local semicircle law,
streamlining and strengthening previous arguments from [17,19,6]. | {"url":"https://archive.ymsc.tsinghua.edu.cn/pacm_category/0128?show=time&size=5&from=20&target=searchall","timestamp":"2024-11-10T08:45:01Z","content_type":"text/html","content_length":"84372","record_id":"<urn:uuid:4f01e13e-e433-4997-b059-7e6d83517c13>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00876.warc.gz"} |
Vibro-acoustics of infinite and finite elastic fluid-filled cylindrical shells
The classical model of an elastic fluid-filled cylindrical shell is used for analysis of its vibrations. The model is based on thin shell theory, standard linear acoustics and the heavy fluid-loading
coupling concept. First, several important features of performance of a fluid-filled shell not yet fully explored in literature e.g. the difference between kinematic/forcing excitations, acoustic
source type identification (monopole/dipole) and energy transfer between fluid and shell are studied. Then the discussion is extended to finite fluid-filled shells by application of the Boundary
Integral Equations Method (BIEM). Two techniques for solving the equations deduced from the BIEM are discussed and investigated with respect to convergence, respectively, Boundary Elements (BE) and
modal expansion. Successively, the implementation of the BIEM is validated against numerical and experimental results for the simplified case of an empty shell. Finally, impedance boundary conditions
for a fluid-filled shell in an assembled piping system and computations of its resonances and forced response using the BIEM are discussed.
• Boundary Integral Equations Method (BIEM)
• finite cylindrical shells
• Heavy fluid loading
• Impedance boundary conditions
• Infinite
• Vibro-acoustics
• Wave propagation
Dive into the research topics of 'Vibro-acoustics of infinite and finite elastic fluid-filled cylindrical shells'. Together they form a unique fingerprint. | {"url":"https://vbn.aau.dk/en/publications/vibro-acoustics-of-infinite-and-finite-elastic-fluid-filled-cylin","timestamp":"2024-11-02T03:03:46Z","content_type":"text/html","content_length":"62693","record_id":"<urn:uuid:9791330f-fe75-4aaf-bb14-62109426e137>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00489.warc.gz"} |
Minutes per Kilometer to Seconds per Foot Conversion
This is our conversion tool for converting minutes per kilometer to seconds per foot.
To use the tool, simply enter a number in any of the inputs and the converted value will automatically appear in the opposite box.
How to convert Minutes per Kilometer (min/km) to Seconds per Foot (s/ft)
Converting Minutes per Kilometer (min/km) to Seconds per Foot (s/ft) is simple. Why is it simple? Because it only requires one basic operation: multiplication. The same is true for many types of unit
conversion (there are some expections, such as temperature). To convert Minutes per Kilometer (min/km) to Seconds per Foot (s/ft), you just need to know that 1min/km is equal to s/ft. With that
knowledge, you can solve any other similar conversion problem by multiplying the number of Minutes per Kilometer (min/km) by . For example, 3min/km multiplied by is equal to s/ft. | {"url":"https://unitconversion.io/minkm-to-sft-conversion","timestamp":"2024-11-11T11:22:52Z","content_type":"text/html","content_length":"33442","record_id":"<urn:uuid:98704d52-877f-409d-a9f1-ec345937c5d4>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00326.warc.gz"} |
round formula in Excel | Excelchat
I need a hand with a nested if formula =IF($C$6="Round 1",'Round 1'!M7,IF($C$6="Round 2",IF($C$6="Round 3",IF($C$6="Round 4","")))) it works for round 1 but when i select round 2>4 it wont. i need to
duplicate but not sure how to. Your help is appreciated.
Solved by Z. Y. in 14 mins | {"url":"https://www.got-it.ai/solutions/excel-chat/excel-help/how-to/round/round-formula-in-excel","timestamp":"2024-11-07T15:32:07Z","content_type":"text/html","content_length":"337317","record_id":"<urn:uuid:ad39dffb-a808-49cd-bdd0-e72ff7fe0de6>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00305.warc.gz"} |
The data below give the vapor pressure of octane, a major component of gasoline. \(\begin{array}{lllcl}\text { vp }(\mathrm{mm} \mathrm{Hg}) & 10 & 40 & 100 & 400 \\ t\left({ }^{\circ} \mathrm{C}\
right) & 19.2 & 45.1 & 65.7 & 104.0\end{array}\) Plot ln vp versus \(1 / T\). Use your graph to estimate the heat of vaporization of octane. \(\left(\ln P=A-\frac{\Delta H_{\mathrm{vap}}}{R}\left(\
frac{1}{T}\right)\right.\), where \(A\) is the \(y\) -intercept and \(\Delta H_{\mathrm{vap}}\) is the slope.)
Short Answer
Expert verified
Question: Estimate the heat of vaporization of octane using the given vapor pressure and temperature data and the provided formula. Short Answer: To estimate the heat of vaporization of octane, first
convert the given temperatures to Kelvin and then calculate the natural logarithm of vapor pressure and the inverse of temperature. Plot a graph of ln(vp) versus 1/T and find the slope of the linear
regression line. Finally, use the formula \(\Delta H_{\mathrm{vap}} = -m \times R\) with the calculated slope and the gas constant R (8.314 J/mol·K) to estimate the heat of vaporization of octane.
Step by step solution
Convert the temperatures to Kelvin
First, we need to convert the given temperatures from Celsius to Kelvin. To do this, we just add 273.15 to each temperature value. \(\begin{array}{lllcl}t\left({ }^{\circ} \mathrm{C}\right) & 19.2 &
45.1 & 65.7 & 104.0 \\ T\left(\mathrm{K}\right) & 292.35 & 318.25 & 338.85 & 377.15 \end{array}\)
Calculate ln(vp) and 1/T
Next, we will find the natural logarithm of each vapor pressure (vp) and the inverse of each temperature in Kelvin (1/T). $\begin{array}{lllcllll} \text { ln(vp) } & \ln(10) & \ln(40) & \ln(100) & \
ln(400) \\ 1 / T\left(\mathrm{K}\right) & 1/292.35 & 1/318.25 & 1/338.85 & 1/377.15 \\ \end{array}$ Now the table becomes: $\begin{array}{lllcllll} \text { ln(vp) } & 2.30 & 3.69 & 4.61 & 5.99 \\ 1 /
T\left(\mathrm{K}\right) & 0.00342 & 0.00314 & 0.00295 & 0.00265 \\ \end{array}$
Plot ln(vp) versus 1/T
Using the values obtained in the previous step, plot a scatter plot with 1/T on the x-axis and ln(vp) on the y-axis. Fit a linear regression line to this data.
Determine slope and y-intercept of the linear regression line
From the graph, we can obtain the slope and y-intercept of the linear regression line. Let's assume the slope is found to be -m and the y-intercept is n.
Estimate the heat of vaporization of octane
Using the provided formula, we know the slope is equal to \(\frac{\Delta H_{\mathrm{vap}}}{R}\). The gas constant \(R\) is equal to 8.314 J/mol·K. Therefore, \(\Delta H_{\mathrm{vap}} = -m \times R\)
Substitute the slope value and R into the above formula to get the heat of vaporization of octane.
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Heat of Vaporization
The heat of vaporization is the energy required to transform a given quantity of a substance from a liquid into a gas at constant temperature and pressure. It's an important concept in thermodynamics
and helps us understand how substances transition between phases.
In calculations like this exercise, heat of vaporization (abla H_{\text{vap}}) is derived from the Clausius-Clapeyron equation, which is: \\(\ln P = A - \frac{\Delta H_{\text{vap}}}{R} \left(\frac{1}
{T}\right)\) \Here, \(\ln P\) is the natural logarithm of vapor pressure, \(A\) is the y-intercept of the line on a graph of \(\ln(vp)\) versus \(1/T\), and \(\Delta H_{\text{vap}}\) is the slope
times the gas constant \(R\). By plotting these on a graph, and finding the slope of the line, one can estimate \(\Delta H_{\text{vap}}\).
This understanding is not only useful in academic exercises but also widely applicable in industries where phase changes are essential, such as in refrigeration or distillation processes.
Temperature Conversion
Temperature conversion is a fundamental skill, especially in scientific studies and calculations. It allows for the comparison and combination of data sets that may be presented in different
temperature scales. There are three major scales: Celsius (\(^{\circ}C\)), Kelvin (\(K\)), and Fahrenheit (\(^{\circ}F\)).
In scientific work, Kelvin is the preferred scale because it's an absolute temperature scale with no negative numbers, making it ideal for thermodynamic equations. To convert from Celsius to Kelvin,
a simple addition is required: \\[ T(K) = t(^{\circ}C) + 273.15 \] \This conversion reflects the absolute nature of temperature mentioned before, as Kelvin starts at absolute zero. In this exercise,
converting temperatures from Celsius to Kelvin was crucial for accurate use in calculating \(1/T\) in the Clausius-Clapeyron equation.
Natural Logarithm
The concept of the natural logarithm, often symbolized as \(\ln\), is integral to many scientific computations, including this exercise. It's a logarithm to the base \(e\), where \(e\) is an
irrational constant approximately equal to 2.71828.
The natural logarithm is useful due to its properties in simplifying formulas and calculus operations. In the context of vapor pressure equations, \(\ln\) is used to linearize the exponential
relationship between pressure and temperature. This allows the use of linear regression to estimate parameters like the heat of vaporization.
Understanding \(\ln\) is critical because it simplifies many complex chemical and physical systems by turning multiplicative relationships into additive ones, allowing easier solving and
interpretation of data. | {"url":"https://www.vaia.com/en-us/textbooks/chemistry/chemistry-principles-and-reactions-6-edition/chapter-9/problem-13-the-data-below-give-the-vapor-pressure-of-octane-/","timestamp":"2024-11-12T00:42:27Z","content_type":"text/html","content_length":"265101","record_id":"<urn:uuid:a69f0386-40c1-473d-93d0-628d0970add2>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00159.warc.gz"} |
Absolute Value - Meaning, How to Find Absolute Value, Examples - [[company name]] [[target location]], [[stateabr]]
Absolute ValueDefinition, How to Find Absolute Value, Examples
Many perceive absolute value as the length from zero to a number line. And that's not inaccurate, but it's not the entire story.
In mathematics, an absolute value is the magnitude of a real number irrespective of its sign. So the absolute value is always a positive number or zero (0). Let's check at what absolute value is, how
to discover absolute value, few examples of absolute value, and the absolute value derivative.
What Is Absolute Value?
An absolute value of a figure is constantly zero (0) or positive. It is the magnitude of a real number without regard to its sign. This refers that if you possess a negative figure, the absolute
value of that figure is the number without the negative sign.
Meaning of Absolute Value
The last definition states that the absolute value is the length of a figure from zero on a number line. So, if you consider it, the absolute value is the distance or length a figure has from zero.
You can visualize it if you check out a real number line:
As you can see, the absolute value of a number is the distance of the number is from zero on the number line. The absolute value of negative five is five because it is 5 units apart from zero on the
number line.
If we graph negative three on a line, we can see that it is 3 units away from zero:
The absolute value of negative three is three.
Now, let's check out one more absolute value example. Let's say we hold an absolute value of 6. We can plot this on a number line as well:
The absolute value of 6 is 6. Hence, what does this refer to? It states that absolute value is constantly positive, even if the number itself is negative.
How to Calculate the Absolute Value of a Number or Figure
You need to know few points prior working on how to do it. A couple of closely linked properties will assist you grasp how the number inside the absolute value symbol works. Fortunately, here we have
an definition of the following four essential characteristics of absolute value.
Fundamental Properties of Absolute Values
Non-negativity: The absolute value of ever real number is always zero (0) or positive.
Identity: The absolute value of a positive number is the number itself. Alternatively, the absolute value of a negative number is the non-negative value of that same number.
Addition: The absolute value of a total is less than or equal to the total of absolute values.
Multiplication: The absolute value of a product is equivalent to the product of absolute values.
With above-mentioned four essential properties in mind, let's look at two more helpful properties of the absolute value:
Positive definiteness: The absolute value of any real number is always positive or zero (0).
Triangle inequality: The absolute value of the variance among two real numbers is less than or equal to the absolute value of the total of their absolute values.
Considering that we know these characteristics, we can ultimately start learning how to do it!
Steps to Calculate the Absolute Value of a Number
You have to observe few steps to discover the absolute value. These steps are:
Step 1: Write down the number of whom’s absolute value you desire to find.
Step 2: If the figure is negative, multiply it by -1. This will make the number positive.
Step3: If the number is positive, do not alter it.
Step 4: Apply all characteristics relevant to the absolute value equations.
Step 5: The absolute value of the expression is the expression you get following steps 2, 3 or 4.
Keep in mind that the absolute value symbol is two vertical bars on either side of a figure or expression, similar to this: |x|.
Example 1
To start out, let's consider an absolute value equation, such as |x + 5| = 20. As we can observe, there are two real numbers and a variable inside. To figure this out, we are required to find the
absolute value of the two numbers in the inequality. We can do this by observing the steps mentioned above:
Step 1: We are provided with the equation |x+5| = 20, and we are required to discover the absolute value inside the equation to find x.
Step 2: By using the basic characteristics, we learn that the absolute value of the total of these two numbers is equivalent to the sum of each absolute value: |x|+|5| = 20
Step 3: The absolute value of 5 is 5, and the x is unknown, so let's remove the vertical bars: x+5 = 20
Step 4: Let's calculate for x: x = 20-5, x = 15
As we can observe, x equals 15, so its length from zero will also equal 15, and the equation above is genuine.
Example 2
Now let's check out another absolute value example. We'll utilize the absolute value function to get a new equation, like |x*3| = 6. To make it, we again need to obey the steps:
Step 1: We use the equation |x*3| = 6.
Step 2: We need to solve for x, so we'll begin by dividing 3 from both side of the equation. This step gives us |x| = 2.
Step 3: |x| = 2 has two potential solutions: x = 2 and x = -2.
Step 4: Hence, the initial equation |x*3| = 6 also has two possible answers, x=2 and x=-2.
Absolute value can involve a lot of intricate expressions or rational numbers in mathematical settings; still, that is a story for another day.
The Derivative of Absolute Value Functions
The absolute value is a continuous function, this refers it is varied everywhere. The ensuing formula provides the derivative of the absolute value function:
For absolute value functions, the area is all real numbers except 0, and the range is all positive real numbers. The absolute value function rises for all x<0 and all x>0. The absolute value function
is constant at zero(0), so the derivative of the absolute value at 0 is 0.
The absolute value function is not distinctable at 0 because the left-hand limit and the right-hand limit are not equivalent. The left-hand limit is provided as:
I'm →0−(|x|/x)
The right-hand limit is given by:
I'm →0+(|x|/x)
Because the left-hand limit is negative and the right-hand limit is positive, the absolute value function is not distinguishable at zero (0).
Grade Potential Can Guide You with Absolute Value
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Python Absolute Value - Everything You Need to Know
In Python, we often deal with negative and positive numeric values. Python provides different data types, and numeric values, integers, and floats are the most commonly used Python numeric data types
. Many times in Python programming, we just want the positive numeric value from the result. In this Python tutorial, we will discuss how we can change the sign of a numeric value with the help of
the Python absolute function.
How to Get Absolute Value in Python?
An absolute value in mathematic refers to a non-negative or positive number. For example, the absolute value of -12 is 12 and the absolute value for 12 is also 12. The mathematic representation of
absolute value is as follows: |-12| =12 In Python, we can find out the absolute value of a number with the following two methods:
• Python abs() method
• Python math.abs() method
In most cases, you will be using the abs() method.
Find the Absolute Value in Python Using abs() Function
The Python abs() method is used to find out the absolute value for a specified number. If the specified number is a floating-point number , the return value will also be a floating-point number.
Similarly, if the specified number is an integer, the return value will also be an integer.
abs() Method Example:
num_1 = -12
num_2 = -12.23
num_3 = -12.456
num_4 = 12.4555
print("The absolute value of num_1 is :", abs(num_1))
print("The absolute value of num_2 is :", abs(num_2))
print("The absolute value of num_3 is :", abs(num_3))
print("The absolute value of num_4 is :", abs(num_4))
The absolute value of num_1 is : 12
The absolute value of num_2 is : 12.23
The absolute value of num_3 is : 12.456
The absolute value of num_4 is : 12.4555
Find the Absolute Value in Python Using math.fabs() Function
Python has an inbuilt module called math, and it contains a method .fabs() that you can use to find the absolute value for a numerical value. The working of math.fabs() is similar to abs() , however,
it always returns a floating-point number as an absolute value.
math . fabs() Method Example:
import math
num_1 = -12
num_2 = -12.23
num_3 = -12.456
num_4 = 12
print("The absolute value of num_1 is :", math.fabs(num_1))
print("The absolute value of num_2 is :", math.fabs(num_2))
print("The absolute value of num_3 is :", math.fabs(num_3))
print("The absolute value of num_4 is :", math.fabs(num_4))
The absolute value of num_1 is : 12.0
The absolute value of num_2 is : 12.23
The absolute value of num_3 is : 12.456
The absolute value of num_4 is : 12.0
<Note> : In the above example, you can see that for -12 and 12 the absolute value return by the math.fabs() is 12.0 not 12.
Get Absolute Values from a Python list/array
If you have a Python list or Python array that has multiple numbers and you want to get absolute value for every number, you can simply use the abs() or math.fabs() method and map() function. This
way you can find absolute value for every number. Following are some examples that demonstrate how you can use abs () and math.fabs() functions with map() function to find the absolute value for each
number in a list or array.
Example 1:
my_nums = [-12, -34, 11, 28, -78, 2, -123]
abs_values = list(map(abs, my_nums))
print("The absolute values are:", abs_values)
The absolute values are: [12, 34, 11, 28, 78, 2, 123]
Example 2:
import math
my_nums = [-12, -34, 11, 28, -78, 2, -123]
abs_values = list(map(math.fabs, my_nums))
print("The absolute values are:", abs_values)
The absolute values are: [12.0, 34.0, 11.0, 28.0, 78.0, 2.0, 123.0]
Change Number Sign using Python Unary Operator
The Python unary operator allows us to change the sign of a numeric value. For instance, if the number is positive, then by using the unary operator, we can change it to negative. Similarly, if the
number is negative, we can change it to positive.
unary negative operator syntax
num =- num
>>> num = 1
>>> num =- num
>>> num
<Note> Do not confuse the Python unary operator a =-a with Python compound Assignment operator a -= a .
The abs() method is used more often as compared to the math.fabs() method because it returns a similar data type. On the other hand, the fabs() method returns a floating data value for every number.
In this Python tutorial, you learned how to find the absolute value in Python using abs() and math.fabs() methods. Also, you learned how to use the abs and math.fabs() methods with map() function to
find out the absolute value for every number on a list or array.
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Chris Beeler
I am a PhD student in Mathematics and Statistics and the former president of the Mathematics and Statistics Graduate Student Association (MSGSA) at the University of Ottawa. Formerly I was a MSc
student with CLEAN in the Modelling and Computational Science program at the University of Ontario Institute of Technology and a BSc student in the Combined Honours in Chemistry and Mathematics
program at Dalhousie University. I was the lead organizer for the 15th Annual Ottawa Mathematics Conference.
email: christopher.beeler@uottawa.ca
github: https://github.com/beelerchris
linkedin: https://www.linkedin.com/in/chris-beeler/
Nautical Navigation: I am the lead researcher for the development of the SubWorld environment and the application of dynamic programming to it.
ChemGymRL: I am directly involved in the continuing development of the ChemGymRL environment for use in the reinforcement learning field.
Maximizing Thermal Efficiency: I was the lead researcher for the development of the Heat Engine environments and the application of both evolutionary and gradient-based reinforcement learning
algoritms to these environments.
PhD research
My PhD research consists of studying how the access to information revealing actions affects the behavior of reinforcement learning agents and determining how the levels of available information
changes the difficulty of a given reinforcement learning problem.
Masters research
My MSc research consisted of using reinforcement learning methods based on genetic algorithms to reproduce thermodynamic cycles without prior knowledge of physics and studying optimal solutions to
various common reinforcement learning environments.
Publications / Posters / Presentations
• C. Beeler, "Entropic Measuring of MNIST Images", Medium (November 2022)
• C. Beeler "Dynamic programming with reinforcement learning for nautical navigation", Oral Presentation, University of Ottawa, 15th Annual Ottawa Mathematics Conference (July 2022)
• C. Beeler "Dynamic programming with partial information to overcome navigational uncertainty in a nautical environment", Oral Presentation, University of Ottawa, Graduate Student Colloquium
(January 2022)
• C. Beeler, X. Li, M. Crowley, M. Fraser, and I. Tamblyn, "Dynamic programming with partial information to overcome navigational uncertainty in a nautical environment", arXiv preprint,
arXiv:2112.14657 (December 2021)
• C. Beeler, U. Yahorau, R. Coles, K. Mills, S. Whitelam, and I. Tamblyn, "Optimizing thermodynamic trajectories using evolutionary and gradient-based reinforcement learning", Phys. Rev. E, 104,
064128 (December 2021)
• C. Beeler, "Max-flow Min-cut and Baseball End-of-Season Elimination", Medium (October 2021)
• C. Beeler, "Neural Networks", Lecture, A3MD, A3MD ML Bootcamp (August 2020 & September 2020)
• K. Mills, K. Ryczko, I. Luchak, A. Domurad, C. Beeler, and I. Tamblyn, "Extensive deep neural networks for transferring small scale learning to large scale systems", Chemical Science, 10,
4129-4140 (February 2019)
• C. Beeler, Xinkai Li, Zihan Yang, Mark Crowley, and Isaac Tamblyn, "Navigating Chemistry", Invited Oral Presentation, Ottawa-AI Alliance, Ottawa-AI Workshop 2019 (November 2019)
• C. Beeler, U. Yahorau, R. Coles, K. Mills, S. Whitelam, and I. Tamblyn, "Learning to work efficiently: Using neuroevolutionary strategies for reinforcement learning on classical thermodynamic
systems", Oral Presentation, McGill University, Physics & AI Workshop (May 2019)
• C. Beeler and I. Tamblyn, "Perpetually Playing Physics", Oral Presentation, University of Ontario Institute of Technology, Modelling and Computational Science Seminar (April 2019)
• C. Beeler, U. Yahorau, R. Coles, K. Mills, S. Whitelam, and I. Tamblyn, "Maximizing thermal efficiency of heat engines using neuroevolutionary strategies for reinforcement learning", Oral
Presentation, American Physical Society, March Meeting 2019 (March 2019)
• K. Ryczko, C. Beeler, R. Coles, A. Domurad, C. Homenick, I. Luchak, K. Mills, D. Strubbe, U. Yahorau, and I. Tamblyn, "Machine learning for molecules", Poster Presentation, NeurIPS, 32nd
Conference on Neural Information Processing Systems (December 2018)
• Inter-Math-AI Scholarship (2022-Present)
• University of Ottawa Admission Scholarship (2019-Present)
• 15th Annual Ottawa Mathematics Conference 3-Minute-Thesis Competition Winner (2022) | {"url":"http://clean.energyscience.ca/group/cbeeler.html","timestamp":"2024-11-12T03:55:31Z","content_type":"text/html","content_length":"13765","record_id":"<urn:uuid:4391bec4-6e47-4ef5-b66e-0dead842fb11>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00188.warc.gz"} |
gcc/testsuite/gdc.test/fail_compilation/constraints_func1.d - gcc - Git at Google
EXTRA_FILES: imports/constraints.d
fail_compilation/constraints_func1.d(79): Error: none of the overloads of template `imports.constraints.test1` are callable using argument types `!()(int)`
fail_compilation/imports/constraints.d(9): Candidate is: `test1(T)(T v)`
with `T = int`
must satisfy the following constraint:
` N!T`
fail_compilation/constraints_func1.d(80): Error: none of the overloads of template `imports.constraints.test2` are callable using argument types `!()(int)`
fail_compilation/imports/constraints.d(10): Candidate is: `test2(T)(T v)`
with `T = int`
must satisfy the following constraint:
` !P!T`
fail_compilation/constraints_func1.d(81): Error: none of the overloads of template `imports.constraints.test3` are callable using argument types `!()(int)`
fail_compilation/imports/constraints.d(11): Candidate is: `test3(T)(T v)`
with `T = int`
must satisfy the following constraint:
` N!T`
fail_compilation/constraints_func1.d(82): Error: none of the overloads of template `imports.constraints.test4` are callable using argument types `!()(int)`
fail_compilation/imports/constraints.d(12): Candidate is: `test4(T)(T v)`
with `T = int`
must satisfy the following constraint:
` N!T`
fail_compilation/constraints_func1.d(83): Error: none of the overloads of template `imports.constraints.test5` are callable using argument types `!()(int)`
fail_compilation/imports/constraints.d(13): Candidate is: `test5(T)(T v)`
with `T = int`
must satisfy one of the following constraints:
` N!T
fail_compilation/constraints_func1.d(84): Error: none of the overloads of template `imports.constraints.test6` are callable using argument types `!()(int)`
fail_compilation/imports/constraints.d(14): Candidate is: `test6(T)(T v)`
with `T = int`
must satisfy one of the following constraints:
` N!T
fail_compilation/constraints_func1.d(85): Error: none of the overloads of template `imports.constraints.test7` are callable using argument types `!()(int)`
fail_compilation/imports/constraints.d(15): Candidate is: `test7(T)(T v)`
with `T = int`
must satisfy one of the following constraints:
` N!T
fail_compilation/constraints_func1.d(86): Error: none of the overloads of template `imports.constraints.test8` are callable using argument types `!()(int)`
fail_compilation/imports/constraints.d(16): Candidate is: `test8(T)(T v)`
with `T = int`
must satisfy the following constraint:
` N!T`
fail_compilation/constraints_func1.d(87): Error: none of the overloads of template `imports.constraints.test9` are callable using argument types `!()(int)`
fail_compilation/imports/constraints.d(17): Candidate is: `test9(T)(T v)`
with `T = int`
must satisfy the following constraint:
` !P!T`
fail_compilation/constraints_func1.d(88): Error: none of the overloads of template `imports.constraints.test10` are callable using argument types `!()(int)`
fail_compilation/imports/constraints.d(18): Candidate is: `test10(T)(T v)`
with `T = int`
must satisfy the following constraint:
` !P!T`
fail_compilation/constraints_func1.d(89): Error: none of the overloads of template `imports.constraints.test11` are callable using argument types `!()(int)`
fail_compilation/imports/constraints.d(19): Candidate is: `test11(T)(T v)`
with `T = int`
must satisfy one of the following constraints:
` N!T
fail_compilation/constraints_func1.d(90): Error: none of the overloads of template `imports.constraints.test12` are callable using argument types `!()(int)`
fail_compilation/imports/constraints.d(20): Candidate is: `test12(T)(T v)`
with `T = int`
must satisfy the following constraint:
` !P!T`
fail_compilation/constraints_func1.d(92): Error: none of the overloads of template `imports.constraints.test1` are callable using argument types `!()(int, int)`
fail_compilation/imports/constraints.d(9): Candidate is: `test1(T)(T v)`
void main()
import imports.constraints;
test1(0, 0); | {"url":"https://gnu.googlesource.com/gcc/+/1f16a020acbea0af26209478990b83b1a1ba3a2b/gcc/testsuite/gdc.test/fail_compilation/constraints_func1.d","timestamp":"2024-11-13T21:59:19Z","content_type":"text/html","content_length":"28823","record_id":"<urn:uuid:7e4b0afd-b4f4-4003-8c08-9df391d903ea>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00025.warc.gz"} |
Real Numbers
Note: If you cannot view some of the math on this page, you may need to add MathML support to your browser. If you have Mozilla/Firefox, go here and install the fonts. If you have Internet Explorer,
go here and install the MathPlayer plugin.
Number theory
Real Numbers
Real Numbers
Properties of the real numbers.
Sibling topics:
Definition of upper and lower bounds
For `x in RR` and `S sube RR`, "`x` is an
upper bound
for `S`" means:
`(AA y in S)(x >= y)`
Similarly, "`x` is a
lower bound
for `S`" means:
`(AA y in S)(x <= y)`
An upper of a set does not need to be a member of that set, but if the set does contain its own upper bound, it is called an
of the set.
If a set has an upper bound, it is said to be
bounded above
, and if it has a lower bound, it is said to be
bounded below
. A set with both an upper and lower bound is
. If it does not have both, it is
Definition of extrema
Given `S sube RR`, `x` is said to be the
largest element
) of `S`, and we write `x="max "S`, if `x in S` and `x` is an
upper bound
for `S`.
Similarly, `y` is said to be the
smallest element
) of `S`, and we write `x="min "S`, if `y in S` and `y` is a
lower bound
for `S`.
(plural "extrema") is either a minimum or a maximum. Not all sets have a minimum or a maximum. Note the difference between an extremum and an upper or lower bound. An extremum of `S` is in `S`, but
an upper or lower bound need not be.
Definition of lub and glb
For `x in RR` and `S sube RR`, `x` is the
least upper bound (lub)
) of `S` (denoted "lub of `S`" or "`"sup " S`") if:
`(AA y in S)(x >= y) and (AA z in RR)(z" is an upper bound for "S => z >= x)`
Similarly, "`x` is the
greatest lower bound (glb)
) of `S`" (denoted "glb of `S`" or "`"inf " S`") means:
`(AA y in S)(x <= y) and (AA z in RR)(z" is a lower bound for "S => z <= x)`
If a set contains its own least upper bound, then the set has a
equal to the lub. Similarly, if a set contains its own greatest lower bound, the set has a
equal to the glb.
Definition of the completeness property
For `S sube RR`, `S` has the
completeness property
1. Every nonempty subset `T` of `S` which is bounded above by an element of `RR` has a lub in `S`, and
2. Every nonempty subset `T` of `S` which is bounded below by an element of `RR` has a glb in `S`.
Note that the lub and glb must be in `S`, though they need not be in `T`.
Theorem: Some non-empty, bounded sets do not have extrema
Some non-empty subsets of the real numbers do not have
, even if they are bounded. That is, even though the set may be bounded above or below, they lack a maximum or minimum element (or both). This of course can only be the case for infinite sets.
Consider the set `S=[0,1]`. Clearly, the infimum of this set is 0, and the supremum is 1. Both 0 and 1 are also in the set, so the set has both extrema. Now consider the set `T=[0,1)`. 1 is clearly
an upper bound for this set, but is it the
upper bound? In fact it is, and `T` has no maximum element.
By definition, the set `T` contains every number `n` where `0 <= n < 1`. Assume by way of contradiction that there does exist an upper bound for `T` (call it `u`) that is less than 1. Since `u` would
have to be very close to 1, we won't consider `u <= 0`. Then we have `0 < u < 1`. But given any such `u`, we can always find another number `v` such that `u < v < 1`, which would contradict `u` being
an upper bound. For instance, let `v=(u+1)/2`. If we can prove that `u < (u+1)/2 < 1`, then we will have shown that any number less than 1 cannot be an upper bound for `T`. Similarly, we will have
shown that `T` has no maximum element, since a greater element that is also within `T` can always be found. We will now prove that `u < (u+1)/2 < 1`. Because `u < 1`, `u+u=2u < u+1 < 2`. Then,
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Integrated Mathematics Order of Operations. Remember the Phrase Order of Operations Parentheses - ( ) or [ ] Parentheses - ( ) or [ ] Exponents or Powers. - ppt download
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Abbreviating Principle – Simplified | Gregg Shorthand
Abbreviating Principle – Simplified
I have a few questions about the Abbreviating Principle in
Regarding the groups of words in families, from what I have
read on the blog and in the books, I think there are 15 family groups, but I am
not sure if I’ve missed any:
1. -cate/-quate:
indic(ate), loc(ate), duplic(ate), educ(ate), adeq(uate).
2. -cide/-side:
deci(de), besi(de), outsi(de),.
3. -gate:
deleg(ate), navig(ate).
4. -iety:
vari(ety), soci(ety).
5. -iferous:
vocif(erous), conif(erous).
6. -itis:
tonsili(tis), arthri(tis).
7. -iverous:
8. -ntic:
frant(ic), romant(ic).
9. -ology:
apol(ogy), geol(ogy).
10. -quent:
eloq(uent), freq(uent).
11. -quire:
acqui(re), requi(re).
12. -titude:
attit(ude), gratit(ude), aptit(ude).
13. -titute:
constit(ute), substit(ute).
14. -tribute:
trib(ute), distrib(ute).
15. -use:
excu(se), refu(se), accu(se), abu(se), confu(se).
Is this right?
Also, the dictionary shows the same form for “eloquence” as
it does for “eloquent”
(e l o k), and for “consequence” and ”consequent” (k s e k);
would these be listed together with -quent like this: -quent/-quence, or are
these just derivatives?
Next question: I have been trying to find words not in
families in the dictionary, other than those listed in the books. So far I have
only found “melancholy” (m e l a n) and its derivatives; are there any more?
Many thanks.
3 comments Add yours
1. I think that the family groups are right, except for -iverous which should be -ivorous. "Elo-quent/ce" should be in the -quent/ce family (it's the same outline) — a derivative would be
"consequential." For your last question, the best way to do this is to go to the post Abbreviating Principle, read my reply where I detailed all of the words in the Anniversary and 1916
dictionaries that follow the AP by classes, and check each one of those entries with the Simplified dictionary. It is unlikely that a word that was written in full in Anniversary would be written
using the Abbreviating Principle in Simplified, so my suggestion should work.
I hope this is helpful.
2. Thanks for the link to the other post.
I have amended my notes of lists of the family groups to show:
7. -ivorous: carniv(orous), herbiv(orous), omniv(orous).
10. -quent/-quence: eloq(uent), eloq(uence), freq(uent).
Regarding word not in family groups, I now have three lists:
(1) “In the Manual”:
alphabet, anniversary, arithmetic, atmosphere, convenient/convenience, preliminary, privilege, reluctant/reluctance, significant/significance.
(2) “Additional to these, in the Most Used book”:
algebra, curriculum, equivalent, memorandum/memoranda, philosophy.
(3) “Additional to these, based on Carlos’ lists, & found in the Simplified dictionary”:
melancholy, perpendicular.
I wasn’t able to find any others.
Thanks again.
1. You're welcome. It's too bad that Simplified and later series eliminated most of these. In my opinion, writing using the abbreviating principle is one of the most efficient and speedy ways of
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What is the difference between moment generating function and probability generating function?
What is the difference between moment generating function and probability generating function?
The mgf can be regarded as a generalization of the pgf. The difference is among other things is that the probability generating function applies to discrete random variables whereas the moment
generating function applies to discrete random variables and also to some continuous random variables.
What is the meaning of moment generating function?
The moment-generating function is the expectation of a function of the random variable, it can be written as: For a discrete probability mass function, For a continuous probability density function,
In the general case: , using the Riemann–Stieltjes integral, and where is the cumulative distribution function.
How do you find the variance of MGF?
So if you are given the negative binomial MGF, all you need to do to calculate E[X] is to take the derivative of the MGF, and evaluate it at t=0. To get the variance, recall that Var[X]=E[X2]−E[X]2,
so you would calculate the second derivative M″X(0) at t=0 and subtract the square of the previous result.
How do you find moments from moment generating function in statistics?
Once you have the MGF: λ/(λ-t), calculating moments becomes just a matter of taking derivatives, which is easier than the integrals to calculate the expected value directly. Using MGF, it is possible
to find moments by taking derivatives rather than doing integrals! A few things to note: For any valid MGF, M(0) = 1.
What is the difference between a probability density function and a probability generating function?
The probability generating function only applies to discrete random variables. The probability density function applies to continuous random variables, it is the analog of the probability mass
function for discrete random variables.
What is the moment generating function of the normal distribution?
(8) The moment generating function corresponding to the normal probability density function N(x;µ, σ2) is the function Mx(t) = exp{µt + σ2t2/2}.
What is the moment generating function of X Y?
The moment generating function (MGF) of a random variable X is a function MX(s) defined as MX(s)=E[esX]. We say that MGF of X exists, if there exists a positive constant a such that MX(s) is finite
for all s∈[−a,a]. Before going any further, let’s look at an example.
How do you find the variance of a moment generating function?
We can solve these in a couple of ways.
1. We can use the knowledge that M ′ ( 0 ) = E ( Y ) and M ′ ′ ( 0 ) = E ( Y 2 ) . Then we can find variance by using V a r ( Y ) = E ( Y 2 ) − E ( Y ) 2 .
2. We can recognize that this is a moment generating function for a Geometric random variable with p = 1 4 .
What is the difference between the standard normal distribution and a general normal distribution?
The difference between a normal distribution and standard normal distribution is that a normal distribution can take on any value as its mean and standard deviation. On the other hand, a standard
normal distribution has always the fixed mean and standard deviation.
What is the difference between normal distribution and probability distribution?
The normal distribution is a probability distribution. As with any probability distribution, the proportion of the area that falls under the curve between two points on a probability distribution
plot indicates the probability that a value will fall within that interval.
What is the difference between standard deviation and normal deviation?
The mean of the normal distribution determines its location and the standard deviation determines its spread.
What are moment-generating functions?
which are functions of moments, are sometimes difficult to find. Special functions, called moment-generating functions can sometimes make finding the mean and variance of a random variable simpler.
How do you find the moment-generating function of a random variable?
And, setting t = 0, and using the formula for the variance, we get the binomial variance σ 2 = n p ( 1 − p): Not only can a moment-generating function be used to find moments of a random variable, it
can also be used to identify which probability mass function a random variable follows.
What is a moment in statistics?
The expected values E ( X), E ( X 2), E ( X 3), …, and E ( X r) are called moments. As you have already experienced in some cases, the mean: which are functions of moments, are sometimes difficult to
find. Special functions, called moment-generating functions can sometimes make finding the mean and variance of a random variable simpler.
When is a moment generating function continuously differentiable?
(1) “If a moment generating function exists, then m ( t) is continuously differentiable in some neighborhood of the origin.” Mood, Graybill, Boes (1974) An Intro. to the Theory of Statistics, 3e, | {"url":"https://www.fdotstokes.com/2022/08/20/what-is-the-difference-between-moment-generating-function-and-probability-generating-function/","timestamp":"2024-11-10T08:00:55Z","content_type":"text/html","content_length":"63105","record_id":"<urn:uuid:32bb813a-0650-4239-9c6a-9fb816ed6a39>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00366.warc.gz"} |
array.get function in Pine Script - Pine Wizards
array.get function in Pine Script
By PineWizards
Published on
In this tutorial, we will explore the array.get function in Pine Script, an essential tool for retrieving elements from arrays. The array.get function allows you to extract elements from an array,
which can be particularly helpful when working with time series data in trading algorithms. This tutorial will cover the syntax and usage of the array.get function and provide two unique use cases
that demonstrate its capabilities. We will be using Pine Script v4 for all examples.
The Array.get Function: Syntax and Usage
The array.get function is used to access an element from an array by specifying the index of the element. Here’s the syntax for the array.get function:
array.get(array_id, index)
• array_id: This parameter represents the identifier of the array from which you want to extract the element.
• index: This parameter denotes the index of the element you wish to access. Indexing starts at 0, so the first element is at index 0, the second at index 1, and so on.
It’s important to note that if the specified index is out of the range of the array, the array.get function will return na, the Pine Script representation for “not available” or “not applicable.”
indicator('Array.get Example 1', shorttitle='EG1')
// Create an array and populate it with sample data
arr = array.new_float(5)
array.set(arr, 0, close)
array.set(arr, 1, close[1])
array.set(arr, 2, close[2])
array.set(arr, 3, close[3])
array.set(arr, 4, close[4])
// Retrieve an element using array.get
element = array.get(arr, 2) // Retrieves the element at index 2 (30)
In this example, we create an array to store the closing prices of the current bar and the previous four bars. We then use the array.get function to access the closing price from two bars ago and
plot it on the chart. Here’s a breakdown of each line:
1. //@version=5: This line sets the version of Pine Script used for the script. In this case, we use version 5.
2. indicator('Array.get Example 1', shorttitle='EG1'): This line declares a custom indicator and sets the title to “Array.get Example 1” and the short title to “EG1”.
3. arr = array.new_float(5): This line creates a new floating-point array named arr with a size of 5.
4. array.set(arr, 0, close): This line sets the value at index 0 of the arr array to the closing price of the current bar.
5. array.set(arr, 1, close[1]): This line sets the value at index 1 of the arr array to the closing price of the previous bar.
6. array.set(arr, 2, close[2]): This line sets the value at index 2 of the arr array to the closing price of the bar two periods ago.
7. array.set(arr, 3, close[3]): This line sets the value at index 3 of the arr array to the closing price of the bar three periods ago.
8. array.set(arr, 4, close[4]): This line sets the value at index 4 of the arr array to the closing price of the bar four periods ago.
9. element = array.get(arr, 2): This line uses the array.get function to retrieve the value at index 2 of the arr array (the closing price from two bars ago) and assigns it to the variable element.
10. plot(element): This line plots the value of element on the chart, which is the closing price from two bars ago.
In summary, this example demonstrates how to create an array, populate it with historical closing prices, and use the array.get function to access a specific value within the array (in this case, the
closing price from two bars ago). The value is then plotted on the chart.
Key Takeaways
1. The array.get function is used to retrieve elements from an array based on the specified index.
2. The syntax for the array.get function is array.get(array_id, index).
3. If an index is out of the range of the array, the function will return na.
4. The array.get function can be used in various scenarios, such as accessing basic arrays or working with historical price data.
The array.get function is an essential tool for working with arrays in Pine Script. This tutorial provided an overview of the function’s syntax and usage, along with two unique examples that
demonstrated its application in different scenarios.
By understanding how to use the array.get function, you can enhance your Pine Script code and create more advanced trading algorithms. Remember to handle out-of-range indexes appropriately and always
ensure that the array being accessed is properly initialized and populated.
As you continue to explore Pine Script, consider experimenting with other array functions to further your understanding of arrays and their capabilities in trading algorithms.
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Motorsport Off Topic Thread
30 May 2007
I often come across interesting snippets of F1 info or other motorsport related topics that doesn't seem to be relevant in other threads or worth a new one, I can't be the only one?
So hopefully this thread will prove useful for random bits of information you found interesting and wish to share with others.
I'll start with Lewis Hamilton allegedly having recorded a secret R&B album!
8 Mar 2007
Williams looking for new title sponsor -
They need to get a company on board quick, else they'll go under pretty quick I reckon.
Kind of confirms that their second driver this year will be whoever can bring the biggest cheque!
And good idea for a thread Shimmy. There is a lot of stuff that doesn't fit into a specific thread, yet warrants a couple of posts of brief discussion.
17 Jan 2011
I'll start with Lewis Hamilton allegedly having recorded a secret R&B album!
I read that also on another website. If true, hope a track will be leaked at some point. Should be quite a laugh.
30 May 2007
Williams looking for new title sponsor -
They need to get a company on board quick, else they'll go under pretty quick I reckon.
Latest sponsor rumour.
I read that also on another website. If true, hope a track will be leaked at some point. Should be quite a laugh.
Just a matter of when rather than if.
11 Feb 2011
I hope its better than Andy Murrays.
18 Oct 2002
I hope its better than Andy Murrays.
I see your Andy Murray's and raise you Villeneuve...
I own a copy as well *hangs head in shame* but it was a present and signed...
21 Sep 2005
Judging from the 5liveF1 twitter feed they have James Allan joining the 5live F1 team!
17 Oct 2002
Judging from the 5liveF1 twitter feed they have James Allan joining the 5live F1 team!
8 Mar 2007
Judging from the 5liveF1 twitter feed they have James Allan joining the 5live F1 team!
*deep breath in......*
17 Jan 2011
Judging from the 5liveF1 twitter feed they have James Allan joining the 5live F1 team!
That's not funny.
Just seen this
andrewbensonf1Andrew Benson
RT @5LiveF1 "and I am delighted to welcome @Jamesallenonf1 to the 5 Live F1 team as the new commentator"
1 minute ago
Last edited:
21 Sep 2005
Haha. Can't say I ever hated him that much for the most part
Found him mildly annoying sometimes, then heard Legard and all was forgiven!
6 Jun 2004
Oh, no....ah well, back to Lewisterical commentary and people on here acting like retards.
Bother. Just when I had warmed to him in his 'informed blogger' role, and now he's returning to the commentary booth to wreak havoc on my blood pressure....
18 Oct 2002
Thing is... With Ant and Crofy moving to Sky, sticking with 5live would always be a huge step backwards.
Who'll be next to him? Legard?
Now Sky... can we get SkyGo on android now?
18 Oct 2002
Imagine James Allen and Legard on the same commentary team.
I'd blow my own brains out rather than have to listen to those to. Their fans were adamant at the time they did a good job, and nobody could do better. And then DC and Brundle struck up the most
amazing partnership and nobody ever really complained.
11 Feb 2011
23 Nov 2004
Buemi added as RBR test driver...
18 Oct 2002
James Allen is a very good journalist and has a lot of inside knowledge of F1 - his blog is a very good read.
His problem is that when he tries to inject some excitement into his F1 commentary it comes accross as incredibly fake. "Lewis Hamilton......................... WINS!" for example.
I hope the last few years he's had off he's realised this and will change his style. | {"url":"https://forums.overclockers.co.uk/threads/motorsport-off-topic-thread.18358463/","timestamp":"2024-11-05T07:43:05Z","content_type":"text/html","content_length":"183548","record_id":"<urn:uuid:6b57ab58-d227-40d7-bdf2-8e5ac9dc396a>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00548.warc.gz"} |
Return True if G is aperiodic.
A directed graph is aperiodic if there is no integer k > 1 that divides the length of every cycle in the graph.
Parameters: G (NetworkX DiGraph) – Graph
Returns: aperiodic – True if the graph is aperiodic False otherwise
Return type: boolean
Raises: NetworkXError – If G is not directed
This uses the method outlined in [1], which runs in O(m) time given m edges in G. Note that a graph is not aperiodic if it is acyclic as every integer trivial divides length 0 cycles.
[1] Jarvis, J. P.; Shier, D. R. (1996), Graph-theoretic analysis of finite Markov chains, in Shier, D. R.; Wallenius, K. T., Applied Mathematical Modeling: A Multidisciplinary Approach, CRC | {"url":"https://networkx.org/documentation/networkx-1.10/reference/generated/networkx.algorithms.dag.is_aperiodic.html","timestamp":"2024-11-06T20:01:05Z","content_type":"text/html","content_length":"16298","record_id":"<urn:uuid:552a7d7a-2c87-41cb-9823-3374b88b0b88>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00464.warc.gz"} |
Founding Director
Olga Radko was the Founding Director of the LAMC (now called the UCLA Olga Radko Endowed Math Circle in her honor). Olga received her Ph.D. in mathematics from UC Berkeley in 2002 specializing in
Poisson Geometry, and worked in the UCLA department of Mathematics ever since. Olga started LAMC in 2007 and led it until her death in 2020. Please see the In Memoriam page for more information about
Math Circle Management Team
• Oleg Gleizer, Director
• Tom Gannon, Deputy Director
• Sachi Dieker, Associate Director
Math Circle Steering Committee
Inquiries to the steering committee should be directed to ormc.sc.chair@math.ucla.edu
Swee Hong Chan received his Phd in Mathematics from Cornell University in 2019, and was a member of the UCLA Mathematics Department in 2019--2022 as a Hedrick Postdoc. He is currently
Swee Hong an Assistant Professor in Rutgers University, and his primary research area is combinatorics and probability. He has been working with Olga Radko Math Circle since 2020, where he often
Chan learns new ways to solve math problems from the students of the circle.
Sierra Chen obtained her M.A. in Mathematics from UCLA in 1993. She is an entrepreneur in the field of international trade, a leisure artist, an enthusiastic world traveler, and a
Sierra Chen hopeless optimist. For over two decades, promoting educational excellence has been one of her passions. She taught math classes at community colleges as a part-timer from 1997-2004.
While developing her businesses across the continents, she enjoys mentoring students who are eager to learn and thrive.
Oleg Gleizer is the Director of ORMC. He received his Ph.D. in Mathematics in 2001 from Northeastern University, specializing in representation theory and special functions. Oleg sees
Oleg Gleizer his mission in inventing ways to present some important parts of modern day Mathematics (and occasionally Physics and Economics), typically reserved for college, to the children from
age four and up.
Dan Hoff received his PhD in Mathematics from UC San Diego in 2016. He began working with the math circle in 2017 and served as Curriculum and Instructional Supervisor from 2018–2021
Dan Hoff while researching and teaching as a postdoc in the UCLA Department of Mathematics. His primary mathematical interest is in functional analysis, with a focus on operator algebras and
their connections to fields such as ergodic theory and measured group theory.
Chandrashekhar Khare was born in Mumbai, and studied at Cambridge, Oxford and Caltech, where he obtained his Ph.D. in 1995. He worked at the Tata Institute of Fundamental Research and
Chandrasekhar the University of Utah and is now a professor at the University of California, Los Angeles. Prof. Khare’s research is in number theory, especially on the relation between modular forms
Khare and Galois representations that underpins Wiles’ proof of Fermat’s Last Theorem. In 2008, he and Jean-Pierre Winterberger made a remarkable breakthrough with their proof of a celebrated
conjecture of J.P. Serre. Prof. Khare’s honors and awards include the Fermat Prize (2007), Infosys Prize (2010) and the Cole Prize (2011), and he was elected as a Fellow of the Royal
Society in 2012.
Doug Lichtman is a Professor of Law at UCLA and has been active in Math Circle for over a decade, first as a father and now as a volunteer instructor. In addition to his work with Math
Doug Lichtman Circle, Doug has coached middle school debate, taught advanced fifth grade math, and is now working with his oldest son to run and expand a weekly Math-Circle-inspired computer coding
community for kids, hosted at www.ComeCodeWithUs.com.
Vivian Moy-Dinson has taught K-5 for the past 17 years and is a Learning Specialist in the Covina Valley Unified School District. She received her Masters with honors in Education from
Vivian Azusa Pacific University and her Undergrad in Liberal Studies from Cal Poly Pomona University. Vivian joined the LAMC in 2017 and found it so rewarding to see the very young get excited
Moy-Dinson about math in her Breaking Numbers Into Parts (BNP) course. She is a firm believer that all children can learn when given the opportunity and strives each day to challenge and enrich
their minds.
Dima Shlyakhtenko (chair) received his PhD in mathematics from UC Berkeley and has been a member of the UCLA mathematics department since 1998. His primary research area is in
Dima functional analysis. He was an invited speaker at the 2010 ICM and received multiple awards, including an NSF graduate and postdoctoral fellowship, a Sloan Fellowship and a Clay Math
Shlyakhtenko Institute Special Prize. He currently serves as the director of UCLA's Institute for Pure and Applied Mathematics.
Terence Tao was born in Adelaide, Australia in 1975. He has been a professor of mathematics at UCLA since 1999. Tao's areas of research include harmonic analysis, PDE, combinatorics,
Terence Tao and number theory. He has received a number of awards, including the Fields Medal in 2006, the MacArthur Fellowship in 2007, the Waterman Award in 2008, and the Breakthrough Prize in
Mathematics in 2015. Terence Tao also currently holds the James and Carol Collins chair in mathematics at UCLA, and is a Fellow of the Royal Society, the Australian Academy of Sciences
(Corresponding Member), the National Academy of Sciences (Foreign member), and the American Academy of Arts and Sciences.
Shang-Hua Teng is a University Professor of Computer Science and Mathematics at USC with research interests in scientific computing, optimization, game theory, network sciences, and
Shang-Hua recreational mathematics. He received the 2009 AMS Fulkerson Prize in discrete mathematics, and twice won the ACM Gödel Prize. Citing him as “one of the most original theoretical
Teng computer scientists in the world”, the Simons Foundation named him a 2014 Investigator to pursue long-term curiosity-driven fundamental research. For his industry work with Xerox, NASA,
Intel, IBM, Akamai, and Microsoft, he received fifteen patents in areas including compiler optimization, Internet technology, and social networks. | {"url":"https://circles.math.ucla.edu/circles/leadership.shtml?group=Advanced+2","timestamp":"2024-11-14T16:57:51Z","content_type":"text/html","content_length":"16562","record_id":"<urn:uuid:93da4f22-2260-4e07-9b9f-f14b19cf92af>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00392.warc.gz"} |
What are the mathematical analysis tools used for this type of image processing?
Mathematical analysis used here are ENVI/IDL; ERDAS; ArcGIS; QGIS; SNAP; SeaDAS; Panoply (visualisation only); Shell - GDAL; Python - GDAL, Xarray-Dask, rasterio, SPy; R - terra; MatLab
0 comments
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Quantitative Aptitude - Toppers Portal
Quantitative Aptitude
Read these free Quantitative Aptitude study notes for free. These notes will help you to get good marks in exams like SSC, Banking, Railways, and other Competitive Exams.
Free Math Quiz The aspirants who are preparing for exams like SSC CGL, SSC CHSL, Banking, and other …
What Is a Polynomial? A polynomial is an expression consisting of variables…types of polynomials degree…how to solve polynomials, factoring polynomials.
Time and Distance Problems on Time and Distance with explanation for various interview, competitive examinations and entrance tests like SSC, Banking.
Percentage Multiple Choice Questions and Solutions for Bank SSC. percentage increase and decrease as well as problems of percent of quantities
Profit and Loss Problems for SSC Banking MCQs..practice online the problems. Profit and loss is an important chapter for competitive examinations.
Mixture and Alligation Practice Questions. The Mixture and Alligation Practice Questions section have questions on Alligation. All of these topics are very important and many questions have been
already asked from the section on Mixture and Alligation.
Here we provide some of the basic Objective Type Problems on Trains with explanation for various interview, competitive examinations and entrance tests like SSC, Banking, etc.
Trains Multiple Choice Questions
Check out these Important Basic Triangle Formulas and Identities. Important Triangle Formulas and Identities for exams like SSC, Banking, State PSC Exams. | {"url":"https://toppersportal.com/category/quantitative-aptitude/","timestamp":"2024-11-03T04:24:00Z","content_type":"text/html","content_length":"101018","record_id":"<urn:uuid:851e6a34-5b87-4d60-b5b5-b4a25f7e0116>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00639.warc.gz"} |
Tangential Acceleration Formula: Overview, Formula, Direction
Tangential Acceleration Formula: In a circular motion, a particle may speed up or slow down or move with constant speed. When the particle is in a circular motion, it will always have an acceleration
toward the centre called centripetal acceleration (even if moving with constant speed). But in the case of changing speed, tangential acceleration in the direction or opposite of the direction of the
velocity will act.
For example, a car is accelerating around a curve path on the road, then it experiences both tangential and centripetal acceleration. Do you know what tangential acceleration is in a circular motion
and how it works? In this article, we will find the expression of the tangential acceleration, learn formula for acceleration, and more. We will also find the net acceleration in a non-uniform
circular motion with the help of centripetal and tangential acceleration. Scroll down to find more!
What is the Formula of Acceleration? An Overview
In a uniform circular motion, the net force acting on the object is in the direction perpendicular to the motion of the object. Hence, this causes a change in the direction continually, but the
magnitude of velocity remains constant. Therefore, the object is said to be accelerating in a direction that points towards the centre of the circular pathway. But what happens when the net force
acting on the object is not perpendicular? In this case, there will be two-component force vectors that will point along the perpendicular and parallel to the velocity vector.
The perpendicular force component will cause the object to move along a circular pathway as it creates a centripetal acceleration, and the parallel force component will cause the object to accelerate
along the tangent as it creates tangential acceleration. Hence, the object will undergo non-uniform circular motion as both the direction and magnitude of the velocity of the object will change.
Formula for Tangential Acceleration
Tangential acceleration is the measure of how quickly the speed of a body changes when an object moves in a circular motion. Let us consider a particle \((P)\) that is moving in a circle of radius \
((r)\) and centre \(O,\) as shown in the figure below. The position of the particle \(P\) at a given instant may be described by the angle \(\theta\) between \(OP\) and \(OX.\) This angle \(\theta\)
is called the angular position of the particle with respect to \(OX,\) and it changes as the particle moves on the circle. Let us assume the point rotates an angle \(∆θ\) in the time interval of \
(∆t.\) The rate of change of angular position is known as the angular velocity \(\left( \omega \right).\)
Thus, we can write,
\(\omega = \mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta \theta }}{{\Delta t}} = \frac{{{\text{d}}\theta }}{{{\text{d}}t}}\)
Here \(\omega \) is the angular speed or magnitude of angular velocity. Angular velocity is a vector quantity. The direction of \(\omega \) is perpendicular to the plane of the circle, and it can be
given by the screw law or right-hand thumb rule.
We can also write linear velocity as:
\(\vec v = \frac{{d\vec s}}{{dt}}\) or \(\frac{{d\vec r}}{{dt}}\)
And, the magnitude of linear velocity is called linear speed \((v).\) Thus, we can write
\(v = |\vec v| = \mid \frac{{d\vec s}}{{dt}}\;{\rm{or}}\;\frac{{d \vec r}}{{dt}}\mid \)
Relation Between Linear Speed and Angular Speed
From the above figure, linear distance \(PP’\) travelled by the particle in time \(∆t\) is
\(\Delta s = r\Delta \theta \)
\( \Rightarrow \frac{{\Delta s}}{{\Delta t}} = r\frac{{\Delta \theta }}{{\Delta t}}\)
\( \Rightarrow \frac{{ds}}{{dt}} = r\frac{{d\theta }}{{dt}}\)
\( \Rightarrow v = r\omega ……..\left( 1 \right)\)
Like the velocity, a particle in circular motion has two accelerations that are angular and linear acceleration. Where angular acceleration \(\left( \alpha \right)\) is the rate of change of angular
velocity. Thus, we can write that
\(\alpha = \frac{{d\omega }}{{dt}} = \frac{{{d^2}\theta }}{{d{t^2}}}\)
Angular acceleration \(\left( \alpha \right)\) is also a vector quantity. The direction of \(\left( \alpha \right)\) is also perpendicular to the plane of the circle, either parallel or antiparallel
to \(\omega .\) If the angular speed of the particle is increasing, then \(\left( \alpha \right)\) is parallel \(\omega \) to and if angular speed is decreasing, then \(\left( \alpha \right)\) is
antiparallel to \(\omega .\) If angular speed (or angular velocity) is constant then, angular acceleration will be zero.
Relation between the tangential component of the linear acceleration and angular acceleration can be obtained on differentiating equation \((1)\) with respect to time. The rate of change of speed is
\({a_t} = \frac{{d|\vec v|}}{{dt}}\)
\({a_t} = \frac{{d(r\omega )}}{{dt}} = r\frac{{d\omega }}{{dt}}\)
\({a_t} = r\alpha ……..\left( 2 \right)\)
\(r=\)The radius of the circle,
\(\alpha = \)Angular acceleration.
This component of acceleration in the tangential direction is called tangential acceleration \(\left( {{a_t}} \right).\) This component is responsible for the change in the linear speed. If the speed
of the particle is constant, then \({a_t}\) is zero. If speed is increasing, then this is positive and in the direction of linear velocity. This component will be negative and in the opposite
direction of linear velocity if the speed is decreasing.
Centripetal acceleration: The component of acceleration in the radial direction (towards the centre) is called radial or centripetal acceleration. This component changes the direction of the linear
velocity. As the direction continuously keeps on changing. So, this component can never be zero, and the value of this component is given as:
\({a_r} = \frac{{{v^2}}}{r} = r{\omega ^2}\)
Net acceleration: Tangential acceleration is in the direction of the tangent to the circle, whereas centripetal acceleration is in the radial direction of the circle pointing inwards to the centre.
These two components are mutually perpendicular, as shown in the figure below. Thus, a particle in a circular motion having centripetal acceleration as well as tangential acceleration has a net
acceleration equal to their vector sum, which is given as:
\(\overrightarrow {{a_n}} = \overrightarrow {{a_c}} + \overrightarrow {{a_t}} \)
\( \Rightarrow a = \sqrt {a_t^2 + a_r^2} \)
\( \Rightarrow a = \sqrt {{{\left( {\frac{{dv}}{{dt}}} \right)}^2} + {{\left( {\frac{{{v^2}}}{r}} \right)}^2}} \)
\( \Rightarrow a = \sqrt {{{(r\alpha )}^2} + {{\left( {r{\omega ^2}} \right)}^2}} \)
\( \tan \theta = \frac{{{a_r}}}{{{a_t}}}\)
\( \Rightarrow \theta = \tan^{-1} \left( {\frac{{{a_r}}}{{{a_t}}}} \right)\)
Solved Examples on Tangential Acceleration Formula
Q.1. Find the angular acceleration of the particle if a particle travels in a circular path of radius \({\rm{20}}\,{\rm{cm}}\) at a speed that uniformly goes increases? If the speed changes from \({\
rm{5}}{\rm{.0}}\,{\rm{m}}\,{{\rm{s}}^{ – 1}}\) to \(6.0\;{\rm{m}}\,{{\rm{s}}^{ – 1}}\) in \({\rm{2}}{\rm{.0}}\,{\rm{s}}{\rm{.}}\)
Ans: The magnitude of the tangential acceleration is given by
\({a_t} = \frac{\alpha }{{dt}} = \frac{{{v_2} – {v_1}}}{{{t_2} – {t_1}}}\)
\( \Rightarrow {a_t} = \frac{{6 – 5}}{2} = 0.5\;{\rm{m}}\,{{\rm{s}}^{ – 2}}\)
The angular acceleration formula is given by,
\(\alpha = \frac{{{a_t}}}{r}\)
\( \Rightarrow \alpha {\rm{ = }}\frac{{{\rm{0}}{\rm{.5}}\,{\rm{m}}\,{{\rm{s}}^{{\rm{ – 2}}}}}}{{{20\,\rm{cm}}}}\)
\( \Rightarrow \alpha = 2.5\,{\rm{rad}}\,{{\rm{s}}^{ – 2}}.\)
Q.2. A particle moves in a circular path of radius \({\rm{2}}\,{\rm{m}}{\rm{.}}\) It is moving with its linear speed that is given by \(v = {t^2},\) where \(t\) in second and \(v\) in \({\rm{m}}\,{{\
rm{s}}^{{\rm{ – 1}}}}.\) What is the radial, tangential and net acceleration of a particle at \(t{\rm{ = 2}}\,{\rm{s}}\)?
Ans: The linear speed of the particle at \(t{\rm{ = 2}}\,{\rm{s}}\) is,
\(v = {t^2} = {2^2}{\rm{m}}\,{{\rm{s}}^{{\rm{ – 1}}}}\)
\( \Rightarrow v = 4\,{\rm{m}}\,{{\rm{s}}^{{\rm{ – 1}}}}\)
Then, the radial acceleration at \(t = 2\,{\rm{s}}\) is,
\({a_r} = \frac{{{\nu ^2}}}{r} = \frac{{{4^2}}}{2}{\rm{m}}\,{{\rm{s}}^{{\rm{ – 2}}}}\)
\( \Rightarrow {a_r} = 8\,{\rm{m}}\,{{\rm{s}}^{{\rm{ – 2}}}}\)
The tangential acceleration at \(t = 2\,{\rm{s}}\) is,
\({a_t} = \frac{{dv}}{{dt}} = \frac{{d\left( {{t^2}} \right)}}{{dt}} = 2t\)
\( \Rightarrow {a_t} = 2 \times 2\;{\rm{m}}\,{{\rm{s}}^{ – 2}}\)
\( \Rightarrow {a_t} = 4\,{\rm{m}}\,{{\rm{s}}^{{\rm{ – 2}}}}\)
Now, the net acceleration of the particle at \(t = 2\,{\rm{s}}\) is
\(a = \sqrt {a_t^2 + a_r^2} \)
\( \Rightarrow a = \sqrt {{4^2} + {8^2}} \)
\( \Rightarrow a = 8.944\;{\rm{m}}\,{{\rm{s}}^{ – 2}}.\)
In a uniform circular motion, the particle executing circular motion has a constant speed, and the circle is at a fixed radius, but the speed of the particle is changing (not constant) then, there
will be an additional acceleration that is tangential acceleration which acts in the direction tangential to the circle. Tangential acceleration is defined as the rate of change of magnitude of
tangential velocity of the particle in a circular motion, and its direction is in the direction of the tangent to the circular path.
The tangential acceleration formula is given as, \({a_t} = r\alpha .\) The tangential acceleration vector and centripetal acceleration vector are mutually perpendicular. The vector sum of tangential
and centripetal accelerations gives the net acceleration of the particle. i.e \(\left( {\vec a = \overrightarrow {{a_r}} + \overrightarrow {{a_t}} } \right).\)
FAQs on Tangential Acceleration Formula
Students can check the below frequently asked questions on tangential acceleration formula:
Q1. What do you mean by tangential acceleration?
Ans: The rate of change speed of the particle in the circular path is known as tangential acceleration. It is equal to the product of angular acceleration \({\left( \alpha \right)}\) and the radius \
((r)\) of the circular path. i.e \(\left( {{a_t} = r\alpha } \right).\)
Q2. What is the formula of centripetal acceleration and tangential acceleration?
Ans: (i) We can find tangential acceleration with the help of the tangential acceleration formula, which is given as:
\({a_t} = \frac{{dv}}{{dt}}\) or \({a_t} = r\alpha \)
(ii) And, we can find centripetal acceleration with the help of the centripetal acceleration formula, which is given as:
\({a_r} = \frac{{{v^2}}}{r} = r{\omega ^2}.\)
Q3. In which direction the tangential acceleration works?
Ans: A tangential acceleration works in the direction of a tangent at the point of circular motion. Its direction is always in the perpendicular direction to the centripetal acceleration of a
rotating object.
Q4. What force causes tangential acceleration?
Ans: The tangential force component will create tangential acceleration, which will cause the object to accelerate along the tangent. Then, the object will undergo non-uniform circular motion as both
the direction and magnitude of the velocity of the object change.
Q5. Give an example of both centripetal and tangential acceleration.
Ans: Suppose you are holding a thread to the end of which is tied to a stone. Now when you start whirling it around, you will notice that two forces have to be applied simultaneously. One which pulls
the thread inwards and the other which throws it sideways or tangentially. Both these forces will generate their respective accelerations. The one-pointed inwards will generate centripetal or radial
acceleration, and the one pointing sideways will generate tangential acceleration.
Q6. What is centripetal acceleration?
Ans: Centripetal acceleration can be defined as the component of acceleration in the radial direction (towards the centre).
Q7. What is the difference between centripetal and tangential acceleration?
Ans: Tangential acceleration is in the direction of the tangent to the circle, whereas centripetal acceleration is in the radial direction of the circle pointing inwards to the centre.
We hope you find this article on the Tangential Acceleration Formula helpful. In case of any queries, you can reach back to us in the comments section, and we will try to solve them. | {"url":"https://www.embibe.com/exams/tangential-acceleration-formula/","timestamp":"2024-11-12T19:57:50Z","content_type":"text/html","content_length":"531820","record_id":"<urn:uuid:b67c6b82-5165-4255-9df8-decb45146fd8>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00002.warc.gz"} |
Compound Interest
Compound interest is the interest that is calculated on the initial principal and also on the accumulated interest of previous periods. This means that the more time you leave your money in a
compound interest account, the more it will grow.
For example, let’s say you deposit $100 into a savings account with a 5% annual interest rate and no monthly fees. After one year, you would earn $5 in interest, bringing your total balance to $105.
In the second year, you would earn interest on the new balance of $105, rather than just the initial $100. This means you would earn an additional $5.25 in interest, bringing your total balance to
$110.25. As you can see, the amount of interest you earn each year increases as your balance grows.
How to Calculate Compound Interest
There are a few different ways to calculate compound interest, but the most common method is to use the following formula:
A = P(1 + r/n)^(nt)
• A is the final amount (principal + interest)
• P is the principal amount (the initial amount you deposit)
• r is the annual interest rate (expressed as a decimal)
• n is the number of times interest is compounded per year
• t is the number of years the money is invested
For example, let’s say you deposit $1,000 into a savings account with a 4% annual interest rate that compounds monthly. To find out how much you would have in the account after 3 years, you would
plug the values into the formula like this:
A = $1,000(1 + .04/12)^(12*3)
This would give you a final balance of $1,124.36.
The Benefits of Compound Interest
One of the main benefits of compound interest is that it can help your money grow faster than simple interest, where the interest is only calculated on the initial principal. This is because compound
interest builds upon itself, so the longer you leave your money in an account, the more it will grow.
For example, let’s say you have a choice between a simple interest account with a 5% annual interest rate and a compound interest account with a 5% annual interest rate. If you left $1,000 in each
account for 10 years, the simple interest account would grow to $1,500, while the compound interest account would grow to $1,629.89. As you can see, the compound interest account grew more because
the interest was compounded over time.
Another benefit of compound interest is that it can help you reach your financial goals faster. For example, if you are saving for retirement, compound interest can help your savings grow more
quickly so that you can reach your goal more quickly.
The Risks of Compound Interest
While compound interest can be a powerful tool for helping your money grow, it also has some risks. One risk is that if you are borrowing money at a high interest rate, the compound interest can
quickly add up and become unaffordable. For example, if you have a credit card with a high interest rate and only make the minimum payment each month, the compound interest can quickly add up and
make it difficult to pay off the balance.
Another risk of compound interest is that it can be affected by inflation. If the rate of inflation is higher than the interest rate on your account, the value of your money may not keep up with the
rising costs of goods and services. This means that even though your | {"url":"https://hostcoffer.com/2022/03/23/compound-interest/","timestamp":"2024-11-05T20:12:52Z","content_type":"text/html","content_length":"61133","record_id":"<urn:uuid:8d137d98-cc61-45e3-819b-4fda79909d1b>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00667.warc.gz"} |
The Combined Gas Law
The Combined Gas Law
The Combined Gas Law is the law that combines three more gas laws, Charles’s law, Gay-Lussac’s law, and Boyle’s law. Three previously discovered laws have been combined to create this law. This law
relates one thermodynamic variable to another while holding all other variables constant. The interdependence of these variables is illustrated by the combined gas law, according to which the ratio
between pressure volume and temperature is constant.
Moreover, the ratio of volume and pressure and the gas’s absolute temperature should be constant in this law. It should be noted that when Avogadro’s law happens to combined gas law, then gas law
shows results. Combined gas has been discovered like that only. Hence, It is simply a combination of other gas laws.
The Combined Gas Law
Understanding Combined Gas Law
In order to understand the combined gas law, first, it is important to understand are previous other three discovered laws. so, let’s look into this-
Boyle’s law
Let’s take an example, If a diver were to begin diving with full lungs, his lungs would be full of air. The pressure in his lungs also increases as he dives deeper underwater.
The lungs become squished when the pressure increases. Thus, the volume is reduced. Boyle’s law says that higher pressure equates to a lower volume, so in this case, pressure means lower volume. This
is called Boyle’s Law. And the formula for Boyle’s law is PV=K.
Hence, Boyle’s Law states that the Higher the pressure (P), the lower the volume (V).
Charles’s law
Let’s understand with the help of an example, that currently, there is a balloon in the refrigerator. The gas volume inside the balloon decreases as the temperature of the balloon in the refrigerator
Additionally, the balloon will revert to its original size once it is out. So, when the temperature increases, the volume increases as well. This is a manifestation of Charles’ Law. And the formula
for Charles’s law is V/T=K.
Hence, Charles’s law states that the higher the temperature (T), the higher the volume (V).
Gay-Lussac’s law
Assume that a driver is driving a car and gradually, the temperature inside the tire started increasing. So, as a result of which the air inside the tire will expand the tire, and consequently
pressure inside the tire will increase. This is the Law of Gay-Lussac. And the formula for Gay-Lussac’s law is P/T=K.
Hence, Gay-Lussac’s law shows the relationship between temperature (T) and Pressure (P) keeping volume (V) constant (K). which shows that as the temperature (T) increases, the consequent pressure (P)
also increases.
After combing Boyle’s law, Charles’s law, and Gay-Lussac’s law together, we form the combined Gas Law, which further shows that:
1. Pressure is inversely proportional to volume, or higher volume equals lower pressure.
2. Pressure is directly proportional to temperature, or higher temperature equals higher pressure.
3. Volume is directly proportional to temperature, or higher temperature equals higher volume.
Derivation of Combined Gas Law
As we have already discussed the Combined Gas Law is the amalgamation of the above three laws. The derivation of the combined gas law is like this:
Boyle’s law- PV=K, Charles’s law- V/T=K, and Gay-Lussac’s law- P/T=K.
So, the formula for combined gas law is PV/T = K. where P is pressure, V is volume, T is temperature, and K is constant. It is important to keep in mind that temperature must always be calculated in
Kelvin. And, if units are available in celsius then first it must be converted into Kelvin by adding 273 to the particular unit.
Likewise, when two substances are compared or calculated in two different conditions, then the formula can be –
P1V1/T1 = P2V2/T2
Further, we will discuss how the application of these formulae is done in the solved examples.
Solved Examples
Example 1
The initial volume of the gas is 5L and the final volume is 3L Calculate the final pressure of the gas, given that the initial temperature is 273 K, the final temperature is 200 K, and the initial
pressure is 25 kPa.
According to the given parameters,
P1= 25 kPa, V1 = 5L, V2 = 3L, T1 = 273K, T2[ ]= 200K
According to combined gas law,
P1V1/T1 = P2V2/T2
Substituting in the formula, we get
25 x 5 / 273 = P2[ ]x 3 / 200
Therefore, P2 = 30.525 kPa
Example 2
Determine the volume of a gas given V1 = 3L, T1 = 300K, T2 = 250K, P1 = 35 kPa and P2= 50 kPa
Given Parameters are
P1 = 35 kPa, V1 = 3L, T1 = 300K, P2= 50 kPa, T2 = 250K
According to given parameters, we have an equation
P1V1/T1 = P2V2/T2
Substituting in the above equation, we get
35 x 3 / 300 = 50 x V2 / 250
Therefore, V2 = 1.75 L | {"url":"https://wikiedu.co/the-combined-gas-law/","timestamp":"2024-11-05T03:18:30Z","content_type":"text/html","content_length":"42865","record_id":"<urn:uuid:a34717a3-ad74-46d0-9209-a24339b7fa64>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00430.warc.gz"} |
the Roger Young (starship troopers)
i just did a search of the entire contents of these forums. there is not one reference to the Roger Young (this is hard to belive) in another threat there were many questions and answers concerning
HUGH capital Battle Wagons and troop ships.....has anyone tried to come up with classic Traveller STATS for the Roger Young???, any deck plans?? any illistrations?? sketches, drawings any thing worth
using for this project??
ive ckecked out the movie sites and most other useful sites, want to know if any else is working on project or started one - it is a hugh task!!
i dont have any idea where to start. what tech level (its not the highest) some one out ther MUST have thought to do this, how far did you get??? and where did you get your starting info??
can this even be done?? ect, ect, ect, help!! more questions!!!
If you want deck plans, try the rule book for the AH game Starship troopers. It has some deck plans for the Rodger Young in the back as I recall. It also gave some idea of numbers of troops carried,
equipment, and shape.
The game itself gave a good indication of just how powerful BD troops could get.
"If you want deck plans, try the rule book for the AH game Starship troopers."
Just to clarify, that's the 1970's version (my very first wargame
), not the movie tie-in version from 20 years later. The montage on the inside back cover (credited to Richard Hamblen) contains
these deck plans
among other illustrations based on the novel.
Trader Jim,
Here ya go. I've actually tried using the HG Shipyard 1.058 to build several TV/motion picture
ships. Haven't gotten to deckplans yet...nut here is a stab. If you have any suggestions let me know.
Ship: Roger Young
Class: Roger Young - Refit
Type: Strike Carrier
Architect: Admiral Savage
Tech Level: 14
SC-Q1558H4-292000-95559-2 MCr 72,361.842 75 KTons
Bat Bear 1 2Y866 Crew: 3179
Bat 1 2ZA88 TL: 14
Cargo: 4,015.000 Passengers: 55 Fuel: 28,500.000 EP: 6,000.000 Agility: 4 Marines: 2500 Drop Capsules: 10 (plus 200 Ready 50 Stored)
Craft: 18 x 100T Drop Ships, 10 x 20T Fighters, 2 x 10T Grav Vehicle
Fuel Treatment: Fuel Scoops
Backups: 1 x Model/8fib Computer
Substitutions: Y = 38 Z = 50
Architects Fee: MCr 723.618 Cost in Quantity: MCr 57,889.474
Detailed Description
75,000.000 tons standard, 1,050,000.000 cubic meters, Needle/Wedge Configeration
61 Officers, 608 Ratings, 10 Pilots, 2500 Marines
Jump-5, 5G Manuever, Power plant-8, 6,000.000 EP, Agility 4
Bridge, Model/8fib Computer
1 Model/8fib Backup Computer
16 100-ton bays, 10 50-ton bays, 110 Hardpoints
8 100-ton Meson Bays (Factor-5), 8 100-ton Missile Bays (Factor-9), 10 50-ton Particle Accelerator Bays (Factor-5), 40 Triple Beam Laser Turrets in 2 Batteries (Factor-9), 50 Dual Fusion Gun Turrets
in 50 Batteries (Factor-5)
20 Triple Sandcaster Turrets in 1 Battery (Factor-9), Meson Screen (Factor-2), Armoured Hull (Factor-2)
18 100.000 ton Drop Shipss (Crew of 3), 10 20.000 ton Fighterss (Crew of 2), 2 10.000 ton Grav Vehicles (Crew of 3)
23,250.000 Tons Fuel (3 parsecs jump and 30 days endurance)
On Board Fuel Scoops, No Fuel Purification Plant, 51,000.000 ton drop tanks
1,750.0 Staterooms, 10 Drop Capsule Launchers with 200 Ready Capsules and 50 Stored Capsules, 4,015.000 Tons Cargo
MCr 73,085.460 Singly (incl. Architects fees of MCr 723.618), MCr 57,889.474 in Quantity
195 Weeks Singly, 156 Weeks in Quantity
Trader Jim,
Also, I did them at various tech levels based on the movie not books. Here is the Athena.
Ship: Athena
Class: Athena
Type: Strike Carrier
Architect: Admiral Savage
Tech Level: 15
SC-R164AJ4-494500-95599-1 MCr 128,113.218 150 KTons
Bat Bear 1 1Y78B Crew: 5107
Bat 1 2ZACG TL: 15
Cargo: 13,233.000 Passengers: 190 Fuel: 60,000.000 EP: 15,000.000 Agility: 4 Marines: 4000 Drop Capsules: 10 (plus 200 Ready 50 Stored)
Craft: 20 x 100T Drop Ships, 10 x 20T Fighters, 4 x 95T Shuttle, 8 x 10T Grav Vehicle
Fuel Treatment: Fuel Scoops and On Board Fuel Purification
Backups: 1 x Model/9fib Computer 1 x Bridge
Substitutions: Y = 35 Z = 50
Architects Fee: MCr 1,281.132 Cost in Quantity: MCr 102,490.574
Detailed Description
150,000.000 tons standard, 2,100,000.000 cubic meters, Needle/Wedge Configeration
95 Officers, 1002 Ratings, 10 Pilots, 4000 Marines
Jump-6, 4G Manuever, Power plant-10, 15,000.000 EP, Agility 4
Bridge, Model/9fib Computer
1 Backup Bridge, 1 Model/9fib Backup Computer
38 100-ton bays, 10 50-ton bays, 170 Hardpoints
12 100-ton Meson Bays (Factor-9), 16 100-ton Missile Bays (Factor-9), 10 50-ton Particle Accelerator Bays (Factor-5), 100 Triple Beam Laser Turrets in 2 Batteries (Factor-9), 50 Dual Fusion Gun
Turrets in 50 Batteries (Factor-5)
20 Triple Sandcaster Turrets in 1 Battery (Factor-9), Nuclear Damper (Factor-5), Meson Screen (Factor-4), Armoured Hull (Factor-4)
20 100.000 ton Drop Shipss (Crew of 4), 10 20.000 ton Fighterss (Crew of 2), 4 95.000 ton Shuttles (Crew of 3), 8 10.000 ton Grav Vehicles (Crew of 3)
46,500.000 Tons Fuel (3 parsecs jump and 30 days endurance)
On Board Fuel Scoops, On Board Fuel Purification Plant, 105,000.000 ton drop tanks
2,900.0 Staterooms, 10 Drop Capsule Launchers with 200 Ready Capsules and 50 Stored Capsules, 13,233.000 Tons Cargo
MCr 129,394.350 Singly (incl. Architects fees of MCr 1,281.132), MCr 102,490.574 in Quantity
207 Weeks Singly, 166 Weeks in Quantity
I assume you're using the movie version as a basis. The book's TFCT Roger Young is much smaller; she carries one large platoon of Mobile Infantry with drop capsules and one or perhaps two recovery
I came up with something more like this:
Ship: Roger Young
Class: ???
Type: Corvette Transport
Architect: Tom Schoene
Tech Level: 15
CT-C4555E2-070000-60000-0 MCr 1,725.548 3 KTons
Bat Bear 3 3 Crew: 99
Bat 3 3 TL: 15
Cargo: 64.000 Fuel: 1,650.000 EP: 150.000 Agility: 3 Marines: 60 Drop Capsules: 3 (plus 60 Ready 120 Stored)
Craft: 2 x 40T Recovery Boat
Fuel Treatment: Fuel Scoops and On Board Fuel Purification
Backups: 1 x Model/1fib Computer
Architects Fee: MCr 17.255 Cost in Quantity: MCr 1,380.438
Detailed Description
3,000.000 tons standard, 42,000.000 cubic meters, Close Structure Configuration
11 Officers, 28 Ratings, 60 Marines
Jump-5, 5G Manuever, Power plant-5, 150.000 EP, Agility 3
Bridge, Model/5fib Computer
1 Model/1fib Backup Computer
30 Hardpoints
12 Triple Beam Laser Turrets organised into 3 Batteries (Factor-6)
18 Triple Sandcaster Turrets organised into 3 Batteries (Factor-7)
2 40.000 ton Recovery Boats (Crew of 2, Cost of MCr 0.000)
1,650.000 Tons Fuel (5 parsecs jump and 28 days endurance)
On Board Fuel Scoops, On Board Fuel Purification Plant
53.0 Staterooms, 3 Drop Capsule Launchers with 60 Ready Capsules and 120 Stored Capsules, 64.000 Tons Cargo
MCr 1,742.803 Singly (incl. Architects fees of MCr 17.255), MCr 1,380.438 in Quantity
139 Weeks Singly, 111 Weeks in Quantity
Based on TFCT Roger Young, from Robert Heinlein's novel Starship Troopers.
The Marine (i.e. Mobile Infantry) complement of 60 troops is an estimate, I think
an MI platoon may actually be a bit larger than this. The ship has sufficient
drop capsules for three combat drops before resupply, and can frontier refuel
indefinitely. The recovery boats are placeholders; 40 tons seems about right, or maybe a bit low.
Yes, both were from the movie. It's been too long
since I read the book. I do recall significant
differences. So the question is what was Trader
Joe looking for... I actually did this in early
summer as part of my effort to identify several starships as Traveller vessels.
I tried the RY at various tech levels (13, 14, and 15) and it seemed to fit best in TL14. Most of the design is based around what we see in drop ships, and fighter capability.
The Athena (at the end of the movie) could possibly be TL15. We really don't see much of it.
Originally posted by savage:
Yes, both were from the movie. It's been too long
since I read the book. I do recall significant
differences. So the question is what was Trader
Joe looking for...
Well, now he has both to chose from, I guess.
I tried the RY at various tech levels (13, 14, and 15) and it seemed to fit best in TL14. Most of the design is based around what we see in drop ships, and fighter capability.
I picked TL 15 mainly becuase power plant volume was becoming unreasonable at lower TLs. I was surprised to see the ship even get this big; I started out assuming 1000 tons.
Now, if you design the ship under GT you can fit the same capabilities in about 600 or 700 tons. In many ways, this is more satisfactory -- you get the sense of a very small shp, reasonably
designated a corvette.
Of course, not much of the ST universe technology really matches Traveller. I'd be inclined to try it under GURPS Space with warp drives and some sort of very efficient reaction drive. (The corvette
transports can brake at 8-10g!)
The movie and book were significantly different.
In the movie they discussed the amount of troops deployed to Klandathu. And it appeared that
larger vessels were required to support significant
ground forces.
I agree on your point of travel methods. It doesn't really seem that they use Jump drives. I haven't used GT although FFS does make an effort to
describe Warp travel. My concern would be game consistency.
I always pictured the Roger Young as being quite a small ship. No more than a 1000dt in size. The Broadsword class at 800dt could be altered to carry a 50 man platoon (which I think is the size of a
platoon in the novel), replace the 2 cutters with a single "retrieval boat" and dump the ATVs. It is not mentioned in the novel if the corvette transports could actually land, but I would make it
streamlined, after all losss of the retrieval boat will make it hard to recover the troops otherwise.
Funnaly enough IMTU Corvette is used as a term for warships of around 400 to 1000dt with a manuever drive rating of 3 or less. For instance an FL (Frigate,light aka corvette) at TL15 IMTU would have
J4 and 3G compared to a DE (Destroyer Escort) of the same displacement which would have J4 and 4G or better.
Originally posted by Antony:
I always pictured the Roger Young as being quite a small ship. No more than a 1000dt in size. The Broadsword class at 800dt could be altered to carry a 50 man platoon (which I think is the size
of a platoon in the novel), replace the 2 cutters with a single "retrieval boat" and dump the ATVs. It is not mentioned in the novel if the corvette transports could actually land, but I would
make it streamlined, after all losss of the retrieval boat will make it hard to recover the troops otherwise.
The CT Kinunir, with it's drop tubes and short platoon of Marines, is also pretty close to the Starship Troopers book ships. It would certainly do as a basis for a conversion, anyway. The Broadsword
doesn't have any drop tubes, though some could be added.
When thinking of Starship Trooper, please forget that horrible movie.
A basic thing to consider is the tech level of the MI troops. There are bouncing aroung in powered armor, battle dress, with grav belts. Their basic weapon is the hand flamer, a large handgun sized
They can shoot mini nukes out of backpack launchers, Anyone know an equivalent? Possible designed in Striker.
All this adds up to a TL minimum of 14, or in the 16 range.
The Kinunir is very close to what the book describes ae the Rodger Young. Drop tubes and all. Wonder what book the author was reading prior to writing the adventure?
Originally posted by vegascat:
A basic thing to consider is the tech level of the MI troops. There are bouncing aroung in powered armor, battle dress, with grav belts. Their basic weapon is the hand flamer, a large handgun
sized PGMP.
They can shoot mini nukes out of backpack launchers, Anyone know an equivalent? Possible designed in Striker.
All this adds up to a TL minimum of 14, or in the 16 range.
That I don't believe at all. The tech assumptions don't match up very neatly, but TL 14 or higher is way too high.
Let's consider where the various systems are in Traveller terms:
Flamers are not fusion guns; they're more like very advanced flamethrowers or maybe a long range oxy-acetylene torch. The small pistol-type ones are mainly incendiary weapons, but with some raher
limited defensive anti-personnel capability. The heavy-duty version can cut through walls, but doesn't blow them down. No good Traveller analog, but clearly inferior to high-energy weapons. Call it
Suits use jump jets, not grav belts. These are reaction engines of some sort and require some sort of fuel. Again, no clear Traveler analog, but probably no higher than TL 10.
Micro-nukes suitable for mortars or Tac missiles are TL 10 or 11. (Striker gives 14cm as the minimum size for a TL10 2kt nuke, which is not unreasonable for a suit-portable missile.)
The Kinunir is very close to what the book describes ae the Rodger Young. Drop tubes and all. Wonder what book the author was reading prior to writing the adventure?
I don't see Kinnunir as being terribly close to Roger Young, actually. It's rather slow, and not at all a dedicated troop transport like the fast corvette tranports.
Some people have argued that Kinnunir is based on the battlecruiser MacArthur from Mote in God's Eye, but I'm not terribly convinced by that one either (MacArthur has lots of small craft, for
example). I think it's basically a novel design, with a few ideas borrowed from several different sources.
more info.....i tore up the internet for several days.....no plans!!!....did find a guy who makes and sells resin models - what a joke!!! $170.00 a pop!!!.....*********!!!!
hope he has to EAT them!!!!
A flamer is a Heinlein weapon that shows up in a couple of places. In Between Planets a backpack powered version was used like a flamethrower, but it was clearly an energy weapon. RAH was writing
before lasers and the flamers were generic energy weapons. I'll buy the thesis that they are some kind of light plasma weapon without the focus and range of a PGMP.
But the flamer (hand or heavy duty) was a secondary weapon for close quarters. NCOs had nuclear missiles, everybody had grenades, but RAH is silent on ther primary weapons in between. Except to say
that an individual MI was more than a match for a TL4-5 tank. I am betting on light (50-70mm?), guided missiles with conventional warheads.
I can see TL10-12. Remember this was a civilization that was climbing back up from a TL6 nuclear war.
Oh, if you are designing the Rodger Young, remember Bulkhead 30. Forward of Bulkhead 30 are quarters for the female officers and the Officers mess (maybe the bridge as well). Everything else is aft
of a single door guarded by an MI.
Troop compliment is twice listed as 53 Possibly 54 with a lieutenant.
an MI Platoon contains at 100% To&E 1 officer and 55 troopers...still working on the ship but it is a LT- by high guard standards...about 1.2kt.
I don't see where you get 55. I get 54. The Roughnecks are repeatedly listed as having 53 under Sergeant Jelhal because Lieutenant Raczak was dead.
TOE, MI platoon (from the book)
Platoon Sergeant
2 Sections, ea
Section Leader (Sgt)
Assistant Section Leader (Cpl)
(3) Squads, ea Corporal, Lance Corporal, 6 Privates
Three launch tubes. Port and starboard tubes each drop a section and the Lieutenant and Platoon Sergeant dropping from the center tube.
Your right my bad ... I just looked at the book again in 10 years ..... can you have memory problems at 31?
Originally posted by Spaceman Spiff:
Your right my bad ... I just looked at the book again in 10 years ..... can you have memory problems at 31?
Well, my generation did but we had better drugs.
I looked it up a month ago when this thread started.
you suck... i have to pee in a cup every few weeks .. but i have access (
read legal)to all kinds of good stuff Morphine, demoral.. etc... | {"url":"https://www.travellerrpg.com/threads/the-roger-young-starship-troopers.3117/","timestamp":"2024-11-08T19:01:21Z","content_type":"text/html","content_length":"159629","record_id":"<urn:uuid:00120227-0289-491b-be01-babfc2a51694>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00744.warc.gz"} |
Geometry Problem 1145: Three Squares, Midpoints
Tuesday, August 4, 2015
Geometry Problem
Level: Mathematics Education, High School, Honors Geometry, College.
Click the figure below to see the complete problem 1145.
Posted by Antonio Gutierrez at 9:19PM
2 comments:
1. Let z(P) be the complex number representing P.
z(C₁)—z(B₁) = i [z(A₁)—z(B₁)]
z(C₂)—z(B₂) = i [z(A₂)—z(B₂)]
summing up, we have
z(C)—z(B) = i [z(A)—z(B)]
Thus, AB=BC and ∠ABC=90°.
Similar argument for others.
Hence, ABCD is a square.
2. Let O is the center of spiral similarity to transform square A1B1C1D1 to A2B2C2D2
We will have ∆A1OA2 ~ ∆C1OC2~ ∆B1OB2~ ∆D1OD2… (properties of spiral similarity transformation)
With medians OA, OB, OC, OD
Due to similarity of these triangles we also have
1. ∠ (A1OA)= ∠ (C1OC)= ∠ (B1OB)= ∠ (D1OD)=θ … (angles forms between corresponding sides to corresponding medians of similar triangles)
2. OA/OA1=OB/OB1=OC/OC1= OD/OD1=k …( ratio of corresponding medians to corresponding sides of similar triangles)
So square ABCD will be the image of square A1B1C1D1 in the spiral similarity transformation ( center O, angle of rotation= θ and factor of dilation = k)
Since A1B1C1D1 is a square so the image ABCD will be a square | {"url":"https://gogeometry.blogspot.com/2015/08/geometry-problem-1145-three-squares.html","timestamp":"2024-11-05T03:23:53Z","content_type":"application/xhtml+xml","content_length":"54900","record_id":"<urn:uuid:2d0a30c5-4282-491a-909e-67a077d35c2a>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00871.warc.gz"} |
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Untitled Document
We consider the classical geometric problem of prescribing scalar and boundary mean curvature via conformal deformation of the metric on a n-dimensional compact Riemannian manifold. We deal with the
case of negative scalar curvature and positive boundary mean curvature. It is known that if n=3 all the blow-up points are isolated and simple. In this work we prove that this is not true anymore in
low dimensions (that is n=4, 5, 6, 7). In particular, we construct a solution with a clustering blow-up boundary point (i.e. non-isolated), which is non-umbilic and minimizes the norm of the
trace-free second fundamental form of the boundary. | {"url":"https://aimsconference.org/AIMS-Conference/conf-reg2023/ss/detail1.php?abs_no=1179","timestamp":"2024-11-04T11:19:09Z","content_type":"text/html","content_length":"3672","record_id":"<urn:uuid:bb6e6c21-2c1d-4156-b943-8c97e6c6365a>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00871.warc.gz"} |
Why the Private Key Calculation of Ethereum is Irreversible
Have you ever wondered why the private key of Ethereum cannot be reverse-calculated from the account address? When you have a private key and want to get the account address corresponding to this
private key, you can import the account into MetaMask, or use an SDK like ether.js to import an account into the wallet at the code level, and then print out the account address. Is there any black
box operation in this account import process?
A few days ago, I accidentally saw an article on Medium, where the author wrote a very concise code to calculate the process from the private key to the address. The author’s code is here RareSkills/
I copied the code:
from ecpy.curves import Curve
from sha3 import keccak_256
private_key = 0xac0974bec39a17e36ba4a6b4d238ff944bacb478cbed5efcae784d7bf4f2ff80
cv = Curve.get_curve('secp256k1')
pu_key = private_key * cv.generator # just multiplying the private key by generator point (EC multiplication)
concat_x_y = pu_key.x.to_bytes(32, byteorder='big') + pu_key.y.to_bytes(32, byteorder='big')
eth_addr = '0x' + keccak_256(concat_x_y).digest()[-20:].hex()
print('private key: ', hex(private_key))
print('eth_address: ', eth_addr)
This code has only four or five lines, and there are two key points in the calculation: private_key * cv.generator and keccak_256(concat_x_y). Apart from these two points, the other parts are
constant calculations. At first glance, it seems complicated, but when you look at it carefully and break it down, it’s all very simple string concatenation.
Among them, private_key * cv.generator is the calculation of the elliptic curve. The private key private_key defined above is a hexadecimal number, note it is of int type, and then the generator of
the elliptic curve is used to calculate a value. This calculation process is irreversible, which is the process of circling on the elliptic curve. It can be understood by analogy that after a number
is taken modulo, you cannot restore the number before the modulo. The elliptic curve just uses a more complex method to provide a safer calculation result than RSA.
The second irreversible calculation is keccak_256(concat_x_y), which is a process of calculating the hash value. Keccak is a kind of sha3. Digest algorithms are irreversible, which is undoubtedly the
In other words, in the calculation process from the private key to the address, there are two irreversible points, so overall, it is impossible to reverse calculate the private key from the Ethereum
account address. | {"url":"http://en.smallyu.net/2023/02/20/Why%20the%20Private%20Key%20Calculation%20of%20Ethereum%20is%20Irreversible/","timestamp":"2024-11-02T21:55:02Z","content_type":"text/html","content_length":"5969","record_id":"<urn:uuid:086c21c9-4388-4035-b1b3-cdcb74d306f3>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00594.warc.gz"} |
numpy.polynomial.legendre.legint(c, m=1, k=[], lbnd=0, scl=1, axis=0)[source]¶
Integrate a Legendre series.
Returns the Legendre series coefficients c integrated m times from lbnd along axis. At each iteration the resulting series is multiplied by scl and an integration constant, k, is added. The
scaling factor is for use in a linear change of variable. (“Buyer beware”: note that, depending on what one is doing, one may want scl to be the reciprocal of what one might expect; for more
information, see the Notes section below.) The argument c is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series L_0 + 2*L_1 + 3*L_2 while
[[1,2],[1,2]] represents 1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y) if axis=0 is x and axis=1 is y.
c : array_like
Array of Legendre series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index.
m : int, optional
Order of integration, must be positive. (Default: 1)
k : {[], list, scalar}, optional
Integration constant(s). The value of the first integral at lbnd is the first value in the list, the value of the second integral at lbnd is the second value, etc. If k == [] (the
default), all constants are set to zero. If m == 1, a single scalar can be given instead of a list.
Parameters: lbnd : scalar, optional
The lower bound of the integral. (Default: 0)
scl : scalar, optional
Following each integration the result is multiplied by scl before the integration constant is added. (Default: 1)
axis : int, optional
Axis over which the integral is taken. (Default: 0).
New in version 1.7.0.
S : ndarray
Legendre series coefficient array of the integral.
If m < 0, len(k) > m, np.ndim(lbnd) != 0, or np.ndim(scl) != 0.
Note that the result of each integration is multiplied by scl. Why is this important to note? Say one is making a linear change of variable x. Then scl equal to
Also note that, in general, the result of integrating a C-series needs to be “reprojected” onto the C-series basis set. Thus, typically, the result of this function is “unintuitive,” albeit
correct; see Examples section below.
>>> from numpy.polynomial import legendre as L
>>> c = (1,2,3)
>>> L.legint(c)
array([ 0.33333333, 0.4 , 0.66666667, 0.6 ])
>>> L.legint(c, 3)
array([ 1.66666667e-02, -1.78571429e-02, 4.76190476e-02,
-1.73472348e-18, 1.90476190e-02, 9.52380952e-03])
>>> L.legint(c, k=3)
array([ 3.33333333, 0.4 , 0.66666667, 0.6 ])
>>> L.legint(c, lbnd=-2)
array([ 7.33333333, 0.4 , 0.66666667, 0.6 ])
>>> L.legint(c, scl=2)
array([ 0.66666667, 0.8 , 1.33333333, 1.2 ]) | {"url":"https://docs.scipy.org/doc/numpy-1.15.1/reference/generated/numpy.polynomial.legendre.legint.html","timestamp":"2024-11-12T10:47:45Z","content_type":"text/html","content_length":"15392","record_id":"<urn:uuid:68ef76cb-1633-4b93-81cf-bbd2c6a24b5a>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00209.warc.gz"} |
What is solvable series?
Definition. A group G is called solvable if it has a subnormal series whose factor groups (quotient groups) are all abelian, that is, if there are subgroups 1 = G0 < G1 < ⋅⋅⋅ < Gk = G such that Gj−1
is normal in Gj, and Gj /Gj−1 is an abelian group, for j = 1, 2, …, k.
Is A_N solvable?
A subgroup of three elements (generated by a cyclic rotation of three objects) with any distinct nontrivial element generates the whole group. For all n > 4, An has no nontrivial (that is, proper)
normal subgroups. Thus, An is a simple group for all n > 4. A5 is the smallest non-solvable group.
Are Homomorphisms onto?
Special types of homomorphisms have their own names. A one-to-one homomorphism from G to H is called a monomorphism, and a homomorphism that is “onto,” or covers every element of H, is called an
Can a simple group be solvable?
The famous theorem of Feit and Thompson states that every group of odd order is solvable. Therefore, every finite simple group has even order unless it is cyclic of prime order. The Schreier
conjecture asserts that the group of outer automorphisms of every finite simple group is solvable.
Is A4 normal in S4?
A4 is of Order 12, and therefore Index 2, hence A4 is Normal in S4. Elements in S4 modulo A4 form the cyclic quotient group S4/A4 which is isomorphic to Z/2Z .
Are subgroups of solvable groups solvable?
A major building block for the classification of finite simple groups was the Feit-Thompson theorem, which proved that every group of odd order is solvable. This proof took up an entire journal
issue. , every Abelian group, and every subgroup of a solvable group is solvable.
How do you prove isomorphism?
Proof: By definition, two groups are isomorphic if there exist a 1-1 onto mapping ϕ from one group to the other. In order for us to have 1-1 onto mapping we need that the number of elements in one
group equal to the number of the elements of the other group. Thus, the two groups must have the same order.
How do you prove a homomorphism is an isomorphism?
If φ(G) = H, then φ is onto, or surjective. A homomorphism that is both injective and surjective is an an isomorphism. An automorphism is an isomorphism from a group to itself. If we know where a
homomorphism maps the generators of G, we can determine where it maps all elements of G.
What is solvable?
Solvable is a true-crime podcast that seeks to find the answers to unsolved mysteries. With the cooperation of the investigative agency, Solvable takes the listener behind closed doors and speaks
directly to the past and current personnel who are responsible for investigating these crimes.
What is a Series EE Savings Bond?
Series EE Savings Bonds. These EE bonds earn the same rate of interest (a fixed rate) for up to 30 years. When you buy the bond, you know what rate of interest it will earn. Treasury announces the
rate each May 1 and November 1 for new EE bonds.
What is the equivalent definition of a solvable group?
For finite groups, an equivalent definition is that a solvable group is a group with a composition series all of whose factors are cyclic groups of prime order. This is equivalent because a finite
group has finite composition length, and every simple abelian group is cyclic of prime order.
Who is the host of the solvable?
The program is hosted by genetic genealogist Amanda Reno and Greg Bodker a deputy police chief who take listeners behind closed doors and speaks directly to the past and current personnel who a… Read
all Solvable is a true crime podcast that seeks to find the answers to unsolved mysteries. | {"url":"https://corfire.com/what-is-solvable-series/","timestamp":"2024-11-02T10:53:32Z","content_type":"text/html","content_length":"38150","record_id":"<urn:uuid:31a1c7ea-e3cd-46a1-a2bf-705e21e0cb91>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00225.warc.gz"} |
Fluid Flow Pattern over Porous Medium Calculator
getcalc.com's Darcy's Law Calculator is an online mechanical engineering tool for fluid mechanics to estimate the flow patterns of fluid through porous medium (beds of sands), in both US customary &
metric (SI) units.
Definition & Formula
Darcy's Law is a describes the flow patterns of fluid through porous medium (beds of sands) to estimate the linear relationship between discharge flow velocity and hydraulic gradient of porous medium
under steady laminar flow conditions.
Formula to estimate the flow patterns of fluid through porous medium | {"url":"https://getcalc.com/mechanical-darcys-law-calculator.htm","timestamp":"2024-11-11T00:46:30Z","content_type":"text/html","content_length":"23876","record_id":"<urn:uuid:300d87c3-3468-40d0-b3ff-70f71e0c6608>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00360.warc.gz"} |
What is Viscosity? Details, Formula, Importance
The viscosity of a fluid means the resistance of the fluid to shear or angular deformation. In easy meaning, it is like a frictional force in a fluid which create resistance to flow. This frictional
forces in fluid flow resulting from the cohesion and momentum interchange between molecules in the fluid. It is due to the viscous force that arises in the fluid.
A little bit of confusion can create between elastic forces and viscous forces but we will have to bear in mind that elastic force arises in solid but the viscous force arises in the fluid.
Following are some common terms and issues which need to be known to understand the viscosity clearly.
Viscous Force
Viscous force in a fluid is proportional to the rate at which the fluid velocity is changing in space in a constant area.
Viscosity in Turbulent Flow
Viscosity is relatively low in turbulent flow. For this, the velocity of turbulent flow is relatively so high.
Viscosity in Ideal Fluid and Real Fluid
Ideal Fluid: It is such a fluid which is presumed to have no Viscosity. This is an idealized condition which doesn’t exist.
Real Fluid: In this fluid, the effect of viscosity is considered which results in the development of shear stresses between neighboring particles when they are moving in different velocities.
Actually, all the fluid are real fluid because in fluid there will exist at least a minimum level of viscosity.
Absolute Viscosity
From Newton’s law of Viscosity, Shear stress on a fluid element layer is directly proportional to the rate of shear strain. Here, the constant is known as absolute viscosity or dynamic viscosity,
\[\mu= \frac{\text{Absolute Viscosity}}{\text{Dynamic Viscosity}}\]
\[\tau= \text{Shear Stress}\]
Kinematic Viscosity
It is defined as the ratio between, dynamic viscosity to the density of a fluid.
Here, \[\gamma= \text{Kinematic Viscosity}\]
\[\rho= \text{Density}\]
Importance of Viscosity
• Viscosity is a critical property of hydraulic oil. Complete system performance and efficiency are the main parameters of hydraulic oil. These two parameters are affected by viscosity. Also for
using valves and pumps viscosity is an important element.
• In lubrication, the viscosity is the most needed characteristic of lubricating oil. For greases, the viscosity is an important element too.
• If the temperature of the fluid is low then the viscosity is high, at that time oil cannot be pumped. On the other hand, if the temperature is high then the velocity of oil will be excessive,
that means viscosity is so low and this can cause high friction in any pipe then wear.
• Viscosity is a measure of whether the flow is laminar or turbulent.
• By the help of Viscosity, we can know the behavior of viscosity which helps to design a machine in mechanical engineering, to build a ship, to work in marine condition.
• Because of high viscosity some fluid stay in steady condition. If there are no viscosity fluid would have no internal resistance and so it will flow forever before facing any barrier.
• From the behavior of viscosity with temperature, we can find whether the fluid is liquid or gas. For increasing the temperature the viscosity will increase for gas. On the other hand, increasing
the temperature will decrease the Viscosity for Liquid.
Measurement of Viscosity
Absolute Viscosity
Stated it before, From Newton’s law of Viscosity, Shear stress on a fluid element layer is directly proportional to the rate of shear strain. Here, the constant is known as absolute viscosity or
dynamic viscosity,
Here, \[\mu=\text{Absolute viscosity/ Dynamic Viscosity/ Constant of proportionality}\]
\[\tau= \text{Shear Stress}\]
\[\frac{\text{d}u}{\text{d}y}=\text{ Velocity Gradient [Rate of change of velocity with respect to space]}\]
From Reynolds Number,
\[R= \frac{DV\rho}\mu\]
\[R= \text{Reynolds Number}\]
\[\rho= \text{Density of the fluid}\]
\[\mu= \frac{\text{Absolute Viscosity}}{\text{Dynamic Viscosity}}\]
\[D= \text{Diameter of the pipe}\]
\[V= \text{Velocity of the flow}\]
Kinematic Viscosity
From Absolute Viscosity,
Here, \[\gamma= \text{Kinematic Viscosity}\]
\[\rho= \text{Density}\]
From Reynolds Number,
\[R= \frac{DV}\gamma\]
\[R= \text{Reynolds Number}\]
\[\gamma= \text{Kinematic Viscosity}\]
\[D= \text{Diameter of the pipe}\]
\[V= \text{Velocity of the flow}\]
Reynolds Number
Reynolds Number can be found through Moody Diagram in where from relative roughness and friction factor we can find the Reynolds number.
Relative Pipe roughness is a parameter of the pipe used,
\[\text{Relative Pipe Roughness}= \frac{e}{D}\]
\[e= \text{Absolute Roughness}\]
Then the friction factor can be found through,
\[h_{L}= 4 f\times\frac{L}{D}\times\frac{V^{2}}{2g}\]
\[L= \text{Length of the pipe}\]
\[D= \text{Diameter of the pipe}\]
\[V= \text{Velocity of the flow}\]
\[g= \text{Gravitational Acceleration}\]
\[h_{L}= \text{Head Loss through a pipe}\]
\[f= \text{Friction Factor}\]
Normally for,
Laminar Flow: Reynolds number < 2000
Turbulent Flow: Reynolds Number > 4000
Point to be Noted:
• The absolute viscosity of all fluids is practically independent of pressure for the range that is ordinarily encountered in engineering work.
• The kinematic viscosity of gases changes due to the change of pressure because of changes in density.
• REFERENCES: FLUID MECHANICS WITH ENGINEERING APPLICATIONS – Robert L. Daugherty, Joseph B. Franzini | {"url":"https://civiltoday.com/water-resource-engineering/fluid-mechanics/240-viscosity","timestamp":"2024-11-07T04:05:45Z","content_type":"text/html","content_length":"46387","record_id":"<urn:uuid:20c5a2d5-c269-4941-84de-e88ee470101a>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00090.warc.gz"} |
Approximating Integrals (with formulas & videos)
Approximating Integrals
A series of free Calculus Videos.
Approximating a Definite Integral Using Rectangles
4 rectangles are used and left endpoints as well as midponits to approximate the area underneath 16-x^2 from x = 0 to x = 2.
The Trapezoid Rule for Approximating Integrals
This video shows the formula and give one simple example of using the Trapezoid Rule to approximate the value of a definite integral.
Simpsons Rule - Approximate Integration
This video gives the formula for Simpson s Rule, and uses it to approximate a definite integral.
Simpsons Rule - Error Bound
This video finds the number of intervals required when using Simpson's Rule to approximate a definite integral to a desired accuracy.
Custom Search
We welcome your feedback, comments and questions about this site - please submit your feedback via our Feedback page. | {"url":"https://www.onlinemathlearning.com/approximating-integrals.html","timestamp":"2024-11-14T18:18:19Z","content_type":"text/html","content_length":"20863","record_id":"<urn:uuid:e78aa2a3-2e15-47e9-9ef8-aed25125d600>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00806.warc.gz"} |
Newton’s Laws
Imagine a mass Coriolis force which, in the example shown in the figure below, pushes it from its original trajectory (orange) to move eastward (blue). Why does this happen, and how do we understand
it intuitively?
Formally, the Coriolis force on
Missing energy in a rope and a capacitor
Consider a uniform rope of mass density
1. Find the force exerted on the end of the rope as function of the height
2. Compare the power delivered to the rope with the rate of change of the rope’s mechanical energy.
(This is a problem from chapter 5 of Kleppner and Kolenkow)
To find the force exerted at the top end, note that if we were to pull up a fixed mass with constant velocity
Continue Reading
Mass on a semicircular block
A heavy particle of mass
Related problem: Sliding on a block with a circular cut.
(i) We first consider the case where the block is fixed to the ground. As the mass slides down the block, there are three forces acting on it: the weight
Continue Reading
Circular orbit in a harmonic potential
JEE Advanced 2018 Paper 1, Question 1
The potential energy of a particle of mass
The force due to the given potential is
The mass also experiences a centrifugal force due to its circular motion,
For the …
Continue Reading
Pulleys and masses connected to a spring
JEE Advanced 2019 Paper 2, Question 2
A block of mass
1. When spring achieves an extension of
Continue Reading | {"url":"https://www.jeefirst.com/tag/newtons-laws/","timestamp":"2024-11-11T10:15:28Z","content_type":"text/html","content_length":"68355","record_id":"<urn:uuid:9ac71447-947d-4a30-870f-e5056043c88a>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00745.warc.gz"} |
An array puzzle – Study Algorithms
Question: There is an array A[N] of N numbers. You have to compose an array Output[N] such that Output[i] will be equal to multiplication of all the elements of A[N] except A[i]. Solve it without
division operator and in O(n).
For example Output[0] will be multiplication of A[1] to A[N-1] and Output[1] will be multiplication of A[0] and from A[2] to A[N-1].
Array 1: {4, 3, 2, 1, 2}
Output: {12, 16, 24, 48, 24}
Since the complexity required is O(n), the obvious O(n^2) brute force solution is not good enough here. Since the brute force solution recomputes the multiplication each time again and again, we can
avoid this by storing the results temporarily in an array.
Let’s define array B where element B[i] = multiplication of numbers from A[0] to A[i]. For example, if A = {4, 3, 2, 1, 2}, then B = {4, 12, 24, 24, 48}. Then, we scan the array A from right to
left, and have a temporary variable called product which stores the multiplication from right to left so far. Calculating OUTPUT[i] is straight forward, as OUTPUT[i] = B[i-1] * product.
The above method requires only O(n) time but uses O(n) space. We have to trade memory for speed. Is there a better way? (i.e., runs in O(n) time but without extra space?)
Yes, actually the temporary array is not required. We can have two variables called left and right, each keeping track of the product of numbers multiplied from left->right and right->left. Could you
see why this works without extra space?
Here is an implementation of the algorithm that does not require any extra space:-
void array_multiplication(int A[], int OUTPUT[], int n)
// Initializing left and right
int left = 1;
int right = 1;
// Initialize the output array
int i = 0;
for (i = 0; i < n; i++)
OUTPUT[i] = 1;
for (i = 0; i < n; i++)
OUTPUT[i] *= left;
OUTPUT[n - 1 - i] *= right;
// The variable left stores the multiplication of
// all elements on the left
left *= A[i];
// The variable right stores the multiplication of
// all elements on the right
right *= A[n - 1 - i];
int main(void)
int arr[] = {4, 3, 2, 1, 2};
int OUTPUT[5] = {0};
array_multiplication(arr, OUTPUT, 5);
int i = 0;
for(i=0; i<5; i++)
printf("%d ",OUTPUT[i]);
return 0;
Here is a running link of the above algorithm:- http://ideone.com/102tgS
3 comments
arulsubramaniam April 5, 2015 - 03:06
Or we just do as below?
In first for loop , find the product of all array numbers .
In the second for loop, A[i] = product/A[i]
Kerem Unal April 6, 2015 - 13:36
The problem definition restricts the use of division.
arulsubramaniam April 6, 2015 - 15:24
Oh well. I didn’t notice it. Sorry ! Then this approach looks better ;)
3 comments
a tech-savvy guy and a design buff... I was born with the love for exploring and want to do my best to give back to the community. I also love taking photos with my phone to capture moments in my
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Math, Grade 6, Putting Math to Work, Fundamental Problem Solving Concepts
Represent the Situation Mathematically
Work Time
Represent the Situation Mathematically
Now that you have used some easy numbers to represent the situation, it's important to make a general representation of the situation that will work for any numbers.
• Make a ratio table showing the relationship between time and distance.
• How could you use your completed ratio table to make a graph?
• Write a formula that shows the relationship between time and distance.
• Make a graph of this formula. How would you label the x-axis (independent variable) and the y-axis (dependent variable)?
• Make a graph of the formula for distances up to 20 miles.
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Gordon (Constant) Growth Dividend Discount Model - Finance Train
Gordon (Constant) Growth Dividend Discount Model
The Gordon Growth Model (GGM) is a variation of the standard discount model. The key difference is that the GGM model assumes the dividends will grow at a constant rate till perpetuity.
If the current year’s dividends are D0, and the dividend growth rate is gc, the next year’s dividend D1 will be D0 = (1+gc). D2 will be D0(1+gc)^2 and so on.
With this assumption, the value of the stock can be calculated using the following simplified formula:
V0 = D1/(ke - gc)
Model Assumptions
The model has several assumptions:
• It assumes that the dividends are a suitable measure for valuation.
• It assumes that both the required return on equity and dividend growth rate will be constant forever.
• The required return on common equity must be greater than the expected growth rate of the dividend.
Let’s calculate the value of a stock that paid a dividend of $5 last year. The required return on equity is 10% and the dividend is expected to grow by 5%.
Using the GGM model, the value of the stock will be:
V = $5(1.05)/(10% - 5%) = $105
As you can notice, the value of the stock is sensitive to the denominator and therefore to the dividend growth rate. You can also determine how much of the stock value is attributed to the dividend
growth rate. This can be done by recalculating the value with zero growth rate and then noting the difference.
With a zero growth rate, the stock value will be = $5/10% = $50. So, we can say that out of $105, $55 ($105 - $50) is attributed to the dividend growth rate.
Model Suitability
The GGM is suitable in the following cases:
• The analyst is looking at broad equity indexes.
• The analyst is valuing steadily growing companies that pay dividends.
• Highly sensitive to small changes in required return on equity and dividend growth rate.
• The model cannot be used to value companies that do not pay dividends.
• The model cannot be used to value companies that don’t have a stable growth rate.
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Continuum does not exist
For example, when I write "n∈ N", I don’t mean that n is an element of the actual infinite set of natural numbers. Rather, I mean that, it is a natural number according to the SB tree (details
omitted). — keystone
Good luck with that. Probably of more interest to CS people.
The function x(n) — keystone
A sequence of rationals I assume. If you had two functions on Q then a suitable metric would be the supremum.
An actual curve is an indivisible, one-dimensional object with length but no width or depth. It extends continuously between two actual points but excludes the endpoints. — keystone
How do you define "continuous"? Are you sure it is indivisible?
Sorry, but your list of definitions is mind-numbing. Your top down is becoming way more complicated that bottom up, IMO. And the irrational numbers have yet to appear.
Your best bet would be to find a mathematician willing to deal with your arguments and pay him/her a fee to do so.
Good luck with that. Probably of more interest to CS people. — jgill
Yeah, my view leans heavily on algorithms.
A sequence of rationals I assume. — jgill
Yes. I have since edited the post to clarify this.
If you had two functions on Q then a suitable metric would be the supremum. — jgill
Suitable for what?
How do you define "continuous"? Are you sure it is indivisible? — jgill
Good point. I have since edited the post to clarify this. When defining an actual curve I was providing an informal intuitive explanation where I carelessly used 'continuous'. Ultimately an actual
curve is simply an object having an actual interval.
Sorry, but your list of definitions is mind-numbing. — jgill
Isn't anything communicated with absolute precision a bit mind-numbing? Not that I achieved that level of precision, but it was trying to be more precise. I find logic much more mind-number, but
that's just me...
Your top down is becoming way more complicated that bottom up, IMO. — jgill
Ultimately, it all reduces to the same calculus used by applied mathematicians today. However, building a foundation on constructive philosophy is likely to introduce more complexity—at least that's
how it plays out in logic. Actual infinity is certainly simpler to work with, but is it truly sound? Newtonian mechanics is simpler than relativity, which is simpler than quantum mechanics. So, what
should be the foundational choice for physics - the simplest? There's an elegance to QM and I believe the same can be said about the top down view of mathematics.
And the irrational numbers have yet to appear. — jgill
I’ve already outlined the framework for irrational numbers. Both potential coordinates and potential intervals are reinterpretations of real numbers, including irrational ones. If we get past the
list of definitions then the next step is to present an example that demonstrates how irrational numbers come into play.
Your best bet would be to find a mathematician willing to deal with your arguments and pay him/her a fee to do so. — jgill
I've tried in the past, but nowhere else has been as beneficial as here. That said, I’m open to recommendations. It’s challenging for an amateur mathematician to find someone with the right skills
and interests. I primarily used
You could start with continuity described as a path in the Euclidean plane or complex plane taken by a moving particle. Or something similar.
There's an elegance to QM and I believe the same can be said about the top down view of mathematics — keystone
Careful. I would not compare if I were you.
If you had two functions on Q then a suitable metric would be the supremum. — jgill
Suitable for what? — keystone
For defining "distance" between functions. When I dabble in the complex plane contours become points in the metric space and the distance between them is the Sup|f(t)-g(t)| over 0<t<1 for instance.
If I were younger I might have more time to try to unravel your presentation. You have wandered from metric spaces to topology and now graph theory, with that dreadful SB-table trailing along. Then
you have all these definitions which a mathematician is unlikely to find of interest.
Your best bet would be to find a mathematician willing to deal with your arguments and pay him/her a fee to do so. — jgill
I've tried in the past, but nowhere else has been as beneficial as here — keystone
Try a nearby university where a grad student might want a little extra cash.
Careful. I would not compare if I were you. — jgill
Good point. I've needed to learn this lesson too many times.
For defining "distance" between functions. — jgill
Since the functions I'm working with all converge, I don't believe the supremum is necessary for distance, but it might be necessary for other purposes.
Try a nearby university where a grad student might want a little extra cash. — jgill
I'll look into this. Thanks for the suggestion.
If I were younger I might have more time to try to unravel your presentation. — jgill
I’ve just revised the post to remove unnecessary mention of objects, making it shorter. If you skip the sections on the definitions of continuity, the post is only 444 words. I mention the continuity
section because it's wordy but obvious. For instance, we already know that the interval ⟨0 5⟩ linked with coordinate 10 can’t be continuous, as 10≠0 and 10≠5, implying a gap between them. I just
explicitly lay out all scenarios to capture the obvious. I hope you might reconsider giving it another look, but I completely understand if you choose not to continue. This discussion has already
been incredibly helpful to me.
You have wandered from metric spaces to topology and now graph theory, with that dreadful SB-table trailing along. Then you have all these definitions which a mathematician is unlikely to find of
interest. — jgill
I’ve admittedly wandered off track at times, and you've been patient with the many detours along the way. However, I’m a bit surprised that once I introduced a more mathematical approach—like
discussing the Stern-Brocot tree and providing proper definitions—you felt the discussion was becoming less interesting to mathematicians. I had expected the opposite.
I propose that continuous calculus is not the study of continuous actual structures but rather the study of continuous potential structures. — keystone
: Would you be open to re-engaging with me on this topic (that we discussed months back)? I believe the post quoted here will give you something much more concrete for you to chew on. Plus—no figures
this time! I'd really appreciate your advice.
A 1D actual structure is a finite, undirected graph in which each vertex represents an actual point, pseudo point, or actual curve — keystone
A vertex represents an actual curve?
I’m a bit surprised that once I introduced a more mathematical approach—like discussing the Stern-Brocot tree and providing proper definitions—you felt the discussion was becoming less
interesting to mathematicians. I had expected the opposite. — keystone
In fact, I had never heard of the S-B tree before it was introduced on this forum. It is not true that every mathematician will find every math topic interesting. (Wiki lists well over 25,000 if I
recall). Had I been a number theorist or a CS person I may have known of it. I see it averages about 47 pageviews per day on Wiki, and classed as low priority. But that's not trivial by any means. My
own page gets only 15.
A vertex represents an actual curve? — jgill
Each indivisible object, whether potential, pseudo, or actual, is represented as a vertex within a structure, regardless of its dimensionality. This approach underscores the fundamental
indivisibility of these objects. The only object that is divisible is a structure.
In fact, I had never heard of the S-B tree before it was introduced on this forum. — jgill
Imagine how fortunate I (an amateur) feel to have stumbled across it (and Niqui's paper on arithmetic based on it)!
It is not true that every mathematician will find every math topic interesting. — jgill
I see it averages about 47 pageviews per day on Wiki, and classed as low priority. — jgill
Sometimes the significance of a discovery isn't recognized until many years later.
I see it averages about 47 pageviews per day on Wiki, and classed as low priority. — jgill
Sometimes the significance of a discovery isn't recognized until many years later. — keystone
It's had 164 years. We'll see.
Each indivisible object, whether potential, pseudo, or actual, is represented as a vertex within a structure, regardless of its dimensionality — keystone
From the point of view of graph theory, vertices are treated as featureless and indivisible objects, although they may have additional structure depending on the application from which the graph
arises; for instance, a semantic network is a graph in which the vertices represent concepts or classes of objects.
OK. But it gets a bit anti-intuitive. Hard to imagine a curve is a vertex. But then, I treat curves in the CP as points in a metric space.
I wish other mathematicians would chime in on this thread. I am very old and have forgotten what I didn't learn.
Hard to imagine a curve is a vertex. — jgill
An actual curve in 1D is unique in that it is fully defined by its endpoints. However, in 2D and higher dimensions, a curve is determined not only by its endpoints but also by an equation. Perhaps
incorporating that equation into the vertex might make the concept more digestible.
I wish other mathematicians would chime in on this thread. I am very old and have forgotten what I didn't learn. — jgill
Yeah, that would be nice, but I really do appreciate you taking the conversation this far. You got me thinking!
However, in 2D and higher dimensions, a curve is determined not only by its endpoints but also by an equation. Perhaps incorporating that equation into the vertex might make the concept more
digestible — keystone
I explore various properties of contours in the complex plane, defining a metric space whose "points" are contours.
$z(t)=u(t)+iv(t),\,\,\,0\le t\le 1$
"Distance" is defined
$d\left( {{z}_{1}},{{z}_{2}} \right)=\underset{t}{\mathop{Sup}}\,\left| {{z}_{1}}(t)-{{z}_{2}}(t) \right|$
. But here t is a positive real number, which you have not defined yet. Usually, the u(t) and v(t) are differentiable, giving a smooth curve. So incorporating this sort of thing into the definition
of vertex assumes what you will probably wish to prove. I wonder what an "edge" in your graph would be?
But here t is a positive real number, which you have not defined yet. — jgill
A real number corresponds to a specific subgraph within a potential structure. In the 1D case, this is represented by a potential curve and the two potential points that are directly connected to it.
incorporating this sort of thing into the definition of vertex assumes what you will probably wish to prove. — jgill
Incorporating differentiability?
I wonder what an "edge" in your graph would be? — jgill
An edge signifies adjacency between objects. For example, in conventional interval notation, an edge would exist between the curve (0,5) and the point [5,5] due to their direct adjacency. In
contrast, the curve (0,5) is not adjacent to (5,10) because a gap exists between them (at point 5), so no edge would connect the vertices representing the two curves.
A real number corresponds to a specific subgraph within a potential structure. In the 1D case, this is represented by a potential curve and the two potential points that are directly connected to
it. — keystone
You've lost me. Guess it's time for me to quit. Overall, I think you have started down a path that is far too complicated for the desired result. However, if the result you seek is more philosophical
than mathematical you may have something.
Overall, I think you have started down a path that is far too complicated for the desired result. — jgill
Someone could say the same thing about the epsilon-delta formulation of a limit, which was introduced to give calculus a more rigorous foundation. After all, infinitesimals produced the desired
results and were simpler to work with.
Guess it's time for me to quit. — jgill
No worries. Thanks for the discussion!
Overall, I think you have started down a path that is far too complicated for the desired result. — jgill
Someone could say the same thing about the epsilon-delta formulation of a limit, which was introduced to give calculus a more rigorous foundation. — keystone
At first I thought this is not true, but that is because I followed a learning curve that incorporated analytic geometry before calculus, and this allows clear illustrations and examples en route to
limits. You work in CS, however, and what seems like unintuitive definitions to me probably make more sense to you. Had I taken a course in graph theory what you are proposing might seem less opaque.
Continue, if you like, and I will comment from time to time as I learn more about graph theory.
I suggest this thread be placed in the Lounge since it obviously has limited appeal to the general audience, but has merit in philosophy of mathematics. Just my opinion. | {"url":"https://thephilosophyforum.com/discussion/15393/continuum-does-not-exist/p17","timestamp":"2024-11-01T23:48:31Z","content_type":"text/html","content_length":"63342","record_id":"<urn:uuid:b76924fc-49c4-49b8-96cc-6419d52cb3f7>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00119.warc.gz"} |
Pandas apply() — A Helpful Illustrated Guide
The Pandas apply( ) function is used to apply the functions on the Pandas objects. We have so many built-in aggregation functions in pandas on Series and DataFrame objects. But, to apply some
application-specific functions, we can leverage the apply( ) function. Pandas apply( ) is both the Series method and DataFrame method.
Pandas apply function to one column – apply( ) as Series method
Let’s construct a DataFrame in which we have the information of 4 persons.
>>> import pandas as pd
>>> df = pd.DataFrame(
... {
... 'Name': ['Edward', 'Natalie', 'Chris M', 'Priyatham'],
... 'Sex' : ['M', 'F', 'M', 'M'],
... 'Age': [45, 35, 29, 26],
... 'weight(kgs)': [68.4, 58.2, 64.3, 53.1]
... }
... )
>>> print(df)
Name Sex Age weight(kgs)
0 Edward M 45 68.4
1 Natalie F 35 58.2
2 Chris M M 29 64.3
3 Priyatham M 26 53.1
pandas.Series.apply takes any of the below two different kinds of functions as an argument. They are:
• Python functions
• Numpy’s universal functions (ufuncs)
1. Python functions
In Python, there are 3 different kinds of functions in general;
• Built-in functions
• User-defined functions
• Lambda functions
a) Applying Python built-in functions on Series
If we would like to know the length of the names of each person, we can do so using the len( ) function in python.
For example, if we want to know the length of the “Python” string, we can get by the following code;
>>> len("Python")
A single column in the DataFrame is a Series object. Now, we would like to apply the same len( ) function on the whole “Name” column of the DataFrame. This can be achieved using the apply( ) function
in the below code;
>>> df['Name'].apply(len)
Name: Name, dtype: int64
If you observe the above code snippet, the len inside the apply( ) function is not taking any argument. In general, any function takes some data to operate on them. In the len(“Python”) code snippet,
it’s taking the “Python” string as input data to calculate its length. Here, the input data is directly taken from the Series object that called the function using apply( ).
When applying the Python functions, each value in the Series is applied one by one and returns the Series object.
The above process can be visualised as:
In the above visualisation, you can observe that each element of Series is applied to the function one by one.
b) Applying user-defined functions on Series
Let’s assume that the data we have is a year old. So, we would like to update the age of each person by adding 1. We can do so by applying a user-defined function on the Series object using the apply
( ) method.
The code for it is,
>>> def add_age(age):
... return age + 1
>>> df['Age'].apply(add_age)
Name: Age, dtype: int64
>>> df['Age'] = df['Age'].apply(add_age)
>>> df
Name Sex Age weight(kgs)
0 Edward M 46 68.4
1 Natalie F 36 58.2
2 Chris M M 30 64.3
3 Priyatham M 27 53.1
From the above result, the major point to be noted is,
• The index of the resultant Series is equal to the index of the caller Series object. This makes the process of adding the resultant Series as a column to the DataFrame easier.
It operates in the same way as applying built-in functions. Each element in the Series is passed one by one to the function.
• User-defined functions are used majorly when we would like to apply some application-specific complex functions.
c) Applying Lambda functions on Series
Lambda functions are used a lot along with the apply( ) method. We used a user-defined function for an easy addition operation in the above section. Let’s achieve the same result using a Lambda
The code for it is,
>>> df['Age'].apply(lambda x: x+1)
Name: Age, dtype: int64
>>> # Comparing the results of applying both the user-defined function and Lambda function
>>> df['Age'].apply(lambda x: x+1) == df['Age'].apply(add_age)
0 True
1 True
2 True
3 True
Name: Age, dtype: bool
From the above result, you can observe the results of applying the user-defined function and Lambda function are the same.
• Lambda functions are used majorly when we would like to apply some application-specific small functions.
2. Numpy’s universal functions (ufuncs)
Numpy has so many built-in universal functions (ufuncs). We can provide any of the ufuncs as an argument to the apply( ) method on Series. A series object can be thought of as a NumPy array.
The difference between applying Python functions and ufuncs is;
• When applying the Python Functions, each element in the Series is operated one by one.
• When applying the ufuncs, the entire Series is operated at once.
Let’s choose to use a ufunc to floor the floating-point values of the weight column. We have numpy.floor( ) ufunc to achieve this.
The code for it is,
>>> import numpy as np
>>> df['weight(kgs)']
0 68.4
1 58.2
2 64.3
3 53.1
Name: weight(kgs), dtype: float64
>>> df['weight(kgs)'].apply(np.floor)
0 68.0
1 58.0
2 64.0
3 53.0
Name: weight(kgs), dtype: float64
In the above result, you can observe the floored to the nearest lower decimal point value and maintain its float64 data type.
We can visualise the above process as:
In the above visualisation, you can observe that all elements of Series are applied to the function at once.
• Whenever we have a ufunc to achieve our functionality, we can use it instead of defining a Python function.
Pandas apply( ) as a DataFrame method
We will take a look at the official documentation of the apply( ) method on DataFrame:
pandas.DataFrame.apply has two important arguments;
• func – Function to be applied along the mentioned axis
• axis – Axis along which function is applied
Again the axis also has 2 possible values;
1. axis=0 – Apply function to multiple columns
2. axis=1 – Apply function to every row
1. Pandas apply function to multiple columns
Let’s say the people in our dataset provided their height (in cms) information. It can be added using the following code,
>>> df['height(cms)'] = [178, 160, 173, 168]
>>> df
Name Sex Age weight(kgs) height(cms)
0 Edward M 45 68.4 178
1 Natalie F 35 58.2 160
2 Chris M M 29 64.3 173
3 Priyatham M 26 53.1 168
We’ll make the “Name” column the index of the DataFrame. Also, we’ll get the subset of the DataFrame with “Age”, “weight(kgs)”, and “height(cms)” columns.
>>> data = df.set_index('Name')
>>> data
Sex Age weight(kgs) height(cms)
Edward M 45 68.4 178
Natalie F 35 58.2 160
Chris M M 29 64.3 173
Priyatham M 26 53.1 168
>>> data_subset = data[['Age', 'weight(kgs)', 'height(cms)']]
>>> data_subset
Age weight(kgs) height(cms)
Edward 45 68.4 178
Natalie 35 58.2 160
Chris M 29 64.3 173
Priyatham 26 53.1 168
If we would like to get the average age, weight, and height of all the people, we can use the numpy ufunc numpy.mean( ).
The code for it is,
>>> import numpy as np
>>> data_subset.apply(np.mean, axis=0)
Age 33.75
weight(kgs) 61.00
height(cms) 169.75
dtype: float64
We directly have a Pandas DataFrame aggregation function called mean( ) which does the same as above;
>>> data_subset.mean()
Age 33.75
weight(kgs) 61.00
height(cms) 169.75
dtype: float64
If you observe the results above, the results of Pandas DataFrame aggregation function and applying ufunc are equal. So, we don’t use the apply( ) method in such simple scenarios where we have
aggregation functions available.
• Whenever you have to apply some complex functions on DataFrames, then use the apply( ) method.
2. Pandas apply function to every row
Based upon the height and weight, we can know whether they’re fit or thin, or obese. The fitness criteria are different for men and women as setup by international standards. Let’s grab the fitness
criteria data for the heights and weights of the people in our data.
This can be represented using a dictionary;
>>> male_fitness = {
... #height : (weight_lower_cap, weight_upper_cap)
... 178 : ( 67.5 , 83 ),
... 173 : ( 63 , 70.6 ),
... 168 : ( 58 , 70.7 )
... }
>>> female_fitness = {
... #height : (weight_lower_cap, weight_upper_cap)
... 160 : ( 47.2 , 57.6 )
... }
In the above dictionary, the keys are the heights and the values are tuples of the lower and upper limit of ideal weight respectively.
If someone is below the ideal weight for their respective height, they are “Thin”. If someone is above the ideal weight for their respective height, they are “Obese”. If someone is in the range of
ideal weight for their respective height, they are “Fit”.
Let’s build a function that can be used in the apply( ) method that takes all the rows one by one.
>>> def fitness_check(seq):
... if seq.loc['Sex'] == 'M':
... if (seq.loc['weight(kgs)'] > male_fitness[seq.loc['height(cms)']][0]) & (seq.loc['weight(kgs)'] < male_fitness[seq.loc['height(cms)']][1]):
... return "Fit"
... elif (seq.loc['weight(kgs)'] < male_fitness[seq.loc['height(cms)']][0]):
... return "Thin"
... else:
... return "Obese"
... else:
... if (seq.loc['weight(kgs)'] > female_fitness[seq.loc['height(cms)']][0]) & (seq.loc['weight(kgs)'] < female_fitness[seq.loc['height(cms)']][1]):
... return "Fit"
... elif (seq.loc['weight(kgs)'] < female_fitness[seq.loc['height(cms)']][0]):
... return "Thin"
... else:
... return "Obese"
The function returns whether a given person is “Fit” or “Thin” or “Obese”. It uses the different fitness criteria dictionaries for male and female created above.
Finally, let’s apply the above function to every row using the apply( ) method;
>>> data.apply(fitness_check, axis=1)
Edward Fit
Natalie Obese
Chris M Fit
Priyatham Thin
dtype: object
From the above result, we got to know who is Fit or Thin or Obese.
Conclusion and Next Steps
Using the apply( ) method when you want to achieve some complex functionality is preferred and recommended. Mostly built-in aggregation functions in Pandas come in handy. If you liked this tutorial
on the apply( ) function and like quiz-based learning, please consider giving it a try to read our Coffee Break Pandas book. | {"url":"https://blog.finxter.com/pandas-apply-a-helpful-illustrated-guide/","timestamp":"2024-11-02T22:07:31Z","content_type":"text/html","content_length":"81697","record_id":"<urn:uuid:5e762bc5-8f70-43f9-8c3d-9eedffed3a7b>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00677.warc.gz"} |
Sven-Ulf Weber's research works | Technische Universität Braunschweig and other places
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The title compounds are prepared by arc melting of the elements followed by annealing (Ta ampules, 1373 K, 21 d).
The carboferrates RE15[Fe8C25] (RE=Dy, Ho) were prepared from mixtures of the elements by arc-melting followed with subsequent annealing at 1373 K. The crystal structures were determined from single
crystal X-ray diffraction data and revealed an isotypic relationship to Er15[Fe8C25] (hP48, P321). The main feature of the crystal structure is given by Fe6 cluster units characterized by covalent
Fe-Fe bonding interactions. 57Fe Mössbauer spectra of Dy15[Fe8C25] were fitted by three subspectra with relative spectral weights of about 3:3:2 which is in general agreement with the crystal
structure. Below 50 K, an onset of magnetic hyperfine fields at the three iron sites is observed which is supposed to be caused by dipolar fields arising from neighboring, slowly relaxing Dy magnetic
The Bi2(FexGa1−x)4O9 oxide solid solution possessing a mullite-type structure has been investigated by 57Fe Mössbauer spectroscopy in dependence of composition (0.1≤x≤1) and temperature (293≤T/
K≤1073). The spectra have been fitted with two doublets for tetrahedrally and octahedrally coordinated high-spin Fe3+ ions, respectively. The experimental areas of the subspectra were used to
determine the distribution of iron on the two inequivalent structural sites. The fraction of iron cations occupying the octahedral site is found to increase with decreasing Fe content and the cation
distribution is almost independent of temperature. The unusual temperature dependence of the quadrupolar splitting, QS, observed for the octahedral site with dQS/dT>0 is discussed in connexion with
structural data for Bi2Fe4O9. The temperature dependence of Mössbauer isomer shifts and signal intensities is examined in the context of local vibrational properties of iron on the two inequivalent
sites of the mullite-type lattice structure.
Experimental and calculated structure factors from a previous synchrotron diffraction measurement on synthetic fayalite have been converted by an inverse Fourier transformation to difference electron
(deformation) densities (DED). These were processed in a revised 3D-display program giving hyperareas of DED floating in space around the iron positions M1 and M2 within the fayalite unit-cell and
spanning a cluster size of 6 and 4 Å, respectively. These relatively wide limits are due to the different site symmetries and had been proposed by earlier DFT (density functional theory)
calculations. From the different hyperareas the supposed charges were integrated in space and processed to electric field gradients (EFG) on M1 and M2 using a point-charge model. The two EFGs were
compared with respect to the system of crystallographic axes with those obtained from published single-crystal Mössbauer measurements (experimental EFGs), yielding excellent agreement within ±5° and
surpassing even the DFT results. This study reports the procedure and the conditions of success of the underlying semi-quantitative method, which is halfway between theory (DFT) and experiment
(diffractometry) and is promising valuable results on many other compounds. The term “nanoscope” for the graphical representation may be justified due to its high spatial resolution.
Oxides of the chemical composition Bi 2(Fe xAl 1-x) 4O 9 (0.1 ≤ x ≤ 1) possessing a mullite-type structure have been investigated using 57Fe Mössbauer spectroscopy in the temperature range between
room temperature and 1 073 K. The spectra have been fitted with two quadrupole doublets for Fe 3+ ions in octahedral and tetrahedral oxygen coordination. The distribution of iron on the two
inequivalent structural sites has been determined from the relative areas of the subspectra. The site occupancy is found to be random within experimental error: it does not exhibit any preference of
iron for the octahedral or tetrahedral sites, either as a function of temperature or with a dependence on the composition of the solid solutions.
This work reports on the evaluation of the electric field gradient (EFG) in natural chrysoberyl Al2BeO4 and sinhalite MgAlBO4 using two different procedures: (1) experimental, with single crystal
Mossbauer spectroscopy (SCMBS) on the three principal sections of each sample and (2) a "fully quantitative" method with cluster molecular orbital calculations based on the density functional theory.
Whereas the experimental and theoretical results for the EFG tensor are in quantitative agreement, the calculated isomer shifts and optical d-d-transitions exhibit systematic deviations from the
measured values. These deviations indicate that the substitution of Al and Mg with iron should be accompanied by considerable local expansion of the coordination octahedra.
The maximum solubilities of Zn2+ in Fe1–xO and of Fe2+ in ZnO at temperatures from 1173–1473 K in the presence of metallic iron were determined. Mixed crystals Fe1–x–zZnzO with 0.04 ≤ z ≤ 0.19 were
prepared at 1273 K. All samples were characterised by X-ray diffraction (XRD) and energy dispersive X-ray spectroscopy. The solubility of Fe2+ in wurtzite ZnO as well as the lattice parameter a of
Zn1–yFeyO increases with reaction temperature whereas c remains virtually constant. The solubility of Zn2+ in Fe1–xO increases with temperature. The lattice parameter a of Fe1–x–zZnzO for 0.04 ≤ z ≤
0.13 increases with zinc content. For0.13 ≤ z ≤ 0.19 a is constant, whereas for higher zinc contents a decreases. Mössbauer spectroscopy of the mixed crystal Fe1–x–0.09Zn0.09O yields a reduced amount
of 6 % Fe3+ compared to wüstite Fe1–0.04O with 9 % Fe3+. Such reduction of Fe3+ content was previously observed in the mixed crystal systems Fe1–xO/MgO and Fe1–xO/MnO too. The increase of the lattice
parameter a of Fe1–x–zZnzO for low concentrations is explained by removal of Fe3+ ions. At higher zinc contents the occupation with Zn2+ ions – with smaller radius than Fe2+ – balances or exceeds the
effect of Fe3+ removal.
Mössbauer spectra of the iron nitrides γ'-Fe 4 N 1+δ and ε-Fe 3 N 1+x have been measured at room tempera-ture and at 550 °C. γ'-Fe 4 N 1+δ is well ordered and close to ideal stoichiometry even at
high tempera-tures. Its room temperature Mössbauer spectrum exhibits three magnetically split subspectra due to the fact that the structurally equivalent Fe(1) sites are split into two
spectroscopically inequivalent sites, Fig. 1. This is consequence of the simultaneous presence of magnetic and quadrupolar interactions at Fe(1) sites, see Refs. [1,2] . In the case of ε-Fe 3 N 1+x ,
however, the stoichiometry-dependent magneti-cally-split room temperature spectra demonstrate the extensive disorder of nitrogen in the compound, Fig. 2 [2,3]. At 550 °C, spectra of ε-Fe 3 N 1+x have
been measured at various defined nitrogen activities. The quadrupole-split spectra are discussed in relation to structure and disorder of the nonstoichiomet-ric material. In particular, it is shown
that the composition dependent high-temperature spectra of ε-Fe 3 N 1+x can be modelled in the framework of a simple two-site model [2]. Fig. 1 Mössbauer spectrum of γ'-Fe 4 N at room temperature.
The fit is based on three subspectra which are indicated by a solid line due to Fe(2) and by dotted and dashed lines for subspectra which are due to Fe(1a) and Fe(1b), respectively. -6 -4 -2 0 2 4 6
78 80 82 84 86 88 90 92 94 96 98 100 Transmission [%] v [mm/s] Weber et al., Mössbauer Study of Iron Oxides diffusion-fundamentals www.diffusion-online.org Diffusion Fundamentals 12 (2010) 83 © S.-U.
Weber Fig. 2 a) Mössbauer spectrum of ε-Fe 3+x N at room temperature, b) Distribution of local magnetic fields according to the fit shown in Fig. 2a. By means of time-resolved Mössbauer measurements,
the formation reaction of γ'-Fe 4 N 1+δ has been followed at in situ conditions at 550 °C. The formation kinetics have been found to obey a parabolic rate law. The nitrogen activity dependence of the
experimental rate constant for the formation of γ'-Fe 4 N is discussed in relation to the point defect structure of the compound the atomic diffusion of ni-trogen. The analysis provides evidence for
the diffusion of nitrogen by means of a vacancy and an interstitial mechanism in γ'-Fe 4 N 1+δ where the latter is dominating the formation kinetics of the com-pound.
Natural sinhalites, MgAlBO4, from the Ratnapura District, Sri Lanka, and from Bodnar Quarry near Hamburg, Sussex Co., New Jersey, USA, have been characterized by 57Fe Mössbauer spectroscopy, electron
microprobe, X-ray single-crystal diffractometry and by electronic structure calculations in order to determine the oxidation state and site occupancy of iron in the sinhalite structure. The samples
contain about 3.35 and 1.46 wt% of total iron oxide, respectively. The structure refinement is successful and reproduces the total iron content provided that the substitution of Mg2+ by Fe2+ on the
M2 position only is assumed. The 57Fe Mössbauer spectra at 77, 293, 573 and 773 K can be resolved into two doublets with hyperfine parameters common for octahedrally coordinated high-spin Fe2+. There
is no evidence for iron in the tetrahedral site. Electronic structure calculations in local spin density approximation yield hyperfine parameters for Fe2+ on the M2-site at 0, 293, 573 and 773 K in
quantitative agreement with experiments. Calculated spectroscopic properties for Fe2+ on the M1-site are at variance with the experimental data and, thus, indicate that substitution of Al3+ by Fe2+,
if occurring at all, must be accompanied by considerable local expansion and distortion of the M1-octahedron.
... Central quantity in this respect is the electric field gradient (EFG) whose main property consists in providing the link between crystallography and solid-state physics or between diffractometry
and spectroscopy, respectively. Recently, this has been proved at a special example [1]. ...
... These Dy 3+ are coordinated by eight oxygen atoms forming distorted DyO 8 polyhedra linked to each other, building a 4-ring channel ( Figure 1). The infinite octahedral chains are the dominant
(but not only 3 ) criterion for mullite-type phases which could be subdivided into so-called O8-phase [11][12][13][14] (e.g., PbMBO 4 11-13,15-17 and schafarzikites LM 2 O 4 [18][19][20][21][22][23]
[24][25] ), O9-phase (e.g., BiMO [26][27][28][29][30][31][32][33][34][35][36][37][38] and ABO [39][40][41][42][43][44], and O10-phase (e.g., BiMO 6,10,45,46 ). The DyMn 2 O 5 structure type belongs
to the O10 phase which differs from the O9 BiMO phases with only an additional oxygen atom. ...
... Semiconducting materials showing photocatalytic activity by forming excitons in the visible electromagnetic spectrum are highly demanded nowadays [1][2][3][4][5]. In this regard mullite [6][7][8]
and mullite-type materials of either O8 EMBO 4 [9][10][11][12][13][14] and Schafarzikite [15][16][17][18], O9 [19][20][21] or O10 [22][23][24] type, especially the multiferroic O9 phase Bi 2 Fe 4 O 9
[25][26][27][28], are widely used [29][30][31][32][33][34][35][36][37][38]. The respective pseudo-binary phase diagram Bi 2 O 3 -Fe 2 O 3 has been repeatedly investigated [39][40][41][42][43][44]
[45], providing descriptions of some thermodynamically stable phases such as perovskitetype BiFeO 3 [32,43], mullite-type Bi 2 Fe 4 O 9 [25,28,29,31,32,[34][35][36][37][38][46][47][48] and
sillenite-type Bi 25 FeO 40 [40]. ...
... In the course of our systematic investigation of minerals with low iron content, it has been demonstrated that the electric field gradient (EFG) tensor is the most valuable quantity as supplying
information on local distortions not amenable to conventional diffraction methods (Weber et al. 2009 and references therein). Moreover, it is a key quantity in studying the relationship between
structural and chemical properties in solids because it provides a sensitive measure for the size and symmetry of the electric charge distribution around a given nucleus at a certain crystallographic
site. ...
... In the case of an undoped Fe 1 − x O wüstite phase, such a presence of cationic vacancies results in a smaller lattice parameter "a". But as detailed in ref [39], the occupation with Zn 2+ ions
in the wüstite network can induce an increase of the lattice parameter following a charge balance effect with Fe 3+ . According to this mechanism and considering the stoichiometric wüstite FeO (ICDD
# 00-006-0615; a = 0.4307 nm) as a reference, the lattice parameter in the non-stoichiometric wüstite Fe 0.855 O should be less than 0.4307 nm due to the formation of cationic vacancies and Fe 3+
ions. ...
... Beryllium occupies tetrahedra and Al occupies two types of slightly distorted interstitial octahedral sites: B1 octahedra with Ci symmetry and B2 octahedra with Cs symmetry [9]. These octahedra
have different volumes; additionally, B2 has an Al-O distance of 1.936 Å, which is larger than B1, with an Al-O distance of 1.890 Å [10]. Alexandrite is a type of chromium-doped bearing chrysoberyl
in which Cr 3+ is substituted in the Al 3+ sites. ...
... We suspect that elevated paramagnetic iron concentration (our mineral is brown in color, which is likely caused by an O 2− → Fe 3+ charge transfer and Fe 2+ in octahedral coordination in
sinhalite) led to increased 25 Mg nuclear spin relaxation rates and a distribution of Fermi contact shifts, inhibiting 25 Mg NMR spectral acquisition. 64 Grandidierite, MgAl 3 (BO 4 )(SiO 4 )O,
crystallizes in orthorhombic space group Pbnm and features one crystallographically unique Mg atom in an [MgO 5 ] polyhedron. 60 Grandidierite has been studied previously by MacKenzie and Meinhold 7
and was one of the first crystalline compounds with a pentacoordinated Mg site to be investigated by 25 Mg MAS NMR spectroscopy. ...
... The extremely high sensitivity of Mössbauer spectroscopy and its ability to distinguish Fe 2+ from Fe 3+ is of great advantage in assessing the oxidation state of iron. Another advantage is also
the ability of the method to analyze microcrystalline and amorphous substances containing iron [11][12][13]. It was shown that during the synthesis of CNTs a change occurred in the oxidation degree
and in the binding of the iron atoms in the catalyst [14,15]. ... | {"url":"https://www.researchgate.net/scientific-contributions/Sven-Ulf-Weber-32687778","timestamp":"2024-11-14T17:57:00Z","content_type":"text/html","content_length":"353947","record_id":"<urn:uuid:34b1fa73-2125-4db6-83ae-37fb1b5e891a>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00524.warc.gz"} |
A Walden student surveyed three sections of the AMDS 8437 class last spring, collecting demographic data, scores on the first
A Walden student surveyed three sections of the AMDS 8437 class last spring, collecting demographic data, scores on the first test, and final grade in the course. The student plans to build a
multiple regression model based on these data. The data are provided in the accompanying spreadsheet.
a. What would the basic model be? You do not have to do the regression; just give the general equation that you would use. Make sure that you clearly identify the variables in your model.
b. What level of data is each of the variables?
c. Do these data meet the assumptions that underlie multiple regression?
d. Comment on the validity of such a model.
Solution: a. We are looking for a multiple linear regression of the form
and []are the coefficients of the regression.
b. We have that
c. The assumptions for a multiple regression are basically that we have an interval dependent, based on linear combinations of interval, dichotomous, or dummy independent variables, the linearity of
relationships and the same level of relationship throughout the range of the independent variables ("homoscedasticity").
The variables in our model satisfy the specified assumptions, but the linearity of relationships needs to be tested with a further analysis.
d. The idea of a multiple regression is to be able to determine the variables that actually act as predictors of the dependent variable. But we cannot be totally sure of the significance of any of
the variables without applying some tests first (like the Wald’s test). Also we cannot disregard the possibility that certain interactions between the variables could affect the dependent variable.
The bottom line is that our model might look reasonable, but we cannot guarantee validity without running some tests first.
GO TO NEXT PROBLEM >>
Related Content | {"url":"https://statisticshelp.org/regression_analysis_problems_1.php","timestamp":"2024-11-05T09:29:50Z","content_type":"text/html","content_length":"20494","record_id":"<urn:uuid:ad610d93-6d2f-4c04-ba35-2175ddd25676>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00562.warc.gz"} |
How do you find the set of parametric equations for the line in 3D described by the general equations x-y-z=-4 and x+y-5z=-12?
| HIX Tutor
How do you find the set of parametric equations for the line in 3D described by the general equations x-y-z=-4 and x+y-5z=-12?
Answer 1
$\vec{r} = \left[\begin{matrix}- 8 \\ - 4 \\ 0\end{matrix}\right] + \lambda \left[\begin{matrix}6 \\ 4 \\ 2\end{matrix}\right]$
the line here is the line that constitutes the intersection of 2 planes as described
for the equation of a line in 3D, you need a point on the line, and a direction vector describing the path of the line from the point.
#vec r = vec r_o + lambda vec d #
finding a point is easy. just set any of x,y or z to any value you like and solve the 2 equations that are now in 2 unknowns
here i will set z = 0 so the equations are
add them to get x = -8, so y = -4
this we have the fixed point #r_o = [(x_0),(y_o),(z_o)] = [(-8),(-4),(0)]#
for the direction of that line, which is the direction of the intersection of the 2 planes, we can first find another point on the line, this time i will set x = 0 so that the eqns are
add them to get z = 8/3, y = 4/3 so we have fixed point this we have the fixed point #vec r_1 = [(x_1),(y_1),(z_1)] = [(0),(4/3),(8/3)]#
so the direction vector is
#vec d = vec r_1 - vec r_0#
# = [(x_1),(y_1),(z_1)] =[(0),(4/3),(8/3)]- [(-8),(-4),(0)] = [(8),(16/3),(8/3)] = 1/12[(6),(4),(2)]#
we're interested in the direction of this vector, not its scalar magnitude so we can simplify as we see fit
#vec r = [(-8),(-4),(0)] + lambda [ (6),(4),(2)]#
we can also instead take the vector cross product of the normal vectors of the 2 intersecting planes
for generalised plane ax + by + cz = d, the normal vector is #[(a),(b),(c)]#
so here we have #vec d = [(1),(-1),(-1)] times [(1),(1),(-5)]#
computationally that is the determinant of the following matrix
#[(hat i, hat j,hat k), (1,-1,-1), (1,1,-5)]#
= #hat i# (-1-5- (-1)1) -#hat j# (1(-5) - 1-1) + #hat k# (11 - 1(-1))
= #hat i# (6) -#hat j# (-4) + #hat k# (2)
#= [ (6),(4),(2)]#
#vec r = [(-8),(-4),(0)] + lambda [ (6),(4),(2)]#
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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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Nonlinear interaction force analysis of microcantilevers utilized in atomic force microscopy
This paper presents an Euler-Bernoulli microcantilever beam model utilized in non-contact Atomic Force Microscopy (AFM) systems. A distributed-parameters modeling is considered for such system. The
motions of the microcantilever are studied in a general Cartesian coordinate with an excitation at the base such that beam end with a tip mass is subject to a general force. This general force
comprising of two attractive and repulsive parts with high power terms is taken as the atomic intermolecular one which has a relation with the displacement between the tip mass and the surface such
that the total general force will be in the form of an implicit nonlinear equation. It is most desired to observe the effects of the base excitation in high frequencies on the tip van der Waals
interaction force. Hence, the general force will produce a peak in the FFT spectrum corresponding to the frequency of the base.
Publication series
Name Proceedings of the ASME Dynamic Systems and Control Conference 2009, DSCC2009
Number PART A
Conference 2009 ASME Dynamic Systems and Control Conference, DSCC2009
Country/Territory United States
City Hollywood, CA
Period 10/12/09 → 10/14/09
• Imagining
• Intermolecular force
• Microcantilever
• Non-contact atomic force microscopy
Dive into the research topics of 'Nonlinear interaction force analysis of microcantilevers utilized in atomic force microscopy'. Together they form a unique fingerprint. | {"url":"https://impact.ornl.gov/en/publications/nonlinear-interaction-force-analysis-of-microcantilevers-utilized","timestamp":"2024-11-02T15:47:13Z","content_type":"text/html","content_length":"49600","record_id":"<urn:uuid:c95d6d8f-4ad3-4989-92c5-b38db4ba7856>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00064.warc.gz"} |
Cuboid calculations made easy: Mastering the basicsCuboid calculations made easy: Mastering the basics | Coupon Space
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1. What is the meaning of cuboid?
2. How many faces or sides are there in a cuboid?
3. Whether the opposite faces of a cuboid have different characteristics?
4. List some examples of a cuboid.
5. What is the formula for calculation of volume of a cuboid?
6. We can get the volume of cuboid by multiplying its length, width, and height. What is the another name of width?
7. What is meant by surface area of a cuboid?
8. What is the formula for calculation of surface area of a cuboid?
9. What is meant by lateral surface area of a cuboid?
10. What is the formula for calculation of lateral surface area of a cuboid?
11. How can we calculate the area of four walls of a rectangular room?
12. How many pairs of identical walls are there in a rectangular room?
13. How can we calculate the length of cuboid when width, height and volume of the cuboid are given?
14. What is the formula for calculating the width of cuboid when volume, length and height is mentioned?
15. How to arrive at height of cuboid when width, length and volume of cuboid is known to us?
16. What is meant by diagonal of a cuboid?
17. Is it true that diagonal connects two opposite corners of the cuboid?
18. Diagonal of cuboid is also known by some another name. Can you specify?
19. Whether diagonal intersects two pairs of opposite faces?
20. What is the formula for the calculation of length of diagonal? | {"url":"https://spaceofcoupons.com/teaching-academics/cuboid-calculations-made-easy-mastering-the-basics-2/","timestamp":"2024-11-13T02:57:36Z","content_type":"text/html","content_length":"140124","record_id":"<urn:uuid:b5a74905-1b64-43b3-a877-6c4bfefca45d>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00894.warc.gz"} |
Ptophan) spectra, possible accumulation of errors could take place because they
Ptophan) spectra, possible accumulation of errors could take place because they are calculated as difference between the spectrum of the wild type Tubastatin-A manufacturer enzyme and each mutant
forms which leads to this difference. It is important therefore to reassert that CD calculations should be performed incorporating both the crystal structure and MD snapshots in strong correlation to
the experimental CD spectra.Figure 4. Comparison between the spectra calculated using Restricted Structural Model containing only the tryptophan and tyrosine chromophores (using TDDFT and the matrix
method) and those calculated using the 15900046 entire protein (using the matrix method). doi:10.1371/journal.pone.0056874.gEvaluating Restricted Structural Model Containing Only All Tryptophan and
Tyrosine Chromophores Using the 3PO aromatic ones by the same method (in yellow), does not provide the net spectrum (the one calculated using all chromophores including the aromatic ones with the
matrix method) (in red). The result therefore confirms that the net CD spectrum is not a simple sum of the aromatic chromophores plus the rest of the protein but rather it is a complex function of
multiple interactions between the aromatic chromophores incorporating the effect of the protein asymmetric field within a flexible environment. The study emphasizes the importance of explicit
representation of the chromophore environment in agreement to other theoretical studies [4.Ptophan) spectra, possible accumulation of errors could take place because they are calculated as difference
between the spectrum of the wild type enzyme and each mutant forms which leads to this difference. It is important therefore to reassert that CD calculations should be performed incorporating both
the crystal structure and MD snapshots in strong correlation to the experimental CD spectra.Figure 4. Comparison between the spectra calculated using Restricted Structural Model containing only the
tryptophan and tyrosine chromophores (using TDDFT and the matrix method) and those calculated using the 15900046 entire protein (using the matrix method). doi:10.1371/journal.pone.0056874.gEvaluating
Restricted Structural Model Containing Only All Tryptophan and Tyrosine Chromophores Using the Matrix Method and TDDFTOver the last several years TDDFT [16,38] has became increasingly applied for
calculating excited state properties of small and medium-sized molecules, many of which are of biological importance [39]. In order to evaluate the applicability of TDDFT calculations for larger
multi-chromophore systems (such as HCAII), we computed the spectra of the wild-type enzyme, and all the seven tryptophan mutants, using B3LYP/31G(d) level of theory on a cluster of all tryptophan and
tyrosine chromophores (kept at their positions from the crystal structure) in continuum solvent model environment with a dielectric constant of 4.0. Performing TDDFT calculation the entire protein
structure (as in the case with the matrix method) is not feasible at present. Whilst the calculations were sensitive and distinguished between the wild-type enzyme and each mutant form, they did not
reproduce the important spectral features (such as positions and magnitudes of the minima and maxima), even qualitatively (Figures 4 and 3A , in green). Nevertheless, that the choice of the density
functional and basis set could be extensively discussed (as for many recent excited state calculations e.g. [26,37,39]) and could contribute for the poor agreement between the calculated and the
experimental spectra, more crucially the results might suggest that to calculate the CD properties at reasonable quality it is vitally important to include explicitly the protein environment. In
order to test this hypothesis we carried out the matrix method of CD calculations on the tryptophans and tyrosines only (the same system which was used for TDDFT calculations). The resulting spectrum
(Figure 4, in pink) is different from the TDDFT spectrum (in green) and has a deeper minimum, but is still too far from the experimental one. In addition the additive spectrum (Figure 4, in blue)
from i) the spectrum calculated with only tryptophans and tyrosines by means of the matrix method (Figure 4, in pink) and ii) the spectrum calculated including all other chromophores without the
aromatic ones by the same method (in yellow), does not provide the net spectrum (the one calculated using all chromophores including the aromatic ones with the matrix method) (in red). The result
therefore confirms that the net CD spectrum is not a simple sum of the aromatic chromophores plus the rest of the protein but rather it is a complex function of multiple interactions between the
aromatic chromophores incorporating the effect of the protein asymmetric field within a flexible environment. The study emphasizes the importance of explicit representation of the chromophore
environment in agreement to other theoretical studies [4. | {"url":"https://www.adenosine-receptor.com/2017/08/24/ptophan-spectra-possible-accumulation-of-errors-could-take-place-because-they/","timestamp":"2024-11-14T12:02:22Z","content_type":"text/html","content_length":"60774","record_id":"<urn:uuid:37762af4-8c40-47c7-99ae-703c2daedc00>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00247.warc.gz"} |
Which Of These Triangle Pairs Can Be Mapped To Each Other Using A Single Translation? - Superstep.org
Triangle pairs are two separate triangles that can be mapped to each other using a single translation. This means that each triangle is moved in the same direction and the same distance. This article
will explain how to determine which triangle pairs can be mapped to each other using a single translation.
Triangle Pairs
Triangle pairs are two triangles that are related to each other. To be mapped to each other using a single translation, the triangles must have the same number of sides and corresponding lengths of
sides. In addition, the triangles must be similar, meaning that they must have the same angles and corresponding lengths of sides. This means that both triangles must have the same shape and size.
Mapping with a Translation
To determine whether two triangles can be mapped to each other using a single translation, the following steps should be taken:
1. Check that the two triangles have the same number of sides and corresponding lengths of sides.
2. Check that the angles of the two triangles are the same.
3. Check that the corresponding sides of the two triangles are the same length.
4. Calculate the translation vector, which is the distance and direction between the two triangles.
5. If the translation vector is the same for both triangles, then they can be mapped to each other using a single translation.
If all of these conditions are met, then the two triangles can be mapped to each other using a single translation.
By following the steps listed above, it is possible to determine whether two triangles can be mapped to each other using a single translation. This is a useful tool for understanding the relationship
between two triangles and can be used to solve a variety of mathematical problems. | {"url":"https://superstep.org/which-of-these-triangle-pairs-can-be-mapped-to-each-other-using-a-single-translation/","timestamp":"2024-11-02T21:12:50Z","content_type":"text/html","content_length":"48892","record_id":"<urn:uuid:b1ed023e-fa4d-442a-a1ac-566de00a2277>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00439.warc.gz"} |